Mojtaba Vaezi · Zhiguo Ding H. Vincent Poor Editors Multiple Access Techniques for 5G Wireless Networks and Beyond Multiple Access Techniques for 5G Wireless Networks and Beyond Mojtaba Vaezi Zhiguo Ding H. Vincent Poor • Editors Multiple Access Techniques for 5G Wireless Networks and Beyond 123 Editors Mojtaba Vaezi Villanova University Villanova, PA USA H. Vincent Poor Princeton University Princeton, NJ USA Zhiguo Ding The University of Manchester Manchester UK ISBN 978-3-319-92089-4 ISBN 978-3-319-92090-0 https://doi.org/10.1007/978-3-319-92090-0 (eBook) Library of Congress Control Number: 2018941989 © Springer International Publishing AG, part of Springer Nature 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface Waveform design, multiple access, and random access techniques for fifth-generation (5G) wireless networks and beyond are cutting-edge research topics that motivate a very wide range of research problems. Despite being different, these three areas are intertwined and lie at the heart of wireless communication systems. They allow multiple users to effectively share a communication medium. The previous generations of cellular networks have adopted radically different multiple access techniques with one common theme in mind: to have orthogonal signals for different users at the receiver side. As an example, the fourth-generation (4G) cellular networks have adopted orthogonal frequency division multiplexing (OFDM). In view of emerging applications such as the Internet of Things (IoT), and in order to fulfill the need for massive numbers of connections with diverse requirements in terms of latency and throughput, 5G and beyond cellular networks are experiencing a paradigm shift in design philosophy: shifting from orthogonal to non-orthogonal design in waveform, multiple access, and random access techniques. This book provides a comprehensive and intensive examination of multiple access, random access, and waveform design techniques for 5G and beyond systems. It contains numerous state-of-the-art techniques and experimental results to address the challenges in building 5G and beyond wireless networks. The book will be of interest to readers from the communications, signal processing, and information theory communities. It will serve as a reference for graduate students, researchers, and engineers involved in the design and standardization of wireless communication systems. It can also serve as a reference book for graduate-level courses for students in electrical engineering. The book is organized into four parts and twenty-one chapters, each meant to be self-contained. The contents of different chapters in each part are chosen so that they reinforce and complement each other. Part I is focused on waveform design for 5G and beyond and includes four chapters outlining several advanced multicarrier waveforms designs. Parts II through IV cover several related topics in multiple access and random access. These parts include various non-orthogonal multiple access (NOMA) techniques in the power domain, code domain, and other domains as well as topics on random access. Power domain NOMA is mainly covered in v vi Preface Part II. Several code domain NOMA and other NOMA techniques as well as random access schemes are discussed in Part III. Part IV includes experimental trials and applications of NOMA in certain fields other than cellular communications. Part I (Chaps. 1–4) addresses waveform design. Chapter 1 introduces the reader to 1G to 4G cellular systems and motivates the need for new multiple access and waveforms for 5G and beyond systems. Chapter 2 presents various waveform designs envisioned for the 5G new radio (NR). It first outlines the shortcomings of conventional OFDM in serving the diverse use cases envisioned for 5G systems. It then discusses design principles for new waveforms and introduces three new waveforms considered in the standardization process of 5G NR, all developed from OFDM. The new variants of OFDM are focused on reducing the out-of-band (OOB) emission of cyclic-prefix OFDM by signal processing techniques such as time-domain windowing and subband-based filtering. Chapter 3 introduces another advanced multicarrier waveform, namely filter bank multicarrier modulation (FBMC). FBMC can perform much better than windowed and filtered OFDM in reducing the OOB emissions of conventional OFDM and can be considered as a potential waveform for next generation wireless networks. Chapter 4 studies yet another advanced multicarrier waveform, generalized frequency-division multiplexing (GFDM). GFDM is a multicarrier waveform technique that encapsulates windowed and filtered OFDM techniques of 5G while providing an additional design space reserved for forward comparability beyond 5G. Part II (Chaps. 5–11) is dedicated to NOMA relying on the power domain. Chapter 5 studies NOMA from an information-theoretic perspective. This chapter reviews the basic premise behind NOMA in single- and multi-cell networks both in the downlink and uplink. It also introduces various information-theoretic channels that can be used to model physical layer security in NOMA. Chapter 6 investigates power allocation for downlink NOMA under different performance metrics, such as fairness, sum rate, and energy efficiency. The design principles of multiple-antenna NOMA systems, including user clustering, channel state information acquisition, and transmit beamforming are studied in Chap. 7. Chapter 8 is focused on applying NOMA to millimeter wave networks with three transmissions schemes, namely unicast, multicast and cooperative multicast. Chapter 9 is dedicated to full-duplex NOMA, a technology that has the potential to double the spectral efficiency via simultaneous transmissions in the uplink and downlink. Resource allocation in heterogeneous NOMA with energy cooperation is discussed in Chap. 10. Chapter 11 evaluates the performance of NOMA in vehicle-to-vehicle massive MIMO channels. Part III (Chaps. 12–17) introduces several code domain NOMA schemes as well as non-orthogonal random access. All multiple access techniques presented in this part have a common philosophy: to exploit efficient and low-complexity multiuser detection. In particular, Chap. 12 studies sparse code multiple access (SCMA), a code domain NOMA scheme that exploits the sparsity of the multi-dimensional codewords to apply the low-complexity message passing algorithm for multiuser detection. Chapter 13 discusses interleave division multiple access (IDMA). Preface vii IDMA applies a low-complexity iterative technique for multiuser detection and can achieve near-capacity sum rate with proper power allocation. Chapter 14 introduces pattern division multiple access (PDMA) which is a NOMA technique in which a pattern defines the mapping of transmitted data to a group of time, frequency, and spatial resources. In Chap. 15, low-density spreading (LDS), a variant of code division multiple access (CDMA) in which spreading sequences have low density, is studied. Owing to this, a near optimal message passing algorithm receiver with practically feasible complexity can be exploited. Chapter 16 discusses a grant-free multiple access scheme that enables lower transmission latency and savings in device energy. Chapter 17 provides a comprehensive survey of random access schemes that are suited to support IoT. These schemes are commonly based on time, frequency, and code division multiple access techniques and different variants of NOMA described in the previous chapters. Part IV (Chaps. 18–21) outlines experimental trials, challenges, and future trends of NOMA. To evaluate the performance of NOMA using real-world hardware and in realistic radio environments, a test-bed is described in Chap. 18. Indoor and outdoor experimental trials are then conducted which confirm that NOMA improves user throughput as compared to orthogonal multiple access. Applications and extension of NOMA to visible light communication networks and terrestrial-satellite networks are studied in Chaps. 19 and 20, respectively. Finally, Chap. 21 provides future research directions for NOMA in 5G wireless networks and beyond as well as other fields. We would like to extend our thanks to the people and organizations who made this book possible. Our sincere thanks go to the chapter authors; it has been an honor and a privilege to work with such a dedicated and talented group of authors and researchers. Princeton University and the universities, research laboratories, and corporations with which the authors are affiliated deserve credit for providing facilities and intellectual environments for this project. It is also a pleasure to acknowledge Springer and its team: Mary James, Brian Halm, and Zoe Kennedy. Finally, we offer our deepest appreciation and gratitude to our families for their patience and support during the months we were immersed in this project. Villanova, PA, USA Manchester, UK Princeton, NJ, USA Mojtaba Vaezi Zhiguo Ding H. Vincent Poor Contents Part I Orthogonal Multiple Access Techniques and Waveform Design 1 Introduction to Cellular Mobile Communications . . . . . . . . . . . . . . Joseph Boccuzzi 2 OFDM Enhancements for 5G Based on Filtering and Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rana Ahmed, Frank Schaich and Thorsten Wild 3 Filter Bank Multicarrier Modulation . . . . . . . . . . . . . . . . . . . . . . . Ronald Nissel and Markus Rupp 4 Generalized Frequency Division Multiplexing: A Flexible Multicarrier Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ahmad Nimr, Shahab Ehsanfar, Nicola Michailow, Martin Danneberg, Dan Zhang, Henry Douglas Rodrigues, Luciano Leonel Mendes and Gerhard Fettweis Part II 3 39 63 93 Non-Orthogonal Multiple Access (NOMA) in the Power Domain 5 NOMA: An Information-Theoretic Perspective . . . . . . . . . . . . . . . . 167 Mojtaba Vaezi and H. Vincent Poor 6 Optimal Power Allocation for Downlink NOMA Systems . . . . . . . . 195 Yongming Huang, Jiaheng Wang and Jianyue Zhu 7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Xiaoming Chen, Zhaoyang Zhang, Caijun Zhong and Derrick Wing Kwan Ng ix x Contents 8 NOMA for Millimeter Wave Networks . . . . . . . . . . . . . . . . . . . . . . 257 Zhengquan Zhang and Zheng Ma 9 Full-Duplex Non-Orthogonal Multiple Access Networks . . . . . . . . . 285 Mohammed S. Elbamby, Mehdi Bennis, Walid Saad, Mérouane Debbah and Matti Latva-aho 10 Heterogeneous NOMA with Energy Cooperation . . . . . . . . . . . . . . 305 Bingyu Xu, Yue Chen and Yuanwei Liu 11 NOMA in Vehicular Communications . . . . . . . . . . . . . . . . . . . . . . 333 Yingyang Chen, Li Wang, Yutong Ai, Bingli Jiao and Lajos Hanzo Part III NOMA in Code and Other Domains 12 Sparse Code Multiple Access (SCMA) . . . . . . . . . . . . . . . . . . . . . . 369 Zheng Ma and Jinchen Bao 13 Interleave Division Multiple Access (IDMA) . . . . . . . . . . . . . . . . . . 417 Yang Hu and Li Ping 14 Pattern Division Multiple Access (PDMA) . . . . . . . . . . . . . . . . . . . 451 Shanzhi Chen, Shaohui Sun, Shaoli Kang and Bin Ren 15 Low Density Spreading Multiple Access . . . . . . . . . . . . . . . . . . . . . 493 Mohammed Al-Imari and Muhammad Ali Imran 16 Grant-Free Multiple Access Scheme . . . . . . . . . . . . . . . . . . . . . . . . 515 Liqing Zhang and Jianglei Ma 17 Random Access Versus Multiple Access . . . . . . . . . . . . . . . . . . . . . 535 Riccardo De Gaudenzi, Oscar del Río Herrero, Stefano Cioni and Alberto Mengali Part IV Challenges, Solutions, and Future Trends 18 Experimental Trials on Non-Orthogonal Multiple Access . . . . . . . . 587 Anass Benjebbour, Keisuke Saito and Yoshihisa Kishiyama 19 Non-Orthogonal Multiple Access in LiFi Networks . . . . . . . . . . . . . 609 Liang Yin and Harald Haas 20 NOMA-Based Integrated Terrestrial-Satellite Networks . . . . . . . . . 639 Xiangming Zhu, Chunxiao Jiang, Linling Kuang, Ning Ge and Jianhua Lu 21 Conclusions and Future Research Directions for NOMA . . . . . . . . 669 Zhiguo Ding, Yongxu Zhu and Yan Chen Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 About the Editors Mojtaba Vaezi received the Ph.D. degree in Electrical Engineering from McGill University in 2014. From 2015 to 2018, he was with Princeton University as a Postdoctoral Research Fellow and Associate Research Scholar. He is currently an Assistant Professor of ECE at Villanova University and a Visiting Research Collaborator at Princeton University. Before joining Princeton, he was a researcher at Ericsson Research in Montreal, Canada. His research interests include the broad areas of information theory, wireless communications, and signal processing, with an emphasis on physical layer security and radio access technologies. Among his publications in these areas is the book Cloud Mobile Networks: From RAN to EPC (Springer, 2017). He has served as the president of McGill IEEE Student Branch during 2012–2013. He is an Associate Editor of IEEE Communications Magazine and IEEE Communications Letters. He has co-organized four international NOMA workshops at VTC-Spring’17, Globecom’17, ICC’18, and Globecom’18. He is a recipient of a number of academic, leadership, and research awards, including the McGill Engineering Doctoral Award, IEEE Larry K. Wilson Regional Student Activities Award in 2013, the Natural Sciences and Engineering Research Council of Canada (NSERC) Postdoctoral Fellowship in 2014, and Ministry of Science and ICT of Korea’s best paper award in 2017. Zhiguo Ding received his B.Eng. in Electrical Engineering from the Beijing University of Posts and Telecommunications in 2000, and the Ph.D. degree in Electrical Engineering from Imperial College London in 2005. From July 2005 to April 2018, he was working in Queen’s University Belfast, Imperial College, Newcastle University and Lancaster University. Since April 2018, he has been with the University of Manchester as a Professor in Communications. From September 2012 to September 2018, he has also been an academic visitor in Princeton University. His research interests are 5G networks, game theory, cooperative and energy harvesting networks and statistical signal processing. He is serving as an Editor for IEEE Transactions on Communications and IEEE Transactions on Vehicular Technology. He served as an Editor for IEEE Wireless Communication Letters, IEEE Communication Letters, and Journal of Wireless Communications xi xii About the Editors and Mobile Computing. He received the best paper award in IET International Communication Conference on Wireless Mobile and Computing, 2009, and the IEEE WCSP 2015, IEEE Transactions on Vehicular Technologies Top Editor 2017, and the EU Marie Curie Fellowship 2012–2014. H. Vincent Poor received the Ph.D. degree in EECS from Princeton University in 1977. From 1977 until 1990, he was on the faculty of the University of Illinois at Urbana-Champaign. Since 1990, he has been on the faculty at Princeton, where he is currently the Michael Henry Strater University Professor of Electrical Engineering. During 2006 to 2016, he served as Dean of Princeton’s School of Engineering and Applied Science. His research interests are in the areas of information theory and signal processing, and their applications in wireless networks, energy systems, and related fields. He is a member of the National Academy of Engineering and the National Academy of Sciences and is a foreign member of the Chinese Academy of Sciences, the Royal Society, and other national and international academies. Other recognition of his work includes the 2017 IEEE Alexander Graham Bell Medal, and honorary doctorates and professorships from a number of universities. Acronyms 1G 2G 3G 3GPP 4G 5G 5GC ACEP ACK ACLR ACO-OFDM ACRDA ADC AF AGC AMC AMF AMI AMPS AN AoA AoD APD APM APP APSK AR AWGN BC BEP First generation Second generation Third generation 3rd Generation Partnership Project Fourth generation Fifth generation 5G core Average codeword error probability Acknowledgement Adjacent channel leakage rejection Asymmetrically clipped optical OFDM Asynchronous contention resolution diversity ALOHA Analog-to-digital converter Application function Automatic gain control Adaptive modulation and coding Access and mobility management function Averagemutual information Advanced mobile phone services Artificial noise Angle of arrival Angle of departure Avalanche photodiode Amplitude-phase modulation A posteriori probability Amplitude phase-shift keying Augmented reality Additive white Gaussian noise Broadcast channel Bit error probability xiii xiv BER BF BICM BICM-ID BLER BP BPSK BRAM BRU BS BSA BSC BSS BTS C2X CA CB CBRS CCDF CCI CDD CDF CDMA CEP CFO CIM CIR CLT CM CMT CN CN CNR CoF CoMP CP CPICH CP-OFDM CPRI CQI CR CRA C-RAN CRC CRDSA Acronyms Bit error rate Beamforming Bit-interleaved coded modulation BIMC with iterative decoding Block error rate Belief propagation Binary phase-shift keying Block RAM Basic resource unit Base station Binary switching algorithm Base station controller Base station subsystem Base transceiver stations Car to anything Carrier aggregation Coordinated beamforming Citizens broadband radio service Complementary cumulative distribution function Co-channel interference Cyclic delay diversity Cumulative distribution function Code division multiple access Channel estimation preamble Carrier frequency offset Color intensity modulation Channel impulse response Central limit theorem Cubic metric Cosine-modulated multitone Core network Channel observation node Channel-to-noise ratio Cycle-of-four Coordinated multipoint Cyclic prefix Common pilot channel Cyclic prefix OFDM Common public radio interface Channel quality indicator Cognitive radio Contention resolution ALOHA Cloud radio access network Cyclic redundancy check Contention resolution diversity slotted ALOHA Acronyms CR-NOMA CRS CS CS CS CSA CsDMA CSI CSIR CSIT CSK CSMA CSMA/CA CSMA/CD CTS CU CW-SIC D/A D2D DA DAC DACE DC DC DCCC DCI DCO-OFDM DCS DD DEC DFE DL DMRS DoF DPC DPD DRAM DS-ALOHA DS-CDMA DSP DS-SS DTFT DU DVB DVB-T xv Cognitive radio inspired NOMA Cell-specific reference signal Circuit switched Cyclic suffix Coordinated scheduling Coded Slotted ALOHA Code-shift division multiple access Channel state information Channel state information at the receiver Channel state information at the Transmitter Color shift keying Carrier sense multiple access Carrier sense multiple access/collision avoidance Carrier sense multiple access/collision detection Clear to send Centralized unit Codeword level SIC Digital-to-analog Device-to-device Deferred acceptance Digital-to-analog converter Data-aided channel estimation Direct current Difference of convex functions Discrete codebook-constrained capacity Downlink control information Direct current-biased optical OFDM Dynamic cell selection Direct detection Decoder Decision feedback equalizer Downlink Demodulation reference signal Degree of freedom Dirty-paper coding Digital predistortion Dynamic random access memory Diversity slotted ALOHA Direct-sequence CDMA Digital signal processor Direct-sequence spread spectrum Discrete-time Fourier transform Distributed unit Digital video broadcasting Digital video broadcasting terrestrial xvi DZT EDGE EE eMBB eMBMS eMTC eNB EP EPC ES ESE E-SSA ETU EV-DO EVM EXIT FBMC FD FDD FDE FDM FDMA FEC FER FET FFT FIFO FIR FMT FN FO f-OFDM FOV FPA FPGA FSPA FTPA GA GA GB GEO GEVD GF GFDM GGSN Acronyms Discrete Zak transform Enhanced data rates for GSM evolution Energy efficiency Enhanced mobile broadband Evolved multimedia broadcast multicast service Enhanced machine type communications Enhanced NodeB Expectation propagation Equal power control Exhaustive search Elementary signal estimation Enhanced spread spectrum ALOHA Extended typical urban Evolution-data optimized Error vector magnitude Extrinsic information transfer Filter bank multicarrier modulation Full-duplex Frequency division duplex Frequency domain equalization Frequency division multiplex Frequency division multiple access Forward error control Frame error rate Field-effect transistor Fast Fourier transform First in first out Finite impulse repose Filtered multitone Function Node Frequency offset Filtered OFDM Field of view Fixed power allocation Field programmable gate array Full search power allocation Fractional transmission power allocation Genetic algorithm Gaussian approximation Grant-based Geosynchronous earth orbit Generalized eigenvalue decomposition Grant-free Generalized frequency division multiplexing Gateway GPRS support node Acronyms GMSC GMSK gNB GOCA GP GPRS GSM GSVD GT HARQ HD HetNet HK HLR HOM HOS HPPP HSDPA HSPA HSTRN HSUPA i.i.d. IA IAI IBFD IBI IC IC ICI ICI ICT IDD IDFT IDMA IFFT IFPI IFS IGCH IGMA IM I-MRC INI IoT IR IRSA xvii Gateway mobile switching center Gaussian minimum shift keying Next generation NodeB Group-orthogonal coded access Guard period General packet radio services Global system for mobile communications Generalized singular value decomposition Guard tone Hybrid automatic repeat request Half-duplex Heterogeneous network Han-Kobayashi Home location register Higher-order modulation Hierarchy of orthogonal sequences Homogeneous Poisson point process High speed downlink packet access High speed packet access Hybrid satellite terrestrial relay network High speed uplink packet access identically independently distributed Interference alignment Inter-antena interference In-band full-duplex Inter-block interference Interference cancellation Interference channel Inter-carrier interference Inter-cell interference Information and communication technology Iterative detection and decoding Inverse discrete Fourier transform Interleave division multiple access Inverse fast Fourier transform Interference-free pilot insertion Inter-frame spacing Information-guided channel hopping Interleave grid multiple access Intensity modulation Iterative maximum ratio combining Inter-numerology interference Internet of Things Infrared Irregular repetition slotted ALOHA xviii ISI ISM ITS ITU IUI JP JRA JT J-TACS KKT KPI LDM LDPC LDS LED LEO LiFi LLR LMMSE LOS LP LPWAN LS LTE LTE-A M2M MA MAC MAC MAC MACA MAI MAP MARSALA MBS MC-CDMA MC-LDSMA MC-NOMA MCS MEC MEO ME-SSA MF MF-CRDSA MI Acronyms Inter-symbol interference Industrial scientific and medical Intelligent transportation systems International telecommunications union Inter-user interference Joint processing Joint resource allocation Joint transmission Japan TACS Karush-Kuhn-Tucker Key performance indicator Layer division multiplex Low-density parity-check Low-density spreading Light-emitting diode Low Earth orbit Light fidelity Log-likelihood ratio Linear minimum mean squared error Line-of-sight Linear program Low power wide area networks Least squares Long term evolution LTE advanced Machine to machine Multiple access Multiple access channel Medium access control Medium access layer Multiple access collision avoidance Multiple access interference Maximum a posterior Multi-replica decoding using correlation based Iocalization Macro base station Multicarrier CDMA Multicarrier low density spreading multiple access Multi-channel NOMA Modulation coding scheme Multi-access edge computing Medium Earth orbit MMSE enhanced spread spectrum ALOHA Matched filter Multi-frequency CRDSA Mutual information Acronyms MIIT MIMO MINLP MISO ML MM MMF MMSE mMTC mmWave MPA MPF MPR MRC MRT MSC MSE MTC MTSO MU MUD MUI MU-MIMO MUSA MuSCA MUST NAICS N-AMPS NB NB-IoT NCMA NEF NFV NLOS NMSE NMT NOCA NOMA NP NR O/E OFDM OFDMA OMA OOB xix Ministry of industrial and information technology Multiple-input and multiple-output Mixed integer nonlinear programming Multiple-input and single-output Maximum likelihood Metameric modulation Maximin fairness Minimum mean squared error Massive machine type communication Millimeter wave Message passing algorithm Marginal product of functions Multi-packet reception Maximum ratio combining Maximum ratio transmission Mobile switching center Mean squared error Machine type communication Mobile telephone switching office Mobile user Multi-user detection Multi-user interference Multi-user MIMO Multi-user shared access Multi-slots coded ALOHA Multi-user superposition transmission Network-assisted interference cancellation and suppression Narrowband AMPS NodeB Narrow band IoT Non-orthogonal coded multiple access Noise enhancement factor Network function virtualization Non-line-of-sight Normalized mean-squared error Nordic mobile telephone Non-orthogonal coded access Non-orthogonal multiple access Non-deterministic polynomial-time New radio Optical to electrical Orthogonal frequency division multiplexing Orthogonal frequency division multiple access Orthogonal multiple access Out-of-band xx OOK OQAM O-QPSK OSS OSTBC OVSF P/S PA PAM PAPR PBS PC PCCC PCM PD PDCCH pdf PDMA PDR PEP PER PHY PIA-ASP PIC p-i-n PL PLR PN PPM PPP PRB PS PS PSD PSK PSTN PWM QAM QoS QPSK RA RACH RAN RAR RAT Acronyms On-off keying Offset quadrature amplitude modulation Offset quadrature phase shift keying Operation and support subsystem Orthogonal space-time block coding Orthogonal variable spreading factor Parallel-to-serial Power amplifier Pulse amplitude modulation Peak-to-average power ratio Pico BS Power control Parallel concatenated convolutional code Policy control function Photodiode Physical downlink control channel Probability density function Pattern division multiple access Packet drop rate Pairwise error probability Packet error rate Physical layer Prior-information aided adaptive subspace pursuit Parallel interference cancellation Positive-intrinsic-negative Primary layer Packet loss rate Pseudo noise Pulse position modulation Poisson point process Physical resource block Packet switched Phase shifter Power spectral density Phase-shift keying Public switched telephone network Pulse width modulation Quadrature amplitude modulation Quality of service Quadrature phase shift keying Random access Random access channel Radio access network Random access response Radio access technology Acronyms RB RC RDMA RE RF RMS RNC RNTI RPC RPMA RR RRC RRC RRH RS RSMA RSRP RTS RV RV RX S-ALOHA SAMA SA-SCMA SC SC SCA SC-FDMA SCM SCMA SCR SCS SDMA SDN SDR SE SER SFBC SG SGSN SIC SINR SIR SISO SL xxi Resource block Raised-cosine Repetition division multiple access Resource element Radio frequency Root mean square Radio network controller Radio network temporary identifier Randomized power control Random phase multiple access Round robin Radio resource control Root-raised cosine Remote radio head Rate-splitting Resource spread multiple access Reference signal received power Request to send Redundancy version Random variable Receiver Slotted ALOHA Successive interference cancellation amenable multiple access Spread asynchronous scrambled coded multiple access Sub-carrier Superposition coding Successive convex approximation Single carrier frequency division multiple access Superposition coded modulation Sparse code multiple access Signal-to-clipping-noise ratio Sub-carrier spacing Space division multiple access Software defined networking Software defined radio Spectral efficiency Symbol error rate Space frequency block coding Scheduling grant Serving GPRS support node Successive interference cancellation Signal-to-interference-plus-noise ratio Signal-to-interference ratio Single-input and single-output Secondary layer xxii SLA SLL SLS SL-SIC SM SMF SMS SMT SNR SPS SR SR SrCMA SS-ALOHA SSD SSK STC STO SUD SVD Tx TACS TB TCP TDD TDL TDM TDMA TO TO TR TR-STC TS TTI TX UA UDM UDN UE UFMC UF-OFDM UL UMTS UN UNB Acronyms Side lobe attenuation Side lobe level System level simulation Symbol level SIC Spatial modulation Session management function Short message service Staggered multitone Signal-to-noise ratio Semi-persistent scheduling Scheduling request Sum rate Scrambled coded multiple access Spread-spectrum ALOHA Signal-space diversity Space shift keying Space-time coding Symbol time offset Single-user detection Singular value decomposition Transmit antenna Total access communication system Transport block Transmission control protocol Time division duplex Tapped delay line Time division multiplexing Time division multiple access Transmission occasion Time offset Technical report Time-reversal space-time coding Time-sharing Transmission time interval Transmitter User association Unified data management Ultra dense network User equipment Universal filtered multicarrier Universal filtered OFDM Uplink Universal mobile telephone system User node Ultra narrow band Acronyms U-OFDM UPC UPF URLLC UTRAN V2I V2N V2P V2V V2X VANET VBLAST VLC VLR VN VR WAVE WBE WCDMA WOLA WSR ZF ZFBF ZP ZP-OFDM ZT-DFT xxiii Unipolar OFDM Unequal power control User plane function Ultra-reliable low latency communications UMTS terrestrial radio access network Vehicle-to-infrastructure Vehicle-to-network Vehicle-to-pedestrian Vehicle-to-vehicle Vehicle-to-everything Vehicular ad hoc network Vertical Bell laboratories layered space-time Visible light communication Visitor location register Variable node Virtual reality Wireless access for vehicular environments Welch-bound equality Wideband CDMA Weighted overlap and add Weighted sum rate Zero-forcing Zero-forcing beamforming Zero-padding Zero prefix OFDM Zero tail DFT Part I Orthogonal Multiple Access Techniques and Waveform Design Chapter 1 Introduction to Cellular Mobile Communications Joseph Boccuzzi 1.1 Introduction This chapter provides an overview of the evolution of the cellular mobile communication systems. We begin with a quote from a conversation held over a mobile cellular network from Martin Cooper on 3 April 1973 [1]. I’m calling you from a cell phone, a real handheld portable cell phone. The mobile device used during this conversation was a Motorola DynaTAC weighing approximately 2.5 lbs with a cost of approximately $9,000 USD. This historic event ignited a movement which would change the lives of so many people. This life change is much, much more than supporting mobile users, it snowballed into creating highly complex devices (presently called smart phones) that help us remain connected to the world. These devices not only perform our much needed voice and data communication needs, but they also take on a very wide array of supporting applications such as keeping our friends informed on social media, competing with online gaming, consuming and producing video content, performing medical measurements, utilizing location-based services, etc. As these wireless devices benefited from Moore’s law, the cellular mobile technologies were able to remain a focal point to introduce such new and exciting features, and benefits, to the end user. This chapter is intended to address important driving technologies behind the 5G new radio (NR) system designs, which focuses on solutions to supporting 5G new services in uplink (UL) transmissions with requirements such as low-latency and high-reliability, energy-saving, and small packet applications. Grant-free (GF) resources in NR UL is termed as “a configured grant,” which means that the J. Boccuzzi (B) Intel Corporation, San Diego, CA, USA e-mail: Joseph.Boccuzzi@intel.com © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_1 3 4 J. Boccuzzi pre-configured UE-specific resources will be used for UE UL transmission without dynamic scheduling/grant. Also, the base station (BS) in 5G NR network is referred to as “next generation NodeB” or “gNB.” 1.2 Cellular Mobile Communication: A Primer The cellular standards use a variety of multiple access (MA) techniques, which we highlight in Table 1.1. These techniques include frequency division multiple access (FDMA), time division multiple access (TDMA), code division multiple access (CDMA), and orthogonal frequency division multiple access (OFDMA). We also describe the relevant duplex method used for two-way communication and the actual physical resources available to be assigned to each user. The duplex methods are time division duplex (TDD) and frequency division duplex (FDD). All of the above multiple access techniques can be viewed as a form of “orthogonal” multiple access (OMA), where the access of users, theoretically, do not interfere with one another as they share the wireless medium. They are, however, limited to the number of resources available that make them orthogonal to each other. An exception to this would be CDMA, where the transmission from the wireless device to the base station is inherently non-orthogonal. In FDMA, the frequency is divided into channels to be utilized by various users. In TDMA, time is divided into time slots as a means to allow various users to access the cellular system. In CDMA, users are separated by PN codes and transmit over the entire frequency channel, all at the same time. In OFDMA, users are allocated to various frequency channels (groups of sub-carriers) at different time slots. For the next generation digital cellular system called 5G, OFDMA is still used where the sub-carrier spacing and time slot durations are flexible and scalable to support wide-varying requirements and use cases. It is also expected to utilize NOMA in 5G. In Fig. 1.1, we provide an overview to showcase various multiple access techniques that will be discussed in this section. They are compared in three dimensions or domains: power, time, and frequency. Table 1.1 Multiple access in different generations of cellular networks Cellular MA technique Duplex method Physical generation resources 1G 2G 3G FDMA TDMA CDMA FDD FDD FDD/TDD 4G 5G OFDMA OFDMA FDD/TDD FDD/TDD Frequency Time slots Time slots/PN Codes Time/Frequency Time/Frequency Notable examples AMPS, NMT GSM, IS-54 WCDMA LTE, LTE-A 5G-NR 1 Introduction to Cellular Mobile Communications 5 Fig. 1.1 Overview of various multiple access techniques: a FDMA, b TDMA, c CDMA, d OFDMA-4G Spectral Efficiency (bps/Hz) 80 70 60 50 GSM WCDMA/HSPA 5G LTE 5G 40 LTE/ LTE-A 30 20 10 0 1.E+04 WCDMA/ HSDPA GSM/ EDGE 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10 Data Rate (bps) Fig. 1.2 Spectral efficiencies (bps/Hz) of the digital cellular evolution A system performance metric that continues to be improved in every generation is spectral efficiency, bps/Hz. Figure 1.2 shows the DL spectral efficiencies of the 2G, 3G, 4G, and 5G digital cellular standards versus the peak theoretical data rates. Notice that with each new standard the demand for higher and higher data rates along with an increased demand for spectral efficiency becomes more pronounced. With every cellular generation, there is not only an expectation of increased performance, but also the addition of new features. Figure 1.3 shows how the cellular 6 J. Boccuzzi Fig. 1.3 User capabilities as a function of cellular generations User Capability 5G 4G 3G ~2020 2G ~2010 1G ~1980 ~1990 ~2000 Cellular GeneraƟons user capabilities (and expected features) have increased exponentially over the evolution of the cellular generations. We started with voice only and then moved on to voice and short message service (SMS) capabilities in 2G. The data capabilities were improved in 3G to include packet-switched services. 4G provided mobile Internet with expanded use cases for the Internet of things (IoT), vehicle-to-everything (V2X), device-to-device (D2D), etc. The next generation cellular system, 5G, is expected to only increase the use case possibilities, thus opening many doors for innovative products to be delivered. The DL is the communication direction from the BS to the handset or user equipment (UE). The UL is the communication direction from the UEs to the BS. The UL also consists of random access where UEs attempt to access the communication systems resources, either from power on state or initiating a new transaction. The method used to separate the DL and UL communication is called duplex. For example, this operation can be performed in the time (TDD) and/or the frequency (FDD) domains. In TDD, certain time slots are allocated to the DL and other time slots to the UL. In FDD, the UL and DL transmission occur simultaneously in different frequency bands. The benefits of TDD are a single spectrum is needed and shared (no paired spectrum is necessary), and there are symmetrical channel views (UL measurements can be used for DL communications and vice versa). The benefit of FDD is the need for less timing synchronization requirements; however, due to the frequency separation between the DL and UL, the UL measurements may not be useful for DL communications as reciprocity cannot be guaranteed. Whichever method is used, latency (time duration to access the networks resources) is becoming more and more critical as a system performance indicator. 1 Introduction to Cellular Mobile Communications 7 1.2.1 The Evolution of Mobile Technologies In this section, we will introduce the mobile radio access technologies (RATs) and comprehend their evolutionary benefits and advantages. Figure 1.1 shows the cellular standard evolution from 1G to 4G. We notice as 2G and 3G evolved, there was an increase in system complexity across multiple standards. This changed as the industry converged to a single 4G standard, where now there is an increase in complexity within a single standard. Orthogonal Multiple Access Techniques • FDMA (frequency division multiple access)  Difficult to assign multiple carriers in the same channel  Narrowband channels (less than the coherence bandwidth of the wireless channel) are desirable  Guard bands in frequency domain are needed to reduce spectral emissions into adjacent frequency bands  Finite number of orthogonal resources. • TDMA (time division multiple access)  Inter-symbol interference compensation (equalization) is needed  Uses guard bands in time domain to allow for time delay variations of UL transmissions  Synchronization of time slots across all uses is critical to not destroy the OMA principle  Finite number of orthogonal resources. • CDMA (code division multiple access)  Uses the entire bandwidth at the same time utilizing spreading codes  Finite number of orthogonal resources. • OFDMA (orthogonal frequency division multiple access)  Assigns different sub-carriers to different users (at different time slots)  Finite number of orthogonal resources. Spectrum is very precious to the operators and remains necessary to deliver increased system and user throughput. There is an industry-wide movement to not only use the traditional licensed spectrum, but also embrace the unlicensed (traditionally used by WiFi devices) and the shared spectrum whenever and wherever possible. 8 J. Boccuzzi 1.2.2 First-Generation Cellular Systems The first-generation (1G) mobile cellular system was created to enable voice communications and support mobile users when a voice call would “hand off” to another base station (or cell) as the mobile user physically traversed the cellular environment. The technology used was analog frequency modulation (FM), and the spectrum was divided into 30 kHz segments, called channels. A single user utilized the entire channel for the duration of its call. This system is called advanced mobile phone services (AMPS) and is referred to as 1G [2]. In order to support a wide coverage area, a frequency reuse technique was introduced. Here the same frequency channels were allowed to be reused by other users, at the same time, as long as the distance was large enough to cause minimal interference. This interference is called co-channel interference or inter-cell interference. In an effort to increase overall system capacity, a new technology was introduced called narrowband AMPS (N-AMPS). Here channel spacing was reduced to 10 kHz. Similarly, in an effort to introduce data services (which were not supported in AMPS), cellular digital packet data was proposed which utilized frequency channels when voice users were not present. However, it was quickly determined that an integrated voice and data wireless network is needed to effectively and efficiently deliver such services. Simple and robust discriminator detectors were used which were implementable, yet susceptible to random FM and deep fades from multipath observed in the radio environment. Forcing the mobile cellular system community to move to a different modulation technique [3, 4]. A typical cellular network architecture for 1G is provided in Fig. 1.4. Where a cell is denoted as a hexagonal shape. To be able to increase capacity, the cells can be divided into smaller cells, also called sectors. The mobile telephone switching office (MTSO) connects to base transceiver stations (BTS) and the public switched telephone network (PSTN). It also controls handovers, call routing, registration, authentication, etc. This was a circuit switched (CS)-based network. The network BTS BTS PSTN BTS BTS MTSO BTS BTS BTS Fig. 1.4 1G network architecture block diagram 1 Introduction to Cellular Mobile Communications 9 used licensed spectrum to deliver the voice services, spectrum that operators purchased from the relevant governing bodies. The 1G analog cellular standards globally deployed are listed below. Note a single global standard did not exist. • Advanced mobile phone services (AMPS)—US based  Analog FM modulation, FDD duplex, FDMA-based multiple access  Supports N-AMPS for narrowband, the channel bandwidth was decreased from 30 to 10 kHz. • Nordic mobile telephone (NMT)—The Nordic countries  Analog FM modulation, FDD duplex, FDMA-based multiple access  Channel bandwidth was dependent on the frequency band deployed: either 25 kHz or 12.5 kHz  Supported roaming in European countries. • Total access communication system (TACS)—UK based  Variant for Japan available (J-TACS)  Analog FM modulation, FDD duplex, FDMA-based multiple access  Channel bandwidth 30 kHz. 1.2.3 Second-Generation Cellular Systems The second-generation (2G) mobile cellular systems were created to expand the voice user capacity as well as to offer an integrated data services capability. The technology moved away from analog and toward digital modulation. This shift to digital enabled better quality voice communications via usage of voice coders (vocoders), support of data services, initially through short messaging services (SMS), enabled encryption to support security, and increased system capacity. This generation created a shift from FDMA to TDMA and CDMA. These were very interesting times the cellular users were facing; by this we mean being exposed to incompatible 2G cellular systems. The European community was backing global system for mobile communications (GSM), while the USA was struggling with two competing standards: IS-54 (later renamed IS-135) based on TDMA and IS-95 (later renamed CDMA-One) based on CDMA. All three of these cellular standards had technical merit. In order to increase system capacity, not only was the frequency band divided into channels, but also time was divided into time slots for TDMA. In the CDMA case, each user’s information was scrambled and frequency spread by a pseudonoise (PN) sequence; all users transmitted at the same time over the entire channel. 10 J. Boccuzzi These standards used licensed spectrum purchased by network operators from the local spectrum governing body. Receiver complexity was growing exponentially especially when considering data rates, modulation scheme, and number of antennas involved have increased. The 2G digital cellular standards globally deployed are listed below. Note a single global cellular standard did not exist. • GSM—single standard in Europe  TDMA based  Digital modulation (GMSK), FDD duplex  Channel bandwidth = 200 kHz  Frame duration = 4.615 ms  Time slot duration = 0.557 ms (8 slots/frame)  Data Rate = 270.833 Kbps  Evolved to general packet radio services (GPRS), also considered 2.5G  Evolved to enhanced data rates for GSM evolution (EDGE), also considered 2.75G. • IS-54 (also called IS-136)—standard in US  TDMA based  Digital modulation (π /4-DQPSK), FDD duplex  Channel bandwidth = 30 kHz  Frame duration = 40 ms  Time slot duration = 6.67 ms (6 slots/frame)  Data rate = 48.6 Kbps. • IS-95 (also called CDMA-One)—standard in US and Korea  CDMA based, developed by Qualcomm  Digital modulation (QPSK, O-QPSK), FDD duplex  Frame duration = 20 ms  Data rate = 115 Kbps. These standards were all circuit switched (CS)-based networks, which over time, had extensions (e.g., evolving from 2G → 2.5G → 2.75G) which allowed interfacing to packet switched (PS)-based networks. Due to economies of scale, deployment costs, patent policies, and global backing, GSM held the largest piece of the cellular market share. The users’ appetite increased thus forcing 2G to take incremental evolutionary steps such as 2.5G (GPRS) and 2.75 (EDGE). Both of which were created to increase the user data rate beyond the baseline GSM capability as well as add packet services capability. These systems are very much in use today [5]. 1 Introduction to Cellular Mobile Communications 11 OSS Core Network VLR BTS GMSC MSC BTS HLR BSC BTS SGSN Voice PSTN AUC GGSN Data PDN BSS Fig. 1.5 2G GSM network architecture block diagram The GSM network architecture block diagram is shown in Fig. 1.5 and is made up of the following network elements: • Base station subsystem (BSS) which is composed of two parts: BTSs and base station controller (BSC) • Operation and support subsystem (OSS) which controls and monitors the overall GSM network • Mobile switching center (MSC) which provides registration, authentication, call location, call routing, etc. • Home/visitor location register (HLR/VLR), a database of subscriber information • Gateway mobile switching center (GMSC) obtains subscriber information from HLR to route calls to correct MSC • Serving GPRS support node (SGSN) for packet routing and mobility management • Gateway GPRS support node (GGSN) organizes the GPRS network and external packet-switched internetworking. 1.2.4 Third-Generation Cellular Systems This third-generation (3G) digital cellular system was created to increase system user capacity and satisfy the increasing data rate appetite. This generation provided users the ability to surf the Internet and have simultaneous voice and data services. It also was the ecosystem catalyst to introduce video applications to the cellular user’s devices. Both CS and PS services were supported from its initial definition. At this point, in the cellular evolution, mobile access to the Internet was becoming more and 12 J. Boccuzzi more important. The MA technique shifted from using both TDMA and CDMA to standardizing on CDMA. CDMA-One evolved into CDMA2000, and GSM/IS-136 evolved into Wideband CDMA (WCDMA). CDMA is a multiple access technique where multiple users are separated by PN codes and transmit at the same time over the whole bandwidth allocated. It is well known as more users transmit, intra-cell interference grows called multiple access interference. A power control mechanism was used in the system to not only improve performance in a multipath fading environment, but also control the interference introduced by each additional user in the system. Power control was the solution to the near-far problem, with its goal of having the UE transmission flexible so that all users received by the NodeB would have comparable energy. This created a solution where all users were able to equally interfere with each other. The international telecommunications union (ITU) provided 3G goals in the form of IMT-2000 requirements. The 3GPP standards body was formed and created specifications to support implementations which satisfied these ITU requirements. The 3G cellular system continued to use the licensed spectrum. The small cells concept was introduced in the standard and was called HomeNB. Carrier aggregation (CA) was a seed planted into the 3G system as a method to evolve and support higher user data rates. This seed grew and is presently benefiting the modern 4G systems. Multiple-input and multiple-output (MIMO) spatial multiplexing was also a seed planted into 3G where multiple streams or layers were transmitted to the user (provided the channel matrix rank requirement was satisfied). Higher-order modulation (HOM) was also standardized; a movement from 16-QAM to 256-QAM in a land mobile cellular system was very new during these times. An example of a rake receiver, designed to counter the effects of multipath fading, used for the reception of the WCDMA downlink signal is provided in Fig. 1.6. A key WCDMA system design parameter is to have the transmission bandwidth be larger than the coherence bandwidth of the wireless channel so that multipath (or echos) can be used to exploit time diversity of the channel. The rake receiver consists of N fingers which individually track multipath and demodulate the respective waveforms. Each finger is assumed to demodulate the common pilot channel (CPICH) to support channel estimation [6, 7]. Receiver complexity grows linearly with data rate, modulation scheme used and number of antennas supported. In the WCDMA standard, both FDD and TDD duplex options were provided for paired and un-paired spectrum, respectively. To aid receiver digital signal processing, both common and dedicated pilot symbols were inserted into the waveforms. A complete shift from non-coherent detection to coherent detection was recognized by the cellular industry. 1 Introduction to Cellular Mobile Communications 13 Searching P() Candidate List Echo Profile Manager 1 2 AcƟve List 1  ... Finger #1 Rx Signal N C O M B I N E R 2 Finger #2 . . . N RAKE Output Finger #N CPICH Despreader Finger Input X S*(N) DPCH Despreader De-Pilot & Chan. Est. X C-CPICH(N) X X MA ( )* p* Delay X Finger Output Channel CompensaƟon C-DPCH(N) Fig. 1.6 3G WCDMA rake receiver block diagram The 3G WCDMA network architecture is shown in Fig. 1.7. The NodeB replaced the BTS functions, and the radio network controller (RNC) replaced the BSC functions. WCDMA is also called universal mobile telephone system (UMTS). The UMTS terrestrial radio access network (UTRAN) consists of NodeB and RNC groupings [8]. 14 J. Boccuzzi UTRAN NodeB Core Network NodeB RNC NodeB MSC & VLR GMSC Voice PSTN HLR & AUC SGSN NodeB NodeB NodeB NodeB: o Power Control o ModulaƟon o Spreading o Error CorrecƟon GGSN Data PDN RNC RNC: o Seƫng Power Parameters o Common Channel Scheduling o Dedicated Channel Scheduling o Handover Control o QoS o Outer Loop Power Control o Radio Resource Control o Sets OperaƟng Frequencies o Admission control Fig. 1.7 3G WCDMA network architecture block diagram The 3G cellular standards globally deployed are listed below. Note that a single global cellular standard did not exist. • WCDMA (also called UMTS)  Digital modulation (QPSK, 16-QAM, 64-QAM, etc.), FDD/TDD duplex  Channel bandwidth = 5 MHz (with a chip rate = 3.84 Mcps)  Frame duration = 10 ms  Time slot duration = 0.667 ms (15 time slots/frame)  Data rates up to 1 Mbps  Defined by the 3GPP standards body. • CDMA2000  Digital modulation (QPSK, 16-QAM, 64-QAM), FDD duplex  Channel bandwidth = 1.25 MHz × 3  Frame duration = 10 ms  Time slot duration = 0.667 ms (15 time slots/frame)  Data rates up to 1 Mbps  Defined by 3GPP2 standards. A WCDMA high-level functional block diagram of the downlink transmitter is shown in Fig. 1.8. Each cell has a unique scrambling code, whereas the same spreading codes (orthogonal variable spreading factor (OVSF)) are reused in every cell. The spreading codes were also called channelization codes. The block diagram of the uplink transmitter is also shown in Fig. 1.9. Each cell has a unique scrambling 1 Introduction to Cellular Mobile Communications I P-CCPCH, S-CCPCH, CPICH, etc. QPSK 15 X OVSFk Q X SCH ... I DPCHj QPSK Complex Spreading X OVSFj Q + X X LPF LPF + Quad Transmit Mod Signal ... SDL HS-DSCH QPSK 16QAM 64QAM Etc. I X OVSFn Q X Fig. 1.8 WCDMA downlink transmitter block diagram DPDCH (1,3,5) HS-DPCCH E-DPDCH DPDCH (2,4,6) DPCCH HS-DPCCH E-DPDCH E-DPCCH Cd d X X Chs hs X X Ced ed X X Cd d X X Cc c X X Chs hs X X Ced ed X X S U M Complex Spreading X LPF LPF Quad Transmit Mod Signal SUL S U M Fig. 1.9 WCDMA uplink transmitter block diagram code, spreading codes are also reused in every cell. The difference is the uplink also uses quadrature multiplexing between the I and Q channels [9, 10]. The WCDMA cellular system evolved to what is called high speed packet access (HSPA)1 which consisted of both the downlink (HSDPA) and uplink (HSUPA) components. HSPA was created because an efficient way to deliver packet services was 1 The CDMA2000 cellular system evolved to what is called evolution-data optimized (EV-DO) to support data only extension. 16 J. Boccuzzi Data CRC Turbo Encode HARQ PHY Segment Intlv. ModulaƟon AdapƟve ModulaƟon & Coding (AMC) MulƟ-Carrier Spreading Tx IQ Samples UE Feedback Info Fig. 1.10 HSDPA transmitter block diagram UTRAN NodeB Core Network NodeB RNC NodeB MSC & VLR GMSC Voice PSTN HLR & AUC SGSN NodeB NodeB NodeB NodeB: o Power Control o ModulaƟon o Spreading o Scheduling o Dynamic Resource AllocaƟon GGSN Data PDN RNC RNC: o Seƫng Power Parameters o Handover Control o QoS o Outer Loop Power Control o Radio Resource Control o Sets OperaƟng Frequencies o Admission control Fig. 1.11 HSDPA network architecture block diagram needed. HSPA introduced the shared channel concept and adaptive modulation and coding (AMC) supporting hybrid automatic repeat request (HARQ). Also, in HSPA • Whole frequency band was used (no frequency reuse greater than 1). • Users scheduled on time slot (referred to a transmission time interval (TTI) with a duration of 2 ms) basis and used PN codes as physical resources. • Network architecture flattening concept was introduced to support low-latency communications. An HSDPA block diagram is shown in Fig. 1.10. Each user packet is protected and transmitted to the UE; an acknowledgement (ACK) is expected to ensure error-free communication. In the event of negative acknowledgements (NACK), the scheduler will decide which combination of coding, modulation, and physical resources should be used to increase the likelihood of error-free communication [11]. The 3G HSDPA network architecture block diagram is shown in Fig. 1.11. Certain functionality (highlighted in the figure) previously performed in the RNC are now performed closer to the edge of the access network within the NodeB—supporting the network flattening initiative. Evolving WCDMA further became a great concern to cellular system designers. Every known tool was being used to increase the user data rate. HOM was used to 1 Introduction to Cellular Mobile Communications 17 increase the data rate within an allowed spectral bandwidth. MIMO, in the form of spatial multiplexing, was used to increase the data rate within an allowed spectral bandwidth. The spectral bandwidth was also increased in the form of aggregating carriers, to increase the data rate; however, the spectral efficiency remained unchanged. Increasing the single-carrier bandwidth brought along increased concerns. The baseline WCDMA system used a rake receiver which performs better when the processing gain is larger, rather than smaller. This bandwidth expansion factor (signal spreading operation), coupled with the desired high user data rates, prohibited the conventional WCDMA system from evolving further. 1.2.5 Fourth-Generation Cellular Systems This fourth-generation (4G) digital cellular system was created to support the exponential system capacity and data rate appetite. Much higher data rates were required to enable mobile Internet access and video applications. Long-term evolution (LTE) is also known as 4G and only supports PS-based networking. The standard is also evolving to use licensed, unlicensed, and shared spectrum options—all with a common goal of increasing the user data rate, increasing system capacity, lowering latency, and improving the user experience. The ITU provided 4G goals in the form of IMT-2010 requirements. At this point in the cellular evolution, the industry converged to a single standard, LTE. The LTE cellular system is based on OFDMA where the TTI has been reduced from 2 ms (used in the 3G cellular system) to 1 ms. This TTI reduction improved performance by being able to more quickly react to changing channel conditions, so more efficient scheduling algorithms can be used. The reduced TTI also provided a reduction in the end-to-end latency. The frequency bandwidth options have also increased: 1.4, 3, 5, 10, 15, and 20 MHz to provide flexible bandwidth deployments. To efficiently support FDMA multiple access, OFDMA (via inverse fast Fourier transform (iFFT) and FFT operations) was chosen which divided the frequency band into sub-channels (or sub-carriers) of 15 kHz spacing. To keep receiver signal processing complexity to a minimum, it was desirable to have the sub-carrier spacing less than the coherence bandwidth of the wireless channel. To deliver higher data rate services, MIMO support is mandatory to accommodate multiple layers through spatial multiplexing [12]. Recall that with a TDMA system, the increased data rate (or decreased symbol time duration) caused the receiver to use an equalizer to combat inter-symbol interference (ISI). The higher data rates, higher-order modulation, and longer delay spreads caused a significant increase in equalizer complexity. With a WCDMA system, the increased data rate (or decreased chip time duration) forced the receiver to make use of the time diversity of the wireless channel but required a large processing gain to adequately combat ISI. With WCDMA, the motivation was to have the transmission bandwidth larger than the coherence bandwidth of the wireless channel; however, for OFDM, the opposite holds true. OFDM addresses the higher data rate demand 18 J. Boccuzzi Fig. 1.12 Time/frequency representation of the OFDM signal for LTE standard. There are four different symbols (QPSK) each represented by one color by generating many narrowband channels, where each narrowband channel can be seen to experience frequency flat disturbance. This observation coupled with the fact the frequency domain signal processing is possible, made OFDM a very attractive multi-carrier technique to mitigate a frequency selective fading environment. In OFDMA, users are multiplexed in both the frequency and time domains, as depicted in Fig. 1.12 for LTE system. On the LTE air interface, the unit of allocation is a physical resource block (PRB). A PRB is 12 sub-carriers by 7 OFDM symbols which is equal to 84 modulation symbols. The minimum allocation to a single UE during a subframe (1 ms) is 2 PRBs with one PRB in each slot of the subframe. Thus, a UE will get a total of 2 PRBs/subframe which equals to 168 modulation symbols/subframe. Note that not all these 168 modulation symbols can be used to transmit user information, but some of these modulation symbols are used for synchronization or as pilot for channel estimation. Each PRB contains 12 sub-carriers, and thus have a bandwidth of 12 × 15 kHz = 180 kHz. Figure 1.12 represents 2 PRB (2 × 7 symbols × 12 sub-carrier). Assuming QPSK modulation, there are four different symbols represented by four different colors. Each color represents one resource element (RE) and carries two bits with QPSK modulation. A block diagram providing an example of the OFDMA waveform generation is provided in Fig. 1.13. We also highlight the various points on the processing chain that can significantly impact system performance. The number of sub-carriers (SCs) has a direct impact on the data rate and user capacity of the system. From a system perspective, this value should be as large as possible; however, the occupied bandwidth needs to be controlled via the spectral shaping function. The OFDM symbol’s peak to average power also impacts the occupied bandwidth and imposes linearity requirements to be met to minimize any increased spectral growth. Lastly, note the addition of cyclic prefix (CP) removes ISI from the wireless channel. The CP time duration should be large enough to exceed the length of the 1 Introduction to Cellular Mobile Communications Combat ISI User1 Data … UserK Data IFFT … P/S CP 19 Reduce Spectral Emissions Spectral Shaping RF SecƟon … *Not Standardized by 3GPP. (Vendor Specific) fs = SCS x Nŏ # of SC SCS & Nŏ Spectral re-growth due to PAPR and Non-linearity. Fig. 1.13 OFDMA waveform generation with K sub-carrier (SC) wireless channel time dispersion, but also as small as possible to maximize the user data information during the subframe duration. There are certain disadvantages with OFDM that should be addressed in future systems, such as: • CP overhead: The need for adding the CP introduces redundancy to the transmitted and thus results in a loss in spectral efficiency. This loss is larger when long CP is used or when the sub-carrier spacing (SCS) is small. • Sensitivity to frequency and timing offsets: In order to keep the orthogonality in OFDM, the transmitter and receiver must have exactly the same reference frequency. Any frequency offsets will ruin the orthogonality, causing sub-carrier leakage known as inter-carrier interference (ICI). • High out-of-band (OOB) emission: OFDM assumes rectangular pulse in time domain which is equivalent to sinc in the frequency domain which has infinite bandwidth theoretically and cause relatively high (OOB) emissions. The lack of spectral shaping (either filtering or windowing) is creating large spectral side lobes in the transmit spectrum. • High peak-to-average power ratio (PAPR): The envelope of the OFDM waveform has a large variation which causes problems when encountering a nonlinear device such as a transmit power amplifier. The high PAPR in OFDM compared to the single-carrier transmission technique is due to the summation of the many individual sub-carriers with different phases which can results in a high PAPR when added together. In terms of occupied bandwidth, 3GPP did not specify any spectral shaping technique in LTE and such each equipment and device vendor implements their own solution. OFDM sub-carriers are treated as sin(x)/x, so applying spectral shaping will help produce a more spectrally efficient waveform with minimal or no impact to orthogonality performance. These spectral side lobes are relatively high in power due 20 J. Boccuzzi EUTRAN Control Plane EPC MME eNodeB HSS eNodeB eNodeB S-GW Data Plane eNodeB: o Packet Scheduling o Dynamic Resource Allocation o Load Control o Radio Resource Management o Admission Control P-GW Voice & Data PDN PCRF EPC: o Data routing o Mobility Handling o Sets Config Parameters Fig. 1.14 4G LTE network architecture block diagram to the assumed rectangular shaping. These high side lobes require a large guard band to reduce the out of band interference. Applying spectral shaping techniques, such as filter bank multi-carrier, universal filtered multi-carrier, etc., which are discussed in Chap. 2 through Chap. 4 of this book, will help reduce the side lobes. The other component can be found when viewing adjacent transmitted OFDM symbols in the time domain; there will be phase discontinuities that also cause spectral emissions. The peak to average power concern of OFDM can be viewed as a weighted sum of sinusoids, which helps explain the large PAPR of the generated OFDM symbol (as high as 12 dB). A high PAPR can be problematic if the waveform encounters nonlinarites. Crest factor reduction is a technique used to reduce PAPR and a technique used to compensate for nonlinear distortion is digital pre-distortion. The LTE uplink waveform uses the single-carrier FDMA (SC-FDMA) method to reduce the PAPR impact on portable devices. Lastly, to minimize ISI and provide the property of cyclic convolution, a small part of the end of each symbol is added to the beginning of each transmitted OFDM symbol. CP size depends on delay spread and LTE uses a short and a long CP. For LTE the short CP has a value of 4.7 µs, which is approximately 8% of the symbol time. Generally speaking, if a large delay spread is not expected to be encountered in a particular deployment, then a lower or shortened duration CP should be used. The 4G LTE network architecture block diagram is shown in Fig. 1.14. Note that we now have a single global cellular standard. The evolved packet core (EPC) replaced the core network (CN) functions, and the eNodeB replaced the NodeB functions. The EUTRAN consists of eNodeB and EPC groupings. The EUTRAN to EPC connection consists of both control plane and user plane signaling. This was the beginning of an effort to separate user and control planes to allow for different evolution rates and network deployment scenarios/options [13]. In LTE, the SCS is set to 15 kHz which translates to an OFDM symbol duration of 66.67 µs. There are 14 data symbols per time slot (1 ms), and every OFDM symbol 1 Introduction to Cellular Mobile Communications 21 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 Release 8 LTE-Advanced Release 9 LTE MBMS Release 10 LTE-Advanced Pro 2xCA Relays MBMS DL8x8 Release 11 Relays CoMP CA MBMS [3GPP.org] Release 12 CA UL8x8 MTC MBMS CoMP 256QAM MU-MIMO D2D DC Release 13 UL64QAM LAA MTC D2D LWA LAA MTC 1024QAM CA NB-IoT V2X 5G Phase 1 Release 14 5G Phase 2 Release 15 Release 16 Fig. 1.15 3GPP release feature timeline requires a CP. Including the time durations of all the CPs, results in an equivalence of 15 symbols (data + CP) in a time slot. The largest FFT size is 2048 which creates a sampling frequency of 30.72M samples/s. Below we list some LTE features in the 3GPP standards body release schedule. • Data rates increased by employing HOM, MIMO, and CA • New features are added: DC, V2X, IoT, D2D, etc. • Advanced technology to support performance improvement: coordinated multipoint (CoMP), latency reduction, beamforming • Spectral and RAT flexibility: licensed, shared, unlicensed and LTE-WiFi aggregation. As discussed earlier, certain seeds were planted in the 3G cellular system to observe how beneficial they would become to the later generations. For example, CA continues to be useful, MIMO has become more and more essential, and HOM is effective. In fact, all three techniques have been successfully tested and are commercially deployed; they are required to achieve the greater than 1 Gbps data rate in LTE [14]. Figure 1.15 also reveals a departure from the typical cellular system evolution which has been given to increasing data rates, increasing user capacity, and lowering latency. This new trend clearly shows the additions of new services (or features, use cases) that the industry is recognizing are required as society demands. These new services were not really intended to be addressed when 4G was created in 2006. The width of new expected services is growing very rapidly (as depicted in Fig. 1.3). The network is also experiencing its own evolutionary growth. The softwaredefined networking (SDN) and network function virtualization (NFV)-based “wave front” is departing the data center [15], making its way through the core network and on to the wireless access network. These CN and RAN workloads have 22 J. Boccuzzi started to be implemented on a homogeneous, general-purpose CPU-based platform (instead of the traditional dedicated logic + digital signal processor + microcontroller approaches). This has sparked the cloud-RAN movement utilizing the Information and communication technology (ICT) industry benefits. 4G will deploy these technologies, and when they are successful, the expectations are that 5G will be a network upgrade. 1.3 5G Drivers, Technologies, and Spectrum The network architecture block diagram of fifth-generation (5G) of cellular is shown in Fig. 1.16. The 5G core (5GC) replaced EPC; next-generation radio access network (NG-RAN) consists of distributed unit (DU) and centralized unit (CU) grouping; the gNodeB replaced the eNodeB. As we will discuss, providing a flexible and scalable network architecture is essential for 5G. In this theme, the combination of DU and CU were introduced to support various RAN split options to extract the above-said benefits [16]. The 5GC elements consists of the following: • Access and mobility management function (AMF): performs ciphering and integrity protection, mobility management, authentication, and authorization, etc. • Session management function (SMF): performs UE IP address allocation and management, selection and control of UPF, roaming, etc. • Unified data management (UDM): performs subscription management, user data, registration and mobility management, etc. • Policy control function (PCF): performs policy rules for CP functions, etc. • User plane function (UPF): performs the external interconnect point to data network, QoS handling of UP, etc. • Application function (AF): interacts with policy framework for policy control, etc. 5GC NG-RAN Control Plane gNodeB DU PCF AF gNodeB CU gNodeB DU UDM AMF SMF gNodeB DU gNodeB DU gNodeB CU Voice UPF PDN Data Data Plane Fig. 1.16 5G network architecture block diagram 1 Introduction to Cellular Mobile Communications 23 1.3.1 5G Drivers 5G cellular systems need to make a significant jump in features and performance over LTE as incremental improvements are not wanted and do not justify the significant capital investments operators need to commit to deploy 5G services. It is important to note that we have maintained a single global cellular standard. The 5G driving factors are [17]: • • • • • • • • Increased user data rate Increased system capacity Massive number of connections Reduction in end-to-end latency Heterogenous mix of services Flexible bandwidth deployments Network flexibility Move to more energy-efficient communications. The ITU provided 5G goals in the form of IMT-2020 requirements, they are provided in Table 1.2. For comparison purposes, we have also included the IMTadvanced requirements. 5G NR will support both non-stand-alone and stand-alone modes of deployment. The NSA deployment will use LTE to provide wide area coverage, control and data planes and connection into an evolving EPC. 5G services will provide high-speed data via a dual connectivity scenario. The stand-alone deployment will provide control and data planes as well as a connection into a 5G CN. The ITU published the diagram shown in Fig. 1.17 to identify 5G services. The three significant use cases (corners of the triangle) are meant to encapsulate the expected usages of 5G in the future: • Enhanced mobile broadband (eMBB) • Massive machine to machine communication (mMTC) • Ultra-reliable low-latency communications (URLLC). These 5G use cases range from smart home, connected drones, ehealth, connected energy, autonomous cars, real-time virtual reality/augmented reality gaming, etc. The introduction of low latency techniques has started in LTE to aid the transition to transforming the network in preparations for the widely varying 5G services [17]. The 5G cellular system is expected to support these usage scenarios by employing the following technologies: • Flexible spectrum deployments: licensed, unlicensed and shared spectrum, larger and contiguous bandwidth, multi-RAT, etc. • Improved network architecture: support of the ICT industry cloud trend, SDN/NFV, network slicing, multi-access edge computing, lower latency, etc. 24 J. Boccuzzi Table 1.2 Comparison of IMT-2010 and IMT-2020 requirements System metrics IMT-2010 IMT-2020 Peak data rate DL: 1 Gbps UL: 0.5 Gbps DL: 20 Gbps UL: 10 Gbps Area traffic capacity (Mbps/m2 ) Network energy efficiency (bit/Joule) 0.1 10 1x 100x (less) Connection density (devices/km2 ) 104 106 Latency (ms) 10 1 Mobility (kmph) 350 500 Spectral efficiency (bps/Hz) 1x 3x (more) User expected data rate (Mbps) 10 100 Fig. 1.17 IMT-2020 usage scenarios Comments Maximum achievable data rate under ideal conditions Total traffic served per geographic area Quantity of info bits per unit of energy consumption Total number of connected devise per unit area Time from when source sends a packet to when the destination receives it (end-to-end one way) Maximum speed a defined QoS can be achieved Average data throughput per unit of spectrum and per cell Achievable data rate ubiquitously available across the coverage area 1 Introduction to Cellular Mobile Communications 25 • Flexible numerology: Support a wide variety of use cases and spectral deployments (below and above 6 GHz bands), flexible time slots and transmission bandwidths, etc. • Modulation and coding: QAM modulation continues to provide a reasonable spectral and power efficiency trade-off, polar and other forward error correcting codes. • Advanced techniques: NOMA, full-duplex, spectral shaping, etc. 1.3.2 5G Technologies Expected to be commercialized around 2019/2020 time frame, 5G mobile networks are under intense reach and development activities. Compared to the current 4G mobile networks, 5G networks are expected to support enormous system capacity, much less latency, and about 1000 times more devices per squared kilometer, among other requirements. To satisfy these requirements, several new technologies have been suggested and are being developed for 5G networks. These technologies include but are not limited to: massive MIMO, software-defined networking, mm-Wave, cloud radio access network (cloud-RAN), non-orthogonal multiple access, M2M communications, mobile edge computing, wireless caching, ultra-dense networks, and full-duplex communication. In the following, we briefly describe some of these technologies. 1.3.2.1 Massive MIMO In discussing massive MIMO, let us first address the term “massive.” It is used to denote the large number of antenna elements that are used in the antenna signal processing. The number of antennas to be considered massive should be greater than 64 elements. Massive MIMO relies on the law of large numbers to make sure that the channel and hardware imperfections (e.g., noise, fading, and hardware) average out when signals from a large number of antennas are combined in the air together [18]. Multiple antennas afford two options in which the antennas can be used: First is to provide an array gain by focusing energy in desired directions and nulling in unwanted signal directions (forming a beam). Second, is to provide spatial multiplexing gain by sending independent data streams on each antenna. Either technique can be used to increase the overall user or system data rate. Both options are shown in Fig. 1.18 [18]. First, consider using massive MIMO for beamforming; here the antenna arrays can be arranged in either linear, rectangular, or circular arrays that can also be stacked. Massive MIMO will be deployed for 4G and 5G; in fact, high-frequency bands lead to more compact, large-scale antenna arrays due to the smaller wavelength. Massive MIMO can be deployed in either FDD or TDD duplex methods, TDD systems allow 26 J. Boccuzzi U E M I M O M I M O B F U E U E Fig. 1.18 Massive MIMO examples: spatial multiplexing (left) and single-/multi-user beamforming (right) the users to invoke the theorem of reciprocity to apply what is observed on the UL to the DL. Next, consider using massive MIMO for spatial multiplexing, which has been widely used for 4G and will continue in 5G deployments. Spatial multiplexing can be achieved provided the rank of the channel matrix between the transmit and receive antennas is greater than 1. In fact, for a 4 × 4 MIMO system, full capacity is only observed if the channel matrix rank is full (in this case, a value of 4). Due to the success of spatial multiplexing in LTE, it would be logical to assume this continues for 5G and on a larger scale. This is true; however, should come with a warning. The larger the MIMO dimension, the less likely one would experience full rank. This means designing a 256 × 256 MIMO array and expecting to send 256 layers to a single user, all the time is a difficult assumption to make. This is one of the reasons 5G has limited the number of DL layers per user to 8. The implementation complexity involved in implementing massive MIMO in the digital domain is significant. Hybrid beamforming has been proposed to provide a compromise in performance/capability with complexity. This brings forth an interesting question: Assuming a maximum number of layers of 8, what can one do with the remaining degrees of freedom? Some can be used to create (or form) beams and some can be used to multiplex other users over the antenna array. This last comment is known as multi-user MIMO (MU-MIMO). Here, multiple users transmit and their collective transmissions are treated as though they came from a single source of multiplexing. The beamforming weights can create a beam in the azimuth and elevation directions. When considering beamforming, array gain can be used in a variety of ways. It can be used to the extend coverage area, reduce the transmit power of devices on the UL, improve signal-to-interference-plus-noise ratio (SINR) resulting in high user throughput and to reduce the transmit power on the DL thus improving overall power efficiency. The number of antenna elements needed depends on a few items: • • • • Array gain (coverage area, power relief, etc.) Multiplexing layers needed Multi-users expected to be serviced Frequency band used (form factor, etc.) 1 Introduction to Cellular Mobile Communications 27 • Signal processing complexity (CSI estimation, analog vs. digital domain, etc.) • System performance gains (SINR, capacity, data rate, etc.). One of the benefits of using multiple antenna techniques, for either transmitting or receiving, is the significant reduction in channel variation. This behavior is essential in combating multipath fading, and having at least 64 antennas in the antenna array significantly reduces the channel variations. Multiple 5G deployment scenarios proposed by 3GPP have varying use cases for eMBB, URLLC, and mMTC services. In these deployment scenarios, the maximum number of DL antennas discussed was 256 and maximum number of UL antennas discussed was 32. 1.3.2.2 Software-Defined Networking Network functions virtualization (NFV) and software-defined networking (SDN) are supporting the movement to a software-centric network. These capabilities offer great technical (in the form of system performance) and financial (in the form of CAPEX and OPEX) improvements to the network operators. This movement provides the network operators with tremendous benefits such as: a more manageable means to monitor the network, better support of new feature roll-outs, network relocation, etc. However, it also opens the doors for new market players (such as Internet service giants, cable service providers, etc.) who wish to establish wireless network presence. The adoption has been to initially virtualize the less timing critical functions, such as in the EPC (also called vEPC) and then transition down the protocol software stack toward the physical layer [15]. Moving to a SDN allows network operators to become nimble in deployments of various use cases. One benefit is called network slicing. Here the network will be able to dynamically pull together the access and core network functions necessary to satisfy the requirements of a specific use case (latency, bandwidth, etc.). We have seen a trend that started in 4G where a diverse set of services have emerged, and 3GPP is addressing this demand as part of LTE’s evolution. We expect this demand to increase and continue to create diverse requirements. The LTE network architecture (at its conception) has been called monolithic and needs to be more flexible and scalable as we introduce 5G services. Network slicing is a technique proposed to support these wide variety of use cases. Network slicing creates virtual network architectures based on SDN and NFV principles. These virtual networks (or slices) are created on top of a common shared physical infrastructure and can be “optimized” to meet requirements of applications, services, or operators. The virtual networks consist of a set of network functions instantiated to provide a complete end-to-end logical (or virtual) network to meet the targeted performance requirements. For example, mMTC communications rely on user capacity and not necessarily low latency, whereas autonomous cars rely on low latency and not necessarily the highest throughput eMBB services would require. Figure 1.19 provides a block diagram example of how the network may be sliced to support the various 5G services discussed above. 28 J. Boccuzzi uR-LLC Services Data Center Core Network Access Network Front Haul eMBB Services Back Haul Network Transport Network mMTC Services Fig. 1.19 Network slicing example supporting UR-LLC, eMBB and mMTC services 1.3.2.3 Multi-Access Edge Computing To support demands for lower latency, optimizations in the 5G air interface alone are not sufficient, we must also optimize the network. Multi-access edge computing (MEC) is a method of moving core network or data center centric functions closer to the edge of the network (toward the antenna) where the data will be operated upon. It has been shown using this Principle of Relocation; the user end-to-end latency can be significantly reduced. Additionally, the backhaul traffic can also be reduced since the “back-and-forth” traffic has been significantly minimized by this move [19]. MEC enables cloud computing capability to be within the access network, which is closer to the user devices. This will also be supported by fog computing. The edge of the network is considered to be the antenna within the remote radio heads (RRHs) which are connected to the radio access network (RAN). There are a number of reasons to place the computing capability at the edge of the network. The most significant reason is to reduce latency (or delay) a mobile application encounters when trying to connect to a server. This eliminates the time a packet needs to enter the wireless network before being acted upon. The closer the MEC server is to the edge, the smaller the delay the applications would encounter. Examples of the expected delays are: latencies <1 ms are needed to support industrial robots and autonomous driving applications, latencies <10 ms are needed to support augmented reality applications, and latencies <100 ms are needed to support-assisted driving applications. Figure 1.20 shows the concept of distributing the functionality which is typically located in the CN and data center (cloud computing) to the edge (fog computing). Besides lower application latency, we can observe lower backhaul traffic by not sending large packets all the way into the network to be processed and then sent all the way back to the edge [20]. MEC will perform compute and storage functionality with some market drivers for MEC deployments being: • Reduce total cost of ownership (OPEX and CAPEX) 1 Introduction to Cellular Mobile Communications C-RAN NR 29 DistribuƟng the funcƟonality. MEC New Front Haul Device Back Haul C-RAN 5G PC MEC Core Network C-RAN MEC Fig. 1.20 Network diagram displaying distributing computing functionality to the edge • Increase revenue by providing ability to create new services utilizing new technology such as artificial intelligence, content distribution network, etc. • Natural migration as virtualization proliferates out to the access network (edge and fog) • Improve performance (lower latency, reduce backhaul traffic). A point also worth discussing. Why does the network edge need to be at the antennas? We should move away from the black and white viewpoint of network/device (also known as a cell-centric view) toward a more colorful viewpoint (also known as a user-centric view) where the edge is more blurred. Many reports reveal the total wireless devices are expected to be greater than 20B devices around the 2025 time frame. We should be cognizant that the number of devices is exceeding the number of people in the world. Also, since the computing performance of devices (handheld, laptop, etc.) is becoming more and more complex and capable, devices should be considered as an extension of the network—in other words the network edge. 1.3.2.4 RAN Split The traditional and most commonly deployed fronthaul technology is based on fiber using the common public radio interface (CPRI) protocol. CPRI carries the IQ samples between the RAN and RRHs [21]. The CPRI capabilities are being stressed to support the evolution of LTE, especially when CA and massive MIMO deployments are required. This stress is due to the larger bandwidth required to transport the IQ waveform samples to the RRH, and only becomes more problematic when 5G enters the picture. Hence, next generation front haul technology is needed to support the expected 5G services [16]. A few front haul options exist: One solution is to standardize on another protocol that can use higher bandwidth technologies such as ethernet-based protocols (e.g., 25, 100 GB) while another component of the solution is to use a different RAN split options (with lower bandwidth requirements). A few RAN split options exist 30 J. Boccuzzi (proposed by the 3GPP) that can reduce the front haul bandwidth requirements as well as latency, and potentially trade-off performance [16]. One RAN split option transports modulated symbols, which is a point in the processing chain that is prior to being converted to the time domain by the iFFT operation on the transmit side. The frequency domain sampling rate is much lower, thus allowing more carrier-antenna combinations to be supported. This technique still maintains a centralized processing capability to allow for more complicated scheduling across cells. Another RAN split option transports user data packets, for example PDCP packets. These packets have had their headers compressed and properly ciphered and protected to address any security concerns. This results in a much lower data rate, but loses the centralized processing ability. In addition to splitting the RAN functions, control and user planes are migrating to become separable to allow for separate evolution rates, lower latency, and support new deployment scenarios. This will, for example, provide the ability to have a control plane supplied by a wide area LTE macrocell while the user plane supplied by a small cell 5G. The 5G cellular system will also be based on OFDMA where the time slots have been defined to be variable to handle the widely varying requirements across all the expected services. As noticed in 4G, spectrum is extremely important to provide higher data rates. The OFDMA parameters (sub-carrier spacing, time slot duration, iFFT/FFT size, etc.) have been made flexible to support various spectral deployments. 1.3.3 5G Spectrum and mm-Wave Band LTE has a maximum bandwidth of 20 MHz, as previously discussed user data rates have been increasing due to the use of HOM, MIMO spatial layers, and CA techniques. While present solutions support up to 5 CA, it is worth mentioning the 3GPP LTE specifications can support up to 32 carriers. This means if we sacrifice the complexity in supporting many carriers, there is plenty of room to further increase the data rates. In many cases, operators need to aggregate licensed and unlicensed spectrum (via license-assisted access) to reach the Gbps data rates. In fact, band number 46 (B46), whose spectral range is 5.15–5.925 GHz. is defined for that intention [22]. 5G is defined to have a maximum bandwidth of 100 MHz for frequency bands below 6 GHz. Note that large bandwidth delivers high data rates, but lower bandwidth can also provide 5G services. This coupled with the fragmented spectral band allocations is a reason to support the need for flexibility in the OFDMA parameters discussed above. Another option besides traditional licensed and unlicensed (5–5.9, 64–71 GHz) spectrum usage is to use the citizens broadband radio service (CBRS) spectrum. The CBRS spectrum range is 3.55–3.7 GHz (totaling 150 MHz of bandwidth) and is governed by a three-tiered spectrum authorization framework to accommodate users on a shared basis with incumbent federal and non-federal users of this band. A summary of the items that need to be considered in using 5G frequency bands is provided in Fig. 1.21. A point worth mentioning, in these new 1 Introduction to Cellular Mobile Communications 1 GHz 31 6 GHz o Large Range/Coverage o Indoor PenetraƟon o Mobile o IoT 100 GHz o Medium Range/Coverage o Mobile o Mission CriƟcal o IoT o Limited Range/Coverage o Mobile o Mission CriƟcal Support A Flexible & Scalable soluƟon to handle the wide variety of Use Cases is needed. High Mid Low Low Mid High Frequency Band Fig. 1.21 5G frequency band considerations frequency bands the availability of paired spectrum to support FDD is minimal forcing the industry to focus more on TDD deployments. Hence, not only do we expect bandwidth availability to vary across the low (<1 GHz), medium (<6 GHz), and high (>6 GHz) frequency bands, but we should also expect the duplex method to also vary. Some operators are focusing on fixed wireless access to deliver high-speed 5G services (approximately 1Gbps) in place of cable/fiber deployments as initial 5G deployments in mm-Wave bands instead of, and in addition to, supporting mobile broadband applications. This approach will help develop a mm-wave-based ecosystem that will enable 5G technologies which need to be used for battery operated devices. The heterogeneous spectrum usages discussed so far assumed the licensed spectrum is always used; there is an initiative to support services which only use the unlicensed spectrum (like WiFi today). The MulteFire alliance allows LTE technology (and 5G) to be exclusively used (in a stand-alone fashion) in shared and unlicensed spectrum to enable private services, neutral host network architecture, industrial networks, etc. Spectrum for 5G service will be challenging. Some of the new frequency bands being considered in 5G NR by region is provided in Table 1.3. Operators and equipment manufacturers are faced with various options to identify spectrum (re-farm, acquire new, partner, etc.). We see reasonable convergence (toward Global Harmonization) around the 3–4 GHz frequency bands around the world and, at the moment, less so in the USA. 32 J. Boccuzzi Table 1.3 New 5G frequency bands Region Freq. band (<6 GHz) Europe China Japan Korea United States 3.4–3.8 3.3–3.6 3.6–4.2 3.4–3.7 3.55–3.7 <6 GHz bandwidth Freq. band (>6 GHz) >6 GHz bandwidth 400 MHz 300 MHz 800 MHz 300 MHz 150 MHz 24.25–27.35 3.1 GHz 27.5–29.5 26.5–29.5 27.5–28.35 2 GHz 3 GHz 0.85 GHz 1.4 Waveform Design for 5G As discussed in Sect. 1.2.5, CP-OFDM has certain limitations that makes it not the most suitable waveform for all 5G applications. However, due to its advantages and for backward compatibility reasons, OFDM will still be the main waveform for 5G systems. On the other hand, due to its limitations, certain modifications have been proposed in the literature to make it suitable for 5G application. Among these limitations, fixed SCS (in 4G LTE), CP overhead, and high OOB emission are the most important. Before listing these new waveforms, in the following, we discuss these limitations one by one. Internet of Things (IoT) is a main contributor to the exponential growth of users in 5G. IoT devices, e.g., sensors, usually send sporadic short data packets and have a limited power. On the other hand, for eMBB a large volume of data should be transmitted in a short amount of time. Such varying characteristics of the bursts to be transported makes CP-OFDM with a fixed SCS an inefficient waveform. For IoT applications, 5G waveform is required to support a transmission mode with very low air interface latency enabled by very short frames [23]. To enable low-latency transmissions, very short TTIs are required, for energy-efficient communications by minimizing on times of low-cost devices. OOB emission can be reduced by applying time domain windowing that smooths the transition from one symbol to another. As discussed earlier, the OFDM parameters have been made flexible to support various spectral deployments. Specifically, the SCS numerology is now 15, 30, 60, 120, 240, and 480 kHz. The maximum FFT size is now set to 4096, and the maximum number of resource blocks (RBs) that can be transmitted was also increased to 275 (or 3300 sub-carriers). Besides spectral deployment advantages, these options allow more spectrally efficient transmissions to occur. For example, in LTE, we utilize 18 MHz of the available 20 MHz of spectrum, with the adoption of the new numerology we are capable to utilize up to 99 MHz of the available 100 MHz of spectrum. In considering an example 100 MHz deployment, a set of parameters can include SCS = 30 kHz and FFT size = 4096 thus resulting in a sampling frequency of 122.88 MHz (which is 4 times greater than LTE while utilizing 5 times greater spectrum). Having a flexible OFDMA system is critical to efficiently deploying the wide variety of 5G services [24]. Based on propagation characteristics, it is expected the 1 Introduction to Cellular Mobile Communications 33 lower frequency bands will be used for large-area deployments with smaller SCS and the associated larger subframe time durations, while higher frequency bands are expected to be used for the dense deployments with larger SCS and their associated smaller subframe time durations. These are examples and other others can surely exist. As can be seen, this deployment capability can be easily derived from a flexible numerology system. To reduce OOB emission, various filtering and windowing-based solutions are applied to OFDM [23]. Filtered OFDM (F-OFDM), windowed OFDM (also known as weighted overlap and add or WOLA-OFDM), universal filtered OFDM (UF-OFDM), filter bank multi-carrier (FBMC), and other candidates have been suggested for new waveform in 5G and beyond. These candidates will be studied Chap. 2 through Chap. 4. 1.5 Multiple Access Techniques in 1G to 5G Let us recall the multiple access techniques deployed in the cellular systems so far. In the first-generation, cellular systems employed FDMA where the frequency band was divided up into frequency channels and users were assigned channels. In the second-generation, TDMA and CDMA were used and in both cases the frequency band was divided into smaller frequency channels. In TDMA, the new dimension of time was used as a resource (time slot), and in CDMA, the new dimension in the code domain (PN sequence) was used. TDMA receiver complexity grew exponentially as the data rate was increased, modulation order increased and number of antennas increased. In the third-generation, CDMA was deployed which utilized larger bandwidth and more importantly introduced the concept of a shared channel. Here the physical resources allocated to users are: time slots and PN codes. CDMA technology complexity increased as the data rate increased. The resulting WCDMA spread bandwidth required a larger processing gain to have reasonable inter-path interference suppression capabilities. The fourth-generation of cellular systems deployed OFDMA and kept the shared channel concept. Here the physical resources were time slots and frequency subcarriers. OFDMA technology maintained the flexibility of resources and kept the available information bandwidth at the desired value. Due to the use of the cyclic prefix and frequency domain signal processing, the receiver complexity is manageable. It is also a reason why the fifth-generation has decided to continue with OFDMA. We would like to briefly discuss the differences between the DL and UL communication links; this is shown in Fig. 1.22. The DL starts with a common signal transmitted which consists of the aggregate sum of all UEs in that cell. Each UE is physically located in a different cell position and thus experiences different multipath fading, denoted by h i . Each UE has its own additive noise, denoted by n i . The UL starts with individual signal transmissions that encounter different fading, due to the physical locations within a cell. These individual signals are summed at the base 34 J. Boccuzzi n1 + h1 n h1 UE1 UE1 n2 h2 . . . BS hk + h2 + UE2 . . . UE2 nk hk BS + UEk UEk Fig. 1.22 Downlink and uplink communications R2 User 2 rate User 2 rate R2 User 1 rate R1 User 1 rate R1 Fig. 1.23 The rate region for two-user DL and UL station receive antenna, where the base station adds its additive noise. The rate regions of the DL and UL multiple access are shown in Fig. 1.23, for the two-user cases. The figures compare OMA, denoted by the solid line, against superposition coding, denoted by the dashed line. The left curve is used to showcase the DL capacity while the right curve is used to showcase the UL capacity [25–27]. 1.6 What is Non-Orthogonal Multiple Access? In an orthogonal multiple access (OMA) system, such as TDMA and FDMA, orthogonal resource allocation is used among users to avoid intra-cell (inter-user) interference. The number of users that can be supported is then limited by the number of orthogonal resources available. Non-orthogonal multiple access (NOMA) allows and utilizes intra-cell interference in the resource allocation of users. Interference cancelation techniques, such as success interference cancelation (SIC) or multi-user detection (MUD) are used to mitigate this interference. NOMA is a technique being considered by 3GPP in Release 16. 1 Introduction to Cellular Mobile Communications 35 OMA Transmit Power Transmit Power NOMA Frequency Frequency Fig. 1.24 OMA and NOMA power and spectral allocations NOMA refers to non-orthogonal MA that can support multiple users within a single resource and thus can improve user and overall system throughput. It can be realized within the power domain, code domain or other domains. Power domain NOMA (see Part II of this book) exploits the channel strength differences between users and is the optimal capacity-achieving multiple access technique in a single-cell network, as shown in Figs. 1.22 and 1.23. More details and the multi-cell cases can be found in Chap. 5 and [25–27]. The spectrum and power allocation for power domain NOMA is graphically compared with that of OMA in Fig. 1.24. In the NOMA-based systems, two users can share the same spectral band, where each user has a different power allocated to it. Code domain NOMA schemes (see Part III of this book) usually exploit low-complexity multi-user detection schemes. Sparse code multiple access (SCMA), interleave division multiple access (IDMA), and low-density spreading (LDS)-CDMA are notable examples of code domain NOMA. Some of the possible benefits when using NOMA are [27]: • Massive connectivity: While OMA is limited by the number of orthogonal resources, NOMA is not. Theoretically, NOMA can support an unlimited number of users. • Lower latency: OMA waits for available resource blocks to transmit which is accomplished by waiting for an access grant whereas NOMA can support a flexible scheduling and grant-free transmission. • Improved spectral efficiency (bps/Hz): Every NOMA user can utilize the entire bandwidth, whereas an OMA user can utilize a limited amount. The data rates of properly grouped users can be increased when compared to OMA. 36 J. Boccuzzi The NOMA cellular system components are • Multi-user grouping, i.e., deciding which users should be grouped together to deploy NOMA. • Resource allocation (power, code, etc.), e.g., for the power domain NOMA case, users with large power differences are favorable. • SIC or MUD interference cancelation techniques to remove the controlled NOMA additions. With SIC or MUD, NOMA can support this multiple access concept. As we have seen, we expect to support an exponential growth in system capacity and user throughput in future systems. This amount of growth presents challenges that forces us to investigate new solutions. The choice of radio access technology plays an important role. NOMA is a proposed scheme to address the future system demands. 1.7 Conclusion In this chapter, we have reviewed the evolution of 1G to 5G cellular networks. 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Bubley, Mobile/Multi-access edge computing: how can telcos monetise this cloud? (2017) (Online), https://stlpartners.com/research/mobilemulti-access-edge-computing-howcan-telcos-monetise-this-cloud/ M. Vaezi, Y. Zhang, Radio access network evolution, in Cloud Mobile Networks (Springer, 2017), pp. 77–86 B. Ayvazian, H. Sarkissian, Spectrum Strategies for 5G. Wireless 20/20 Report (2017) (Online), http://www.wireless2020.com/media/articles.html F. Schaich, T. Wild, Waveform contenders for 5GOFDM vs. FBMC vs. UFMC, in Proceedings of 6th International Symposium on Communications, Control and Signal Processing (ISCCSP) Y. Liu, X. Chen, Z. Zhong, B. Ai, D. Miao, Z. Zhao, J. Sun, Y. Teng, H. Guan, Waveform design for 5g networks: analysis and comparison. IEEE Access 5, 19282–19292 (2017) D. Tse, P. Viswanath, Fundamentals of Wireless Communication (Cambridge University Press, 2005) A. Goldsmith, Wireless Communications (Cambridge University Press, 2005) W. Shin, M. Vaezi, B. Lee, D.J. Love, J. Lee, H.V. Poor, Non-orthogonal multiple access in multi-cell networks: theory, performance, and practical challenges. IEEE Commun. Mag. 55(10), 176–183 (2017) Chapter 2 OFDM Enhancements for 5G Based on Filtering and Windowing Rana Ahmed, Frank Schaich and Thorsten Wild 2.1 Motivation One of the main drivers of new radio (NR) is the huge market opportunity in the Internet of Things (IoT) applications [1]. It is predicted that such verticals will evolve as well as their needs. New services will emerge which cannot be efficiently served with any of the available dedicated solutions. The killer application (service) in 2030 cannot be easily predicted 10 years in advance; hence, forward compatibility is a design principle in NR. Such a killer application will not be most efficiently served by a fixed design, a configurable multiservice air interface is therefore the solution. In contrast to previous long-term evolution (LTE) releases, which mainly target serving broadband users and to which serving verticals came only as an afterthought, e.g., narrow band IoT (NB-IoT) in Releases 13 and 14 [2, 3], 5G NR aims at serving verticals as a basic system capability in addition to broadband users. Consequently, the use cases considered by 5G NR are more diverse. Beyond enhanced mobile broadband (eMBB), massive machine communication (mMTC) and ultra-reliable low latency communication (URLLC) have to be supported. For example, NR targets applications with limited battery capability, which demand less stringent time synchronization requirements, and at the same time NR targets applications which are very sensitive to time delay and thus require shorter symbol transmission time. To make it possible, various considerations for the radio access in general, and for the design of the waveform in particular, have to be accounted for, as will be discussed in the upcoming sections. R. Ahmed (B) · F. Schaich · T. Wild Nokia Bell Labs Stuttgart, Lorenzstrasse 10, Stuttgart, Germany e-mail: rana.ahmed_salem@nokia-bell-labs.com F. Schaich e-mail: frank.schaich@nokia-bell-labs.com T. Wild e-mail: thorsten.wild@nokia-bell-labs.com © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_2 39 40 R. Ahmed et al. 2.1.1 Multi-carrier Transmission The selection of the applied waveform is one of the most fundamental design decisions to be taken. It defines the temporal and spectral characteristics of the transmit signal. The resulting time domain peak-to-average power ratio will impact the power amplifier design and hence is responsible for the energy efficiency of communication. The resulting transmit signal spectrum impacts spectral efficiency and the coexistence with other communication systems. Waveforms will carry modulated symbols, so their design will impact the multiple access possibilities and frame structure design for multiplexing data symbols, pilot symbols, and blocks of control symbols. With conventional single carrier transmission, each transmitted symbol occupies the whole transmission bandwidth. As the transmission bandwidth increases, and thus the symbol length Ts gets shorter, the channel delay spread τmax becomes more significant. This leads to a distortion caused by inter-symbol interference (ISI), where the sum of several delayed replicas of the transmitted symbol is received. To compensate for this effect, multi-tap equalizers have to be employed at the receiver side, e.g., nonlinear decision feedback equalizer (DFE) [4]. In general, if the symbol duration Ts is much larger than the maximum delay spread Ts ≫ τmax or alternatively the signal bandwidth is much smaller than the coherence bandwidth Bs ≪ Bc , the channel is considered as a “flat” channel; i.e., the received signal can be considered to be a version of the transmitted signal weighted by a complex scalar factor. Thus very low equalization effort is required. Indeed, this is exactly the basic idea of a multi-carrier system. The idea is to divide the total bandwidth of the signal Bs into smaller subchannels, referred to as subcarriers. Such that each subchannel bandwidth is equal to Bsc = BNs , referred to as subcarrier spacing ∆f . The information is transmitted in parallel over these subcarriers. If the number of subcarriers, N , is large enough for a given overall bandwidth such that ∆f ≪ Bc , the channel experienced over every subcarrier is “flat,” and hence, a single-tap equalizer is sufficient to compensate the channel distortion in the frequency domain. The idea of multi-carrier transmission has emerged a long time ago as early as the 1966 paper of Chang [5]. However, it was only considered for practical implementation when an efficient implementation using the fast Fourier transform (FFT) [6] was proposed. Multi-carrier modulation is therefore a favorable choice in channels with long delay spread, since it avoids the high computational complexity needed with the single carrier equalization. The inherent serial-to-parallel conversion of multi-carrier modulation naturally offers a basic delay spread protection, which can be extended by using a cyclic prefix or zero postfix. Furthermore, multi-carrier modulation allows frequency-selective channel access, which exploits high gain links while avoiding fading dips. The decoupling into narrowband subchannels is very appealing for multiple input multiple output (MIMO) antenna processing techniques. The multi-carrier flexibility in time–frequency multiplexing allows good design properties for frame structures, including the multiplexing of pilot symbols and control information. Pilot symbols design can be flexibly 2 OFDM Enhancements for 5G Based on Filtering and Windowing 41 tailored to the coherence bandwidth and coherence time of the radio propagation channel. All those benefits made multi-carrier modulation the technique of choice in 4G LTE, which uses orthogonal frequency division multiplexing (OFDM) in the downlink (DL). The merits of OFDM, however, come at the price of an increased peakto-average power ratio (PAPR), compared to single carrier transmission. In fact, as the number of subcarriers increases, the PAPR increases as well. That is why single carrier frequency division multiple access (SC-FDMA) is used in the uplink (UL) transmission of LTE, instead of OFDM. Note, however, that via a discrete Fourier transform (DFT) precoding, OFDM can be transformed into SC-FDMA, so the established multi-carrier processing techniques can be reapplied to single carrier modulation. To provide the required orthogonality1 between the different subcarriers: • Every OFDM symbol is appended at the beginning by a guard interval of NG I samples, where NG I is designed to be larger than the channel delay spread. The guard interval contains either: 1. A duplicate of the last NG I samples of the OFDM symbol and hence is referred to as cyclic prefix (CP). 2. Or NG I zero samples and hence is referred to as zero prefix (ZP). The guard interval is important to avoid inter-block interference (IBI) between successive OFDM symbols. Therefore, the symbol time (in samples) is equal to Ns = N + Nc in case of CP-OFDM or Ns = N + Nz in case of ZP-OFDM, where Nc and Nz are the number of samples in the cyclic prefix and the zero prefix, respectively. • The subcarriers are arranged in the frequency domain such that the frequency spacing between the subsequent subcarriers is ∆f = 1 1 = Tu NT (2.1) where T is the sampling period between two successive samples in the time domain and Tu = N T is the useful symbol period in time domain. Such an arrangement guarantees that, at the frequency sampling point of any subcarrier q, no other ′ contribution from any other subcarrier q = q exists after removal of the CP, i.e., orthogonality between the different subcarriers. Assuming an UL transmitter to which Q subcarriers are allocated, transmitting at a symbol rate T1s , where Ts = Ns T , the output of a multi-carrier CP-OFDM transmitter 1 Orthogonality subcarriers. here means that no crosstalk occurs in the detection process between the different R. Ahmed et al. Ɵ Ɵ 42 Fig. 2.1 Transmit–receive chain for one subband in an OFDM system at time instant n can be written as [7] (after dropping the DFT spreading for ease of representation) x (n) = Q−1  i sq,i w (n − i Ns ) e j2π fq (n−Nc −i Ns ) , (2.2) q=0 where w (n) is a rectangular window function, holds the value of 1 over the interval [0, N + Nc ]. sq,i are the i.i.d. complex-valued symbols transmitted at subcarrier frequency f q and symbol i. Figure 2.1 shows the block diagram of the OFDM transceiver for one subband. In frequency domain, the window function for one subcarrier in Eq. 2.2, w (n), is written as  sin (π f (N + Nc )) W ( f ) = (N + Nc )e− j2π f (N +Nc ) , (2.3) π f (N + Nc ) m As shown in Fig. 2.2, the nulls of W ( f ) for subcarrier q occur at f = f q + (N +N , c) where m is an integer not equal to zero. It is worth noting that in ZP-OFDM, the nulls of W ( f ) for subcarrier q occur at f = f q + mN . Therefore, one can find that the power spectral density of ZP-OFDMbased waveforms have “true” nulls in the frequency domain, since the subcarrier spacing of the transmitted subcarriers, according to Eq. 2.1, coincides exactly with the nulls of W ( f ). The first notable multi-carrier modulation technique came up even before OFDM; filter-bank multi-carrier (FBMC) [8]. FBMC is a consequent application of Gabor signaling [9] generated by an orthogonal train of time–frequency shifted pulses. In 2 OFDM Enhancements for 5G Based on Filtering and Windowing Fig. 2.2 Windowing function in frequency domain W ( f ) with CP-OFDM 43 1 0.8 sinc(fN s) 0.6 0.4 0.2 0 -0.2 -0.4 -3 -2 -1 0 1 2 3 fNs order to make it spectrally efficient, offset-QAM is used [10]. The overlapping symbols generated by long filters create an impressive spectral containment of signals. However, this comes at the price of suitability for short bursts and loss in multiplexing flexibility [11]. 2.2 5G Waveform Requirements and Scenarios NR is targetting a diverse set of use cases [12, 13]. It is foreseen that NR will have to support a wide range of user velocities, data rates, reliability, and power efficiency requirements. In order to be able to configure the waveform parameters to match the requirement of every use case, NR is supposed to support subcarrier spacing scaling principle of ∆f = 15 × 2 S kHz, where S is an integer, unlike LTE which supports only ∆f = 15 KHz. The coexistence of all these services (waveform configurations) together is essential for an efficient use of resources and to be able to adapt to traffic load changes. Since the waveform is a fundamental component in the design of the air interface, NR waveform should be designed to facilitate that coexistence. In other words, NR waveform should be robust enough against possible inter-carrier interference (ICI) distortions caused by the support of different services, which will be detailed in Sects. 2.2.1 and 2.2.2 As mentioned earlier, in CP-OFDM with perfect synchronization, only a frequency domain one tap zero-forcing (ZF) equalizer is sufficient at the receiver side to equalize the effect of the channel. However, in reality, ICI can occur at the receiver side due to Doppler distortions caused by temporal channel variations or synchronization errors, etc. In such case, the relatively slow decay of the sinc waveform in Fig. 2.2 is especially problematic and the overall performance is not adequate. Therefore, in NR, it is desirable to design waveforms with faster decay rates in frequency, i.e., better frequency localization. In addition, waveforms with higher frequency 44 R. Ahmed et al. localization than baseline OFDM can achieve higher spectrum utilization than 90%, which was the maximum achieved in LTE [14]. A further driver for higher spectral confinement is the created forward compatibility; i.e., any kind of signals/waveforms which are favorable for a future use case can be inserted in the evolution of the 5G standard, as well-defined in-band requirements allow for “cleaning-up” the spectrum from sidelobes much better than regular OFDM could do. As will be shown in Sects. 2.3.1–2.3.3, the design is a trade-off between time and frequency localization. 2.2.1 Mixed Numerology In LTE, OFDM parameters, namely cyclic prefix length and subcarrier spacing, are selected as a reasonable compromise for different transmission scenarios (e.g., Doppler spread vs channel delay spread). In NR, because of the extreme use cases, more configuration options are available to serve each use case most efficiently [15]. For example, on one hand with use cases requiring URLLC, in order to save on the latency part, one option is to reduce the symbol lengths. This corresponds to a wider subcarrier spacing. Similarly, users who travel at a very high speed (e.g., high-speed trains), can benefit from having a large subcarrier spacing, reducing the interference arising from Doppler spread. On the other hand, for low-end devices, to enhance the coverage, or for users in a high channel spread environments, longer symbol durations (consequently longer CP) and smaller subcarrier spacings are more favored. For carrier frequencies below 6 GHz, Release 15 supports subcarrier spacings of 15, 30 and 60 kHz [14]. NR supports the multiplexing of these numerologies in UL and DL [16]. On one carrier, NR supports mixed numerologies in frequency division multiplex (FDM) or time domain multiplex manner. As discussed in [15], FDM of different numerologies in neighboring subbands generates ICI at the edge between the two different numerologies, which can be explained as follows: Assuming two neighboring allocations in FDM with two numerologies, namely (∆f 1 , Ts1 ) and (∆f 2 , Ts2 ), such that ∆f 2 = 2∆f 1 and Ts2 = T2s1 . 1. As shown in Fig. 2.3, allocation 1 suffers from ICI because the neighboring allocation (with larger subcarrier spacing) have now nonzeros contributions at its own subcarrier positions. 2. The ICI on the larger numerology can be better understood in time domain, where the symbol of allocation 2 is half the duration of the symbol of allocation 1, Ts2 = T2s1 . Therefore, the demodulator of allocation 2 collects only half the samples of allocation 1. The effect is equivalent to multiplying with a time domain window, which is equivalent to a convolution with the frequency response of the window [15]. 2 OFDM Enhancements for 5G Based on Filtering and Windowing 45 1 sinc(fT s), sinc(2fTs) 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -5 0 5 fT s Fig. 2.3 ICI on smaller numerology 2.2.2 Asynchronous Uplink Transmission The OFDM orthogonality in LTE UL requires that uplink transmissions from various users are synchronized at the BS; i.e., the different symbols from the UEs arrive at the BS within a certain time window which does not exceed the CP length. To compensate for different propagation delays, a user equipment (UE) would apply a timing advance [2] depending on how far it is from the BS, which means that UE devices which are far from the BS send their UL signals earlier than those close to the BS. Therefore, any device wanting to transmit a few bits of data has to enter the network via the random access procedure, wherein Msg 2 of the random access procedure, the BS informs the UE about its timing advance value (Fig. 2.4). The whole random access procedure includes a closed loop timing advance control, which in some cases may not be suitable for low-end machine-type communication (MTC) devices, dealing with sporadic traffic and stringent requirements on energy efficiency. The lack of UL synchronization, as shown in Fig. 2.5, creates ICI between neighboring UL users as explained in [17, 18]. The effect is actually similar to the windowing effect mentioned in Sect. 2.2.1. 2.3 Candidate 5G Waveforms As discussed in Sects. 2.1–2.2.2, there is a strong need for a waveform design in NR, which is robust against time–frequency misalignments. NR has selected CPOFDM/DFT-s-OFDM as the baseline waveform including the optional addition of a windowing or a filtering functionality [16, 19]. However, such improvements should 46 R. Ahmed et al. 0.05 2 'short' symbols 0.045 CP CP 0.04 N2 L2 N2 L2 envelope 0.035 0.03 1 'long' symbol 0.025 0.02 0.015 CP 0.01 N1 L1 0.005 0 0 100 200 300 400 500 600 700 800 900 1000 time index Fig. 2.4 ICI on larger numerology Fig. 2.5 Multi-user scenario (FDMA) be agnostic to the UE/BS in Release 15 [20], meaning that a baseline CP-OFDM/DFTs-OFDM receiver should work seamlessly without prior knowledge of the method used at the transmitter to reduce the out-of-band leakage (OOB). To enable a seamless multiplexing of different services, the used waveform should abide by in-band requirements, which are currently discussed in RAN4 [21]. These requirements can be met either by using gaps between the different subband allocations (guard 2 OFDM Enhancements for 5G Based on Filtering and Windowing 47 Fig. 2.6 Options for modifying the OFDM transceiver subcarriers) or by using a frequency localized waveform. Figure 2.6 shows the options for modifying the baseline CP-OFDM transceiver in Fig. 2.1 to achieve this goal. In this section, we discuss waveform examples which can be used in NR. Two main classes appear in this context; subband filtering and windowing, where the latter can be mapped to subcarrier filtering [7]. Subband filtering is motivated by the fact that ICI typically occur at the edge between neighboring subband allocations (blocks of subcarriers). For example, different uplink users having different waveform configurations/requirements. Therefore, the idea is to apply a well frequency localized filter, the bandwidth of which is close to the subbands bandwidth. As a result, only a few subcarriers close to the edges of the subband in frequency are affected by the filter, as the filter suppresses their out-of-subband sidelobes. By adapting filter parameter, the distortion on subband edges can be alleviated with little negative impact, depending on the use case. We discuss two waveforms which belong to this category, namely UF-OFDM (a.k.a UFMC) and f-OFDM. Windowing, on the other hand, is applied in the time domain, by modifying the rectangular pulse shape of CP-OFDM waveform to have smoother transitions in time at both ends. As an example of this category, we discuss weighted overlap and add (WOLA) waveform. Other candidate windowing techniques exist [22], but will not be discussed here. Other waveforms which gained a lot of attention, but were not considered for Release 15 due to incompatibility with CP-OFDM, are FBMC, mentioned at the beginning of this chapter, and zero tail discrete Fourier transform (ZT-DFT-s) [23]. All the methods mentioned in this chapter are compatible with DFT spreading as shown in Fig. 2.6. 48 R. Ahmed et al. Fig. 2.7 WOLA waveform Fig. 2.8 WOLA transmitter operation 2.3.1 Weighted Overlap and Add (WOLA) In CP-OFDM with WOLA, the windowing function w (n) in (2.2) is replaced by a pulse  function with soft edges at both sides, the length of the window is extended to − L2wt , N + L2wt [24, 25], where L wt is the length of the extension beyond the CP-OFDM length. The soft edges at the beginning and end of the window function result in better localization of the WOLA waveform in the frequency domain. In [24, 25], the CP-OFDM symbol is first extended by a cyclic extension in the time domain, and both edges are shaped by a weighting function, as shown in Fig. 2.7. As shown in Fig. 2.8, the resulting symbol is overlapped and added to the next symbol, and hence the overhead remains the same as in CP-OFDM. It is worth noting that, although the WOLA symbols overlap within one burst, when considering TDD, the tails from the end of the last WOLA symbol of one burst and from the first WOLA symbol of the following burst would already extend into the guard period (GP). In TDD, the GP is placed between two successive bursts when DL/UL switching is made. Therefore, the window length should be chosen carefully so as not to hinder TDD transmission. At the receiver side, a WOLA receiver can optionally be applied to suppress ICI, leaked from a neighboring allocation [24, 25]. The WOLA receiver processing is shown in Fig. 2.9. 2 OFDM Enhancements for 5G Based on Filtering and Windowing 49 Fig. 2.9 WOLA receiver operation The design of the soft edges of w (n), including the overlap length L wt , determines the frequency domain behavior of the WOLA symbol. In [24, 25], a raised cosine window design is used, but this does not exclude other possible window shapes. In general, the design is a trade-off between time and frequency localization, i.e., between ISI and ICI. The longer the window length L wt , the better the ICI localization of the WOLA waveform is, but the longer the overlap between the successive WOLA symbols, and the less robust the WOLA symbol is to channels with long delay spread. Figure 2.10 shows the power spectral density (PSD) of WOLA for a subband of 2 PRBs with two different window lengths L wt = 144, 72 samples. Compared to CPOFDM (L wt = 0), we can see that windowing reduced the OOB leakage of the waveform. 2.3.2 Universal Filtered OFDM (UF-OFDM) UF-OFDM is a 5G candidate waveform, also known as universal filtered multi-carrier (UFMC), where blocks of subcarriers (subbands) are filtered. As shown in Fig. 2.6, this is done by passing the OFDM signal output for user k through a subband filter f k (n) with filter order L f . f k (n) is built by shifting a prototype filter f (n) to the center of subband of user k. The modified subband filtered OFDM signal for user k can be written as 50 R. Ahmed et al. Fig. 2.10 PSD of WOLA waveform for different window lengths at a subband allocation size of 2 PRBs 10 WOLA Lwt =144 WOLA Lwt =72 0 CPOFDM PSD -10 -20 -30 -40 -50 0 10 20 30 40 50 60 freq index xk (n) =  i ⎡ f k (n) ∗ ⎣ Q−1  q=0 ⎤ sk,q,i w (n − i Ns ) e j2π fq (n−Nc −i Ns ) ⎦. (2.4) The input to the UF-OFDM subband filter can be either a ZP-OFDM signal or a CP-OFDM signal (depending on the value of Nc in Eq. 2.4). The advantage of applying the subband filter on a ZP-OFDM signal is that the resulting overall symbol can be limited in time domain, such that no overlapping between the successive symbols occur in an ISI-free environment (if the subband filter order is smaller than the guard interval L f < NG I ). The prototype filter of choice in UF-OFDM is the Dolph–Chebychev filter, but it is not restricted to this selection. If the subband filter is applied at the transmitter on a ZP-OFDM signal, at the receiver side, the L f samples from the tail of the received signal are simply added to the beginning of the symbol before applying the FFT. Hence, even if a ZP-OFDM signal is used, no extra complexity is needed at the receiver side to demodulate the signal as explained in [26]. In [7], it was shown that subband filtering applied on CPOFDM signal has near identical rate-versus-SNR performance to subband filtering applied on a ZP-OFDM signal. In [27], it was proposed to use Dolph–Chebyshev-based subband filtering in combination with variable CP/ZP. The summation of the CP plus ZP parts was assumed to be constant, and the filter order matches the CP length. Each part (CP/ZP) can be tuned depending on the channel environment, i.e., to optimize the trade-off between ISI and ICI rejection. However, in Release 15, the NR waveform is based on CPOFDM [16], therefore only CP-based UF-OFDM implementation can be supported in Release 15. Dolph–Chebychev filters are optimal in the sense that for a given side lobe level (SLL) the main lobe width is minimized. They are adjustable by the tuning parameter for the side lobe attenuation (SLA) as well as by the filter length L. The optimum filter choice of L and SLA depends on the use case. For example, on the one hand, in high ICI use cases with asynchronous transmission, it makes sense to use filters 2 OFDM Enhancements for 5G Based on Filtering and Windowing Fig. 2.11 Time domain impulse response for Dolph–Chebyshev filter with L = 72 and S L A = 35, 60 dB 51 1 SLA=35dB SLA=60dB 0.8 0.6 0.4 0.2 0 -3 -2 -1 0 1 2 3 t[ s] Fig. 2.12 Frequency domain response for Dolph–Chebyshev filter with L = 72 and S L A = 35, 60 dB SLA=35dB SLA=60dB 0 -20 -40 -60 -80 -100 -120 -1000 -500 0 500 1000 frequency [kHz] which are longer than the guard interval L > NG I , at the price of higher vulnerability to delay spreads [7]. On the other hand, in environments with high delay spread, a shorter filter length is used to protect against ISI. The SLA controls the trade-off between the main lobe width and the SLL. As shown in Figs. 2.11 and 2.12, as the SLA increases, the main lobe width increases and the SLL decreases. In general, for any subband filtering technique, when the allocation size is larger than a single PRB, a subband filter which has a broader passband offers a steeper side lobe level decay at the pass-band edge. For equalization at the receiver side, the receiver has to be aware of the filter coefficients. In order to avoid that, pre-equalization of a UF-OFDM signal is proposed in [28] for reference signals, but can also be applied to data symbols [26]. In addition, equal transmit power per subcarrier prevents an increased number of error events at the pass-band edge, as those errors from weaker edge subcarriers, would be detrimental for bit error rate (BER) performance. 52 R. Ahmed et al. To further reduce the effect of ICI leaked from a neighboring allocation due to mixed numerology or asynchronous transmission, etc., windowing or matched filtering can be used at the receiver side of the UF-OFDM [7, 18, 29]. 2.3.3 Filtered OFDM (f-OFDM) Filtered OFDM (f-OFDM) is a 5G candidate waveform based on subband filtering of a CP-OFDM signal [30, 31]. Therefore, the f-OFDM signal can also be built according to (2.4). Compared to UF-OFDM, the key property of f-OFDM is that the filter length, L f , can well exceed the guard interval length, which enables it to provide very good frequency localization. As shown in [30], soft truncation of a prototype filter is used, which in this case is a sinc impulse response p B (n). The sinc impulse response should have a bandwidth B in the frequency domain equal to the subband allocation size. The subband filter is obtained by applying a time windowing mask to p B (n) (2.5) f k (n) = w(n) p B (n), where w(n) is the windowing mask with duration Tw . The windowing mask has to have smooth transitions to zero on its both ends so that it avoids abrupt jumps at the beginning and end of the truncated filter. An example of such a windowing function is the Hanning window as proposed in [30] or the raised cosine window as proposed in [32]. Tw is usually chosen as Tw = T2u . In Figs. 2.13 and 2.14, the frequency domain response and the time domain impulse response of the designed filter for f-OFDM with bandwidth equal to 2 PRBs (360 kHz) and 3 PRBs (540 ‘kHz) are depicted, where the Hanning window function is used [30]. The filter length should be long in order to achieve desirable frequency localization. With half OFDM symbol filter length for example, one f-OFDM symbol extends into 25% of each of the previous and following f-OFDM symbols. However, most of the energy of the time domain impulse response, for allocation sizes greater than 3 PRBS, is limited to the CP part which is eventually dropped at the receiver side. Similar to WOLA, the long tails of the f-OFDM is also a problem in TDD transmission. Especially with long filter lengths, Tw = T2u . That is why in [32], signal burst tail reduction is proposed, where a hard truncation is suggested to reduce the tail overhead at both ends. In [32], one design criteria is to choose the sinc filters bandwidth B to be larger than the subband bandwidth W by a small excess bandwidth ∂ W , called tone offset (TO), on each side, i.e., B = W + 2 × ∂ W . The motivation behind using the TO is to guarantee a flat passband across all used subcarriers. The TO in f-OFDM is then analogous in use to the pre-equalization used on UF-OFDM, as mentioned in Sect. 2.3.2. The benefits of the TO in f-OFDM and the pre-equalization in UFOFDM come at the expense of a larger mainlobe (or higher SLL), and thus, a weaker frequency localization. 2 OFDM Enhancements for 5G Based on Filtering and Windowing Fig. 2.13 Frequency domain response of designed filter for f-OFDM with 2 PRB and 3 PRBs 53 B=540kHz B=360kHz 0 -20 -40 -60 -80 -100 -120 -1000 -500 0 500 1000 frequency [kHz] Fig. 2.14 Impulse response of designed filter for f-OFDM with 2 PRB and 3 PRBs versus sampling time 1 B=540kHz B=360kHz 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -20 -15 -10 -5 0 5 10 15 20 t[ s] f-OFDM is often described in conjunction with a matched filter, which suppresses the ICI leaked into the UE subband from neighboring UEs. In addition, it maximizes the received signal-to-noise ratio (SNR) of the UE. 2.3.4 Comparison Between the Different Waveforms In this section, we compare the performance of the four different waveforms discussed in this chapter, namely CP-OFDM, WOLA, UF-OFDM (CP based), and f-OFDM. Table 2.1 depicts the waveform parameters which were assumed for the different preprocessing approaches applied on top of the baseline CP-OFDM signal. 54 R. Ahmed et al. Table 2.1 Waveform parameters Waveform Baseline CP-OFDM WOLA CP-UFOFDM f-OFDM Parameters N G I = 72 samples, FFT size N = 1024, one subband = 1 and 4 PRBs L wt = 144 SLA=25dB and 75dB for subband sizes 1 PRB and 4 PRBs, respectively, L f = 73 samples, pre-equalization applied L f = 513, TO=0 and 2 and 4, Hanning window used for soft truncation of sinc impulse response Fig. 2.15 UL polynomial power amplifier model: gain and phase distortion 2.3.4.1 Power Amplifier Model The power amplifier, at the RF frontend, introduces signal nonlinearities in the final transmitted signal. Hence, the effectiveness of the OOB emission reduction by subband processing is eventually limited by the spectral regrowth due to these nonlinearities. As the PAPR increases, the spectral regrowth increases for the same power amplifier efficiency. Third-generation partnership project (3GPP) uses modified Rapp model and polynomial model to model the effect of the power amplifier in NR DL and UL, respectively [33]. Figure 2.15 shows the UL polynomial power amplifier model gain and phase distortion curves. In order to operate at an output power of 22 dBm, an input power operating point of −5.12 dBm is assumed, as shown in Fig. 2.16. As shown in Fig. 2.17, the error vector magnitude (EVM), used to describe the signal quality [34], reaches a minimum point at a phase compensation value of −77.1◦ . As mentioned earlier, one of the drawbacks of multi-carrier OFDM is the increase in the PAPR. The subband processing, based on windowing or filtering, leads to a further increase in the PAPR of the input signal to the power amplifier, which can also be observed in Fig. 2.17, where CP-OFDM has the lowest EVM value. 2 OFDM Enhancements for 5G Based on Filtering and Windowing 55 Fig. 2.16 Output power in dBm of amplifier versus input power in dBm for UL power amplifier Fig. 2.17 EVM versus phase compensation value for all waveforms for UL power amplifier 7 CPOFDM UFOFDM 6.5 WOLA FOFDM FOFDM TO=4 Averaged EVM [%] 6 5.5 5 4.5 4 3.5 -79 -78.5 -78 -77.5 -77 -76.5 -76 -75.5 -75 phaseComp[degrees] 2.3.4.2 Performance Comparison In Figs. 2.18 and 2.19, we can see the PSD of all waveforms before and after the UL power amplifier. As we can see, with the given waveform configurations, subband filtering (f-OFDM followed by CP-UFOFDM) have better spectral localization compared to WOLA and is therefore more effective in dealing with frequency misalignments. After the power amplifier, the relative performance of the different waveforms remain the same, but the gap between them is significantly reduced. A waveform with high spectral localization translates into higher spectral efficiency, 56 R. Ahmed et al. Fig. 2.18 PSD with 1 PRB allocation: solid lines are PSD without PA, dashed lines are PSD with PA 10 CPOFDM CPOFDM WOLA WOLA CP-UFOFDM CP-UFOFDM FOFDM FOFDM FOFDM-TO2 FOFDM-TO2 0 PSD -10 -20 -30 -40 -50 -60 0 10 20 30 40 50 60 freq index since with high spectral localization, relatively smaller number of in-band guards are needed to satisfy the in-band requirements of NR. Figures 2.18 and 2.19, however, do not show the performance in a synchronous environment, especially with high delay spread. Results for this case in [35, 36] show an opposite ranking for the three waveforms in a synchronous environment with high delay spread, where WOLA shows the best performance followed by CP-UFOFDM and then f-OFDM. In [37], it is shown that by taking into consideration, the reduced guard band overhead required in case of well spectrally localized f-OFDM, f-OFDM without using a matched filter can have a higher spectral efficiency than than of WOLA. No comparison was made in [37] against UF-OFDM. In [7], it is shown that both windowing and filtering techniques are comparably effective in combating channel time–frequency misalignments depending on the selected waveform configuration, with filtering techniques showing a slightly superior performance. 2.3.4.3 Implementation Aspects WOLA requires an additional complexity over baseline CP-OFDM depending on the window length, which is L 2wt multiplications, as shown in Fig. 2.7. Similarly, if a WOLA receiver L 2wr is applied to suppress ICI from a neighboring allocation, extra L 2wr multiplications are needed on top [36]. Therefore, the total added complexity is that of the overlap operation and L 2wt +L 2wr . For subband filtering, higher computational complexity is required, and therefore, a range of low-complex solutions exists in the literature. Two main directions that will be discussed here are: • Multi-window approximation discussed in [38, 39] • Fast convolution discussed in [40, 41] 2 OFDM Enhancements for 5G Based on Filtering and Windowing 57 10 0 PSD -10 CPOFDM CPOFDM WOLA WOLA CP-UFOFDM CP-UFOFDM fOFDM fOFDM fOFDM-TO4 fOFDM-TO4 -20 -30 -40 -50 40 42 44 46 48 50 52 54 56 58 60 freq index Fig. 2.19 PSD with 4 PRB allocation: solid lines are PSD without PA, dashed lines are PSD with PA Recalling Eq. 2.4, the overall signal x (n) composed of all subbands can be written as x (n) = K −1  xk (n) e j2πk Qn N . (2.6) k=0 The complexity of generating the overall signal (e.g., in the DL case) for one symbol scales linearly with the number of subcarriers per subband Q (to generate one subband k) and with the number of allocated subbands K . The multi-window approximation is based on the observation that subband filtering is equivalent to subcarrier windowing, with the result that Eq. 2.4 can be rewritten, for one symbol duration, as [38] xk (n) = Q−1  f q (n)sk,q w (n) (2.7) q=0 where f q (n) is the effective filter that is used to modulate the qth subcarrier in each subband. One can observe that for close subcarriers, the value of f q (n) is not so different, depending mainly on the bandwidth for which the prototype filter f (n) was designed. Therefore, the idea is to divide the subband allocation into multiple subcarrier groups, every subcarrier group g consists of Q g subcarriers and is windowed using one effective filter f g (n), with the result that, the complexity scales only with the number of subcarrier groups G [39]. The overall signal for one symbol duration can be approximated as 58 R. Ahmed et al. Fig. 2.20 Overlap-save processing flow used for fast convolution x (n) ≈ G−1  g=0 f g (n) Q−1 K −1   sk,q w (n) e j2πk Qn+qn N , (2.8) k=0 q=0 where the strength of the approximation thus depends on the number of subcarrier groups G. In the extreme case of using G = 1, this approximation has the same complexity as a windowing operation. With G = 3, the spectral localization is improved, coming at the price of roughly 3 times baseline OFDM complexity [38]. The main idea behind fast convolution algorithms is that filtering through a higherorder impulse response can be implemented effectively through multiplication in FFT-domain. This is done by taking the DFT of the input sequence as well as the DFT of the filter impulse response. The time domain output signal is finally obtained by IDFT [40]. As shown in Fig. 2.20, overlap-save processing is applied for long sequences to combine all processed blocks into the output signal. 2.4 Summary In this chapter, we have discussed the requirements on the waveform design in 5G NR is driven by the new and challenging use cases in NR. To enable a seamless multiplexing of different services and to fulfill the forward comparability vision for NR, it is highly desirable to have a frequency localized waveform design in NR without sacrificing too much time localization. CP-OFDM has been proven as a powerful and flexible waveform already for 4G, and hence in 5G, it is the dominant candidate 2 OFDM Enhancements for 5G Based on Filtering and Windowing 59 solution, while dealing with its weak points such as frequency localization by some modifications To this end, three candidate waveform preprocessing techniques for the 5G NR are discussed, namely WOLA, UF-OFDM, and f-OFDM, where each is based on either time domain windowing or subband filtering. The design principle and implementation aspects of each waveform are outlined. The design parameters of each waveform can be tuned to fit the target use case, e.g., the window length or the window shape in WOLA and the subband filter length in subband filtering techniques, mainly optimizing a trade-off between time domain localization and frequency domain localization. Depending on the choice of these parameters, the performance of windowing and subband filtering techniques is found to be comparable, with subband filtering offering a slightly better performance, at the price of higher complexity. Several techniques can be used to reduce the implementation complexity of the subband filtering technique. One interesting technique, besides frequency domainbased fast convolution, is to approximate the subband filtering by a multi-windowing operation, which in essence yields a hybrid between a subband filtering technique and a windowing technique, with a trade-off between frequency localization (and hence performance) versus implementation complexity cost. References 1. Machina research, Technical report, Aug 2016 2. 3rd Generation Partnership Project; TS 36.211, E-UTRA, Physical Channels and Modulation (Release 13) (2016) 3. 3rd Generation Partnership Project; TS 36.211, E-UTRA, Physical Channels and Modulation (Release 14) (2017) 4. J. Proakis, Digital Communications. 4th edn. (Mc Graw-Hill Book Company, 2001) 5. R.W. Chang, High-speed multichannel data transmission with bandlimited orthogonal signals. Bell Sys. Tech. 45, 1775–96 (1966) 6. S.B. Weinstein, The history of orthogonal frequency-division multiplexing [History of Communications]. Commun. Mag. IEEE 47(11), 26–35 (2009) 7. S. Venkatesan, R.A, Valenzuela, OFDM for 5G: cyclic prefix versus zero postfix, and filtering versus windowing, in 2016 IEEE International Conference on Communications (ICC), pp. 1–5, May 2016 8. B. Farhang-Boroujeny, OFDM versus filter bank multicarrier. IEEE Signal Process. Mag. 28(3), 92–112 (2011) 9. G. Wunder, P. Jung, M. Kasparick, T. Wild, F. Schaich, Yejian Chen, S. Brink, I. Gaspar, N. Michailow, A. Festag, L. Mendes, N. Cassiau, D. Ktenas, M. Dryjanski, S. Pietrzyk, B. Eged, P. Vago, F, Wiedmann, 5GNOW: non-orthogonal, asynchronous waveforms for future mobile applications. IEEE Commun. Mag. 52(2), 97–105 (2014) 10. M. Bellanger, FBMC Physical Layer: A Primer (2010) 11. F. Schaich, T. Wild, Y. Chen, Waveform Contenders for 5G—suitability for short packet and low latency transmissions, in 2014 IEEE 79th Vehicular Technology Conference (VTC Spring), pp. 1–5, May 2014 12. NGMN Alliance, NGMN 5G White Paper (2015), http://www.ngmn.org/5g-white-paper.html 60 R. Ahmed et al. 13. A. Osseiran, V. Braun, T. Hidekazu, P. Marsch, H. Schotten, H. Tullberg, M.A. Uusitalo, M. Schellman, The foundation of the mobile and wireless communications system for 2020 and beyond: challenges, enablers and technology solutions, in 2013 IEEE 77th Vehicular Technology Conference (VTC Spring), pp. 1–5, June 2013 14. 3GPP TR 38.912 Study on New Radio (NR) access technology (Release 14) (2017) 15. F. Schaich, T. Wild, Subcarrier spacing–a neglected degree of freedom? in 2015 IEEE 16th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), pp. 56–60, June 2015 16. R1-167963 Way forward on waveform RAN1#86 (2016) 17. Y. Chen, F. Schaich, T. Wild, Multiple access and waveforms for 5G: IDMA and universal filtered multi-carrier, in 2014 IEEE 79th Vehicular Technology Conference (VTC Spring), pp. 1–5, May 2014 18. F. Schaich, T. Wild, Relaxed synchronization support of universal filtered multi-carrier including autonomous timing advance, in 2014 11th International Symposium on Wireless Communications Systems (ISWCS), pp. 203–208, Aug 2014 19. R1-163615 WF on overview of NR RAN1#84bis (2016) 20. RAN1#86bis chairmans notes 21. R4-1610921 Way forward on in-band requirements for NR RAN4#81, Nov 2016 22. Ericsson. R1-163224 Waveform candidates RAN1#84bis (2016) 23. G. Berardinelli, F.M.L. Tavares, T.B. Strensen, P. Mogensen, K. Pajukoski, Zero-tail DFTspread-OFDM signals, in 2013 IEEE Globecom Workshops (GC Wkshps), pp. 229–234, Dec 2013 24. Qualcomm, 5G Waveform & Multiple Access Techniques, https://www.qualcomm.com/ documents/5g-research-waveform-and-multiple-access-techniques (2015) 25. Qualcomm, R1-162199 Feasibility of Mixing Numerology in an OFDM System RAN1#84bis (2016) 26. Nokia, R1-165014 Subband-wise filtered OFDM for New Radio below 6 GHz RAN1#85 (2016) 27. LG. R1-162516 Flexible CP-OFDM with variable ZP RAN1#84bis, Apr 2016 28. X. Wang, T. Wild, F. Schaich, S. ten Brink, Pilot-aided channel estimation for universal filtered multi-carrier, in 2015 IEEE 82nd Vehicular Technology Conference (VTC2015-Fall), pp. 1–5, Sept 2015 29. R. Ahmed, T. Wild, F. Schaich, Coexistence of UF-OFDM and CP-OFDM, in 2016 IEEE 83rd Vehicular Technology Conference (VTC Spring), pp. 1–5, May 2016 30. J. Abdoli, M. Jia, J. Ma, Filtered OFDM: a new waveform for future wireless systems, in 2015 IEEE 16th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), pp. 66–70, June 2015 31. X. Zhang, M. Jia, L. Chen, J. Ma, J. Qiu, Filtered-OFDM—enabler for flexible waveform in the 5th generation cellular networks, in 2015 IEEE Global Communications Conference (GLOBECOM), pp. 1–6, Dec 2015 32. Huawei, R1-164033 f-OFDM scheme and filter design RAN1#85 (2016) 33. R1-166004, R4-164542, Response LS on realistic power amplifier model for NR waveform evaluation, May 2016 34. E. Dahlman, S. Parkvall, J. Skold, 4G: LTE/LTE-Advanced for Mobile Broadband. Academic Press, 1st ed. (2011) 35. R1-1609564 Implementation-specific UF-OFDM for New Radio (2016) 36. R1-164685, OFDM based waveform single user evaluation RAN1#85 (2016) 37. R1-166093, Waveform evaluation updates for case 1a and case 1b, Aug 2016 38. M. Matthe, D. Zhang, F. Schaich, T. Wild, R. Ahmed, G. Fettweis, A reduced complexity time-domain transmitter for UF-OFDM, in 2016 IEEE 83rd Vehicular Technology Conference (VTC Spring), pp. 1–5, May 2016 2 OFDM Enhancements for 5G Based on Filtering and Windowing 61 39. Samsung, R1-166746 Discussion on multi-window OFDM for NR waveform RAN1#86 (2016) 40. M. Renfors, J. Yli-Kaakinen, T. Levanen, M. Valkama, T. Ihalainen, J. Vihriala, Efficient fastconvolution implementation of filtered CP-OFDM waveform processing for 5G, in 2015 IEEE Globecom Workshops (GC Wkshps), pp. 1–7, Dec 2015 41. J. Yli-Kaakinen, T. Levanen, S. Valkonen, K. Pajukoski, J. Pirskanen, M. Renfors, M. Valkama, Efficient fast-convolution-based waveform processing for 5G Physical Layer. CoRR, abs/1706.02853 (2017) Chapter 3 Filter Bank Multicarrier Modulation Ronald Nissel and Markus Rupp 3.1 Why FBMC? Future mobile systems will be characterized by a large range of different use cases, ranging from enhanced mobile broadband (eMBB) over enhanced machine type communications (eMTC) to ultra-reliable low-latency communications (URLLC) [2, 41, 53, 63]. To efficiently support such diverse use cases, a flexible time–frequency allocation becomes necessary. In particular, the out-of-band (OOB) emissions must be sufficiently low in order to efficiently support different use cases within the same band. Furthermore, low OOB emissions reduce the synchronization requirements. Conventional orthogonal frequency-division multiplexing (OFDM) with cyclic prefix (CP) performs poorly in this context because of the underlying rectangular prototype filter, which causes large OOB emissions. To improve spectral properties in OFDM, the 3rd Generation Partnership Project (3GPP) is therefore considering windowing and filtering [45, 52, 63]. The windowed OFDM scheme is called OFDM with weighted overlap and add (WOLA) and the filter-based methods are called universal filtered OFDM (UF-OFDM) and filtered OFDM (f-OFDM). While windowing and filtering can indeed reduce the OOB emissions of conventional OFDM, filter bank multicarrier modulation (FBMC) with offset quadrature amplitude modulation (OQAM) [41] still performs much better, as shown in Fig. 3.1. Additionally, FBMCOQAM has a maximum symbol density, that is, a time–frequency spacing of TF = 1 for complex-valued symbols. In OFDM-based schemes, on the other hand, the symbol density is lower, as indicated by T F > 1, additionally worsening the spectral efficiency. Besides the better support of different use cases, FBMC also increases the throughput of legacy Long Term Evolution (LTE) transmissions because fewer R. Nissel (B) · M. Rupp TU Wien, Gusshausstraße 25, 1040 Vienna, Austria e-mail: rnissel@nt.tuwien.ac.at M. Rupp e-mail: mrupp@nt.tuwien.ac.at © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_3 63 R. Nissel and M. Rupp Power Spectral Density [dB] 64 0 −20 CP-OFDM −40 WOLA LTE like: T F=1.07 24 subcarriers T F=1.09 −60 UF-OFDM f-OFDM −80 FBMC OQAM T F=1 (complex) −100 −50 −40 −30 −20 −10 0 10 20 30 40 50 Normalized Frequency, f /F Fig. 3.1 FBMC has much better spectral properties compared with CP-OFDM. Windowing (WOLA) and filtering (UF-OFDM, f-OFDM) can improve the spectral properties of CP-OFDM. However, FBMC still performs much better and has the additional advantage of a maximum symbol c density, T F = 1 (complex). 2017 IEEE, [41] FBMC-OQAM, T F = 1 (complex) mean +30% 87 Subcarriers 95 % confidence interval obtained by bootstrapping 1.4 MHz LTE Like OFDM, T F = 1.07 72 Subcarriers 6 Throughput [Mbit/s] Fig. 3.2 Real-world testbed measurements at 2.5 GHz show that FBMC has a higher throughput than OFDM (1.4 MHz LTE resembling SISO signal) because of a higher available bandwidth and no CP overhead [38, 41]. The channel estimation in FBMC is based on the data spreading approach [21, 35] 4 2 0 −5 0 5 10 15 20 25 Signal-to-Noise Ratio for OFDM [dB] guard subcarriers are required and no CP is needed. Figure 3.2 shows real-world testbed measurements and compares FBMC with an 1.4 MHz LTE single-input and single-output (SISO) signal (including pilots but ignoring signaling overhead). For high signal-to-noise ratio (SNR) values, the throughput of FBMC is approximately 30% higher than for OFDM. Even compared with f-OFDM, FBMC would still be approximately 20% better, as indicated by the time–frequency efficiency calculations in [41]. However, one has to keep in mind that the potential improvement strongly depends on the number of subcarriers and the required guard band. In particular, once the number of subcarriers is very high, windowed OFDM and filtered OFDM will perform close to FBMC. Unfortunately, all the nice features of FBMC-OQAM come at a price, namely, the complex orthogonality condition is replaced by the less strict real orthogonality condition. While this limitation has in many cases either no or only a minor influence 3 Filter Bank Multicarrier Modulation 65 on the performance, some important methods, such as channel estimation and some multiple-input and multiple-output (MIMO) techniques, become more challenging. There exist different variants of FBMC, but we will mainly focus on OQAMbased schemes because they provide the highest spectral efficiency. Different names are used to describe OQAM, such as OFDM/OQAM [8], fbmc-pulse-amplitude modulation (PAM) [26], staggered multitone (SMT) or Cosine Modulated Multitone (CMT) [12], which, however, are essentially all the same. One can easily transform one of those schemes into another by appropriately tuning the underlying parameters. For example, FBMC-PAM is a conventional FBMC-OQAM scheme for which the subcarrier spacing is reduced by a factor of two, the number of subcarriers is increased by two, and the offset is applied in the frequency domain instead of the time domain. In general, all those “different” FBMC schemes are characterized by: • A prototype filter which is localized in time and frequency. • Only real-valued information symbols can be transmitted at a given time–frequency position. • A time–frequency spacing of T F = 0.5 for real-valued symbols (equivalent to T F = 1 for complex-valued symbols). • Intrinsic imaginary interference. Although FBMC has been considered as a strong contender for replacing OFDM in the fifth-generation (5G) of wireless systems [4, 5, 59], 3GPP eventually decided that they will stick to OFDM [3]. While such decision makes sense in terms of backward compatibility to fourth-generation (4G) wireless systems, it is not the most efficient technique for all possible use cases, especially if the number of subcarriers is low. Thus, if the envisioned concept of different use cases within the same band turns out to be successful in 5G, we expect that FBMC will again gain momentum for beyond 5G communications. 3.2 Multicarrier Modulation Multicarrier modulation has a long-standing history in wireless communications [9, 51, 58]; however, widespread practical applications have only been realized in the latest versions of wireless systems in the form of OFDM, enabled by advances in the field of integrated circuits. Current applications of OFDM include LTE, WiFi and digital video broadcasting-terrestrial (DVB-T). In multicarrier systems [50], information is commonly transmitted over orthogonal pulses which overlap in time and frequency. The big advantage is that these pulses usually occupy only a small bandwidth, so that frequency-selective broadband channels transform into multiple, virtually frequency flat, sub-channels (subcarriers). Mathematically, the transmitted signal, s(t), of a multicarrier system in the time domain can be expressed as 66 R. Nissel and M. Rupp s(t) = K −1  L−1  gl,k (t) xl,k , (3.1) k=0 l=0 where xl,k denotes the transmitted symbol at subcarrier position l and time position k, and is chosen from a symbol alphabet, usually a QAM or a PAM signal constellation. The total number of subcarriers is denoted by L and the total number of symbols in time by K . The basis pulse gl,k (t) in (3.1) is defined by gl,k (t) = p(t − kT ) ej2π l F (t−kT ) e jθl,k , (3.2) and is, essentially, a time and frequency shifted version of prototype filter p(t), with T denoting the time spacing and F the frequency spacing (subcarrier spacing). The choice of phase shift θl,k becomes relevant later in the context of FBMC-OQAM. After transmission over a channel, the received symbols are decoded by projecting the received signal, r (t), onto the basis pulses, gl,k (t), that is, yl,k = r (t), gl,k (t) = ∞ ∗ (t) dt. r (t) gl,k (3.3) −∞ In (3.3), we implicitly apply a matched filter if the channel perturbation is additive white Gaussian noise (AWGN), maximizing the SNR. In a doubly selective channel, it might be better to choose the transmit and receive prototype filters slightly different, as, for example, suggested in pulse-shaped multicarrier transmissions [27] or in the practically more relevant case of CP-OFDM. However, employing an AWGN matched filter is usually close to the optimum because the channel induced interference is often dominated by noise, see Sect. 3.5. One of the biggest advantages of orthogonal multicarrier systems is that the transmission in (3.3) can be modeled by a one-tap channel, that is, yl,k = H (kT, l F) xl,k + n l,k + zl,k , (3.4) where H (t, f ) denotes the time-variant transfer function and represents the one-tap channel. The noise in (3.4) is described by n l,k and the channel induced interference by zl,k . Often, the delay spread and the Doppler spread are low enough so that the channel induced interference is dominated by noise, that is, E{|n l,k |2 } ≫ E{|zl,k |2 }. Thus, the channel induced interference can be neglected and the employment of low-complexity one-tap equalizers correspond to the maximum likelihood symbol detection in case of Gaussian noise. Multicarrier systems are mainly characterized by prototype filter p(t) as well as time spacing T and frequency spacing F, so that the ambiguity function [12, 50], 3 Filter Bank Multicarrier Modulation A(τ, ν) = ∞ 67  τ  j2πνt τ ∗ p t+ e p t− dt, 2 2 −∞ (3.5) captures the main properties of a multicarrier system in a compact way. The projection of the transmitted basis pulses gl1 ,k1 (t) onto the received basis pulses gl2 ,k2 (t) can then be expressed by the ambiguity function according to −k2 ) j(θl1 ,k1 −θl2 ,k2 ) A( T (k1 − k2 ), F(l1 − l2 ) ) . e gl1 ,k1 (t), gl2 ,k2 (t) = e−jπ T F(l1 +l2 )(k1   only a phase shift ambiguity function (3.6) There exist some fundamental limitations of multicarrier systems, as formulated by the Balian–Low theorem [13], which states that it is mathematically impossible that the following four desired properties are fulfilled at the same time: 1. Maximum symbol density, T F = 1, (3.7) 2. Time localization,  ∞ (t − t¯)2 | p(t)|2 dt < ∞, (3.8) ( f − f¯)2 |P( f )|2 d f < ∞, (3.9) gl1 ,k1 (t), gl2 ,k2 (t) = δ(l1 −l2 ),(k1 −k2 ) (3.10) σt = −∞ 3. Frequency localization, σf =  ∞ −∞ 4. Orthogonality, A(T (k1 − k2 ), F(l1 − l2 )) = δ(l1 −l2 ),(k1 −k2 ) , (3.11) ∞ with δ denoting the Kronecker delta function. The pulse P( f ) = −∞ p(t) e−j2π f t dt ∞ in (3.9) represents the Fourier transform of p(t) while t¯ = −∞ t | p(t)|2 dt corre∞ sponds to the mean time and f¯ = −∞ f |P( f )|2 d f the mean frequency of the pulse. Furthermore, we assume that p(t) is normalized to preserve unit energy. The localization measures in (3.8) and (3.9) can be interpreted as standard deviation, with | p(t)|2 and |P( f )|2 representing the probability density function (pdf). This relates the Balian-Low condition to the Heisenberg uncertainty relationship [57, Chap. 7]. The Balian-Low theorem implies that at least one of the four desired properties has to be sacrificed when designing multicarrier waveforms. For example, CP-OFDM sacrifices frequency localization while FBMC-OQAM the complex orthogonality condition. 68 R. Nissel and M. Rupp 3.2.1 CP-OFDM CP-OFDM is the most prominent multicarrier technique and is applied, for example, in Wireless LAN and LTE [24, 29]. CP-OFDM employs a rectangular transmit and receive pulse, which greatly reduces the computational complexity. Furthermore, the CP guarantees orthogonality in frequency-selective channels. The transmitter (TX) and receiver (RX) prototype filter can be expressed by pTX (t) = pRX (t) = √1 T0 0 if − T0 2  + TCP ≤ t < T0 2 (3.12) otherwise T0 2 √1 T0 if 0 otherwise ≤t < T0 2 (3.13) for which (Bi)-Orthogonal : T = T0 + TCP ; F = 1/T0 → T F = 1 + √ CP ; Localization : σt = T02+T σf = ∞ 3 TCP T0 , (3.14) with T0 representing a time-scaling parameter which depends on the desired subcarrier spacing (or time spacing). Note that, in contrast to FBMC, the prototype filter is differently at the TX and RX side. Figure 3.3 shows the ambiguity function, see (3.5), for CP-OFDM (without CP, that is, TCP = 0). Orthogonality is guaranteed for a time spacing of T = T0 and a frequency spacing of F = 1/T0 , leading to T F = 1. This is also indicated by the rectangular grid (the small circles) inside of Fig. 3.3. The ambiguity function decays very slowly in frequency because of the underlying rectangular pulse. Additionally, the CP simplifies equalization in frequency-selective channels but also reduces the spectral efficiency. In order to reduce the OOB emissions in OFDM, 3GPP is currently considering windowing [45] and filtering [52, 63]. As already shown in Fig. 3.1, such |A( , )|2 Time Shift, /T0 Fig. 3.3 The ambiguity function, 10 log10 |A(τ, ν)|2 , for OFDM (without CP) shows a good localization in the time domain, but a poor localization in the frequency domain. The orthogonal time–frequency spacing is T F = 1, indicated by the small circles in the figure 4 3 2 1 0 −1 −2 −3 −4 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 [dB] Frequency Shift, T0 0 −20 −40 −60 −80 −100 3 Filter Bank Multicarrier Modulation 69 methods can reduce the OOB emissions in OFDM, but still do not perform as good as FBMC and have the additional disadvantage of a reduced symbol density, that is, T F > 1, even in an AWGN channel. Note that for all the windowed- and filteredbased OFDM techniques, receive windowing and filtering is of utmost importance [41]. Very often, people forget this crucial aspect and only focus on reducing the OOB emissions at the transmitter. 3.3 FBMC-OQAM FBMC-OQAM replaces the complex orthogonality condition with the less strict real orthogonality condition, ℜ{gl1 ,k1 (t), gl2 ,k2 (t)} = δ(l2 −l1 ),(k2 −k1 ) , and works, in principle, as follows: 1. Design a prototype filter with p(t) = p(−t) which is orthogonal for a time spacing of T = T0 and a frequency spacing of F = 2/T0 , leading to T F = 2. 2. Reduce the time–frequency spacing by a factor of two each, that is, T = T0 /2 and F = 1/T0 , leading to T F = 0.5. 3. The so induced interference is shifted to the purely imaginary domain by the phase shift θl,k = π2 (l + k) in (3.2). Let us take a closer look at the intrinsic interference. With θl,k = 0.5, the inner product in (3.6) transforms to π π gl+∆l,k+∆k (t), gl,k (t) = e j 2 (∆l+∆k) e−j 2 ∆k(2l+∆l) purely imaginary for odd ∆k, ∆l π (l 2 + k) and T F = A(∆k T, ∆l F).   0 if both ∆k =0, ∆l =0 are even (3.15) The ambiguity function in (3.15) approaches zero if ∆k and ∆l are even because the prototype filter is designed to be orthogonal for those cases. If, on the other hand, either ∆k or ∆l is odd, A(·) no longer approaches zero, leading to interference. The main idea of FBMC is to shift this interference to the imaginary domain. To be specific, the exponential function in (3.15) becomes purely imaginary valued if either ∆k or ∆l is odd. Furthermore, the ambiguity function is always real-valued because of p(t) = p(−t), so that it has no influence on the imaginary part. Note that we consider a phase shift of θl,k = π2 (l + k), but other phase shifts are also possible, for example, θl,k = j π2 (l + k) − jπlk. Similar as in orthogonal multicarrier systems, FBMC also allows the employment of low-complexity one-tap equalizers. To be specific, the transmission can be modeled by a one-tap channel H (kT, l F), similar as in (3.4), according to yl,k = H (kT, l F) (xl,k + j vl,k ) + n l,k + zl,k . (3.16) Compared with orthogonal multicarrier systems, the data symbols xl,k are real-valued and there exists an imaginary interference term, described by jvl,k , which depends 70 R. Nissel and M. Rupp on the adjacent symbols. The big advantage of FBMC compared with other nonorthogonal schemes is that the imaginary interference can easily be canceled, simply by taking the real part after equalization.1 Thus, computational demanding equalization and cancellation methods are not necessary. Note that the imaginary interference does not carry any useful information for L → ∞ and K → ∞, so that, by taking the real part, we do not lose any useful information; see Sect. 3.6 for more details. Although the time–frequency spacing in FBMC-OQAM is equal to T F = 0.5, only real-valued information symbols can be transmitted in such as system, leading to an equivalent time–frequency spacing of T F = 1 for complex-valued symbols. Very often, the real part of a complex-valued symbol is mapped to the first time slot and the imaginary part to the second time slot, thus the name offset-QAM. However, such self-limitation is not necessary. We can equivalently perform this mapping over subcarriers or directly consider PAM symbols instead of “staggered” QAM symbols. As already mentioned in the beginning of this section, the prototype filter has to be an even function and orthogonal for a time–frequency spacing of T F = 2. All prototype filters which satisfy this condition can be utilized in FBMC-OQAM. Let us discuss two prominent prototype filters, namely the Hermite filter and the PHYDYAS filter. The Hermite prototype filter is based on Hermite polynomials Hn (·), as proposed in [16], and can be expressed as 1 −2π p(t) = √ e T0  t T0 2   √ t ai Hi 2 π , T0 i={0,4,8,  (3.17) 12,16,20} for which the coefficients can be found to be [35] a0 = 1.412692577 a12 = −2.2611 × 10−9 a4 = −3.0145 × 10−3 a16 = −4.4570 × 10−15 a8 = −8.8041 × 10 −6 a20 = 1.8633 × 10 −16 (3.18) , leading to the following properties of (3.17), Orthogonal : T = T0 ; F = 2/T0 → TF =2 Localization : σt = 0.2015 T0 ; σ f = 0.403/T0 . (3.19) Note that in practical systems, the pulse in (3.17) will be truncated so that it fits within the time interval − O2T0 ≤ t < O2T0 , with O denoting the overlapping factor. Figure 3.4 shows the ambiguity function for the Hermite prototype filter. The filter has the same shape in time and frequency, allowing us to exploit symmetries. Furthermore, the filter is based on a Gaussian function and therefore has a good joint time–frequency localization of σt σ f = 1.02 × 1/4π , almost as good as the bound of σt σ f ≥ 1/4π ≈ 0.08 (attained by the Gaussian pulse). In Fig. 3.4, we also observe 1 The channel must be known at the receiver. Channel estimation itself becomes more challenging in FBMC, as we will discuss later in this section. 3 Filter Bank Multicarrier Modulation 71 Time Shift, /T0 |A( , )|2 4 3 2 1 0 −1 −2 −3 −4 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 [dB] 0 −20 −40 −60 −80 −100 Frequency Shift, T0 Fig. 3.4 The ambiguity function for the Hermite prototype filter shows good localization in both, time and frequency. The orthogonal time–frequency spacing is T F = 2. To improve the spectral efficiency, the time–frequency spacing is reduced to T F = 0.5, indicated by the black markers. The so induced interference is shifted to the purely imaginary domain that the pulse is orthogonal for a time spacing of T = T0 and a frequency spacing of F = 2/T0 , indicated by the small circles. In FBMC-OQAM, the time–frequency spacing is reduced to T = T0 /2 and F = 1/T0 . This causes interference, indicated by the black markers in Fig. 3.4, which, however, is purely imaginary valued, see (3.15). Another prominent filter is the PHYDYAS prototype filter [6, 7], constructed by: p(t) = ⎧  O−1   ⎨ 1+2 i=1 bi cos O2πtT0 √ O T0 ⎩0 if − O2T0 ≤ t < O T0 2 . (3.20) otherwise The coefficients bi were calculated in [28] and depend on overlapping factor O. For example, for an overlapping factor of O = 4, the coefficients become, b1 = 0.97195983; b2 = √ 2/2; b3 = 0.23514695, (3.21) and lead to the following properties of (3.20) F = 2/T0 → TF =2 Orthogonal : T = T0 ; Localization : σt = 0.2745 T0 ; σ f = 0.328/T0 . (3.22) Figure 3.5 shows the ambiguity function for the PHYDYAS prototype filter (O = 4). Compared to the Hermite prototype filter, the PHYDYAS filter has a better frequency localization but a worse time localization. The joint time–frequency localization of σt σ f = 1.13 × 1/4π is also worse. Note that the PHYDYAS filter is not perfectly orthogonal, as shown in Fig. 3.5 for time position ±3 and frequency position {−2, 0, 2}, However, this interference is very low and thus be neglected for typical working points of wireless communications. 72 |A( , )|2 Time Shift, /T0 Fig. 3.5 The PHYDYAS prototype filter shows good localization in time and frequency. Compared with the Hermite prototype filter, the time localization is worse but the frequency localization is better. The small circles indicate the orthogonal time–frequency spacing, while the black markers correspond to the imaginary interference R. Nissel and M. Rupp 4 3 2 1 0 −1 −2 −3 −4 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 [dB] 0 −20 −40 −60 −80 −100 Frequency Shift, T0 3.3.1 Latency For our latency considerations, we focus on the underlying pulse duration but ignore other sources of delays, such as channel delays and processing delays. The transmission time of FBMC-OQAM then depends on subcarrier spacing F, the number of symbols in time K and overlapping factor O, according to, Tblock = 0.5 O + (K − 1). F F (3.23) In particular, the subcarrier spacing can be utilized to reduce the latency. This, however, comes at the expense of an increased sensitivity to frequency-selective channels (but improves the robustness in time-variant channels). When compared to OFDM, the overlapping factor plays an important role in FBMC. A common value is O = 4, leading to very low OOB emissions for the PHYDYAS prototype filter, see Fig. 3.1. However, FBMC allows more flexibility than that. For example, the Hermite prototype filter can be used with a lower overlapping factor. One can also design a prototype filter that is specifically tailored for low-latency scenarios, such as a time domain root-raised-cosine pulse with overlapping factor O = 1. An LTE subframe requires a transmission time of 1 ms and consists of K = 14 1 and F = 15 kHz). Because each FBMC symbol only OFDM symbols (TCP = 14F carries half the information of that of an OFDM symbol (same number of subcarriers), we need in total K = 28 FBMC symbols for a fair comparison to LTE. For an overlapping factor of O = 1, this implies that FBMC has a transmission time of Tblock ≈ 0.97 ms and is therefore faster than LTE. For an overlapping factor of O = 1.5, the transmission time is exactly 1 ms, same as in LTE. An overlapping factor of O = 4, on the other hand, performs relatively poor and requires 1.2 ms, 20% longer than LTE. Note that the ramp-up and ramp-down period in FBMC increases the latency, but not necessarily the sum throughput of the whole system, because different blocks can overlap in time. This works as long as the required phase pattern, which shifts the 3 Filter Bank Multicarrier Modulation 73 intrinsic interference to the purely imaginary domain, is fulfilled, as typically the case in downlink transmissions. However, in multi-user uplink transmissions, different users experience different phase shifts. Thus, the necessary phase pattern is violated, leading to interference, even for a perfectly time and frequency synchronized system. One then has to include a guard time in order to avoid interference. In such cases, the ramp-up and ramp-down period not only increases the latency, but also reduces the sum throughput. 3.3.2 Channel Estimation The main idea of FBMC-OQAM is to equalize the phase, followed by taking the real part in order to get rid of the imaginary interference. This, however, only works once the phase is known, thus only after channel estimation. The channel estimation itself has to be performed in the complex domain, affected by the imaginary interference, and one observes an SIR of 0 dB. Thus, additional processing becomes necessary. Preamble-based channel estimation was, for example, discussed in [20]. However, LTE employs pilot-aided channel estimation because it has a low overhead and allows a simple tracking of the channel in time. A straightforward method for pilot-aided channel estimation in FBMC was proposed in [19], where one data symbol per pilot, the so-called auxiliary symbol, is sacrificed to cancel the imaginary interference at the pilot position. The big disadvantage of such method is the high power of the auxiliary symbols, worsening the PAPR and wasting signal power. Subsequently, different methods have been proposed to mitigate these harmful effects [10, 21, 35, 60]. In particular, the data spreading approach of [21] is promising because no energy is wasted, there is no noise enhancement, and the performance is close to OFDM [33]. The idea of [21] is to spread data symbols over several time–frequency positions, close to the pilot symbol, in such a way, that the imaginary interference at the pilot position is canceled. The drawback is a slightly higher computational complexity [35]. To reduce the computational complexity and to improve the applicability in doubly selective channels, one can combine the data spreading approach with the auxiliary symbol method, as proposed in [11]. Note, however, that also the classical spreading approach performs well in doubly selective channels [43]. 3.4 Discrete-Time System Model The continuous-time representation, discussed so far, provides analytical insights and gives physical meaning to multicarrier systems. However, such representation becomes analytically hard to track in doubly selective channels because double integrals have to be solved. Furthermore, in practice, the signal is generated in the discrete-time domain. Thus, we will switch from the continuous-time domain to the discrete-time domain. In contrast to many other authors, we employ a matrix-based 74 R. Nissel and M. Rupp system model instead of a discrete-time filter representation because it simplifies analytical investigations and provides a more compact description. If one is interested in the conventional discrete-time filter representation, we refer to [47, 55]. In our matrix-based system model, the basis pulses in (3.2) are sampled at rate f s = 1/∆t = F NFFT and stacked in a basis pulse vector gl,k ∈ C N ×1 according to [gl,k ]n = √   ∆t gl,k (t) t=n∆t−O T , (3.24) for n = 0, 1, . . . , N − 1, where the total number of samples is given by N = O NFFT + NFFT (K − 1). The interpretation of overlapping factor O and fast Fourier 2 transform (FFT) size NFFT ≥ L becomes more clear later in this section, when we discuss an efficient FFT implementation. Practical systems will never operate at a critically sampling rate (NFFT = L) because this would lead to large OOB emissions, caused by the repetition of the spectrum in the frequency domain. We strongly advise to never use a critically sampled system, which is only useful for emulating the asymptotic case of infinitely many subcarriers, L → ∞. Unfortunately, many authors ignore this important aspect. By stacking all basis pulse vectors from (3.24) in a large transmit matrix G ∈ C N ×L K ,   G = g0,0 · · · g L−1,0 g0,1 · · · g L−1,K −1 , (3.25) and all data symbols in a large transmit symbol vector x ∈ C L K ×1 , ⎧⎡ ⎤⎫ ⎪ ⎨ x0,0 · · · x0,K −1 ⎪ ⎬ ⎢ ⎥ .. x = vec ⎣ ... . . . ⎦ . ⎪ ⎪ ⎩ ⎭ x L−1,0 · · · x L−1,K −1 T  = x0,0 · · · x L−1,0 x0,1 · · · x L−1,K −1 , (3.26) (3.27) we can express the sampled transmit signal s ∈ C N ×1 in (3.1) by: s = Gx. (3.28) Because of linearity, matrix G can easily be found even if the underlying modulation format is not known in detail. For that, all transmitted symbols have to be set to zero, except xl,k = 1. Vector s then provides immediately the l + Lk-th column vector of G. Repeating this step for each time–frequency position delivers transmit matrix G. At the receiver, we perform matched filtering by GH , so that the whole transmission system simplifies to y = GH H G x + GH n ≈ diag{h} GH G x + GH n, (3.29) (3.30) 3 Filter Bank Multicarrier Modulation 75 with y ∈ C L K ×1 denoting the received symbols, H ∈ C N ×N the banded time-variant convolution matrix ([H]i, j = h conv. [i, i− j] with time-variant impulse response h conv. [i, m τ ]) and n ∼ C N (0, Pn I N ) the additive white Gaussian noise in the time domain with zero mean and variance Pn . In most practical scenarios, the delay spread and the Doppler spread are low enough so that the channel induced interference can be neglected [36, 41]. This allows us to factor out the channel in (3.29) according to (3.30), for which h ∈ C L K ×1 describes the one-tap channels (frequency domain), that is, the diagonal elements of GH H G. In particular, the l + Lk-th element of h is given by H H gl,k ≈ H (kT, l F), (3.31) h l,k = gl,k and represents the one-tap channel at subcarrier position l and time position k. FBMC experiences imaginary interference, described by the off-diagonal elements of GH G and only the real orthogonality condition holds true, that is, ℜ{GH G} = I L K . 3.4.1 IFFT Implementation Practical systems must be much more efficient than the matrix multiplication in (3.28). It was, for example, shown in [55] that FBMC-OQAM can be efficiently implemented by an inverse FFT (IFFT) together with a polyphase network. Unfortunately, the authors of [55] rely on a filter bank representation which is very different to the conventional OFDM description. We therefore consider an alternative interpretation, more closely related to conventional OFDM systems. A similar representation was, for example, suggested in [27] for pulse-shaping multicarrier systems. To simplify the exposition and without losing generality, we consider only time position k = 0. Any other time position can easily be obtained by time shifting this . The main idea is to factor out the prototype special case by T = T20 , respectively NFFT 2 filter p(t) from (3.1), so that the sampled transmit signal can be expressed by s0 (n ∆t) = p(n ∆t) L−1  e j2π l N n FFT e jθl,0 xl,0 , (3.32) l=0 for n = − O N2FFT , . . . , O N2FFT − 1. The summation in (3.32) corresponds to an NFFT point IFFT with the input arguments {e jθ0,0 x0,0 , e jθ1,0 x1,0 , . . . , e jθL−1,0 x L−1,0 , 0, 0, . . .}. Furthermore, because l is an integer, the summation in (3.32) is NFFT periodic with respect to n. Thus, the IFFT has to be calculated only for NFFT samples. Those samples can then be copied O-times, followed by an element-wise multiplication with prototype filter p(n ∆t). By stacking the transmitted samples in a vector s0 ∈ C O NFFT ×1 , we can therefore express (3.32) by 76 R. Nissel and M. Rupp ⎜ ⎜ ⎜ s0 = p ◦ ⎜ ⎜ 1 O×1 ⎜ ⎝   ⎤⎞ e jθ0,0 x0,0 ⎥⎟ ⎢ .. ⎥⎟ ⎢ . ⎢ ⎥⎟ H jθ L−1,0 ⎢ ⎟ x L−1,0 ⎥ ⊗ W NFFT ⎢e ⎥ ⎟, ⎢ ⎟ ⎥ 0 ⎣ ⎦⎠ .. .   IFFT  ⎡ ⎛ (3.33) repeat O -times  element-wise multiplication where ◦ denotes the element-wise Hadamard product, ⊗ the Kronecker product and W NFFT ∈ C NFFT ×NFFT a Discrete Fourier Transform (DFT) matrix. Note that, by circular shifting the IFFT input in (3.33), we can shift the signal in frequency by multiples of F. The sampled prototype filter p ∈ C O NFFT ×1 in (3.33) is given by,  √  [p]n = ∆t p(t) for n = 0, 1, . . . , O NFFT − 1. t=n ∆t−O T (3.34) Figure 3.6 illustrates the efficient FBMC-OQAM implementation and compares it to windowed OFDM. Both modulation schemes apply the same basic operations, that is, IFFT, repetition and element-wise multiplications. However, windowed OFDM has overall a lower complexity because the element-wise multiplication is limited to a window of size 2 TW and time symbols are further apart, that is, T = TW + TCP + T0 in windowed OFDM versus T = T0 /2 in FBMC-OQAM. Thus, FBMC needs to apply the IFFT more than two times (exactly two times if TW = TCP = 0). Of course, the overhead TW + TCP in windowed OFDM reduces the spectral efficiency. Because Windowed CP-OFDM FBMC-OQAM IFFT IFFT IFFT IFFT IFFT IFFT × |p(t)|2 + IFFT IFFT p(t) t t + t T0 TW IFFT × t T0 2 T0 2 TCP Fig. 3.6 From a conceptional point of view, the signal generation in windowed OFDM and FBMCOQAM requires the same basic operations, namely, an IFFT, copying the IFFT output, element-wise c multiplication with the prototype filter and, finally, overlapping. 2017 IEEE, [41] 3 Filter Bank Multicarrier Modulation 77 the signal generation for both modulation formats is very similar, FBMC-OQAM can utilize the same hardware components as windowed OFDM. The receiver works in a similar way, but in reversed order, that is, element-wise multiplication, reshaping the received symbol vector to NFFT × O followed by a row-wise summation and, finally, an FFT. In matrix notation, this can be expressed by ( )   y0 = diag e−jθ0,0 . . . e−jθL−1,0 0 L×(NFFT −L) W NFFT (11×O ⊗ I NFFT ) (p ◦ r0 ), (3.35) where r0 ∈ C O NFFT ×1 represents the received samples and y0 ∈ C L×1 the received symbols, both at time position k = 0. WOLA requires at the receiver the same basic operations as FBMC. However, in contrast to FBMC, WOLA employs a different transmit and receive prototype filter. 3.5 One-Tap Equalizers in Doubly Selective Channels The biggest advantage of multicarrier systems is that the transmission over a doubly selective channel can be approximated by one-tap channels. In this section, we calculate the approximation error by considering the signal-to-interference ratio (SIR). In orthogonal multicarrier systems, the SIR can be calculated by SIRQAM = E{|h l,k xl,k |2 } E{|zl,k |2 } (3.36) with interference zl,k given by, H H G x − h l,k xl,k . zl,k = gl,k (3.37) In FBMC-OQAM, on the other hand, the SIR cannot be calculated as easily as in (3.36) because directly applying (3.36) leads to an SIR of approximately 0 dB due to the inherent self-interference. We have to equalize the phase followed by taking the real part, before calculating the SIR. Thus, the SIR can be expressed by SIROQAM = E{|ℜ{e−jϕl,k h l,k xl,k }|2 } , E{|ℜ{e−jϕl,k zl,k }|2 } (3.38) where e jϕl,k = hl,k/|hl,k | represents the phase of the one-tap channel. The SIR is very helpful because it determines the point at which interference starts to dominate the noise, that is, SNR > SIR, leading to a saturation of the bit error rate (BER), see [36]. As long as the SNR is approximately 10 dB lower than the SIR, interference is completely dominated by noise and can thus be neglected. Once the SNR approaches 78 Optimal Subcarrier Spacing SIR Fig. 3.7 The SIR depends on the subcarrier spacing. For a fair comparison of different modulation schemes, we consider an optimal subcarrier spacing R. Nissel and M. Rupp Limiting factor: Doppler spread (time-variant channel) Limiting factor: delay spread (multipath delays) Subcarrier Spacing the SIR, that is, SNR = SIR, one observes a performance degeneration equivalent to an SNR shift of approximately 3 dB. The matrix representation in Sect. 3.4 can be utilized to calculate the SIR in (3.36). For example, the channel power can be T H T H H ⊗ gl,k )Rvec{H} (gl,k ⊗ gl,k ) , with channel correlacalculated by E{|h l,k |2 } = (gl,k 2 2 H N ×N tion matrix Rvec{H} = E{vec{H}vec{H} } ∈ C . For OQAM, on the other hand, additional processing becomes necessary, see [41] for more details. Note that the SIR can also be calculated with the ambiguity function, as, for example, demonstrated in [15, 50]. In [36], we showed that FBMC (Hermite prototype filter) outperforms CP-OFDM in high-velocity scenarios. This, however, was only true because interference from the Doppler spread dominated interference from the delay spread. By increasing the subcarrier spacing, the overall SIR could be improved, as illustrated in Fig. 3.7. The big question is then, does FBMC still outperform CP-OFDM if both modulation schemes apply an optimal subcarrier spacing? Because 5G will include a flexible subcarrier spacing [3], our considerations here are also relevant for future wireless systems. As a rule of thumb, the subcarrier spacing should be chosen so that [12] τrms σt ≈ , σf νrms (3.39) where time localization σt and frequency localization σ f are given by (3.19) for the Hermite pulse and by (3.22) for the PHYDYAS pulse. For FBMC-OQAM, this leads to the following optimal subcarrier spacings: Fopt,PHYDYAS * νrms , τrms * νrms ≈ 0.91 × . τrms Fopt,Hermite ≈ 0.71 × (3.40) (3.41) For a Jakes Doppler spectrum, the root mean square (RMS) Doppler spread is given by νrms = √12 vc f c , with v denoting the velocity, c the speed of light and f c the carrier frequency. Note that (3.39) represents only an approximation. The exact relationship can be calculated, as, for example, done in [17] for the Gaussian pulse, and depends on the underlying channel model and the prototype filter. However, for our chosen numerical parameters, the differences between the optimal SIR (exhaustive search) 3 Filter Bank Multicarrier Modulation 79 and the SIR obtained by applying the rule in (3.39) is less than 0.1 dB for FBMCOQAM and less than 1 dB for FBMC-QAM. For the rest of this section, we always find the optimal subcarrier spacing in FBMC through an exhaustive search. As a reference, we also consider an optimal subcarrier spacing in CP-OFDM. The rule in (3.39), however, cannot be applied because the underlying rectangular pulse is not localized in frequency. Instead, we assume, for a fixed CP overhead of κ = TTCP0 = TCP F = T F − 1, that the subcarrier spacing is chosen as high as possible, while still satisfying the condition of no Inter Symbol Interference (ISI), that is, κ TCP = τmax . This leads to F = τmax . For a Jakes Doppler spectrum, the SIR can be expressed by a generalized hypergeometric function 1 F2 (·) [48], which, together with an optimal subcarrier spacing, leads to [41], CP−OFDM SIRopt., noISI = 1 F2  1 − 1 F2 1 3 ; , 2; − 2 2  τmax π νTmaxF−1 1 3 ; , 2; − 2 2 2  τmax π νTmaxF−1 2  . (3.42) For our numerical example, we consider a TDL-B channel model, as proposed by 3GPP [1, Sect. 7.7.3], and a carrier frequency of 2.5 GHz. Furthermore, we assume a long delay spread of 300 ns. We expect that in future wireless systems, the “typical” delay spread will be much lower than 300 ns [41]. Nonetheless, such a long delay spread allows robustness considerations. The optimal subcarrier spacing for a velocity of zero approaches F → 0 Hz, not feasible in practice. We therefore assume that the subcarrier spacing is lower bounded by F ≥ 15 kHz. For our channel parameters, the SIR is illustrated in Fig. 3.8 and allows the following conclusions: 1. The SIR in FBMC is high enough, so that the channel induced self-interference is usually dominated by noise. Thus, self-interference can be neglected.2 2. For an optimal subcarrier spacing, the Hermite prototype filter outperforms the PHYDYAS prototype filter, but only by approximately 0.6 dB. 3. For low velocities, on the other hand, the PHYDYAS prototype filter becomes better than the Hermite filter because of a better frequency localization in combination with a fixed subcarrier spacing (lower bound). 4. For a maximum symbol density of T F = 1 (complex), FBMC performs much better than OFDM without CP, especially for low velocities. 5. CP-OFDM, see (3.42), performs best, but also has a lower data rate than FBMC because of the CP overhead. κ and TCP = 0) has a lower SIR than CP-OFDM 6. WOLA (TW,TX = TW,RX = 2F and performs close to FBMC in high-velocity scenarios. 7. In general, OFDM-based schemes perform good in low-velocity scenarios, while in high-velocity scenarios, they lose most of their advantages when compared with FBMC. 2 The SNR is often below 20 dB. Wireless systems are interference limited and what we call “noise” is in practice often interference from other users and real-world hardware effects. 80 R. Nissel and M. Rupp Channel Model: TDL-B, Long delay spread: 50 rms = 300 ns, fc = 2.5 GHz Signal-to-Interference Ratio [dB] T F = 1.07 CP-OFDM, T F = 1.07, F=51 kHz, no ISI 40 WOLA, T F = 1.07 30 FBMC-OQAM (Hermite) 20 TF = 1 (complex) FBMC-OQAM (PHYDYAS) OFDM (no CP) CP-OFDM: F=51kHz WOLA: F=44kHz Hermite: F=37kHz PHDYAS: F=51kHz OFDM (no CP): F=22kHz 10 Subcarrier Spacing, F=15kHz Subcarrier Spacing, F>15kHz 0 0 50 100 150 200 250 300 350 400 450 500 Velocity [km/h] Fig. 3.8 The SIR is high enough so that the channel induced interference can usually be neglected in FBMC because it is dominated by noise Note that Fig. 3.8 represents a typical behavior. Different channel models and RMS delay spread values only shift the curves to some other point [41]. The main conclusion here is that one-tap equalizers are in most practical cases sufficient. For all of our testbed measurements in [34, 38, 42], low-complexity one-tap equalizers were sufficient. Feedback delays and repeated handovers are usually more problematic than a small, channel induced, interference. For the sake of completeness, we also consider a highly double-selective channel, where we assume an RMS delay spread of 720 ns and a carrier frequency of 60 GHz, as suggested in [1]. However, we want to emphasize that such extreme channel condition will rarely happen in practice and a system should therefore not be optimized for it, but it should be able to cope with those scenarios, at least to some extent. There exist several ways of dealing with such harsh channel environments: Firstly, the employment of computationally demanding equalizers, as, for example, proposed in [40, 49]. Secondly, we can treat interference as noise and accept a (small) throughput loss, see Fig. 3.2. Thirdly, spectral efficiency can be sacrificed in order to gain robustness. Let us discuss the last method in more detail, where we utilize the underlying orthogonality of the prototype filter in FBMC to transmit complex-valued symbols. This is enabled by setting some symbols, corresponding to the black markers in Figs. 3.4 and 3.5, to zero, so that a time–frequency spacing of T F = 2 is achieved. We call this transmission scheme FBMC-QAM but there does not exist a unique definition and different authors use this term differently. In FBMC-QAM, complex-valued symbols are transmitted and no intrinsic imaginary interference appears, allowing us to straightforwardly apply all known methods of 3 Filter Bank Multicarrier Modulation 81 Channel Model: TDL-B, Very long delay spread 50 rms = Hermite: F=204kHz PHDYAS: F=212kHz WOLA: F=234kHz CP-OFDM: F=292kHz Signal-to-Interference Ratio [dB] CP-OFDM, T F = 2, F=292 kHz, no ISI WOLA, T F = 2 40 720 ns, fc = 60 GHz FBMC-QAM (Hermite) 30 TF = 2 FBMC-QAM (PHYDYAS) 20 FBMC-OQAM (PHYDYAS) 10 TF = 1 (complex) Subcarrier Spacing, F=60kHz Subcarrier Spacing, F>60kHz 0 0 50 100 150 200 250 300 350 400 450 500 Velocity [km/h] Fig. 3.9 In the rare case of a highly double-selective channel, it is possible to sacrifice spectral efficiency (T F = 2) in order to improve robustness. FBMC-QAM then even outperforms CPOFDM OFDM. Figure 3.9 shows the SIR for a highly double-selective channel. The following important observations can be identified. 1. The Hermite prototype filter performs much better than the PHYDYAS prototype filter thanks to a better joint time–frequency localization. In FBMC-OQAM, this effect is largely lost because of the time–frequency squeezing. 2. FBMC with a Hermite prototype filter outperforms even CP-OFDM for velocities larger than 60 km/h. 3.6 Block Spread FBMC: Enabling All MIMO Methods The loss of complex orthogonality is the main obstacle in FBMC-OQAM and seriously hampers some important transmission techniques, such as channel estimation [35], Alamouti’s space-time block code [23] or maximum likelihood MIMO detection [62]. In particular, the limited MIMO compatibility3 is a major issue, preventing a widespread application of FBMC. In this section, we investigate a method which restores complex orthogonality in FBMC-OQAM, so that all known techniques for OFDM can be straightforwardly 3 Only some specific MIMO techniques become more challenging in FBMC. Many other MIMO methods, such as receive diversity or spatial multiplexing based on Zero-Forcing (ZF) or MMSE equalization, can be straightforwardly employed in FBMC. 82 R. Nissel and M. Rupp employed in FBMC. This is enabled by adding an additional code dimension (besides time and frequency). In contrast to conventional FBMC, the data symbols no longer belong to a certain time–frequency position, but are rather spread over several time or frequency positions. Such spreading typically increases the sensitivity to doubly selective channels. However, if the delay spread and the Doppler spread are sufficiently low, the channel induced interference can still be neglected. Spreading is very beneficial in FBMC because it can solve the underlying problem of Alamouti’s space-time block code and maximum likelihood (ML) MIMO detection in FBMC. For example, authors in [46] proposed a block-Alamouti scheme (over time) which can be seen as a special kind of spreading (distributing symbols in time). The same method was recently applied by [30] in the frequency domain. However, Walsh–Hadamard spreading [23, 34] offers more flexibility because it restores complex orthogonality, so that it not only works for Alamouti transmissions (as in [30, 46]), but additionally allows to straightforwardly employ all other methods known in OFDM, such as channel estimation, other space-time block codes or low-complexity maximum likelihood symbol detection. Similar to Walsh–Hadamard spreading, authors in [62] propose FFT spreading in time to restore (quasi)-orthogonality. Let us describe the spreading approach in more detail. In a first step, we assume an AWGN channel, that is, H = I N , so that (3.30) transforms to y = GH G x + GH n. (3.43) Note that (3.43) describes a block transmission of L subcarriers and K symbols in time. Several of those blocks must be concatenated in time and frequency to achieve a desired bandwidth and transmission time. LK In spread FBMC, complex-valued data symbols x̃ ∈ C 2 ×1 are precoded by a LK coding/spreading matrix C ∈ C L K × 2 , so that the transmitted symbols x ∈ C L K ×1 can be expressed by x = C x̃. (3.44) A priori, the size of C and x̃ is unknown. We will explain later in this section why LK the size was chosen that way. The received data symbols ỹ ∈ C 2 ×1 are obtained by decoding of the received symbols according to ỹ = CH y. (3.45) To restore complex orthogonality, the coding matrix must be chosen so that the following condition is fulfilled, CH GH G C = IL K/2 . (3.46) A straightforward way to find coding matrix C is based on an eigenvalue decompoLK sition of GH G = UΛUH , so that coding matrix C ∈ C L K × 2 becomes, 3 Filter Bank Multicarrier Modulation 83 ⎡ −1/2 Λ1 ⎢ ⎢ 0 ⎢ ⎢ C = U⎢ 0 ⎢ 0 ⎢ ⎣ : 0 0 .. . 0 ⎤ ⎥ 0 ⎥ ⎥ 1 − /2 0 ΛL K/2 ⎥ ⎥, ... 0 ⎥ ⎥ : : ⎦ ... 0 (3.47) Fig. 3.10 The eigenvalues of GH G for an FBMC-OQAM system. Similar as for the derivation of the MIMO channel capacity, the eigenvalues in combination with eigenvector precoding can be utilized to determine the optimal precoding matrix Eigenvalues of GH G where Λi represents the i-th eigenvalue (sorted) of GH G and U the unitary eigenvector matrix. Figure 3.10 shows the eigenvalues of a typical FBMC-OQAM transmission matrix GH G. For the limit case of K → ∞ and L → ∞, the eigenvalues are Λ1 = Λ2 = · · · = ΛL K/2 = 2 and Λi = 0 for i > L2K . Thus, (3.47) implicitly applies water filling [56], so that C becomes the optimal spreading matrix in terms of maximizing the information rate. In particular, it shows that the optimal size of LK the spreading matrix is L K × L2K and that any matrix, C ∈ C L K × 2 , which satisfies (3.46), is optimal for K → ∞ and L → ∞ (the SNR is always the same). Moreover, it also shows that the intrinsic imaginary interference does not consist of any useful information and can thus be canceled by taking the real part. For a limited number of subcarriers and time symbols, the spreading matrix in (3.47) no longer corresponds to the optimal solution. Instead, water filling could improve the performance, where the column size of matrix C will usually be larger than LK . However, a spreading matrix of size L K × L2K , which satisfies (3.46), still 2 performs close to the optimum, as indicated by the eigenvalues in Fig. 3.10. For example, for L = 36 and K = 30, the suboptimal spreading matrix performs only 3.6% worse in terms of achievable rate than the optimal spreading matrix (water filling) for SNR values smaller than 20 dB. Furthermore, the optimal spreading matrix requires different code rates for layers close to the eigenvalue of ΛL K/2 . This increases the overall complexity, while the possible improvement is rather low, so that, employing a slightly suboptimal spreading matrix makes sense in practical systems.(Note that precoding power by a factor of two, that ) ( reduces the ( ) ) average transmit is, tr GCCH GH = tr CH GH GC = L2K = 21 tr GH G . Thus, for the same SNR as without precoding, the data symbol power has to be increased by a factor of two. It is also interesting that the noise after despreading is white, ñ ∼ C N (0, Pn IL K/2 ), 2 L = 12, K = 10 1.5 L = 36, K = 30 1 Cyclic repetition, corresponds to L ,K 0.5 0 0 0.2 0.4 0.6 i-th position / LK 0.8 1 84 R. Nissel and M. Rupp Code Time Frequency Fig. 3.11 In conventional FBMC-OQAM, real-valued symbols are transmitted over a rectangular time–frequency grid (T F = 0.5). Two real-valued symbols are required to transmit one complexvalued symbol. Thus, the name “offset”-QAM, where we apply the offset not in time (as often in c literature) but in frequency. 2017 IEEE, [39] even though the spreading matrix itself is not necessarily semi-unitary. The reason behind this is again the orthogonalization condition in (3.46), implying that Rñ = CH GH Rn GC = Pn IL K/2 . While the spreading matrix in (3.47) provides analytical insight, it is not very practical because of a high computational complexity and the fact that the spreading is performed in both, time and frequency, which only works for a doubly flat channel. Walsh–Hadamard spreading [22, 23, 34, 39], on the other hand, is a much more practical solution because it requires almost no additional complexity and the spreading is performed in only one dimension, either in time or in frequency. In conventional FBMC-OQAM, see Fig. 3.11, each time–frequency position can only carry real-valued symbols, so that two time–frequency positions are required to transmit one complex-valued data symbol, indicated by the color in Fig. 3.11. In block spread FBMC-OQAM, on the other hand, data symbols no longer belong to a specific time– frequency position, but are spread over several subcarriers, see Fig. 3.12. To keep the spectral efficiency the same as in FBMC-OQAM (ignoring possible guard symbols), several data symbols are transmitted over the same time–frequency resources, but differentiated by their spreading/coding sequence. To be specific, L/2 complex-valued data symbols are spread over L subcarriers. This leads to the same information rate as in conventional FBMC-OQAM (again, ignoring possible guard symbols). Although complex orthogonality can be perfectly restored within one block, there still exists interference between different blocks. Thus, if we spread in frequency, a guard subcarrier might be necessary. If we spread in time, on the other hand, a guard symbol in time might be required. Such guard symbols typically reduce the spectral efficiency by a few percent, see [34, 39]. Spreading in frequency can be described by frequency spreading matrix C f ∈ L× L2 R for which we take every second column out of a sequency-ordered [25] Walsh– Hadamard matrix H ∈ R L×L , that is, 3 Filter Bank Multicarrier Modulation 85 Code Time Guard subcarrier Channel is approximately frequency-flat Frequency Fig. 3.12 In block spread FBMC-OQAM, complex-valued symbols are spread over several subcarriers (or time positions), allowing to restore complex orthogonality within one block. To improve c the SIR between different blocks, a guard symbol might be necessary. 2017 IEEE, [39] [C f ]l,m = [H]l,2m−1 ; for l = 1, 2, . . . , L; m = 1, 2, . . . , L . 2 (3.48) Note that matrix C f in (3.48) could equivalently be defined by [H]l,2m . Utilizing the underlying structure of our matrix notation (vectorization) and the fact that we LK spread in frequency only, allows us to express overall spreading matrix C ∈ R L K × 2 by, C = IK ⊗ C f , (3.49) where Kronecker product ⊗ together with identity matrix I K map coding matrix C f to the correct time slots. Spreading in time can be described in a similar way except that we have to alternate K K between spreading with Ct ′ ∈ R K × 2 and spreading with Ct ′′ ∈ R K × 2 for adjacent subcarriers. The spreading matrices itself are again found by taking every second column out of a sequency-ordered [25] Walsh–Hadamard matrix H ∈ R K ×K , that is, [Ct ′ ]k,m = [H]k,2m−1 [Ct ′′ ]k,m = [H]k,2m ; for k = 1, 2, . . . , K ; m = 1, 2, . . . , K . 2 (3.50) LK To find the overall spreading matrix C ∈ R L K × 2 , we have to map the individual spreading matrices Ct ′ and Ct ′′ to the correct subcarrier positions. For the vectorized system model in (3.26), this implies that, + , , 00 10 C = Ct ′ ⊗ IL/2 ⊗ , + Ct ′′ ⊗ IL/2 ⊗ 01 00 + (3.51)     where the matrices 01 00 and 00 01 are necessary to alternate between spreading with Ct ′ and Ct ′′ for adjacent subcarriers. 86 R. Nissel and M. Rupp It can easily be checked by numerical evaluations that (3.49) and (3.51) satisfy the complex orthogonalization condition in (3.46). For a formal proof that Walsh– Hadamard spreading restores complex orthogonality in FBMC, we refer to [22]. Authors of [22] left the question open whether it is possible to find a spreading matrix that has more than L2K columns while still satisfying (3.46). Our investigations in (3.47) show that this is not possible (ignoring any edge effects which become negligible for a large number of K and L). A small disadvantage of Walsh–Hadamard spreading is the fact that the spreading length has to be a power of two. This makes the integration into existing systems problematic, but has almost no impact if a system is designed from scratch. The big advantage of Walsh–Hadamard spreading, on the other hand, is that only additions, but no multiplications are needed. Thus, the additional computational complexity is very low. Moreover, a fast Walsh–Hadamard transformation can be used, further reducing the computational complexity. For example, spreading in time only requires log2 (K ) − 1 extra additions/subtractions per data symbol at the transmitter and log2 (K ) extra additions/subtractions per data symbol at the receiver. For spreading in frequency, it is log2 (L) − 1, respectively log2 (L). Similar as in (3.29), it is possible to include a doubly selective channel into our transmission model. The input output relationship between the transmitted data symbols x̃ and the received data symbols ỹ can then be modeled by ỹ = CH GH H GC x̃ + CH GH n H H ≈ diag{h̃} x̃ + C G n. (3.52) (3.53) Similar as in (3.29), if the delay spread and the Doppler spread are sufficiently low, the transmission can be approximated by a one-tap channel, see (3.53). If we spread in time, block spread FBMC becomes more sensitive to time-variant channels. At the same time, it becomes slightly more robust to multipath delays. For spreading in frequency, the opposite holds true. In [34, 39], we discuss the effect of doubly selective channels on block spread FBMC. Precoding by C can also be interpreted  as transforming the underlying basis pulses according to G̃ = GC = g̃1 · · · g̃L K/2 . Thus, instead of modulating data symbols with gl,k (t), as in (3.1), we modulate them with g̃i (t). In contrast to conventional multicarrier systems, however, the transformed basis pulses g̃i (t) no longer all employ the same underlying prototype filter p(t). Instead, many basis pulses have their own, unique, prototype filter pi (t). Thus, we cannot directly implement G̃ in an efficient way. However, by interpreting G̃ as a precoded FBMC system, the advantage of an efficient signal generation are preserved. Moreover, such interpretation offers overall a high flexibility. We have validated the block spread FBMC approach by real-world testbed measurements in [34] for outdoor-to-indoor scenarios (150 m link distance, 2.5 GHz carrier frequency) and in [42] for indoor-to-indoor scenarios (5 m link distance, 60 GHz carrier frequency). For both measurements, the assumption of a low delay spread and a low Doppler spread was fulfilled, so that (3.53) accurately described 3 Filter Bank Multicarrier Modulation 87 2.5 GHz, 16-QAM, F=15 kHz Bit Error Ratio 10 −1 2×2 MIMO, ML detection 60 GHz, 4-QAM, F=500 kHz, 10 −2 2×1 Alamouti 2×1 Alamouti CP−OFDM Spread FBMC 10 −3 −5 0 5 10 15 20 25 Signal-to-Noise Ratio for OFDM [dB] Fig. 3.13 Real-world testbed measurements [34, 42] show that MIMO works in FBMC once symbols are spread in time. The spreading process itself has a low computational complexity because of a fast Walsh–Hadamard transformation. FBMC and OFDM experience both the same c BER, but FBMC has lower OOB emissions. 2017 IEEE, [41] the true physical behavior. Because of a time-invariant channel, we spread in time instead of frequency, see (3.50) and (3.51). For the 60 GHz measurement setup, we employed a high subcarrier spacing of F = 500 kHz, as often considered in millimeter wave transmissions [44]. This implicitly reduces the latency, so that even though we spread symbols in time, the overall transmission time was less than 40 µs, satisfying the low-latency condition of 100 µs [14]. Figure 3.13 shows the measured BER over SNR. Alamouti’s space-time block code and low-complexity ML MIMO detection performs in FBMC as good as in OFDM, but FBMC has the advantage of much lower OOB emissions. 3.7 Pruned DFT-Spread FBMC-OQAM: Reducing the PAPR Besides the intrinsic imaginary interference, nonlinearities, such as a limited Digitalto-Analog Converter (DAC) resolution or a nonlinear power amplifier, are even more problematic in practical systems because they destroy the superior spectral properties of FBMC [41, 50]. Thus, the concept of sharp digital filters to enable a flexible time– frequency allocation, as discussed in [41], only works as long as FBMC operates in the linear regime. In multicarrier systems, this is, in general, hard to achieve because of the poor peak-to-average power ratio (PAPR). To reduce the PAPR in practical systems, LTE employs single carrier frequency division multiple access (SC-FDMA) in the uplink [54], a DFT precoded OFDM system. The same technique will also be used in 5G [3]. Unfortunately, simply combining FBMC and a DFT, as done in SC-FDMA for OFDM, performs poorly in FBMC [18, 31, 61]. This motivated us to develop pruned DFT-spread FBMC [32, 37], a novel transmission technique R. Nissel and M. Rupp Fig. 3.14 Precoding matrix C shapes the transmitted signal in such a way, that the average transmit power (diagonal elements of E{s sH } = GC CH GH ) shows an almost perfect rectangular shape, with many beneficial properties. For the illustration we consider only one FBMC symbol, that is, K =1 Transmit Power, E{|s(t)|2 } 88 1 0.8 diag{GC CH GH } Pruned DFT-s FBMC 0.6 0.4 Conventional FBMC-OQAM diag{GGH } 0.2 0 −1 −0.5 0 0.5 1 Normalized Time, t/T0 with the remarkable properties of a low PAPR, low-latency transmissions and a high spectral efficiency. The idea of pruned DFT-spread FBMC is closely related to Walsh–Hadamard spreading, see Sect. 3.6. In particular, we again spread L2 data L symbols over L subcarriers, described by frequency spreading matrix C f ∈ C L× 2 , C f = W L×L/2 diag{b}, L (3.54) where W L×L/2 ∈ C L× 2 describes a pruned DFT matrix, that is, a conventional DFT L matrix for which L2 column vectors are canceled. Scaling vector b ∈ R 2 ×1 , on the other hand, guarantees that the diagonal elements of CH GH G C are exactly one, with C = IK ⊗ C f . Figure 3.14 shows the expected transmit power over time for one FBMC symbol. In conventional FBMC, there exists a large overlapping of symbols in time and the transmission requires a long ramp-up and ramp-down period. In pruned DFT-spread FBMC, on the other hand, precoding by C f shapes the transmitted signal in such a way, that the overlapping in time is very low and the ramp-up and ramp-down period dramatically reduced. This reduces the overall latency. The pruned DFT matrix in (3.54) is found by canceling those column vectors of a conventional DFT matrix, so that the main energy is concentrated within the time interval − T40 ≤ t ≤ T40 , see Fig. 3.14. The OOB emissions of pruned DFT-spread FBMC are comparable to conventional FBMC transmissions, leading to a high spectral efficiency. From a conceptional point of view, the key difference between pruned DFTspread FBMC and block spread FBMC, discussed in Sect. 3.6, is that pruned DFT-spread FBMC spreads the data symbols over the whole bandwidth, while for block spread FBMC the bandwidth is split into smaller chunks. Those small chunks can then be equalized by a simple one-tap equalizer, so that Alamouti’s space-time block code and ML MIMO detection become feasible. In pruned DFT-spread FBMC, on the other hand, low-complexity ML detection is often not possible and one has to rely on minimum mean squared error (MMSE) equalization before despreading by CHf , same as in SC-FDMA. Another small drawback of pruned DFT-spread FBMC is 3 Filter Bank Multicarrier Modulation 10 4QAM, L = 128 Subcarriers 0 CP-OFDM 10 −1 FBMC CCDF Fig. 3.15 Pruned DFT-spread FBMC [32, 37] has the same PAPR as SC-FDMA, but the additional advantages of a higher spectral efficiency. Note that a simple DFT-spread FBMC transmission scheme performs relatively poor [18, 31] 89 10 10 −2 −3 SC-FDMA Simple DFT-s FBMC Pruned DFT-s FBMC 5 6 7 8 9 10 11 Peak-to-Average Power Ratio (PAPR) [dB] that orthogonality is only approximately restored, that is, CH GH G C ≈ IL K/2 , leading to some (small) interference. By restricting the time domain of p(t) to approximately − 43 T0 ≤ t ≤ 43 T0 , the interference can be reduced, so that it becomes neglectable in most cases. Moreover, a frequency CP can further reduce the interference [32, 37], if necessary. Figure 3.15 shows the complementary cumulative distribution function (CCDF) of the PAPR for a 4-QAM signal constellation and L = 128 subcarriers. Conventional FBMC has the same poor PAPR as OFDM. A simple DFT-spread FBMC transmission scheme, as proposed in [18], only slightly improves the PAPR. Even an π π optimal phase condition [31], that is, e j 2 (l+k) → e j 2 (l+k) e−jπlk , hardly reduces the PAPR. In pruned DFT-spread FBMC, on the other hand, the PAPR is as good as in SC-FDMA and approximately 3 dB better than in OFDM and FBMC. 3.8 Summary FBMC has the best spectral properties among all 5G waveform candidates. This is especially useful if the number of subcarriers is low, for example, in eMTC, and to support different use cases within the same band. To efficiently generate an FBMC signal, many hardware components from windowed OFDM can be reused. The main drawback of FBMC is that orthogonality only holds in the real domain, which makes some techniques, such as channel estimation or some MIMO transmission methods, more challenging. However, there exist several solutions to overcome those limitations. Future wireless systems will be characterized by a relatively low delay spread, so that the channel induced self-interference in FBMC is very low and can often be neglected. This is even more true if an optimal subcarrier spacing is employed. To restore complex orthogonality in FBMC, one can spread data symbols over several time or frequency positions. In this context, we presented block spread FBMC which 90 R. Nissel and M. Rupp allows to straightforwardly employ all known methods from OFDM in FBMC (if the delay spread and the Doppler spread are sufficiently low). If, on the other hand, the focus lies on reducing the PAPR, pruned DFT-spread FBMC is a better option. 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Zakaria, D. Le Ruyet, A novel filter-bank multicarrier scheme to mitigate the intrinsic interference: application to MIMO systems. IEEE Trans. Wirel. Commun. 11(3), 1112–1123 (2012) 63. X. Zhang, M. Jia, L. Chen, J. Ma, J. Qiu, Filtered-OFDM-enabler for flexible waveform in the 5th generation cellular networks, in IEEE Global Communications Conference (GLOBECOM) (2015), pp. 1–6 Chapter 4 Generalized Frequency Division Multiplexing: A Flexible Multicarrier Waveform Ahmad Nimr, Shahab Ehsanfar, Nicola Michailow, Martin Danneberg, Dan Zhang, Henry Douglas Rodrigues, Luciano Leonel Mendes and Gerhard Fettweis 4.1 Introduction to GFDM Modulator In this section, we give an overview of generalized frequency division multiplexing (GFDM) waveform. We start from the continuous model representation, from which an analytical expression of power spectral density (PSD) is derived. Based on that, a discrete signal model representation is derived and expressed in terms of matrix model. We show the structure of the modulation matrix by mean of decomposition. A. Nimr (B) · S. Ehsanfar · M. Danneberg · D. Zhang · G. Fettweis Vodafone Chair Mobile Communication Systems, Technische Universität Dresden, Dresden, Germany e-mail: ahmad.nimr@ifn.et.tu-dresden.de S. Ehsanfar e-mail: shahab.ehsanfar@ifn.et.tu-dresden.de M. Danneberg e-mail: martin.danneberg@ifn.et.tu-dresden.de D. Zhang e-mail: dan.zhang@ifn.et.tu-dresden.de G. Fettweis e-mail: gerhard.fettweis@ifn.et.tu-dresden.de N. Michailow National Instruments Corp., 11500, Mopac Expwy, Austin, TX 78759, USA e-mail: nicola.michailow@ni.com H. D. Rodrigues · L. L. Mendes Intituto Nacional de Telecomunicações (Inatel), Santa Rita do Sapucai, MG, Brazil e-mail: henry@inatel.br L. L. Mendes e-mail: lucianol@inatel.br © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_4 93 94 A. Nimr et al. From this model, the design requirements and performance indicators are derived. Finally, we show the ability of GFDM representation to generate other state-of-the-art waveforms. 4.1.1 Continuous Signal Model GFDM is a block-based multicarrier modulation technique that employs circular filtering [1]. For a better understanding of the GFDM structure, we represent the modulation technique using the continuous time model. Consider a time-frequency resource block defined by time duration T and frequency bandwidth B. The target is to use this resource to convey a data message of maximum length of N data symbols. For this purpose, the available bandwidth is divided into K equally spaced subcarriers with subcarrier spacing Δf = KB , and the available time is divided into M subsymbols with subsymbol spacing Tsub = MT . The subcarrier spacing is related , where to the subsymbol spacing with the relation ΔfTsub = 1. Hence, T = NB = Δf M N = KM . Each pair (k, m)-(subcarrier, subsymbol) can be used to transmit one data symbol dk,m modulated by a pulse shape gk,m (t) given by gk,m (t) = wT (t)gT (t − mTsub )ej2πkΔft , (4.1) where wT (t) is a rectangular window of duration T , namely wT (t) = 1, t ∈ [0, T ] and 0 elsewhere. gT (t) is a prototype periodic pulse shape of period T , which can be expressed using Fourier series. This means that the pulse shapes are generated by time and frequency shifts of a periodic prototype pulse shape in addition to multiplication with a finite time window to form the GFDM block. In conventional GFDM, the number of the frequency components of gT is limited to 2M such that, gT (t) = M −1  q=−M q g̃T [q]ej2π M Δft , (4.2) where g̃T [q] are the nonzero coefficients of the Fourier series. This allows each subcarrier to span at maximum two subcarrier spacing. The frequency response of the prototype pules shape is given by G T (f ) = M −1  q=−M g̃T [q]δ(f − q M Δf ). (4.3) Here, δ(·) is the Dirac pulse. GFDM commonly adopts a cyclic prefix (CP) of duration Tcp ≥ τmax to tackle the impact of fading channel with maximum excess delay spread τmax . In addition, a cyclic suffix (CS) with duration Tcs may be added to the end of the block. The CP and CS can be simply introduced by extending the window to 4 Generalized Frequency Division Multiplexing … 95 Fig. 4.1 Pulse shape generation using periodic Raised-Cosine prototype pulse shape, with roll-off factor α = 0, M = 4 for k = 0, m = 0, . . . , 3 wTs (t) where Ts = T + Tcp + Tcs , as depicted in Fig. 4.1. Mathematically, it can be written as (t) gk,m (t) = wTs (t − Tcp )gT (t − mTsub )ej2πkΔft . (4.4) Therefore, the frequency domain representation is given by     −j2πTcp (f −kΔf ) ∗ G T (f )e−j2πmTsub f , G (t) k,m (f ) = WTs (f − kΔf )e (4.5) where ∗ denotes the convolution operator. Replacing G T (f ) from (4.3), we get (t) G k,m (f ) = e−j2π Tcp (f −kΔf ) M −1  q=−M q q Δf g̃T [q]WTs (f − (kM + q) M )e−j2π m M e+j2π Tcp Δf M . (4.6) In practice, not all subcarriers and subsymbols are used, thus, we define Kon and Mon as the sets of active subcarriers and subsymbols, respectively. Therefore, the signal corresponding to the i-th GFDM block that modulates the data symbols {dk,m,i } is generated as   (t) dk,m,i gk,m (t). (4.7) xi (t) = m∈M on k∈K on Moreover, the signal of a frame that contains Ns blocks can be expressed as x(t) = N s −1  i=0 xi (t − iTs ) = N s −1    i=0 m∈M on k∈K on (t) dk,m,i gk,m (t − iTs ). (4.8) From (4.8) and (4.6), the PSD  of the GFDM signal assuming uncorrelated data symbols with unit power, i.e., E dk∗1 ,m1 ,i1 dk2 ,m2 ,i2 = δ(k1 − k2 )δ(m1 − m2 )δ(i1 − i2 ), can be computed as, 96 A. Nimr et al. Fig. 4.2 PSD of subsymbols using a periodic raised-cosine (RC) prototype pulse with roll-off factor α = 0, M = 4 for k = 0, m = 0, . . . , 3 Sx (f ) = 1 Ts 1 = Ts  m∈M on 2   (t)  G k,m (f ) k∈K on  −1 q   M −j2π m M +j2π Tcp Δf  g̃T [q]WTs (f − (kM + q) Δf e  M )e m∈M on k∈K on q=−M  q M 2 (4.9)    .   According to (4.9), the PSD is obviously influenced by the prototype pulse shape coefficients {g̃T [q]}, the window WTs (t) and the number of subsymbols M . In addition, the active subsymbol set Mon plays an important role throughout the phase term q e−j2πm M , which depends on the subsymbol index m. To clarify that, Fig. 4.2 shows the individual PSD of each subsymbol using a rectangular window and no CP nor CS. As can be seen, the first subsymbol, namely m = 0, is the source of high out-ofband (OOB) emission for the selected prototype pulse shape. This can be intuitively understood via the discontinuity between successive blocks, which happens when dk,0,i and dk,0,i+1 are not identical. 4.1.2 Discrete Signal Model The discrete time signal representation can be derived from the sampling of the analog signal with frequency Fs = B. With that, we get K, N , Lcp , and Lcs samples per subsymbol, symbol, CP and CS, respectively. The discrete prototype pulse shape results from (4.2) with 4 Generalized Frequency Division Multiplexing … g[n] = M −1  q=−M 97 qn g̃T [q]ej2π N , n = 0, . . . , N − 1. (4.10) Let g̃ = N-DFT {g} be the N -point finite discrete Fourier transform (DFT) of g such that N −1 qn 1  g[n] = g̃[q]ej2π N , (4.11) N q=0 then, the relation between the Fourier series coefficient g̃T [q] of the continuous model and the frequency bins {g̃[q]} of the discrete model can be expressed as, g̃[q] = 1 g̃T [< q >N ], N (4.12) where < · >N is the modulo-N operator. Assuming a rectangular window as in (4.1), we get k (4.13) gk,m [n] = g[< n − mK >N ]ej2π K n . Thus, the subcarrier–subsymbol pulse shapes are generated from the circular shift of the prototype pulse shape in the time and frequency domains. In fact, the circularity in time results from the design with periodic pulse shape and in frequency from the sampling. Consequently, the design of the prototype pulse shape g[n] can be carried out in the frequency domain such that only the first and last M samples of g̃ have nonzero values. This ensures limited inter-carrier interference (ICI) to only adjacent subcarriers under the assumption of perfect synchronization. However, in the case of asynchronous subcarriers, we resort to the continuous discrete-time Fourier transform (DTFT) G k,m (ν) = DFT(gk,m [n]), which takes into account the frequency response of the window as well. Actually, g̃k,m [q] = G k,m (ν = Nq ), as illustrated in Fig. 4.3. Fig. 4.3 ICI of adjacent subcarriers using periodic RC prototype pulse with roll-off factor α = 0.5, M = 16 for k = −1, 0, 1, m = 0. The sampling points stand for g̃k,m [q], and the solid line represents G k,m (ν) 98 A. Nimr et al. Next, we focus on the matrix representation of the GFDM block showing that both modulation and demodulation matrices follow the same structure. This structure is determined with the decomposition of the GFDM matrix. 4.1.2.1 Modulation Matrix Model One GFDM block can be represented in a vector x ∈ CN ×1 such that, [x](n) =   m∈M on k∈K on k dk,m g[< n − mK >N ]ej2π K n . (4.14) In the frequency domain, x̃ = N-DFT{x} can be written as     m x̃ (n) = dk,m g̃[< n − kM >N ]e−j2π M n . (4.15) m∈M on k∈K on Adding CP and CS is done by copying the last Lcp to the beginning and the first Lcs samples to the end of x. Let D ∈ CK×M be the matrix representing the data symbols, with [D](k,m) = dk,m and for (k, m) ∈ / Kon × Mon , dk,m = 0. We define the data vector d = vec {D}, namely, [d](k+mK) = [D](k,m) . In addition, the modulation matrix A ∈ CN ×N is defined by k [A](n,k+mK) = g[< n − mK >N ]ej2π K n . (4.16) Thereby, [x](n) = M −1 K−1   [A](n,k+mK) [d](k+mK) , (4.17) m=0 k=0 and thus, x = Ad. (4.18) Taking into account the active sets of subcarriers and subsymbols, we can define a compact representation for the active resource set Non = {n = k + mK, (k, m) ∈ Kon × Mon } (4.19) x = A(on) d(on) , (4.20) as where A(on) = [A](:,N on ) stands for the active modulation matrix and d(on) = [d](N on ) is the vector of the active data symbols. 4 Generalized Frequency Division Multiplexing … 4.1.2.2 99 Demodulation Matrix Model An equalized received signal y[n], represented by the vector y ∈ CN ×1 , is to be demodulated using a receive prototype filter γ [n]. The estimated data symbols can be expressed as  k d̂k,m = γ ∗ [−n] ⊛ y[p]e−j2π K n |n=mK d̂ (k+mK) = N −1  n=0 k y[n]γ ∗ [< n − mK >N ]e−j2π K n . (4.21) Then, d̂ = BH y, where (4.22)  H k B (k+mK,n) = γ ∗ [< n − mK >N ]e−j2π K n , so that k [B](n,k+mK) = γ [< n − mK >N ]ej2π K n . (4.23) Comparing with (4.16), we conclude that the demodulation matrix has the same structure as the modulation matrix. 4.1.3 GFDM Matrix Decomposition As the GFDM matrix is generated by circular shift of a prototype pulse shape in the time and frequency domains, it has a well-defined structure and its properties can be derived from the prototype pulse shape. To investigate the structure, first, we define several auxiliary matrices to facilitate the derivation. For a vector a ∈ CPQ×1 , we define the following matrices T  (a) = unvecQ×P {a} , VP,Q (a) (a) ZP,Q = FP VP,Q = (a) ṼP,Q , (4.24) (4.25) where unvecQ×P {a} denotes the inverse of vectorization operation. Thus, (a) VP,Q (p,q) = [a](q+pQ) . (4.26) ij FP is the P-point DFT matrix, where [FP ](i,j) = e−j2π P . The matrix V(a) P,Q represents (a) the polyphase components generated by the sampling of a with factor Q, while ZP,Q is known as discrete Zak transform (DZT) transform [2]. Figure 4.4 visualizes these matrices by mean of an example. 100 A. Nimr et al. Fig. 4.4 DZT example. a of size 6 × 1, P = 3, Q = 2 Recall the GFDM block Eq. (4.14) with the consideration of full allocation and defining n = q + pK, q = 0, . . . , K − 1 and p = 0, . . . , M − 1, then [x](q+pK) = M −1 K−1   m=0 k=0 k dk,m g[< q + pK − mK >N ]ej2π K q . (g) Based on (4.26), [x](q+pK) = V(x) M ,K (p,q) and VM ,K (p,q) = g[< q + pK >N ]. Fol- lowing this, it is easy to show that g[< q + pK − mK >N ] = V(g) M ,K Putting all notations together we get V(x) M ,K (p,q) = = M −1  (g) VM ,K m=0 M −1  (g) VM ,K m=0 (<p−m>M ,q) (<p−m>M ,q) K−1  (4.27) (<p−m>M ,q) . k dk,m ej2π K q k=0 (4.28)  T H D FK (m,q) . (g) The second line represents circular convolution between the q-th column of VM ,K and the q-th column of DT FH K . By representing the circular decomposition in the frequency domain we get, V(x) M ,K (:,q) = 1 H F diag M M (g) ṼM ,K (:,q)   FM DT FH K (:,q) . (4.29) Finally, by stacking the columns according to the q index and using (4.25), we get V(x) M ,K =   1 H  (g) FM ZM ,K ⊙ FM DT FH . K M (4.30) 4 Generalized Frequency Division Multiplexing … 101 Here, ⊙ denotes the element-wise multiplication operator. Thus, x = vec Following the same approach on x̃ defined in (4.15), we get V(x̃) K,M (q,p) = K−1  (g̃) VK,M k=0 (<q−k>K ,p) M −1   m dk,m e−j2π M p . (x) VM ,K T . (4.31) m=0 Thereby, V(x̃) K,M = Similarly, x̃ = vec  V(x̃) K,M 1 H  (g̃) F ZK,M ⊙ [FK DFM ] . K K (4.32) T . Finally, using the vectorized output of (4.30) and (4.32) with respect to (4.14), we can express the modulation matrix A as 1 T P UH L(g) UK,M PM ,K UH M ,K , M M ,K K,M 1 1 H (g̃) T = F PM ,K UH M ,K L UM ,K PM ,K UK,M PM ,K . NK N A= (4.33) (4.34) Here, PP,Q ∈ ℜPQ×PQ is the permutation matrix that fulfills for any Q × P matrix X   vec XT = PP,Q vec {X} , (4.35) UP,Q = IP ⊗ FQ , (4.36) where ⊗ is the Kronecker product, and    (g) L(g) = diag vec ZM ,K ,    (g̃) L(g̃) = diag vec ZM ,K . (4.37) (4.38) Either of these diagonal matrices can be used to analyze the GFDM performance (g) (g̃) indicators and derive the demodulator matrix. In other words, ZM ,K or ZM ,K is the key of GFDM design and analysis. 4.1.3.1 Demodulation Prototype Filter While the GFDM-based demodulator matrix has the same structure as the modulation matrix, by using the decomposition (4.33) or (4.34), the prototype filter of the demodulator can be easily computed from the inverse of DZT as γ =  1 (γ ) vec FMH ZM ,K M T . (4.39) 102 A. Nimr et al. First, we need to find B. Using the representation (4.33), the product BH A can be calculated as 1 (γ )H (g) L UK,M PM ,K UH UK,M PTM ,K UH K,M L M ,K , M = KUH L(γ )H L(g) U, BH A = where 1 U = √ UK,M PM ,K UH M ,K , N (4.40) (4.41) is a unitary matrix. Thus, the diagonal elements of KL(γ )H L(g) represent the eigenvalues of BH A. Obviously, the matched filter demodulator uses γ [n] = g[n]. On the other hand, the zero forcing (ZF) is determined when KL(γ )H L(g) = IN . Conse(γ ) quently, γzf can be computed form the DZT inverse of ZMzf,K , defined by (γ ) ZMzf,K (m,k) = 1 K (g) ∗ ZM ,K (4.42) . (k,m) The minimum mean square error (MMSE) demodulator, assuming additive white Gaussian noise (AWGN) of noise power PN and uncorrelated data symbols with power PD , can be computed via the matrix  BMMSE = AAH + = PN I PD N −1 A  1 T (g) (g)H PM ,K UH + K,M KL L M PN PD −1 (4.43) L(g) UK,M PM ,K UH M ,K . Comparing with (4.33), we find that (γ ) ZMMMSE ,K (m,k) = (g) ZM ,K (k,m)    (g) K  ZM ,K 2   + (k,m)  PN PD −1 . (4.44) Then, γMMSE can be found form the DZT inverse using (4.39). It is worth noting that in the case of non-full allocation, an MMSE or least squares (LS) receiver can be derived based on the compact model x = A(on) d(on) . However, the obtained matrix does not necessary have a GFDM structure and may complicate the implementation. 4.1.4 Performance Indicators In order to evaluate the design with a given prototype pulse shape, we study three performance indicators of the modulation matrix A under the full allocation assumption. 4 Generalized Frequency Division Multiplexing … 103 (g̃) All these indicators can be computed from ZK,M using the decomposition represented in (4.34), which can be reformulated as 1 A = √ WH L(g̃) V, K (4.45) with WH = √ 1 H FH N PM ,K UM ,K , NK 1 V = √ UM ,K PTM ,K UK,M PM ,K , N (4.46) (4.47) are unitary matrices. Conditional Number 2 , then {σk,m = |zk,m |2 } correspond K maxk,m {σk,m } maxk,m {|zk,m |} = . mink,m {σk,m } mink,m {|zk,m |} (4.48) (g̃) Defining the short-hand notation zk,m = ZK,M (k,m) to the singular values of A. The conditional number of A is given by cond (A) = When cond (A) = 1, i.e., |zk,m | = 1, ∀(k, m), then A is orthogonal, and when there is at least (k0 , m0 ) such that zm0 ,k0 = 0, A becomes singular. The rank of A is reduced (g̃) by the number of zero elements in ZK,M . The conditional number is important in all receiver processing steps that require the computation of the inverse of A. Thus, a well-conditioned modulation matrix with smaller conditional number is preferred. Although we can always design GFDM with an orthogonal matrix, some other requirements cannot be achieved. More details on that are introduced in Sect. 4.1.5. Noise Enhancement Factor Considering the received signal in AWGN channel, y = Ad + w, (4.49)     with E ddH = PD IN and E wwH = PN IN , the noise enhancement factor (NEF) is defined by the ratio of the average signal-to-noise ratio (SNR) before and after applying the ZF demodulator; −1        trace E AddH AH trace E ddH       ξ= trace E vvH trace E A−1 vvH A−1H     1 = 2 trace AAH trace A−1 A−1H N 104 A. Nimr et al.      (g̃) 2  (g̃)−1 2  L  L F ⎛ ⎞ ⎞⎛ F  1 1 ⎝ ⎠. = 2 |zk,m |2 ⎠ ⎝ N |zk,m |2 = 1 N2 k,m (4.50) k,m Noting that after applying the ZF demodulator, we get A−1 y = d + A−1 w and because the rows of A−1 are generated from circular shift of the prototype filter γ zf , the noise enhancement is equal on each of the elements of d in the case of white noise. Contrariwise, it depends on the subcarrier–subsymbol index for colored noise. Self-interference Ratio Because the columns of A are generated from the prototype pulse shape g[n] by circular shift in time and frequency, then and hence,   AH A (n,n) = g 2 , (4.51)   1  |zk,m |2 = N g 2 . trace AH A = K (4.52) k,m After matched filter we get   AH y = g 2 d + AH A − g 2 IN d + AH w  = g 2 d + VH K1 L(g̃)H L(g̃) − g 2 IN Vd + AH w. (4.53) The signal-to-interference ratio (SIR) is defined by the ratio of the signal power to the self-interface SIR =   1 k,m = K N g |zk,m |2 − g N   |zk,m |2 k,m ⎡ 4 K g 2 −1  2 2 2 ⎛ (4.54) 2 ⎞2 ⎤−1 ⎢ 1  ⎜ |zk,m | ⎟ ⎥ =⎣ − 1⎠ ⎦ ⎝1  2 N |z | k,m N k,m . k,m The computed self-interference is averaged over all data symbols. Nevertheless, when full allocation is considered and all data symbols  have the same power, the SIR is identical for each data symbol. This is because [V](i,j)  = √1N , ∀i, k ∈ {0, . . . , N − 1}. 4 Generalized Frequency Division Multiplexing … 105 The later discussion shows the dependency of the modulation on the prototype pulse shape, where all indicator can be expressed in terms of its DZT. Although we use DZT of g̃, the same results hold with respect to the DZT of g. In the next section, we introduce a method to design the prototype filter g̃. 4.1.5 GFDM Pulse Shaping Filter Design Starting from the DTFT of a preselected basis filter h[n] of practical interests, e.g., RC or root-raised cosine (RRC), which is denoted as H (ν). Here, ν is the normalized frequency and thus the period of H (ν) is equal to 1. Then, we compute g̃[n] = H ( Nn ). With such design, it has been shown in [3] that A becomes singular for even M , K and (g̃) = 0. The requirement a real symmetric filter h[n]. This is caused by ZK,M (K/2,M /2) of odd M or K impedes an efficient implementation in terms of low-complexity radix-2 FFT operations. In [4], we present a design approach that overcomes this restriction for any basis filter h[n] fulfilling the following conditions, 1. h[n] is real-valued, i.e., H (ν) = H ∗ (1 − ν) = H ∗ (−ν). 2. H (ν) spans two subcarriers within each period, namely H (ν) = 0, ∀ν ∈ [ K1 , 21 ]. 3. |H (ν)| is decreasing from 1 to 0 for ν ∈ [0, K1 ]. The idea is to introduce a fractional shift λ ∈ [0, 1] when sampling H (ν), as shown in Fig. 4.5. Accordingly, the samples of g̃ are defined by  n+λ  ⎧ ⎫ 0≤n<M −λ ⎨ H N , ⎬   [g̃]n (λ) = H ∗ N −n−λ , N − M − λ < n ≤ N − 1 . N ⎭ ⎩ 0, otherwise (4.55) Moreover, g̃ can be reshaped as in (4.24) to (g̃) VK,M (λ) (k,m)  m+λ  ⎧ ⎫ k=0 ⎬ ⎨ H N ,   = H ∗ M −m−λ , k = K − 1 . N ⎭ ⎩ 0, elsewhere (4.56) With this design restriction, the same frequency taps can be used with different values of K ≥ 2. Then, we compute the DZT from (4.25), so that zk,m (λ) = H ( m+λ N ) + H∗ ( ) M − m − λ j2π k e K. N (4.57) Due to the symmetry of H (ν), we have k ∗ j2π K , zk,m (1 − λ) = zk,M −1−m (λ)e |zk,m (1 − λ)|2 = |zk,M −1−m (λ)|2 . (4.58) 106 A. Nimr et al. Fig. 4.5 Filter design with sampling. Basis filter is Raised-Cosine with roll-off factor α = 0.5, M = 8. The samples are shown for 0 ≤ n < M − λ (g̃) Fig. 4.6 Filter design with ISI free with and without matched filtering. Here, VK,M (λ) is shown Thereby, all results regarding conditional number, NEF and SIR are symmetric around λ = 0.5. Moreover, Eq. (4.57) shows that zk,m (1 + λ) = zk,m+1 (λ). Therefore, it suffices to study the range 0 ≤ λ ≤ 0.5. We subsequently focus on particular design cases for K = 2x , x > 1 and different values of M . Namely, design with the families of H (ν) that fulfill the inter-symbolinterference (ISI)-free criterion without or with matched filtering, Fig. 4.6. 4.1.5.1 ISI-Free Without Matched Filter In this case, H (ν) additionally satisfies the condition K−1  k=0 ) ( k = 1. H ν− K (4.59) 4 Generalized Frequency Division Multiplexing … 107 From the symmetry and limited band of H (ν), it follows that H (ν) + H ∗ ( 1 −ν K ) + * 1 . = 1, ∀ν ∈ 0, K (4.60)     + H ∗ M −m−λ = 1. From that, there exists a function f (ν) = As a result, H m+λ N N   jφ(ν) ∗ 1 r(ν)e with f (ν) = −f K − ν and H (ν) = + * 1 1 . (1 + f (ν)) , ∀ν ∈ 0, 2 K (4.61) Let us assume a real-valued f (ν), i.e., φ(ν) = 0 and f (ν) = r(ν). A complex-valued f (ν) as in Xia filters is treated in the following section. Due to the constraint of the decreasing amplitude of H (ν), r(ν) must be decreasing from 1 to −1 for ν ∈ [0, K1 ]. From (4.60) and (4.61) we get |zAk,m (λ)|2 = ) ( (1 + fm2 (λ) (1 − fm2 (λ)) k , + cos 2π 2 2 K (4.62)   where fm (λ) = f ( m+λ − 1. The singular values are symmetric with ) = 2H m+λ N N respect to k, and decreasing with k = 0, . . . , K2 . Therefore, |zA0,m (λ)|2 = 1 and |zA K ,m (λ)|2 = fm2 (λ) are the maximum and minimum singular values with respect 2 to k, respectively. Therefore, |zAmax (λ)|2 = 1, because fm2 (λ) ≤ 1, and |zAmin (λ)|2 is 1 , obtained from minm {fm (λ)2 }. Since f (ν) is decreasing and antisymmetric around 2K 1 1 1 2 f (ν) is decreasing ∀ν ∈ [0, 2K ] and increasing ∀ν ∈ [ 2K , K ]. As a result, when M is even and 0 ≤ λ ≤ 0.5, |zAmin |2 is obtained at m = M /2, and when M is odd, it is obtained at m = (M − 1)/2. Consequently, 2 |zAmin | (λ) = f where S(λ) = ) S(λ) 1 . + 2K 2N (4.63) 2λ, M is even . 1 − 2λ, M is odd (4.64) 2 ( From the increasing/decreasing intervals of f 2 (ν), |zAmin |2 (λ) increases with 0 ≤ λ ≤ 0.5 for even M and decreases when M is odd. Hence, the condition number can be expressed as 1 . (4.65) cond(AA )(λ) =     1 + S(λ) f 2K 2N  Similarly, cond(AA )(λ) is decreasing for even M and increasing for odd M . Hence, the best condition of A is attained at λ = 0.5 for even M and λ = 0 for odd M . 108 4.1.5.2 A. Nimr et al. ISI-Free After Matched Filtering A filter H (ν) is ISI-free after matched filtering if K−1  k=0 ( )2   H ν − k  = 1.  K  (4.66) By exploiting the symmetry and band limit, we get  ( + )2 *  ∗ 1  1   |H (ν)| + H , − ν  = 1, ∀ν ∈ 0, K K 2 (4.67)   2  ∗  M −m−λ 2  + H  = 1. Furthermore, there exists a realand hence, H m+λ N 1 N valued function f (ν) = −f K − ν , which is decreasing from 1 to −1 in the interval ν ∈ [0, K1 ] with + * 1 1 . (4.68) |H (ν)|2 = (1 + f (ν)), ∀ν ∈ 0, 2 K Adding an (arbitrary) phase φ(ν) yields the original H (ν) by H (ν) = e jφ(ν) , + * 1 1 . (1 + f (ν)), ∀ν ∈ 0, 2 K (4.69) Using (4.67) and (4.69), H H ∗ ( ( m+λ N M −m−λ N ) ) =e jφa,m (λ) =e jφb,m (λ) , , 1 (1 + fm (λ)), 2 (4.70) 1 (1 − fm (λ)), 2  b      where φma (λ) = φ m+λ , φm (λ) = −φ M −m−λ , and fm (λ) = f m+λ . As special N N N 1  cases, we study the phase in the form φ(ν) = −φ K − ν + β π2 , β = 0, 1, 2, 3. Then, ejφa,m (λ) = jβ ejφb,m (λ) . The case of no ISI with and without (MF), as the Xia filters [5] provide, is obtained with f (ν) = cos(2φ(ν)) and β = 2 or, equivalently, φ(ν) = 21 acos(f (ν). From (4.57), we get σB2k,m (λ) = 1 + -  k − β K4 1 − fm2 (λ) cos 2π K  . (4.71) The maximum singular value with respect to k is located at kmax = β K4 and the minimum one at kmin =< β + 2 >4 K4 . This requires that K is a multiple of 4 for β = 1, 3. 4 Generalized Frequency Division Multiplexing … 109 1 − fm2 (λ), |zBkmin ,m (λ)|2 = 1 − 1 − fm2 (λ). |zBkmax ,m (λ)|2 = 1 + (4.72) Following the same arguments as in the previous subsection and based on the properties of f (ν), both |zBmin (λ)|2 and |zBmax (λ)|2 are obtained at m = M /2 for even M and m = M 2−1 for odd M . Accordingly, . ) S(λ) 1 , + =1+ 1−f 2K 2N . ( ) 1 S(λ) 2 2 + , |zBmin (λ)| = 1 − 1 − f 2 2K 2N (4.73) and the conditional number can then be written as      1 + S(λ) f 2K 2N  cond(AB )(λ) = . ,  1 + S(λ) 1 − 1 − f 2 2K 2N (4.74) |zB2 max (λ)|2 2 ( It can be seen that cond(AB )(λ) is decreasing for even M and increasing for odd M with λ ∈ [0, 0.5]. When using the same function f (ν) in cases A and B, it is notable that that |zBmax (λ)|2 ≥ 1 = |zAmax |2 and |zBmin |2 ≤ fm2max (λ) = |zAmin |2 , and hence, cond(AA )(λ) ≤ cond(AB )(λ), (4.75) proving that the condition number is smaller when using an ISI-free filter, compared to using its square root. 4.1.5.3 Numerical Example In this section, we study the family of prototype filters with roll-off factor α, being obtained with the generator function Fig. 4.7, ⎫ ⎧ 1, 0 ≤ ν ≤ 1−α ⎪ ⎪ ⎪ ⎪ 2K ⎪ ⎪ ⎬ ⎨  2K  1−α 1 1+α a f [ν − ] , < ν ≤ . (4.76) f (ν) = α 2K 2K 2K ⎪ ⎪ ⎪ 1 ⎪ 1+α ⎪ ⎪ <ν≤ −1, ⎭ ⎩ 2K K Here, f a is a real-valued antisymmetric (f a (x) = f a (−x)), and decreasing from 1 to −1 for x ∈ [−1, 1], which produces f (ν) = −f ( K1 − ν). Hence, f (ν) can be used to construct pulse shapes that provide ISI free without matched filter used to generate AA 110 A. Nimr et al. Fig. 4.7 Generator function of filters with roll-off factor α (4.61) or ISI free with matched filter used to generate AB (4.69). From (4.74), (4.65), and (4.76), we get for M α ≤ S(λ), cond(AA ) = cond(AB ) = 1. For S(λ) ≤ M α, 1 , cond(AA )(λ) =    a S(λ)  f αM      a S(λ)  f αM  cond(AB )(λ) = . ,  S(λ) a2 1− 1−f αM (4.77) The condition number is independent of K and, based on the properties of f a , increases with αM . As a particular example, RC and RRC use the function f a (x) = − sin( π2 x). Replacing in (4.77) we get, )) ( ( π S(λ) −1 cond(ARC )(λ) = sin , 2 αM )) ( ( π S(λ) −1 . cond(ARRC )(λ) = tan 4 αM (4.78) Figure 4.8 illustrates the condition number of A for different sampling shift λ and validates the closed-form expressions (4.78) numerically. As shown, λ = 0 is optimal for odd M and λ = 21 for even M when K is also even. In addition, as proven in (4.78), using RC yields a better conditioned A than RRC. Furthermore, numerically obtained values for the NEF as shown in Fig. 4.9 behave similarly as the condition number. This can be explained by the influence of the smaller singular value on the noise enhancement. In both number as well as the smallest  cases, the condition    π S(λ) = sin . Considering the optimum λ, singular value depends on f a S(λ)  αM 2 αM Fig. 4.10 shows the NEF and SIR with different M . The proper choice of λ with 4 Generalized Frequency Division Multiplexing … 111 = 0.5, K = 64 RC-Sim RC-Closed-form RRC-Sim RRC-Closed-form Conditional number 14 12 10 8 M=9 M=9 6 M=8 M=9 4 M=9 M=8 2 0 0.2 0.4 0.6 0.8 1 Sampling offset Fig. 4.8 Conditional number = 0.5, K = 64 8 RC RRC Noise enhancment [dB] 7 6 5 4 3 2 M=9 M=9 1 0 M=9 M=9 M=8 M=8 0.2 0.4 0.6 0.8 1 Sampling offset Fig. 4.9 NEF versus λ respect to M preserves the trend of NEF which increases with M . On the other hand, the SIR is independent of M when M is big enough. In fact, the SIR approaches 01 the interference value that can be directly obtained from SIR = 2 21 |H (ν)|2 d ν, 2K which is independent of λ and K but depends on α. Finally, Fig. 4.11 depicts the dependency of the NEF on K for the case of design with ISI free without matched filter. Although the condition number is independent of the even values of K, the 112 A. Nimr et al. = 0.5, K = 64 NEF NEF/SIR [dB] 0 RC RRC -5 SIR -10 -15 5 10 15 20 25 30 35 40 M Fig. 4.10 NEF and SIR for optimal λ M = 16 8 RC( = 0.9) RC( = 0.6) RC( = 0.3) NEF [dB] 6 4 2 0 5 10 15 20 K Fig. 4.11 NEF versus even values of K NEF is higher for smaller values, especially when α is small, and converges to fixed value when K gets bigger. As a result, K should be bigger than the product of αM . 4 Generalized Frequency Division Multiplexing … 113 4.1.6 Multicarrier Waveforms Generator In this section, we provide a general representation of linear multicarrier modulation techniques inspired from the GFDM model. Then, we show how different stateof-the-art waveforms can be generated from the GFDM block modulator, block multiplexing, windowing, and filtering. This enables the development of a universal waveform generator. 4.1.6.1 General Linear Modulation In general multicarrier modulation, a stream of data symbols can be split into timefrequency substreams {dk,m,i }, with k is the index in the frequency domain denoted as subcarrier, m the index in the time domain as subsymbol, and i stands for the block t [n] with finite index. Each stream is modulated by a transmitter pulse shape gk,m / {0, . . . , Lt − 1}, gk,m [n] = 0. The discrete length Lt , specifically, ∀(k, m) and ∀n ∈ transmitted signal can be written as xt [n] = = ∞    i=0 k∈K on m∈M on ∞  i=0 xit [n t dk,m,i gk,m [n − iLs ] (4.79) − iLs ], where Ls is the block spacing, Kon and Mon are the sets of active subcarriers and subsymbols, respectively, and xit [n] is the i-th multicarrier block, which is given by xit [n] =   k∈K on m∈M on t dk,m,i gk,m [n]. (4.80) Accordingly, xit [n] has a length of Lt samples. The number of available resources per block is denoted as N = MK. The difference between the block length and the block spacing Lo = Lt − Ls determines the overlapping between successive blocks as illustrated in Fig. 4.12. When Lo > 0, the blocks overlap, which means that the last Lo samples of the previous block are added to the first Lo samples of the current block prior to transmission. On the other hand, for Lo ≤ 0, there is a guard interval of Lo zero padding (ZP) samples between successive blocks. Therefore, the multicarrier waveform can be defined by knowing the set of pulse shapes {gk,m }, the set of active resources Kon × Mon , the resource dimensions K, M and the overlapping length Lo . 114 A. Nimr et al. Fig. 4.12 Different cases of block multiplexing 4.1.6.2 Generic Waveforms Generator t [n] can be generated from a prototype In the common modulation techniques, gk,m pulse shape with shift in time and frequency. Moreover, CP and CS can be added afterward. In addition, windowing and subband filtering may be applied. Some waveforms involve more than one prototype pulse shape, which can be seen as superposition of different waveforms. As an example, we list the overall procedure for generating waveforms employing one prototype pulse shape g[n] of length N samples. • Shifting in time and frequency k gk,m [n] = g[< n − mK >N ]ej2π K n , n = 0, . . . , N − 1. (4.81) • CP and CS insertion cp gk,m [n] = gm,k [< n − mK − Lcp >N ], n = 0, . . . , N + Lcp + Lcs − 1. (4.82) • Time domain windowing using a window function w[n] of length N + Lcp + Lcs samples cp,w cp gk,m [n] = w[n] · gm,k [n], n = 0, . . . , N + Lcp + Lcs − 1. (4.83) • Filtering using a filter f [n] with Lf samples cp,w,f cp,w gk,m [n] = f [n] ∗ gk,m [n], n = 0, . . . , N + Lcp + Lcs + Lf − 2. (4.84) t t gk,m [n] results from one or more of that steps. In the most complicated case gk,m [n] = cp,w,f gk,m [n]. Therefore, (4.80) becomes 4 Generalized Frequency Division Multiplexing … 115 Fig. 4.13 Multicarrier waveform generator stages ⎛ xit [n] = f [n] ∗ ⎝w[n] · where   k∈K on m∈M on ⎞ dk,m,i gm,k [< n − mK − Lcp >N ]⎠ (4.85)   = f [n] ∗ w[n] · xi [< n − Lcp >N ] , xi [n] = =   k∈K on m∈M on   k∈K on m∈M on dk,m,i gk,m [n], n = 0, . . . , N − 1 k dk,m,i g[< n − mK >N ]ej2π K n . (4.86) This means that the transmitted block can be obtained starting from a core block of length N samples, which can be generated using the GFDM modulator. Then, the CP and CS can be added to the core block followed by windowing and filtering. Finally, the blocks are multiplexed in the time domain to generate the waveform. This procedure is depicted in Fig. 4.13. 4.1.6.3 Example of the State-of-the-Art Waveforms As discussed, the core block is the essential part of the waveform, and this can be generated with the GFDM modulator under proper setting of the related parameters. OFDM Variants Obviously, orthogonal frequency division multiplexing (OFDM) is a special case of GFDM when M = 1 and g[n] is a rectangular pulse. In the simplest form of OFDM, only a CP is added. Thus Lt = K + Lcp and no overlapping, i.e., Ls = Lt . In the windowed version, a window is applied after adding the CP and CS. Then, Lt = K + Lcp + Lcs , with Lcp > Lcs . In order to reduce the overhead, the blocks are overlapped with Lo = Lcs , so that Ls = K + Lcp , Fig. 4.12. In the filtered variants, either the ones based on subband filtering for multiple users or the others that apply filtering to the whole signal, a filter is applied on the generated CP symbols with the corresponding active subcarriers. Hence, Lt = K + Lcp + Lf − 1, Ls = K + Lcp , and due to the filtering Lo = Lf − 1. 116 A. Nimr et al. DFT-spread OFDM In this modulation, [6] a set of M data symbols are transferred into the frequency domain using FM and then allocated to the subcarrier set of N subcarriers using 1 H F . Thus, the modulation matrix of this waveform is given by N N A= 1 H F UK,M . N N (4.87) with the GFDM modulaComparing with (4.34), this waveform can be   generated tor where the input vector is given by vec DT and using a pulse shape g̃ with (g̃) ZK,M (k,m) = 1, ∀(k, m). This actually corresponds to the Dirichlet pulse given by g[n] = ejπn M −1 N sin(π Kn ) . sin(π Nn ) (4.88) Filtered Multitone (FMT) This waveform does not originally consider subsymbols representation [7]. However, it is easy to transform it to fit in this framework. The prototype pulse gFMT [n] is assumed to have a length of KMo , where Mo is denoted as the overlapping factor. t xFMT [n] = = = ∞ K−1   p=0 k=0 k dk,p gFMT [n − pK]ej2π K n ∞ K−1 d −1  M  i=0 k=0 m=0 ∞  i=0 k dk,m+iMd gFMT [n − (m + iMd )K]ej2π K n (4.89) xi [n − iMd K], where xi [n] = K−1 d −1  M k=0 m=0 k dk,m+iMd gFMT [n − mK]ej2π K n . (4.90) With respect to GFDM notations, we define the number of subsymbols M = Md + Mo − 1, where Md ≥ 1 is the number of active subsymbols, the data symbol dk,m,i = dk,m+iMd and the pulse shape g[n] as g[n] = gFMT [< n + KMo 2 >N ], n = 0, . . . , N − 1. (4.91) Therefore, we get the GFDM block xi [n] = K−1 o /2−1  Md +M  k=0 m=Mo /2 k dk,m,i g[< n − mK >N ]ej2π K n . (4.92) 4 Generalized Frequency Division Multiplexing … 117 Fig. 4.14 FMT pulse shape to GFDM Noting that Mon = {Mo /2, . . . , Md + Mo /2 − 1}, in other words, there are Mo − 1 subsymbols turned off, Fig. 4.14. However, while Ls = Md K and Lt = MK, there is an overlapping of Lo = (Mo − 1)K samples, which compensates the overhead results from the non-active subsymbols. 4.1.6.4 Further Degrees of Freedom Recalling the compact equation   1 H  (g) F ZM ,K ⊙ FM DT FH . K M M   The GFDM core block is generated as x = vec XT and then further processing steps take place. Inspired from that, it is possible to generate three types of other signals; X = V(x) M ,K = 1. By constructing the core block according to the columns, i.e., x1 = vec {X}. This can be seen as a block of stacked K precoded OFDM symbols with M subcarriers. 2. Adding a CP to each column of X to get X(cp) , then x2 = vec X(cp) . This signal can be seen as transmitting precoded OFDM symbols successively.   (cp) (cp) 3. Adding a CP to each column of XT to get X3 , then x3 = vec X3 . This signal can be seen as inserting CPs within the GFDM symbol. (g) Moreover, the matrix  ZM ,K can be populated with unit amplitude and variant phase   (g̃)  = 1, so that the overall modulation matrix is orthogelements, namely,  ZK,M (k,m)    n2 (g̃) onal. For example, with M = 1 and ZK,M = ej2πc N = L(c) (k,k) , we get (k,1) 118 A. Nimr et al. x = L(c) FH K d, (4.93) which represents a chirp-based waveform [8]. In conclusion, the GFDM-inspired waveform generator is a very powerful tool for unified implementation of various standard waveforms, which makes it appropriate for mixed numerology approach. 4.1.7 Channel Estimation for GFDM Detection Consider a multiple-input multiple-output (MIMO) transceiver with Nt transmit and Nr receive antennas where the complex-valued data symbols are spatially multiplexed. Combining the MIMO system with a GFDM-based modulation, we encounter a three-dimensional interference situation due to ISI, ICI as well as inter-antennainterference (IAI). Such an interference-limited scenario challenges the MIMOGFDM receiver design in different stages. A critical functional unit at the receiver side is the channel estimation. Due to broad subcarrier spacing in GFDM, the individual subcarriers become frequency selective and correct detection of the data symbols requires a reliable estimate of the wireless channel transfer function. In the following, we focus on pilot-aided channel estimation techniques, where some reference signals (also referred as pilots), which are known to both transmitter and receiver, are multiplexed with the data symbols within the same time-frequency resource block. Given the knowledge of the pilots at the receiver, the frequency selective channel transfer function can be estimated and utilized for coherent detection of the data symbols. Here, we also assume that the interference autocorrelation matrix of the MIMO-GFDM transceiver is known at the receiver side, and based on such knowledge we derive the two well-known estimation techniques, namely least squares (LS) and linear minimum mean squared error (LMMSE). Later on, we also discuss an alternative approach for the pilot insertion of a MIMO-GFDM system in order to achieve interference-free channel estimation performance. 4.1.7.1 MIMO Wireless Channel We assume an urban scenario with multipath Rayleigh fading MIMO channels. Further, we assume that the individual channels between each antenna pairs are independent and they are block fading, namely the channel impulse response (CIR) does not vary significantly within one GFDM symbol duration and therefore, it can be considered as constant. Hence, we model the CIR between the it -th and ir -th antennas as a linear finite-impulse-response filter given by hit ir [n] = L−1  ℓ=0 hℓ,it ir δ[n − ñℓ ], (4.94) 4 Generalized Frequency Division Multiplexing … 119 where ñℓ is the discrete-time-delay of the ℓ-th path, hℓ,it ir is the complex-valued gain of the ℓ-th path, and it is independent and identically distributed (i.i.d.) zero mean  Gaussian process parameterized by the power-delay profile (PDP) E |hℓ,it ir |2 = Pit ir [ℓ]. Collecting all the L paths in a vector, we have hit ir =   diag pit ir qit ir , (4.95) where, qit ir ∼ C(0, IL ) and pit ir ∈ RL is the vector of normalized PDP. Thanks to the utilization of CP in GFDM, the individual channels between each antenna pair are diagonal in the frequency domain. Thus, the receive (Rx) signal in DFT domain at the Rx antenna ir is characterized by the linear expression ỹir = Nt  (x̃) (D(x̃) p,it + Dd ,it )FN ,L hit ir + w̃ir , (4.96) it =1 where w̃ir is the frequency counterpart of AWGN samples with variance σw2 , FN ,L ⊆ FN is the N -DFT matrix with the L columns associated to the discrete path delays ñℓ . (x̃) Moreover, Ds,i = diag(x̃s,it ), s ∈ {p, d } and x̃s,it = FN xs,it , is the diagonal frequency t domain transmitted signal associated with either pilots or data. For the time domain signal, we have xs,it = Ads,it where dp,it and dd ,it are the N -dimensional vectors of the pilots and data symbols at data symbols at transmit (Tx) antenna it , respectively. In addition, the multiplexing of the pilots and data symbols satisfies dp,it ◦ dd ,it = 0. If the number of pilot subcarriers is smaller than K, i.e., the spacing between the pilot subcarriers Δk > 1, only a subset of the observations in the frequency domain with Np = ⌊N /Δk⌋ samples, that contain the knowledge of pilots, are of interest. Therefore, the observed signal ỹ¯ ir of size Np × 1 at the pilot-bearing frequency bins1 follows Nt  ¯ ¯ x̃) ¯i , ỹ¯ ir = (D(p,i + D(dx̃),it )F′N ,L hit ir + w̃ (4.97) r t it =1 ¯ where D(s,ix̃)t = diag(x̃¯ s,it ), s ∈ {p, d }, x̃¯ s,it = F′N xs,it , F′N ,L ⊆ FN ,L of size Np × N , and F′N ⊆ FN of size Np × N includes the rows of FN ,L and respectively FN in which their inner product with the time domain signal xp,it is nonzero. We rearrange the expression (4.97) into a matrix form as Y = (Xp + Xd )F′Nt H + W, (4.98) herein, each of the above parameters is defined as F′Nt = INt ⊗ F′N ,L , Y = (ỹ¯ 1 , . . . , ¯ ¯ ¯ N ) ∈ CNp ×Nr , Xs = (D(x̃) ¯ 1 , . . . , w̃ , . . . , D(x̃) ) ∈ CNp ×Np Nt ỹ¯ N ) ∈ CNp ×Nr , W = (w̃ r 1 We r s,1 s,Nt refer to each inner product of the rows of the DFT matrix with the time domain signal as a frequency bin. 120 A. Nimr et al. Fig. 4.15 Overview of the matrix structure for a 2 × 2 MIMO channel with s ∈ {p, d }. The matrix H ∈ CLNt ×Nr is the Nt × Nr block-matrix of channel impulse responses hit ir . An example of the matrix structures for a 2 × 2 MIMO channel is depicted in Fig. 4.15. Note that due to the structure of H, its vectorization h = vec {H} consists of Nt Nr independent column vectors of channel impulse responses. Thus, by assuming Rayleigh fading channels with  no spatial correlation, the covariance matrix of all channel impulse responses E hhH becomes diagonal. Resorting to the matrix property vec {ABC} = (CT ⊗ A)vec {B}, the associated vectorization of the observed matrix Y yields the expression y = vec {Y} = X̃p h + X̃d h + w, (4.99) where X̃s = (INr ⊗ Xs F′Nt ) ∈ CNp Nr ×LNt Nr and w = vec {W}. 4.1.7.2 Least Squares Estimation The LS estimator of H minimizes Y − Xp H 2 with respect to H. This yields [9]  −1 ĤLS = QLS Y = (Xp F′Nt )H (Xp F′Nt ) (Xp F′Nt )H Y, (4.100) which subjects to estimation errors due to the enhancement of noise as well as interference. Subsequently, the mean squared error (MSE) of the LS estimation is given by MSELS = E = H − ĤLS 2 (4.101)     σw2  1 trace INr ⊗ (QH trace INr ⊗ (QH Q ) R + LS Ψ Ψ LS LS QLS ) Δk Δk where we calculate the interference covariance matrix as H    RΨ Ψ = E vec Xd F′Nt H vec Xd F′Nt H     = EXd (INr ⊗ Xd F′Nt )E hhH |Xd (INr ⊗ Xd F′Nt )H   = EXd (INr ⊗ Xd F′Nt )Rhh (INr ⊗ Xd F′Nt )H . (4.102) 4 Generalized Frequency Division Multiplexing … 121 In the above expression, Rhh is diagonal because independent Rayleigh fading has been considered for the MIMO channels. Thus, the covariance matrix is given by   Rhh = diag [pT11 , . . . , pTNt 1 , . . . , pT(Nt −1)Nr , pTNt Nr ]T . (4.103) RΨ Ψ = RΨ Ψ (1) ⊕ · · · ⊕ RΨ Ψ (ir ) ⊕ · · · ⊕ RΨ Ψ (Nr ) , (4.104) Further, due to the block-diagonal structure of (INr ⊗ Xd F′Nt ) in (4.102), the resulting interference covariance matrix RΨ Ψ follows the form where ⊕ is the direct sum of matrices. Moreover, we calculate the interference covariance matrix RΨ Ψ (ir ) at Rx antenna ir as RΨ Ψ (ir ) = Nt  RΨ Ψ (it ,ir ) , (4.105) it =1 wherein for each antenna pair it -ir , we have    ′H H  RΨ Ψ (it ,ir ) = EXd ,it Xd ,it F′N ,L Eh hit ir hH it ir FN ,L Xd ,it = Rhhf (it ir ) ◦ RXd Xd ,it , (4.106)   where Rhhf (it ir ) = F′N ,L diag pit ir F′H N ,L is the channel covariance in the frequency domain. Furthermore, the covariance RXd Xd ,it is given by RXd Xd ,it = F′N ARdd ,it AH F′H N , (4.107)   herein, Rdd ,it = E dd ,it dH d ,it is the covariance matrix of the data symbols transmitted on antenna it . Assuming the data symbols are i.i.d. with unit variance, Rdd ,it becomes diagonal with zero entries at the pilot positions. 4.1.7.3 Linear Minimum Mean Squared Error Estimation The LMMSE estimation calculates the coefficients of a linear filter aiming at minimizing the MSE. In accordance with (4.98) and the corresponding vectorization in (4.99), we have H H ĥLMMSE = Rhh X̃p (X̃p Rhh X̃p + RΨ Ψ + σw2 INp Nr )−1 y. 23 4 1 23 4 1 Rhy (4.108) Ryy Here, ĥLMMSE ∈ CLNt Nr is a column vector that contains Nt Nr individual columns of size L associated to the LMMSE estimate of each channel impulse response. The resulting MSE performance of the LMMSE estimation follows 122 with A. Nimr et al.   MSELMMSE = trace RHH − RĤĤ , RHH = (INr ⊗ F′Nt )Rhh (INr ⊗ F′Nt )H , RĤĤ = (INr ⊗ 4.1.7.4 H F′Nt )Rhy R−1 yy R hy (INr ⊗ (4.109) (4.110) F′Nt )H . (4.111) Interference-Free Pilot Insertion In this section, we slightly modify the GFDM modulation at the pilot subcarriers2 in order to insert orthogonal pilots. The low complexity frequency domain processing of the GFDM modulation can be written as in [10]. x = FH N K−1  P(k) Gδ T(δ) FM d, (4.112) k=0 where T(δ) is δ-fold repetition matrix which concatenates δ identity matrices IM of size M , i.e., T(δ) = (IM IM . . .)T . The value of δ is based on the number of nonzero values in the filter frequency response, e.g., if a filter spans over two subcarriers δ is typically selected  as δ =2. In (4.112), due to the circular filtering, the subcarrier (δ) filter G(δ) = diag FδM g(δ) is diagonal  infrequency domain. The circulant filter g is the down-sampled version of g = g[n] n=0,...,N −1 by factor K/δ. The permutation matrix P(k) shifts the DC signals to their corresponding subcarriers (i.e., k) and is given by * +T I 0 0 P(k) = Cn′ M δ/2 M δ/2 M δ/2×(N −δM ) , (4.113) IM δ/2 0M δ/2 0M δ/2×(N −δM ) where n′ = kM − M δ/2. The circulant matrix Cn′ follows Cn′ = circ([0Tn′ T mod N , 1, 0(N −n′ −1)modN ]), (4.114) here, circ(·) returns a circulant matrix associated to its input row vector. Note that in (4.112) the M -point DFT matrix FM can be considered as a special form of precoding. Hence, by slightly modifying such precoder we can reserve some frequency bins specifically for the pilots without any influence from the data symbols [11]. Thus, at the pilot subcarriers k ∈ Kp , we modify the expression (4.112) by replacing FM with CP x K p = FH N 2 Pilot  P(k) G(δ) T(δ) CP dk , (4.115) k∈K p subcarrier is referred to a subcarrier in which Mp pilots are multiplexed with Md data subsymbols while M = Mp + Md . 4 Generalized Frequency Division Multiplexing … 123 Null bin Data Cy clic Cy clic Data Pilot bin Pre fix f Pre fix f Pilot bin Null bin t t (a) Antenna 1 (b) Antenna 2 Fig. 4.16 Pilots and data subsymbols in time-frequency resources 1 0.8 Magnitude Fig. 4.17 DFT domain of the signal for M = 15, K = 4, P′ = IM 0.6 0.4 Pilot on Antenna 1 Pilot on Antenna 2 Pilot subcarrier Data Subcarrier 0.2 0 0 1 2 3 normalized frequency f /M 4 where CP = P′ (λINt ⊕ F(M −Nt ) ). Here, λ is a scaling factor that normalizes the pilots energy to one. P′ can be any permutation matrix of compatible size which allocates the pilots to any frequency bin within the pilot subcarriers. For instance, the choice P′ = IM allocates the pilots on the center frequency bins of the pilot subcarriers. Furthermore, INt in CP ensures that the first Nt subsymbols of the pilot subcarriers (i.e., dk [0], dk [1], . . . , dk [Nt − 1] for k ∈ Kp which are filled with pilots) are processed directly in the frequency domain being orthogonal to the rest of subsymbols (i.e., dk [Nt ], . . . , dk [M − 1] for k ∈ Kp ). Nevertheless, such orthogonality holds if and only if the pilots are located at the frequency bins where no inter-carrier interference is present. Moreover, reserving each orthogonal subsymbol for a specific Tx antenna, the Nt × Nr MIMO channel can be processed in terms of Nt Nr single-input singleoutput (SISO) channels for channel estimation. The approach can be considered as a variation of cell-specific reference signal mapping in Long-Term Evolution (LTE) [12]. Figure 4.16 shows an example how the pilot subsymbols in the GFDM data block are mapped into the time-frequency grid of the resources for a 2 × 2 MIMO channel. Here, two frequency bins of the pilot subcarriers are reserved only for the pilots while at each Tx antenna only one pilot is being transmitted. Thus, the pilot is being transmitted during the whole GFDM symbol, while also the energies of the data subsymbols are no longer concentrated at equally spaced M peaks. Figure 4.17 124 A. Nimr et al. shows an example of the signal filtering in frequency domain, where the pilots at different antennas are orthogonal to one another as well as to the data bins within the pilot subcarrier. 4.1.7.5 Simulation Results In this section, we verify the validity of the closed-form expressions of the channel estimation MSE by simulation and numerical results while we also compare them with the channel estimation performance of OFDM. Later on, we evaluate the performance of MIMO-GFDM with an interference-free pilot design, where we adopt 2 × 2 MIMO block fading multipath channel with Rayleigh distribution. Since the interference-free pilot insertion (IFPI) in the GFDM block might modify the original signal characteristics, we analyze the Tx signal in terms of peak-to-average power ratio (PAPR) and OOB emission via Monte Carlo simulations. Consider a sequence of 16-QAM symbols with energy per symbol Es being transmitted through a multipath MIMO channel with noise energy N0 and with Nt = {2, 3, 4} and Nr = {2, 3, . . . , 8} antennas. A single block of GFDM contains M = 7 subsymbols, and it is filtered by an RC pulse with roll-off factor α = 0.3. For comparison purpose, we configure OFDM to have K ′ = MK subcarriers. Assuming both signals have an identical bandwidth, the subcarrier spacing of GFDM becomes M times broader with respect to the OFDM one and therefore, each GFDM subcarrier consists of M bins while OFDM has a single frequency bin per subcarrier. Figure 4.18 illustrates the MSE evaluations for theoretical analysis as well as simulation results. Here, each channel is chosen to have L = 9 taps with exponential PDP. As expected from the theoretical expressions, the channel estimation for GFDM contains an error floor due to the interference from data symbols while for OFDM, the MSE decreases linearly with the increase of the SNR. Moreover, comparing the 100 OFDM, 2 × 2, LS, Sim. OFDM, 2 × 2, LS, Theo. GFDM, 2 × 2, LS, Sim. GFDM, 2 × 2, LS, Theo. GFDM, 2 × 8, LS, Sim. GFDM, 2 × 8, LS, Theo. GFDM, 4 × 4, LS, Sim. GFDM, 4 × 4, LS, Theo. GFDM, 4 × 8, LS, Sim. GFDM, 4 × 8, LS, Theo. MSE 10−1 10−2 10−3 10−4 0 5 10 15 20 25 30 OFDM, 2 × 2, LMMSE, Sim. OFDM, 2 × 2, LMMSE, Theo. GFDM, 2 × 2, LMMSE, Sim. GFDM, 2 × 2, LMMSE, Theo. GFDM, 2 × 8, LMMSE, Sim. GFDM, 2 × 8, LMMSE, Theo. GFDM, 4 × 4, LMMSE, Sim. GFDM, 4 × 4, LMMSE, Theo. GFDM, 4 × 8, LMMSE, Sim. GFDM, 4 × 8, LMMSE, Theo. 35 Es /N0 (dB) Fig. 4.18 MSE results of channel estimation versus Es /N0 for simulation and theoretical calculations in Rayleigh fading MIMO channel with a pilot spacing of Δk = 2 and K = 128 subcarriers 4 Generalized Frequency Division Multiplexing … 125 10 0 −20 10 −2 −40 PSD [dB] CCDF 0 10 −1 10 −3 OFDM IFPI GFDM Basic GFDM 10 −4 10 −5 4 6 −60 OFDM IFPI GFDM Basic GFDM W-IFPI GFDM W-GFDM −80 −100 8 10 12 PAPR [dB] (a) Signal PAPR for M = 21 , K = 4 vs. its CCDF −120 −40 −20 0 20 40 subcarrier index f /F (b) OOB for K = 96, M = 15 and the total number of frequency samples F. Fig. 4.19 Transmitted signal characteristics GFDM channel estimation results for various number of Tx and Rx antennas, we notice that the error does not directly depend on the number of receive antennas, e.g., the MSE curves for 2 × 2 versus 2 × 8 antennas are overlapped (as well as 4 × 4 vs. 4 × 8). This is due to the fact that, by linearly increasing the number of Rx antennas we increase the number of observations while the number of estimation parameters (i.e., channel taps) also increases linearly, e.g., doubling the number of Rx antennas, we also double the number of channel taps while their ratio remains identical. As a consequence, no analytical difference should be expected in this case. On the other hand, as we increase the number of Tx-Rx antennas, the estimation performance for both LS and LMMSE estimators degrades, because by linearly increasing the number of Tx-Rx antennas, the number of channel taps increases quadratically and thus, the estimation performance degrades. The PAPR of the IFPI GFDM is compared to the original GFDM beside OFDM in Fig. 4.19a. One can see that due to orthogonal pilot insertion, the PAPR of IFPI GFDM increases with respect to the basic GFDM. However, it still has more than one dB difference with the PAPR of an OFDM signal. On the other hand, comparing the power spectral densities of the signals in Fig. 4.19b, we observe that, despite IFPI GFDM has slightly larger OOB compared to the original GFDM signal, the windowed case achieves almost the same OOB radiation as in original windowed GFDM (W-GFDM). The window function is configured in form of an RC window with a ramp length of a quarter subsymbol. For further details regarding the windowing process, we refer the interested readers to [1]. The coded performances of the three receiver types are provided in Fig. 4.20. Here, a GFDM block has M = 7 subsymbols and K = 96 subcarriers. The received signal constellations are detected via GFDM zero forcing demodulation and MMSE frequency domain channel equalization. The channel codes are chosen as parallel concatenated convolutional codes (PCCCs) (1, 15/13), and they provide a gain in spectral efficiency leading to energy per bit Eb /N0 = Es /N0 − 10 log10 (μr) where μ and r denote the modulation order and the code rate, respectively. The detected data 126 A. Nimr et al. IFPI GFDM, Genie-aided OFDM, Genie-aided Basic GFDM, Genie-aided IFPI GFDM, LS OFDM, LS Basic GFDM, LS IFPI GFDM, LMMSE OFDM, LMMSE Basic GFDM, LMMSE 100 BER 10−1 10−2 10−3 10−4 10−5 −2 0 2 4 Eb /N0 [dB] (a) 1/3 code-rate 6 85 10 15 20 25 30 Eb /N0 [dB] (b) 5/6 code-rate Fig. 4.20 Bit error rate performance with 5% pilots overhead over 2 × 2 MIMO channel (M = 7, K = 96) symbols are transferred into maximum likelihood (ML) symbol log-likelihoods and they are inserted into the soft demapper with 8 turbo decoder iterations. In Fig. 4.20, the channel estimation performance for IFPI GFDM and OFDM is identical, although the basic pilot insertion for GFDM has an error floor at high SNR regions as was shown before. Note that in Fig. 4.20, employing a robust code rate of 1/3, OFDM, IFPI GFDM, and basic GFDM receivers obtain almost similar BER, though, basic GFDM with LS estimation has 2–3 dB worse BER performance than the rest of receivers with imperfect channel knowledge. Here, due to around 1 dB gap of the genie-aided receivers of OFDM and IFPI GFDM, the latter receiver stays around 0.5 dB behind OFDM when the channel is estimated through pilot transmission. On the other hand, Fig. 4.20b shows that the basic GFDM channel estimation with non-orthogonal pilots has appreciable performance loss for a high code rate of 5/6 which is due to its large error floor in the channel estimation. Comparing OFDM and IFPI GFDM, we observe that the performance loss in GFDM which is a non-orthogonal waveform is not significant compared to OFDM. Furthermore, in Fig. 4.20b the BER for LS and LMMSE estimations in OFDM as well as IFPI GFDM are identical due to identical channel estimation performances at high SNR regions. In short, if the SNR region is low and a robust code rate is utilized, with normal pilot insertion in GFDM the performance has slight degradation in comparison with OFDM. Although for higher SNR regions when faster code rates are in favor, it is necessary to insert interference-free pilots in order to fully exploit the capacity of the GFDM system while the advantages of its signal characteristics are also preserved. 4 Generalized Frequency Division Multiplexing … 127 4.1.8 Transmission Diversity for GFDM Robustness against time-variant and frequency-selective channel is an important feature for the fifth-generation (5G) networks. Transmit diversity can be exploited to improve coverage, reduce the need for retransmissions, and improve the reliability of the system. Although the space-time coding (STC) as proposed in [13] can be applied to GFDM with the help of wide linear equalization, the complexity of the receiver can hinder its application in some 5G scenarios, such as Internet of Things (IoT) and machine-type communication (MTC). A simple and elegant solution for this issue is based on employing the time-reversal space-time coding (TR-STC) [14] applied to GFDM in order to achieve maximum diversity gain and low implementation complexity without any performance loss. In this case, two antennas are used to transmit two subsequent GFDM blocks xi [n] and xi+1 [n], building the STC codeword as Block i Block i + 1 Antenna 1 Antenna 2 ∗ xi [n] −xi+1 [− < n >N ] . xi+1 [n] xi∗ [− < n >N ] (4.116) After CP removal, the received signals at the j-th receiving antenna, in the frequency domain, for the i-th and (i + 1)-th time instants are given by ỹi,ir = D(irh̃),1 x̃i − Di(rh̃),2 x̃∗i+1 + w̃i,ir ỹi+1,ir = D(irh̃),1 x̃i+1 + D(irh̃),2 x̃∗i + w̃i+1,ir , (4.117) where D(irh̃),it = diag(h̃ir ,it ) with h̃ir ,it = FN hir ,it , x̃i = FN xi and w̃i,ir is the noise vector on the i-th time instant and ir -th receive antenna. Assuming that Nr receiving antennas are employed by the receiver, the received signals can be combined in the frequency domain as Nr −1  (h̃) ∗ Dir ,1 ỹi,ir + D(irh̃),2 ỹ∗i+1,ir x̃ˆ i = D(eqh̃) ir =1 x̃ˆ i+1 = D(eqh̃) −1 Nr  (4.118) D(irh̃),1 ir =1 where D(eqh̃) = Nr 2   ∗ ỹi+1,ir − D(irh̃),2 ỹ∗i,ir , ∗ Di(rh̃),it D(irh̃),it . (4.119) it =1 ir =1 The combined GFDM blocks in the time domain are given by x̂i = 1 Hˆ F x̃i , N N (4.120) 128 A. Nimr et al. which can be used to recover the data symbols, using the demodulation matrix B, as presented in Sect. 4.1. TR-STC can achieve full diversity gain of order 2J, which means that the approximated expression for the symbol error rate derived for the maximum-ratio combiner (MRC) can be adapted for the TR-STC-GFDM. Assuming that a V -QAM constellation is used to map the data bits into each subcarrier and a non-orthogonal prototype pulse that leads to an NEF of ξ = N1 tr(BH B) is employed, the symbol error probability for the TR-STC-GFDM is approximately given by pe ≈ 4μ 2N r −1 (  i=0 2Nr − 1 + i i )( 1+η 2 )i , (4.121) where √ ( ) 1 − η 2J V −1 μ= and √ 2 V 5 6 3σe2 Es 6 V −1 ξ N 0 η=7 , 3σe2 Es 2 + V −1 ξ N0 (4.122) (4.123)  with σe2 = n E[|hn |2 ], ES and N0 denote the average symbol energy and the noise power, respectively. Figure 4.21 shows the TR-STC-GFDM symbol error rate (SER) performance assuming the parameters presented in Table 4.1 and the average channel impulse response based on the Extended Pedestrian A model from LTE, which is described Fig. 4.21 TR-STC-GFDM SER performance under time-variant and frequency-selective channel 4 Generalized Frequency Division Multiplexing … Table 4.1 Simulation parameters Parameter Symbol Mapper Transmit filter # subcarriers # subsymbols CP length [samples] CS length [samples] Detector Noise enhancement factor # receiving antennas V -QAM g[n] K M Lcp Lcs – ξ J 129 GFDM OFDM 16-QAM RC, α = 0.25 128 7 9 3 ZF 1.02 1 16-QAM Rect 128 1 9 3 ZF 1 1 Table 4.2 Channel power-delay profile used in the simulations Tap (nth sample) 0 1 2 3 4 Tap gain hn (dB) 0 −1 −2 −3 −8 5 −17.2 6 −20.8 by Table 4.2. The taps of the channel model are multiplied by i.i.d. complex Gaussian variable with zero mean and unitary variance, resulting in independent Rayleigh fading channels between each transmit and receive antennas. The results presented in Fig. 4.21 assumes two different situations. The first one considers SISO, where just one transmit antenna is employed to send data to the receiver. In the second scenario, both transmit antennas are active, providing full diversity gain. It is also assumed that the receiver knows the state information of all channels. We can see from Fig. 4.21 that the approximation presented in (4.121) can be used to predict the TR-STC-GFDM under time-variant frequency-selective channels. Also, it is possible to conclude that the TR-STC is able to provide full diversity gain to GFDM, introducing a considerable SER performance gain when compared to the SISO-GFDM. Therefore, the high SER performance gain introduced by the TR-STC comes with a very small complexity increment on the receiver side, once the received signals can be easily combined in the frequency domain. 4.2 Link-Level Waveform Comparison In this section, we compare the link-level performance of advanced multicarrier waveforms under MIMO wireless communication channels. The baseline waveform is CPOFDM. In the last decade, it has evolved as a popular multicarrier scheme in different standards, including 3GPP LTE and Wi-Fi families. However, with new and even more stringent requirements in 5G and beyond, OFDM faces its limitations, such as sensitivity to time-frequency misalignments, high OOB emission, limited flexibility, 130 A. Nimr et al. and high PAPR [15, 16]. To overcome these limitations, advanced alternatives have been intensively investigated in recent years. Two groups will be examined in this section. One group of waveforms attempts to improve OFDM while mostly keeping its orthogonality. Filtered OFDM (F-OFDM) [17] linearly filters a set of contiguous subcarriers that form a subband. It is evident that filtering is effective to limit the OOB emission. On the other hand, the presence of inter-block-interference (IBI) is a new issue as the linear filter tail will spread outside the duration of each OFDM block. One pragmatic solution is to insert one or several guard tones (GTs) between adjacent subbands. A widened subband helps reducing the filter length, thereby alleviating IBI. Another approach is to use ZP instead of CP as ZP has zero energy and minimizes IBI. This yields the second waveform in this group, namely universal-filtered OFDM (UF-OFDM) [18]. The other group of waveforms, consisting of filter bank multicarrier (FBMC) [19] and GFDM [1], completely discards the orthogonality requirement of OFDM to achieve better temporal and spectral characteristics. Comparing with OFDM and its variants in the first group, FBMC and GFDM have three core different features that are in common. They are: (1) filtering on a subcarrier basis, (2) permission of more than one data symbol per subcarrier, and (3) being subject to ISI and ICI arising from their non-orthogonality. Between FBMC and GFDM, they both also have several distinct features.3 FBMC adopts linear filtering to achieve ultra-low OOB emission [19, 20]. On the other hand, the long filter length makes it more suitable for continuous rather than burst transmission, considering the usage efficiency of time resources. FBMC does not use a CP and relies on the soft transition of its filter tail to combat multipath fading. In GFDM, a unique feature initially adopted by it is circular filtering. This ensures a block-based waveform with no filter tails, but at the cost of an increased OOB emission. CP has been suggested as a default setting for GFDM, but a single one can protect multiple data symbols for the sake of temporal efficiency. The self-introduced interference of non-orthogonal waveforms often results in much-increased receiver complexity, e.g., [21–23] and references therein. In MIMO communications, IAI will additionally takes place. To tackle such three-dimensional interference, we contributed an innovative way in [24] to perform MMSE equalization such that they can be jointly resolved with complexity in the same polynomial order as that of the (quasi-)orthogonal waveforms in the first group. The focus of this section is to study the link-level performance of the abovementioned waveforms, including OOB emission, PAPR and coded frame error rate (FER) achieved by the MMSE receiver mentioned before. Challenging channel conditions in terms of large delay spread and time-varying fading with imperfect synchronization and channel estimation are under the consideration. 3 Here we consider the features that were suggested when FBMC and GFDM were invented. As the recent progress in waveform engineering, these features start becoming mutually usable. Therefore, they may no longer be regarded as distinct. 4 Generalized Frequency Division Multiplexing … 131 4.2.1 System Configurations Taking OFDM as the baseline, we use K to denote the total number of subcarriers, where the indices of the active ones are recorded in the set Kon . The symbol duration without CP and ZP is denoted as T . At the sampling rate K/T, the CP and ZP respectively contain Lcp and Lzp samples. Here we assume each frame carries a codeword, producing Ns OFDM blocks per frame. For the linear filter adopted by F-OFDM and UF-OFDM, we use Lf to denote its length that is normalized by the sampling period T /K. Since both GFDM and FBMC permit multiple data symbols per subcarrier, let M denote this number and T be the time spacing between two consecutive data symbols. Table 4.3 summarizes the representative configurations of the waveform candidates, deriving from the baseline CP-OFDM. Their configurations attempt to fulfill the constraint that each frame uses the same bandwidth and carries the same number of data symbols.4 Exploiting the additional degree of freedom in the time domain, we particularly examine two configuration types of GFDM and FBMC. The first type sets the time spacing to be the same as the duration of the OFDM symbol, i.e., yielding identical subcarrier spacing 1/T. Each frame consists of a single block to optimize the temporal efficiency. On contrary, the second type makes the subcarrier spacing of GFDM and FBMC M times wider than that of OFDM, i.e., M times shorter time spacing between two consecutive data symbols. In doing so GFDM-II has the same block length as OFDM, thereby yielding the same number of blocks per frame. For the sake of temporal efficiency, FBMC-II still has one block per framework, resulting M times more data symbols than FBMC-I per subcarrier.5 Now, let us set the parameters of the baseline CP-OFDM as: K = 1536, |Kon | = 36, T = 66.67 µs, and Ns = 7. With the subcarrier spacing T −1 = 15 kHz, the occupied subband out of the total 23.04 MHz band has about 0.5 MHz bandwidth. In accordance with Table 4.3, the corresponding configurations for the other waveforms can be readily computed. For the type-II configuration of GFDM and FBMC in Table 4.3, we additionally set M = 12. Furthermore, following the suggestions in the literature, the filters adopted by the waveform candidates except CP-OFDM are set as follows. UF-OFDM adopts the Dolph–Chebyshev filter with length Lf = 74 and the side-lobe attenuation −51 dB [25]. The filter used by F-OFDM is a Hanning windowed sinc-function with length Lf = K/2 + 1 [26]. The PHYDYAS filter of FBMC has the longest filter length equal to 4 K [19]. GFDM adopts a periodic RC function with the roll-off factor α = 1. 4 Due to different choices of the filter, it is difficult to achieve the same frame duration without violating the bandwidth constraint. Considering the strict regulation on the spectrum, identical bandwidth is our primary constraint. 5 For FBMC, the guard time interval to accommodate the filter tail between blocks can be too large. Therefore, in both configuration types, we only consider one block per framework to ensure a good temporal efficiency. 132 Table 4.3 Frame parameterization and modulation complexity Waveform Nr. dat. symbs per subcar. Nr. blks per Nr. subcar. frame per blk. Nr. act. subcar. Subcar. spacing. Sampling rate CP-OFDM − Ns |Kon | 1/T K/T (F/UF)OFDM − Ns K |Kon | 1/T K/T GFDM-I Ns 1 K |Kon | 1/T K/T |Kon | 1/T FBMC-I Ns 1 K K K/T GFDM-II M Ns K/M |Kon |/M M/T K/T FBMC-II Ns M 1 K/M |Kon |/M M/T K/T For UF-OFDM, its Lzp takes on the same value as Lcp O(1) represents the arithmetic complexity per multiplication Frame length (K+Lcp )Ns T K (K+Lcp )Ns +Lf −1 T K Lcp (Ns + K )T (Ns + 27 )T (K+Lcp )Ns T K 7 (Ns + 2M )T Arithmetic complexity per data symb. O O O O O O  K  |K on | K  |K on | K  |K on | K  |K on | K  |K on | K |K on | log K log K + log K + log K + KLf |K on | KM |K on | 4K |K on | log(K/M ) + log(K/M ) + KM |K on | 4K |K on | A. Nimr et al. 4 Generalized Frequency Division Multiplexing … 133 Fig. 4.22 Power spectral densities (PSDs) of the waveform candidates in their baseband signal form (per transmit antenna) 4.2.2 OOB Emission Figure 4.22 depicts the PSD of the waveform candidates in their baseband signal form. Note that the impairment from the RF front-end is not considered here. As expected, the OOB emission of CP-OFDM, namely outside the allocated subband ranging from 5.5 to 6 MHz, is very high, because of the disruptive change from one OFDM block to another in the time domain. UF-OFDM, F-OFDM, and FBMC rely on linear filtering to smoothen the transition between blocks, thereby achieving lower OOB emission. The longer the filter is, the lower is the achieved OOB emission. For GFDM, circular filtering however keeps the disruptive change between blocks. GFDM-I achieves slightly improved OOB emission performance by having longer block duration and reducing the number of blocks per frame. On the other hand, GFDM-II with the same number of blocks per frame as OFDM achieves nearly identical OOB emission performance. Note that the soft transition of its PSD on the shoulder of the occupied spectrum is due to (1) its wider subcarrier spacing than CP-OFDM; and (2) the large roll-off factor. Such soft transition also appears with FBMC-II for the same reasons. Compared to linear filtering, time domain windowing can be an attractive solution to reduce the OOB emission of block-based waveforms as well. GFDM-I uses the minimum time resource among all evaluated waveforms, even 6Lcp shorter than the baseline CP-OFDM. Targeting the same temporal efficiency, we can extend the CP of GFDM-I by 3Lcp samples and add a CS with identical length to window each GFDM block without impairing the data transmission plus one CP to combat the multipath fading channel. Here, the time domain window takes the frequency domain expression of an RC function, whose ramp up and down are contained within the extended part of the CP and CS. Figure 4.22 shows that such RC windowing is very efficient in reducing the OOB emission of GFDM-I, making it competent with 134 A. Nimr et al. Pr(PAPR > PAPR0 ) 100 CP-OFDM DFT-s-OFDM (spreading factor 12) UF-OFDM F-OFDM GFDM-I FBMC-I FBMC-I w./o. ramp up and down GFDM-II FBMC-II w./o. ramp up and down GFDM-II (OQAM) FBMC-II (OQAM) w./o. ramp up and down 10−1 10−2 10−3 10−4 6 7 8 9 10 11 PAPR0 (dB) 12 13 Fig. 4.23 PAPRs of the waveform candidates in their baseband signal form (per transmit antenna), where the CCDF is empirically constructed from 106 frames and the oversampling factor is 4. The default modulation scheme is 16-QAM (gray). The explicit label OQAM represents offset-16-QAM (gray) the outer waveforms using linear filters. Besides filtering and windowing, it is also possible to reduce the OOB emission via preprocessing the transmitted data symbols, e.g., [27–29], but it is beyond the scope of our discussion here. 4.2.3 PAPR Figure 4.23 depicts the PAPRs achieved by the waveforms. For those using linear filter, we note that the ramp up and down of the filtered signal reduce the average power without affecting the peak power. This causes the PAPR increment of UFOFDM, F-OFDM, and FBMC-I compared to CP-OFDM. It is more noticeable when the filter length is longer. Note that such PAPR increment will not impose additional challenge on designing the power amplifier. So, it is not a concern. Since FBMC has the longest linear filter length among all waveforms, we compute its PAPR of FBMC excluding its ramp up and down phase on purpose. For GFDM and FBMC, their additional degree of freedom in the time domain can be leveraged to improve the PAPR performance. By having fewer subcarriers and more data symbols per subcarrier, the configuration type-II achieves much lower PAPRs than the type-I. Figure 4.23 also shows that offset quadrature amplitude modulation (OQAM) is beneficial to further reduce the PAPR of GFDM-II and FBMC-II. If necessary, other PAPR reduction techniques, such as tone reservation and active constellation extension, are applicable on top of these waveforms. 4 Generalized Frequency Division Multiplexing … 135 4.2.4 FER Under a Doubly Dispersive Channel In this following, the waveforms are evaluated in a 4 × 4 spatial multiplexing MIMO system with spatially uncorrelated multipath Rayleigh fading channels. On top of the waveforms, two modulation coding schemes (MCSs) are applied. Namely, the turbo code with the generator polynomial {1, 15/13}o can operate at rate 1/2 and 3/4,6 which are respectively modulated with 16- and 64-QAM (gray). Unless otherwise stated, QAM is the default choice. Its comparison with OQAM is always under the same modulation order. The metric Es /N0 denotes the energy per data symbol to noise ratio. As for the channel model, we choose the extended typical urban (ETU) model specified by 3GPP and with the total power of the path gains normalized to one. Given its large delay spread, we accordingly choose the long CP mode in LTE, specifically, Lcp = 16.67 µs. Following the Jakes’ model, the maximum Doppler frequency fd reflects the channel varying rate. For coherently equalizing each block, the used CIR is obtained by averaging the continuously time-variant CIR over each block duration. Additionally, we assume perfect synchronization. Generally speaking, the time-varying channel can affect the FER performance from two conflicting aspects. First, a continuous time-varying channel introduces ICI that increases along with the maximum Doppler frequency. Second, the time selectivity across the blocks is desirable for the decoder to exploit the code diversity in order to improve the decoding performance. When the additive noise dominates over ICI, the second aspect plays the determinant role. Therefore, lower FERs are attained at a higher maximum Doppler frequency, e.g., CP-OFDM, UF-OFDM, FOFDM, and GFDM-II in Fig. 4.24a, c. However, when the ICI becomes the dominant factor, we observe higher FERs as the maximum Doppler frequency increases, e.g., in Fig. 4.24b, d. For a similar reason, the IBI introduced by linear filtering of UF-OFDM and F-OFDM becomes particularly harmful at the higher MCS, which requires higher operating Es /N0 for a satisfactory FER performance. UF-OFDM benefits from the use of ZP and a shorter filter to suffer from less IBI than F-OFDM. As mentioned at the beginning of this section, we can alleviate the IBI issue encountered by F-OFDM through GT insertion. Figure 4.24b, d depict the performance achieved by having one GT on each side of the subband, which however costs spectral efficiency, i.e., (2/38) ≈ 5% loss. For FBMC and GFDM-I, they suffer from the need of a long block length. With long blocks, the time-variant CIR cannot be well approximated by its average value and the resulting mismatched channel knowledge can severely degrade the performance of equalization and subsequent decoding. Therefore, they perform poorly with fd = 300 Hz. Even at a lower maximum Doppler frequency fd = 70 Hz, such impairment is non-negligible at high SNRs, i.e., Fig. 4.24b. 6 For every six information bits input to the turbo code, we keep all information bits plus two parity bits respectively generated by the two identical component convolutional codes. 136 A. Nimr et al. FER (after 10 turbo iter.) CP-OFDM GFDM-I FBMC-II UF-OFDM GFDM-II FBMC-I (OQAM) 100 100 10−1 10−1 10−2 10−2 F-OFDM FBMC-I FBMC-II (OQAM) F-OFDM with one GT 10−3 5 6 7 8 9 Es /N0 (dB) 10 11 12 10−3 16 FER (after 10 turbo iter.) (a) Code Rate 1/2, 16 QAM, fd = 70 Hz 18 20 22 24 Es /N0 (dB) 26 28 30 (b) Code Rate 3/4, 64 QAM, fd = 70 Hz 100 100 10−1 10−1 10−2 10−2 F-OFDM with one GT 10−3 5 6 7 8 9 Es /N0 (dB) 10 11 (c) Code Rate 1/2, 16 QAM, fd = 300 Hz 12 10−3 16 18 20 22 24 Es /N0 (dB) 26 28 30 (d) Code Rate 3/4, 64 QAM, fd = 300 Hz Fig. 4.24 FERs achieved by the waveforms under perfect synchronization and channel knowledge Only in Fig. 4.24a, the benefit of FBMC becomes appreciable. Non-orthogonality not only introduces interference, but also spreads the information of each data symbol over more than one channel observations in the frequency domain. Particularly, the type-II has larger subcarrier spacing than the type-I to ensure the necessary frequency selectivity among the channel observations. Analogous to FBMC-II, GFDM-II is also equipped with such feature. Between them, FBMC-II in this case has more channel observations per data symbol to achieve a lower FER. Between GFDM-II and FBMC-II, the former permits short block lengths without considerably reducing the temporal efficiency. Therefore, it outperforms the latter in higher mobility case, e.g., Fig. 4.24c, d. Last but not least, as shown in our work [24], QAM can more efficiently exploit the frequency selectivity of the channel than OQAM, therefore outperforming in Fig. 4.24. 4 Generalized Frequency Division Multiplexing … 137 4.2.5 FER with Imperfect Synchronization and Channel Estimation This part investigates the impact of synchronization and channel estimation error on the FER performance of the waveforms. The channel is generated by following another 3GPP channel model termed extended vehicular A model (EVA) with the maximum Doppler frequency equal to 30 Hz and with the sum of the average path gains normalized to one. Due to the reduced maximum delay spread, the CP length accordingly decreases to 4.69 µs, namely the normal mode in LTE. For data-aided channel estimation, we insert a preamble consisting of one baseline CP-OFDM block before each payload frame. It consists of Nt orthogonal pilot vectors that are periodically modulated onto the subcarriers belonging to the occupied subband plus Nt subcarriers on each side for achieving sufficient channel estimation quality also on the edge of the subband. Using such preamble, the receiver performs LMMSE channel estimation by assuming a uniform power-delay profile with maximum delay length equal to Lcp [30]. The obtained channel estimates will be used as the true one for coherently equalizing the whole frame. One important reason behind this setup is to assure that the MMSE equalizer of each waveform works with the same quality of channel knowledge. Figure 4.25a shows that FBMC-II and GFDM-II both are robust against the channel estimation error, achieving up to 5 dB gain in comparison with CP-OFDM and its variants. This observation indicates that it is possible to harness the benefits of the waveform-induced interference instead of only suffering from it. To this end, we CP-OFDM GFDM-II UF-OFDM FBMC-I GFDM-I 10−1 100 FER (after 10 turbo iter.) F-OFDM FBMC-II 10−2 10−1 10−3 −10 −8 −6 −4 −2 100 10−2 0 2 4 6 8 10 STO (sample) 10−1 10−2 10−3 10 12 14 16 18 Es /N0 (dB) 20 22 24 (a) Perf. Sync. and Imperf. Channel Knowledge 10−3 0 10 20 30 40 50 abs. CFO (ppm) at Carrier Freq. 2.6 GHz (b) Sensitivity to STO and CFO at Es /N0 = 18 dB and with Imperf. Channel Knowledge Fig. 4.25 FERs achieved by the waveforms with the maximum Doppler frequency 30 Hz and relying on imperfect synchronization and channel knowledge, where the 16 QAM and code rate 1/2 are the default MCS and the contrastive OQAM has the same modulation order 138 A. Nimr et al. need to exploit the data information conveyed by the interference rather than treating it as a part of the background white noise. We further investigate the sensitivity of the waveforms against symbol time offset (STO) and carrier frequency offset (CFO), respectively. Specifically, the use of guard intervals, no matter in the form of CP, ZP with overlap-and-add or soft termination of the filter, provides protection against negative STO estimates, i.e., the estimated frame arrival being earlier than the true one. The performance degradation appears once we have a positive STO estimate. How severe the degradation is depends on the power of the first non-negligible paths of the channel in its discrete-time model.7 From Fig. 4.25b, we can infer that the initial 4 paths of the channel are insignificant. According to Fig. 4.25c, the waveforms can work under the CFO up to ±20 ppm. For FBMC-II and GFDM-II, even with ±30 ppm CFO, their performances are similar to that of OFDM working with ±20 ppm CFO. 4.2.6 Section Summary In this section, we have analyzed the link-level performance of advanced waveforms that are being intensively researched as alternatives to CP-OFDM for future systems. There is no single waveform that can outperform the others in all examined aspects, i.e., OOB emission, PAPR and FER under different channel conditions. By observing the 3GPP RAN1 discussion through publicly available materials, the waveforms targeted by this section have all appeared in the proposals from different organizations. At this moment, the OFDM-based waveforms are more interested and supported by the main industrial players to ensure a good backward capability. Nevertheless, non-orthogonal waveforms, i.e., GFDM and FBMC, are definitely worth investigation. We believe their benefits can be exploited with a complexity that is affordable by today’s hardware. Furthermore, non-orthogonality is not necessary to be a curse in the system design. Further research on non-orthogonal waveforms is no doubt valuable for a wide range of communications systems using multicarrier waveforms, including but not limited to mobile systems. 4.3 Multiple Access with GFDM In multicarrier-based multiple access, the time and frequency resources are distributed among users. The basic resource element corresponds to transmitting one t [n]. However, in practice, the data symbol dk,m per block using the pulse shape gk,m 7 Given the power-delay profile, the discrete-time channel model is obtained by sampling the lowpass filtered CIR, where the bandwidth equals the sampling rate. The discrete-time model very often have more resolvable paths than the power-delay profile. This is because the delays specified by the power-delay profile are not integer multiples of the sampling period. 4 Generalized Frequency Division Multiplexing … 139 Fig. 4.26 GFDM resource allocation. In this example, M = 5, K = 8, Mu = 4 and Ku = 2 smallest physical resource block (PRB) consists of a number of basic resources and several blocks. For example, in LTE standard, which uses OFDM in the physical layer (PHY), the smallest PRB consists of 12 subcarriers and 7 or 6 OFDM symbols, for long and short CP, respectively. In addition, the scheduling, where the resources can be reassigned to different users, takes place after certain time, for example, in LTE after 12 or 14 symbols [31]. In GFDM, the resource allocation is done by setting the entries of the data matrix D corresponding to the allocated {(k, m)} pairs and the other entries are set to zero. (u) ⊂ Kon × Mon be the set of allocated {(k, m)} to the u-th user. Let Kon(u) × Mon (u) Practically, we fix Mon = Mon and vary Kon(u) . Therefore, we define a PRB that have Mu = |Mon | subsymbols and Ku subcarriers. The scheduling can take place after certain number of GFDM blocks. Figure 4.26 shows an example of GFDM resource allocation. 4.3.1 Signal Model We consider a network that consists of a base station (BS) and U users. In the downlink (DL), as seen in Fig. 4.27, the BS multiplexes the data symbols of all users in a data vector d = vec {D} using the set of indexes Non(u) = {n = k + mK, (k, m) ∈ (u) Kon(u) × Mon }, with d(u) = [d]N (u) , and generates the modulated signal as dison cussed in Sect. 4.1.6.2. Each user receives the multiplexed signal given by y(u) [n] = ej2πfu n h(u) [n] ∗ x(u) [n] + v[n], (4.124) where h(u) [n] is the fading channel and fu represents the CFO between the u-th user and the BS. The STO is implicitly included in h(u) [n]. Under perfect synchronization, i.e., fu is perfectly estimated and the first channel tap is detected. By employing a sufficiently long CP, we get the following signal model after block demultiplexing, 140 A. Nimr et al. Fig. 4.27 DL signal model Fig. 4.28 UL signal model y(u) = H(u) Ad + v, (4.125) where H(u) is the circular channel matrix. This model can be used directly in a simple receiver, where the u-th user demodulates y(u) to get d̂ and then applies resource demapping to extract its allocated data symbols. More advanced receiver can work on the model, y (u) (u) (u) (u) =H A d +H U  (u) v=1,v=u A(v) d(v) + v. (4.126) Here, the second term corresponds to the inter-user-interference (IUI). Actually, the interference in this case is inherited from the self-interference of the matrix A. Thus, the study of the DL performance is similar to that of point-to-point link. In the uplink (UL), Fig. 4.28, each user generates its modulated signal using the truncated modulation matrix A(u) = [A]:,N (u) , as discussed in Sect. 4.1.2.1. The on BS receives a superposition of the signals from all users, which is expressed as y[n] = U  u=1 ej2πfu n h(u) [n] ∗ x(u) [n] + v[n]. (4.127) In the UL scenario, we distinguish two multiple access (MA) schemes • Synchronous MA: the users are strictly synchronized with the BS, i.e., fu = 0 and no time offset, which can be achieved via closed-loop synchronization, then 4 Generalized Frequency Division Multiplexing … y = H(u) A(u) d(u) + U  v=1,v=u 141 H(v) A(v) d(v) + v. (4.128) In this case, the IUI is inherited from the self-interference properties of the GFDM waveform. For example, when the self-interference is limited to adjacent subcarriers, the IUI can be null if a sufficiently large guard subcarrier is used between the adjacent users. • Asynchronous MA: the synchronization is coarse that there is a remaining CFO modeled by fu and remaining STO which leads to the increase of the channel delay spread [32]. Nevertheless, the CP can be extended to take into account the maximum possible STO in additional to the channel excess delay. With that, and assuming the BS is able to perfectly estimate fu and H(u) , we get the signal corresponding to each user after the CFO compensation as  H y(u) = diag φ (u) y = H(u) A(u) d(u) + U  v=1,v=u   diag φ (v,u) H(v) A(v) d(v) + v, (4.129)     where φ (u) (n) = ej2π(fu n+cu ) and φ (v,u) (n) = ej2π(Δfv,u n+cv,u ) are constant phase and Δfv,u = fv − fu is the relative CFO. In this case, the IUI arises from the relative CFO among users. In the next subsections, we study the IUI using the frequency domain processing. 4.3.2 Frequency Domain Processing Consider GFDM with prototype pulse shape that has a maximum discrete frequency response within two subcarrier spacing. Without loss of generality, we let ⎡ g̃T1 ⎤ (g̃) VK,M = ⎣ 0K−2,M ⎦ , g̃T2 (4.130)     where g̃1 = g̃ (0:M −1) ∈ CM ×1 and g̃1 = g̃∗ (N −M :N −1) ∈ CM ×1 . Recalling (4.31), V(x̃) K,M (q,p) = K−1  m=0 (g̃) VK,M (<q−k>K ,p) M −1  m=0 m dk,m e−j2π M p , 142 A. Nimr et al. Fig. 4.29 Frequency domain signal processing model we get V(x̃) K,M V(x̃) K,M (q,:) (q+1,:) = K−1  (g̃) VK,M k=0 (<q−k>K ,:) = g̃1 ⊙ D̃ (q,:) = g̃1 ⊙ D̃ (q+1,:)   ⊙ [D](k,:) FM + g̃2 ⊙ D̃ (<q+1>K ,:) + g̃2 ⊙ D̃ , (<q>K ,:) where D̃ = DFM (4.131) represents the M -DFT-spread data matrix. Thus, due to the M -DFT spreading, any allocated data symbol in the k-th subcarrier produces samples in the k-th and (k + 1)th subband, as demonstrated in Fig. 4.29. For the set of allocated subcarriers Kon(u) , the set of occupied frequency subbands can be defined by K (u) = K1(u) ∪ K2(u) , (4.132) with K1(u) = Kon(u) and K2(u) = < Kon(u) − 1 >K . In addition, let M1 and M2 be the indexes of the nonzero elements of g̃1 and g̃2 , respectively, we define the set of occupied frequency indexes as S (u) = {(k, m) ∈ Ki × Mi , i = 1, 2}. Accordingly, we adapt a masking matrix U(u) of size |K (u) (4.133) | × M defined as  (u)  U (k,m) = 1, (k, m) ∈ S (u) , and 0 elsewhere. (4.134) 4 Generalized Frequency Division Multiplexing … 143 This matrix can  be used  as q frequency domain window at the receiver. Recalling (x̃) T that, x̃ = vec VK,M , this window can be applied to the received vector using the form  (u)T  w(u) . (4.135) F = vec U so that w(u) F (N (u) ) = 1, and 0, elsewhere, (4.136) and N (u) = {n = m + kM : (k, m) ∈ S (u) }. (4.137) Moreover, by taking the N -DFT of (4.129) and applying the frequency domain windowing we get * H +  ỹ(u) = FN diag φ (u) y ⊙ w(u) F ⎡ =D  h̃([) =D  h̃([) u]Ã u]Ã (u) (u) d (u) (u) d (u) + wF + ⊙ ⎣F N U  v=1,v=u U  v=1,v=u  diag φ (v,u)  (v) H A d   (u) Z(v→u) d(v) + diag wF ṽ. ⎤ (v) (v) ⎦ (u) + wF ⊙ ṽ (4.138) where Z̃ (v→u)   1   (v,u)  H h̃(v) (v) j2πcv,u e F diag φ F D = diag w(u) Ã . N N F N (4.139)    h̃(x) = diag h̃(x) . Equation (4.139) defines the signal model after synchroand D nization, so we can compute the SIR assuming uncorrelated data as SIR(u) =     h̃(u) (u) 2  Ã  Pu D  F 2 ,  U  (v→u)   Pv Z̃  v=1,v=u (4.140) F   where · F is the Frobenius norm, and Px = E |d (x) |2 is the x-th user power per symbol. The SIR depends on the modulation matrix, resource and power allocation, CFO and the CIR. The SIR metric is useful to evaluate the IUI; however, it is also important to examine the overall performance after channel equalization and demodulation. Let (u) B̃ be the receiver matrix involving channel equalization and demodulation, then 144 A. Nimr et al. d̂ (u) (u)H (u) = B̃ (u)H = B̃ ỹ D  h̃(u) (u) (u)H Ã d(u) + B̃ U  (v→u) (v) Z̃ d v=1,v=u   (u) ṽ. + B̃ diag w(u) F (4.141) The average signal-to-interference-plus-noise ratio (SINR) is considered as a performance metric. It is given by (u) SINRB = 2     (u)H h̃(u) (u) D Ã − I Pu   + B̃ F  Actually, SINR(u) B symbols. −1 U     (u)  Pu Non  v=1,v =u .       (u)  (u) (v→u) 2 (u) 2 Pv B̃ Z̃  + N0 B̃ diag wF  F F (4.142) is the normalized mean squared error (NMSE) of the data 4.3.3 Asynchronous MA Evaluation For the purpose of comparing the asynchronous MA with different waveform parameters, we consider two users, i.e., U = 2, with identical power allocation P1 = P2 and evaluate the SIR (4.140) and the SINR (4.142) in different scenarios. 4.3.3.1 AWGN Channel This evaluation is useful to comprehend the interference due to the CFO. Let C(1,2) = then SIR(u) =   1 FN diag φ (1,2) FH N, N    g̃k,m 2  2 k∈K on(1) m∈M on  2 .       (1) (1,2) diag w C g̃  k,m  F (4.144) 2 k∈K on(2) m∈M on Noting that (4.143)   (1,2) g̃k,m (q) = G k,m (ν = C q N − Δf1,2 ), (4.145) where G k,m (ν) is the DTFT of gk,m [n], then  2        (1) G k,m ( q − Δf1,2 )2 . diag wF C(1,2) g̃k,m  = N 2 q∈N (1) (4.146) 4 Generalized Frequency Division Multiplexing … Therefore, 145  (2→1) 2    S2 ( Nq − Δf1,2 ), Z̃  =N F q∈N (4.147) (1) where Su (ν) is the PSD of the signal of the u-th user without CP expressed as Sx (ν) = 2 1    G k,m (ν) . N (x) k∈K on As a result, we can write the SIR in the form  SIR(1) = q∈N  q∈N (1) (4.148) m∈M on (1) S1 ( Nq ) S2 ( Nq − Δf1,2 ) . (4.149) This form is intuitively comprehensive as N (1) represents the frequency band of the first user. Additionally, the interference of the second user is the integral over the leakage within the first user’s band, as illustrated in Fig. 4.30. From this, it can be shown how important to study the OOB of the waveform, which in the case of GFDM depends on the prototype filter. This equation can be simply extended to U > 2. Figure 4.31 justifies the closed-form solution with the numerical simulation. In addition, it can be seen that for the same allocated bandwidth and the same guard band, GFDM is significantly less sensitive to CFO compared to OFDM. It is important to highlight that the first subsymbol is turned off. Additionally, the worst situation for the used prototype filter happens when the CFO is half the frequency sample, which is exactly half the subcarrier spacing in OFDM. On the other hand, a shift by complete sample retains the orthogonality if a sufficient guard band is used. Fig. 4.30 IUI due to CFO using. In the synchronized case no interference as the sampling points are at zero crossing. With CFO, the samples are not equal to zeros 146 A. Nimr et al. Fig. 4.31 SIR(1) simulation versus closed-form for GFDM and OFDM 4.3.3.2 Fading Channel In the existence of fading channel, we define the modified pulse shape as 1 (u) FN (h̃ ⊙ g̃m,k ). N gk,m|h(u) = (4.150) Following the same discussion, the PSD of the u-th user can be written as Su|h(u) (ν) = 2 1    G k,m|h(u) (ν) , N (u) k∈K on and then SIR(1) =  q∈N  q∈N (1) (4.151) m∈M on (1) S1|h(1) ( Nq ) S2|h(2) ( Nq − Δf1,2 ) . (4.152) By averaging over different channel realization, we achieve the SIR for certain PDP. Figure 4.32 shows the effect of the fading channel considering the TGn channel model in comparison with AWGN channel for different roll-off factors of RC filter. As expected, while larger roll-off factor produces well-localized filter and reduces the side lobes, the SIR is higher for larger roll-off. However, this gain may be reduced after the receive filter. For example, if ZF receiver is used, then the gain will be reduced by the NEF, which is higher for larger roll-off. On the other hand, with fading channel the average SIR increases, that is because the channel gain may attenuate the interfering signal. This can be beneficial if a proper equalization such as MMSE is used. 4 Generalized Frequency Division Multiplexing … 147 Fig. 4.32 SIR(1) for different roll-off in AWGN and TGn fading channel. K = 64, M = 16, Mon = {1, . . . , M − 1} 4.3.4 Mixed-Numerology with GFDM The supported services in the 5G are diverse, and each service has different requirements [33]. For example, to support the low latency of tactile Internet, shorter symbol duration needs to be used. To tackle the effect of Doppler shift in high speed vehicular communications, larger subcarrier spacing is required. The massive machine-type communications (mMTCs) employ fewer subcarriers to ensure lower PAPR. In order to multiplex all these services at one BS, the need of mixed-numerology arises. As discussed in Sect. 4.1.6, GFDM can be seen as a universal multicarrier waveform generator, with several reconfigurable parameters that can be altered on the fly to generate the corresponding signal. This feature promotes the GFDM framework to be a candidate for mixed-numerology. The main issue arises here is the inter-numerology-interference (INI), which can be seen as IUI if we consider different users per service. In all cases, we need a model to evaluate the interference in order to optimize the design and the allocation of the services in the time and frequency domains. 4.3.4.1 General Inter-user-Interference Model Without loss of generality, we consider two users u = 1, 2 transmitting their signals with GFDM modulation using Ku subcarriers, Mu subsymbols and a pulse shape (u) g (u) [n] with subcarrier allocation indexes Kon,u , whose entries are subset of Kon,u (u) and Mon,u = Mon,u . Note that, the sub-index indicates the numerology index, while the sup-index represents the user index in that numerology. Let y[n] be the discrete received signal (4.153) r[n] = r (1) [n] + r (2) [n] + v[n], 148 A. Nimr et al. where r (u) [n] is the signal contribution from the u-th user and v[n] is the additive noise. In order to decode the u-th signal, the receiver performs its operation based on r (u) [n] configurations, while the signal of the other user is the interference. After removing the CP, we get the block of the u-th user as zu,i [n] = r[n + Lcp,u + iLs,u ], n = 0, . . . , Nu − 1, (4.154) where Lcp,u and Ls,u are the CP size and block spacing used by the u-th user, respectively. Afterward, Nu -DFT is computed, where = Ku Mu . Therefore, we get (2→1) + ṽ1,i ∈ CN1 ×1 , z̃1,i = z̃(1) i + z̃i (4.155) z̃2,i = (4.156) z̃(2) i + z̃(1→2) i N2 ×1 + ṽ2,i ∈ C . Here, z̃i(u1 ) is the i-th block of the u-th user containing all samples, z̃i(u1 →u2 ) is the interfering block from u1 to u2 and ṽu1 ,i the additive noise. First, we consider the case of AWGN. Using the multicarrier representation, we have 1) z̃(u = i  (u ) k∈K on,u11  (u1 ) (u1 ) g̃k,m dk,m,i , (4.157) m∈M on,u1 (u1 ) where g̃k,m is the Nu1 -DFT of the pulse shapes of the u1 -th user as originally generated by the waveform. Moreover, the interfering signal can be expressed as z̃i(u2 →u1 ) =    (u2 →u1 ) (u2 ) g̃k,m,j dk,m,I j (i) . (4.158) j∈J i k∈K (u2 ) m∈M on,u2 on,u2 (u2 →u1 ) In this notation, g̃k,m,j is the Nu1 -DFT of the pulse shapes seen by the receiver of u1 . This pulse shape can be computed from the original pulse shape g (u2 ) and the multiplexing parameters, as seen in Fig. 4.33. The set Ji represents a set of indexes depending on the index i and Ij (i) is the block index of u2 that contributes to the interference. Note that Ij1 (i1 ) = Ij2 (i2 ), i1 = i2 and j1 = j2 . This is the key difference in mixed-numerology. If the same numerology is used, then Ji disappears and Ij (i) = i. But in mixed-numerologies, the interference may depend on the block index. However, by a proper design we can control the pattern, such that JPi+p = Jp , in order to have P interference patterns applied to different block indexes, such that    (u2 →u1 ) (u2 ) (u2 →u1 ) g̃k,m,j dk,m,I j (i) . (4.159) = z̃Pi+p j∈J p k∈K (u2 ) m∈M on,u2 on,u2 Following the same approach in (4.149), we get the SIR for each pattern 4 Generalized Frequency Division Multiplexing … 149 Fig. 4.33 INI example. N2 = 2N1 , Lcp,2 = 2Lcp,1 . The i-th block of u = 2 interferes with the 2i-th and (2i + 1)-th blocks of u = 1 in two different patterns. Each pattern has one pulse shape. On the other side, two blocks of u = 1 interfere with one block of u = 2. In the latter case, there is only one pattern but with two pulse shapes SIRp(u1 ) =  (u ) q∈N u1 1  (u ) q∈N u1 1 S1 ( Nq ) Sp,u2 →u1 ( Nq ) . (4.160) Here, Nu1(u1 ) is the set of nonzero frequency samples of u1 in its numerology, and Sp,u2 →u1 (ν) = 1  Nu1    (u →u ) 2 2 1 (ν) . G k,m,j (4.161) j∈J p k∈K (u2 ) m∈M on,u2 on,u2 The average SIR can be computed by averaging the SIR overall patterns. This can be extended in a similar way to the case of fading channel as in (4.152). 150 A. Nimr et al. Fig. 4.34 INI for OFDM with K1 = 32, Lcp,1 = 8, K2 = 64, Lcp,2 = 8, Δf = Δf1 = 2Δf2 . The users are allocated the same bandwidth, so that, Ku,1 = 1, Ku,2 = 2 4.3.4.2 Numerical Examples To verify the closed-form solution in (4.160), consider OFDM systems with different subcarrier spacing. The first system uses K1 subcarriers and the second system uses K2 = 2K1 , in other word, the second system employs half the subcarrier spacing. The CP length in both cases is 1/4 of the symbol length, and both systems transmit data with unit power. Under perfect synchronization, we get the frame structure presented in Fig. 4.33. As illustrated in Fig. 4.34, we compare the SIR of one allocated subcarrier of the first system with two subcarriers from the other system, so we get the same bandwidth. The guard band is an integer number of subcarriers which is normalized to the largest subcarrier spacing Δf . As expected, the interference from user 2 to user 1 decreases with larger guard band. In addition, the guard band can be controlled by the smaller subcarrier spacing of system 2, where Δf2 = Δf /2. Nevertheless, it can be observed that the behavior of the interference from user 1 to the user 2 fluctuates and is influenced by the guard interval. Next, we compare GFDM with different settings. In this example, the target is to have short and long blocks as in Fig. 4.33. This can be attained by changing either the subcarrier spacing or the subsymbol spacing. The configurations and results are shown in Fig. 4.35, where the pair (U1, U2) corresponds to changing the subcarrier spacing, while the pair (U3, U2) realizes the different block lengths via different subsymbol spacing. It can be seen that with the used RC pulse shape, the interference in both directions follows the same behavior. Interestingly, it can be shown also that both settings achieve similar performance with slight gain when different subsymbol spacing is applied. Another advantage is that keeping smaller subcarrier spacing allows fine control of the guard band. Finally, GFDM outperforms OFDM by a gain up to 10 dB in the achieved SNR for the same subcarrier spacing. 4 Generalized Frequency Division Multiplexing … 151 Fig. 4.35 INI for GFDM with different settings for user 1 and user 2 4.4 GFDM Implementation The path to meeting all requirements of 5G is a challenging endeavor. The demand to achieve higher data rates for the enhanced media broadband (eMBB) scenario and novel use cases like ultra-reliable and low-latency communication (URLLC) and mMTC drive researchers and engineers to consider new concepts and technologies for future wireless communication systems. The goal is to identify promising candidate technologies among a vast number of new ideas and decide which are suitable to be implemented in future products. Figure 4.36 gives a rough overview of the development process. New ideas and concepts typically first undergo extensive software simulations, which allow to make early predictions on the expected performance. After selection of the best candidates, individual aspects of the envisioned system can be implemented on a hardware-accelerated platform, e.g., software-defined radio (SDR), in order to learn about real-time behavior and over-the-air performance with real radio frequency (RF) components. Technologies that prove promising at this stage can be further evaluated in test beds, where the focus shifts toward the interaction of different technology building blocks and the realization of complete end-to-end applications. New concepts and technologies that have been proven to exhibit improved performance in practically relevant environments will ultimately find their way into new standards and lastly, industry will adopt them in future products. 152 A. Nimr et al. New ideas and concepts Simulations Predict expected performance based on suitable models and reasonable assumptions. Prototypes Find reasonable simplifactions, algorithm partitioning and estimate implementation complexity. Testbeds Standards Demonstrate the interworking of different components under real-world conditions and in real-time. Decide which technologies should be included in future products. Products Fig. 4.36 From theory to practice This section presents a field-programmable gate array (FPGA)-based, real-time implementation of a modulator/demodulator for multicarrier waveforms [34]. This proof-of-concept design provides a large number of degrees of freedom to the user and offers the flexibility for practical evaluation of new algorithms that aim to address various 5G aspects. Example applications include experiments with flexible numerology, which is a key differentiator of 5G new radio (NR) compared to fourth-generation (4G) LTE, as well as the design and implementation of a corresponding scheduler that utilizes the additional flexibility. The presented platform is a building block for test beds that will assist the design of 5G radio interface and network architecture [35, 36]. 4.4.1 Modem Implementation For the sake of simplicity, consider a communication system that consists of two nodes. Each node is realized by a control PC that is connected to a software-defined radio platform. This hardware setup is depicted in Fig. 4.37. Note that the design that is described in this section is tailored for the USRP-RIO hardware [37]. However, the basic principles of the implementation are valid for other platforms in general. The block diagram in Fig. 4.38 shows how the components of the overall system are mapped to the hardware platform. 4 Generalized Frequency Division Multiplexing … 153 Fig. 4.37 Hardware setup Fig. 4.38 Block diagram of the transceiver In order to be able to support the various requirements of 5G wireless systems, the individual blocks need to be implemented with flexibility in mind. This section will focus on the PHY aspects of a 5G NR transceiver. The corresponding signal processing algorithms have to be implemented on FPGA, in order to be able to meet throughput and latency requirements. As medium access layer and higher layers have more relaxed requirements w.r.t. timing, when operating in sub-6 GHz bands, those components rarely require specialized hardware acceleration. Hence, the assumption is made that they are implemented using software running on standard PC hardware. 4.4.1.1 Baseband Modem The baseband signal processing is performed by the resource mapper and modulator blocks on the transmit path, and demodulator and resource demapper on the receive path, respectively. Note that the block encoder, QAM mapper, decoder, and QAM demapper are standard implementations taken from a 4G LTE library [38], and hence will not be discussed here. As seen in Fig. 4.39, the resource mapper takes complex symbols from various input sources, e.g., payload data, control channel data, and reference signals, and 154 A. Nimr et al. Fig. 4.39 Block diagram of the resource mapper and the modulator maps them to a two-dimensional time-frequency resource grid. The mapping pattern, i.e., the resource map, is fully programmable. This mechanism allows to support any user-defined resource grid that can be adapted during the run-time. The resource demapper performs the inverse operation. The modulation operation can be separated into two main functional blocks. The first block is an inverse discrete Fourier transform (IDFT) that transforms the data from frequency domain into time domain. The second block, which will be called core modem, applies a pulse shaping filter to each subcarrier of the transmit signal. The demodulation operation consists of the same processing blocks in reversed order, where the only differences are the direction of the Fourier transformation and the filter coefficients that are used in the core modem. The first task of the core modem is to split the incoming IDFT output stream δm into M subsymbols with K samples in each, which are stored in block RAM (BRAM) 0, . . . , M − 1. Each individual subsymbol has to be repeated M times, such that the filter gm with N samples can be applied. Therefore, the K samples of each subsymbol are stored inside an independent subsymbol memory bank. All M memories are read sample-by-sample in parallel. The filter gm is also stored in M parallel subsymbol filter banks. Each of the filter memory banks contains K coefficients that represent a different part of the pulse shaping filter in the time domain. The filter needs to be applied to the data in a circularly shifted way. This is implemented with a circular pattern that dynamically selects which filter BRAM is connected to which subsymbol BRAM. The last step in the core modem is to accumulate the contributions from all M parallel branches to get the transmit signal. 4.4.1.2 Post Modem Processing After the signal is modulated, CP, CS, time-windowing, and preamble are added. Figure 4.40 depicts the general frame format. The CP and CS can be applied on both preamble and data block. NP,CP defines the length of the CP for the preamble, NCP for the data block, NP,CS defines the length of the CS for the preamble, and NCS for 4 Generalized Frequency Division Multiplexing … NP,CP NP NP,CS NP,W 155 NCP N NCS NP,W NW CP Preamble CS NW CP GFDM block CS Fig. 4.40 Supported frame structure with one preamble and one data block Fig. 4.41 CP, CS, windowing and preamble insertion the data block. It is assumed that the window is symmetric, thus, the length of one half is given by NP,W and NW . In addition, NP denotes the length of the preamble. The preamble is calculated in advanced on the host computer and written to the BRAM memory during the configuration phase of the transceiver. The CP and CS of the modulated GFDM data block are added via first in, first out (FIFO) memory as depicted in Fig. 4.41, which can be seen as a variable delay to shift the data samples into the correct output order. The windowing unit follows, where only the rising half is stored inside a memory. An integrated counter in the control logic counts up until NW is reached to trigger the memory for the appropriate samples. During the main data block, the unit is disabled. Finally, the same counter is decreased to create the falling flank. Whenever the controller has finished reading in the first data block of a frame into the data-FIFO, the preamble insertion unit is triggered to push the preamble samples to the digital-to-analog (D/A) converter. 4.4.2 Complete Transceiver Chain and Extension for MIMO A complete transceiver was implemented in FPGA as a proof of concept with its block diagram shown in Fig. 4.42. It includes all required processing functions for a real-world wireless communication system. 156 A. Nimr et al. PA Rate Adaption Polar Encoder CP/CS & Windowing TR-STC Encoder Frame Multiplex Digital PreDistortion TX Front-End PA Synch. Preamble Transmit Message Resource Mapper GFDM Modulator Transmitter Receiver RX Front-End Imparments Correction RX Front-End Channel Estimation Preamble Automatic Gain Control Synchronization Channel Estimation STC Combiner Polar Decoder Stuffing Removal Synch. Preamble Channel Estimation Preamble GFDM Demodulator Resource De-mapper Received Message Fig. 4.42 Transceiver block diagram A flexible frame structure was designed to cope with many design aspects such as synchronization, channel estimation, codeword length, resource allocation, MCS, channel multipath protection, OOB emission, MIMO operation, etc. Figure 4.43 depicts the adopted frame structure. The number of GFDM symbols within a frame, NG , is chosen such that there is an integer number of codewords carried by an integer number of GFDM symbols. Therefore, the receiver can synchronously start decoding a codeword at the beginning of the frame. Synchronization and channel estimation were implemented using a preamble-based scheme, i.e., the transmitter inserts waveforms known by the receiver, multiplexed with the GFDM symbols. The periodicity of the synchronization preambles is proportional to frequency precision and stability difference between transmitter and receiver time base, likewise, the channel estimation preambles periodicity is inversely proportional to the mobile velocity. The BSs operate in continuous transmission mode, where the carrier is always on the air independent if useful data is available for transmission. Considering the modulator is a wireless pipeline, it requires a constant input data rate. As the useful transmit message rate varies over time, a rate adaption scheme is required. The simplest solution is to fill dummy data in order to maintain a constant rate, and remove this stuffing data at the receiver. Polar code was derived from the channel polarization theory and introduced in [39]. The code presents explicit construction, hardware efficient coding and decoding algorithms, and high flexibility, namely the code rate can continuously vary from N1C to NNC −1 , where NC is the codeword length. It makes it a strong candidate for the future C wireless networks, such as 5G. The implemented encoding algorithm is systematic [40] with semi-parallel architecture [41] for improved throughput. The implemented decoding algorithm is based on successive cancellation in the logarithm domain [42]. 4 Generalized Frequency Division Multiplexing … (a) 157 (b) Fig. 4.43 Frame structure for a data encapsulation and; b waveforms multiplexing R = 1/2 FP R = 3/4 FP R = 7/8 FP R = 1/2 HW R = 3/4 HW R = 7/8 HW 10−1 Bit error rate 10−2 10−3 10−4 10−5 10−6 10−7 1 2 3 4 5 6 Eb /N0 [dB] Fig. 4.44 Polar code bit error rate for different code rates in floating-point simulation (FP) and hardware fixed-point implementation (HW) A comparison between a floating-point simulation and fixed-point implementation for the polar code is presented in Fig. 4.44. The signal to be transmitted needs to be amplified by a power amplifier (PA). The majority of today’s BSs employ the high electrical efficient Doherty topology [43], which is intrinsically nonlinear due to its class-C branch responsible for amplifying signal peaks. The nonlinearity distorts the amplified signal generating spectral regrowth, also known as intermodulation, presented at both in-band and OOB frequencies. The in-band intermodulation affects the signal quality, and, therefore, the receiving threshold. The OOB intermodulation results in OOB emissions, interfering in adjacent channels, and losing the benefit of using a waveform with 158 A. Nimr et al. low-OOB emissions such as GFDM. Linearization using digital predistortion (DPD) has been widely employed to mitigate the intermodulation [44]. The DPD system design procedure is subdivided into three distinct tasks: (1) to choose a behavioral model equation which is able to represent the PA characteristics, e.g., nonlinearity and memory effects, with the minimum number of coefficients; (2) to design a real-time DPD block which is able to generate the distortion according to the model equation and its coefficients; and (3) to select an algorithm which identifies the optimum values for the model coefficients in order to compensate for the PA distortion. Our chosen behavioral model equation is based on the memoryless orthogonal baseband polynomial for Gaussian distributed signals [45] and modified to include the memory as y(n)= M KD  k D −1 m=0 k=1 l=1 √ ( ) hm,k k k − 1 |x[n − m]|2(l−1) x[n − m], l!(−1)l−k l − 1 (4.162) where x[n] and y[n] are the input and output model signals, respectively, KD and MD are the polynomial order and memory length, respectively, and {hm,k } are the DPD model coefficients at the m-th tap and k-th order. The DPD performance was tested with a gallium nitride device operating at 3 W average, amplifying a 20 MHz wide GFDM signal centered at 723 MHz. The PA linearization results are shown in Fig. 4.45. Table 4.4 shows the DPD performance in terms of adjacent channel leakage rejection (ACLR). There are impairments on the received signal caused by imperfections in the receiver front-end analog components. It causes interference and need to be digitally compensated in the real and imaginary parts of the received signal in three steps: (1) remove the average value, μ; (2) remove the average correlation between parts, β; Fig. 4.45 DPD intermodulation reduction performance measured with a spectrum analyzer −30 Power [dBm] −40 Without DPD With DPD −50 −60 −70 −80 −90 673 693 713 733 Frequency [MHz] 753 773 4 Generalized Frequency Division Multiplexing … Table 4.4 DPD performance in terms of ACLR Lower ACLR (dB) Without DPD With DPD Improvement −36.6 −53.2 16.6 159 Upper ACLR (dB) −37.9 −53.1 15.2 Fig. 4.46 Receiver impairments correction and (3) equalize the power (or variance σ 2 ) difference between parts. This process is shown in Fig. 4.46. At the receiver side, the signal peak-to-peak voltage needs to comply with the analog-to-digital converter (ADC) input range. If the voltage swing is too low, the SNR is compromised by the quantization error. If the voltage swing is too high, clipping effects will also affect the SNR. Assuming a Gaussian distribution and an ADC with a given number of bits, it is possible to find the optimum signal amplitude which maximizes the SNR [46]. The automatic gain control (AGC) goal is to keep the voltage level at this optimum point as shown in Fig. 4.47. The existing preamble-based receiver synchronization techniques for OFDM are applicable for GFDM. For instance, a preamble with two repeated halves may be used, where the autocorrelation ρ[n] = n+N P −1  k=n r[k]∗ r[k + NP /2], (4.163) can be calculated between the halves, where NP is the synchronization preamble length. Normalizing the autocorrelation by the signal energy leads to 2 |ρ[n]|2 μS [n] = n+NP −1 . |r[k]|2 k=n (4.164) 160 A. Nimr et al. Fig. 4.47 SNR due to quantization and clipping, and optimal operating point 120 Quantization SNR (14 bits) Clipping SNR Total SNR SNR (dB) 100 80 (11.8 dB → 74 dB) 60 40 8 9 10 11 12 13 14 Clipping limit (above average power) [dB] The presence of CP and CS produces the plateau effect which can be mitigated by integrating (4.164) and resulting in n  1 μM [n] = μS [n], L+1 (4.165) k=n−L where L = NCP + NCS . The STO can also be estimated through the cross-correlation, which is given by ρC [n] = NP −1 1  r ′ [n + k]px∗ [k], NP (4.166) k=0 where px [n] is the known preamble waveform. Finally, (4.165) and (4.166) can combined for an enhanced performance given by μA [n] = |ρC [n]| · μM [n], (4.167) where the multiplication suppresses the cross-correlation side peaks, which appear due to the repeated halves. All discussed synchronization metrics are depicted in Fig. 4.48. The STO is estimated as the sample time index where the metric peak is. The preamble-based channel estimation scheme is accomplished in the frequency domain. It is straightforward since the channel estimation preamble (CEP) in the frequency domain is known to the receiver. Considering the CEP length is usually shorter than the GFDM symbol, the estimated channel needs to be interpolated. However, when some subcarriers are muted, the IFFT/FFT interpolation method 4 Generalized Frequency Division Multiplexing … 161 μ S [n] μ M [n] | ρ C [n] | μ A [n]= | ρ C [n] | · μ M [n] 1 Magnitude 0.8 0.6 0.4 0.2 0 50 100 200 150 250 300 Samples [n] Fig. 4.48 Synchronization metrics FFT × IFFT Zero Padding Windowing N-DFT Smoothing 1 FFT{CEP} Estimation Interpolation Fig. 4.49 Preamble-based channel estimation block diagram fails because all the frequency components are required to calculate the interpolated response. In order to solve this problem, a time-windowing is done on the channel impulse response being interpolated. 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(2014), https://doi.org/10.1109/COMST.2014.2383691 32. M. Matthé et al., Asynchronous multi-user uplink transmission with generalized frequency division multiplexing, in Communication Workshop (ICCW) (2015), https://doi.org/10.1109/ ICCW.2015.7247519 33. A.A. Zaidi, Gross et al., Waveform and numerology to support 5G services and requirements. IEEE Commun. Mag. (2016), https://doi.org/10.1109/MCOM.2016.1600336CM 34. GFDM on GitHub, Jan 2018, https://github.com/ewine-project/Flexible-GFDM-PHY 35. The ORCA Project, Jan 2018, https://www.orca-project.eu/ 36. The eWINE Project, Jan 2018, https://ewine-project.eu/ 37. National Instruments reconfigurable software defined radio (USRP-RIO), Jan 2018, http://sine. ni.com/nips/cds/view/p/lang/en/nid/212174 38. LabVIEW Communications LTE Application Framework, Jan 2018, http://sine.ni.com/nips/ cds/view/p/lang/en/nid/213083 39. E. 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Wood, System-level design considerations for digital pre-distortion of wireless base station transmitters. IEEE Trans. Microw. Theory Tech. (2017), https://doi.org/10.1109/TMTT.2017. 2659738 45. R. Raich, G.T. Zhou, Orthogonal polynomials for complex Gaussian processes. IEEE Trans. Signal Process. (2004), https://doi.org/10.1109/TSP.2004.834400 46. C. Berger, Y. Benlachtar, R. Killey, Optimum clipping for optical OFDM with limited resolution DAC/ADC. Advanced Photonics, OSA Technical Digest, paper SPMB5 (2011) Part II Non-Orthogonal Multiple Access (NOMA) in the Power Domain Chapter 5 NOMA: An Information-Theoretic Perspective Mojtaba Vaezi and H. Vincent Poor 5.1 What Is Non-Orthogonal Multiple Access (NOMA)? Multiple access lies at the heart of cellular communication systems. It refers to a technique that allows multiple users to share a communication channel. The firstgeneration (1G) to the fourth-generation (4G) of cellular networks have adopted radically different multiple access schemes with one common theme in mind—to have orthogonal signals for different users at the receiver side [1]. In particular, 1G to 4G cellular networks has adopted one or more of the following multiple access methods: • • • • • Frequency division multiple access (FDMA) Time division multiple access (TDMA) Code division multiple access (CDMA) Orthogonal frequency division multiple access (OFDMA) Space division multiple access (SDMA) For example, in OFDMA which has widely been used in 4G systems, different users’ signals are orthogonal in the frequency and/or time domains. In other words, one orthogonal frequency division multiplexing (OFDM) resource block (180 kHz) cannot be allocated to more that one user. Non-orthogonal multiple access (NOMA) [2], in contrast, allows multiple users to share the same resource elements, be it in the time, frequency, space, or code domain. NOMA is currently a hot research topic for 5G and beyond systems, both in academia and industry. While it is concerned with “non-orthogonality” of multiple access, it appears that the research community is perceiving this term in somewhat M. Vaezi (B) Villanova University, Villanova, PA, USA e-mail: mvaezi@villanova.edu H. Vincent Poor Princeton University, Princeton, NJ, USA e-mail: poor@princeton.edu © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_5 167 168 M. Vaezi and H. Vincent Poor different ways. Due to the different interpretations, there is not a consensus about applying this term to some well-known existing techniques such a CDMA. While the majority of recent works see CDMA as an orthogonal multiple access (OMA) technique, there are a group of other papers that categorize it as a NOMA technique. In the following, we present and discuss different viewpoints used to define nonorthogonality in NOMA.1 NOMA Definitions and Viewpoints • Superposition Coding with Successive Interference Cancellation: A large body of papers consider NOMA to be equivalent to superposition coding and successive interference cancellation (SC-SIC), respectively, at the transmitter and receivers. This is partly because the first paper using the term NOMA considered the problem of downlink transmission with SIC [2] and partly due to the fact that SC-SIC is the capacity-achieving technique for the downlink channel in single-cell single-input single-output (SISO) transmission, as we discuss later in this chapter. In fact, the key term is SIC as it also appears in the uplink transmission. With this in mind, SC-SIC is also applied to several different cases, such as multiple-input multiple-output (MIMO) networks and multi-cell networks (SISO or MIMO), in which SC-SIC is suboptimal. This method is also known as power domain NOMA. • Overloading: A second important view is to distinguish NOMA and OMA based on the system loading. In this setting, NOMA refers to overloaded systems and “overloading” means to have more than one user per available resource element in the time, frequency, code, or space domain. This point of view is rooted in CDMA systems. With this definition, a CDMA system will be seen as a NOMA scheme if it is overloaded (i.e. when there are more users than the number of codes) and will be considered as an OMA scheme if there are more codes than the number of users in the systems. Examples of the NOMA schemes developed with this view are low-density spreading (LDS) CDMA, LDS-OFDM, sparse code multiple access (SCMA), and multi-user shared access (MUSA), which are collectively known as code domain NOMA. This definition (overloading) can be applied to other settings such as SDMA systems. That is, similar to the CDMA case, a multi-user MIMO (MU-MIMO) system can be seen as a NOMA or OMA scheme. The former happens when the number of transmit antennas n t is less than the number of users K , whereas in the latter case n t ≥ K . • Linear Transform Decoding: Some even define NOMA based on the complexity of multi-user detection. With this point of view, in an OMA scheme, the signals of different users can be separated in orthogonal subspaces using a linear transform. Then, any scheme that does meet this definition can be 1 We should highlight that in this chapter we are discussing non-orthogonal multiple access methods. This should not be confused with non-orthogonal random access which is another related topic and will be discussed in Chap. 17 of this book. 5 NOMA: An Information-Theoretic Perspective 169 categorized as NOMA. For example, CDMA systems theoretically can be constructed using independent random coding for different users. Such CDMA systems are naturally non-orthogonal. Since the 1990s, there has been a vast amount of research on the capacity of multiple access systems based on CDMA (involving power control, serial interference cancellation, dirty-paper coding, etc.)2 However, historically, CDMA implies directsequence CDMA (DS-CDMA) operation as used in 3G wideband CDMA (WCDMA) systems. Since the 2000s, other types of CDMA schemes have been developed, such as interleave division multiple access (IDMA) [3]. In [3], NOMA is used to cover both DS-CDMA and IDMA. In this respect, NOMA has a broader meaning than DS-CDMA. Therefore, when comparing CDMA and NOMA, we need to distinguish between CDMA in the general sense, i.e. CDMA defined by information theory, and CDMA in a narrow sense such as DS-CDMA. • Information-Theoretic View: Looking from an information-theoretic perspective, NOMA may refer to any technique in which concurrent transmission is allowed over the same resources in time/frequency/code/space. This achieves a better rate region when compared to orthogonalization of one or some of the resources and includes SC-SIC but is not limited to that. In this context, other techniques such as rate-splitting (RS) and dirty-paper coding (DPC) can be seen as NOMA. On the other hand, CDMA techniques that are based on orthogonality in the code domain will be seen as OMA techniques. This definition is very broad and includes many existing techniques as a subset, and in general, its concern is to promote optimal strategies in various uplink/downlink communication strategies. In this chapter, NOMA refers to the last sense.3 We should highlight that the theory of NOMA has been around for many years. In effect, the basic premise behind NOMA is to reap the benefits promised by information theory for the downlink and uplink transmission of wireless systems, modelled by the broadcast channel (BC) and multiple access channel (MAC), respectively. Capacity regions of the BC and MAC have been established several decades ago [1, 4, 5], and concurrent non-orthogonal transmission is the optimal transmission strategy in both cases. That is, in general, to achieve the capacity region, the users must transmit at the same time and frequency. In particular, the capacity region of the BC is achieved using superposition coding at the base station (BS). For decoding, the user with the stronger channel gain (usually the one closer to the BS) uses successive interference cancellation (SIC) to decode 2 For example, CDMA with power control is capacity achieving for the single-antenna broadcast channel. 3 However, in the SDMA case, we assume that the system is overloaded. This is relevant in the MIMO (and MU-MIMO) case where “space” comes in as a new resource. We note that when the number of transmit antennas n t is greater than the number of users K , different users can be served in different (orthogonal) spaces, reducing the underlying system to an OMA one. 170 M. Vaezi and H. Vincent Poor User 2 signal detecƟon SIC of User 2 signal Fig. 5.1 users User 1 signal detecƟon Illustration of single-cell downlink NOMA using power domain multiplexing for two its signal free of interference, while the user with the weaker channel gain treats the signal of the stronger users as noise. This is illustrated in Fig. 5.1. Similarly, to get the highest achievable region in multi-cell systems, concurrent non-orthogonal transmission is still required [5–7], and orthogonal transmission is suboptimal. 5.2 What Drives NOMA? The next generation of wireless networks must support very high throughput, low latency, and massive connectivity. According to the international telecommunication union (ITU) [8], 5G networks must fulfill several requirements including: (1) a minimum peak data rate of 10 Gbps (100 times more than that in the 3rd Generation Partnership Project (3GPP) Long-Term Evolution (LTE)) (2) a latency of 1ms (ten times lower than that in 4G networks), and (3) a connection density of 1,000,000 devices per km2 (100 times more than 4G networks). Effective multiple access is a key enabler in achieving these requirements. As noted above, the first to the fourth-generations of cellular networks have adopted different multiple access schemes with one common theme in mind—to have orthogonal signals for different users at the receiver side. By allowing multiple users to share all domains (e.g. time, frequency, space), NOMA can address the above challenges of the next generation of wireless networks more efficiently than the conventional orthogonal multiple access schemes. NOMA can increase spectral efficiency and userfairness by exploiting a capacity-achieving scheme in the downlink. It can support 5 NOMA: An Information-Theoretic Perspective 171 more connections in the uplink by letting multiple users simultaneously access the same wireless resources, which, in turn, can reduce latency [9–12]. Code domain NOMA4 uses user-specific sequences for sharing the entire radio resource. On the other hand, power domain NOMA exploits the channel gain differences between the users for multiplexing via power allocation. In this chapter, we study NOMA in the power domain, a technique that can improve wireless communication through the following benefits: • Massive Connectivity: There appears to be a reasonable consensus that NOMA is essential for massive connectivity. This is because the number of served users in all OMA techniques is inherently limited by the number of resources (e.g. the number of codes in CDMA and the number of resource blocks in OFDMA). In contrast, by superimposing all users’ signals, NOMA theoretically can serve an arbitrary number of users even within one resource block. In this sense, NOMA can be tailored to Internet of Things (IoT) applications where a large number of devices sporadically transmit a small number of packets. In fact, allocating an entire resource block (180 kHz in LTE) to one device, as is done in OMA, is extremely inefficient. • Low Latency: Latency requirements for 5G applications are rather diverse and very stringent in some cases. For example, ITU requires a user plan latency of 4ms and 1ms for enhanced mobile broadband (eMBB) and ultra-reliable and lowlatency communications (URLLC), respectively [8]. With OMA, it is very difficult to guarantee such stringent delay requirements because no matter how many bits a device wants to transmit the device must wait until an unoccupied resource block becomes available. In contrast, NOMA supports flexible scheduling since it can accommodate a variable number of devices depending on the application that is being used and the perceived quality of service (QoS) of the device. • High Spectral Efficiency: According to ITU requirements for IMT-2020, downlink peak spectral efficiency should be 30 bits/s/Hz. NOMA offers a higher spectral efficiency and a better user-fairness compared to OMA. As will be seen in Sect. 5.3, NOMA is the theoretically optimal way of using the spectrum for both uplink and downlink communications in a single-cell network. Such better performance is achieved due to the fact that every NOMA user can enjoy the whole bandwidth, whereas OMA users are limited to a smaller fraction of the spectrum which is inversely proportional to the number of users. NOMA can also be combined with other emerging technologies, such as massive MIMO and millimeter wave (mmWave) technologies, to further improve spectral efficiency and support higher throughput. 4 Sparse code multiple access (SCMA) is an important variant of NOMA in the code domain. 172 M. Vaezi and H. Vincent Poor In consideration of the above benefits, NOMA has drawn significant attention from both academia and industry during past a few years. However, as we briefly mentioned earlier and will see in detail in this chapter, from an information-theoretic perspective the theory behind NOMA has been established for several decades. Nevertheless, despite this insight from information theory, orthogonal multiple access techniques have been used in the cellular networks from 1G to 4G. This was mainly to avoid interuser interference cancellation which would have resulted in unacceptably complex receivers. Today, with the advances in processing power, the implementation of interference cancellation at user equipment has been made practical. For example, a category of relatively complex user terminals, known as network-assisted interference cancellation and suppression (NAICS) terminals, has recently been adopted in 3GPP LTE-A. Such technological advances, in conjunction with the need to support exponentially increasing numbers of devices and better spectral efficiency, have motivated a new wave of research on NOMA. Saito et al. first showed that NOMA can improve system throughput and user-fairness over a SISO channel using OFDMA [2]. The spectral efficiency of NOMAbased systems is further boosted when combined with MIMO communication [13–15]. Successful operation of this technique, however, depends on knowledge of the channel state information (CSI) between the BS and the end-users. More practical solutions, e.g. those with limited and delayed CSI, are crucial in making NOMA workable. Today, a variation of NOMA, known as multi-user superposition transmission [12], is considered for the 3GPP LTE-A systems. While recent advances in processor capabilities have made SIC, and consequently NOMA, feasible, significant research challenges remain to be addressed before NOMA can be deployed. In addition to the above practical issues, NOMA-based transmission introduces new security and privacy challenges. This is because in NOMA-based transmission a user with a better channel is able to decode the other user’s signal. Even, a user with a weaker channel can also partly decode the stronger user’s signal. On top of that, wireless transmission is naturally vulnerable to external eavesdroppers. Although upper-layer security approaches (e.g. cryptography) can be used to secure transmissions, there are numerous risks in cryptographic methods due to the rapid advancement of computing technologies. Also, cryptographic approaches require a key management infrastructure which should be secured, in turn. Moreover, traditional key agreement algorithms are not suitable for many existing and emerging wireless networks, such as ad hoc networks and IoT, since they consume scarce resources such as bandwidth and battery power. 5 NOMA: An Information-Theoretic Perspective 173 5.3 Theory Behind NOMA Analysis of cellular communication can generally be classified as either downlink or uplink. In the downlink channel, the BS simultaneously transmits signals to multiple users, whereas in the uplink channel multiple users transmit data to the same BS. As noted above, from an information-theoretic perspective, the downlink and uplink are modelled by the broadcast channel and multiple access channel, respectively. The basic premise behind single-cell NOMA in the power domain is to reap the benefits promised by the theory of multi-user channels [1, 5]. As such, we review what information theory promises for these channels, both in the single-cell and multi-cell settings. In particular, we seek to answer the following two questions in this section: (1) What are the highest achievable throughputs for these multi-user channels? and (2) how can a system achieve such rates? 5.3.1 Single-Cell NOMA For simplicity of illustration, we first consider a network consisting of a single cell. In addition, we assume that there are only two users in that cell. Later, we discuss the general case with multiple cells each consisting of multiple users. For OMA, we consider a TDMA technique5 where a fraction α of the time (0 ≤ α ≤ 1) is dedicated to user 1 and a fraction ᾱ  1 − α of the time is dedicated to user 2. The capacity regions of the two-user MAC and BC are achieved via NOMA, where both users’ signals are transmitted at the same time and in the same frequency band [5]. To gain more insight, we describe how these regions are obtained. Throughout the chapter, we use C (x)  1 log2 (1 + x), 2 (5.1) and γi = |h i |2 P is the received signal-to-noise ratio (SNR) for user i, where h i is the channel gain, P is the transmitter power, and the noise power is normalized to unity. 5.3.1.1 Two-User MAC (Uplink) The capacity regions of the two-user MAC is achieved via non-orthogonal transmission in which both users’ signals are transmitted at the same time and in the same frequency band [5]. The curves labelled NOMA in Fig. 5.2 represent the capacity regions of the MAC for different values of P1 and P2 , which are the respective transmit powers of the two users. From these figures, it can be seen that except for a few points OMA is strictly suboptimal. One of these points is the sum capacity of 5 FDMA has exactly the same performance [5]. 174 M. Vaezi and H. Vincent Poor NOMA OMA−PC OMA R 2 R2 NOMA OMA−PC OMA R R 1 1 (a) P1 = 5, P2 = 10 (b) P1 = 10, P2 = 10 Fig. 5.2 Best achievable regions by OMA and NOMA in the two-user MAC (uplink) for different values of the two users’ transmit powers P1 and P2 the MAC. This means that both OMA and NOMA can achieve the sum capacity of MAC. To gain more insight, we describe how the above regions are obtained. Using OMA, each user sees a single-user channel in its dedicated fraction of the time. As a result R1 = αC (γ1 ), R2 = ᾱC (γ2 ), (5.2a) (5.2b) are achievable. If power control (PC) is applied, these rates can be increased to R1 = αC γ  1 ,  γα  2 R2 = ᾱC . ᾱ (5.3a) (5.3b) This region is labelled NOMA-PC in Fig. 5.2a, b. In the case of NOMA, both users concurrently transmit, and their signals interfere with each other at the BS. The BS can use SIC to achieve any point in the NOMA region, which is the capacity region of this channel [1]. Theorem 1 The capacity region of the two-user MAC is the set of nonnegative (R1 , R2 ) such that R1 ≤ C (γ1 ), R2 ≤ C (γ2 ), (5.4a) (5.4b) R1 + R2 ≤ C (γ1 + γ2 ). (5.4c) 5 NOMA: An Information-Theoretic Perspective 175 Particularly, to achieve the right corner point of the capacity region (in which R1 = C (γ1 )), the BS first decodes user 2’s signal, treating the other signal as noise. 2 ). The BS then removes user 2’s signal and decodes This results in R2 = C ( γ1γ+1 user 1’s signal free of interference; i.e. R1 = C (γ1 ). As a result, the sum capacity R1 + R2 = C (γ1 + γ2 ) is achievable. To achieve the other corner point, the order of decoding needs to be changed. From Fig. 5.2, it is seen that the gap between the NOMA and OMA regions becomes larger if power control is not used in OMA. 5.3.1.2 Two-User BC (Downlink) Similar to the two-user MAC, the capacity region of the two-user BC is known and is achieved via non-orthogonal transmission in which both users’ signals are transmitted at the same time and in the same frequency band [5]. The curves in Fig. 5.3a–c represent the capacity regions of the BC as well as the best achievable regions obtained by OMA for different values of channel gains. From these figures, it can be seen that except for a few points, OMA is strictly suboptimal in the downlink. In fact, OMA can only achieve (5.5a) (5.5b) R1 = αC (γ1 ), R2 = ᾱC (γ2 ). However, making use of a NOMA scheme can strictly increase this rate region as shown in Fig. 5.3. In particular, the capacity region of this channel is known and can be achieved using superposition coding at the BS and successive interference cancellation at the receiver. For decoding, the user with the stronger channel uses SIC to decode its signal free of interference at a rate of R1 = C (βγ1 ) while the other NOMA OMA R1 (a) P1 = 1, P2 = 10 2 NOMA OMA R R 2 R2 NOMA OMA R1 (b) P1 = 5, P2 = 10 R 1 (c) P1 = 9, P2 = 10 Fig. 5.3 Best achievable regions by OMA and NOMA in the BC (download) for different values of P1 and P2 176 M. Vaezi and H. Vincent Poor 2 user is capable of decoding at a rate of R2 = C ( βγβ̄γ2 +1 ) where β is the fraction of the BS power allocated to user 1’s data and β̄ = 1 − β. By varying β from 0 to 1, any rate pair (R1 , R2 ) on the boundary of the capacity region of the BC (NOMA region) can be achieved. That is, Theorem 2 The capacity region of the two-user BC is the set of nonnegative (R1 , R2 ) such that R1 ≤ C (βγ1 ),   β̄γ2 , R2 ≤ C βγ2 + 1 (5.6a) (5.6b) in which β ∈ [0 1] and β̄ = 1 − β. The fact that the capacity region of downlink NOMA is known enables us to find the optimum power allocation corresponding to any point (R1 , R2 ) on the boundary of the capacity region. In fact, all we need to know to achieve such a rate pair is to find what fraction of the BS power should be allocated to each user. Corresponding to each (R1 , R2 ), there is a 0 ≤ β ≤ 1 such that β P and β̄ P are the optimal powers for user 1 and user 2, respectively, where P is the BS power. Conversely, every β generates a point on the boundary of the capacity region. The above argument implies that NOMA can improve user-fairness smoothly and in an optimal way by flexible power allocation. Suppose that a user has a poor channel or it has not been served for a long time (in OMA). To boost this user’s rate and improve user-fairness, the BS can simply increase the fraction of power allocated to this user. We can look at this problem from yet another perspective. To increase the rate of such a user, we can maximize the weighted sum-rate µR1 + R2 where a high weight (µ) is given to such a user. This is because, to maximize µR1 + R2 for any µ ≥ 0, there exists an optimal power allocation strategy, determined by β. Seeing that µ > 1 (µ < 1) corresponds to the case where user 1 has higher (lower) weight than user 2, to improve user-fairness, we can assign an appropriate weight to the important user and find the corresponding β. 5.3.1.3 K -User Uplink/Downlink In the above, we described coding strategies for the two-user uplink/downlink channels. Interestingly, very similar coding schemes are still capacity achieving for the K -user MAC and BC, as described in the following. K -User MAC: To achieve the capacity region of the K -user MAC, the users transmit their signals concurrently and the BS applies SIC decoding. Specifically, we have [5]. Theorem 3 The capacity region of the K -user MAC is the set of nonnegative (R1 , . . . , R K ) such that 5 NOMA: An Information-Theoretic Perspective  j∈S Rj ≤ C  γj j∈S  177 for every S ⊆ [1 : K ]. (5.7) For example, for K = 3, the capacity region is the set of nonnegative (R1 , R2 , R3 ) such that R1 ≤ C (γ1 ), (5.8a) R2 ≤ C (γ2 ), R3 ≤ C (γ3 ), R1 + R2 ≤ C (γ1 + γ2 ), (5.8b) (5.8c) (5.8d) R1 + R3 ≤ C (γ1 + γ3 ), R2 + R3 ≤ C (γ2 + γ3 ), (5.8e) (5.8f) R1 + R2 + R3 ≤ C (γ1 + γ2 + γ3 ). (5.8g) The capacity-achieving scheme is based on non-orthogonal transmission that allows multiple users to transmit at the same time and frequency. To be specific, the capacity region is achieved by point-to-point codes, successive cancellation decoding, and time-sharing. Again, OMA is strictly suboptimal [1]. K -User BC: To achieve the capacity region of the K -user BC, the users’ signals are superimposed at the BS and transmitted altogether. Without loss of generality assume that γ1 ≥ γ2 ≥ · · · ≥ γ K . At the receiver sides, receiver k ∈ [1, . . . , K ] first decodes the signal of users k + 1, . . . , K and removes them from the received signals. It (receiver k) then decodes its own signal treating users’ 1, . . . , k − 1 signals as noise. As a result, we obtain Theorem 4 The capacity region of the K -user BC is the set of nonnegative (R1 , . . . , R K ) such that Rk ≤ C  1+ in which k ∈ [1, . . . , K ], β j ≥ 0 ∀ j and βk γk k−1 j=1 K j=1 β j γk , (5.9) β j = 1. Remark 1 To achieve the above rates, it is important to note that the single-antenna K -user Gaussian BC is a set of degraded channels. This implies that the users can be ordered based on their channel strengths (for example, in Theorem 4, we have assumed that γ1 ≥ γ2 ≥ · · · ≥ γ K ). We note that due to the SIC decoding, the user with the strongest channel (user 1) is able to decode its own signal free of interference, whereas the user with the weakest channel (user K ) has to treat all other users’ signals as noise. For example, for K = 3 the capacity region is the set of nonnegative (R1 , R2 , R3 ) such that 178 M. Vaezi and H. Vincent Poor R1 ≤ C (β1 γ1 ),   β2 γ2 R2 ≤ C , 1 + β1 γ2   β3 γ3 R3 ≤ C . 1 + (β1 + β2 )γ3 (5.10a) (5.10b) (5.10c) So far, we have assumed a network with a single cell. In fact, most of the work on NOMA is limited to single-cell analysis, where there is no co-channel interference caused by an adjacent BS. However, to verify the benefits of NOMA in a more realistic setting, it is necessary to consider a multi-cell network. Specifically, as wireless networks get denser and denser, inter-cell interference (ICI) becomes a major obstacle to achieving the benefits of NOMA. In the next section, we discuss the theory behind NOMA in multi-cell networks. 5.3.2 Multi-Cell NOMA In a multi-cell setting, finding the best achievable region is much more involved than in the single-cell case, and simple channel models are insufficient. A few models, including the interference channel, interfering MAC, and interfering BC can be used to model multi-cell networks. Unfortunately, the capacities of these channels are unknown in general. However, the known achievable rate regions for these channels indicate the superiority of NOMA to OMA. 5.3.2.1 Interference Channel (IC) The capacity region of the two-user IC is not known in general; however, it is known that OMA is strictly sub-optimal. Han and Kobayashi introduced an achievable region in 1981 [6], which is still the best known inner bound for the general interference channel. In the Han-Kobayashi (HK) scheme, each user can split its message to be sent into two submessages of smaller rates and power. These are known as private and common messages; the former is intended to be decoded only at the respective receiver, whereas the latter can be decoded at both receivers. The rationale behind this coding scheme is to decode part of the interference (the common message) and treat the rest as noise. The optimal input distributions are not known for the HK region. As such, commonly a subset of the HK region with Gaussian codebooks is used to represent the HK region for the Gaussian channel; see, e.g. [16–19]. The basic HK scheme employs rate-splitting (RS) and superposition coding (SC) at each transmitter and SIC at the receiver. Since both transmitters send their signals concurrently at the same frequency, the HK scheme is a NOMA scheme. Flexibility in splitting each user’s transmission power into the common/private portions of information and time-sharing between them make the HK scheme very strong, but 5 NOMA: An Information-Theoretic Perspective 179 complicated. Not surprisingly, though, the optimal HK strategy is not well-understood, in general. Nevertheless, the HK scheme is known to be within 21 bit of the capacity region of the two-user Gaussian interference channel [16]. Although this gap could be due to a suboptimal scheme, loose outer bounds, or both, the outer bounds seem to be the most crucial. In general, the HK scheme applies time-sharing to improve the basic HK region and can be seen as a combination of NOMA and OMA [7]. Although the basic HK scheme is optimal for the strong and very strong interference regimes, when one interference link is strong and the other one is weak, time-sharing can enlarge the basic HK region in general. Specifically, we have [7, 20] Lemma 1 The HK achievable region for the one-sided IC is the set of rate pairs (R1 , R2 ) satisfying R1 ≤ λ1 R11 , R2 ≤ λ1 R21 + λ2 R22 , (5.11a) (5.11b) in which P1 λ1 R11 ≤ γ  1 + aβ1 P21 R21 ≤ γ  P1 λ1 a β̄1 P21 1+ (5.12a) , + aβ1 P21 + γ (β1 P21 ), R22 ≤ γ (P22 ), (5.12b) (5.12c) where λ1 + λ2 = 1, λ1 P21 + λ2 P22 = P2 , 0 ≤ β1 ≤ 1, and β̄1 = 1 − β1 . It is worth mentioning that Lemma 1 characterizes a strictly better region when compared with the HK region without time-sharing. In particular, for λ1 = 1, this lemma reduces to the HK region without time-sharing (i.e., the basic HK region). These regions are compared in Fig. 5.4. Also, substituting P21 = 0, Lemma 1 reduces to   P1 , (5.13a) R 1 ≤ λ1 γ λ1   P2 , (5.13b) R2 ≤ (1 − λ1 )γ 1 − λ1 for 0 ≤ λ1 ≤ 1. This region is known as the TDMA (or FDMA) region. The main difference between the TDMA and time-sharing regions is in the fact that in the TDMA approach only one user is allowed to transmit during each subband, while in time-sharing method, both users can transmit in the same subband (e.g. during λ1 in the region defined by Lemma 1). M. Vaezi and H. Vincent Poor R R 2 2 180 TS NOMA OMA R1 (a) P1 = 1, P2 = 3 TS NOMA OMA R1 (b) P1 = 3, P2 = 3 Fig. 5.4 Best achievable regions by OMA, NOMA, TS (applying time-sharing to OMA and NOMA in different time slots) in the IC (multi-cell network) for different values of P1 and P2 In other words, the HK scheme that combines NOMA and OMA gives the largest rate region [7], as shown in Fig. 5.4 for different values of P1 and P2 . In these plots, OMA refers to TDMA, whereas NOMA refers to the basic HK scheme in which time-sharing is not applied. In this case, both transmitters send their messages using the HK strategy but in one time-slot only. The third curve, labelled TS, is based on the HK scheme with time-sharing (TS) in which two time slots are used: in one time-slot, both users are active, while in the other time-slot, only one of them is transmitting. As can be seen from this figure, both NOMA and OMA are suboptimal when compared with the case where NOMA and OMA are combined. We should highlight that the rate region obtained via the HK scheme without time-sharing (NOMA) is however very close to that with time-sharing (OMA + NOMA). That is, NOMA yields a good approximation of the best achievable region for this channel, for the two-user IC. 5.3.2.2 Interfering MAC and BC Consider a mutually interfering two-cell network in the uplink, where each cell includes one MAC. Assume that only one of the transmitters of each MAC (typically the closest one to the cell-edge) is interfering with the BS of the other MAC. In this network, the interfering transmitters can employ HK coding, similar to that used in the IC, while the non-interfering transmitters in each MAC employ single-user coding. This NOMA-based transmission results in an inner bound which is within a one-bit gap of the capacity region [21]. Likewise, one can use the interfering BC to model a mutually interfering two-cell downlink network. 5 NOMA: An Information-Theoretic Perspective 181 Despite years of intensive research in information theory, finding optimal uplink and downlink transmit/receive strategies for multi-cell networks remains rather elusive. In fact, as discussed earlier, even for a much simpler case of the two-user IC, the optimal coding strategy is still unknown. Nonetheless, fundamental results from information theory as a whole suggest that NOMA-based techniques result in a superior rate region when compared with OMA. It should be highlighted that, despite the above insight from information theory, OMA techniques have been used in the cellular networks from 1G to 4G, mainly to avoid interference and due to its simplicity. In addition, the lack of understanding of optimal strategies for multi-cell networks has motivated pragmatic approaches in which interference is simply treated as noise. 5.3.3 NOMA in MIMO Networks With the rapid advancement of multi-antenna techniques, today wireless nodes (particularly BSs) are often equipped with more than one antenna. Multi-antenna systems create spatial dimension which, in turns, opens the door to SDMA. In SDMA, multiple users can communicate at the same time-frequency but are distinguished in space. That is, each user has a different beam. Then, in the downlink and uplink of singlecell networks, we will have the MIMO-BC and MIMO MAC. In addition, similar to the multi-cell SISO networks, we can use MIMO IC to model a multi-antenna multi-cell network. Although in some cases the capacity region of these channels is unknown in general, similar to the SISO case, it is known from information theory that TDMA (OMA) is suboptimal and concurrent transmission (NOMA) can result in a better rate region. We discuss this in the following sections. 5.3.3.1 MIMO BC Unlike the Gaussian SISO BC, the Gaussian MIMO BC is non-degraded in general. That is, in general, the MIMO BC users cannot be ordered based on their channel strengths. This is because the users’ channels are matrices (or vectors), and matrices cannot be ordered in general, unlike the scalars in the SISO BC. This has been the main difficulty in proving the capacity of the MIMO BC. However, the capacity region of the MIMO BC has recently been established in [22] in which it is proved that the capacity region can be achieved by DPC [23]. Because of the non-degradedness of the MIMO BC, SC-SIC which is the capacity-achieving technique for the SISO BC is not capacity achieving for the MIMO BC. While in SC-SIC interference cancellation is performed at the receiver side, DPC constitutes interference cancellation at the transmitter side and consequently the transmitter requires knowledge of CSI. The same strategy is also the only technique that 182 M. Vaezi and H. Vincent Poor achieves the capacity region of the multiple-input single-output (MISO) BC. DPC is a non-orthogonal transmission (NOMA) strategy, and it is well-understood that TDMA (OMA) is suboptimal for the MIMO BC and the MISO BC. DPC is, however, prohibitively complex for practical systems. Practical MIMO systems usually use linear precoding to simplify the transmitter design. This technique creates different beams for different users (or sets of users) and allocates a fraction of the total transmit power to each beam. At the receiver side, the interference from other users is treated as noise. Due to its simplicity, SDMA is usually implemented using linear precoding. This access technique is the basic principle behind several well-established techniques in 4G and upcoming 5G networks. The examples include but are not limited to multi-user MIMO, network MIMO, coordinated multipoint (CoMP), millimeter-wave MIMO, and massive MIMO [24]. Another line of research seeks to extend the SISO NOMA principles to MIMO NOMA transmission. In fact, the performance of NOMA can be further boosted in multi-antenna networks. MIMO NOMA solutions exploit multiplexing and diversity gains to improve outage probability and throughput, by converting the MIMO channel into multiple parallel channels. One approach is to directly apply SISO NOMA methods by making the MIMO NOMA networks degraded. That is to order the users based on an effective scalar channel and decode their messages using SIC. A second approach combines SDMA with superposition coding at the receiver and SIC at the transmitter. Such solutions try to allocate different beams to each group of users and then to use SISO NOMA within each group. By allocating different beams to each cluster (group), the interference between the clusters can be managed and removed. Within each cluster, the SISO NOMA solutions are then applied. SDMA and MIMO NOMA can both be viewed as superpositions in the power domain with different approaches at the receiver side. In the former, the users are separated spatial beamformers, whereas in the latter case SIC is used at the receivers to separate the users [14]. In other words, SDMA fully treats the interference as noise whereas NOMA with SIC fully decodes interference. Recalling the flexible rate-splitting in the HK scheme for the IC, in which a user’s transmission power is split between the common/private portions of information and part of the interference (the common message) is decoded while treating the remainder as noise, makes the achievable region larger, researchers have applied the same concept to the MIMO BC channel. RS obviously can bridge between the SDMA and NOMA with SIC by enabling a receiver to decode part of the interference and treat the remaining part of that as noise. In this sense, SDMA and NOMA with SIC can be seen as two extreme cases of RS. All of these approaches suggested by information theory are based on the concurrent transmission of multiple users signal without making them orthogonal and thus can be seen as different variants of NOMA. To summarize, from information-theoretic results, it is clear that OMA is suboptimal in MIMO networks too. This is valid both for single-cell and multi-cell (network MIMO) cases. The best techniques are either based on NOMA or a combination of NOMA and OMA using time-sharing. Several other techniques such as superposition coding, SIC, RS, and time-sharing fall within these NOMA schemes. 5 NOMA: An Information-Theoretic Perspective 5.3.3.2 183 MIMO IC The capacity region of the MIMO IC, similar to many other multi-user networks, is unknown. Finding the exact capacity region has been quite challenging for those channels. Because of this, approximations are widely used to get insight into the behavior of these channels. One commonly used approximation metric is degrees of freedom (DoF), or multiplexing gain. The DoF gives the pre-log of the capacity of a given channel in the high SNR regime; i.e. DoF = lim S N R→∞ C (S N R) . log(S N R) (5.14) For example, the DoF of the MIMO channel with M and N antennas at the transmitter and receiver is min(M, N ) which is its multiplexing gain. The DoF of the MIMO broadcast channel with M antennas at the transmitter and N1 and N2 antennas at the receivers is min(M, N1 + N2 ). Interference alignment (IA) is the main technique used to achieve the degrees of freedom of interferences channels. IA was introduced by Maddah-Ali et al. in the context of the MIMO X channel [25, 26], where an iterative achievable scheme for this channel built upon dirtypaper coding and successive decoding was introduced. IA refers to a mechanism for aligning (overlapping) interference spaces. It was then applied to the K -user SISO IC in [27] leading to the surprising conclusion that wireless networks are not essentially interference limited. The DoF of the K -user SISO IC is equal to K2 [27], meaning that each user can enjoy half of the spectrum in the high SNR regime. The DoF of the K -user user MIMO Gaussian IC with M antennas at each transmitter and N antennas at each receiver is also known [28, 29] and is obtained using IA. All these results show the suboptimality of orthogonalization. For example, in the three-user MIMO IC with two antennas at each node, IA allows a total of three DoF, whereas the schemes based on orthogonalization can allow a maximum of two DoF per channel use. In the orthogonalized solutions, e.g. TDMA, only one user will be active at a time and can send two symbols on its 2 × 2 MIMO channel, whereas IA allows each user to send only one information symbol, but all the users are active at all times. These results show that the DoF of the K -user MIMO IC is achieved when all the users are active at all times and orthogonalization is suboptimal. In other words, NOMA performs better than OMA. 5.4 Moving from Theory to Practice As explained in Sect. 5.3, the basic theory behind NOMA has been around for many years. Specifically, the capacity region of SISO NOMA in the single-cell setting has been known for several decades. Moreover, it is known that to get the highest achievable region in multi-cell systems concurrent non-orthogonal transmission is 184 M. Vaezi and H. Vincent Poor required [5–7], and orthogonal transmission is suboptimal. This conclusion is based on the exact and approximated (DoF) analysis, as discussed in the previous section. Furthermore, the theoretical results for MIMO networks (MIMO BC, MIMO MAC, and MIMO IC) on the whole indicate that OMA is suboptimal and NOMA, with sophisticated schemes such as DPC and IA, can achieve better rates. Despite these insights from information theory, orthogonal multiple access techniques have been used in the cellular networks from 1G to 4G. This has mainly been to avoid inter-user interference cancellation which would have resulted in unacceptably complex receivers. In fact, schemes such as SIC and DPC used for interference cancellation are very complex to implement in user equipment. Today, with the advances in processing power driven by Moore’s law,6 the implementation of interference cancellation at user equipment has been made practical, as discussed in Sect. 5.4.1. Such technological advances, in conjunction with the need to support exponentially increasing numbers of devices and better spectral efficiency, have motivated a new wave of research on NOMA. 5.4.1 SIC in 4G Networks Owing to advances in processing power, interference cancellation on user equipment has become more practical and is realized in LTE-A networks in a few different settings. Some of these are listed below. • Network-Assisted Interference Cancellation and Suppression (NAICS): NAICS refers to a category of relatively complex user terminals that has recently been adopted by 3GPP LTE-A. Network-assisted interference cancellation/suppression can enable more effective interference cancellation/suppression at the user-side with possible network coordination [30]. In developing NAICS, an extensive study was done on advanced receivers with various capabilities of interference cancellation/suppression. As an example, single-user MIMO with minimum mean square error successive interference cancellation (MMSE-SIC) has been designed in LTE Release 8 [31]. • Multi-User Superposition Transmission (MUST): MUST is a recent proposal for 3GPP for downlink mobile broadband (MBB) services [12]. MUST has different categories corresponding to different transmitting schemes [32]. Although the interference scenarios are not the same in NAICS and MUST, many of the receivers proposed for NAICS can also be used for MUST. 6 Moore’s Law, hypothesized by Intel founder Gordon Moore in 1965, states that the number of transistors in a dense integrated circuit will double approximately every two years. This enables a larger number of transistors to be concentrated in a given area which, in turn, results in a faster processor that can operate at lower power. 5 NOMA: An Information-Theoretic Perspective 185 Saito et al. first showed that NOMA can improve system throughput and user-fairness over a SISO channel using OFDMA [2]. The spectral efficiency of NOMA-based systems is further boosted when combined with MIMO communication [13–15]. Successful operation of this technique, however, depends on knowledge of the CSI between the BS and the end-users. More practical solutions, e.g. those with limited and delayed CSI, are crucial to making NOMA workable. Today, a variation of NOMA, known as MUST [12], is being considered for use in 3GPP LTE-A systems, as noted above. 5.4.2 Multi-Cell NOMA Solutions Inter-cell interference reduces a cell-edge user’s performance and is the main issue in multi-cell networks. ICI management approaches are used to improve cell-edge users’ performance. Depending on the availability of the data messages desired at the users among multiple BSs, multi-cell techniques can be categorized into coordinated scheduling/beamforming (CS/CB) and joint processing (JP) [33]. Specifically, in CS/CB, data for a user is only available at and transmitted from a single BS, whereas in JP, the data is shared among multiple BSs. ICI management approaches can be combined with NOMA resulting in multi-cell NOMA solutions, e.g. NOMA-JP and NOMA-CS/CB. In NOMA-JP, the users’ data symbols are available at more than one BS. NOMAJP has two main categories: NOMA-joint transmission (JT) and NOMA-dynamic cell selection (DCS). In the former, multiple BSs are active and simultaneously serve a cell-edge user using a shared wireless resource. NOMA-JP can significantly improve the quality of the signals received by cell-edge users as the two BSs cooperate instead of interfering with each other. In NOMA-DCS, although the user’s data is shared among multiple BSs, this data is transmitted only from one BS at a time. The transmitting BS can be dynamically changed over time. In NOMA CS/CB, the users’ data is not shared among two or more cooperating BSs. Specifically, in NOMA CB, the beamforming decision is made with coordination of other BSs. As an example, IA-based CB is applied in [34] in which two BSs jointly optimize their beamforming vectors in order to improve the data rates of cell-edge users by removing ICI. On the other hand, in NOMA-CS multiple BSs coordinate scheduling to serve NOMA users with less ICI. The cooperating BSs in NOMA CS/CB need to exchange global CSI and cooperative scheduling information via a standardized interface named X2 which may result in considerable overhead especially when the users are highly mobile. A review of different multi-cell NOMA techniques can be found in [11]. While recent advances in processing capabilities have paved the way for SIC, and consequently NOMA, significant research challenges remain to be addressed before NOMA can be deployed. In addition to the above practical issues, NOMA-based transmission may introduce new security and privacy challenges. We introduce the channel modes relevant to the physical layer security of NOMA in the next section. 186 M. Vaezi and H. Vincent Poor 5.5 Physical Layer Security in NOMA In a NOMA-based transmission, a user with a better channel is capable of decoding the other user’s signal. Even, a user with a weaker channel can also partly decode the stronger user’s signal. This may introduce new security and privacy challenges. Moreover, wireless transmission is naturally vulnerable to external eavesdroppers. Although upper-layer security approaches (e.g. cryptography) are still relevant since only the legitimate user has a key to decode its message, there are numerous risks in cryptographic methods due to the rapid advancement of computing technologies. Besides, cryptographic approaches require a key management infrastructure which should be secured, in turn. Moreover, traditional key agreement algorithms are not suitable for many existing and emerging wireless networks, such as ad hoc networks and IoT, since they consume scarce resources such as bandwidth and battery power. Considering these challenges, physical layer security schemes are of interest. 5.5.1 Description of the Channel Models The goal of this section is to identify and leverage information-theoretic channel models for securing communication in the context of NOMA. Three basic channel models are considered as depicted in Figs. 5.5, 5.6 and 5.7. In these figures, n t , n 1 , n 2 , and n e are the number of antennas at the transmitter (Tx), legitimate receivers (Rx1 and Rx2), and eavesdropper (Eve), respectively. For the purpose of illustration, we only consider the case of two legitimate receivers. These models can be used for NOMA with two users. A more general setting is the case with K (K ≥ 2) legitimate receivers. It should be highlighted that these channels also can be seen as multi-user MIMO BCs. Strictly speaking, with these channel models, NOMA is relevant only in cases in which the number of transmit antennas (n t ) is less than the number of users (K ), i.e. when the system is overloaded. This is because for n t < K , there is more than one user per resource block (time/frequency/space), while for n t ≥ K , this figure is less than or equal to one, implying that each user’s signal can be transmitted in a dedicated resource block orthogonal to the other users’ resources. In the latter case, there is at least one spatial degree of freedom per user which is the meaningful operating regime for SDMA. It should be highlighted that regardless of the relation between n t and K , optimal beamforming provides interesting open problems. In the conventional multi-user MIMO BC, n t users are selected out of K to satisfy the constraint n t ≤ K . However, in 5G networks, due to the explosion of connected devices, K is usually very large. In addition, many of these connections are from low-rate IoT devices, and a dedicated resource block may allocate more resources than needed.7 In view of these factors, 7 In LTE one resource block is 180 kHz. 5 NOMA: An Information-Theoretic Perspective 187 1 W1 1 Rx1 H1 n1 W2 W1, W2 nt H2 1 W2 Rx2 n2 W1 Fig. 5.5 MIMO BC with confidential messages NOMA is a useful technology for accommodating a higher number of connections and improving spectral efficiency. 5.5.1.1 MIMO BC with Confidential Messages A MIMO BC, similar to any other BC, has an inherent vulnerability in terms of security and privacy, even if there is no external eavesdropper. This is due to the fact that each legitimate user can, partly or wholly, decode the other legitimate user’s message. To study this issue, the MIMO BC with confidential messages (see Fig. 5.5) has been introduced, in which independent messages W1 and W2 are intended for their respective receivers but need to be kept secret from the other receiver. This model is important in NOMA transmission as each legitimate user is a potential threat to security or privacy of its counterpart. 5.5.1.2 MIMO BC with an External Eavesdropper In the above channel model, there is no external eavesdropper, but each legitimate user is seen as a potential adversary (eavesdropper) to the other legitimate user. In Fig. 5.6, another channel that models a class of BCs with an external eavesdropper is depicted. In this model, the messages are to be secured from the eavesdropper but not necessarily from other legitimate users. This channel is also known as the MIMO multi-receiver wiretap channel whose capacity was established in [35], under a matrix power constraint. This channel model is relevant for secure NOMA transmission when external eavesdropping is the only security issue but the legitimate users are not security threats to each other. It can be used, for example, for the case where the users are resource-limited single-antenna devices such as sensors and, thus, are not capable of or interested in decoding the other users’ data. Therefore, the confidentiality of 188 M. Vaezi and H. Vincent Poor 1 W1 1 Rx1 H1 n1 W1, W2 H2 1 W2 nt Rx2 G n2 1 W2 Eve ne W1 Fig. 5.6 MIMO BC with an external eavesdropper a message from other legitimate users is not an issue. Obviously, in such cases, the channel model reduces to a MISO BC. 5.5.1.3 MIMO BC with Confidential Messages and an External Eavesdropper A third and more general channel modeling secure communication in a NOMA-based transmission is the MIMO BC with confidential messages and an external eavesdropper, depicted in Fig. 5.7. This channel consists of a transmitter who wishes to communicate two messages to their respective receivers, each needing to be kept secret from the other legitimate receiver and a third unauthorized receiver (eavesdropper). This configuration models NOMA with two users and one external eavesdropper. While the capacity region of the channel in Fig. 5.7 is unknown, the capacity regions of the channel models in Fig. 5.5 and Fig. 5.6 are established in [35] and [36], respectively. In [36], using a combination of DPC and stochastic encoding, known as secret DPC, it is shown that both messages can be simultaneously transmitted at their respective maximal secrecy rates under a matrix power constraint. Similar to the channel model in Fig. 5.5, secret DPC achieves the capacity region of the channel in Fig. 5.6, under a matrix power constraint. However, under the more practical total average power constraint, a computable secrecy capacity expression is not known for any of those channels, in general. Because of this and also due to the fact that the secret DPC is prohibitively complex for practical implementations, it is important to find simple solutions, e.g. based on linear precoding, that maximize the achievable secret rate and/or can achieve the secret capacity region in practice. 5 NOMA: An Information-Theoretic Perspective 189 1 W1 1 Rx1 H1 n1 W1, W2 H2 W2 1 W2 nt Rx2 G n2 W1 1 W2 Eve ne W1 Fig. 5.7 MIMO BC with confidential messages and an external eavesdropper 5.5.2 Physical Layer Security via Beamforming Developing practical physical layer security techniques that address the security issues of NOMA in the downlink is an important research topic. In particular, securing communication for the three basic channel models illustrated in Figs. 5.5, 5.6 and 5.7 and their extension to K -user cases leads to practically important open problems. Specifically, secure transmission strategies based on linear beamforming techniques are of interest. In the following, we summarize different types of beamforming that have been proposed for physical layer security. Beamforming is one of the most widely studied approaches to physical layer security for multi-antenna systems. Beamforming approaches can be categorized as follows: • Zero-forcing beamforming: One intuitive approach to secure transmission is to send the signal as orthogonal to the eavesdropper’s channel as possible. When the transmitter has a larger number of antennas than the eavesdropper, it is possible to transmit the information in the null-space of the eavesdropper’s channel. This approach, which is called zero-forcing beamforming, is a simple but suboptimal beamforming scheme for multi-user MIMO systems [37]. • GSVD-based beamforming: Generalized singular value decomposition (GSVD)based precoding has been widely used for security and confidentiality purposes in the MIMO wiretap and BC channels [38, 39] as well as in multicasting [40]. Simplicity is the main advantage of GSVD-based transmission as it decouples the MIMO channel between the transmitter and the receivers into several parallel subchannels which can be selected independently and then be encoded separately. However, even in the case of the MIMO wiretap channel, GSVD-based precoding is neither capacity achieving nor very close to capacity in general [41]. Nevertheless, 190 • • • • M. Vaezi and H. Vincent Poor this approach is useful in that it can usually result in reasonably high achievable rates with low complexity. GEVD-based transmission: Linear beamforming based on the generalized eigenvalue decomposition (GEVD) is known to be optimal for both the MISO wiretap channel [39] and MISO BC with confidential messages [42]. Despite being theoretically appealing for its optimality, in practice, there is no guarantee to have single-antenna eavesdroppers. It is worth noting that under a matrix power constraint, GEVD-based precoding also achieves the secrecy capacity of the MIMO wiretap channel [43]. Trigonometric approach: This is another simple linear beamforming approach which was recently introduced in the context of the MIMO wiretap channel in [41, 44] and shown to be optimal for any numbers of antennas at the eavesdropper and the legitimate receiver when the transmitter has two antennas. Convex-optimization-based precoding: Numerical solutions based on convex optimization have also been proposed to compute a transmit covariance matrix for the secrecy capacity maximization in the MISO and MIMO channels [45, 46]. These methods solve the underlying non-convex optimization problem in an iterative way and thus are very complex because the objectives are matrices. Artificial Noise (AN)-aided transmission: When the eavesdropper’s CSI is not available at the transmitter, an AN-aided transmission is useful for providing security at the physical layer. In this method, multiple antennas at the transmitter and legitimate receiver are used to inject artificial noise into directions orthogonal to those of the main channel [47]. Due to its simplicity and practicality against passive eavesdroppers, AN-based beamforming is widely used for secure transmissions [48, 49], particularly when the eavesdropper’s CSI is not known. In some cases, e.g. for the fading MISO wiretap channel with no eavesdropper’s CSI, it is shown that the optimal solution converges to transmitting AN in all null-space dimensions of the main channel [50]. 5.5.3 Research Directions It is of considerable interest to develop physical layer security for NOMA transmission with both in-network and external eavesdroppers. The former is specifically important in a NOMA-based transmission since a user with a stronger channel is able to decode a weaker user’s signal, at the physical layer, and compromise their privacy. Therefore, security with respect to external eavesdroppers and confidentiality (privacy) with respect to legitimate users are both critical in NOMA. The case with multiple-antenna transmitters should be the main focus of such a study, as those are now ubiquitous in many wireless networks, including cellular networks. In this case, space-time signal processing such as linear beamforming with/without artificial noise can be used to put physical layer security into practice and to improve secure transmission rates. The key is to use signal processing techniques to increase the signal strength difference between the legitimate user and eavesdropper. 5 NOMA: An Information-Theoretic Perspective 191 One set of problems in this setting is to identify linear beamforming techniques that maximize the secrecy rate region of the models shown in Figs. 5.5, 5.6 and 5.7 assuming the availability of the legitimate users’ instantaneous perfect CSI at the transmitter (CSIT). Another set of problems may consider the above security problems with more realistic assumptions, i.e. developing linear beamforming techniques that maximize physical layer security with imperfect CSIT. References 1. D. Tse, P. Viswanath, Fundamentals of Wireless Communication (Cambridge university press, 2005) 2. Y. Saito, Y. Kishiyama, A. Benjebbour, T. Nakamura, A. Li, K. 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Motahari, A.K. Khandani, Communication over MIMO X channels: interference alignment, decomposition, and performance analysis. IEEE Trans. Inf. Theory 54(8), 3457–3470 (2008) 27. V.R. Cadambe, S.A. Jafar, Interference alignment and degrees of freedom of the K -user interference channel. IEEE Trans. Inf. Theory 54(8), 3425–3441 (2008) 28. C.M. Yetis, T. Gou, S.A. Jafar, A.H. Kayran, On feasibility of interference alignment in MIMO interference networks. IEEE Trans. Signal Process. 58(9), 4771–4782 (2010) 29. T. Gou, S.A. Jafar, Degrees of freedom of the K user M × N MIMO interference channel. IEEE Trans. Inf. Theory 56(12), 6040–6057 (2010) 30. 3GPP TR 36.866 V12.0.1, Study on Network-Assisted Interference Cancellation and Suppression (NAIC) for LTE (Release 12), Mar 2014 31. Q. Li, G. Li, W. Lee, M.-i. Lee, D. Mazzarese, B. Clerckx, Z. Li, MIMO techniques in WiMAX and LTE: a feature overview. IEEE Commun. Mag. 48(5), (2010) 32. Y. Yuan, Z. Yuan, G. Yu, C.-H. Hwang, P.-K. Liao, A. 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Swindlehurst, Optimal power allocation for GSVD-based beamforming in the MIMO Gaussian wiretap channel, in Proceedings of the IEEE International Symposium on Information Theory (2012), pp. 2321–2325 39. A. Khisti, G.W. Wornell, Secure transmission with multiple antennas I: the MISOME wiretap channel. IEEE Trans. Inf. Theory 56(7), 3088–3104 (2010) 40. W. Mei, Z. Chen, J. Fang, GSVD-based precoding in MIMO systems with integrated services. IEEE Signal Process. Lett. 23(11), 1528–1532 (2016) 41. M. Vaezi, W. Shin, V. Poor, Optimal beamforming for Gaussian MIMO wiretap channels with two transmit antennas. IEEE Trans. Wirel. Commun. 16, 6726–6735 (2017) 42. R. Liu, H.V. Poor, Secrecy capacity region of a multiple-antenna Gaussian broadcast channel with confidential messages. IEEE Trans. Inf. Theory 55(3), 1235–1249 (2009) 43. R. Bustin, R. Liu, H. V. Poor, S. Shamai, An MMSE approach to the secrecy capacity of the MIMO Gaussian wiretap channel. EURASIP J. Wirel. Commun. Netw. (1) (2009) 44. M. Vaezi, W. Shin, H. V. Poor, J. Lee, MIMO Gaussian wiretap channels with two transmit antennas: optimal precoding and power allocation, in Proceedings of the IEEE International Symposium on Information Theory (2017), pp. 1708–1712 5 NOMA: An Information-Theoretic Perspective 193 45. Q. Li, W.-K. Ma, Optimal and robust transmit designs for MISO channel secrecy by semidefinite programming. IEEE Trans. Signal Process. 59(8), 3799–3812 (2011) 46. Q. Li, M. Hong, H.-T. Wai, Y.-F. Liu, W.-K. Ma, Z.-Q. Luo, Transmit solutions for MIMO wiretap channels using alternating optimization. IEEE J. Sel. Areas Commun. 31(9), 1714– 1727 (2013) 47. S. Goel, R. Negi, Guaranteeing secrecy using artificial noise. IEEE Trans. Wirel. Commun. 7(6), 2180–2189 (2008) 48. Z. Li, P. Mu, B. Wang, X. Hu, Optimal semiadaptive transmission with artificial-noise-aided beamforming in MISO wiretap channels. IEEE Trans. Veh. Technol. 65(9), 7021–7035 (2016) 49. T.-X. Zheng, H.-M. Wang, J. Yuan, D. Towsley, M.H. Lee, Multi-antenna transmission with artificial noise against randomly distributed eavesdroppers. IEEE Trans. Commun. 63(11), 4347–4362 (2015) 50. S. Gerbracht, C. Scheunert, E.A. Jorswieck, Secrecy outage in MISO systems with partial channel information. IEEE Trans. Inf. Forensics Secur. 7(2), 704–716 (2012) Chapter 6 Optimal Power Allocation for Downlink NOMA Systems Yongming Huang, Jiaheng Wang and Jianyue Zhu 6.1 Introduction With the popularity of smartphones and Internet of Things, there is an explosive demand for new services and data traffic for wireless communications. The capacity of the fourth-generation (4G) mobile communication system is insufficient to satisfy such a demand in the near future. The development of the fifth-generation (5G) mobile communication system has been placed on the agenda with higher requirements in data rates, latency, and connectivity [1]. In order to meet the new standards, some potential technologies, such as massive multiple-input–multiple-output (MIMO) [2], millimeter wave [3], and ultra densification [4, 5], will be introduced into 5G. Meanwhile, new multiple access technologies, which are flexible, reliable, and efficient in terms of energy and spectrum, are also considered for 5G communication. Conventionally, cellular systems have adopted orthogonal multiple access (OMA) approaches, in which wireless resources are allocated orthogonally to multiple users. The common OMA techniques include frequency-division multiple access (FDMA), time division multiple access (TDMA), code-division multiple access (CDMA), and orthogonal frequency-division multiple access (OFDMA). Ideally, in OMA, the intracell interference does not exist as result of dedicated resource allocation. Also, for this reason, the information of multiple users can be retrieved at a low complexity. Nonetheless, the number of served users is limited by the number of orthogonal resources, which is generally small in practice. Consequently, it is difficult for OMA systems to support a massive connectivity. Recently, non-orthogonal multiple access (NOMA) technologies are developed and proposed for 5G, which will contribute to disruptive design changes on radio access and alleviate the scarcity of suitable spectra. By using superposition coding Y. Huang (B) · J. Wang · J. Zhu Southeast University, Nanjing 210096, China e-mail: huangym@seu.edu.cn J. Wang e-mail: jhwang@seu.edu.cn J. Zhu e-mail: zhujy@seu.edu.cn © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_6 195 196 Y. Huang et al. at the transmitter with successive interference cancellation (SIC) at the receiver, downlink NOMA allows one (frequency, time, code, or spatial) channel to be shared by multiple users simultaneously [6, 7], thus leading to better performance in terms of spectral efficiency, fairness, or energy efficiency [8]. Therefore, NOMA has received much attention recently. Its combinations with MIMO and multi-cell technologies were studied in [9, 10] and [11], respectively. NOMA was also considered to be used in, e.g., visible light communication [12] and millimeter wave communication [13]. The principle of NOMA is to implement multiple access in the power domain [14]. Hence, allocation is critical for NOMA systems. In the literature, there are a number of works on power allocation for NOMA. In particular [15, 16] focused on power allocation in a two-user NOMA system and [17–21] investigated power allocation for multiple users (more than two) sharing one channel, which is referred as multi-user NOMA (MU-NOMA). There were also some works, e.g., [12, 22–28], studying the resource allocation problems in multi-channel NOMA (MC-NONA) systems, where multiple channels are available for NOMA. Different criteria, such as maximin fairness [18–20, 22], sum rate [15–17, 22, 28–30], and energy efficiency [21–23, 26], were considered. This chapter focuses on power allocation for downlink NOMA. We first briefly review the basic concepts of downlink NOMA transmission and introduce the twouser NOMA, MU-NOMA, and MC-NOMA schemes. Then, we investigate the optimal power allocation strategies for these NOMA schemes under different performance criteria such as the maximin fairness, sum rate, and energy efficiency along with user weights and quality-of-service (QoS) constraints. We show that the optimal NOMA power allocation can be analytically characterized in most cases, otherwise it can be numerically computed via convex optimization methods. This chapter is organized as follows. Section 6.2 introduces the fundamentals of downlink NOMA and the two-user NOMA, MU-NOMA, and MC-NOMA schemes. In Sects. 6.3–6.5, we investigate the optimal power allocation for two-user NOMA, MU-NOMA, and MC-NOMA schemes, respectively. The performance of the NOMA power allocation strategies is evaluated in Sect. 6.6 via simulations, and the conclusion is drawn in Sect. 6.7. 6.2 Fundamentals of Downlink NOMA In this section, we review the basic concepts of downlink NOMA transmission in a single-cell network.1 To begin with, we start from the simplest two-user case, where a base station (BS) serves two users, namely UE1 and UE2 , on the same frequency band with bandwidth B. The BS transmits a signal sn for user n (UEn , n = 1, 2) with transmission power pn . The total power budget of the BS is P, i.e., p1 + p2 ≤ P. Such a simple downlink NOMA system is displayed in Fig. 6.1 [31]. 1 For multi-cell NOMA, the reader is referred to [11]. 6 Optimal Power Allocation for Downlink NOMA Systems 197 Fig. 6.1 A downlink NOMA system with one base station and two users Superposition Coding: According to the NOMA principle, the BS exploits the superposition coding and broadcasts the signal x= √ p 1 s1 + √ (6.1) p 2 s2 to both users. The received signal at UEn is yn = h n −ρ √ p 1 s1 + √  p 2 s2 + z n , (6.2) where h n = gn dn is the channel coefficient from the BS to UEn , gn follows a Rayleigh distribution, dn is the distance between the BS and UEn , ρ is the path loss exponent, and z n is the additive white Gaussian noise with zero mean and variance  σn2 , i.e., z n ∼ C N 0, σn2 . Successive Interference Cancellation (SIC): In NOMA systems, each user exploits SIC at its receiver. Let Γn = |h n |2 /σn2 be the channel-to-noise ratio (CNR) of UEn . Assume without loss of generality (w.l.o.g.) that the users are ordered by their normalized channel gains as Γ1 ≥ Γ2 , i.e., UE1 and UE2 are regarded as the strong and weak users, respectively. It is expected that more power is allocated to the weak user UE2 and less power is allocated to the strong user UE1 , i.e., p1 ≤ p2 [14, 25]. Then, UE1 first decodes the message of UE2 and removes it from its received signal, while UE2 treats the signal of UE1 as interference and decodes its own message. Achievable Rate: Suppose that the channel coding is ideal and UE1 is able to decode the message of UE2 successfully. Then, the achievable rates of UE1 and UE2 are given respectively by  R1 = B log (1 + p1 Γ1 ) , R2 = B log 1 + p 2 Γ2 1 + p 1 Γ2  (6.3) 198 Y. Huang et al. which are often used as the design objectives of NOMA systems. Multi-User NOMA (MU-NOMA): Consider a more general case where a BS serves N ≥ 2 users on the same spectrum, which are indexed by n = 1, . . ., N. The broadcast signal by the BS is then given by x= N  √ pi si (6.4) i=1 and then the received signal at each UEn is given by yn = h n N  √ i=1 pi si + z n . (6.5) Similarly, suppose that the users are ordered by their normalized channel gains as Γ1 ≥ Γ 2 ≥ · · · ≥ Γ N (6.6) and the NOMA protocol allocates higher powers to the users with lower CNRs, leading to p1 ≤ p2 ≤ · · · ≤ p N . Therefore, UEn is able to decode the message of UEl for l > n and remove it from the received signal so that UEn is only interfered by UE j for j < n. Therefore, after SIC, the achievable rate of UEn is  p n Γn Rn = log 1 + n−1 j=1 p j Γn + 1 (6.7) for n = 1, . . . , N . Multi-Channel NOMA (MC-NOMA): The frequency band shared by the users could be viewed as a channel, which may also be a time slot, spread code, or resource block. In cellular systems, there are often multiple channels available, which leads to a more general NOMA scheme called multi-channel NOMA (MC-NOMA), where multiple users share multiple channels. Specifically, in a downlink MC-NOMA network, the BS serves N users through M channels and the total bandwidth B is equally divided to M channels so the bandwidth of each channel is Bc = B/M. Let Nm ∈ {N1 , N2 , ..., N M } be the number of users using channel m for m = 1, 2, . . ., M and UEn,m denotes user n on channel m for n = 1, 2, . . . , Nm . The signal transmitted by the BS on each channel m can be expressed as xm = Nm  √ pn,m sn , (6.8) n=1 where sn is the symbol of UEn,m and pn,m is the power allocated to UEn,m . The received signal at UEn,m is 6 Optimal Power Allocation for Downlink NOMA Systems yn,m = Nm  √ i=1 199 pi,m h n,m si + z n,m . (6.9) It is easily seen that on each channel m is an MU-NOMA scheme. Similarly, assume w.l.o.g. that the CNRs of the users on channel m are ordered as Γ1,m ≥ · · · ≥ Γn,m ≥ · · · ≥ Γ Nm ,m , (6.10) which will lead to p1,m ≤ · · · ≤ pn,m ≤ · · · ≤ p Nm ,m . Then, the achievable rate of UEn,m using SIC is  Rn,m = Bc log 1 + pn,m Γn,m n−1 1 + i=1 pi,m Γn,m . (6.11) The basic idea of NOMA is to implement multiple access in the power domain [14]. Hence, power allocation is the key to achieve the full benefit of NOMA transmission. In the following parts, we will investigate the optimal power allocation strategies for different NOMA schemes, including the simplest two-user case, the MU-NOMA scheme, and the MC-NOMA scheme, under different performance measures. 6.3 Two-User NOMA In this section, we investigate the optimal power allocation for the two-user NOMA scheme. Although the two-user scheme is the simplest case of NOMA, the results and insights obtained in this case will serve the more complicated MU-NOMA and MC-NOMA schemes. 6.3.1 Optimal Power Allocation for MMF The NOMA scheme enables a flexible management of users’ achievable rates and provides an efficient way to enhance user fairness. A widely used fairness metric is the maximin fairness (MMF), which is achieved by maximizing the worst (i.e., minimum) user rate. According to (6.3), the power allocation to achieve the MMF is given by the solution to the following optimization problem: T U MMF : max min {R1 ( p1 , p2 ), R2 ( p1 , p2 )} p1 , p2 s.t. 0 ≤ p1 ≤ p2 , p1 + p2 ≤ P This problem admits a closed-form solution as follows. 200 Y. Huang et al. Proposition 1 Suppose that Γ1 ≥ Γ2 . Then, the optimal solution to T U MMF is given by p1⋆ = Λ and p2⋆ = P − p1⋆ , where Γl  |h l |2 /σl2 and Λ − (Γ1 + Γ2 ) + (Γ1 + Γ2 ) 2 + 4Γ1 Γ22 P 2Γ1 Γ2 . (6.12) Proof Please refer to the proof of Proposition 1 in [22]. Remark 1 It can be verified that at the optimal point R1 ( p1⋆ , p2⋆ ) = R2 ( p1⋆ , p2⋆ ), i.e., UE1 and UE2 achieve the same rate. This indicates that, under the MMF criterion, the NOMA system will provide absolute fairness for two users on one channel. To elaborate another important insight, we introduce the following definition. Definition 1 A NOMA system is called SIC stable if the optimal power allocation satisfies p1 < p2 on one channel. Remark 2 In NOMA systems, SIC is performed according to the order of the CNRs of the users on one channel [14, 25], which is guaranteed by imposing an inverse order of the powers allocated to the users, i.e., p1 ≤ p2 . Specifically, UE1 (the stronger user) first decodes the signal of UE2 (the weaker user) and then subtracts it from the superposed signal. Therefore, from the SIC perspective, a large difference between the signal strengths of UE2 and UE1 is preferred [32]. However, even with the power order constraint p1 ≤ p2 , the power optimization may lead to p1 = p2 ; i.e., UE1 and UE2 have the same signal strength, which is the worst situation for SIC. In this case, SIC may fail or has a large error propagation and thus is unstable. Indeed, the authors in [33] pointed out that the power of the weak user must be strictly larger than that of the strong user, otherwise the users’ outage probabilities will always be one. Definition 1 explicitly concretizes such a practical requirement in NOMA systems. Lemma 1 The NOMA system is SIC stable for T U MMF . Proof Please refer to the proof of Lemma 1 in [22]. Indicated by Lemma 1, the two-user NOMA system is always SIC stable under the MMF criterion, as in this case the optimal power allocation always satisfies p1⋆ < p2⋆ . On the other hand, in the subsequent subsections, we will show that a NOMA system may not always be SIC stable under different criteria. 6.3.2 Optimal Power Allocation for SR Maximization In this subsection, we seek the optimal power allocation for maximizing the sum rate (SR). In SR maximization, to take user priority or fairness into account, user weights or quality-of-service (QoS) constraints are often adopted. 6 Optimal Power Allocation for Downlink NOMA Systems 6.3.2.1 201 Weighted SR Maximization (SR1) According to (6.3), the problem of maximizing the weighted SR (WSR) is given by T U SR1 : max W1 R1 ( p1 , p2 ) + W2 R2 ( p1 , p2 ) p1 , p2 s.t. 0 ≤ p1 ≤ p2 , p1 + p2 ≤ P where Wi denotes the weight of UEi for i = 1, 2. Note that T U SR1 is a nonconvex problem due to the interference between UE1 and UE2 . Nevertheless, its optimal solution can be found as follows. Proposition 2 Suppose that Γ1 ≥ Γ2 , W1 < W2 and P > 2Ω, with Ω W 2 Γ2 − W 1 Γ1 . Γ1 Γ2 (W1 − W2 ) (6.13) Then, the optimal solution to T U SR1 is given by p1⋆ = Ω and p2⋆ = P − p1⋆ . Proof Please refer to the proof of Proposition 2 in [22]. Remark 3 In Proposition 2, the conditions W1 < W2 and P > 2Ω are both to avoid a failure of SIC. Indeed, if W1 ≥ W2 , the solution to T U SR1 is p1⋆ = p2⋆ = P/2 ; i.e., the NOMA system is unstable according to Definition 1. SIC may also fail if P ≤ 2Ω, which will lead to p1⋆ = p2⋆ = P/2 as well. Therefore, the two-user NOMA system is SIC stable for the WSR maximization if and only if W1 < W2 and P > 2Ω. 6.3.2.2 SR Maximization with QoS (SR2) Now, we consider maximizing the SR with QoS constraints. In this case, the power allocation problem is given by max R1 ( p1 , p2 ) + R2 ( p1 , p2 ) p1 , p2 T U SR2 : s.t. 0 ≤ p1 ≤ p2 , p1 + p2 ≤ P Ri ≥ Rimin , i = 1, 2. where Rimin is the QoS threshold of UEi . The optimal solution to T U SR2 is provided in the following result. Proposition 3 Suppose that Γ1 ≥ Γ2 , A2 ≥ 2, and P ≥ Υ , with min Al = 2 Rl , Υ  A2 − 1 A2 (A1 − 1) + , Γ1 Γ2  Then, the optimal solution to T U SR2 is given by p1⋆ = Γ2 P − A 2 + 1 . A 2 Γ2 and p2⋆ = P − p1⋆ . (6.14) 202 Y. Huang et al. Proof Please refer to the proof of Proposition 3 in [22]. Remark 4 Similarly, in Proposition 3, the conditions A2 ≥ 2 and P ≥ Υ are to guarantee the SIC stability. Indeed, if A2 < 2, then > P/2 and the optimal solution will be p1⋆ = p2⋆ = P/2, which may lead a failure of SIC. At the same time, SIC may also fail if P < Υ , which will lead to p1⋆ = p2⋆ = P/2 as well. Therefore, the NOMA system is SIC stable in this case if and only if A2 ≥ 2 and P ≥ Υ . According to Proposition 3, if the NOMA system is SIC stable, the optimal solution will be p1⋆ = and p2⋆ = P − p1⋆ . Hence, we have R2 ( p1⋆ , p2⋆ ) = R2min , implying that the user with a lower CNR (i.e., UE2 ) receives the power to meet its QoS requirement exactly, while the remaining power is used to maximize the rate of the user with a higher CNR (i.e., UE1 ). 6.3.3 Optimal Power Allocation for EE Maximization In this subsection, we investigate the optimal power allocation for maximizing the energy efficiency (EE), which is defined as the ratio between the rate and the consumed power. Similarly, user weights and QoS constraints are considered. 6.3.3.1 Weighted EE Maximization (EE1) The problem of maximizing the weighted EE is formulated as follows: T UaEE1 : max η = W1 R1 ( p1 , p2 )+W2 R2 ( p1 , p2 ) p1 , p2 PT + p1 + p2 s.t.0 ≤ p1 ≤ p2 , p1 + p2 ≤ P where PT is the power consumption of the circuits. Given the fraction form of the objective, T UaEE1 is more complicated than T U SR1 . In the following, we show that this problem can also be optimally solved. We introduce an auxiliary variable q with p1 + p2 = q. Then, T UaEE1 can be equivalently written into T UbEE1 : max η = W1 R1 ( p1 )+W2 R2 ( p1 ,q) p1 ,q PT +q s.t. q ≥ 2 p1 , q ≤ P where the rate of UE2 can be expressed as R2 ( p1 , q) = B log  1 + qΓ2 1 + p 1 Γ2  . (6.15) 6 Optimal Power Allocation for Downlink NOMA Systems 203 To deal with the fractional form, let us introduce the following objective function: H ( p1 , q, α)  W1 R1 ( p1 ) + W2 R2 ( p1 , q) − α (PT + q) , (6.16) where α is a positive parameter. Then, we consider the following problem for given α: max H ( p1 , q, α) T UcEE1 : p1 ,q s.t. q ≥ 2 p1 , q ≤ P. The relation between T UcEE1 and T UbEE1 is stated in the following result. Lemma 2 ([34, pp. 493–494]) Let H ⋆ (α) be the optimal objective value of T UcEE1 and p⋆ (α) be the optimal solution of T UcEE1 . Then, p⋆ (α) is the optimal solution to T UbEE1 if and only if H ⋆ (α) = 0. According to Lemma 2, the optimal solution to T UbEE1 can be found by solving T UcEE1 parameterized by α such that H ⋆ (α) = 0. Since H ⋆ (α) is monotonic in α, one can use any line search method, e.g., the bisection method, to find α such that H ⋆ (α) = 0. Then, the left question is how to solve T UcEE1 with given α. Theorem 1 Suppose that Γ1 ≥ Γ2 . Then, T UcEE1 is a convex problem if one of the following conditions hold C1 : W1 ≥ W2 ; (Γ1 +Γ1 Γ2 P)2 2 C2 : 1 < W . ≤ (Γ W1 +Γ Γ P)2 2 1 2 Proof See Appendix A. Theorem 1 reveals that T UcEE1 is in fact a convex problem if condition C1 or C2 holds. Consequently, T UcEE1 can be efficiently solved via convex optimization methods. The optimal solution to T UcEE1 can further be analytically characterized. Proposition 4 Suppose that Γ1 ≥ Γ2 , C2 holds, and P ≥ 2Ω with Ω= W 2 Γ2 − W 1 Γ1 . Γ1 Γ2 (W1 − W2 ) (6.17) Then, the optimal solution to T UcEE1 is p1⋆ = Ω and q⋆ = 1 W2 B − α ln 2 Γ2 Proof See Appendix B. P 2Ω = max 2Ω, min 1 W2 B − ,P α ln 2 Γ2  . (6.18) 204 Y. Huang et al. Remark 5 Similarly, in Proposition 4, the conditions C2 and P > 2Ω are to avoid a failure of SIC. Although T UcEE1 is convex under C1, condition C1 will lead to p1⋆ = p2⋆ , which, according to Definition 1, is SIC-unstable. 6.3.3.2 EE Maximization with QoS Constraints (EE2) Then, we consider maximizing the EE with QoS constraints. In this case, the power allocation problem is given by max p1 , p2 R1 ( p1 , p2 )+R2 ( p1 , p2 ) PT + p1 + p2 T UaEE2 : s.t. 0 ≤ p1 ≤ p2 , p1 + p2 ≤ P Ri ≥ Rimin , i = 1, 2. This problem can be optimally solved following the similar steps as in the previous subsection. Specifically, using p1 + p2 = q, T UaEE2 can be equivalently transformed into max p1 ,q R1 ( p1 )+R2 ( p1 ,q) PT +q T UbEE2 : s.t. q ≥ 2 p1 , q ≤ P Ri ≥ Rimin , i = 1, 2. Then, we consider the following problem with given α: T UcEE2 max Q ( p1 , q, α) p1 ,q : s.t. q ≥ 2 p1 , q ≤ P Ri ≥ Rimin , i = 1, 2 where Q ( p1 , q, α)  R1 ( p1 ) + R2 ( p1 , q) − α (PT + q) . (6.19) According to Lemma 2, the optimal solution to T UbEE2 can be found by solving T UcEE2 for a given α and updating α until the optimal objective value of T UcEE2 , denoted by Q ⋆ (α), satisfies Q ⋆ (α) = 0. From Theorem 1, under condition C1, the objective of T UcEE2 is concave and T UcEE2 is a convex problem too. Therefore, T UcEE2 can be efficiently solved. In fact, the optimal solution to T UcEE2 can also be analytically characterized. Proposition 5 Suppose that Γ1 ≥ Γ2 , A2 ≥ 2, and P ≥ Υ , with Al = 2 Rlmin B , Υ  A2 − 1 A2 (A1 − 1) + . Γ1 Γ2 (6.20) 6 Optimal Power Allocation for Downlink NOMA Systems 205 Then, the optimal solution to T UcEE2 is 1 + q ⋆ Γ2 − A 2 , A 2 Γ2 A2 A2 − 1 1 − + q⋆ = α Γ1 Γ2 p1⋆ = (6.21) P = max Υ, min Υ A2 A2 − 1 1 − + ,P α Γ1 Γ2  . (6.22) Proof See Appendix C. Similarly, it can be verified that the power allocation obtained from T UcEE2 (or T UaEE2 ) is SIC stable if and only if P ≥ Υ and A2 ≥ 2. 6.4 MU-NOMA In this section, we consider the more general MU-NOMA scheme, where a BS serves N ≥ 2 users on the same channel. Similarly, the optimal MU-NOMA power allocation is investigated under the MMF, SR, and EE criteria with user weights or QoS constraints. 6.4.1 Optimal Power Allocation for MMF According to (6.7), the power allocation problem under the MMF criterion is formulated as min {Ri } max i=1,...,N MMF p N : MUa  s.t. 0 < p1 < · · · < p N , pi ≤ P i=1 N where p = { pi }i=1 denotes the powers allocated to the users. It has been shown in [20] that, though nonconvex, MUaMMF is a quasi-convex problem. Thus, the optimal solution to MUaMMF can be found by solving a sequence of convex problems. Specifically, MUaMMF is equivalent to max t p,t MUbMMF N : s.t. 0 < p1 < . . . < p N ,  pi ≤ P i=1 Ri ≥ t, i = 1, . . . , N . The constraint Ri ≥ t can be rewritten into 206 Y. Huang et al. ⎞ ⎛ i−1 2 t − 1 ⎝ p j Γi + 1⎠ . pi ≥ Γi j=1 (6.23) Hence, for fixed t, MUbMMF is a linear program (LP) and can be efficiently solved by a number of LP solvers. Then, one can exploit the bisection method to search the optimal t. Note that the optimal solution to MUbMMF for fixed t can be analytically characterized if there is no power order constraint. Proposition 6 In the absence of power order constraint, the solution to MUbMMF is given by ⎞ ⎛ i−1 2 t − 1 ⎝ pi = (6.24) p j Γi + 1⎠ , i = 1, · · · , N . Γi j=1 Proof Please refer to the proof of Theorem 1 in [20]. The solution in (6.24) implies that all users achieve the same data rate equal to t. Hence, in this case, the NOMA system will provide absolute fairness for all users. Note that, however, the solution in (6.24) is obtained without the power order constraint. One may wonder if this solution is still optimal if the power order constraint is not omitted. The following result provides a sufficient condition to characterize the optimality of (6.24). Theorem 2 The solution in (6.24) is optimal for MUbMMF if P ≥ χ , where χ =  N 2 N −i i=1 Γi . Proof See Appendix E. Theorem 2 indicates that the power order constraints can be omitted under some conditions. In this case, the solution in (6.24) is indeed optimal for MUbMMF . On the other hand, it is unknown if the solution in (6.24) is optimal if the condition in Theorem 2 is not satisfied. Nevertheless, in this case, one can always numerically solve the linear problem MUbMMF for fixed t. 6.4.2 Optimal Power Allocation for SR Maximization In this subsection, we investigate the SR maximization problems in MU-NOMA systems with user weights or QoS constraints. 6.4.2.1 Weighted SR Maximization (SR1) The weighted SR maximization for MU-NOMA is formulated as 6 Optimal Power Allocation for Downlink NOMA Systems max Rsum = p MUaSR1 : s.t. N  i=1 N  207 Wi Ri i=1 pi ≤ P p1 ≤ p2 ≤ · · · ≤ p N . Unlike MUaMMF for MMF, MUaSR1 in its original form is neither a convex nor quasiconvex problem, making it difficult to solve it. Nevertheless, we show that MUaSR1 can be transformed into a convex problem via a linear transformation of the optimization variables.  Introduce the following variable transformation: qi = ij=1 p j for i = 1, 2, . . ., N; and conversely pi = qi − qi−1 for i = 2, . . ., N and p1 = q1 . In this way, we have R1 = log (q1 Γ1 + 1) and ⎛ ⎞   i   qi Γi + 1 j=1 p j Γi + 1 ⎠ = log (qi Γi + 1) − log qi−1 Γi + 1 Ri = log ⎝ i−1 = log qi−1 Γi + 1 j=1 p j Γi + 1 (6.25) for i = 2, . . ., N. Therefore, the weighted sum rate can be expressed as N  i=1 Wi Ri = W1 log (q1 Γ1 + 1) + N  i=2 N     Wi log (qi Γi + 1) − log qi−1 Γi + 1 = f i (qi ) , i=1 (6.26) where f i (qi ) = Wi log (qi Γi + 1) − Wi+1 log (qi Γi+1 + 1) (6.27) for  N i = 1, . . ., N − 1 and f N (q N ) = W N log (q N Γ N + 1). The power constraint i=1 pi ≤ P is equal to q N ≤ P. The power order constraint p1 ≤ p2 ≤ · · · ≤ p N is equal to q1 ≤ q2 − q1 ≤ · · · ≤ q N − q N −1 . Consequently, problem MUaSR1 can be equivalently transformed into the following problem: max N i=1 f i (qi ) MUbSR1 : s.t. q N ≤ P 0 ≤ q1 ≤ q2 − q1 ≤ · · · ≤ q N − q N −1 q Then, the following result identifies the convexity of MUaSR1 (or MUbSR1 ). Theorem 3 MUaSR1 (or MUbSR1 ) is a convex problem if one of the following conditions hold for i = 1, . . ., N − 1: T1 : Wi ≥ Wi+1 ; 2 i Γi+1 P) . ≤ (Γ(Γi +Γ T2 : 1 < WWi+1 +Γ Γ P)2 i i+1 i i+1 Proof Please refer to the proof of Theorem 1 in [17]. Remark 6 Theorem 3 indicates that MUaSR1 (or MUbSR1 ) is a convex problem under some conditions of the user weights. From T1, if the user weights are in the same 208 Y. Huang et al. order as the channel gains, i.e., W1 ≥ W2 ≥ · · · ≥ W N , then the objective function is concave and the problem is convex. Note that this situation includes the most common sum rate as a special case. On the other hand, the user weights can also be in the inverse order of the channel gains, i.e., W1 ≤ W2 ≤ · · · ≤ W N , but in this case the ratio between Wk+1 and Wk cannot be too large according to T2. Consequently, one can find the optimal power allocation via standard convex optimization methods, e.g., the interior point method. 6.4.2.2 SR Maximization with QoS (SR2) Then, we consider the SR maximization problem with QoS constraints for MUNOMA, which is given by max p MUaSR2 : s.t. N  i=1 N  i=1 Ri pi ≤ P, p1 ≤ p2 ≤ · · · ≤ p N Ri ≥ Rimin , i = 1, . . . , N Similarly, although MUaSR2 is nonconvex in its original formulation, it can be transformed into a convex problem.  In particular, we exploit the same variable transformation: qi = ij=1 p j for i = 1, 2, . . ., N. Then, MUaSR2 is transformed into max q N k=1 gi (qi ) MUbSR2 : s.t. q N ≤ P (a1 − 1) /Γ1 ≤ q1 ≤ q2 − q1 ≤ · · · ≤ q N − q N −1 qi−1 ≤ ai qi − εi , i = 2, . . . , N where gi (qi ) = log (qi Γi + 1) − log (qi Γi+1 + 1) (6.28) for i = 1, . . ., N − 1 and g N (q N ) = log (q N Γ N + 1), ai = 2−Ri and εi = (1 − ai ) /Γi . According to condition T1 in Theorem 3, the objective in MUbSR2 is concave and thus MUbSR2 is a convex problem. Therefore, MUbSR2 can also be efficiently solved via convex optimization methods. Moreover, we show that if the power order constraint p1 ≤ p2 ≤ · · · ≤ p N is absent, the optimal solution to MUbSR2 can be analytically characterized. N ϕi , where ςi = 2 Ri − 1/Γi , Proposition 7 Suppose that P ≥ i=1 ϕi = ⎧ ⎪ ⎨ς1 ,   ⎪ ⎩max ϕi−1 , ςi 1 + Γi i−1  j=1 ϕj  i =1 , i = 2, . . . , N (6.29) 6 Optimal Power Allocation for Downlink NOMA Systems 209 and the power order constraint is absent in MUaSR2 and MUbSR2 . Then, the solution to MUbSR2 is  ai+1 q̃i+1 − εi+1 , k = 1, . . . , N − 1 q˜i = (6.30) P, k=N and the solution to problem MUaSR2 is  q˜1 , k=1 p̃i = (1 − ai ) q̃i + εi , k = 2, . . . , N . (6.31) Proof Please refer to the proof of Proposition 3 and Lemma 2 in [17]. Then, a natural question is when the solution in Proposition 7 is indeed optimal with the power order constraint. The answer is given below. Theorem 4 The solution in (6.31) is optimal for problem MUaSR2 with the power order constraint if T3: R2min ≥ 1 and   min Rimin ≥ log 2 − 2−Ri+1 , i = 3, . . . , N . (6.32) Proof Please refer to the proof of Theorem 2 in [17]. Corollary 1 Condition T3 in Theorem 4 holds if Rimin ≥ 1 for i = 2, . . ., N. Theorem 4 indicates that the power order constraint can be omitted without loss of optimality if the QoS thresholds of the last N − 1 users are not small. Corollary 1 specifies that the QoS thresholds are only required to be no less than 1bps/Hz, which is usually satisfied in practice. Therefore, for the SR maximization with QoS constraints, the optimal power allocation is given by Proposition 7 in practical MUNOMA systems. 6.4.3 Optimal Power Allocation for EE Maximization In this subsection, we investigate the EE maximization for MU-NOMA systems. 6.4.3.1 Weighted EE Maximization (EE1) The EE maximization with user weights in an MU-NOMA system is formulated as max η = p MUaEE1 : s.t. N i=1 Wi Ri N pi PT + i=1 N  i=1 pi ≤ P p1 ≤ p2 ≤ · · · ≤ p N . 210 Y. Huang et al. To address this problem, we follow the similar steps for MUaSR1 to simplify MUaEE1 .  Specifically, using the variable transformation: qi = ij=1 p j for i = 1, 2, . . ., N, MUaEE1 can be reformulated as MUbEE1 : max η = q N k=1 f i (qi ) PT +q N qN ≤ P 0 ≤ q1 ≤ q2 − q1 ≤ · · · ≤ q N − q N −1 s.t. where f i (qi ) is defined in (6.27). Then, we introduce the following objective function: H (q, α)  N  i=1  f i (qi ) − α PT + N  pi (6.33) i=1 and consider the following problem parameterized by α: max H (q, α) MUcEE1 q : s.t. q N ≤ P 0 ≤ q1 ≤ q2 − q1 ≤ · · · ≤ q N − q N −1 . According to Lemma 2, the optimal solution to MUbEE1 can be found by solving MUcEE1 with α chosen such that H ⋆ (α) = 0, where H ⋆ (α) is the optimal objective value of MUcEE1 . The desirable α can be found via a line search method by exploring the monotonicity of H ⋆ (α). To solve MUcEE1 , we provide the following result. Theorem 5 Given Γ1 ≥ Γ2 ≥ · · · ≥ Γ N , MUcEE1 is a convex problem if T1 or T2 in Theorem 3 holds for i = 1, . . ., N − 1. Proof Please refer to the proof of Theorem 1 in [17]. Theorem 5 indicates that, under the same condition in Theorem 3, MUcEE1 is a convex problem. Therefore, one can efficiently compute its optimal solution via optimization tools, e.g., the interior method. 6.4.3.2 EE Maximization with QoS Constraints (EE2) Then, we focus on maximizing EE with QoS constraints for MU-NOMA and the corresponding optimization problem is given by max η = p MUaEE2 : s.t. N  i=1 N i=1 Ri N PT + i=1 pi pi ≤ P p1 ≤ p2 ≤ · · · ≤ p N Ri ≥ Rimin , i = 1, . . . , N . 6 Optimal Power Allocation for Downlink NOMA Systems By using the same variable transformation: qi = MUaEE2 can be transformed into max η = q 211 i j=1 p j for i = 1, 2, . . ., N, N k=1 gi (qi ) PT +q N MUbEE2 : s.t. q N ≤ P (a1 − 1) /Γ1 ≤ q1 ≤ q2 − q1 ≤ · · · ≤ q N − q N −1 qi−1 ≤ ai qi − εi , i = 2, . . . , N where gi (qi ) is given in (6.28). Similarly, we introduce the following objective function  N N   (6.34) pi , gi (qi ) − α PT + Q (q, α)  i=1 k=1 and consider the problem parameterized by α: max Q (q, α) q MUcEE2 : s.t. q N ≤ P (a1 − 1) /Γ1 ≤ q1 ≤ q2 − q1 ≤ · · · ≤ q N − q N −1 qi−1 ≤ ai qi − εi , i = 2, . . . , N . Similarly, to obtain the optimal solution to MUbEE2 , one can solve MUcEE2 for given α and search α such that the optimal objective value of MUcEE2 satisfies Q ⋆ (α) = 0, for which we refer the reader to the previous subsection. To solve MUcEE2 , we provide the following result. N ϕi , where ςi = 2 Ri − 1/Γi and Proposition 8 Suppose that P ≥ i=1 ϕk = ⎧ ⎪ ⎨ς1 ,   ⎪ ⎩max ϕi−1 , ςi 1 + Γi i−1  j=1 ϕj  i =1 , i = 2, . . . , N , (6.35) then MUcEE2 is feasible and convex. Proof Please refer to the proof of Theorem 1 and Proposition 3 in [17]. Proposition 8 indicates that if the power budget of BS is not too small, MUcEE2 can be solved by convex optimization methods, e.g., the interior point method. 6.5 MC-NOMA In this section, we consider the MC-NOMA scheme, where multiple users share multiple channels. In this case, the resource optimization includes power allocation and channel assignment. However, the joint optimization results in a mixed 212 Y. Huang et al. integer problem and finding its solution requires exhaustive search [35], which causes prohibitive computational complexity. Therefore, in practice, power allocation and channel assignment are often separately and alternatively optimized [26, 28, 35]. In this section, we focus on seeking the optimal power allocation for given channel assignment. Note that using SIC at each user’s receiver causes additional complexity, which is proportional to the number of users on the same channel. Thus, in the multi-channel case, each channel is often restricted to be shared by two users [25, 26, 36], which is also beneficial to reduce the error propagation of SIC. In this section, we would also like to focus on this typical situation. In this case, suppose w.l.o.g. that the CNRs of UE1,m and UE2,m are ordered as Γ1,m ≥ Γ2,m . Then, the rates of UE1,m and UE2,m on channel m are given, respectively, by    R1,m = Bc log 1 + p1,m Γ1,m , R2,m = Bc log 1 +  p2,m Γ2,m . p1,m Γ2,m + 1 (6.36) 6.5.1 Optimal Power Allocation for MMF In MC-NOMA systems, the power allocation problem under the MMF criterion is given by MCaMMF : max p1 ,p2 min m=1,...,M  R1,m ( p1,m , p2,m ), R2,m ( p1,m , p2,m ) s.t. 0 ≤ p1,m ≤ p2,m , m = 1, . . . , M, M  m=1  p1,m + p2,m ≤ P M M and p2 = { p2,m }m=1 . Note that MCaMMF is a nonconvex probwhere p1 = { p1,m }m=1 M , where qm lem. To address it, we first introduce auxiliary variables q = {qm }m=1 represents the power budget for channel m with p1,m + p2,m = qm . Suppose that M the channel power budgets {qm }m=1 are given. Then, MCaMMF is decomposed into a group of subproblems and each subproblem is same with T U MMF in the two-user case with P replaced by qm . With given channel power budget qm , the optimal power allocation for the two users on channel m has been provided in Proposition 1. We can use this result to further optimize the power budgets {qm }. According to MCaMMF and T U MMF , the corresponding power budget optimization problem is max MCbMMF : q s.t. min m=1,...,M M  m=1 f mMMF⋆ (qm ) qm ≤ P, q ≥ 0 where f mMMF⋆ (qm ) is the optimal objective value of T U MMF for each channel m. Using Proposition 1, we obtain 6 Optimal Power Allocation for Downlink NOMA Systems ⎛ f mMMF⋆  Bc log ⎝ Γ2,m − Γ1,m + 213 ⎞   2 Γ1,m + Γ2,m 2 + 4Γ1,m Γ2,m qm ⎠. 2Γ2,m (6.37) Then, we show that MCbMMF has a closed-form solution. Theorem 6 The optimal solution to MCbMMF is given by qm⋆  Z (λ) Γ2,m + Γ1,m (Z (λ) − 1) , ∀m, = Γ1,m Γ2,m  (6.38) where Z (λ)  X +  X2 + Bc 2λ M m=1 and λ is chosen such that M m=1 1/Γ1,m , X M m=1 qm⋆ = P.     Γ2,m − Γ1,m / Γ1,m Γ2,m , M 4 m=1 1/Γ1,m (6.39) Proof Please refer to the proof of Theorem 1 in [22]. Consequently, the optimal MC-NOMA power allocation under the MMF criterion is fully characterized by Theorem 6 and Proposition 1. It follows from (6.38) that qm⋆ M is monotonically decreasing in λ, so the optimal λ satisfying m=1 qm⋆ = P can be efficiently found via a simple bisection method. 6.5.2 Optimal Power Allocation for SR Maximization In this subsection, we investigate the SR maximization problem with weights or QoS constraints in MC-NOMA systems. 6.5.2.1 Weighted SR Maximization (SR1) With given channel assignment, the problem of maximizing the weighted sum rate is formulated as the following power allocation problem: max MCaSR1 : M   p1 ,p2 m=1 W1,m R1,m ( p1,m , p2,m ) + W2,m R2,m ( p1,m , p2,m ) s.t. 0 ≤ p1,m ≤ p2,m , m = 1, . . . , M, M   m=1   p1,m + p2,m ≤ P M that represent the To solve it, similarly we introduce auxiliary variables q = {qm }m=1 power budgets on each channel m with p1,m + p2,m = qm . Then, MCaSR1 is decom- 214 Y. Huang et al. posed into a group of subproblems, where each subproblem is the same with T U SR1 except P replaced by qm and its solution has been provided in Proposition 2. Next, we further optimize the power budget qm for each channel m. According to Remark 3, to guarantee that the NOMA  Msystem is SIC stable, it is reasonable to Θm for some positive Θm . Then, from assume that qm ≥ Θm > 2Ωm and P ≥ m=1 MCaSR1 and T U SR1 , the corresponding power budget optimization problem is given by M  f SR1⋆ (qm ) max q m=1 m SR1 MCb : M  s.t. qm ≤ P, qm ≥ Θm , ∀m m=1 where f mSR1⋆ (qm ) is the optimal objective value of each subproblem. Using Proposition 2, we obtain     qm Γ2,m + 1 . (6.40) f mSR1⋆ (qm ) = W1,m log 1 + Ωm Γ1,m + W2,m log Ωm Γ2,m + 1 It is easily seen that f mSR1⋆ (qm ) is a concave function, so MCbSR1 is a convex problem, whose solution is provided in the following result. Theorem 7 The optimal solution to MCbSR1 is given by qm⋆ = where λ is chosen such that M m=1 W2,m Bc 1 − λ Γ2,m ∞ (6.41) , Θm qm⋆ = P. Proof The solution to MCbSR1 is given by the well-known waterfilling form. Consequently, the optimal power allocation for the weighted sum rate maximization in MC-NOMA systems is jointly characterized by Theorem 7 and Proposition 2 under the SIC stability. 6.5.2.2 SR Maximization with QoS (SR2) Now, we consider maximizing the SR with QoS constraints. In this case, the power allocation problem is given by max MCaSR2 : M   p1 ,p2 m=1 R1,m ( p1,m , p2,m ) + R2,m ( p1,m , p2,m ) s.t. 0 ≤ p1,m ≤ p2,m , m = 1, . . . , M, M  m=1  ( p1,m + p2,m ) ≤ P min Rn,m ≥ Rn,m , n = 1, 2, m = 1, . . . , M. 6 Optimal Power Allocation for Downlink NOMA Systems 215 We use the similar method to address MCaSR2 . By introducing the power budget qm on each channel m, MCaSR2 decomposes into several subproblems and each of them has the same structure as T U SR2 . Thus, the optimal solution to each subproblem is given in Proposition 3 with P replaced by qm . Then, we focus on optimizing the power budget qm for each channel. Similarly, is SIC stable, we assume that according to Remark 4,Mto guarantee the NOMA system qm ≥ Υm and P ≥ m=1 Υm . According to MCaSR2 and T U SR2 , the corresponding power budget optimization problem is as follows MCbSR2 : max q s.t. M  m=1 M  m=1 f mSR2⋆ (qm ) qm ≤ P, qm ≥ Υm , ∀m where f mSR2⋆ (qm ) is the optimal objective value of each subproblem and given by f mSR2⋆ (qm ) = Bc log  A2,m Γ2,m − A2,m Γ1,m + Γ1,m Γ2,m qm + Γ1,m A2,m Γ2,m  min + R2,m . (6.42) Since f mSR2⋆ (qm ) is a concave function, MCbSR2 is a convex problem, whose solution is also given in a waterfilling form. Theorem 8 The optimal solution to MCbSR2 is given by qm⋆ = where λ is chosen such that A2,m Bc A2,m 1 − + − λ Γ1,m Γ2,m Γ2,m M m=1 ∞ , (6.43) Υm qm⋆ = P. Proof The proof is simple and thus omitted. Therefore, the optimal power allocation for the SR maximization with QoS constraints in MC-NOMA systems is jointly characterized by Proposition 3 and Theorem 8. 6.5.3 Optimal Power Allocation for EE Maximization In this subsection, we investigate the optimal power allocation for maximizing the EE with weights or QoS constraints in MC-NOMA systems. 216 Y. Huang et al. 6.5.3.1 EE Maximization with Weights (EE1) With given channel assignment, the problem of maximizing the weighted EE is formulated as the following power allocation problem: MCaEE1 : max p1 ,p2 M m=1  W1,m R1,m ( p1,m , p2,m ) + W2,m R2,m ( p1,m , p2,m )  M  PT + m=1 p1,m + p2,m s.t. 0 ≤ p1,m ≤ p2,m , m = 1, . . . , M, M   m=1   p1,m + p2,m ≤ P. The difficulties in solving MCaEE1 lie in its nonconvex and fractional objective. In the following, we will show that this problem can also be optimally solved. We use the similar trick to address this problem, i.e., introducing the auxiliary variM with p1,m + p2,m = qm for each channel m. Then, MCaEE1 is decomables {qm }m=1 posed into a group of subproblems. Each subproblem is the same with T U SR1 except P replaced by qm , and thus, its solution is provided in Proposition 1. Then, we concentrate on searching the optimal power budget qm for each channel. Similarly, to guarantee NOMA system is SIC stable, it is assumed that qm ≥ the M Θm > 2Ωm and P ≥ m=1 Θm for some positive Θm . According to Proposition 1 and MCaEE1 , the power budget optimization problem is formulated as MCbEE1 : max η(q)  q s.t. M  m=1 M f mSR1⋆ (qm ) M PT + m=1 qm m=1 (6.44) qm ≤ P, qm ≥ Θm , ∀m where f mSR1⋆ (qm ) is given in (6.40). Although f mSR1⋆ (qm ) is a concave function, MCbEE1 is nonconvex due to the fraction form. To solve it, we introduce the following objective function: H (q, α)  M  m=1 = f mSR1⋆ (qm ) − α PT + M   m=1  R̃1,m  M  qm m=1 qm Γ2,m + 1 + W2,m log Ωm Γ2,m + 1   − α PT + M  qm , (6.45) m=1   where R̃1,m  W1,m log 1 + Ωm Γ1,m and α is a positive parameter. Then, we consider the following convex problem with given α: 6 Optimal Power Allocation for Downlink NOMA Systems 217 max H (q, α) MCcEE1 q : s.t. M  m=1 qm ≤ P, qm ≥ Θm , ∀m. According to Lemma 2, the optimal solution to MCbEE1 can be found by solving MCcEE1 with given α and then updating α until H ⋆ (α) = 0. Hence, we first solve MCcEE1 with given α, whose solution is provided in the following result. Theorem 9 The optimal solution to MCcEE1 is qm⋆ = where λ is chosen such that M m=1 W2,m Bc 1 − α+λ Γ2,m ∞ (6.46) , Θm qm⋆ = P. Proof The solution is obtained by exploiting the KKT conditions of MCcEE1 . After the optimal solution to MCcEE1 is obtained, we shall find an α such that H ⋆ (α) = 0. Since H ⋆ (α) is monotonic in α, one can use the bisection method to find α. Thereby, the optimal power allocation for the weighted EE maximization in MCNOMA systems is provided Proposition 2 and Theorem 9. 6.5.3.2 EE Maximization with QoS (EE2) In this part, we consider maximizing the EE with QoS constraints. The corresponding power allocation problem is given by max p1 ,p2 MCaEE2 : M m=1      R1,m p1,m , p2,m + R2,m p1,m , p2,m  M  PT + m=1 p1,m + p2,m s.t. 0 ≤ p1,m ≤ p2,m , m = 1, . . . , M, M  m=1 ( p1,m + p2,m ) ≤ P min , l = 1, 2, m = 1, . . . , M. Rl,m ≥ Rl,m M We can use the similar method to solve MCaEE2 . Briefly, we also adopt {qm }m=1 EE2 with p1,m + p2,m = qm and decompose MCa into a group of subproblems, whose solution is coincided with T U SR2 and provided in Proposition 3. Next, we optimize the channel power budget qm for each channel. First, we assume M that qm ≥ Υm and P ≥ m=1 Υm to guarantee the SIC stability. Then, according to EE2 Proposition 3 and MCa , the power budget optimization problem is given by 218 Y. Huang et al. MCbEE2 : max η(q)  q s.t. M  m=1 M f mSR2⋆ (qm ) M PT + m=1 qm m=1 qm ≤ P, qm ≥ ϒm , ∀m where f mSR2⋆ (qm ) is given in (6.42). To solve MCbEE2 , we introduce the objective function parameterized by α: Q(q, α)  M  f mSR2⋆ (qm ) m=1 = M   W1,m log m=1  −α PT +  − α PT +  M  M  qm m=1 A2,m Γ2,m − A2,m Γ1,m + Γ1,m Γ2,m qm + Γ1,m A2,m Γ2,m  min + R2,m  (6.47) qm , m=1 and formulate the following problem with given α: max Q (q, α) MCcEE2 q : s.t. M  m=1 qm ≤ P, qm ≥ ϒm , ∀m. Then, from Lemma 2, we shall solve MCcEE2 , which is a convex problem since Q(q, α) is concave in q. The optimal solution to MCcEE2 is provided below. Theorem 10 The optimal solution to MCcEE2 is qm⋆ = A2,m A2,m 1 W1,m Bc − + − λ+α Γ1,m Γ2,m Γ2,m where λ is chosen such that M m=1 ∞ , (6.48) ϒm qm⋆ = P . Proof The solution is obtained by exploiting the KKT conditions of MCcEE2 . Then, we can exploit the bisection method to find an α such that the optimal objective value of MCcEE2 satisfies Q ⋆ (α) = 0. Consequently, the optimal power allocation for the EE maximization with QoS constraints in MC-NOMA systems is obtained by using Theorem 10 and Proposition 3. 6 Optimal Power Allocation for Downlink NOMA Systems 219 6.6 Numerical Results This section evaluates the performance of the optimal power allocation investigated in this chapter. In simulations, the BS is located in the cell center and the users are randomly distributed in a circular range with a radius of 500 m. The minimum distance between users is set to be 40 m, and the minimum distance between the users and the BS is 50 m. Each channel coefficient follows an i.i.d. Gaussian distribution as g ∼ CN (0, 1) and the path loss exponent is ρ = 2. The total power budget of the BS is P = 41 dBm and the circuit power consumption is PT =30 dBm. The noise power is σ 2 = B N0 /M, where the bandwidth is B = 5 MHz and the noise power spectral density is N0 = −174 dBm. First, we evaluate the performance of the proposed optimal power solutions for two-user NOMA (N = 2) and MU-NOMA (N = 6) systems. The user weights satisfy Wi+1 /Wi = 0.5 for i = 1, . . ., N − 1 and the QoS thresholds to be Rimin = 2 bps/Hz for i = 1, . . ., N. In addition, we compare the NOMA schemes with OFDMA and the DC (difference of two convex functions) approach in [26], where the power allocation is optimized via waterfilling and via DC programming, respectively. Figure 6.2 shows the minimum user rates of the two-user NOMA and MU-NOMA schemes using the optimal power allocation under the MMF criterion and the minimum user rate of the OFDMA scheme for different total power budgets and user numbers. The minimum user rate in the NOMA system is higher than that in the OFDMA system especially in the two-user case, implying that NOMA provides better fairness than OFDMA. 106 2 1.8 NOMA N=2 NOMA N=6 OFDMA N=2 OFDMA N=6 Minimum User Rate (Mbps) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 2 3 4 5 6 7 8 9 10 Power of BS (Watt) Fig. 6.2 Minimum user rate for different number of users versus BS power 11 12 220 Y. Huang et al. 106 16 NOMA N=6 NOMA N=2 NOMA DC N=2 OFDMA N=2 1.8 1.6 12 Sum Rate (Mbits/Joule) Weighted Sum Rate (Mbits/Joule) 14 107 2 10 8 6 1.4 NOMA N=6 NOMA N=2 NOMA DC N=2 OFDMA N=2 1.2 1 0.8 0.6 4 2 0.4 0 5 10 Power of BS (Watt) 15 0.2 0 5 10 15 Power of BS (Watt) Fig. 6.3 Sum rate versus BS power In Fig. 6.3 the left subfigure shows the weighted sum rate and the right subfigure shows the sum rate with QoS constraints. Here, in each subfigure, we compare the proposed methods with the OFDMA scheme and the NOMA scheme using DC programming in the two-user case. While NOMA outperforms OFDMA, NOMA with the optimal power allocation also achieves a higher sum rate than the DC approach, as the DC approach generally leads to a suboptimal power allocation. Meanwhile, as expected, the (weighted) sum rate increases with the user number, implying the potential of NOMA. In Fig. 6.4, the similar phenomenon can be observed, i.e., NOMA using the optimal power allocation outperforms OFDMA as well as the (suboptimal) DC approach in terms of energy efficiency. Then, we show the performance of the optimal power allocation in MC-NOMA systems. The user weights are set to be W1,m = 0.9 and W2,m = 1.1 for ∀m and min the QoS thresholds are set to be Rl,m = 2 bps/Hz for l = 1, 2, ∀m. In Fig. 6.5, we compare the joint resource allocation (JRA) method, which uses the optimal power allocation and the matching algorithm [22, 26] for channel assignment, with the exhaustive search (ES), which provides the jointly optimal solution but has high complexity. We set the number of users N = 6 and the power budget of the BS ranges from 2 to 12 W. From Fig. 6.5, the performance of JRA is very close to the 6 Optimal Power Allocation for Downlink NOMA Systems 106 NOMA N=6 NOMA N=2 NOMA DC N=2 OFDMA N=2 3 NOMA N=6 NOMA N=2 NOMA DC N=2 OFDMA N=2 3.5 2.5 2 1.5 1 3 2.5 2 1.5 1 0.5 0 106 4 Energy Efficiency (Mbits/Joule) Weighted Energy Efficiency (Mbits/Joule) 3.5 221 0.5 0 5 10 0 15 0 Power of BS (Watt) 5 10 15 Power of BS (Watt) 140 Sum rate (Mbps) 20 15 10 MMF ES MMF JRA 5 2 4 6 8 10 120 100 SR1 ES SR1 JRA SR2 ES SR2 JRA 80 60 12 2 4 Power of BS (Watt) 50 EE1 ES EE1 JRA 40 30 20 10 2 4 6 8 10 6 8 10 12 Power of BS (Watt) 12 Energy efficiency (Mbps/Joule) Weighted Energy efficiency (Mbps/Joule) Minimum User Rate (Mbps) Fig. 6.4 Energy efficiency versus BS power Power of BS (Watt) Fig. 6.5 Comparison with the exhaustive search (ES) 50 EE2 ES EE2 JRA 40 30 20 10 2 4 6 8 10 Power of BS (Watt) 12 222 Y. Huang et al. globally optimal value and the maximum gap is less than 5%. Therefore, the optimal power allocation method along with efficient (suboptimal) matching algorithm is able to achieve near-optimal performance with low complexity. 6.7 Conclusion In this chapter, we discussed a promising multiple access technology, i.e., NOMA, for 5G networks and focused on the key problem of power allocation in NOMA systems. We have investigated the optimal power allocation for different NOMA schemes, including the two-user MU-NOMA, and MC-NOMA schemes. The optimal power allocation was derived under different performance measures, including the maximin fairness, weighted sum rate, and energy efficiency, wherein user weights or QoS constraints were also considered. We showed that in most cases the optimal NOMA power allocation admits an analytical solution, while in other cases it can be numerically computed via convex optimization methods. Appendix A. Proof of Theorem 1 Since the constraints in T UcEE1 are all linear, it suffices to investigate the concavity of H ( p1 , q, α). The second-order derivative of H ( p1 , q, α) with respect to p1 is 1 ∂2 H = ln 2 ∂ p12  ΥΘ ( p1 Γ1 + 1)2 ( p1 Γ2 + 1)2  (6.49) , √ √ √ where√ Υ = W√ + 1) + √W1 BΓ 2 BΓ2 ( p1 Γ1√  1 ( p1 Γ2 + 1) and Θ = W2 B Γ2 − W1 BΓ1 + BΓ1 Γ2 p1 W2 − W1 . Given Γ1 ≥ Γ2 , if W1 ≥ W2 , then ∂2 H ≤ 0. On the other hand, with q ≤ P and if C2 holds, we have ∂ p2 1  W2 BΓ2 −  W1 BΓ1 + √ BΓ1 Γ2 p1  W2 −  W1  W2 BΓ2 −  W1 BΓ1 +  W2 −  W1 √ BΓ1 Γ2 P ≤ 0, (6.50) 2 2 also implying ∂∂ pH2 ≤ 0. Following the similar manner, it can be verified that ∂∂qH2 ≤ 0, ∂2 H ∂q∂ p1 = 0 and 1 2 ∂ H ∂ p1 ∂q = 0. Therefore, the Hessian matrix  ∂2 H ∂2 H ∂ p12 ∂q∂ p1 ∂2 H ∂2 H ∂ p1 ∂q ∂q 2 is a negative semidefinite matrix, indicating that H ( p1 , q, α) is a concave in ( p1 , q). 6 Optimal Power Allocation for Downlink NOMA Systems 223 B. Proof of Proposition 4 The Lagrange of T UcEE1 is given by L = W1 R1 ( p1 ) + W2 R2 ( p1 , q) − α (PT + q) + μ (q − 2 p1 ) − λ (q − P) (6.51) with Lagrange multipliers μ and λ ≥ 0. According to Theorem 1, T UcEE1 is a convex problem under condition C1 or C2. Therefore, its optimal solution is characterized by the following Karush–Kuhn–Tucker (KKT) conditions: W2 BΓ2 ∂L W1 BΓ1 = − − 2μ = 0, ∂ p1 ln 2 (1 + p1 Γ1 ) ln 2 (1 + p1 Γ2 ) (6.52) W2 BΓ2 ∂L = − α + μ − λ = 0, ∂q ln 2 (1 + qΓ2 ) (6.53) μ (q − 2 p1 ) = 0, (6.54) λ (q − P) = 0. (6.55) According to Definition 1, if p1 = q/2, then the NOMA system is SIC-unstable. Therefore, from (6.54), considering the SIC stability, we have μ = 0. Hence, from (6.52) we obtain the optimal p1⋆ = Ω. It follows from (6.55) that if q < P, then λ = 0. Then, from (6.53) we obtain 2Ω ≤ q = 1 W2 B − < P. α ln 2 Γ2 (6.56) On the other hand, if q = P, then from (6.53) we have λ= which leads to W2 BΓ2 − α ≥ 0, ln 2 (1 + PΓ2 ) 1 W2 B − ≥ P. α ln 2 Γ2 Therefore, the optimal q is given by q ⋆ =  W2 B α ln 2 − (6.57) (6.58) P 1 . Γ2 2Ω 224 Y. Huang et al. C. Proof of Proposition 5 The Lagrange of T UcEE2 is given by L = R1 ( p1 ) + R2 ( p1 , q) − α (PT + q) + μ (q − 2 p1 ) − λ (q − P)   A1 − 1 + σ2 (1 + qΓ2 − A2 − A2 p1 Γ2 ) , + σ1 p1 − Γ1 (6.59) where μ, λ, σ1 , and σ2 are the Lagrange multipliers. The optimal solution is characterized by the following KKT conditions: BΓ1 BΓ2 ∂L − − 2μ + σ1 − σ2 A2 Γ2 = 0, = ∂ p1 ln 2 (1 + p1 Γ1 ) ln 2 (1 + p1 Γ2 ) (6.60) BΓ2 ∂L = − α + μ − λ + σ2 Γ2 = 0, ∂q ln 2 (1 + qΓ2 ) (6.61) μ (q − 2 p1 ) = 0, (6.62) λ (q − P) = 0, (6.63)  (6.64) σ1  A1 − 1 p1 − Γ1 = 0, σ2 (1 + qΓ2 − A2 − A2 p1 Γ2 ) = 0. (6.65) In (6.62), considering the SIC stability, we have q > 2 p1 and hence μ = 0. Note that σ2 = 0. To see this, if σ2 = 0, according to (6.60), we have BΓ1 BΓ2 − + σ1 = 0 ln 2 (1 + p1 Γ1 ) ln 2 (1 + p1 Γ2 ) (6.66) BΓ2 BΓ1 − ln 2(1+ + σ1 > 0 with Γ1 ≥ Γ2 . which, however, does not hold since ln 2(1+ p1 Γ1 ) p1 Γ2 ) We consider two cases: (1) σ1 = 0, σ2 = 0; and (2) σ1 = 0, σ2 = 0. First, if σ1 = 0, σ2 = 0, the optimal solution can be easily obtained as p1⋆ = 1 + q ⋆ Γ2 − A 2 ⋆ , q =Υ A 2 Γ2 (6.67) from (6.64) and (6.65). Then, if σ1 = 0, σ2 = 0, according to (6.60) and (6.61), we have   1 1 A 2 Γ2 = (α + λ) A2 . + (6.68) − 1/Γ1 + p1 1/Γ2 + p1 (1 + qΓ2 ) 6 Optimal Power Allocation for Downlink NOMA Systems From (6.65), we obtain p1 = 1+qΓ2 −A2 , A 2 Γ2 q⋆ = 225 which along with (6.68) leads to A2 A2 − 1 1 − + . α+λ Γ1 Γ2 (6.69) It follows from (6.63) that if q < P, then λ = 0. 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Technol. 65(8), 6010–6023 (2016) Chapter 7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access Xiaoming Chen, Zhaoyang Zhang, Caijun Zhong and Derrick Wing Kwan Ng 7.1 Introduction Current wireless communications in general adopt various types of orthogonal multiple access (OMA) technologies for serving multiple users, such as time division multiple access (TDMA), frequency division multiple access (FDMA), and code division multiple access (CDMA), where one resource block is exclusively allocated to one mobile user (MU) to avoid possible multiuser interference. In practice, the OMA technologies are relatively easy to implement, albeit at the cost of low spectral efficiency. Recently, with the rapid development of mobile Internet and proliferation of mobile devices, it is expected that future wireless communication systems should be able to support massive connectivity, which is an extremely challenging task for the OMA technologies with limited radio resources. Responding to this, non-orthogonal multiple access (NOMA) has been recently proposed as a promising access technology for the fifth-generation (5G) mobile communication systems, due to its potential in achieving high spectral efficiency and supporting massive access [1–4]. X. Chen (B) · Z. Zhang · C. Zhong College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, China e-mail: chen_xiaoming@zju.edu.cn Z. Zhang e-mail: ning_ming@zju.edu.cn C. Zhong e-mail: caijunzhong@zju.edu.cn D. W. K. Ng School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney, NSW, Australia e-mail: w.k.ng@unsw.edu.au © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_7 229 230 X. Chen et al. The principle of NOMA is to exploit the power domain to simultaneously serve multiple MUs utilizing the same radio resources [5–7], with the aid of sophisticated successive interference cancellation (SIC) receivers [8, 9]. Despite the adoption of SIC, inter-user interference still exists except for the MU with the strongest channel gain, which limits the overall system performance [10]. To address this issue, power allocation has been considered as an effective method to harness multiuser interference [11, 12]. Since the overall performance is limited by the MUs with weak channel conditions, it is intuitive to allocate more power to the weak MUs and less power to the strong MU in order to enhance the effective channel gain and minimize the interference to the weak MUs [13]. For the specific two-user case, the optimal power allocation scheme was studied in [14], and [15] proposed two sub-optimal power allocation schemes exploiting the Karush–Kuhn–Tucker (KKT) conditions, while the issue of quality of service (QoS) requirements of NONA systems was investigated in [16]. For the case with arbitrary number of users, the computational complexity of performing SIC increases substantially and the design of the optimal power allocation becomes intractable. To facilitate an effective system design, clustering and user pairing have been proposed [17, 18]. Generally speaking, multiple MUs with distinctive channel gains are selected to form a cluster, in which SIC is conducted to mitigate the interference [19, 20]. In general, a small cluster consisting a small number of MUs implies low complexity of SIC, but leads to high intercluster interference. Thus, it makes sense to dynamically adjust the size of a cluster according to performance requirements and system parameters, so as to achieve a balance between implementation complexity and interference mitigation [21]. However, dynamic clustering is not able to reduce the inter-cluster interference, indicating the necessity of carrying out dynamic clustering in combination with efficient interference mitigation schemes. It is well known that the multiple-antenna technology is a powerful interference mitigation scheme [22–25], hence, can be naturally applied to NOMA systems [26, 27]. In [28], the authors proposed a beamforming scheme for combating inter-cluster and intra-cluster interference in a NOMA downlink, where the base station (BS) was equipped with multiple antennas and the MUs have a single antenna each. A more general setup was considered in [29], where both the BS and the MUs are multiple-antenna devices. By exploiting multiple antennas at the BS and the MUs, a signal alignment scheme was proposed to mitigate both the intra-cluster and inter-cluster interference. It is worth pointing out that the implementation of the two above schemes requires full channel state information (CSI) at the BS, which is usually difficult and costly in practice. To circumvent the difficulty in CSI acquisition, random beamforming was adopted in [30], which inevitably leads to performance loss. Alternatively, the work in [31] suggested to employ zero-forcing (ZF) detection at the multiple-antenna MUs for inter-cluster interference cancelation. However, the ZF scheme requires that the number of antennas at each MU is greater than the number of antennas at the BS, which is in general impractical. To effectively realize the potential benefits of multiple-antenna techniques, the amount and quality of CSI available at the BS play a key role [32, 33]. In practice, the CSI can be obtained in several different ways. For instance, in time duplex 7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access 231 division (TDD) systems, the BS can obtain the downlink CSI through estimating the CSI of uplink by leveraging the channel reciprocity [34]. While in frequency duplex division (FDD) systems, the downlink CSI is usually first estimated and quantized at the MUs, and then is conveyed back to the BS via a feedback link [35]. For both practical TDD and FDD systems, the BS has access to only partial CSI. As a result, there will be residual inter-cluster and intra-cluster interference. To the best of the authors’ knowledge, previous works only consider two extreme cases with full CSI or no CSI, the design, analysis and optimization of multiple-antenna NOMA systems with partial CSI remains an uncharted area. Motivated by this, we present a comprehensive study on the impact of partial CSI on the design, analysis, and optimization of multiple-antenna NOMA downlink communication systems. The rest of this chapter is organized as follows: Sect. 7.2 gives a brief introduction of the considered NOMA downlink communication system and designs the corresponding multiple-antenna transmission framework. Section 7.3 first analyzes the average transmission rates in presence of imperfect CSI and then proposes three performance optimization schemes. Section 7.4 derives the average transmission rates in two extreme cases through asymptotic analysis and presents some system design guidelines. Section 7.5 provides simulation results to validate the effectiveness of the proposed schemes. Finally, Sect. 7.6 concludes this chapter. 7.2 System Model and Framework Design Consider a downlink communication scenario in a single-cell system, where a base station (BS) broadcasts messages to multiple MUs, cf. Fig. 7.1. Note that the BS is equipped with M antennas, while the MUs have a single antenna each due to the size limitation. 7.2.1 User Clustering To strike a balance between the system performance and computational complexity in NOMA systems, it is necessary to carry out user clustering. In particular, user clustering can be designed from different perspectives. For instance, a signalto-interference-plus-noise ratio (SINR) maximization user clustering scheme was adopted in [36] and quasi-orthogonal MUs were selected to form a cluster in [37]. Intuitively, these schemes perform user clustering by the exhaustive search method, resulting in high implementation complexity. In this chapter, we design a simple user clustering scheme based on the information of spatial direction.1 Specifically, the MUs in the same direction but with distinctive propagation distances are arranged 1 The spatial direction of users can be found via various methods/technologies such as GPS or user location tracking algorithms. 232 X. Chen et al. Fig. 7.1 A multiuser NOMA communication system with 4 clusters into a cluster. On one hand, the same direction of the MUs in a cluster allows the use of a single beam to nearly align all MUs in such a cluster, thereby facilitating the mitigation of the inter-cluster interference and the enhancement of the effective channel gain. On the other hand, a large gap of propagation distances avoids severe inter-user interference and enables a more accurate SIC at the MUs [38–40]. If two MUs are close to each other with almost equal channel gains, it is possible to assign them in different clusters by improving the spatial resolution via increasing the number of spatial beams and the number of BS antennas. Without loss of generality, we assume that the MUs are grouped into N clusters with K MUs in each cluster. To 1/2 facilitate the following presentation, we use αn,k hn,k to denote the M-dimensional channel vector from the BS to the kth MU in the nth cluster, where αn,k is the largescale channel fading, and hn,k is the small-scale channel fading following zero mean complex Gaussian distribution with unit variance. It is assumed that αn,k remains constant for a relatively long period, while hn,k keeps unchanged in a time slot but varies independently over time slots. 7.2.2 CSI Acquisition For the TDD mode, the BS obtains the downlink CSI through uplink channel estimation. Specifically, at the beginning of each time slot, the MUs simultaneously send pilot sequences of τ symbols to the BS, and the received pilot at the BS can be expressed as YP = K  N   P αn,k hn,k Φ n,k + N P , τ Pn,k n=1 k=1 (7.1) 7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access 233 P where Pn,k is the transmit power for the pilot sequence of the kth MU in the nth cluster, N P is an additive white Gaussian noise (AWGN) matrix with i.i.d. zero mean and unit variance complex Gaussian distributed entries. Φ n,k ∈ C 1×τ is the pilot sequence sent from the kth MU in the nth cluster. It is required that τ > N K , such that the H = 1, ∀(n, k) = (i, j), can be pairwise orthogonality that Φ n,k Φ i,Hj = 0 and Φ n,k Φ n,k guaranteed. By making use of the pairwise orthogonality, the received pilot can be transformed as  H H P αn,k hn,k + N P Φ n,k . (7.2) = τ Pn,k Y P Φ n,k Then, by using minimum mean squared error (MMSE) estimation, the relation between the actual channel gain hn,k and the estimated channel gain ĥn,k can be expressed as hn,k =  √ ρn,k ĥn,k + 1 − ρn,k en,k , (7.3) where en,k is the channel estimation error vector with i.i.d. zero mean and unit variance complex Gaussian distributed entries, and is independent of ĥn,k . Variable ρn,k = P τ Pn,k αn,k P 1+τ Pn,k αn,k = 1 − 1+τ P1P α is the correlation coefficient between hn,k and ĥn,k . A n,k n,k large ρn,k means a high accuracy for channel estimation. Thus, it is possible to P or the length τ of improve the CSI accuracy by increasing the transmit power Pn,k pilot sequence. For the FDD mode, the CSI is usually conveyed from the MUs to the BS through a feedback link. Since the feedback link is rate-constrained, CSI at the MUs should first be quantized. Specifically, the kth MU in the nth cluster chooses an optimal (1) (2 Bn,k ) , . . . , h̃n,k } codeword from a predetermined quantization codebook Bn,k = {h̃n,k ( j) Bn,k of size 2 , where h̃n,k is the jth codeword of a unit norm and Bn,k is the number of feedback bits. Mathematically, the codeword selection criterion is given by    H ( j) 2 j ⋆ = arg max hn,k h̃n,k  . B 1≤ j≤2 (7.4) n,k Then, the MU conveys the index j ⋆ to the BS with Bn,k feedback bits, and the BS ( j ⋆) recoveries the quantized CSI h̃n,k from the same codebook. In other words, the BS only gets the phase information by using the feedback scheme based on a quantization codebook. However, as shown in below, the phase information is sufficient for the design of spatial beamforming. Similarly, the relation between the real CSI and the obtained CSI in FDD mode can be approximated as [41, 42] h̃n,k =  √ ⋆ ρn,k h̃n,k + 1 − ρn,k ẽn,k , (7.5) n,k ⋆ is the phase of the channel hn,k , h̃n,k is the quantized phase where h̃n,k = hhn,k  information, ẽn,k is the quantization error vector with uniform distribution, and 234 X. Chen et al. Bn,k ρn,k = 1 − 2− M−1 is the associated correlation coefficient or CSI accuracy. Thus, it is possible to improve the CSI accuracy by increasing the size of quantization codebook for a given number of antennas M at the BS. 7.2.3 Superposition Coding and Transmit Beamforming Based on the available CSI, the BS constructs one transmit beam for each cluster, so as to mitigate or even completely cancel the inter-cluster interference. To strike balance between system performance and implementation complexity, we adopt zero-force beamforming (ZFBF) at the BS. We take the design of beam wi for the ith cluster as an example. First, we construct a complementary matrix H̄i 2 as: H̄i = [ĥ1,1 , . . . , ĥ1,K , . . . , ĥi−1,K , ĥi+1,1 , . . . , ĥ N ,K ] H . (7.6) Then, we perform singular value decomposition (SVD) on H̄i and obtain its right singular vectors ui, j , j = 1, . . . , Nu , with respect to the zero singular values, where we can design the beam as wi = Nu is the number of zero singular values. Finally,  Nu Nu θ u , where θ > 0 is a weight such that i, j j=1 i, j i, j j=1 θi, j = 1. Thus, the received signal at the kth MU in the nth cluster is given by yn,k = = √ H αn,k hn,k N  i=1 √ wi si + n n,k H αn,k hn,k wn sn +  H αn,k (1 − ρn,k )en,k N  i=1,i=n wi si + n n,k , (7.7)   where si = Kj=1 Pi,Sj si, j is the superposition coded signal with Pi,Sj and si, j being transmit power and transmit signal for the jth MU in the ith cluster, and n n,k is the AWGN with unit variance. In general, Pi,Sj should be carefully allocated to distinguish the MUs in the power domain, which we will discuss in detail below. Note that √ H H H wi = ρn,k ĥn,k wi + 1 − ρn,k en,k wi = Eq. (7.7) holds true due to the fact that hn,k  H 3 1 − ρn,k en,k wi for ZFBF in TDD mode. With perfect CSI at the BS, i.e., ρn,k = 1, the inter-cluster interference can be completely canceled. 2 In ⋆ , . . . , h̃⋆ , . . . , FDD mode, the complementary matrix is given by H̄i = [h̃1,1 1,K ⋆ ⋆ ⋆ ⋆ H h̃i−1,1 , . . . , h̃i−1,K , h̃i+1,1 , . . . , h̃ N ,K ] .  3 In FDD mode, we have h H w = √ρ h (h̃⋆ ) H w + 1 − ρ h ẽ H w = i n,k n,k n,k n,k n,k i n,k n,k i  d d  H H 1 − ρn,k hn,k ẽn,k wi = 1 − ρn,k en,k wi , where = denotes the equality in distribution. If ρn,k = ρn,k , Eq. (7.7) also holds true in FDD mode. In the sequel, without loss of generality, we no longer distinguish between TDD and FDD. 7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access 235 7.2.4 Successive Interference Cancellation Although ZFBF at the BS can mitigate partial inter-cluster interference from the other clusters, there still exists intra-cluster interference from the same cluster. In order to improve the received signal quality, the MU conducts SIC according to the principle of NOMA. Without loss of generality, we assume that the effective channel gains in the ith cluster have the following order: √ √ H H wi |2 ≥ · · · ≥ | αi,K hi,K wi |2 . | αi,1 hi,1 (7.8) It is reasonably assumed that the BS may know MUs’ effective gains through the channel quality indicator (CQI) messages, and then determines the user order in (7.8). Thus, in the ith cluster, the jth MU can always successively decode the lth MU’s signal, ∀l > j, if the lth MU can decode its own signal. As a result, the jth MU can subtract the interference from the lth MU in the received signal before decoding its own signal. After SIC, the signal-to-interference-plus-noise ratio (SINR) at the kth MU in the nth cluster can be expressed as γn,k = H αn,k |hn,k wn |2   k−1 j=1 H w |2 P S αn,k |hn,k n n,k , K N S Pn, j + αn,k (1 − ρn,k ) |e H wi |2 P S + 1 i=1,i=n n,k l=1 i,l   AWGN Intra-cluster interference Inter-cluster interference (7.9) where the first term in the denominator of (7.9) is the residual intra-cluster interference after SIC at the MU, the second one is the residual inter-cluster interference after ZFBF at the BS, and the third one is the AWGN. For the 1st MU in each cluster, there is no intra-cluster interference, since it can completely eliminate the intra-cluster interference. Note that in this chapter, we assume that perfect SIC can be performed at the MUs. In practical NOMA systems, SIC might be imperfect due to a limited computational capability at the MUs. Thus, there exists residual intracluster interference from the weaker MUs even after SIC [43]. However, the study of the impact of imperfect SIC on the system performance is beyond the scope of this chapter and we would like to investigate it in the future work. Moreover, the transmit power has a significant impact on the SIC and the performance of NOMA [44]. Thus, we will quantitatively analyze the impact of transmit power and then aim to optimize the transmit power for improving the performance in the following sections. 7.3 Performance Analysis and Optimization In this section, we concentrate on performance analysis and optimization of multiantenna NOMA downlink with imperfect CSI. Specifically, we first derive closed- 236 X. Chen et al. form expressions for the average transmission rates of the 1st MU and the other MUs, and then propose separate and joint optimization schemes of transmit power, feedback bits, and transmit mode, so as to maximize the average sum rate of the system. 7.3.1 Average Transmission Rate We start by analyzing the average transmission rate of the kth MU in the nth cluster. First, we consider the case k > 1. According to the definition, the corresponding average transmission rate can be computed as  Rn,k = E log2 1 + γn,k    k N K S S H w |2 H 2 αn,k |hn,k n j=1 Pn, j + αn,k (1 − ρn,k ) i=1,i =n |en,k wi | l=1 Pi,l + 1 = E log2    K N k−1 S S H H w |2 2 αn,k |hn,k n l=1 Pi,l + 1 i=1,i =n |en,k wi | j=1 Pn, j + αn,k (1 − ρn,k )    K N k H H = E log2 αn,k |hn,k Pi,lS + 1 |en,k wi |2 Pn,S j + αn,k (1 − ρn,k ) wn |2 i=1,i =n j=1  N k−1 H Pn,S j + αn,k (1 − ρn,k ) wn |2 −E log2 αn,k |hn,k  i=1,i =n j=1 l=1 H wi |2 |en,k K l=1 Pi,lS + 1  . (7.10) Note that the average transmission rate in (7.10) can be expressed as the difference of two terms, which have a similar form. Hence, we concentrate on the derivation of the we use to denote the term  first term. For notational convenience, N W K S H H 2 wn |2 kj=1 Pn,S j + αn,k (1 − ρn,k ) i=1,i αn,k |hn,k =n |en,k wi | l=1 Pi,l . To compute the first expectation, the key is to obtain the probability density function (pdf) of W . H wn |2 in W , since wn of unit norm is designed Checking the first random variable |hn,k H 2 2 independent of hn,k , |hn,k wn | is χ distributed with 2 degrees of freedom [45]. H Similarly, |en,k wi |2 also has the distribution χ 2 (2). Therefore, W can be considered as a weighted sum of N random variables with χ 2 (2) distribution. According to [46], W is a nested finite weighted sum of N Erlang pdfs, whose pdf is given by f W (x) = where q ηn,k = N  i=1 q i N g(x, ηn,k ), Ξ N i, {ηn,k }q=1 ⎧ k  ⎪ ⎪ Pq,S j ⎨ αn,k if q = n j=1 K  ⎪ ⎪ S ⎩ αn,k (1 − ρn,k ) Pq,l if q = n l=1 , (7.11) 7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access i g(x, ηn,k ) q N Ξ N i, {ηn,k }q=1 237   1 x = i exp − i , ηn,k ηn,k −1  N −1 i  (−1) N −1 ηn,k 1 1 − s+U(s−i) , = N l i ηn,k ηn,k l=1 ηn,k s=1 and U(x) is the well-known unit step function defined as U(x ≥ 0) = 1 and zero othq N erwise. It is worth pointing out that the weights Ξ N are constant for given {ηn,k }q=1 . Hence, the first expectation in (7.10) can be computed as E[log2 (1 + W )] = =  ∞ 0 N  i=1 =− log2 (1 + x) f W (x)d x   1 x log2 (1 + x) i exp − i dx ηn,k ηn,k 0     1 1 q N exp i, {ηn,k }q=1 Ei − i , (7.12) i ηn,k ηn,k q N Ξ N i, {ηn,k }q=1 N 1  ΞN ln(2) i=1  ∞ x where Ei (x) = −∞ exp(t) dt is the exponential integral function. Equation (7.12) t  H wn |2 k−1 follows from [47, Eq. (4.3372)]. Similarly, we use V to denote αn,k |hn,k j=1 K N S H 2 Thus, w | P in the second term of (7.10). |e Pn,S j + αn,k (1 − ρn,k ) i=1,i i t=1 i,t =n n,k the second expectation term can be computed as     N  1 1  1 v N E[log2 (1 + V )] = − Ξ N i, {βn,k }v=1 exp Ei − i , (7.13) i ln(2) βn,k βn,k i=1 where v βn,k = ⎧ k−1  S ⎪ ⎪ Pv, j ⎨ αn,k if v = n j=1 K  ⎪ ⎪ S ⎩ αn,k (1 − ρn,k ) Pv,l if v = n . l=1 Hence, we can obtain the average transmission rate for the kth MU in the nth cluster as follows     N 1 1  1 v N Rn,k = Ξ N i, {βn,k }v=1 exp Ei − i i ln(2) i=1 βn,k βn,k     N 1 1 1  q N exp Ξ N i, {ηn,k }q=1 Ei − i . (7.14) − i ln(2) i=1 ηn,k ηn,k 238 X. Chen et al. Then, we consider the case k = 1. Since the first MU can decode all the other MUs’ signals in the same cluster, there is no intra-cluster interference. In this case, the corresponding average transmission rate reduces to Rn,1     N −1 1 1  1 N −1 v = Ξ N −1 i, {βn,1 }v=1 exp Ei − i i ln(2) i=1 βn,1 βn,1     N 1 1  1 q N Ξ N i, {ηn,1 }q=1 exp − Ei − i , i ln(2) i=1 ηn,1 ηn,1 where q ηn,1 = and v βn,1 = ⎧ S ⎨ αn,1 Pq,1 ⎩ αn,1 (1 − ρn,1 ) K  l=1 if q = n S Pq,l if q = n ⎧ K  ⎪ S ⎪ Pv,l ⎨ αn,1 (1 − ρn,1 ) ⎪ ⎪ ⎩ αn,1 (1 − ρn,1 ) l=1 K  l=1 S Pv+1,l (7.15) , if v < n . if v ≥ n Combing (7.14) and (7.15), it is easy to evaluate the performance of a multipleantenna NOMA downlink with arbitrary system parameters and channel conditions. In particular, it is possible to reveal the impact of system parameters, i.e., transmit power, CSI accuracy, and transmission mode. 7.3.2 Power Allocation From (7.14) and (7.15), it is easy to observe that with imperfect CSI, transmit power has a great impact on average transmission rates. On one hand, increasing the transmit power can enhance the desired signal strength. On the other hand, it also increases the interference. Thus, it is desired to distribute the transmit power according to channel conditions. To maximize the sum rate of the considered multiple-antenna NOMA system subject to a total power constraint, we have the following optimization problem: 7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access J1 : max S Pn,k s.t. C1 : C2 K N   239 Rn,k n=1 k=1 N  K  S S Pn,k ≤ Ptot n=1 k=1 S :Pn,k > 0, (7.16) S where Ptot is the maximum total transmit power budget. It is worth pointing out that in certain scenarios, user fairness might be of particular importance. To guarantee user fairness, one can replace the objective function of J1 with the maximization of a weighted sum rate, where the weights can directly affect the power allocation and thus the MUs’ rates. Unfortunately, J1 is not a convex problem due to the complicated expression for the objective function. Thus, it is difficult to directly provide a closedform solution for the optimal transmit power. As a compromise solution, we propose an effective power allocation scheme based on the following important observation of the multiple-antenna NOMA downlink system: Lemma 1 The inter-cluster interference is dependent of power allocation between the clusters, while the intra-cluster interference is determined by power allocation among the MUs in the same cluster. N Proof A close observation of the inter-cluster interference αn,k (1 − ρn,k ) i=1,i =n K K H Pi,lS in (7.9) indicates that l=1 wi |2 l=1 |en,k Pi,lS is the total transmit power for the ith cluster, which suggests that inter-cluster power allocation does not affect the inter-cluster interference.  Inspired by Lemma 1, the power allocation scheme can be divided into two steps. In the first step, the BS distributes the total power among the N clusters. In the second step, each cluster individually carries out power allocation subject to the power constraint determined by the first step. In the following, we give the details of the two-step power allocation scheme. First, we design the power allocation between the clusters from the perspective of minimizing inter-cluster interference. For the ith cluster, the average aggregate interference to the other clusters is given by ⎡ Ii = E ⎣ ⎛ =⎝ N K   n=1,n=i k=1 N K   n=1,n=i k=1 H αn,k (1 − ρn,k )|en,k wi |2 ⎞ αn,k (1 − ρn,k )⎠ PiS , K  l=1 ⎤ Pi,lS ⎦ (7.17) K where PiS = l=1 Pi,lS is the total transmit power of the ith cluster. Equation (7.17) H follows the fact that E[|en,k wi |2 ] = 1. Intuitively, a large interference coefficient K N k=1 αn,k (1 − ρn,k ) means a more severe inter-cluster interference caused n=1,n=i 240 X. Chen et al. by the ith cluster. In order to mitigate the inter-cluster interference for improving the average sum rate, we propose to distribute the power proportionally to the reciprocal of interference coefficient. Specifically, the transmit power for the ith cluster can be computed as PiS =  N n=1,n=i  N  N K k=1 n=1,n=l l=1 αn,k (1 − ρn,k ) −1 K k=1 αn,k (1 − ρn,k ) −1 S Ptol . (7.18) Then, we allocate the power in the cluster for further increasing the average sum rate. According to the nature of NOMA techniques, the first MU not only has the strongest effective channel gain for the desired signal, but also generates a weak interference to the other MUs. On the contrary, the K th MU has the weakest effective channel gain for the desired signal and also produces a strong interference to the other MUs. Thus, from the perspective of maximizing the sum of average rate, it is better to allocate the power based on the following criterion: S S S ≥ · · · ≥ Pn,k ≥ · · · ≥ Pn,K . Pn,1 (7.19) On the other hand, in order to facilitate SIC, the NOMA in general requires the transmit powers in a cluster to follow a criterion below [31]: S S S ≤ · · · ≤ Pn,k ≤ · · · ≤ Pn,K . Pn,1 (7.20) Under this condition, the MU performs SIC according to the descending order of the user index, namely the ascending order of the effective channel gain. Specifically, the kth MU cancels the interference from the K th to the (k + 1)th MU in sequence. Thus, the SINR for decoding each interference signal is the highest, which facilitates SIC at MUs [44]. To simultaneously fulfill the above two criterions, we propose to equally distribute the powers within a cluster, namely S = PnS /K . Pn,k (7.21) Substituting (7.18) into (7.21), the transmit power for the kth MU in the nth cluster can be computed as S Pn,k =  N i=1,i=n K   N  N l=1 K j=1 i=1,i=l αi, j (1 − ρi, j ) K −1 j=1 αi, j (1 − ρi, j ) −1 S  Ptol . (7.22) Thus, we can distribute the transmit power based on (7.22) for given channel statistical information and the CSI accuracy, which has a quite low computational complexity. 7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access 241 Remark 1 We note that path loss coefficient αn,k , ∀n, k, remain constant for a relatively long time, and it is easy to obtain at the BS via long-term measurement. Hence, the proposed power allocation scheme incurs a low system overhead and can be implemented with low complexity. 7.3.3 Feedback Distribution For the FDD mode, the accuracy of quantized CSI relies on the size of codebook 2 Bn,k , where Bn,k is the number of feedback bits from the kth MU in the nth cluster. As observed in (7.14) and (7.15), it is possible to decrease the interference by increasing feedback bits. However, due to the rate constraint on the feedback link, the total number of feedback bits is limited. Therefore, it is of great importance to optimize the feedback bits among the MUs for performance enhancement. According to the received signal-to-noise ratio (SNR) in (7.9), the CSI accuracy only affects the inter-cluster interference. Thus, it makes sense to optimize the feedback bits to minimizing the average sum of inter-cluster interference given by ⎡ Iinter = E ⎣ = N  K  n=1 k=1 K N   αn,k (1 − ρn,k ) αn,k N  N  i=1,i=n H |en,k wi |2 Bn,k PiS 2− M−1 . K  l=1 ⎤ Pi,lS ⎦ (7.23) i=1,i=n n=1 k=1 Hence, the optimization problem for feedback bits distribution can be expressed as J2 : min Bn,k s.t. C3 : N  K  n=1 k=1 N  K  n=1 k=1 αn,k N  Bn,k PiS 2− M−1 i=1,i=n Bn,k ≤ Btot , C4 :Bn,k ≥ 0, (7.24) where Btot is an upper bound on the total number of feedback bits. J2 is an integer programming problem, hence is difficult to solve. To tackle this challenge, we relax the integer constraint on Bn,k . In this case, according to the fact that 242 X. Chen et al. N  K  αn,k N  i=1,i=n n=1 k=1 ⎛ B n,k PiS 2− M−1 ≥ NK ⎝ N  K  = NK 2  αn,k = NK 2 N K k=1 Bn,k − n=1 M−1 Btot − M−1 B i=1,i=n n=1 k=1  N  ⎛ 1 NK ⎝  ⎞ 1 NK n,k PiS 2− M−1 ⎠ 1 NK N  K  ⎛ ⎝ N  K  αn,k i=1,i=n n=1 k=1 αn,k n=1 k=1 N  N  i=1,i=n ⎞ 1 NK PiS ⎠ ⎞ 1 NK PiS ⎠ , (7.25) Bn,k N S − M−1 , ∀n, k are equal. In where the equality holds true only when αn,k i=1,i =n Pi 2 other words, the objective function in (7.24) can be minimized while satisfying the following condition: αn,k N  PiS 2− Bn,k M−1 i=1,i=n  Btot = 2− M−1 1 NK ⎛ ⎝ K N   n=1 k=1 αn,k N  i=1,i=n ⎞ N1K PiS ⎠ . (7.26) Hence, based on the relaxed optimization problem, the optimal number of feedback bits for the kth MU in the nth cluster is given by Bn,k ⎛ ⎞ ⎛ ⎞ N N N K   Btot 1  S S = Pl ⎠ . Pl ⎠ + log2 ⎝αn,k log2 ⎝αi, j − NK NK i=1 j=1 l=1,l=i (7.27) l=1,l=n Given channel statistical information and transmit power allocation, it is easy to determine the feedback distribution according to (7.27). Note that there exists an integer constraint on the number of feedback bits in practice, so we should utilize the maximum integer that is not larger than Bn,k in (7.27), i.e., ⌊Bn,k ⌋, ∀n, k. Remark 2 The number of feedback bits distributed to the kth MU in the nth cluster is determined by the average inter-cluster interference generated by the kth MU in the nth cluster with respect to the average inter-cluster interference of each MU. In other words, if one MU generates more inter-cluster interference, it would be allocated with more feedback bits, so as to facilitate a more accurate ZFBF to minimize the total interference. 7.3.4 Mode Selection As discussed above, the performance of the multiple-antenna NOMA system is limited by both inter-cluster and intra-cluster interference. Although ZFBF at the BS and SIC at the MUs are jointly applied, there still exists residual interference. Intuitively, 7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access 243 the strength of the residual interference mainly relies on the number of clusters N and the number of MUs in each cluster K . For instance, increasing the number of MUs in each cluster might reduce the inter-cluster interference, but also results in an increase in intra-cluster interference. Thus, it is desired to dynamically adjust the transmission mode, including the number of clusters and the number of MUs in each cluster, according to channel conditions and system parameters. For dynamic mode selection, we have the following lemma: Lemma 2 If the BS has no CSI about the downlink, it is optimal to set N = 1. On the other hand, if the BS has perfect CSI about the downlink, K = 1 is the best choice. Proof First, if there is no CSI, namely ρn,k = 0, ∀n, k, ZFBF cannot be utilized to mitigate the inter-cluster interference. If all the MUs belong to one cluster, interference can be mitigated as much as possible by SIC. In the case of perfect CSI at the BS, ZFBF can completely the interference. Thus, it is optimal to arrange one MU in one cluster.  In above, we consider two extreme scenarios of no and perfect CSI at the BS, respectively. In practice, the BS has partial CSI through channel estimation or quantization feedback. Thus, we propose to dynamically choose the transmission mode for maximizing the sum of average transmission rate, which is equivalent to an optimization problem below: J3 : max N ,K N  K  Rn,k n=1 k=1 s.t. C5 : N K = Nu , C6 : N > 0, C7 : K > 0, (7.28) where Nu is the number of MUs in the multiple-antenna NOMA system. J3 is also an integer programming problem, so it is difficult to obtain the closed-form solution. Under this condition, it is feasible to get the optimal solution by numerical search and the search complexity is O(N K ). In order to control the complexity of SIC, the number of MUs in one cluster is usually small, e.g., K = 2. Therefore, the complexity of numerical search is acceptable. 7.3.5 Joint Optimization Scheme In fact, transmit power, feedback bits and transmission mode are coupled, and determine the performance together. Therefore, it is better to jointly optimize these variables, so as to further improve the performance of the multiple-antenna NOMA systems. For example, given a transmission mode, it is easy to first allocate transmit 244 X. Chen et al. power according to (7.22), and then distribute feedback bits according to (7.27). Finally, we can select an optimal transmission mode with the largest sum rate. The complexity of the joint optimization is mainly determined by the mode selection. As mentioned above, if the number of MUs in one cluster is small, the complex of mode selection is acceptable. 7.4 Asymptotic Analysis In order to provide insightful guidelines for system design, we now pursue an asymptotic analysis on the average sum rate of the system. In particular, two extreme cases are studied, namely interference limited and noise limited. 7.4.1 Interference Limited Case S S With loss of generality, we let Pn,k = θn,k Ptot , ∀n, k, where 0 < θn,k < 1 is a power  N v=1,v =n allocation factor. For instance, θn,k is equal to K   N  N l=1 K j=1 v=1,v =l αv, j (1−ρv, j ) K j=1 −1 αv, j (1−ρv, j ) −1  in S the proposed power allocation scheme in Sect. 7.3.2. If the total power Ptot is large enough, the noise term of SINR in (7.9) is negligible. In this case, with the help of [47, Eq. (4.3311)], the average transmission rate of the kth MU (k > 1) in the nth cluster reduces to N Rn,k = 1  q N i ln(ηn,k ) Ξ N i, {ηn,k }q=1 ln(2) i=1 N − 1  i v N ln(βn,k ), Ξ N i, {βn,k }v=1 ln(2) i=1 (7.29) where we have also used the fact that N  i=1 q N = Ξ N i, {ηn,k }q=1 N  i=1 v N = 1. Ξ N i, {βn,k }v=1 (7.30) Similarly, the asymptotic average transmission rate of the 1st MU in the nth MU can be obtained as 7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access 245 N Rn,1 = 1  q i N ln ηn,1 Ξ N i, {ηn,1 }q=1 ln(2) i=1 − N −1 1  N −1 v i ln βn,1 . Ξ N −1 i, {βn,1 }v=1 ln(2) i=1 (7.31) Combining (7.29) and (7.31), we have the following important result: Theorem 1 In the region of high transmit power, the average transmission rate is S S independent of Ptot , and there exists a performance ceiling regardless of Ptot , i.e., S once Ptot is larger than a saturation point, the average transmission rate will not increase further even the transmit power increases. i i i i S Proof According to the definitions, ηn,k and βn,k can be rewritten as ηn,k = ωn,k Ptot i i S and βn,k = ψn,k Ptot , where i ωn,k = and i ψn,k = ⎧ k  ⎪ ⎪ θi, j ⎨ αn,k if i = n j=1 K  ⎪ ⎪ ⎩ αn,k (1 − ρn,k ) θi,l if i = n ⎧ k−1  ⎪ ⎪ θi, j ⎨ αn,k , l=1 if i = n j=1 K  ⎪ ⎪ ⎩ αn,k (1 − ρn,k ) θi,l if i = n , l=1  q N N v S respectively. Thus, Ξ N i, {ηn,k }q=1 and Ξ N i, {βn,k }v=1 are independent of Ptot . Hence, Rn,k in (7.29) can be transformed as N Rn,k 1  q S i N (ln(Ptot ) + ln(ωn,k )) Ξ N i, {ηn,k }q=1 = ln(2) i=1 N 1  N i v S )) }v=1 ) + ln(ψn,k Ξ N i, {βn,k (ln(Ptot − ln(2) i=1 N = N 1  1  q i i N v N ln(ωn,k )− ln(ψn,k ), Ξ N i, {ηn,k }q=1 Ξ N i, {βn,k }v=1 ln(2) i=1 ln(2) i=1 (7.32)  N q N = i, {ηn,k }q=1 Ξ where Eq. (7.32) follows the fact that N i=1 N N v i=1 Ξ N i, {βn,k }v=1 = 1. Similarly, we can rewrite Rn,1 in (7.31) as 246 X. Chen et al. N Rn,1 = 1  q i N ln ωn,1 Ξ N i, {ηn,1 }q=1 ln(2) i=1 − N −1 1  N −1 v i ln ψn,1 , Ξ N −1 i, {βn,1 }v=1 ln(2) i=1 where i = ωn,1 and i = ψn,1 ⎧ S ⎨ αn,1 θi,1 ⎩ αn,1 (1 − ρn,1 ) K  l=1 if i = n θi,lS if i = n ⎧ K  ⎪ ⎪ ⎨ αn,1 (1 − ρn,1 ) θi,lS ⎪ ⎪ ⎩ αn,1 (1 − ρn,1 ) l=1 K  l=1 (7.33) , if i < n . S θi+1,l if i ≥ n S , which proves Theorem 1.  Note that both (7.32) and (7.33) are regardless of Ptot Now, we investigate the relation between the performance ceiling in Theorem 1 and the CSI accuracy ρn,k . First, we consider Rn,k with k > 1. As ρn,k asymptotically approaches 1, the inter-cluster interference is negligible. Then, Rn,k can be further reduced as ⎡ ⎛ ideal = E ⎣log ⎝α |h H w |2 Rn,k n,k n,k n 2 ⎛ ⎞ k j=1 ωn, j ⎠ = log2 ⎝ k−1 . j=1 ψn, j k  j=1 ⎡ ⎞⎤ ⎛ S ⎠⎦ − E ⎣log ⎝α |h H w |2 Pn, n,k n,k n 2 j k−1  j=1 ⎞⎤ S ⎠⎦ Pn, j (7.34) It is found that even with perfect CSI, the average transmission rate for the (k > 1)th   MU is still upper bounded. The bound log2 k j=1 k−1 j=1 ωn, j ψn, j is completely determined by channel conditions, and thus cannot be increased via power allocation. Differently, for the 1st MU, if the CSI at the BS is sufficiently accurate, the SINR γn,1 becomes high. As a result, the constant term 1 in the rate expression is negligible, and thus the average transmission rate can be approximated as 7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access Rn,1   247 S H αn,1 |hn,1 wn |2 Pn,1 ≈ E log2 N K H αn,1 (1 − ρn,1 ) i=1,i=n |en,1 wi |2 l=1 Pi,lS  H S = E log2 αn,1 |hn,1 wn |2 Pn,1    Ideal average rate ⎡ ⎛ − E ⎣log2 ⎝αn,1 (1 − ρn,1 )  N  i=1,i=n  H |en,1 wi |2 K  l=1 Rate loss due to imperfect CSI ⎞⎤ Pi,lS ⎠⎦ . (7.35) In (7.35), the first term is the ideal average transmission rate with perfect CSI, and the second one is rate loss caused by imperfect CSI. We first check the term of the ideal average transmission rate, which is given by ideal S H Rn,1 = E log2 αn,1 Ptot θn,1 |hn,1 wn |2 C S . = log2 αn,1 Ptot θn,1 − ln(2)  (7.36) Note that if there is perfect CSI at the BS, the average transmission rate of the S 1st MU increases proportionally to log2 (Ptot ) without a bound. However, as seen in (7.34), the (k > 1)th MU has an upper bounded rate under the same condition, which reconfirms the claim in Lemma 2 that it is optimal to arrange one MU in each cluster in presence of perfect CSI. Then, we investigate the rate loss due to imperfect CSI, which can be expressed as    N K   H S loss |en,1 wi |2 θi,t Rn,1 = E log2 αn,1 (1 − ρn,1 )Ptot i=1,i =n  S − = log2 αn,1 (1 − ρn,1 )Ptot where μvn,1 = 1 ln(2) t=1 N −1  i=1 ⎧ K  ⎪ ⎪ ⎨ θv,l l=1 K ⎪ ⎪ ⎩ l=1    N −1 C − ln μin,1 Ξ N −1 i, {μvn,1 }v=1 , (7.37) if v < n . θv+1,l if v ≥ n loss S enlarges as the total transmit power Ptot increases. Given a ρn,1 , the rate loss Rn,1 ideal In order to keep the same rate of increase to the ideal rate Rn,1 , the CSI accuracy ρn,1 should satisfy the following theorem: S Theorem 2 Only when (1 − ρn,1 )Ptot is equal to a constant ε, the average transmission rate of the 1st MU in the nth cluster with imperfect CSI remains a fixed gap with respect to the ideal rate. Specifically, the transmit power for training sequence 248 X. Chen et al. p P S /ε−1 should satisfy Pn,1 = totαn,1 τ in TDD systems, while the number of feedback bits S /ε) in FDD systems. should satisfy Bn,1 = (M − 1) log2 (Ptot Proof The proof is intuitively. By substituting ρn,1 = 1 − 1 P 1+τ Pn,1 αn,1 Bn,1 M−1 into (1 − S S ρn,1 )Ptot into (1 − ρn,1 )Ptot = ε for = ε for TDD systems and ρn,1 = 1 − 2− S Ptot /ε−1 p S FDD systems, we can get Pn,1 = αn,1 τ and Bn,1 = (M − 1) log2 (Ptot /ε), which  proves Theorem 2. p Remark 3 For the CSI accuracy at the BS, Pn,1 τ (namely transmit energy for training Bn,1 sequence) in TDD systems and M−1 (namely spatial resolution) in FDD systems are two crucial factors. Specifically, given a requirement on CSI accuracy, it is possible to shorten the length of training sequence by increasing the transmit power, so as to leave more time for data transmission in a time slot. However, in order to keep the pairwise orthogonality of training sequences, the length of training sequence τ must be larger than the number of MUs. In other words, the minimum value of τ is N K . Similarly, in FDD systems, it is possible to reduce the feedback bits by increasing the number of antennas M. Yet, in order to fulfill the spatial degrees of freedom for ZFBF at the BS, M must be not smaller than (N − 1)K + 1. This is because the beam wi for the ith cluster should be in the null space of the channels for the (N − 1)K MUs in the other N − 1 clusters. Furthermore, substituting (7.36) and (7.37) into (7.35), we have Rn,1 ≈ − log2 (1 − ρn,1 ) + log2 (θn,1 ) − N −1  i=1   N −1 log2 μin,1 . (7.38) Ξ N −1 i, {μvn,1 }v=1 Given a power allocation scheme, it is interesting that the bound of Rn,1 is independent of channel conditions. As analyzed above, it is possible to improve the average rate by improving the CSI accuracy. Especially, for FDD systems, we have the following lemma: Lemma 3 At the high power region with a large number of feedback bits, the average rate of the 1st MU increases linearly as the numbers of feedback bits increase. Bn,1 Proof Replacing ρn,1 in (7.38) with ρn,1 = 1 − 2− M−1 , Rn,1 is transformed as Rn,1 ≈ N −1  Bn,1 N −1 + log2 (θn,1 ) − Ξ N −1 i, {μvn,1 }v=1 log2 μin,1 , (7.39) M −1 i=1 which yields Lemma 3.  7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access 249 7.4.2 Noise-Limited Case If the interference term is negligible with respect to the noise term due to a low transmit power, then the SINR γn,k , ∀n, k is reduced as H S wn |2 Pn,k , γn,k = αn,k |hn,k (7.40) H wn |2 is which is equivalent to the interference-free case. As discussed earlier, |hn,k χ 2 (2) distributed, then the average transmission rate can be computed as  ∞ S log2 1 + Pn,k αn,k x exp(−x)d x     1 1 = − exp Ei − S . S Pn,k αn,k Pn,k αn,k Rn,k = 0 (7.41) Note that Eq. (7.41) is independent of the CSI accuracy, thus it is unnecessary to carry out channel estimation or CSI feedback in this scenario. Since both intra-cluster interference and inter-cluster interference are negligible, ZFBF at the BS and SIC at the MUs are not required, and all optimization schemes asymptotically approach the same performance. 7.5 Simulation Results To evaluate the performance of the proposed multiple-antenna NOMA technology, we present several simulation results under different scenarios. For convenience, we set M = 6, N = 3, K = 2, Btot = 12, while αn,k and ρn,k are given in Table 7.1 for all simulation scenarios without extra specification. In addition, we use SNR (in dB) S . to represent 10 log10 Ptot First, we verify the accuracy of the derived theoretical expressions. As seen in Fig. 7.2, the theoretical expressions for both the 1st and the 2nd MUs in the 1st cluster well coincide with the simulation results in the whole SNR region, which confirms the high accuracy. As the principle of NOMA implies, the 1st MU performs better than Table 7.1 Parameter Table for (αn,k , ρn,k ), ∀n ∈ [1, 3], and k ∈ [1, 2] n k 1 2 1 2 3 (1.00, 0.90) (0.95, 0.85) (0.90, 0.80) (0.10, 0.70) (0.20, 0.75) (0.15, 0.80) 250 X. Chen et al. Average Rate (b/s/Hz) 2 1.8 Theoretical (R1,1) 1.6 Simulation (R1,1) Theoretical (R1,2) 1.4 Simulation (R1,2) 1.2 1 0.8 0.6 0.4 0.2 0 −10 −5 0 5 10 15 20 25 30 35 SNR (dB) Fig. 7.2 Comparison of theoretical expressions and simulation results the second MU. At high SNR, the average rates of the both MUs are asymptotically saturated, which proves Theorem 1 again. Secondly, we compare the proposed power allocation scheme with the equal power allocation scheme and the fixed power allocation scheme proposed in [5]. Note that the fixed power allocation scheme distributes the power with a fixed ratio 1:4 between the two MUs in a cluster so as to facilitate the SIC. It is found in Fig. 7.3 that the proposed power allocation scheme offers an obvious performance gain over the two baseline schemes, especially in the medium SNR region. Note that practical communication systems, in general, operate at medium SNR, thus the proposed scheme is able to achieve a given performance requirement with a lower SNR. As the SNR increases, the proposed scheme and the equal allocation scheme achieve the same saturated sum rate, but the fixed allocation scheme has a clear performance loss. Next, we examine the advantage of feedback allocation for the FDD-based NOMA system with equal power allocation, cf. Fig. 7.4. As analyzed in Sect. 7.4.2, at very low SNR, namely the noise-limited case, the average rate is independent of CSI accuracy, and thus the two schemes asymptotically approach the same sum rate. As SNR increases, the proposed feedback allocation scheme achieves a larger performance gain. Similarly, at high SNR, both the two schemes are saturated, and the proposed scheme obtains the largest performance gain. For instance, at SNR = 30 dB, there is a gain of more than 0.5 b/s/Hz. Furthermore, we investigate the impact of the total number of feedback bits on the average rates of different MUs at SNR = 35 dB. As shown in Fig. 7.5, the performance of the 1st MU is clearly better than that of the 2nd MU. Moreover, the average rate of the 1st MU is nearly a linear function of the number of feedback bits, which reconfirms the claims of Lemma 3. 7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access 251 6 Sum of Average Rate (b/s/Hz) 5 4 3 2 Adaptive Power Allocation Equal Power Allocation Fixed Power Allocation 1 0 -10 -5 0 5 10 15 20 25 30 35 SNR (dB) Fig. 7.3 Performance comparison of different power allocation schemes Sum of Average Rate (b/s/Hz) 3 2.5 2 1.5 1 Adaptive Feedback Allocation Equal Feedback Allocation 0.5 0 −10 −5 0 5 10 15 20 25 30 35 SNR (dB) Fig. 7.4 Performance comparison of different feedback allocation schemes Then, we investigate the impact of the transmission mode on the performance of the NOMA systems at SNR = 10 dB with equal power allocation in Fig. 7.6. To concentrate on the impact of transmission mode, we set the same CSI accuracy of all downlink channels as ρ. Note that we consider four fixed transmission modes under the same channel conditions in the case of six MUs in total. Consistent with the claims in Lemma 2, mode 4 with N = 1 and K = 6 achieves the largest sum rate at low CSI accuracy, while mode 1 with N = 6 and K = 1 performs best at high CSI accuracy. In addition, it is found that at medium CSI accuracy, mode 2 with N = 3 and K = 2 is optimal, since it is capable to achieve a best balance between intra-cluster 252 X. Chen et al. 1.4 Average Rate (b/s/Hz) 1.2 1 0.8 Rideal 1,2 0.6 R1,1 R 1,2 0.4 0.2 0 0 5 10 15 20 25 30 35 Total Number of Feedback (bits) Fig. 7.5 Asymptotic performance with a large number of feedback bits Sum of Average Rate (b/s/Hz) 7 Mode 1 (N=6, K=1) Mode 2 (N=3, K=2) Mode 3 (N=2, K=3) Mode 4 (N=1, K=6) Dynamic Mode 6 5 4 3 2 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ρ Fig. 7.6 Performance comparison of different transmission modes interference and inter-cluster interference. Thus, we propose to dynamically select the transmission mode according to channel conditions and system parameters. As shown by the red line in Fig. 7.6, dynamic mode selection can always obtain the maximum sum rate. Finally, we exhibit the superiority of the proposed joint optimization scheme for the NOMA systems at SNR = 10 dB. In addition, we take a fixed scheme based on NOMA and a time division multiple access (TDMA) based on OMA as baseline schemes. Specifically, the joint optimization scheme first distributes the transmit power with equal feedback allocation, then allocates the feedback bits based on the distributed power, finally selects the optimal transmission mode. The fixed scheme always adopts the mode 2 (N = 3, K = 2) with equal power and feedback allocation. 7 On the Design of Multiple-Antenna Non-Orthogonal Multiple Access 253 The TDMA equally allocates each time slot to the six MUs and utilizes maximum ratio transmission (MRT) based on the available CSI at the BS to maximize the rate. For clarity of notation, we use ρ to denote the CSI accuracy based on equal feedback allocation. In other words, the total number of feedback bits is equal to Btot = −K ∗ N ∗ (M − 1) ∗ log2 (1 − ρ). As seen in Fig. 7.7, the fixed scheme performs better than the TDMA scheme at low and high CSI accuracy, and slightly worse at the medium regime. However, the proposed joint optimization scheme performs much better than the two baseline schemes. Especially at high CSI accuracy, the performance gap becomes substantially large. For instance, there is a performance gain of about 3 b/s/Hz at ρ = 0.8, and up to more than 5 b/s/Hz at ρ = 0.9. As analyzed in Lemma 2 and confirmed by Fig. 7.6, when ρ is larger than 0.8, which is a common CSI accuracy in practical systems, mode 2 is optimal for maximizing the system performance. Thus, the joint optimization scheme is reduced to joint power and feedback allocation, which requires only a very low complexity. Thus, the proposed NOMA scheme with joint optimization can achieve a good performance with low complexity, and it is a promising technique for future wireless communication systems. 7.6 Conclusion This chapter provided a comprehensive solution for designing, analyzing, and optimizing a NOMA technology over a general multiuser multiple-antenna downlink in both TDD and FDD modes. First, we proposed a new framework for multipleantenna NOMA. Then, we analyzed the performance and derived exactly closed-form expressions for average transmission rates. 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K.K. Mukkavilli, A. Sabharwal, E. Erkip, B. Aazhang, On beamforming with finite rate feedback in multiple-antenna systems. IEEE Trans. Inf. Theory 49(10), 2562–2579 (2003) 46. G.K. Karagiannidis, N.C. Sagias, T.A. Tsiftsis, Closed-form statistics for the sum of squared Nakagami-m variates and its application. IEEE Trans. Commun. 54(8), 1353–1359 (2006) 47. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, USA, 2007) Chapter 8 NOMA for Millimeter Wave Networks Zhengquan Zhang and Zheng Ma 8.1 Introduction Millimeter wave (mmWave) communications [1–3] are one of the most important technologies for 5G wireless networks and beyond, due to the rich spectrum resources in the mmWave band from 30 to 300 GHz. According to Shannon’s capacity theorem, increasing the system bandwidth is an effective way to achieve high data rate transmissions. However, compared with conventional low-frequency cellular networks working in 900 MHz to 3.5 GHz, mmWave networks suffer from high path loss and directional transmissions, and are sensitive to blockage. To promote the application of mmWave communications in 5G and beyond, some efforts from both academic and industry have been devoted. Conventionally, it is considered that mmWave communications will be mainly used to achieve high data rate transmissions for enhanced mobile broadband (eMBB) scenario. Recently, some works also discussed the potential that mmWave communications are used to other application scenarios, such as Internet of Things (IoT) cloud-enabled autonomous vehicles [4]. Therefore, to further improve spectrum efficiency and support massive connectivity in mmWave networks, the combination of mmWave communications with other key technologies, e.g., non-orthogonal multiple access (NOMA) [5–7] and multicast transmission [8], is still very important. The works [9–14] studied NOMA for mmWave networks. The multicast transmission for mmWave networks was also studied in [14, 15]. Z. Zhang (B) · Z. Ma Southwest Jiaotong University, West Section, High-tech Zone, Chengdu, Sichuan, China e-mail: zhang.zhengquan@hotmail.com Z. Ma e-mail: zma@swjtu.edu.cn © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_8 257 258 Z. Zhang and Z. Ma In this chapter, the application of NOMA to mmWave networks is studied, and unicast, multicast, and cooperative multicast transmissions for mmWave-NOMA networks are discussed. Their performance in terms of coverage probability, outage probability, and sum rate is also given. In Sect. 8.2, the fundamentals of mmWave communications including path loss model, directivity gain model, and user association are briefly discussed. Then, unicast transmissions for single-tier mmWaveNOMA networks are studied in Sect. 8.3, which enable a user pair by NOMA to be transmitted on the same radio resources. Next, to further improve the network performance, multicast transmissions for single-tier mmWave-NOMA networks are studied in Sect. 8.4, which distribute media data to all interested users in a non-orthogonal manner. Furthermore, considering two-tier mmWave heterogeneous networks (HetNets), cooperative multicast transmissions for mmWave-NOMA HetNets are further studied in Sect. 8.5, followed by summary in Sect. 8.6. 8.2 Fundamentals of mmWave Communications 8.2.1 Path Loss and Small-Scale Fading mmWave communications exhibit obvious line-of-sight (LOS) and non-LOS (NLOS) propagations due to the very short wavelength, which is different from conventional low-frequency cellular networks. To simply characterize LOS and NLOS links, different path loss exponents can be used. Besides, it is assumed that the type of link observed by users is probabilistic. This means that for a link with length d, the probability that it is a LOS link is p L (d), while it is a NLOS link with probability (w.p) p N (d) = 1 − p L (d). Therefore, the path loss model can be given by [16, 17] L S (d) =  C S,L d −αS,L , w.p p L (d), C S,N d −αS,N , w.p p N (d), (8.1) where for s ∈ {L (LOS), N (NLOS)}, α S,s is the path loss exponent for the s link. The intercept of path loss formula for the s link, C S,s , is a function of reference distance and wavelength (or carrier frequency) and is 10−2 log10 (4π/λc ) for the close-in reference distance dref = 1 m [18]. According to [16, 17], the small-scale fading of mmWave link is assumed to be Nakagami-m fading. The Nakagami-m fading parameters, N S,L and N S,N , are used to characterize the fading of the mmWave LOS and NLOS links, respectively. Let h S,i be the channel coefficient of the link between the i-th mmWave small cell and the user. Then, HS,i = |h S,i |2 follows a normalized Gamma distribution. Similar to [16, 17], shadowing is ignored. 8 NOMA for Millimeter Wave Networks 259 8.2.2 Directivity Gain Beamforming [2] is a key technique to overcome high path loss in mmWave communications by using antenna arrays to form directional beams. Especially, analog beamforming is a simple way to form directional beams by using low-cost phase shifters (PSs). The deployment of antenna arrays at both BSs and user equipments (UEs) can be considered. According to 3GPP TR 38.913, up to 256 Tx and Rx BS antenna elements and 32 Tx and Rx UE antenna elements are assumed. To approximate the beamforming pattern for tractable analysis, sectored antenna model is available. In this case, the directivity gain can be given by [16, 17, 19] G(φ) =  G M , |φ| ≤ θ, G m , |φ| > θ, (8.2) where G M and G m are the main and side lobe gains, respectively, φ is a certain angle, and θ is the beamwidth of the main lobe. Furthermore, according to [19], G M and G m can be equal to 2π−(2π−θ)ε and ε, respectively. At the mmWave base station θ side, G S,M , G S,m , and θ S denote the main lobe gain, side lobe gain, and beamwidth, respectively, which are denoted by G U,M , G U,m , and θU at the user side. Therefore, the directivity gain of the communication link between the user and the i-th mmWave base station is G i = G S (φ S )G U (φU ), where φ S and φU are the angle of departure (AoD) and the angle of arrival (AoA) of the signal, respectively. Further, the user is always assumed to be aligned with its serving base station, B0 , such that the directivity gain is G 0 = G U,M G S,M . According to [16, 17], the directivity gain of the i-th interference mmWave link is assumed to be a discrete random variable (RV), whose probability distribution is G S,i = ak with probability bk , k ∈ {1, 2, 3, 4}, where ak and bk are constants defined in Table 8.1. āk = G U,MakG S,M is the normalized directional gain. Note that for macro BSs, the omnidirectional antennas are considered, i.e., θ = 2π . As a result, there is no directivity gain. 8.2.3 User Association Maximum average received power-based user association scheme enables users to be associated with a base station having the minimum average path loss, which averages the effect of fading and can provide robust performance. Note that this user Table 8.1 Probability mass function of G S,i (i ≥ 1) k 1 2 ak bk 3 4 G U,M G S,M G U,M G S,m G U,m G S,M G U,m G S,m θU θ S (2π )2 θU (2π −θ S ) (2π )2 (2π −θU )θ S (2π )2 (2π −θU )(2π −θ S ) (2π )2 260 Z. Zhang and Z. Ma association strategy will not necessarily result in the maximum average user performance due to unbalanced load distribution. However, load balancing technique can be available to change connected users’ distribution between adjacent base stations by optimizing their handover parameters. In mmWave networks, each user can associate with one LOS or NLOS mmWave base station, which depends on the user location and the distance between user and mmWave base station. According to [16, 17], the probability that user associates with one LOS base station is  ∞  ψ L (x) p N (t)tdt (8.3) g L (x)d x, e−2πλS 0 A L = BL 0 where the probability that the user has at least one LOS base station is B L = 1 − e−2πλS ∞ 0 r p L (r )dr (8.4) , and given the user observes at least one LOS base station, the conditional probability density function (PDF) of the distance to its nearest LOS base station is g L (x) = 2π λ S x p L (x)e−2πλS x 0 r p L (x)dr (8.5) /B L . The association probability to a NLOS base station is A N = 1 − A L . The probability that the user has at least one NLOS base station is B N = 1 − e−2πλS ∞ 0 r (1− p L (r ))dr (8.6) , and given the user observes one or more LOS base stations, the conditional PDF of the distance to its nearest LOS base station is g N (x) = 2π λ S x(1 − p L (x))e−2πλS x 0 r (1− p L (x))dr /B N . (8.7) 8.3 Unicast Transmissions for mmWave-NOMA Networks In this section, we will study unicast transmissions for mmWave-NOMA networks. Conventionally, unicasting employs point-to-point mechanism to achieve data transmissions, which has been widely used in cellular networks. However, with NOMA, the messages of multiple users can be multiplexed in the power domain by superposition coding at the transmitter side, and then each user decodes its desired message by successive interference cancelation (SIC). This non-orthogonality overcomes the loss of degree-of-freedom (DoF) caused by orthogonal transmission at the cost of increasing processing complexity. 8 NOMA for Millimeter Wave Networks 261 8.3.1 System Model Figure 8.1 illustrates the system model of single-tier mmWave-NOMA networks with downlink unicast transmissions. The mmWave base stations are located according to a homogeneous Poisson point process (HPPP) Φ S with density λ S and are assumed to have same transmit power PS . We consider an M-user NOMA scenario, which means that a NOMA user pair with M users sorted by ascending order. The M users are randomly distributed in an mmWave service area. After channel ordering, we have H1 L(r1 ) I1 +σ 2 H2 L(r2 ) ≤ I2 +σ 2 M satisfies m=1 ≤ ··· ≤ HM L(r M ) . I M +σ 2 The power allocated to the m-th user is Pm > 0 and Pm = PS . According to the principle of NOMA, the weak users are allocated to more power in order to ensure that they can decode successfully. Therefore, we have P1 ≥ P2 ≥ · · · ≥ PM . NOMA transmissions enable the messages of all users from a NOMA user pair to be multiplexed in the power domain to form a superposed signal, and then this superposed signal is transmitted on the same radio frequency. Therefore, the signal transmitted by the base station to users is x= M   Pm xm , (8.8) m=1 where xm is the message of the m-th user and satisfies E[|xm |] = 1. The m-th user not only receives the signal from its serving base station, B0 , but also suffers from co-channel interference (CCI) from neighboring base stations and can be expressed as Fig. 8.1 System model of single-tier mmWave-NOMA networks with downlink unicast transmissions 262 Z. Zhang and Z. Ma ym1 =  √ h m G 0 L(rm )x ,  h m,i G i L(rm,i )xi +n m (8.9) Bi ∈Φ\B0 ICCI,m where n m is the additive Gaussian noise with power σ 2 for the m-th user. When the users receive signal from base stations, they employ successive SIC to decode their desired messages. According to the principle of SIC, the m-th user first decodes and cancels the messages of UEi , i = 1, . . . , m − 1 from the received sum signal orderly. The signal-to-interference-plus-noise ratio (SINR) that UEm detects UEi can be expressed as SINR1m,i =  M Hm G 0 L(rm )Pi SINR1m =  M Hm G 0 L(rm )Pm 2 j=i+1 Hm G 0 L(r m )P j + ICCI,m + σ . (8.10) Then, UEm decodes its own message after SIC with detecting SINR given by j=m+1 Hm G 0 L(rm )P j + ICCI,m + σ 2 . (8.11) Note that for the M-th user, its SINR after SIC is given by SINR1M = HM G 0 L(r M )PM . ICCI,M + σ 2 (8.12) 8.3.2 Performance Analysis G 0 L(r ) In mmWave networks, for a random variable (RV), X = HICCI , according to [16, +σ 2 17], its complementary cumulative distribution function (CCDF) can be expressed as (8.13) F̄(T ) = A L F̄L (T ) + A N F̄N (T ), while its CDF is F(T ) = 1 − F̄(T ). For s ∈ {L , N }, F̄s (T ) is the conditional CCDF given that the user is associated with a base station in Φs and can be written as F̄L (T ) ≈ NL  n=1 (−1) n+1 NL n  0 ∞ − e α nη L x S,L T σ S2 C L G S,0 +Q n (T,x)+Vn (T,x) f L (x)d x, (8.14) 8 NOMA for Millimeter Wave Networks 263 and  ∞ NN  − NN F̄N (T ) ≈ e (−1)n+1 n 0 n=1 where B N g N (x) −2πλ 0(C L /C N )1/αL x α N /αL e AN f N (x) = Q n (T, x) = 2π λ S 4  Vn (T, x) = 2π λ S bk Wn (T, x) = 2π λ S 4  bk 8.3.2.1 bk k=1 4   k=1 k=1 Z n (T, x) = 2π λ S +Wn (T,x)+Z n (T,x) f N (x)d x, (8.15) B L g L (x) −2πλ 0(C N /C L )1/α N x αL /α N (1− pL (t))tdt , e AL f L (x) = and α nη N x S,N T σ S2 C N G S,0 4   ∞ F(N L , x ∞ ∞ k=1  ∞ x nη L āk T x αS,L ) p L (t)tdt, N L t αS,L (8.17) (8.18) nC N η L āk T x αS,L ) p N (t)tdt, C L N N t αS,N (8.19) F(N L , nC L η N āk T x αS,N ) p L (t)tdt, C N N L t αS,L (8.20) F(N N , nη N āk T x αS,N ) p N (t)tdt. N N t αS,N (8.21) ψ N (x) bk , F(N N , ψ L (x)  p L (t)tdt (8.16) Coverage Probability The coverage probability can be used to characterize the quality of network coverage, which refers to the probability that the SINR received at user exceeds certain threshold T . According to SIC decoding, the coverage probability of the m-th ordered user in a NOMA user pair is defined as 1 (T ) = P[SINR1m,1 > T, . . . , SINR1m,m > T ]. Pc,m (8.22) Since the maximum SINR that the m-th user detects the message of the i-th user from the superposed signal is lim Hm →∞ SINR1m,i =  M Pi P , the coverage probabilj=i+1 j ity is equal to zero when the SINR threshold T is equal or greater than this maximum detecting SINR. When the SINR threshold T is below this maximum detecting SINR, substituting (8.10) and (8.11) into (8.22) and according to the law of total probability, the coverage probability can be rewritten as 264 Z. Zhang and Z. Ma 1 Pc,m (T ) = P[X m > b1 , . . . , X m > bm ] n  n [1 − F̄(max(b1 , . . . , bm ))]i F̄ n−i (max(b1 , . . . , bm )), i i=m (8.23) G 0 L(rm ) 1 , b = , i = 1, . . . , m, and (a) follows (8.13) and where X m = HImCCI,m  i Pi M +σ 2 (a) =1 − T − j=i+1 Pj the property of order statistics [22]. Therefore, the coverage probability of the m-th ordered user in a NOMA user pair with M users in mmWave-NOMA networks can be finally expressed as ⎧ p1 pm ⎪ ⎨0, T >  Mj=2 P j or, . . . , or T >  Mj=m+1 P j , 1 n Pc,m (T ) =  n ⎪ [1 − F̄(max(b1 , . . . , bm ))]i F̄ n−i (max(b1 , . . . , bm )), otherwise. ⎩1 − i=m i (8.24) 8.3.2.2 Outage Probability The outage probability is used to characterize the probability that the user cannot achieve a target rate τ , and is defined as Po  P[R < τ ]. Further, due to R = log2 (1 + SINR) < τ , its form related with SINR can be expressed as Po  P[SINR < 2τ − 1]. We assume that the target rate for the m-th user is τm > 0 and define the outage event E m,i = {SINRm,i < γi }, where γi = 2τi − 1. This means that the m-th user failed to decode the message of the i-th user. Correspondingly, the complementary outage event is Ē m,i = {SINRm,i ≥ γi }. According to the principle of NOMA, the outage probability of the m-th user can be expressed as 1 = P[E m,1 ∪ E m,2 ∪ · · · ∪ E m,m ] = 1 − P[ Ē m,1 ∩ Ē m,2 ∩ · · · ∩ Ē m,m ]. Po,m (8.25) With SIC decoding, when the SINR that the m-th user detects the message of the i-th user is smaller than its maximum SINR, SINR1m,i =  M Pi P , the SIC decoding j=i+1 failed. That is, when γ1 ≥ P M 1 is equal to one. When γ1 < j=2 P j P M 1 j=2 P j or . . . or γm ≥ M Pm j=m+1 and . . . and γm < j , the outage probability Pj Pm M j=m+1 Pj , substituting (8.10) and (8.11) into (8.25), the outage probability can be written as 1 Po,m = 1 − P[X m > c1 , . . . , X m > cm ] n (b)  = i=m n [1 − F̄(max(c1 , . . . , cm ))]i F̄ n−i (max(c1 , . . . , cm )), i (8.26) 8 NOMA for Millimeter Wave Networks where ci = Pi γi−1 − 1 M j=i+1 Pj 265 , and (b) follows the property of order statistics and the complementary property of CCDF and CDF. Therefore, the outage probability of the m-th ordered user in a NOMA user pair with M users can be finally expressed as 1 Po,m ⎧ Pm P1 ⎪ ⎨1, γ1 ≥  Mj=2 P j or . . . or γm ≥  Mj=m+1 P j , n =  n ⎪ [1 − F̄(max(c1 , . . . , cm ))]i F̄ n−i (max(c1 , . . . , cm )), otherwise. ⎩ i=m i (8.27) 8.3.2.3 Sum Rate To the m-th user’s message, xm , it should ensure that all users after it can also decode the message xm in order to perform SIC successfully. Therefore, the data rate for the m-th user is equal to   1 1 , Rm+1,m , . . . , R 1M,m , Rm1 = min Rm,m (8.28) 1 where Rn,m = log2 (1 + SINR1n,m ), n = m, . . . , M is the rate that the n-th user decodes the data xm . Further, we have  Hn G 0 L(rn )Pm 1 Rn,m = log2 1 +  M Hn G 0 L(rn )P j + ICCI,n + σ 2 ⎞ P m ⎠. = log2 ⎝1 +  M ICCI,n +σ 2 P + j=m+1 j Hn G 0 L(rn ) j=m+1 ⎛ 2) 1) ≤ H2 L(r ≤ ··· ≤ Due to H1 L(r I1 I2 Therefore, we have HM L(r M ) , IM  (8.29) 1 1 we have Rm,m ≤ Rm+1,m ≤ · · · ≤ R 1M,m .   1 1 1 . , Rm+1,m , . . . , R 1M,m = Rm,m Rm1 = min Rm,m (8.30) The sum rate for a typical NOMA user pair consisting of M users is defined as the sum of the average rate for each user and is τt1  M  m=1 E[Rm1 ] M 1  = E[ln(1 + SINR1m )]. ln 2 m=1 (8.31) 266 Z. Zhang and Z. Ma For the m-th ordered user, m = 1, . . . , M − 1, the average rate is taken over both the spatial PPP and the fading distribution. We have 1 E[ln(1 + SINR1m )] ln 2 ⎧  ⎨   ln 1+  M Pm P (c) 1 j=m+1 j = P Xm > AL ln 2 ⎩ r >0 0 R̄m = + AN =   r >0  ln 1+  M Pm j=m+1 P j F̄m 0 where (c) follows E[W ] = lim ICCI,m →0,Hm →∞ P Xm > 0 j=i+1 P j   ln 1+  M Pm   Pm et −1 t>0 − 1 M j=m+1 Pm et −1 Pm et −1 Pj  − − 1 M j=m+1 1 M j=m+1 Pj Pj   dt f L (r )dr dt f N (r )dr ⎫ ⎬ ⎭ dt, (8.32) P(W > t)dt for a positive RV, W [20], and ln 1 +  M Hm G 0 L(rm )Pm 2 l=m+1 H M G 0 L(rm )P j + ICCI,m + σ   = ln 1 +  M Pm j=m+1 P j  . (8.33) For the M-ordered user, the integral domain of variable t in (8.32) is (0, ∞). Therefore, the average rate can be written as R̄ M =  ∞ 0 F̄M et − 1 dt. PM (8.34) Combining (8.31), (8.32), and (8.34), the sum rate for a NOMA user pair in mmWave networks can be expressed as    M−1     ln 1+Pm ( Mj=i+1 P j )−1 1 1 τt1 = F̄m dt  Pm ln 2 m=1 0 − M j=m+1 P j et −1  ∞ et − 1 dt . + F̄M PM 0 (8.35) 8.3.3 Numerical Results To evaluate the performance of mmWave-NOMA networks, the parameters are used as follows: the carrier frequency is 28 GHz and the system bandwidth is 100 MHz; the base station transmit power is 30 dBm; the density of mmWave base station is 1 λ S = π200 2 ; the path loss exponents for LOS and NLOS links are set as 2 and 4, 8 NOMA for Millimeter Wave Networks 267 1 SINR Coverage Probability 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Analytical Results Simulations mmWave−NOMA User1 mmWave−NOMA User2 mmWave−OMA User1 mmWave−OMA User2 0.2 0.1 0 −10 −8 −6 −4 −2 0 2 4 6 8 10 SINR Threshold (dB) Fig. 8.2 SINR coverage probabilities of mmWave-NOMA networks with fixed power ratio (0.8, 0.2) respectively, and a NOMA user pair consists of two random users with ascending order. To give a fair comparison to NOMA, OMA employs equal resource allocation for each user. Figure 8.2 shows the coverage probabilities of mmWave-NOMA networks with fixed power ratio (0.8, 0.2). The results show that the coverage probabilities of both NOMA user1 and NOMA user2 are lower than that of OMA. This is because interuser interference introduced by NOMA deteriorates the SINR received at users. The results also show that the coverage probabilities of NOMA users decline to zero when the SINR threshold exceeds a certain value (i.e., PP21 ), while OMA can still achieve some coverage. The reason is that the maximum SINR detecting NOMA user1 is limited by PP21 , according to SIC decoding. Figure 8.3 shows the outage probabilities of mmWave-NOMA networks with fixed power ratios (0.8, 0.2) and (0.9, 0.1). The results show that compared with OMA, the outage probability of the NOMA-weak user (i.e., User1) can be improved, while the NOMA-strong user (i.e., User2) suffers from some loss. The results also show that with the increase of power allocated to the NOMA-weak user, the outage probability of the NOMA-weak user can be further improved, while the NOMA-strong user’s outage probability becomes worse. Figure 8.4 shows the average rates for mmWave-NOMA networks with fixed power ratios (0.8, 0.2) and (0.9, 0.1). The results show that NOMA can achieve higher data rate than conventional OMA. This is because NOMA can enable all users in a NOMA user pair to occupy whole system bandwidth such that there is no loss in DoF, at the cost of the increase of processing complexity and the introduction 268 Z. Zhang and Z. Ma (a) 0 Outage Probability 10 −1 10 Analytical Results Simulations mmWave−NOMA User1 mmWave−NOMA User2 mmWave−OMA User1 mmWave−OMA User2 −2 10 0 0.1 0.2 0.3 0.4 0.5 0.6 Rate Threshold (b/s/Hz) (b) 0 Outage Probability 10 −1 10 Analytical Results Simulations mmWave−NOMA User1 mmWave−NOMA User2 mmWave−OMA User1 mmWave−OMA User2 −2 10 0 0.1 0.2 0.3 0.4 0.5 0.6 Rate Threshold (dB) Fig. 8.3 Outage probabilities of mmWave-NOMA networks with fixed power ratios: a (0.8, 0.2); b (0.9, 0.1) of inter-user interference. However, the results also show that for a certain transmit power range, e.g., [0, 5] dBm, NOMA with power ratio 0.9 can achieve higher data rate for the weak user (i.e, User1), while suffers from a lower data rate for the strong user (i.e., User2) and sum rate. This is because the fixed power ratio for NOMA is not optimal for all transmit powers. Comparing Fig. 8.4a, b, the weak user with power ratio 0.9 can achieve higher average rate than that of power ratio 0.8, while the opposite trend is for the strong user, which results in a lower sum rate. 8 NOMA for Millimeter Wave Networks 269 (a) 9 mmWave−NOMA Sum Rate mmWave−NOMA User1 mmWave−NOMA User2 mmWave−OMA Sum Rate mmWave−OMA User1 mmWave−OMA User2 8 Rate (b/s/Hz) 7 6 5 4 3 2 1 0 0 5 10 15 20 25 30 25 30 Transmit Power (dBm) (b) 9 mmWave−NOMA Sum Rate mmWave−NOMA User1 mmWave−NOMA User2 mmWave−OMA Sum Rate mmWave−OMA User1 mmWave−OMA User2 8 Rate (b/s/Hz) 7 6 5 4 3 2 1 0 0 5 10 15 20 Transmit Power (dBm) Fig. 8.4 Average rates for mmWave-NOMA networks with fixed power ratios: a (0.8, 0.2); b (0.9, 0.1) 8.4 Multicast Transmissions for mmWave-NOMA Networks Compared with unicast transmissions discussed in Sect. 8.3, multicast transmissions employ the point-to-multipoint mechanism to distribute the same media data to multiple interested users on the same radio resources such that higher spectrum efficiency can be achieved. However, conventional multicast transmissions just deliver single data stream with a low data rate in order to ensure that most users can decode the media. With the help of NOMA, multicast transmissions can be enhanced by 270 Z. Zhang and Z. Ma multiplexing multiple data streams with different data rates in the power domain. As a result, users can decode data streams according to their channel conditions. In this section, we will discuss multicast transmissions for mmWave-NOMA networks. 8.4.1 System Model Figure 8.5 illustrates the system model of multicast transmissions in mmWaveNOMA networks. The mmWave base stations are located according to an HPPP Φ S with density λ S . The mmWave base stations are assumed to have same transmit power PS . The users are located according to an HPPP ΦU with density λU . With NOMA, N -layer superposition coded multicast transmission can be achieved by power allocation, which consists of one primary layer and N − 1 secondary layers. The primary layer carries the basic data, while the secondary layers carry the corresponding enhanced data. The fixed data rate for each layer is also assumed. Generally, the scalable media can be encoded into one basic data and N − 1 enhanced data by source layered coding, where the basic data provides the basic service quality, while the enhanced data are used to improve the service quality. Note that the enhanced data cannot work without the basic data. With NOMAenabled multicast transmissions, the basic data and enhanced data are multiplexed in the power domain to form a superposed signal. Then, this superposed signal is distributed on the same radio resources to all users who desire to receive the media. When users receive this superposed signal, they first decode the basic data to obtain the basic service quality directly, then try to decode the enhanced data to achieve better service quality. For the weak users, they can just decode the low-rate basic data, while the strong users can decode both the low-rate basic data and high-rate enhanced data. Therefore, NOMA-enabled multicast transmissions can fully utilize the channel difference between users to improve the performance of the strong users. This technique overcomes the shortage of conventional multicast transmissions that Fig. 8.5 System model of single-tier mmWave-NOMA networks with multicast transmissions 8 NOMA for Millimeter Wave Networks 271 the strong users cannot fully utilize their good channel to obtain better service quality, as the conventional one just transmits single low-rate data to ensure that all users can decode the media successfully. Without loss of generality, two-layer superposition coded multicast transmission is considered as an example. The mmWave base station transmits a superposed signal to all users within its coverage as x=  √ αpxB + 1 − αpxE , (8.36) where 0 < α p < 1 is the power allocation factor (PAF), x B and x E are the transmit messages of the primary and secondary layers, respectively. The signal received at the user with random distance, d0 , from its serving mmWave base station can be expressed as  yS2 = h S,0 G S,0 PS L S (d0 )x +  Xi ∈Φ S \B0   h S,i G S,i PS L S (di )xi +n S . (8.37) IS After receiving the superposed signal, the user first decodes the primary layer, and then cancel it from the received signal before decoding the secondary layer. Therefore, substituting (8.36) into (8.37), the SINRs of detecting the primary and secondary layers can be written, respectively, as SINR2S,P L = α p HS G S,0 L S (d0 ) ,  HS,i G S,i L S (di ) +σ S2 (1 − α p )HS G S,0 L S (d0 ) + X ∈Φ \B  i S 0 IS (8.38) and SINR2S,SL =   (1 − α p )HS G S,0 L S (d0 ) Xi ∈Φ S \B0 HS,i G S,i L S (di ) +σ S2 . (8.39) IS Note that σ S2 is the thermal noise power, normalized by transmit power, PS . 8.4.2 Performance Analysis 8.4.2.1 Coverage Probability The coverage probability that users can decode the primary layer relative to the SINR threshold TP L can be written as 272 Z. Zhang and Z. Ma 2 2 Pc,P L (T P L ) = E R [P[SINR P L > T P L | R = r ]]. (8.40) Considering the maximum SINR for detecting the primary layer, lim Hs →∞ SINR2P L = αp , the integral domain is 1−α p ! " D = (HS ) | SINR2P L > TP L #  (8.41) TP L (I S + σ S2 ) αp . |TP L < = (HS ) | HS > (α p − (1 − α p )TP L )G S,0 L S (r0 ) 1 − αp Therefore, the coverage probability can be expressed as $ $ 2 Pc,S,P P HS > (T , α ) = E P L p R L TP L (I S + σ S2 ) |R=r (α p − (1 − α p )TP L )G S,0 L S (r0 ) %% . (8.42) According to [16, 17] and after some manipulations, the coverage probability of the primary layer can be obtained as 2 Pc,S,P L (T P L , α p ) ⎧   α nη L x S,L T P L σ S2 ⎪ ⎪ ⎪ − +Q n (T P L ,x)+Vn (T P L ,x) N ∞ ⎪ L    C (α −(1−α )T )G  ⎪ p p L PL S ⎪ ⎪ (−1)n+1 NnL e f L (x)d x ⎪ AL ⎪ ⎪ n=1 0 ⎨   α S,N 2 nη N x TP L σS ≈ +Wn (T P L ,x)+Z n (T P L ,x) − N ⎪ ∞ N    ⎪ C (α −(1−α )T )G p p α N P L S ⎪ ⎪ f N (x)d x, T P L < 1−αp , e +A N (−1)n+1 NnN ⎪ ⎪ p ⎪ n=1 ⎪ 0 ⎪ αp ⎪ ⎩ 0, T ≥ . PL 1−α p (8.43) The coverage probability of both the primary and secondary layers relative to the SINR thresholds TP L and TSL can be written as 2 2 2 Pc,P SL (T P L , TSL , α p ) = E R [P[{SINR P L > T P L ∩ SINR SL > TSL } | R = r ]]. (8.44) Its integral domain is & ' D = (HS ) | {SINR2P L > T P L ∩ SINR2S L > TS L } )) ( ( TS L (I S + σ S2 ) T P L (I S + σ S2 ) . = (HS ) | HS > ∩ HS > (α p − (1 − α p )T P L )G S,0 L S (r0 ) (1 − α p )G S,0 L S (r0 ) Let TP L (I S +σ S2 ) (α p −(1−α p )TP L )G S,0 L S (r0 ) = TS L (I S +σ S2 ) , (1−α p )G S,0 L S (r0 ) we can obtain α p = Further, the coverage probability can be expressed as (8.45) TP L (1+TS L ) . TS L +TP L (1+TS L ) 8 NOMA for Millimeter Wave Networks 273 2 Pc,S,P SL (T P L , TSL , α p ) * * ++ ⎧ TP L (I S +σ S2 ) P L (1+TS L ) ⎨ E R P HS > | R = r , α p ≤ TS LT+T , (α p −(1−α p )TP L )G S,0 L S (r0 ) P L (1+TS L ) * * ++ = 2 T (I +σ ) SL S ⎩E P H > P L (1+TS L ) S | R = r , α p > TS LT+T . S R (1−α p )G S,0 L S (r0 ) P L (1+TS L ) (8.46) Let $ $ %% TSL (I S + σ S2 ) 2 (8.47) |R=r . (TSL , α p ) = E R P HS > Pc,S,SL (1 − α p )G S,0 L S (r0 ) 2 Therefore, the coverage probability, Pc,P SL (T P L , TSL , α p ), can finally be written as 2 Pc,S,P SL (T P L , TSL , α p ) = ( TP L (1+TS L ) 2 Pc,S,P L (T P L , α p ), α p ≤ TS L +TP L (1+TS L ) , TP L (1+TS L ) 2 Pc,S,SL (TSL , α p ), α p > TS L +TP L (1+TS L ) , (8.48) 2 2 where Pc,S,P L (T P L , α p ) is expressed as in (8.43) and Pc,S,SL (TSL , α p ) is approximated as 2 Pc,S,SL (TSL , α p )  ∞ NL  − NL ≈ AL e (−1)n+1 n 0 n=1  ∞ NN  − n+1 N N + AN e (−1) n 0 n=1 8.4.2.2 α nη L x S,L TS L σ S2 C L (1−α p )G S +Q n (TS L ,x)+Vn (TS L ,x) α nη N x S,N TS L σ S2 C N (1−α p )G S +Wn (TS L ,x)+Z n (TS L ,x) f L (x)d x f N (x)d x. (8.49) Average Number of Served Users For the mmWave multicast cluster, Bo , the average number of served users by the primary layer can be expressed as ⎡ Eo [N P2 L ]  Eo ⎣  y∈ΦU,Bo ⎤ I(E 2P L (y))⎦ , (8.50) where E 2P L (y) = {SINR2S,P L ≥ 2 R P L − 1}. Given the user density λU , the average number of users covered by a beam with width θ S is λU θ S (2π λ S )−1 . Further, considering the average coverage probability, Eo [N P2 L ] can be finally expressed as 2 −1 Eo [N P2 L ] = λU Pc,S,P L (T P L , α p )θ S (2π λ S ) , (8.51) 274 Z. Zhang and Z. Ma The average number of served users, who can decode the data contained in both the primary and secondary layers, can be written as ⎡ Eo [N P2 SL ]  Eo ⎣  y∈ΦU,Bo ⎤ I(E 2P SL (y))⎦ , (8.52) where E 2P SL (y) = {{SINR2S,P L ≥ 2 R P L − 1} ∩ {SINR2S,SL ≥ 2 R S L − 1}}. Similarly, Eo [N P2 SL ] can be finally expressed as 2 −1 Eo [N P2 SL ] = λU Pc,S,P SL (T P L , TSL , α p )θ S (2π λ S ) . 8.4.2.3 (8.53) Sum Rate The sum rate for NOMA multicast is defined as the mean of the sum rate of all users in coverage of the multicast cluster, who successfully decode the primary layer with data rate, R P L , or both the primary and secondary layers with data rate, R P L + R SL . This is given by 2 = (Eo [N P2 L ] − Eo [N P2 SL ])R P L + Eo [N P2 SL ](R P L + R SL ) R̄sum = Eo [N P2 L ]R P L + Eo [N P2 SL ]R SL . (8.54) Note that Eo [N P2 L ] − Eo [N P2 SL ] is the average number of served users by the mmWave multicast cluster, who can only decode the primary layer. Combining (8.43), (8.49), (8.51), (8.53), and (8.54), the sum rate for the mmWave multicast cluster can be expressed as 2 R̄sum ⎧ (R +R )λ P 2 (T ,α )θ PL S L U c,S,P L PL p S P L (1+TS L ) ⎪ , α p ≤ TS LT+T and TPL < ⎪ 2πλ S P L (1+TS L ) ⎪ ⎪ 2 2 ⎨ R P L Pc,S,P L (TP L ,α p )λU θS R S L Pc,S,S L (TP L ,TS L ,α p )λU θ S + , 2πλ S 2πλ S = α TP L (1+TS L ) ⎪ α p > TS L +TP L (1+TS L ) and TPL < 1−αp p , ⎪ ⎪ ⎪ αp ⎩ 0, TP L ≥ 1−α p , αp , 1−αp (8.55) where, TP L = 2 R P L − 1 and TSL = 2 R S L − 1. 8.4.3 Numerical Results Figure 8.6 depicts the coverage probabilities of multicast transmissions for mmWaveNOMA networks with fixed power ratio (0.8, 0.2). It can be observed that NOMA multicast can provide a similar primary coverage layer as the conventional one and achieve a secondary coverage layer as well. However, when the SINR threshold 8 NOMA for Millimeter Wave Networks 275 0.8 SINR Coverage Probability 0.7 0.6 0.5 0.4 0.3 Analytical results 0.2 Simulations Multicast NOMA multicast (PL) 0.1 0 −10 NOMA multicast (SL) −8 −6 −4 −2 0 2 4 6 8 10 SINR Threshold (dB) Fig. 8.6 SINR coverage probabilities of multicast transmission for mmWave-NOMA networks with fixed power ratio (0.8, 0.2) αP , that users can detect the primary layer, T is larger than the maximum SINR, 1−α P the SINR coverage probabilities of the primary and secondary layers are equal to zero. This is because if users failed to decode the primary layer, they do not further decode the secondary layer through SIC. Figure 8.7 depicts the sum rates for multicast transmission for mmWave-NOMA networks with fixed power ratios (0.8, 0.2) and (0.95, 0.05) and the secondary layer data rate, R SL = 4 b/s/Hz. The results show that NOMA multicast can achieve a significant gain of sum rate, compared with the conventional one in the low multicast rate region. This is because NOMA multicast can fully utilize the channel conditions of strong users. However, the maximum SINR for detecting the primary layer is αP such that NOMA multicast cannot work when the multicast rate for the primary 1−α P αP ). Comparing Fig. 8.7a, b, more power is allocated to layer exceeds log2 (1 + 1−α P the primary layer, a higher multicast rate for the primary layer can be provided, yet a lower sum multicast rate is achieved. This means that NOMA multicast gradually degrades to the conventional one, with the increase of power allocated to the primary layer. Figure 8.8 depicts the sum rates for NOMA multicast with different power allocations, given 0.4, 1 b/s/Hz for the primary layer and 2, 4, 6 b/s/Hz for the secondary layer. It is shown that for each multicast rate pair, as the power ratio grows, the sum rate first experiences a sharp rise to the maximum, then it falls slowly in the medium power ratio region. Finally, it declines rapidly to the lowest point in the high power ratio region. Furthermore, with fixed multicast rate for the secondary layer, the sum rate of the primary layer with rate 1 b/s/Hz is higher than that of the primary layer with rate 0.4 b/s/Hz, which requires more power to be allocated to the primary layer. The sum rate of the secondary layer with rate 6 b/s/Hz is higher than that of the 276 Z. Zhang and Z. Ma (a) 100 90 80 Sum Rate (b/s/Hz) 70 60 50 40 Analytical results Simulations 30 Multicast NOMA multicast 20 10 0 0 1 2 3 4 5 6 7 8 9 10 Multicast Rate (b/s/Hz) (b) 100 90 Sum Rate (b/s/Hz) 80 70 60 50 40 Analytical results Simulations Multicast NOMA multicast 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 Multicast Rate (b/s/Hz) Fig. 8.7 Sum rates for multicast transmission for mmWave-NOMA networks with fixed power ratios. a (0.8, 0.2); b (0.95, 0.05) secondary layer with rates 4 and 2 b/s/Hz, when the multicast rate for the primary layer is fixed. 8.5 Cooperative Multicast Transmissions for mmWave-NOMA HetNets In this section, we will further discuss multicast transmissions in a two-tier mmWaveNOMA HetNet consisting of one low-frequency macro base station (MBS) tier and 8 NOMA for Millimeter Wave Networks 277 90 80 Sum Rate (b/s/Hz) 70 60 50 40 Analytical results Simulations PL0.4−SL2 PL0.4−SL4 PL0.4−SL6 PL1.0−SL2 PL1.0−SL4 PL1.0−SL6 30 20 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Power Ratio Fig. 8.8 Sum rate for multicast transmissions for mmWave-NOMA networks with different power ratios one mmWave base station tier, which is the typical deployment for mmWave wireless networks, and introduce a cooperative multicast scheme for mmWave-NOMA networks to further improve the system performance. The cooperative multicast scheme can increase the success probability of decoding the primary layer with the help of cooperation from MBSs. 8.5.1 System Model Figure 8.9 illustrates the system model of cooperative NOMA-enabled multicast transmissions for a two-tier mmWave HetNet, which consists of one low-frequency MBS tier with transmit power PM and one mmWave base station tier with transmit power PS . Cooperative NOMA-enabled multicast enables the MBS tier to cooperatively transmit the primary layer with a low data rate. With the cooperation of MBSs, the users who failed to decode the primary layer of the superposed signal from the mmWave base station tier will try to decode it from the macro BS tier. If the primary layer is decoded successfully, they cancel it from the superposed signal and further decode the secondary layer. Thus, this increases the success probability that decodes the primary and secondary layers, such that the NOMA multicast performance can be improved. The signal received at the user with random distance, d0 , from its serving and interfering MBSs can be expressed as  −α /2 3 yM = h M,0 PM d0 M x B +  Xi ∈Φ M \B0  h M,i  −α M /2 PM di IM xi,B +n M , (8.56) 278 Z. Zhang and Z. Ma Fig. 8.9 System model of cooperative multicast transmissions for mmWave-NOMA HetNets where n M is the additive Gaussian noise with power σ M2 . Therefore, the SINR of decoding the data can be written as SINR3P L =  HM,0 PM d0−α M Xi ∈Φ M \B0  HM,i PM di−α M +σ M2 . (8.57) IM The SINR of decoding the primary layer, SINR3S,P L , from the mmWave base station tier, can be written as in (8.38), while the SINR of the secondary layer, SINR3S,SL , can be expressed as in (8.39). 8.5.2 Performance Analysis 8.5.2.1 Coverage Probability With cooperative NOMA multicast in a two-tier mmWave HetNet, when users decode the primary layer from a macro or mmWave base station successfully, the basic data can be recovered. Therefore, the coverage probability of the primary layer can be expressed as 3 3 3 Pc,P L (T P L , α p ) = P{{SINR M,P L > T P L } ∪ {SINR S,P L > T P L }} = P{SINR3M,P L > TP L } + P{SINR3S,P L > TP L } − P{SINR3M,P L > TP L }P{SINR3S,P L (8.58) > TP L }, where P{SINR3S,P L > TP L } is obtained as in (8.43) due to SINR3S,P L = SINR2S,P L . 3 And according to [21], Pc,M,P L (T P L ) can be obtained as 8 NOMA for Millimeter Wave Networks 279 3 3 Pc,M,P L (T P L ) = P{SINR M,P L > T P L } $ %  HM PM R −α M P > T | R = r f R (r )dr = PL I M + σ M2 r >0  ∞ 2 −1 α M /2 = π λM e−πλ M (1+ρ(TP L ,α M ))x−TP L (PM /σ M ) x d x, (8.59) 0 where 2/α M ρ(TP L , α M ) = TP L  ∞ −2/α M TP L (1 + t α M /2 )−1 dt. (8.60) The coverage probability of the secondary layer can be expressed as 3 3 3 Pc,P SL (T P L , TSL , α p ) = P{{SINR S,SL > TSL ∩ SINR S,P L > T P L } ∪ {SINR3S,SL > TSL ∩ SINR3M,P L > TP L }}. (8.61) After some manipulations, 3 3 3 Pc,P S L (T P L , TS L , α p ) = P{SINR S,S L > TS L ∩ SINR S,P L > T P L } + P{SINR3S,S L > TS L }P{SINR3M,P L > T P L } − P{SINR3S,S L > TS L ∩ SINR3S,P L > T P L }P{SINR3M,P L > T P L }. (8.62) Since SINR3S,SL = SINR2S,SL and SINR3S,P L = SINR2S,P L , P{SINR3S,SL > TSL ∩ SINR3S,P L > TP L } can be written as in (8.48), and P{SINR3S,SL > TSL } can be expressed as in (8.49). Therefore, combining (8.48), (8.49), (8.58), and (8.62), 3 Pc,P SL (T P L , TSL , α p ) can be finally expressed as 3 2 3 Pc,P SL (T P L , TSL , α p ) = Pc,S,P SL (T P L , TSL , α p )(1 − Pc,M,P L (T P L )) 3 2 + Pc,S,SL (TSL , α p )Pc,M,P L (T P L ). 8.5.2.2 (8.63) Average Number of Served Users For the mmWave multicast cluster, Bo , the average number of served users by the primary layer can be expressed as ⎡ Eo [N P3 L ]  Eo ⎣  y∈ΦU,Bo ⎤ I(E 3P L (y))⎦ , (8.64) where E 3P L (y) = {SINR3M,P L ≥ 2 R P L − 1 ∪ SINR3S,P L ≥ 2 R P L − 1}. Given the user density λU , the average number of users covered by a beam with width θ S is 280 Z. Zhang and Z. Ma λU θ S (2π λ S )−1 . Further, considering the average coverage probability, Eo [N P3 L ] can be finally expressed as 3 −1 Eo [N P3 L ] = λU Pc,P L (T P L )θ S (2π λ S ) . (8.65) The average number of served users by the secondary layer can be expressed as ⎡ Eo [N P3 SL ]  Eo ⎣  y∈ΦU,Bo ⎤ I(E 3P SL (y))⎦ , (8.66) where E 3P SL (y) = {{SINR3M,P L ≥ 2 R P L − 1 ∪ SINR3S,P L ≥ 2 R P L − 1} ∩ {SINR3S,SL ≥ 2 R S L − 1}}. Similarly, Eo [N P3 SL ] can be finally expressed as 3 −1 Eo [N P3 SL ] = λU Pc,P SL (T P L , TSL , α p )θ S (2π λ S ) . 8.5.2.3 (8.67) Sum Rate The sum rate for cooperative NOMA multicast is defined as the mean of the sum rate for all users in coverage of the multicast cluster and is equal to 3 = R P L (Eo [N P3 L ] − Eo [N P3 SL ]) + (R P L + R SL )Eo [N P3 SL ] R̄sum = R P L Eo [N P3 L ] + R SL Eo [N P3 SL ]. (8.68) Combining (8.65), (8.67), and (8.68), the sum rate can be finally expressed as 3 3 3 −1 R̄sum = (R P L Pc,P L (T P L ) + R SL Pc,P SL (T P L , TSL , α p ))λU θ S (2π λ S ) . (8.69) 8.5.3 Numerical Results Figure 8.10 plots the coverage probabilities of cooperative multicast transmissions for mmWave-NOMA HetNets with fixed power ratio (0.8, 0.2). The results show that compared with multicast and NOMA multicast, cooperative NOMA multicast can further improve the coverage probabilities of the primary and secondary layers. More specifically, compared with multicast, cooperative NOMA multicast can achieve superior coverage in the low SINR threshold region, while compared with NOMA multicast, it can achieve better primary coverage, especially in the low and high SINR threshold regions, while it provides better secondary coverage, especially in the high SINR threshold region. This is because cooperative NOMA multicast enables the MBS tier to transmit a copy of the primary layer without power split as well. As a result, users can receive two copies of the primary layer, which increases the success 8 NOMA for Millimeter Wave Networks 281 probability of decoding the primary layer. This also increases the success probability of decoding the secondary layer, as users can try to further decode the secondary layer when they fail to decode the primary layer from a mmWave base station but succeed to decode the primary layer from a MBS. Figure 8.11 plots the sum rates for cooperative multicast transmission for mmWave-NOMA HetNets with fixed power ratios (0.8, 0.2) and (0.95, 0.05). The results show that cooperative NOMA multicast can achieve higher sum rate than NOMA multicast, especially in the medium multicast rate region. This is because in cooperative NOMA multicast scheme, MBS transmits a replica of the primary layer αP , of decoding the primary as well, which overcomes the maximum SINR limit, 1−α P layer caused by NOMA transmission. As a result, this increases the success probability of decoding the primary and secondary layers. Comparing Fig. 8.11a, b, more power is allocated to the primary layer, a higher multicast rate can be provided, yet a lower sum multicast rate is achieved. This means that cooperative NOMA multicast gradually degrades to conventional multicast, with the increase of power allocated to the primary layer. 8.6 Summary This chapter applied NOMA to mmWave networks and discussed unicast, multicast and cooperative multicast transmissions for mmWave-NOMA networks. An analytical framework for performance analysis of large-scale mmWave-NOMA networks by using stochastic geometry was also given. Based on this framework, analytical 1 SINR Coverage Probability 0.9 0.8 0.7 0.6 0.5 0.4 Analytical results Simulations Multicast NOMA multicast (PL) NOMA multicast (SL) Cooperative NOMA multicast (PL) Cooperative NOMA multicast (SL) 0.3 0.2 0.1 0 −10 −8 −6 −4 −2 0 2 4 6 8 10 SINR Threshold (dB) Fig. 8.10 SINR coverage probabilities of cooperative multicast transmission for mmWave-NOMA HetNets with fixed power ratio (0.8, 0.2) 282 Z. Zhang and Z. Ma (a) 100 90 80 Sum Rate (b/s/Hz) 70 60 50 40 Analytical results Simulations Multicast NOMA multicast Cooperative NOMA multicast 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 Multicast Rate (b/s/Hz) (b) 100 90 Sum Rate (b/s/Hz) 80 70 60 50 Analytical results 40 Simulation results Multicast 30 NOMA multicast 20 Cooperative NOMA multicast 10 0 0 1 2 3 4 5 6 7 8 9 10 Multicast Rate (b/s/Hz) Fig. 8.11 Sum rates for cooperative multicast transmission for mmWave-NOMA HetNets with fixed power ratios: a (0.8, 0.2); b (0.95, 0.05) expressions for SINR coverage probability, outage probability, and sum rate were provided to evaluate the performance of the presented schemes. It can be concluded that: (1) NOMA can achieve better performance than OMA, by multiplexing multiple users in the power domain; (2) compared with multicast, NOMA multicast multiplexes multiple data streams with different multicast rates in the power domain such that users can decode data streams according to their channel conditions, which significantly increases the sum rate; (3) the cooperative NOMA multicast increases the success probability of decoding data streams from the superposed signal with the help of cooperation of MBSs such that it further improves the performance of NOMA 8 NOMA for Millimeter Wave Networks 283 multicast. Some further research directions on NOMA for mmWave networks are pointed out as follows: • Power Allocation for mmWave-NOMA Networks: the power ratio is a key factor that NOMA achieves better performance than the orthogonal one. The fixed power ratio is a simple way for NOMA, but it cannot always achieve optimal performance. This is because it does not utilize channel state information (CSI) in real time. The optimization of power ratio to further improve the system performance according to the instantaneous CSI is a great challenge, especially for multicast transmissions. • Cooperation for mmWave-NOMA Networks: HetNet is an important deployment form for mmWave communications. This chapter only discusses cooperative multicast for mmWave-NOMA networks that MBSs cooperate transmission for the primary layer. It is important to exploit other cooperation schemes to further improve the performance of unicast and multicast transmissions for mmWave-NOMA networks. References 1. T.S. Rappaport et al., Millimeter wave mobile communications for 5G cellular: it will work! IEEE Access 1, 335–349 (2013) 2. W. Roh et al., Millimeter-wave beamforming as an enabling technology for 5G cellular communications: theoretical feasibility and prototype results. IEEE Commun. Mag. 52, 106–113 (2014) 3. M. Xiao et al., Millimeter wave communications for future mobile networks. IEEE J. Sel. Areas Commun. 35, 1909–1935 (2017) 4. L. Kong, M.K. Khan, F. Wu, G. Chen, P. Zeng, Millimeter-wave wireless communications for IoT-cloud supported autonomous vehicles: overview, design, and challenges. IEEE Commun. Mag. 55, 62–68 (2017) 5. Z. Ding, Z. Yang, Z. Fan, H.V. Poor, On the performance of non-orthogonal multiple access in 5G systems with randomly deployed users. IEEE Signal Process. Lett. 21, 1501–1505 (2014) 6. Z. Ding, X. Lei, G.K. Karagiannidis, R. Schober, J. Yuan, V.K. Bhargava, A survey on nonorthogonal multiple access for 5G networks: research challenges and future trends. IEEE J. Sel. Areas Commun. 35, 2181–2195 (2017) 7. Z. Zhang, H. Sun, R.Q. Hu, Downlink and uplink non-orthogonal multiple access in a dense wireless network. IEEE J. Sel. Areas Commun. 35, 2771–2784 (2017) 8. G. Araniti, M. Condoluci, P. Scopelliti, A. Molinaro, A. Iera, Multicasting over emerging 5G networks: challenges and perspectives. IEEE Netw. 31, 80–89 (2017) 9. Z. Ding, P. Fan, H.V. Poor, Random beamforming in millimeter-wave NOMA networks. IEEE Access 5, 7667–7681 (2017) 10. D. Zhang, Z. Zhou, C. Xu, Y. Zhang, J. Rodriguez, T. Sato, Capacity analysis of NOMA with mmWave massive MIMO systems. IEEE J. Sel. Areas Commun. 35, 1606–1618 (2017) 11. B. Wang, L. Dai, Z. Wang, N. Ge, S. Zhou, Spectrum and energy-efficient beamspace MIMONOMA for millimeter-wave communications using lens antenna array. IEEE J. Sel. Areas Commun. 35, 2370–2382 (2017) 12. Z. Ding, L. Dai, R. Schober, H.V. Poor, NOMA meets finite resolution analog beamforming in massive MIMO and millimeter-wave networks. IEEE Commun. Lett. 21, 1879–1882 (2017) 13. A.J. Morgado, K.M.S. Huq, J. Rodriguez, C. Politis, H. Gacanin, Hybrid resource allocation for millimeter-wave NOMA. IEEE Wirel. Commun. 24, 23–29 (2017) 284 Z. Zhang and Z. Ma 14. Z. Zhang, Z. Ma, Y. Xiao, M. Xiao, G.K. Karagiannidis, P. Fan, Non-orthogonal multiple access for cooperative multicast millimeter wave wireless networks. IEEE J. Sel. Areas Commun. 35, 1794–1808 (2017) 15. S. Naribole, E. Knightly, Scalable multicast in highly-directional 60-GHz WLANs. IEEE Trans. Netw. 25, 2844–2857 (2017) 16. T. Bai, R.W. Heath, Coverage and rate analysis for millimeter-wave cellular networks. IEEE Trans. Wirel. Commun. 14, 1100–1114 (2015) 17. J.G. Andrews, T. Bai, M. Kulkarni, A. Alkhateeb, A. Gupta, R.W. Heath, Modeling and analyzing millimeter wave cellular systems. IEEE Trans. Commun. 65, 403–430 (2017) 18. T.S. Rappaport, G.R. MacCartney, M.K. Samimi, S. Sun, Wideband millimeter-wave propagation measurements and channel models for future wireless communication system design. IEEE Trans. Commun. 63, 3029–3056 (2015) 19. H.S. Ghadikolaei, C. Fischione, G. Fodor, P. Popovski, M. Zorzi, Millimeter wave cellular networks: a MAC layer perspective. IEEE Trans. Commun. 63, 3437–3458 (2015) 20. J.G. Andrews, F. Baccelli, R.K. Ganti, A tractable approach to coverage and rate in cellular networks. IEEE Trans. Commun. 59, 3122–3134 (2011) 21. Andrews, J.G., Gupta, A.K., Dhillon, H.S.: A Primer on Cellular Network Analysis Using Stochastic Geometry, http://arxiv.org/abs/1604.03183 22. H.A. David, H.N. Nagaraja, Order Statistics, 3rd edn. (Wiley, Hoboken, New Jersey, 2003) Chapter 9 Full-Duplex Non-Orthogonal Multiple Access Networks Mohammed S. Elbamby, Mehdi Bennis, Walid Saad, Mérouane Debbah and Matti Latva-aho 9.1 Introduction In conventional wireless networks, resources are assumed to be allocated exclusively to users by considering half-duplex (HD) transmissions in serving uplink (UL) and downlink (DL) requests. Operating simultaneously in UL and DL over the same frequency band, known as in-band full-duplex (IBFD) or more commonly as fullduplex (FD), has been avoided for a long time, due to the inability to suppress the self-interference in the full-duplex radio to a feasible operating point. Furthermore, orthogonal multiple access techniques, such as orthogonal frequency-division multiple access (OFDMA), is used in multi-carrier settings to allocate subcarriers to users in an exclusive manner. This restriction is imposed to avoid the multi-user interference (MUI) resulting from scheduling multiple users/messages over the same subcarrier. The unprecedented growth in data rate requirement and the number of connected devices mandates going beyond the traditional ways of handling the scarcity of bandwidth in future wireless networks. A fundamental shift in the in the way wireless resources are allocated and managed is thus necessary. M. S. Elbamby (B) · M. Bennis · M. Latva-aho Centre for Wireless Communications, University of Oulu, Oulu, Finland e-mail: mohammed.elbamby@oulu.fi M. Bennis e-mail: mehdi.bennis@oulu.fi M. Latva-aho e-mail: matti.latva-aho@oulu.fi W. Saad Wireless@VT, Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA, USA e-mail: walids@vt.edu M. Debbah Mathematical and Algorithmic Sciences Lab, Huawei France, 92100 Paris, France e-mail: merouane.debbah@huawei.com © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_9 285 286 M. S. Elbamby et al. Full-duplex (FD) is ideally a spectrum efficiency doubler. By relaxing the constraint of orthogonal UL and DL transmissions, transceivers in the user and base station nodes can exploit the non-orthogonality to boost the spectral efficiency [21]. However, HD was adopted as the default setting in wireless networks for multiple reasons. First, FD radios can experience self-interference leaked from the UL transmitter to the DL receiver when operating in the same time–frequency resource. Recent advances in self-interference cancellation techniques have challenged this assumption. By using a combination of analogue and digital cancellation techniques, self-interference can be reduced to a level close to the receiver noise floor. Second, operating in FD results in increased inter-user interference due to the larger number of transmissions per resource. In particular, DL base station to UL base station and UL user to DL user interference will occur when operating simultaneously in UL and DL in the same frequency resource. Furthermore, this interference can occur at the intracell level, unlike the orthogonal resources per cell associated with OFDMA-based HD systems. Another way of boosting the cellular network bandwidth utilization is the use of non-orthogonal multiple access (NOMA) technique to schedule users with potentially overlapping resources [11]. NOMA has been recently discussed as a promising way to boost the network spectral efficiency as compared to orthogonal multiple access (OMA) techniques. NOMA provides diversity in the power domain by transmitting different messages to/from different users using the same time–frequency resource. If the different signals are received with enough power disparity, the signals can be decoded for example using successive interference cancellation (SIC) techniques. Hence, incorporating both FD and NOMA into current wireless networks will pose different challenges in terms of interference and network management. Different solutions are needed to address these challenges, ranging from smart resource scheduling, power allocation, duplex and multiple access mode switching. This chapter sheds light on both FD and NOMA network operation and discusses potential future research directions for this topic. First, preliminaries on FD, self-interference, and the interplay between FD and NOMA are discussed in Sect. 9.2. The objectives and tools used for FD and NOMA networks are discussed and surveyed in Sect. 9.3. Numerical results on the performance of FD-NOMA networks are presented in Sect. 9.4. Finally, Sect. 9.5 concludes the chapter and discusses some open problems. 9.2 Full-Duplex NOMA Networks 9.2.1 Preliminaries Here, we provide the basics of FD and NOMA that can be helpful to the reader. We particularly define the basic concepts of FD and self-interference cancellation and then briefly introduce FD and NOMA network operation challenges. 9 Full-Duplex Non-Orthogonal Multiple Access Networks 9.2.1.1 287 Full-Duplex Conventional wireless networks operate in HD mode, meaning that one direction of transmission is allowed at any given time and frequency resource. Different duplexing techniques have been considered in HD networks to duplex the UL and DL transmissions. In particular, time division duplex (TDD) and frequency division duplex (FDD) are both used commonly in today’s networks. FDD dedicates frequency resources to UL and DL where communication in both directions is orthogonal in the frequency domain. TDD splits the time resources between UL and DL transmissions either in a static manner (fixed UL and DL duty cycles on each time frame) or a dynamic manner (where UL and DL duty cycles can change to match the network UL and DL load conditions [14]. Alternatively, bidirectional transmissions can be used simultaneously, namely full-duplex communication. FD can ideally double the spectrum efficiency by relaxing the constraint on UL and DL orthogonality [21]. This has been an ideal assumption for a long time. However, taking it into practice was hindered by the complexity of removing the self-interference leaked from the transmitter to the receiver of the same device. The transmitted signal is much stronger than the received one, as the latter is significantly weakened by the path and propagation losses. Hence, the transmitted signal saturates the receiver radio chains and prevents signal reception. Self-interference has been seen as the major impairment to FD operation. Ideally, self-interference should be cancelled to the same level as the receiver noise floor such that the received signal is decoded in the same level as in HD. Otherwise, the residual interference is added to the received signal and hence decreases the receive SNR and throughput. 9.2.1.2 Self-Interference Cancellation Cancelling the self-interference from the transmitted signal of a FD radio is not as easy as it might sound. Although the transmitter knows what it is transmitting, this knowledge is of the signal in the baseband level [12, 20]. The baseband signal experiences several linear and nonlinear in the analogue radio chain before it is converted into a transmitted signal. Hence, subtracting the baseband signal does not help in removing the self-interference. Recently, the self-interference cancellation capability has significantly evolved. The study in [20] has shown simultaneous transmission and reception (with a single antenna) is achievable using analogue and digital cancellation techniques to cancel the self-interference. The leaked self-interference is reduced to the noise floor such that the received signal is not degraded. This breakthrough in the self-interference cancellation capability motivated the consideration of full-duplex in future networks for user scheduling [17] as well as relaying [38]. Furthermore, the shift from traditional macrocells to low-power small-cell networks eases the process of integrating FD into future networks. As compared to 46 dBm transmit power of macrocells, femtocells operate in power levels as low as 20 dBm, which makes the self-interference process feasible. 288 M. S. Elbamby et al. Fig. 9.1 An illustration of the different types of interference in multi-cell FD operation (UDI: UL-to-DL interference, DUI: DL-to-UL interference and SI: self-interference.) 9.2.1.3 FD Network Operation Once the self-interference is cancelled, the link throughput of FD can potentially double that of an HD link. However, as FD operates in a network level, another issue arises which is the inter-link interference due to having simultaneous transmission and reception, as illustrated in Fig. 9.1. This interference situation is similar to that of dynamic TDD networks, where base stations operate with different TDD configurations to satisfy their individual UL and DL traffic requirements, resulting in additional inter-link inter-cell interference. In addition to that, FD networks will suffer intra-cell interference between the user pairs operating in UL and DL. Therefore, intelligent user pairing methods are needed to select the appropriate pairs of users to be scheduled in each time–frequency resource such that this interference is avoided. 9.2.1.4 FD-NOMA Network Operation FD-NOMA refers to the concept of operating in IBFD mode and using NOMA to serve multiple requests. It is shown in the literature that NOMA can operate in both UL and DL. To reap the benefits promised by NOMA over OMA, power disparity has to be guaranteed. In UL, decoding the signals of multiple users sharing the time– frequency resource will occur in the base station level. It is intuitive therefore to select users with different receive power levels to be scheduled simultaneously. For example, scheduling a pair of cell-centre and cell-edge users is a convenient approach. In DL, different power levels should be allocated to the messages of different users, according to the decoding order they are going to select. This is essential to ensure a successful SIC process to cancel the intra-cell interference. When FD is combined with NOMA, additional levels of interference are introduced generated from the opposite direction [3]. Different approaches can be considered to mitigate the effect of this unwanted interference. One approach [13] is 9 Full-Duplex Non-Orthogonal Multiple Access Networks 289 to select between operating in FD while using an OMA technique or operating in HD and using NOMA. The scheduling and power allocation schemes play a key role in finding the optimal operating methods given the network conditions and optimization objectives. Prioritizing the serving of users in certain link direction means that NOMA should be selected to operate. On the other hand, FD can be selected to simultaneously serve UL and DL requests. While this approach restricts operation to FD-OMA or HD-NOMA at a certain time instant, it avoids the additional intra-cell interference of FD on NOMA operation and hence allows for more users to be served by NOMA in a certain link direction. Another way to blend the use of FD and NOMA together is to allow the network to operate in FD and NOMA in a given time instant. In this regard, the inter-link interference can be treated as noise [35] and will affect the NOMA performance. In particular, DL users will experience high UL-to-DL interference from multiple users operating in UL-NOMA. This interference can be handled by limiting the maximum number of users allowed to operate in UL-NOMA if the performance of DL users is degraded. 9.2.2 Challenges of FD-NOMA Resource Optimization As discussed above, resource optimization is a key role in reaping the benefits of FD-NOMA operation. Since the challenges of implementing them in practice are similar, there are common tools that can be used to optimize the resource and power allocation when either or both of them are considered. In this chapter, we focus on two main optimization problems that are fundamental for enabling NOMA, which are user pairing/scheduling and power allocation. Furthermore, we discuss the impact of the different objective functions on the resource optimization. 9.2.3 User Pairing and Power Optimization To illustrate the problem of user scheduling FD-NOMA networks, we assume a general multi-cell wireless network that comprises a set B of B FD-capable base stations, with self-interference cancellation capability of ζ , and a set U of U users requesting either UL or DL service. Furthermore, we assume that both base stations and users can operate in NOMA scheme, where the resulting multi-user interference can be cancelled by performing SIC at the receiver side. The user scheduling parameters in UL (t) ∈ X UL , UL and DL at time instant t are defined using the binary parameters xbu DL UL DL xbu (t) ∈ X ∀b ∈ B, u ∈ U . Let f b be the allocated bandwidth, vb (t) and vDL b (t) be the vectors of the UL and DL successive ordering index of all the users in base 290 M. S. Elbamby et al. (.) station b, where vbu (t) ∈ v(.) b (t) is the decoding order of user u when it is scheduled DL UL be the UL and DL data rates between base station and Rbu by base station b, and Rbu b and user u, then for a general network with open-access policy, the UL and DL service rates of user u can be expressed using the Shannon formula as: ruUL (t) =  UL UL xbu (t)Rbu (t), b∈B =  UL UL xbu (t) f b log2 (1 + Γbu (t)), (9.1) b∈B ruDL (t) =  DL xbu (t)RuDL (t), b∈B =  DL DL xbu (t) f b log2 (1 + Γbu (t)). (9.2) b∈B Assuming that the information to be transmitted is encoded based on a Gaussian distribution and a zero-mean additive white Gaussian noise with variance N0 , the UL and DL signal-to-interference-plus-noise ratios (SINRs) between base station b and user u at time instant t are given by: UL (t) = Γbu DL Γbu (t) = puUL (t)h bu (t) , NOMA-UL (t) + pbDL (t)/ζ N0 + IbUL-UL (t) + IbDL-UL (t) + Ibu N0 + IuDL-DL (t) DL (t)h bu (t) pbu , NOMA-DL (t) + IuUL-DL (t) + Ibu (9.3) (9.4)  DL where puUL is the UL transmit power of user u, and pbDL = u∈U pbu is the total DL 2 transmit power of base station b, h x,y (t) = |gx,y (t)| is the channel gain between the two nodes x and y, gx,y (t) is the propagation channel between nodes x and y, and the interference terms in (9.3) and (9.4) are expressed as follows1 : IbUL-UL (t) =  puUL ′ (t)h bu ′ (t), u ′ ∈U \{u} IbDL-UL (t) =  pbDL ′ (t)h b′ b (t), b′ ∈B \{b} IuDL-DL (t) =  pbDL ′ (t)h b′ u (t), b′ ∈B \{b} IuUL-DL (t) =  puUL ′ (t)h u ′ ,u (t), u ′ ∈U 1 Note that the term IUL-DL includes the intra-cell interference, to account for the interference due to FD operation. 9 Full-Duplex Non-Orthogonal Multiple Access Networks  NOMA-UL (t) = Ibu 291 puUL ′ (t)h bu ′ (t), UL u ′ ∈U \{u}|xbu ′ =1, UL UL vbu (t)<vbu ′ (t)  NOMA-DL Ibu (t) = u ′ DL pbu ′ (t)h bu (t), DL ∈U \{u}|xbu ′ =1, DL DL vbu (t)<vbu ′ (t) pbDL (t)/ζ is the leaked self-interference. Remark 1 To guarantee a successful NOMA operation in the DL, a user u ′ should u ′ DL decode the data of user u with an SINR level Γbu (t) that is at least equal to the user DL u received SINR Γbu (t). Otherwise, the data rate of user u is higher than what user u ′ DL DL (t) ≥ Γbu (t) must hold, where: u ′ can decode. Accordingly, the inequality Γbu ′ u DL Γbu (t) = DL pbu ′ (t)h bu ′ (t) DL (t)h bu ′ (t) pbu . NOMA-DL DL-DL (t) (t) + Ibu (t) + IuUL-DL + N 0 + Iu ′ ′ ′ This condition is met if the following metric is greater or equal to zero: ′ DL u DL DL DL ). − Γbu Yuu ′ = 1vbuDL (t)<vDL′ (t) xbu xbu ′ (Γbu bu Note that the above condition is satisfied by default in the UL, since all users’ signals are decoded in the base station’s receiver. A general FD-NOMA resource optimization problem can be defined to be optimizing user scheduling, NOMA decoding ordering, and UL and DL power allocations, UL DL ] and X DL = [xbu ], the decoding orderi.e., the scheduling matrices X UL = [xbu DL UL ing vectors vb and vb , and the power allocation vectors, pUL = [ p1UL , ..., pUUL ] DL DL and pDL b = [ pb1 , ..., pbU ]. Note that the selection of the scheduling parameters essentially includes the problems of user association, UL/DL mode selection, and OMA/NOMA mode selection. Accordingly, a general optimization problem can be cast as follows: P1: max UL DL X ,X , DL {vUL b },{vb } pUL ,{ pDL b }   U {ruUL }, {ruDL } (9.5a) UL subject to puUL ≤ Pmax , ∀u ∈ U , puDL DL Pmax , ≤  UL xbu + ∀b ∈ B, DL xbu ≤ 1, ∀u ∈ U , (9.5b) (9.5c) (9.5d) b∈B  u∈U UL xbu ≤ q UL ,  DL xbu ≤ q DL ∀b ∈ B. (9.5e) u∈U Yuu ′ ≥ 0 ∀u, u ′ ∈ U . (9.5f) 292 M. S. Elbamby et al. Constraints (9.5b) (9.5c) limit the UL and DL transmit powers to their maximum UL DL values Pmax and Pmax , respectively. Constraint (9.5d) restricts the connection of a user to one base station in either UL or DL. The number of users a base station can simultaneously serve in UL or DL-NOMA is limited to a quota of q UL and q DL users, respectively, by constraint (9.5e).2 Constraint (9.5f) guarantees a successful SIC in the DL. The utility in (9.5) should be selected to reflect the objective of the network optimization problem. The work in [35] considers a utility that maximizes the sum rate in a multi-carrier setting. In this regard, the resource allocation includes the subcarrier allocation. In [13], a weighted rate maximization utility was developed from a stochastic optimization problem, where the user weights are derived from the user traffic queue and virtual queue backlogs.3 9.2.4 Optimization Tools Rate-based utility maximizations are often combinatorial and non-convex optimization problems that are computationally complex to solve optimally. In particular, the user scheduling problem is a combinatorial problem whose complexity will increase exponentially as the number of base stations and users increase. In many cases, the optimization problem has to be solved dynamically. For example, if the weighted rate maximization based on user’s queue state is considered as the objective, both the user pairing and power allocation will need to be dynamically adjusted. Efficient and local solutions tools are therefore needed. Matching is a powerful tool that can solve the user scheduling problem [26]. Matching theory is a framework that provides solutions for the combinatorial problem of matching members of two disjoint sets of players in which each player is interested to associate with one or more player the other set. Matching is performed on the basis of preference profiles defined by the players of each side, providing a low-complexity stable matching. Although matching does not necessarily guarantee finding the optimal solution, its suitability for dynamic systems and local implementations made it a popular solution to reduce the complexity of the combinatorial problems [18, 19, 31]. Matching can also be used with other game-theoretic tools, such as cooperative game theory [28, 29], to further solve user grouping problems. Matching should be performed assuming an initial feasible power allocation policy. Subsequently, the power allocation is performed for the selected matching setting. The decoupling of the user scheduling and power allocation problems simplifies both problems, since power allocation is performed to the reduced set of scheduled users [13]. 2 Note that, in theory, the number of users that can be served simultaneously using NOMA is unrestricted. However, we impose a quota to avoid high-complexity SIC in the receiver side if a high number of users are scheduled. 3 Virtual queues result from applying the Lyapunov framework to convert the time-averaged constraints into virtual queues such that the constraints are met as the virtual queues are stabilized. 9 Full-Duplex Non-Orthogonal Multiple Access Networks 293 In addition to the complexity of the user scheduling problem, the power allocation problem is non-convex, due to having interference terms in the denominator of the SINR in the rate expression. Several approaches have been proposed in the literature to deal with the non-convexity of the problem. In [35], the global optimal solution of the joint problem of resource and power allocation is solved using monotonic optimization. The solutions are, however, with high computational complexity. The authors provide a lower complexity solution to solve the problem using successive convex approximation (SCA), which is shown to achieve a close to global optimal solution. SCA and similar tools to convexify the non-convex terms in the optimization problem are used in several works [13, 15, 24, 35, 36]. In this case, the convexified problem is solved iteratively using convex optimization tools until some convergence criterion is met. The NOMA decoding order significantly affects the user performance in both UL and DL. In DL-NOMA, decoding users with the lower channel strength first is optimal in the single-cell scenario [11]. In UL-NOMA, the opposite decoding order based on the user channel strength is shown in [6] to result in a gain over OMA based on users channel gain disparity. The decoding ordering in a multi-cell scenario is a challenging task, especially in the DL where different users might experience different inter-cell interference levels. 9.3 State of the Art in FD and NOMA Resource Optimization This section surveys the recent works in the problem of wireless resource optimization in FD and NOMA networks. First, with the promises of doubling the link throughput, the study of the when and how the potential gains can be achieved in a network level is surveyed. Following that, we overview the studies of resource optimization in NOMA networks in UL and DL. Finally, we highlight the works that combine both FD and NOMA schemes. 9.3.1 FD Resource Optimization In [16], a hybrid HD/FD scheduler for a single-cell network is proposed. The scheduler assigns FD resources only when it is advantageous over HD resource assignment. The joint problem of subcarrier and power allocation are optimized in [36] using an SCA algorithm. An auction-based algorithm to pair UL and DL users in a single-cell IBFD network is proposed in [33]. Subcarrier and power allocation is optimized using a heuristic algorithm with polynomial complexity for a single-cell IBFD network in [37]. The study also considers the case of imperfect self-interference cancellation on the performance of FD as compared to HD. It concludes that a higher number of 294 M. S. Elbamby et al. users can be served using FD as the cancellation capability increases. In [17], the authors extended the work in [16] to multi-cell networks. To reduce the complexity of a centralized solution, a distributed resource allocation scheme is developed where each base station selects locally which user to serve, and then, it coordinates with the neighbouring base stations to coordinate their transmission powers such that the inter-cell interference is minimized. The study shows that FD can achieve up to double the throughput of HD in indoor scenarios and 65% throughput gain in outdoor scenarios. The work in [34] also considered the FD resource and power allocation in multi-cell FD networks but with frequency reuse allowed only once among different cells. Matching theory is leveraged in [7] to develop a resource allocation algorithm. A matching algorithm is proposed to assign subcarriers to UL and DL user pairs. In [5], the user scheduling in FD ultra-dense-networks is optimized using different schemes with and without power optimization. The scheduling is carried out locally assuming no knowledge of the inter-cell interference. Table 9.1-a summarizes the contributions on the FD resource allocation. Remark 2 A common assumption in FD scheduling is that the base station knows the individual channel between its own users. This assumption is necessary to select FD pairs that do not significantly interfere on each other [16]. The assumption should be practical as FD is expected to be feasible in low-power small-cell networks where a low number of users are served by each base station. 9.3.2 NOMA Resource Optimization Recently, several works have looked into the resource optimization in UL and/or DLNOMA networks. Here, we shed light on the papers focusing on the scheduling and power optimization in NOMA networks. The authors in [8] propose a many-to-many algorithm to assign subcarriers to users in a single DL-NOMA network. Many-tomany matching is used in [9] for a multi-cell DL-NOMA scenario. Matching is also used for DL-NOMA resource allocation in [15] with a focus on energy efficiency. The power allocation problems are convexified and solved using difference of convex functions (DC) programming. DC programming is also used in [24] to optimize the power allocation in OFDM-based DL-NOMA networks, whereas a greedy algorithm is proposed for the user selection problem. A greedy approach is also used for the user scheduling in a multiple-input multiple-output (MIMO) DL-NOMA network in [32], and a minimum mean squared error (MMSE)-based power allocation is considered. Several works have looked into NOMA in the UL direction. The performance of NOMA in the UL is investigated in [2] using an iterative channel allocation algorithm. In [30], the problem of user pairing in UL-NOMA for users with single and multi-antennas is optimized. User grouping and power optimization in ULNOMA is studied in [6] where the impact of user ordering and imperfect SIC is 9 Full-Duplex Non-Orthogonal Multiple Access Networks 295 Table 9.1 Summary of existing literature in FD- and NOMA-based resource allocation problems (a) FD References Network Implementation FD scheduling Power allocation scenario [16] [36] Single-cell Single-cell Local Local [33] Single-cell Local [37] [17] Single-cell Multi-cell Local Local [34] [7] Multi-cell Central (subcarrier is reused once) Multi-cell Central [5] Multi-cell Local Link scenario Network scenario Scheduling and power allocation scheme DL DL DL DL DL UL UL UL UL + DL Single-cell Single-cell Single-cell Single-cell Multi-cell Single-cell Single-cell Heterogeneous Multi-cell Matching algorithm Subchannel assignment and power allocation User selection and power optimization User pairing and power allocation Matching algorithm and power allocation User pairing for multi-antenna systems Iterative subcarrier and power allocation User clustering and power allocation Optimal power allocation for a limited number of users (b) NOMA References [8] [15] [24] [32] [9] [30] [2] [6] [4] (c) FD-NOMA References Network scenario [35] [13] Single-cell Multi-cell HD/FD mode selection Joint subchannel and power allocation FD user pairing and channel allocation OFDMA channel allocation Suboptimal HD/FD user selection Mode selection and subcarrier allocation ×  Matching subcarriers to user pairs Local scheduling       Scheduling and power allocation scheme Joint subchannel and powerallocation Joint user scheduling and power allocation investigated. In [4], the authors consider a multi-cell UL and DL-NOMA system where a user grouping and power optimization scheme are proposed. The optimal power allocation is derived for a single macro-cell and a limited number of users. 296 M. S. Elbamby et al. 9.3.3 FD-NOMA Resource Optimization Two recent studies have looked into the incorporation of FD into NOMA networks and the impact on the scheduling and power allocation. In [35], the authors proposed an FD-NOMA approach in which users can be scheduled simultaneously in UL and DL in the same time–frequency resource and NOMA can be used in both directions. SIC is used in UL and DL to decode the messages of different users, whereas the inter-link interference due to FD is treated as noise. The joint subcarrier and power optimization problem is formulated, and the global optimal solution is found using monotonic optimization. Due to the high complexity of finding the global optimal solution, a low-complexity solution based on SCA is found. The results have shown that FD-NOMA improves the spectral efficiency as compared to HD-NOMA. Moreover, the effect of imperfect SIC is shown to impact the performance of the FD scheme. In [13], FD-NOMA is investigated in a dynamic multi-cell scenario where a stochastic optimization problem based on the Lyapunov framework is considered. The benefits of operating in HD or FD, as well as in OMA or NOMA modes, depending on traffic conditions, network density, and self-interference cancellation capabilities are investigated. The optimization problem is decomposed into two subproblems that are solved independently per small-cell base station. User association and mode selection are formulated as a many-to-one matching problem. A distributed matching algorithm aided by an inter-cell interference learning mechanism is proposed which is shown to converge to a pairwise stable matching. The matching algorithm allows small-cells to select between HD and FD and to operate either in OMA or NOMA schemes to serve their users. Subsequently, the UL/DL power optimization problem is formulated as a sequence of convex problems, and an iterative algorithm to allocate the optimal power levels for the matched users and their base stations is proposed. It was shown that using matching theory, the network can dynamically select when to operate in HD or FD and when to use OMA or NOMA to serve different users, which yields significant gains in UL and DL user throughput and packet throughput, as compared to HD-NOMA, FD-OMA, and HD-OMA schemes. 9.4 Numerical Results In this section, we present some numerical results to assess the performance of the queue-aware FD and NOMA resource optimization. We consider a continuous utility function of time-averaged UL and DL service rates. The problem can be decomposed using the Lyapunov framework into an instantaneous weighted rate maximization in which the user weights are their queue backlogs. The network consists of smallcell base stations with a varying self-interference cancellation capability, serving multiple users in an open-access manner. Scheduling can be in HD or FD modes, and in HD mode, users can be scheduled in OMA or NOMA. To cancel the resulting 9 Full-Duplex Non-Orthogonal Multiple Access Networks 297 multi-user interference, base stations or users operating in NOMA can perform SIC at the receiver side. The decoding ordering is assumed to be done in a descending order of channel strength in DL-NOMA, and an ascending order of channel strength in UL-NOMA. We assume that the user’s mean packet arrival rate and mean packet size follow Poisson and exponential distributions, respectively. To satisfy queue stability requirements, base stations need to ensure that user’s traffic queues are mean rate stable. Equivalently, constraints are imposed to ensure that the average service rate is higher or equal to the average arrival rate. The resource optimization problem consists of the scheduling problem (which includes the mode selection) and the power allocation problem. To reduce the complexity of the combinatorial scheduling problem, a many-to-one matching algorithm based on the deferred acceptance (DA) matching [13] is considered to dynamically schedule one or more users to each base station at each time instant. Preference profiles for users and base stations are selected as to maximize their individual weighted rates. The matching algorithm can be performed locally at each base station, which significantly reduce the amount of signalling exchange. After the matching is performed, each base station coordinates with its neighbours to optimize the allocated power, such that a feasible policy is achieved and inter-cell interference is minimized. The multi-cell power optimization is non-convex. Hence, the DC programming is used to convexify the problem, which is guaranteed to converge to a local optimal solution. System level simulations are carried out to show the gains brought by FD and NOMA, as well as to investigate the impact of queue stability constraints on the network performance. Simulation parameters are presented in Table 9.2. For the sake of comparison, the following schemes are considered in the simulation: 1. HD-OMA scheme: users are associated to the nearest base station and are allocated orthogonal resources for UL and DL. Requests are served using a round robin (RR) scheduler. 2. HD-NOMA only scheme: users are associated to the nearest base station, and RR scheduler is used to serve UL and DL queues. If there are multiple users in a scheduling queue, they are ranked according to their channel gains and are served using NOMA if the ratio between their channel gains is at least 2; otherwise, OMA is used. Power is allocated to NOMA users based on their channel ranking, in a uniform descending order for UL-NOMA and a uniform ascending order for DL-NOMA. Base stations operate either in UL or in DL depending on the queue length on each link direction. 3. FD-OMA scheme: users are associated to the nearest base station, and a pair of users is served in FD mode if the channel gain between them is greater than a certain threshold; otherwise, users are served in HD mode using RR scheduler. 4. Uncoordinated scheme: in this scheme, users can be served in either HD or FD and in OMA or NOMA modes. The many-to-one matching algorithm is used for mode selection and user scheduling, with the user queues taken into account in the weighted rate maximization. Power is assumed to be fixed for OMA and is similar to that of scheme 2 for NOMA. No inter-cell interference coordination is considered. 298 Table 9.2 Simulation parameters Parameter System bandwidth Duplex modes Multiplexing mode Subframe duration Network size Number of base stations Avg. number of users per base station Small-cell radius Max. base station transmit power Max. user transmit power Path loss model Shadowing standard deviation Penetration loss Self-interference cancellation Packet arrival rate per user Max. quota of NOMA users q UL , q DL M. S. Elbamby et al. Value 10 MHz TDD HD/ FD OMA/NOMA 1 ms 500 × 500 m2 10 10 40 m 22 dBm 20 dBm Multi-cell pico scenario [1] 4 dB 0 dB 110 dB 10 packet/s 5 5. Proposed scheme: In this scheme, the many-to-one matching algorithm is used for mode selection and user scheduling as in scheme 4. In addition, inter-cell interference coordination is considered by optimizing the UL and DL power allocation using DC programming. We begin by evaluating the performance of the proposed scheme under different traffic intensity conditions. Traffic intensity is varied by changing the mean packet size between 50 and 400 kb. In Fig. 9.2, we show the impact of traffic intensity on the normalized UL and DL user throughput. The normalized user throughput is defined as the user service rate divided by its data arrival rate. Figure 9.2a shows that in the UL, all schemes but the proposed one achieve lower UL throughput as compared to the HD-OMA scheme. The performance drop is due to the DL-to-UL interference that has a significant impact on the uncoordinated schemes due to the higher transmitting power of base stations and the lower path loss between the base station and user. The proposed scheme outperforms the different schemes by mitigating the DL-to-UL interference through power optimization. In Fig. 9.2b, we can see that, in the DL case, the effect of UL-to-DL interference is less significant, as it can be avoided within each cell through the pairing process. By leveraging both matching and the UL and DL power optimization, the proposed scheme outperforms the baseline schemes, with UL and DL gains of 10 and 20% over the HD-OMA scheme. The figure also shows that the coordination gain (over the uncoordinated scheme) is even more evident in the UL as in the DL due to the dominance of the DL-to-UL interference in the uncoordinated scenario. 9 Full-Duplex Non-Orthogonal Multiple Access Networks (a) UL 1 Normalized DL user throughput (Mbps) Normalized UL user throughput (Mbps) 0.98 0.96 0.94 0.92 0.9 0.88 0.86 Proposed Scheme HD-OMA HD-NOMA FD-OMA Uncoordinated 0.84 0.82 0.8 (b) DL 1 0.98 299 1 2 0.96 0.94 0.92 0.9 0.88 0.86 Proposed Scheme HD-OMA HD-NOMA FD-OMA Uncoordinated 0.84 0.82 3 0.8 4 Mean traffic intensity (Mbps) 1 2 3 4 Mean traffic intensity (Mbps) Fig. 9.2 Normalized a UL and b DL user throughput performance for different schemes as the user traffic intensity increases, for a network of ten base station and an average of ten users per base station 101 Average user Queue length (Mbit) Fig. 9.3 Average user queue length performance for different schemes as the user traffic intensity increases, for a user of ten base station and an average of ten users per base station 10 0 10 -1 10 -2 10 -3 10 -4 0.5 Proposed Scheme HD-OMA HD-NOMA FD-OMA Uncoordinated 1 1.5 2 2.5 3 3.5 4 Mean user traffic intensity (Mbps) Next, we investigate the queue behaviour of the different schemes as the traffic intensity varies. In Fig. 9.3, the average queue length is shown as function of the traffic intensity. Figure 9.3 shows that the average queue length grows as the traffic intensifies. In low traffic conditions, FD-OMA and HD-NOMA have smaller queue lengths as compared to the HD-OMA scheme. Also, the uncoordinated FD-NOMA scheme maintains a smaller queue length, since coordination is not crucial in low traffic intensity conditions. As the traffic intensity increases, we can see that the queue length grows significantly for the uncoordinated schemes. The proposed scheme 300 5.5 Proposed Scheme HD-OMA HD-NOMA FD-OMA Uncoordinated 5 Average cell throughput (bit/s/Hz) Fig. 9.4 Average cell throughput (in bit/s/Hz) performance for different schemes as the base station self-interference cancellation capability varies, for a network of ten base stations, an average of ten users per base station, and a mean traffic intensity rate of 3 Mbps per user M. S. Elbamby et al. 4.5 4 3.5 3 2.5 2 1.5 30 40 50 60 70 80 90 100 110 Self-interference cancellation (dB) maintains small queue lengths as it seeks to stabilize the user queues through the weighted rate maximization. Finally, we show the impact of the self-interference cancellation capability on the performance of the FD schemes. Figure 9.4 compares the average cell throughput (in bit/s/Hz) of the different schemes as the self-interference cancellation capability varies from 30 to 110 dB, which is the highest reported value [20]. As shown in the figure, the throughput of the FD schemes degrades with lower self-interference cancellation levels due to the interference leakage on the UL signal. It is also shown that only a slight degradation in the throughput of the proposed FD-NOMA is observed. As the proposed scheme optimizes the mode selection between HD/FD and OMA/NOMA, it can select more frequently the UL-NOMA instead of FD to serve UL users, such that it avoids high interference from the base station’s DL. This shows that enabling both FD and NOMA has the potential to enable higher network spectral efficiency in different network conditions. 9.5 Conclusions and Open Problems This chapter has provided an overview of the topic of full-duplex (FD) nonorthogonal multiple access (NOMA) from a network optimization point of view. Different challenges on the optimization of FD-NOMA networks are discussed. It has highlighted the particular importance of the user pairing and scheduling in both FD and NOMA networks. Several directions are still open for research. As was mentioned throughout the chapter, the decoding ordering is a key factor in the performance of the NOMA systems. Some studies are carried out on the optimal ordering of users, most of which are assuming single-cell operation and focus on the optimal solution from the point of sum rate maximization. Finding the optimal 9 Full-Duplex Non-Orthogonal Multiple Access Networks 301 decoding ordering is a challenging task in a multi-cell scenario and is even challenging in FD networks in which both intra-cell and inter-cell interference impact the network performance. The objective of the decoding order optimization should also take into account the notion of fairness between the users with different channel and queue state conditions. Moreover, enabling NOMA for emerging 5G systems, such as vehicle-to-everything (V2X) networks [10, 25] and networks with unmanned aerial vehicles [22, 23, 27], poses a wide range of open problems. Finally, looking into different objectives beyond the average rate maximization problems is necessary, especially in the context of ultra-reliable and low latency communication (URLLC), which brings further challenges to the system design. Acknowledgements This research was supported in part by the Academy of Finland CARMA Project, in part by the U.S. National Science Foundation under Grant CNS-1513697, and Grant CNS-1617896, and in part by the ERC Starting Grant MORE (Advanced Mathematical Tools for Complex Network Engineering) under Grant 305123. References 1. 3GPP, Further enhancements to LTE TDD for DL-UL interference management and traffic adaptation. TR 36.828, 3rd Generation Partnership Project (3GPP) (2012), http://www.3gpp. org/DynaReport/36828.htm 2. M. Al-Imari, P. Xiao, M.A. Imran, R. Tafazolli, Uplink non-orthogonal multiple access for 5G wireless networks, in 2014 11th International Symposium on Wireless Communications Systems (ISWCS) (2014), pp. 781–785 3. K.S. Ali, H. 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Commun. 16(9), 6032–6046 (2017) Chapter 10 Heterogeneous NOMA with Energy Cooperation Bingyu Xu, Yue Chen and Yuanwei Liu 10.1 Background 10.1.1 Resource Allocation in NOMA HetNets Heterogeneous networks (HetNets) are one of the “big three” technologies which can achieve the system performance requirement of fifth-generation (5G) networks. In conventional one-tier homogeneous networks, there is only one macro-BS in each cell. In HetNets, which are also called multi-tier networks, small BSs such as pico BSs and micro-BSs which have lower transmit powers and smaller coverages are deployed within the coverage of the macrocell. These smaller BSs obtain higher spectrum efficiency and throughput through reusing the spectrum. They are typically deployed in the hotpot zones (e.g., areas with high traffic demand) and dead zones (e.g., cell edge and areas cannot receive signal). Recently, NOMA-enabled HetNets have attracted significant research interests. It is worth introducing NOMA in HetNets due to the following key advantages: (1) in HetNets, users are closer to their associated BSs, which can reduce the interference between users and increase the accuracy of successive interference cancelation (SIC) in NOMA systems. (2) NOMA is capable to deal with the fairness issue among users, which is one of the main challenges of HetNets. Although intensive research contributions have been conducted on the design of NOMA transmission, resource allocation in NOMA-enabled HetNets remains an open problem. Considering the fact that user association (UA) determines that B. Xu (B) · Y. Chen · Y. Liu Queen Mary University of London, Mile End Road, London E1 4NS, UK e-mail: bingyu.xu@qmul.ac.uk Y. Chen e-mail: yue.chen@qmul.ac.uk Y. Liu e-mail: yuanwei.liu@qmul.ac.uk © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_10 305 306 B. Xu et al. users should be connected to a specified BS to form a user group for superposition transmission [1], the number of users associated with a BS has a significant effect on the spectral and energy efficiency in NOMA multi-cell networks [2]. In addition, power control is of great importance in such networks, since the intra-cell interference and inter-cell interference need to be coordinated. Otherwise, the performance of cell edge users will be significantly degraded [3]. A cooperative NOMA scheme in HetNets was proposed in [4] where each user was served by a macro-BS and a pico BS simultaneously. Zhao et al. [5] and [6] focused on downlink joint spectrum and power allocation problem which aims at maximizing the sum rate of small cell users. The macro-BS used conventional OMA protocol while small cells use NOMA protocol. Meanwhile, many works combined the NOMA enabled HetNets with other 5G techniques, such as massive MIMO and cloud radio access. A user association scheme was proposed in [7] for massive MIMO enabled NOMA HetNets to maximize the biased average received power of users. Meanwhile, the downlink power allocation problem in NOMA HetNets with cloud radio access network was investigated in [8]. It analyzed the energy efficiency of the network by finding the optimal number of BSs. 10.1.2 Energy Cooperation In 5G mobile systems, one main goal is to improve energy efficiency significantly compared to today’s networks [9]. Indeed, such large level of connectivity will inevitably give rise to an unprecedented surge in global energy consumption, especially in networks with HetNets, which has a vast number of small cells. The latest analysis shows that the energy demand for information and communications technology already accounts for almost 10% of the world’s total energy consumption [10]. In addition, critical environmental issues such as high carbon emissions are a big concern. Hence, “greener” solutions need to be developed to enhance the network energy efficiency. Among the emerging technologies, energy harvesting is regarded as one viable solution [11]. By allowing base stations (BSs) to harvest energy from renewable energy sources such as solar and wind, the conventional grid energy consumption of wireless networks can be greatly reduced. Although renewable energy harvesting is a viable solution for cutting the conventional grid energy consumption in cellular networks, there are many challenges for integrating energy harvesting capabilities into BSs [12]. In renewable energy harvesting-enabled networks, BSs harvest variable amounts of renewable energy, due to the fluctuating nature of renewable energy sources. When the renewable energy harvested by BSs is insufficient to meet their load conditions, some user equipments (UEs) have to be offloaded to distant BSs with abundant energy and may suffer more from signal degradation. Moreover, some BSs may always have excessive harvested energy (e.g., because of more favorable weather conditions) that will eventually be wasted. Since the deployment of BSs with large energy storage capabilities brings 10 Heterogeneous NOMA with Energy Cooperation 307 high expenditure of networks [13], the energy fluctuation problem cannot be solely solved by using storage. To improve the utilization rate of renewable energy, with the development of smart grid, the definition of energy cooperation is proposed which allows energy transferred through grids [14]. By this way, energy can be shared between BSs with acceptable energy loss during the energy transmission process. Energy cooperation in the point-to-point transmission scenario has been studied in [14–16]. In [15], one-way energy transfer in the Gaussian two-way channel and multiple access channel were considered respectively. This line of work was extended to the two-way case in [16]. The implementation of energy cooperation in multiple access channels and multiple access relay networks were studied in [17] and [18], respectively. In [19], an energy cooperation scheme in cognitive radio networks was proposed to improve both the spectral and energy efficiency. Recently, the potential of energy cooperation in renewable energy-enabled cellular networks has been explored, and various energy-cooperation optimization problems have been studied. In [13], a joint energy and spectrum allocation problem between two neighboring cellular systems was formulated, which aimed to minimize the cost of energy and bandwidth, and the problem was solved by convex optimization. The power control problem between two BSs was considered in [20] under the assumption that the harvested energy, and the energy demand of BSs were deterministic. In [21], the energy cost of cellular networks was minimized with the assumption that BSs traded energy via the smart grid with different prices. The work of [22] aimed to maximize the sum rate through optimizing the transmit powers of BSs in a coordinated multipoint cluster. In [23], the energy trading problem was formulated to minimize the average cost of energy exchange between BSs, and a dynamic algorithm was proposed based on the Lyapunov optimization technique, which did not require the statistical knowledge of the channel and energy. Due to the energy-saving feature of energy cooperation, it is worth to deploy it in NOMA networks. Meanwhile, most of existing NOMA works such as [24– 29] only consider the case consisting of one BS and a group of users. Besides, the practical multi-cell scenario is evaluated in this chapter considering the effect of inter-cell interference which has a substantial impact on the system performance. In this chapter, we study the power control and UA problem in energy-cooperationenabled two-tier NOMA HetNets. The content of this chapter is mainly based on our previous published journal [30]. 10.2 Network Model and Problem Formulation In this section, the system model for energy cooperation in two-tier NOMA HetNets is presented, and the corresponding joint UA and power control problem is formulated. 308 B. Xu et al. 10.2.1 Downlink NOMA Transmission As shown in Fig. 10.1, a two-tier energy-cooperation-enabled HetNet consisting of one macro BS (MBS) and M pico BSs (PBSs) is considered, where the NOMAbased downlink transmission is utilized, and all BSs are assumed to share the same frequency band. In such a network, BSs are powered by both the conventional power grid and renewable energy sources, and energy can be shared between BSs through the smart grid. Let m ∈ {1, 2, 3, ..., M + 1} be the mth BS, in which m = 1 denotes the MBS, and the other values denote PBSs. There are N randomly located user equipments (UEs) in this network, and each UE is associated with only one BS. All BSs and UEs are single-antenna nodes. It is assumed that the global perfect channel state information (CSI) is available. Let j ∈ {1, 2, 3, ..., N } index the jth signal transmitUE. According to the NOMA scheme [31, 32], the superimposed  H   N =1, ∀m, j, where ted by the BS m is sm = j=1 x jm P jm s jm with E s jm s jm x jm ∈ {0, 1} is the binary UA indicator, i.e., x jm = 1 when the jth UE is associated with the mth BS and otherwise it is zero, s jm is the jth user-stream and P jm is the corresponding allocated transmit power. When the jth UE is associated with the mth BS, its received signal can be expressed as Smart Grid MBS PBS 1 PBS 2 UE Renewable energy flow Conventional grid energy flow Superposition coded data flow Solar panel Renewable energy level Fig. 10.1 An example of an energy-cooperation enabled two-tier NOMA HetNet powered by both solar panels and the conventional grid 10 Heterogeneous NOMA with Energy Cooperation y jm = h jm  P jm s jm + h jm ′ 309 N  Pj ′ m s j ′ m x j′ m ′ j =1, j = j Intra−cell interference + ′ M+1  ′ m =1,m =m ′ ⎛ h mjm ⎝ N  j ′ =1 x j ′ m′ ⎞ P j ′ m ′ s j ′ m ′ ⎠ +̟m , (10.1) Inter−cell interference where x j ′ m , x j ′ m ′ ∈ {0, 1}, h jm is the channel coefficient from the associated BS m, ′ ′ h mjm is the interfering channel coefficient from the BS m , and ̟m is the additive white Gaussian noise. The power density of ̟m is σ 2 . In NOMA systems, SIC is employed at UEs, to cancel the intra-cell interference from the stronger UEs’ data signals. Without loss of generality, assuming that there are km (km ≤ N ) UEs constituting a group that is served by the mth BS at the same time and frequency band, the corresponding channel to inter-cell interference plus noise ratios (CINRs) are ordered as     h k m 2 h jm 2 |h 1m |2 m ≥ · · · ≥ (2) , (10.2) ≥ · · · ≥ (2) (2) I1m + σ2 I jm + σ 2 Ik m m + σ 2 (2) where I jm is the inter-cell interference power at the jth UE and σ 2 is the noise power. Based on the principle of multi-cell NOMA [31], the power allocation of the users’ data signals in the mth cell needs to satisfy 0 < P1m ≤ · · · ≤ P jm ≤ · · · ≤ Pkm m , km  P jm = Pm , (10.3) j=1 where Pm is the total transmit power of the mth BS. Such order is optimal for decoding and guarantees the user fairness [31], namely the data signals of users with weaker downlink channels and larger interference need to be allocated more transmit power (2) = 0, (10.3) to achieve the desired QoS. For the special case of single-cell, i.e., I jm reduces to the order based on the channel power gains, as seen in [32]. Therefore, based on (10.1), the data rate after SIC at the jth UE is given by   τ jm = W log2 1 + γ jm , (10.4) where W is the system bandwidth, and γ jm is the signal-to-interference-plus-noise ratio (SINR) given by 310 B. Xu et al. 2 P jm |h jm | γ jm =   M+1 2  j−1 h jm  Pj ′ m + ′ ′ ′ m =1,m =m j =1 (1) I jm = (2) I jm P jm  j−1 ′ j =1 (2) P j ′ m +(I jm +σ 2 )/|h jm | 2 ,  ′ 2  m  h jm  Pm ′ +σ 2 (10.5) j ≤ km (1) in which I jm is the remaining intra-cell interference after SIC, and Pm ′ = N ′ ′ ′ ′ ′ j ′ =1 x j m P j m is the total transmit power of the m th BS. Although this chapter focuses on the single-carrier system, it can be straightforwardly extended to the multi-carrier system by letting W be the subcarrier bandwidth and τ jm multiply the subcarrier indicator to be the data rate of a subcarrier. Thus, the optimal solution over all subcarriers in the multi-carrier case can be iteratively obtained by following the decomposition approach of this chapter. 10.2.2 Energy Model Each BS is powered by both the conventional grid and renewable energy sources. The energy drawn by the mth BS from the conventional grid is denoted as G m . The energy harvested by the mth BS from renewable energy sources is denoted by E m . ′ The energy transferred from BS m to BS m is denoted as Emm ′ , and the energy transfer efficiency factor between two BSs is denoted as β ∈ [0, 1]. Hence (1 − β) specifies the level of energy loss during the energy transmission process. In addition, it is assume that there is no battery to avoid the time-consuming and expensive energy waste during the charging/discharging process, and the energy-cooperation problem in each time slot is independent. The time slot length is normalized as one to simplify the power-to-energy conversion. Therefore, the transmit energy consumption at the mth BS should satisfy. Pm ≤ G m + E m + β M+1  Em ′ m m ′ =1,m ′ =m Energy received from other BSs − ′ M+1  Emm ′ , (10.6) ′ m =1,m =m Energy transferred to other BSs  where Pm = Nj=1 x jm P jm is the total transmit power of the mth BS. From (10.6), it is seen that in energy-cooperation-enabled networks, the grid energy consumption of a BS depends on its harvested renewable energy, transferred energy and transmit power. Given a BS’s transmit power, its grid energy consumption needs to be formulated as a random variable, since the amount of harvested renewable 10 Heterogeneous NOMA with Energy Cooperation 311 energy and transferred energy is uncertain, which is different from the conventional network without energy cooperation. 10.2.3 Problem Formulation Our aim is to maximize the energy efficiency of such networks. The energy efficiency (bits/Joule) is defined as the ratio of the overall network data rate to the overall grid energy consumption; i.e., the network utility is U (x, P, E , G) = N  M+1  m=1 j=1  M+1  Gm . x jm τ jm / (10.7) m=1 In this way, the harvested renewable energy can be maximally utilized to reduce the grid energy consumption [11]. Therefore, our problem can be formulated as follows: P1 : max x,P,E ,G s.t. C1 : M+1  U (x, P, E , G) (10.8) x jm τ jm ≥ τ min , ∀ j, m=1 C2 : M+1  x jm = 1, ∀ j, m=1 C3 : Pm + ′ M+1  Emm ′ ≤ G m + E m + β ′ m =1,m =m C4 : N  M+1  Em ′ m , ∀m, m ′ =1,m ′ =m x jm P jm = Pm , ∀m, j=1 C5 : x jm ∈ {0, 1} , ∀ j, ∀m, C6 : G m ≥ 0, Emm ′ ≥ 0, ∀ j, ∀m, m C7 : 0 ≤ Pm ≤ Pmax , P jm ≥ 0, ∀ j, ∀m,       where x = x jm , P = P jm , E = Emm ′ , G = [G m ], τ min denotes the required m minimum data rate for a UE, Pmax is the maximum transmit power of the BS m. Constraint C1 guarantees the QoS. C2 and C5 ensure that each UE cannot be associated with multiple BSs. C3 is the energy consumption constraint, and C4 is the power allocation under NOMA principle in a cell. C6 indicates that the consumed grid energy and transferred energy are nonnegative values, and C7 is the maximum transmit power constraint. 312 B. Xu et al. From the objective of P1 and its constraint C3, it can be found that when more renewable energy is harvested and shared between BSs, the total grid energy consumption of the network can be reduced, which boosts the energy efficiency. 10.3 Proposed Resource Allocation Scheme 10.3.1 Resource Allocation Under Fixed Transmit Power P1 is a mixed integer nonlinear programming (MINLP) problem, and constitutes a challenging problem. In this section, it is assumed that the transmit power is fixed, and accordingly the original problem P1 can be simplified as P2 : U (x, E , G) max x,E ,G (10.9) s.t. C1, C2, C3, C4, C5, C6. The problem P2 is still a combinatorial problem due to its discrete nature. To efficiently solve it, we adopt a decomposition approach. For a given G and E , the above problem can be rewritten as max U (x) P2.1 : (10.10) x s.t. C1, C2, C4, C5. 10.3.1.1 Lagrangian Dual Analysis Based on P2.1, the Lagrangian function can be written as L (x, λ, θ) = U (x) − N  j=1  λ j τ min − M+1  x jm τ jm m=1  − M+1  θm m=1  N  j=1  x jm P jm − Pm , (10.11) where λ j and θm are the nonnegative Lagrange multipliers. Then, the dual function is given by g(λ, θ ) =  max L(x, λ, θ ) x s.t. C2, C5 , (10.12) and the dual problem of P2.1 is expressed as min g (λ, θ ) . λ,θ (10.13) 10 Heterogeneous NOMA with Energy Cooperation 313 Given the dual variables λ j and θm , the optimal solution for maximizing the Lagrangian w.r.t. x is x ∗jm =  1, if m = m ∗ , 0, otherwise (10.14)   where m ∗ = argmax μ jm with m μ jm = τ jm / M+1  (10.15) G m + λ j τ jm − θm P jm . m=1 The solution of (10.14) can be intuitively interpreted based on the fact that given the grid energy consumption, users select BSs which provide the maximum data rates. Since the objective of the dual problem is not differentiable, we utilize the subgradient  method to obtain the optimal solution λ∗ , θ ∗ of the dual problem, which is given by  λ j (t + 1) = λ j (t) − δ (t)   M+1  x jm τ jm − τ min m=1  θm (t + 1) = θm (t) − δ (t) Pm − N  j=1 x jm P jm +  , (10.16) , (10.17) + where [a]+ = max {a, 0}, t is the iteration index, and δ (t) is the step size. Note that there exist several step size selections such as constant step size and diminishing step size. Here, the nonsummable diminishing step length is used [33].   After obtaining the optimal λ∗ , θ ∗ based on (10.16) and (10.17), the corresponding x is the solution of the primal problem P2.1. Therefore, based on the Lagrangian dual analysis, UA can be determined in a centralized or distributed way. The centralized UA is intuitive and requires a central controller, which has the global CSI and determines which user is connected to a BS in this network. In this chapter, a distributed UA algorithm which does not require any centralized coordination is proposed, as summarized in Algorithm 1. Since our problem satisfies the conditions of the convergence proof in [33], the convergence of the proposed algorithm is guaranteed. The complexity of the proposed algorithm is O ((M + 1)N ) for each iteration and the convergence is fast (less than 40  iterations in the simulation), which is much lower than the brute force algorithm O (M + 1) N . Note that the broadcast operations have a negligible effect on computational complexity (Table 10.1). 314 B. Xu et al. Table 10.1 Algorithm 1 distributed user association Step 1: At user side 1: if t = 0 2: Initialize λ j (t), ∀ j. Each UE measures its received inter-cell interference via pilot signal from all BSs, and feedbacks the CINR values to the corresponding BSs. Meanwhile, each UE selects the BS with the largest CINR value. 3: else 4: User j receives the values of μ jm and τ jm from BSs.   5: Determines the serving BS m according to m ∗ = argmax μ jm . m 6: Update λ j (t) according to (10.16). 7: end if 8: t ← t + 1. 9: Each user feedbacks the UA request to the chosen BS, and broadcasts the value of λ j (t). Step 2: At BS side 1: if t = 0 2: Initialize θm (t), ∀m. 3: else 4: Receives the updated user association matrix x. 6: Updates θm (t) according to (10.17), respectively. 7: Each BS calculates μ jm and τ jm under NOMA principle. 8: end if 9: t ← t + 1. 10: Each BS broadcasts the values of μ jm and τ jm . 10.3.1.2 Genetic Algorithm In this subsection, a genetic algorithm (GA)-based UA is proposed to solve the problem P2.1. Such algorithm will be compared with the proposed Algorithm 1. GA can achieve good performance when the population of candidate solutions is sufficient [34]. Specifically, each feasible chromosome represents a possible solution that satisfies the constraints of problem P2.1, which is defined as Di = {[m 1i ] , [m 2i ] , . . . , [m N i ]} , i ∈ {1, . . . , K }, (10.18) where m ji is the gene representing the index of the BS that the jth UE is associated with, and it has an integer value varying from 1 to M + 1, and K is the population size. During each generation, the fitness of each chromosome is evaluated, to select high fitness chromosomes and produce higher fitness offsprings. Based on the objective of problem P2.1, the fitness value of the chromosome Di is calculated as 10 Heterogeneous NOMA with Energy Cooperation 315 Table 10.2 Algorithm 2 genetic algorithm-based user association algorithm 1: if t = 0 2: Initialize a set of feasible chromosomes {Di } with population size K , and the maximum number of generations tmax . 3: else 4: Rank {Di } based on the fitness values given by (10.19). 5: Based on the selection probability ρs (r ), chromosomes are selected to produce offspring via uniform crossover and mutation operations. 6: if exceed the maximum number of generations ! " ! " 7: x ∗jm := Di∗ , where Di∗ is the feasible chromosome with the highest fitness value. 8: break 9: else 10: t ← t + 1. 11: end if 12: end if Φi (Di ) = U (Di ) . (10.19) Then, all chromosomes are ranked from the best to the worst with ranking r , based on their fitness values. The probability that a chromosome is selected as a parent q(1−q)r −1 to produce offspring is given by ρs (r ) = 1−(1−q) K with a predefined value q [34]. In each generation process, a uniform crossover operation with the probability ρc is utilized to produce offspring by swapping and recombining genes based on the parental chromosomes. In addition, a uniform mutation operation with the probability ρm is employed. Such generation procedure is repeated until reaching the maximum number of generations, and is summarized in Algorithm 2. Given the maximum number of generations Ω and fixed population size K , the complexity of the proposed algorithm is O (Ω K log(K )) [35]. The performance of the GA-based UA algorithm heavily depends on the population size and number of generations, due to the inherent nature of GA [34]. In the simulation results of Sect. 10.5, we will demonstrate that overall, the proposed Algorithm 1 outperforms GA-based Algorithm 2 which is shown in Table 10.2 when the population size of GA is not very large, and thus has lower complexity. The aforementioned approach UA solutions for problem P2.1. After   provides obtaining the UA solution x = x ∗jm , the corresponding pair (G, E ) is obtained by solving the following simple linear programming (LP): 316 B. Xu et al. Table 10.3 Algorithm 3 resource allocation algorithm under fixed transmit power 1: if t = 0 2: For a fixed P, initialize G m , ∀ j, m. 3: else 4: Determine x jm (t) under fixed (E , G) by selecting the user association algorithm from Algorithm 1 or Algorithm 2. 5: Given x jm (t), update the energy allocation policy (E , G) by solving the LP P2.2 via CVX. 6: if convergence 7: Obtain optimal resource allocation policy (x∗ , E ∗ , G∗ ). 8: break 9: else 10: t ← t + 1. 11: end if 12: end if Table 10.4 Algorithm 4 one-dimensional search algorithm 1: if t = 0 l = 0, ν h = ν max , ∀m, calculate F = 2: Given χ j , initialize νm l m m Fh = N  j=1 j-th N  j=1 ∗(l) x jm P jm and $ $ # # ∗(h) ∗(h) ∗(l) x jm P jm , where P jm and P jm are the allocated transmit powers of the l and ν h respectively, which are calculated by using UE’s data stream for the cases of νm m (10.28). 3: else 4: while Fl = ϕm and Fh = ϕm ν l +ν h 5: Let νm = m 2 m , and compute Fm . 6: if Fm = νm 7: The optimal dual variable νm∗ is obtained. 8: break 9: elseif Fm < ϕm 10: νmh = νm . 11: else Fm > ϕm l =ν . 12: νm m 13: end if 14: end while 15: end if 10 Heterogeneous NOMA with Energy Cooperation 317 Table 10.5 Algorithm 5 Joint User Association and Power Control 1: if t = 0 2: Initialize Pm , G m , E m , ∀m 3: else 4: Determine x jm (t) under (P, G, E ) by selecting the user association algorithm from Algorithm 1 or Algorithm 2. 5: Given x jm (t) and the corresponding (G, E ), update the transmit power P based on the following rule: Loop: (2) a) Given Θ jm , loop over UE j: ! " ! " i): Obtain νm∗ using Algorithm 4 given χ j ! " ii): Obtain P jm according to (10.28) with νm∗ , χ j . ! " iii): Update χ j using subgradient method. iv): Update P jm using (10.29). Until convergence. (2) b) Update Θ jm using (10.27). Until convergence. 6: Based on the updated P, update G m and Emm ′ by solving LP problem P2.2 via CVX. 7: if convergence 8: Obtain optimal resource allocation policy (x∗ , P∗ , E ∗ , G∗ ). 9: break 10: else 11: t ← t + 1. 12: end if 13: end if P2.2 : min E ,G M+1  Gm (10.20) m=1 s.t. C3, C6. The problem P2.2 can be efficiently solved by using existing software, e.g. CVX [36]. grid When no energy cooperation is allowed, i.e., Emm ′ = 0, ∀ j, ∀m, the optimal  energy consumption G of problem P2.2 under the UA solution x = x ∗jm is directly obtained as G ∗m = [Pm − E m ]+ , (10.21)  where Pm = Nj=1 x ∗jm P jm . Based on the solutions of subproblems P2.1 and P2.2, we propose an iterative algorithm to solve the problem P2, which is summarized in Algorithm 3 (Table 10.3). 318 B. Xu et al. 10.3.2 Resource Allocation Under Power Control In this subsection, we consider the joint resource allocation and power control design. Specifically, we develop an algorithm to solve the MINLP problem P1 through the decomposition approach. As discussed in the previous section, we first determine the UA indicators given the resource allocation policy (P, E , G), which can be obtained problem P2.1 via Algorithm 1 or Algorithm 2. Then, under a fixed UA " !by solving x jm , the problem for optimizing (P, E , G) is written as P3 : max P,E ,G U (P, E , G) (10.22) s.t. C1, C3, C4, C6, C7. From the utility function, we find that the power allocation vectors P and G are coupled within the objective of problem P3. Thus, given G and E , the above problem can be decomposed into P3.1 : max P N M+1   (10.23) x jm τ jm m=1 j=1 s.t. C1, C3, C4, C7. Problem P3.1 is non-convex. Hence, we provide a tractable suboptimal solution based on the Karush–Kuhn–Tucker (KKT) conditions. The Lagrangian function of problem P3.1 is L (P, ν, χ ) = N M+1   x jm τ jm − m=1 j=1 − M+1  m=1 νm N +1 j=1  N j=1 x jm P jm   M+1  χ j τ min − x jm τ jm m=1  − ϕm , (10.24) $ #   M+1 m ′ ′, P where ϕm = min G m + E m + β mM+1 E ′ ′ ′ =1,m ′  =m Em m − mm max m =1,m =m according to constraints C3 and C7, and χ j and νm are the nonnegative Lagrange multipliers. Without loss of generality, assuming that the jth UE is associated with the BS m, i.e., x jm = 1, based on the KKT conditions, we have ∂L ∂ P jm where Λ jm =    WΛ  (2) − Θ (1) = 1 + χ j 1+P jm jm jm − Θ jm − νm log(2) Λ jm = 0, |h jm |2 (1) (2) I jm +I jm +σ 2 (10.25) is referred to as the channel-to-interference-plus-noise ratio at the jth UE. Based on (10.3) and (10.5), Θ (1) jm resulting from the intra-cell interfer- 10 Heterogeneous NOMA with Energy Cooperation 319 ence is given by Θ (1) jm = km  (1 + χℓ ) ℓ> j W γℓm Λlm , 1 + γℓm (10.26) and Θ (2) jm resulting from the inter-cell interference is given by Θ (2) jm = ′ M+1  ′ m =1,m =m 2      1 + χ j ′ x j ′ m ′ W γ j ′ m ′ h mj′ m ′  .    (1) (2) 2 ′ ′ ′ I + I + σ 1 + γ ′ ′ ′ ′ j m j =1 j m j m N  (10.27) Based on (10.25), the transmit power allocated to the jth user-stream in the mth cell is obtained as ∗ = P jm  (1 + χ j )W (2) Θ (1) jm + Θ jm + νm log(2) − 1 + . Λ jm (10.28) the allocated transmit power is a monotonic function of νm . As such, given !In (10.28), " χ j , we adopt a one-dimensional search over the Lagrange multipliers {νm }, which can efficiently obtain the optimal ν ∗ that satisfies constraints C3 and C7. Accord∗ max ing to (10.28),# we can easily find that νm∗ needs  to satisfy $ 0 ≤ νm ≤ νm , where (2) ∗ (1 + χ j )W Λ jm − Θ (1) νmmax = max j jm − Θ jm / log(2) . Here, νm = 0 represents that there is no limitation on the transmit power of the jth user-stream and νm∗ = νmmax corresponds to the! case " that no transmit power is allocated to the jth user-stream. Thus, by fixing χ j , ν ∗ can be obtained by using Algorithm 4 (Table 10.4). For achieving a specific accuracy ς , the complexity of Algorithm 4 is O (log (1/ς )). After obtaining ν ∗ , the Lagrange multiplier χ j can be updated by using the subgradient method, which is similar to (10.16). To ensure the system stability, we utilize the Mann iterative method to update the transmit power in each iteration [37], which is given by (ℓ) (ℓ+1) ∗ , + η(ℓ)P jm = (1 − η(ℓ))P jm P jm (10.29) where ℓ is the iteration index, 0 < η(ℓ) < 1 is the step size, which is usually chosen as ℓ . After obtaining the optimal solution of problem P3.1, the corresponding η(ℓ) = 2ℓ+1 (G, E ) can be updated by solving the LP problem P2.2 via CVX. As such, the solution of problem P3 can be iteratively obtained. Note that the convergence of the KKTbased algorithm is usually faster than the gradient-based designs [38]. Based on the previous analysis, the proposed joint UA and power control scheme in energy-cooperation enabled NOMA HetNets is summarized in Algorithm 5 (Table 10.5). 320 B. Xu et al. 10.3.3 Comparison with FTPA In 4G networks, fractional transmission power allocation (FTPA) scheme is adopted [31]. The rule of FTPA is that the transmit power will be allocated based on the UEs’ channel conditions, i.e., the data signals of UEs with weaker downlink channels will own more transmit power. Based on the CINR order in (10.30), the transmit power allocated to the jth UE’s data stream in the mth cell under FTPA protocol is expressed as [31] P jm  −α    −α N h jm 2  |h lm |2 / xlm , = Pm (2) (2) I jm + σ2 Ilm + σ2 l=1 (10.30) where 0 ≤ α ≤ 1 is the decay factor. Here, α = 0 represents equal power allocation. For larger α, the transmit power allocated to the data-stream of the user with largest CINR becomes lower, and more power will be allocated to the data-stream of the user with the lowest CINR, in order to achieve the user fairness and the optimal decoding. However, the detrimental effect of using such simple power allocation scheme is that distant users may receive severer inter-cell interference without power control among BSs, due to the fact that each BS has to assign larger transmit power to the faraway users. Therefore, compared to the single-cell NOMA case [32], the inter-cell interference has a significant impact on the power allocation of multi-tier NOMA HetNets. 10.3.4 Comparison with No Renewable Energy When there is no renewable energy harvesting (i.e., E m = 0, ∀m), no renewable ′ energy can be shared between BSs (i.e., Emm ′ = Em ′ m = 0, ∀m, m ), and thus, the required energy can only be supplied by the conventional grid. In this case, Pm = G m , ∀m, and the original problem P1 reduces to P4 : max x,P  M+1  N m=1 j=1  M+1  N m=1 j=1 x jm τ jm x jm P jm (10.31) s.t. C1, C2, C4, C5, C7. The above problem is nonlinear fractional programming and NP-hard, which can be solved by using the proposed Algorithm 5 with E m = 0 and Emm ′ = Em ′ m = 0. 10 Heterogeneous NOMA with Energy Cooperation 321 10.3.5 Comparison with No Energy Cooperation In this case, the energy transfer efficiency β is set to 0, which means that the harvested renewable energy cannot be transferred between BSs. Each BS is powered by the conventional grid and its harvested renewable energy; i.e., the transmit energy consumption at a BS needs to satisfy Pm ≤ G m + E m , ∀m. Then, the proposed Algorithm 5 can still be applied to solve this problem, and during each iteration, the grid energy consumption is updated as G m = [Pm − E m ]+ based on the updated Pm . 10.4 Simulation Results In this section, we present numerical results to demonstrate the effectiveness of the proposed algorithm compared with other schemes as well as the conventional counterpart. Since the renewable energy arrival rate changes slowly in practice and is stationary at each information transmission time slot [39], we consider the amounts of harvested energy at the MBS and PBSs to be constant and each PBS has the same level of renewable energy during each transmission time slot for the sake of simplicity. For simplicity, the amount of harvested energy E m of BS m is modeled as a uniform distribution Um [am , bm ], and varies across different transmission blocks, where am and bm are the minimum and maximum harvested energy values of BS m follows uniform distributions as shown in Table 10.6, respectively [40]. Our analysis and proposed algorithm are independent of the specific renewable energy distribution. For the channel h jm , we focus on the large-scale channel fading condition in low mobility environment, due to the fact that UA is carried out in a large time scale and the small-scale fading can be averaged out [41, 42]. In addition, PBSs and UEs are uniformly distributed in a macrocell geographical area. The basic simulation parameters are shown in Table 10.6. 10.4.1 User Association Under Fixed Transmit Power In this subsection, we study different UA algorithms under fixed transmit power, i.e., power control is unavailable at BSs. Based on the NOMA power allocation condition m , and in (10.3), we consider that the total transmit power at each BS is Pm = Pmax adopt an arithmetic progression power allocation approach for the sake of simplicity, 2j namely the transmit power of the jth user’s data signal is P jm = km (1+k Pm , j ∈ m) {1, 2, 3, ..., km } when km users are multiplexed in the power domain of the mth cell. We also provide the comparison with the conventional reference signal received power (RSRP)-based UA. The aim of this subsection is to show the performance of different UA algorithms under the same fixed power allocation condition. 322 B. Xu et al. Table 10.6 Simulation parameters Parameter Value System bandwidth Noise power density Cell radius Path loss of MBS Path loss of PBS Min harvested energy of MBS Max harvested energy of MBS Min harvested energy of PBS Max harvested energy of PBS Max transmit power of MBS Max transmit power of PBS Fig. 10.2 Energy efficiency versus the number of UEs for different UA algorithms 10 MHz −174 dBm/Hz 500 m 128.1 + 37.6log10 d (km) 140.7 + 36.7log10 d (km) 575 W 660 W 15 W 25 W 46 dBm [43] 30 dBm [43] 5.5 ×10 6 Energy Efficiency (bits/Joule) 5 NOMA, Proposed UA NOMA, GA-based UA, Fixed Population Size=200 NOMA, RSRP-based UA OMA, RSRP-based UA 4 3 2 10 15 20 25 30 Number of UEs Figure 10.2 shows the energy efficiency versus the number of UEs with the number of PBSs M = 6 and the energy transfer efficiency factor β = 0.9. We set the minimum QoS as τ min = 0.1 bits/s/Hz and the amount of energy harvested by MBS and PBS as 37 dBm and 27 dBm, respectively.1 The maximum number of generations for the GA-based UA is 10, q = 0.1, and ρc = ρm = 0.4. The proposed UA scheme with NOMA achieves better energy efficiency than the other cases. The energy efficiency 1 In real networks, the renewable energy generation rate is constant during a certain period, and the time scale of the UA and power control process is much shorter, typically less than several minutes [41, 42]. In addition, the amount of energy harvested by an MBS is usually larger than that at a PBS, since MBS can fit larger solar panel [42, 44]. 10 Heterogeneous NOMA with Energy Cooperation Fig. 10.3 Energy efficiency versus the number of PBSs for different UA algorithms Energy Efficiency (bits/Joule) 7 323 ×10 6 6 5 4 NOMA, Proposed UA NOMA, GA-based UA, Fixed Population Size=600 NOMA, RSRP-based UA OMA, RSRP-based UA 3 5 10 15 20 25 Number of PBSs increases with the number of UEs because of the multiuser diversity gain (i.e., different users experience different path loss, and more users with lower path loss help enhance the overall energy efficiency.) [45]. The use of NOMA outperforms OMA. By using the GA-based UA, the energy efficiency slowly increases with the number of UEs, due to the fact that the efficiency of the GA-based algorithm depends on the population size [34]. In other words, given the population size (e.g., K = 200 in this figure), the GA algorithm may not obtain good solutions when the number of UEs grows large, which indicates that larger populations of candidate solutions are needed [34]. Figure 10.3 shows the energy efficiency versus the number of PBSs with the number of UEs N = 40 and the energy transfer efficiency factor β = 0.9. We set the minimum QoS as τ min = 0.1 bits/s/Hz and the amount of harvested energy at MBS and PBS as 37 dBm and 27 dBm, respectively. The maximum number of generations for GA is 10, q = 0.1, and ρc = ρm = 0.4. NOMA achieves higher energy efficiency than OMA, since NOMA can achieve higher spectral efficiency. The proposed UA algorithm outperforms the other cases and the performance gap between the proposed UA and the conventional RSRP-based UA is larger when deploying more PBSs, due to the fact that the proposed UA can achieve more BS densification gains [9]. For the GA-based UA algorithm with the population size K = 600, solutions are inferior when the number of PBSs is large, as larger populations of candidate solutions are needed [34]. 324 B. Xu et al. 10.4.2 Power Control Under Fixed User Association In, this subsection, we consider three power allocation schemes, namely the power control method proposed in Sect. 10.4, fractional transmission power allocation (FTPA) and the conventional fixed transmit power, to confirm the advantages of our proposal. We adopt the conventional RSRP-based UA in the simulation, and all the considered cases experience the same UA condition. In addition, BSs use their maximum transmit powers in the OMA scenario, and the total transmit power of m , m ∈ {1, 2, 3, ..., M + 1}. each BS for FTPA is set as Pm = Pmax Figure 10.4 shows the energy efficiency versus the number of PBSs with the number of UEs N = 50 and the energy transfer efficiency factor β = 0.9. We set the minimum QoS as τ min = 1 bits/s/Hz and the amount of energy harvested by MBS and PBS as 37 dBm and 27 dBm, respectively. We see that by using NOMA with the proposed power control, energy efficiency rapidly increases with the number of PBS. The proposed algorithm achieves better performance than the other cases. When deploying more PBSs, the performance gap between the proposed solution and the other cases is larger, which indicates that the proposed power control algorithm can achieve more BS densification gains and efficiently coordinate the inter-cell interference. When the number of PBSs is not large, NOMA with FTPA can outperform the conventional OMA case, since NOMA can achieve better spectral efficiency than OMA [32]. However, when adding more PBSs, NOMA with FTPA may not provide higher energy efficiency. The reason is that more UEs will be offloaded to picocells, and the inter-cell interference will become severer, which means that the transmit power of each user-stream needs to be larger to combat the inter-cell interference. As suggested in Sect. 10.3.3, FTPA with α = 0 achieves a higher energy efficiency of the network than the α = 0.7 case, since the data-streams for UEs with poorer Fig. 10.4 Energy efficiency versus the number of PBSs for different power allocation policies 2 ×10 7 NOMA, Proposed Power Control 1.8 NOMA, FTPA with NOMA, FTPA with α =0 α =0.7 Energy Efficiency (bits/Joule) OMA, Fixed Transmit Power 1.4 1 0.6 0.2 1 5 10 Number of PBSs 15 20 10 Heterogeneous NOMA with Energy Cooperation Fig. 10.5 Energy efficiency versus energy transfer efficiency factor for different power allocation policies 2.2 ×10 7 NOMA, Proposed Power Control NOMA, FTPA with α =0 NOMA, FTPA with α =0.7 OMA, Fixed Transmit Power 1.8 Energy Efficiency (bits/Joule) 325 1.4 1 0.6 0.2 0 0.2 0.4 0.6 0.8 1 Energy Transfer Efficiency Factor channel condition (i.e., lower CINR) have to be allocated more power in the case of FTPA with α = 0.7, which reduces the total throughput of the network under the same energy consumption. Figure 10.5 shows the energy efficiency versus the energy transfer efficiency factor β with the number of PBSs M = 3 and the number of UEs N = 40. We set the minimum QoS to τ min = 1 bits/s/Hz and the amount of harvested energy at MBS and PBS to 40 dBm and 35 dBm, respectively. Compared to the no energy-cooperation case (i.e., β = 0), the use of energy cooperation can enhance the energy efficiency, particularly when the energy transfer efficiency factor is large. The implementation of NOMA can achieve higher energy efficiency than the conventional OMA system because of higher spectral efficiency, and the proposed power control algorithm outperforms the other cases. Moreover, the energy efficiency grows at a much higher speed when applying the proposed algorithm. For a specified β, FTPA with α = 0 achieves higher energy efficiency of the network than the α = 0.7 case, as suggested in Fig. 10.4. Figure 10.6 shows the trade-off between the energy efficiency and the minimum QoS with the number of PBSs M = 3 and the number of UEs N = 30. We set the energy transfer efficiency factor to β = 0.9 and the amount of energy harvested by MBS and PBS to 37 dBm and 27 dBm, respectively. For a given minimum QoS, the proposed power control under NOMA achieves higher energy efficiency than conventional OMA. When better QoS is required by the UE, energy efficiency of both NOMA and OMA cases decreases. The reason is that for the proposed solution, more transmit power will be allocated to the UEs with lower CINRs to achieve such minimum QoS, which results in more energy consumption; for conventional OMA, it means that more users cannot obtain the desired QoS and have to experience an outage. We see that energy efficiency decreases significantly in the low minimum QoS regime, because many UEs receive low QoS and increasing the level of the 326 B. Xu et al. Fig. 10.6 Trade-off between the energy efficiency and the minimum QoS for NOMA and OMA 7 ×10 6 Energy Efficiency (bits/Joule) 6 5 4 NOMA, Proposed Power Control OMA, Fixed Transmit Power 3 2 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Minimum QoS (bits/s/Hz) minimum QoS means that these UEs cannot be served. In practice, the minimum QoS can be found in an off-line manner [46]. 10.4.3 Joint User Association and Power Control In this subsection, we examine the benefits of joint UA and power control design in energy-cooperation-enabled NOMA HetNets. We also present comparisons by considering different power allocation schemes with the conventional RSRP-based m in the OMA UA. In the OMA scenario, transmit power at the BS is set to Pm = Pmax scenario. Figure 10.7 shows the energy efficiency versus the number of UEs with the number of PBSs M = 5 and the energy transfer efficiency factor β = 0.9. We set the minimum QoS as τ min = 0.5 bits/s/Hz and the amount of harvested energy at MBS and PBS as 32 dBm and 22 dBm, respectively. We see that the proposed joint UA and power control algorithm achieves higher energy efficiency than the other cases, and significantly improves the performance when more UEs are served in the network. The reason is that the proposed algorithm is capable of obtaining larger multiuser diversity gains. The use of NOMA can obtain higher energy efficiency than the OMA case, due to NOMA’s capability of achieving higher spectral efficiency. Additionally, when equal power allocation is adopted in NOMA HetNets with the conventional RSRP-based UA, energy efficiency decreases with increasing the number of UEs of the network, which can be explained by the fact that given the total transmit power of a BS, the transmit power allocated to the data-streams of the UEs with better channel condition reduces when more UEs are served simultaneously. 10 Heterogeneous NOMA with Energy Cooperation Fig. 10.7 Energy efficiency versus the number of UEs for different joint UA and power allocation designs 12 327 ×10 6 Energy Efficiency (bits/Joule) 10 8 6 NOMA, Proposed Joint UA and Power Control NOMA, RSRP-based UA with Proposed Power Control NOMA, RSRP-based UA with Equal Power Allocation OMA, RSRP-based UA with Fixed Transmit Power 4 2 10 15 20 25 30 Number of UEs Fig. 10.8 Energy efficiency versus the number of PBSs for different joint UA and power allocation designs 2.2 ×10 7 NOMA, Proposed Joint UA and Power Control NOMA, RSRP-based UA with Proposed Power Control NOMA, RSRP-based UA with Equal Power Allocation OMA, RSRP-based UA with Fixed Transmit Power Energy Efficiency (bits/Joule) 1.8 1.4 1 0.6 0.2 2 3 4 5 6 Number of PBSs Figure 10.8 shows the energy efficiency versus the number of PBSs with the number of UEs N = 50 and the energy transfer efficiency factor β = 0.9. We set the minimum QoS as τ min = 0.1 bits/s/Hz and the amount of energy harvested by MBS and PBS as 37 dBm and 27 dBm, respectively. The proposed design outperforms the other cases. By using the proposed joint UA and power control with NOMA, the energy efficiency significantly increases with the PBS number, since the proposed design can obtain more BS densification gains. Again, the use of NOMA achieves better performance than OMA. For the case of RSRP-based UA with NOMA and equal power allocation, energy efficiency decreases with increasing the number of 328 B. Xu et al. PBSs, because the inter-cell interference has a big adverse effect on the NOMA transmission [3]. 10.5 Conclusion and Future Work This chapter studied UA and power control in energy-cooperation- aided two-tier HetNets with NOMA. A distributed UA algorithm was proposed based on the Lagrangian dual analysis, which does not require a central controller. Then, we proposed a joint UA and power control algorithm which achieves higher energy efficiency performance than the existing schemes. The proposed power control algorithm satisfies the KKT optimality conditions. Simulation results demonstrate the effectiveness of the proposed algorithms. The results showed that the proposed algorithm can efficiently coordinate the intra-cell and inter-cell interference and has the capability of exploiting the multiuser diversity and BS densification. The application of NOMA can achieve larger energy efficiency than OMA due to the higher spectral efficiency of NOMA. To further extend this line of work, other UA optimization designs in multi-cell NOMA networks such as proportional fairness or max-min fairness would be of interest, and they are not trivial extensions since the optimization problems involved will be distinct. Moreover, imperfect CSI can have a substantial effect on outage probability and average data rate in NOMA networks, as analyzed in [47]. One of the challenges for optimization designs under imperfect CSI is that error propagation occurs since intra-cell interference cannot be perfectly canceled. Therefore, robust optimization designs need to be developed in multi-cell NOMA networks. In addition, the application of MIMO technology in NOMA networks is another important research area, which can significantly improve the performance gain [32]. In MIMONOMA networks, inter-user pair/group interference can deteriorate the performance, as analyzed in [32, 48]. Therefore, how to mitigate the inter-user pair/group interference is crucial. Currently, UA and power control solutions in multi-cell MIMONOMA networks are not available, and more research efforts need to be made in this area. References 1. D. Liu, L. Wang, Y. Chen, M. Elkashlan, K.K. Wong, R. Schober, L. Hanzo. 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Commun. 15(6), 4438–4454 (2016) Chapter 11 NOMA in Vehicular Communications Yingyang Chen, Li Wang, Yutong Ai, Bingli Jiao and Lajos Hanzo 11.1 Background and Motivation With the rapid development of intelligent transportation systems (ITS), the broad objective of vehicular communications is to improve the travel experience of users. To support a variety ITS applications, the integrated vehicular networking concept termed as ‘vehicle-to-everything’ (V2X) has been proposed. To elaborate a little further, this includes four main types of communications, namely vehicle-to-vehicle (V2V), vehicle-to-pedestrian (V2P), vehicle-to-infrastructure (V2I), and vehicle-tonetwork (V2N) scenarios, where the ultimate objective is that of offering improved road safety, traffic efficiency, and infotainment services [1]. The IEEE 802.11p standard was conceived in support of wireless access for vehicular environments (WAVE), specifically dedicated to vehicular safety applications and to the provision of data rates ranging from 6 to 27 Mbps for short-transmission distances [2]. However, 802.11p fails to support flexible scalability and to provide quality of service (QoS) guarantees. Furthermore, it has a potentially unbounded delay. These characteristics of the IEEE 802.11p standard prevent its employment in demanding V2X services requiring low latency and high reliability [3, 4]. Moreover, Y. Chen · B. Jiao School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China e-mail: chenyingyang@pku.edu.cn B. Jiao e-mail: jiaobl@pku.edu.cn L. Wang (B) · Y. Ai School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China e-mail: liwang@bupt.edu.cn Y. Ai e-mail: ytailiwang@bupt.edu.cn L. Hanzo School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, U.K. e-mail: lh@ecs.soton.ac.uk © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_11 333 334 Y. Chen et al. due to the asynchronous nature of the IEEE 802.11p transmission, its performance is inevitably degraded by the packet collisions imposed by the hidden node problem encountered in carrier sense multiple access with collision avoidance (CSMA/CA) solutions. Finally, the current status of IEEE 802.11p does not support an evolutionary path for improving its reliability, robustness, and coverage [5]. As an alternative to the IEEE 802.11p-based vehicular ad hoc network (VANET) concept, the long-term evolution (LTE)-based V2X solution has been actively developed by the 3rd Generation Partnership Project (3GPP), and it was redefined as LTE V2X within the 3GPP standardization framework so as to provide a beneficial solution for V2X communications [6]. As a benefit of the global deployment and commercialization of LTE systems, LTE V2X serves as an integrated solution for vehicular communications. To be more specific, given an LTE network, both V2I and V2N services can be supported at a high data rate, whilst maintaining an excellent QoS with the aid of the so-called eNodeBs. Meanwhile, LTE can be extended to support V2V and V2P services by invoking direct device-to-device (D2D) communications for satisfying the QoS requirements even in the case of high vehicular densities [7]. Finally, in contrast to 802.11p, the hidden node problem can be avoided in LTE V2X scenario due to its synchronous nature, whilst both the reliability and latency can also be improved. The rest of this section is organized as follows. First, we provide an overview of existing LTE-based V2X systems. Then, we describe the applicability of a range of popular transmission techniques to vehicular communications. A non-orthogonal multiple access (NOMA) and spatial modulation (SM)-based transmission scheme are proposed for supporting the high data rate and high-reliability demands of V2X systems. Finally, we detail the outline of this chapter. 11.1.1 Overview of LTE-Based V2X 11.1.1.1 Typical V2X Services The operational LTE system is already capable of supporting ITS applications. The V2X services can be broadly classified into three typical types, namely road safety, traffic efficiency, and infotainment enhancement applications [1, 8]. Road safety enhancements aim for reducing the risk of accidents and hence have to satisfy stringent reliability and latency specifications. Basically, the road safety services are short messages and periodically broadcast from each vehicle to its neighbours within a particular geographic region. As the second category of vehicular applications, traffic efficiency enhancements aim for optimizing the platooning of vehicles by reducing traffic congestion. The vehicles are required to collect sensed data and send them to the remote management servers for route planning. Although the reliability and delay requirements related to traffic efficiency enhancements are less strict than those of the safety enhance- 11 NOMA in Vehicular Communications 335 ments, it is still necessary to keep the packet loss and latency low in high-velocity environments. In contrast to the previous two categories, infotainment services include a range of traditional and emerging Internet applications, such as popular content download and dissemination, social networking, and Web browsing, with the goal of providing an improved driving experience. 11.1.1.2 LTE-Based V2X Communication Modes To support key applications in vehicular communications, the LTE system offers a pair of communication modes. LTE D2D for V2V/V2P: Although the original system does not support V2V/V2P communications natively, LTE has been extended to support V2V/V2P direct communications based on a device-to-device (D2D) sidelink design through the so-called PC5 interface [7]. The LTE D2D mode allows the terminals in close proximity of each other to communicate directly without involving the base stations. As a benefit, the end-to-end latency can be reduced for satisfying the associated QoS requirements. At the time of writing, the LTE D2D mechanism is considered as the baseline for PC5-based V2V/V2P communications [9]. However, in high user density scenarios, the attributes of V2X services are different from those of the legacy LTE D2D communications, since V2X services are periodic or event-triggered. Hence, efficient resource allocation has to be conceived for dense, high-mobility scenarios. Cellular LTE for V2I/V2N: Cellular LTE refers to the common communication mode between vehicles and infrastructure/network units. Specifically, there are two main cellular LTE mechanisms, namely unicast and multicast. In the case of unicast, the vehicles are addressed individually. By contrast, in the multicast case, all vehicles in the relevant area are collectively addressed. LTE supports high-quality multicast and transmissions through the evolved multimedia broadcast multicast service (eMBMS) capabilities in the radio access network [10]. Compared to unicast services, multicast offers the capability of geocasting the data to a set of users more resource efficiently, although at the cost of longer delays due to the cumbersome eMBMS session set-up, especially in the face of a heavy traffic load. 11.1.1.3 LTE-Challenges in Vehicular Scenarios The LTE system is capable of providing a round-trip delay below 10 ms and a radio access latency of less than 100 ms. It is based on orthogonal frequency-division multiple access (OFDMA) in the downlink and single-carrier frequency-division multiple access (SC-FDMA) in the uplink. It exhibits flexible resource allocation and scheduling. The LTE system also relies on multiple-input multiple-output (MIMO) techniques for improving the diversity and/or multiplexing gain of the previous generations, making LTE attractive in dynamic vehicular wireless propagation environments. 336 Y. Chen et al. However, the ever-growing demands for vehicular communications increase the tele-traffic congestion. Hence, it is desirable to achieve a high bandwidth efficiency, massive connectivity, high reliability, and low latency in V2X communications. One of the limitations of the LTE systems arises from the fact that LTE was designed for supporting the user terminals sharing the wireless resources using orthogonal multiple access (OMA), which can be potentially improved by NOMA schemes in V2X communications. In order to support advanced V2X services, given their stringent reliability and latency requirements, multiple-input multiple-output (MIMO) techniques may be invoked. Traditionally, MIMO schemes have been designed either for enhancing the diversity gain by combating the channel fading (e.g. Alamouti code), or for spatial multiplexing (e.g. Vertical Bell Laboratories Layered Space-Time, termed VBLAST), albeit they are amalgamated by the multi-functional MIMO concept of [11, 12]. To accommodate the ever-increasing demands of multimedia services and applications, the massive MIMO concept emerged [13, 14]. Theoretically, massive MIMO is able to reap all the benefits of conventional MIMO and offers abundant degrees of freedom (DoFs). By exploiting the knowledge of the channel state information at the transmitter (CSIT), a massive antenna array becomes capable of simultaneously serving a large number of users by sharing its multiplexing gain among them, while providing higher data rates and transmission reliability. Furthermore, in contrast to shirt-pocket-sized handsets, the employment of large-scale MIMO schemes becomes realistic in V2X scenarios, since multiple antennas can be realistically accommodated [15, 16]. However, massive MIMOs suffer from various problems, including the interantenna interference (IAI) and the high complexity of the receivers. It would be a particularly costly process to acquire CSIT in frequency-division duplexing (FDD) systems. Moreover, the hardware cost (e.g. a dedicated radio frequency (RF) chain associated with each antenna) becomes excessive for large antenna arrays. In vehicular wireless communications, the gravest challenge is the hostile high-Doppler propagation imposed. For example, the dominant Doppler effect aggravates the intersubcarrier interference of orthogonal frequency-division multiplexing (OFDM), and the strong line of sight (LoS) component of V2V channels would aggravate the spatial correlation between antennas. Therefore, the direct applications of massive MIMO in vehicular transmissions are deemed to be problematic, and another version of massive antenna technology is required to be fit for LTE V2X communications. 11.1.2 The Applicability of NOMA to V2X Communications To mitigate the probability of access collision in V2X environments, a range of novel multiple access techniques has been proposed, such as sparse code multiple access (SCMA), pattern division multiple access (PDMA), and non-orthogonal multiple access (NOMA) to support higher bandwidth efficiency and massive connectivity [17, 18]. Among these techniques, NOMA exhibits an appealing low receiver complexity, 11 NOMA in Vehicular Communications 337 high bandwidth efficiency, and massive connectivity by allowing multiple users to share the same channel resource via power domain multiplexing. Thus, NOMA is considered to be a promising candidate for future wireless access [19]. To mitigate the multiple access interference (MAI), multi-user detection (MUD) techniques such as successive interference cancellation (SIC) [20] can be applied to the end-user receivers for detecting the desired signals. Through power domain multiplexing at the transmitter and SIC at the receivers, NOMA becomes capable of fully exploiting its capacity region hence outperforming the OMA schemes [21]. The specific design aspects of NOMA schemes in cellular environments have been discussed in [22–24]. Explicitly, in [22], the concept of basic NOMA with SIC was introduced and its performance was compared to that of the traditional orthogonal frequency-division multiple access (OFDMA) scheme through a systemlevel evaluation. A beneficial power allocation scheme was designed in [23] for striking compelling tradeoffs between the user fairness and system throughput. Lv et al. [24] studied a new cooperative NOMA transmission scheme and derived the outage probability associated with fixed power allocation. In vehicular environments, NOMA provides a new dimension for V2X services to alleviate the access collisions, thereby improving the bandwidth efficiency as well as supporting massive connectivity. The authors of [25] proposed an contention-based uplink NOMA solution in order to reduce the control signalling overhead. In [26], the NOMA concept was exploited to enhance the transmission of safety information, which required low latency and high reliability within a dense vehicular communication network. The authors of [27] invoked the NOMA principle for boosting the bandwidth efficiency of the infotainment applications in V2X services. In conclusion, NOMA is eminently applicable for supporting V2X services with enhanced bandwidth efficiency and QoS support. 11.1.3 The Applicability of SM to V2X Communications In recent years, spatial modulation (SM) [28] has grown in popularity, because in contrast to the traditional MIMO configurations, it only activates a single transmit antenna (Tx) at every transmission instance. Hence, it only requires a single-RF chain. As a benefit, the inter-antenna interference (IAI) can be completely eliminated. Thus, a reduced implementational cost and complexity are achieved [29, 30]. The basic idea of SM was initially derived from Chau and Yu’s work dating back to 2001 [31], where the receiver decodes the signals transmitted from different antennas. Then, a compelling SM-MIMO solution was proposed by Mesleh et al. in [32]. Since then, SM has been extensively studied in the scenario of point-to-point communications. In [28], the authors studied the channel capacity of the SM system under the parlance of information-guided channel hopping (IGCH). It was shown that IGCH provides better spectral efficiency than orthogonal space-time block coding (OSTBC). In [33], the SM concept was studied by using a low-complexity two-stage demodulator, and the potential advantages of SM-MIMO compared to the exist- 338 Y. Chen et al. ing spatial-multiplexing and Alamouti schemes were shown. In [34], Jeganathan et al. developed the maximum likelihood (ML)-optimum demodulator for SM-MIMO and a range of performance improvements was shown compared to the suboptimal demodulator introduced in [33]. The SM philosophy is that not only the classic quadrature amplitude modulation (QAM) symbols but also the index of the active Tx (spatial constellation) convey information for the sake of achieving bandwidth efficiency enhancements without sacrificing the advantages of a single-RF stage. Consequently, SM was proposed to be combined with massive MIMOs, yielding the novel concept of massive SM-MIMO, where each UE still has one RF chain combined with a massive Tx configuration [35, 36, 38]. Due to the single-RF structure of SM, both the cost and the design complexity of each user terminal remain similar to those of SM-MIMOs, while the data rates can be boosted by conveying more information bits via employing a large Tx array. More specifically, a large-scale multi-user SM-MIMO system was proposed in [35] along with multi-user detection (MUD) schemes. In [36], Wang et al. proposed an uplink transceiver scheme for massive SM-MIMO within frequency-selective fading environments. The authors of [37] investigated the achievable uplink spectral efficiency in a multi-cell massive SM-MIMO scenario, and [38] further investigated the optimal number of Txs at the user equipment. Indeed, a recent survey of SM can be found in [39]. In [40–43], SM and its extensions were considered in vehicular environments. A differential SM scheme was proposed for vehicle communications in [40], exhibiting robustness against timeselective fading and Doppler effects. Fu et al. [41] studied the bit error rate (BER) performance of SM under a three-dimensional V2V channel model. Peppas et al. [42] applied space shift keying (SSK) in inter-vehicular communications and derived a closed-form expression for the pairwise error probability. In [43], the performance of massive SM-MIMO over a spatio-temporally correlated Rician channel was analysed under a high-speed railway scenario. Moreover, Cui and Fang have demonstrated that by activating a single Tx, SM is capable of alleviating the channel correlation. In conclusion, SM has become increasingly appealing for V2V systems. 11.1.4 NOMA-SM Tailored for Vehicular Communications Let us continue by conceiving a novel transmission scheme, termed NOMA-SM, by intrinsically amalgamating NOMA and SM in support of vehicular communications [27]. Specifically, in synergy with the inherent requirement of high bandwidth efficiency, NOMA is invoked for non-orthogonally accessing all the resources combined with the single-RF benefits of SM. The bandwidth efficiency of the proposed NOMA-SM scheme is further boosted by a massive Tx configuration. Against this background, the key points of the proposed scheme are threefold: firstly, the novel NOMA-SM concept is proposed and its link reliability is quantified. Secondly, the capacity of NOMA-SM is derived and verified by Monte Carlo 11 NOMA in Vehicular Communications 339 simulations. Thirdly, a pair of upper bounds on the capacity of NOMA-SM is formulated in closed form and a power allocation optimization is considered. Explicitly, instead of simply combining a pair of popular techniques, their benefits are intrinsically amalgamated. By investigating the BER performance of NOMA in comparison to different MIMO techniques and the bandwidth efficiency of SM combined with distinct multiple access methods, NOMA and SM are shown to cooperatively improve V2V transmissions. 11.1.5 Outline of the Chapter The rest of this chapter is organized as follows. In Sect. 11.2, the system model of NOMA-SM is presented, while Sect. 11.3 provides the capacity analysis and mutual information (MI) evaluation of NOMA-SM. Our capacity upper bound derivations and power allocation problem are considered in Sect. 11.4. Simulation results and discussions of the BER performance are provided in Sect. 11.5, together with the numerical capacity analysis and power allocation optimization. In the final section, we offer the main conclusions of this chapter and discuss some open problems as well as a range of promising potential research directions. For convenience, we list the most important notations here. Notation: Uppercase and lowercase bold-faced letters indicate matrices and vectors, respectively. (·)−1 , (·) H , det (·), and [·] p,q represent inverse, conjugatetranspose, determinant, and the entry in the pth row and q-column of a matrix, respectively. E X {·} denotes the expectation on the random variable X . A ∈ C M×N is a complex-element matrix with dimensions M × N , and I N is an N × N identity matrix. |·| and (·)∗ imply the absolute value and the conjugate of a complex  scalar, while · denotes the Euclidean norm of a vector. Finally, x ∼ C N μ, σ 2 indicates that the random variable x obeys a complex Gaussian distribution with mean μ and variance σ 2 . 11.2 System Model We consider a generic vehicular communication system, where the vehicle-toinfrastructure (V2I), V2V, and intra-vehicle transmissions are all included. As shown in Fig. 11.1, a base station (BS) is located at the roadside while the vehicles V1 and V2 are in motion. There is a mobile user U in V1 who requests to download a file locally cached at the BS. Vehicle V2 also requests to download its own intended signal from BS. We assume that V1 has also acquired the signal of V2 , as a result of the first transmission phase, during which the messages of V1 and V2 are transmitted simultaneously from the BS. For example, BS employs a NOMA technique to multiplex signals of V1 and V2 in the power domain. By involving the classical SIC, V1 extracts the signal of V2 in the spirit of cooperation. Another appropriate interpre- 340 Y. Chen et al. Phase I Phase II (exploiting NOMA-SM) BS V1 V2 U Fig. 11.1 An illustration of the considered vehicular communication system, where NOMA-SM is applied in Phase II tation is related to the distribution of popular multimedia contents in VANET [44], using peer-to-peer protocols for exchanging popular packets through V2V channels. Therefore, as shown in Fig. 11.1, cooperative inter-vehicle transmission is constructed during the second phase to enhance the reception reliability. Specifically, V1 forwards the desired signal to V2 for cooperatively enhancing the reception at V2 . Furthermore, the second phase scenario can be generalized to various situations. For example, user U can be a roadside unit, aiming for exchanging information with the onboard unit of the vehicle V1 . While U may be a vehicle which is much closer to V1 than V2 . Similar to the concept in [45], a VANET is formed among these vehicles for exchanging safety information, or for cooperatively distributing popular multimedia contents within a geographical area of interest. In general, our model is valid in a wide range of vehicular scenarios. In the light of bandwidth scarcity, cognitive radio techniques can be exploited in the second stage to opportunistically exploit the spectrum holes in the licensed spectrum. For example, V1 may be permitted to share the cellular uplink, for which the data traffic is typically lighter than for the downlink, hence resulting in potential spectrum wastage [46]. Basically, underlay cognitive transmission is feasible without traversing through the primary network. However, the interference imposed by V1 on the BS in the second stage should be carefully managed, albeit this is beyond the scope of this article. Our main focus is on the second stage of the cooperative transmission in Fig. 11.1, since the performance in the first phase can be analysed similarly. Particularly, the NOMA-SM strategy is employed in the second stage for both V1 –V2 and V1 –U links. The schematic diagram of NOMA-SM operated in the second stage is presented in Fig. 11.2, where V1 assigns distinct transmit power to V2 and U . The user access is based on NOMA, combined with SM. Although there is the literature proposing 11 NOMA in Vehicular Communications 341 yU Signal for U Signal for V2 Superposition coding l 1 Spatial demodulation SIC of signal for V2 Desired signal detection m U yV V1 Spatial demodulation Desired signal detection V2 Fig. 11.2 The schematic diagram of the proposed NOMA-SM strategy multi-user SM schemes [47, 48], we use a classical SM designed for point-to-point transmission [28, 49] in vehicular environments. In what follows, we first elaborate on the principles of the proposed NOMA-SM scheme. Then, our V2V channel model is detailed. 11.2.1 The Principles of NOMA-SM Let us assume that Nt , Nr , and Nu omnidirectional antennas are employed at V1 , V2 , and U , respectively. As illustrated in Fig. 11.2, the proposed NOMA-SM strategy is applied both for the V1 –V2 and V1 –U links. At the transmitter V1 , two independent bit streams are prepared for transmission. The bit stream for V2 is partitioned into two parts: the first log2 (Nt ) bits are used for Tx activation, activating a specific Tx index n t (n t ∈ {1, . . . , Nt }). The other log2 (M) bits destined for V2 are combined coding. with log2 (L) bits for U , employing superposition √ √ Subsequently, the modulated symbol αγl + 1 − αχm is radiated from the activated Tx n t , where γl and χm are intended for the in-car user U of V1 and for V2 , respectively, satisfying E{|γl |2 } = E{|χm |2 } = E s , where E s is the average energy per transmission at V1 , while α is the power allocation factor. According to the NOMA principle [23], the transmit power of the distant user in Fig. 11.2 must be higher than that of the close-by user, that is (1 − α) E s > α E s . With this, 0 < α < 21 should be guaranteed since the in-car user has a good channel. As a result, a block of identifies the active Tx n t and the superimposed log2 (Nt M L) bits√unambiguously √ complex symbol αγl + 1 − αχm transmitted from it. Hence, a NOMA-SM super symbol can be expressed as x = en t √ αγl +  √ 1 − αχm , where en t is the n t th column of the identity matrix I Nt , indicating that the n t th Tx of V1 is activated, while the other (Nt − 1) Txs are deactivated. Furthermore, χm is 342 Y. Chen et al. the mth symbol in the M-ary amplitude-phase modulation (APM) used for V1 –V2 transmission, while γl is the lth symbol in the L-ary APM for V1 –U transmission. Considering the propagation inside the vehicle V1 , we assume that the in-car user U experiences a frequency-flat Rayleigh channel. For example, the Txs of V1 are installed on the central column of the vehicular dashboard, while the receive antennas (RAs) of U are placed behind the passenger front seat, without LoS from V1 . In [50], this scenario has been shown to be well suited to characterize diffuse scattering. Thus, we let G ∈ C Nr ×Nt denote the channel matrix between V1 and U , and assume that all entries of G are independent identically distributed (i.i.d), obeying the distribution C N (0, 1). The signal vector received at U and V2 can be written as yU = gn t yV = √ √ p0 hn t αγl + √  √ 1 − αχm + wU , αγl +  √ 1 − αχm + wV , (11.1) (11.2) respectively, where p0 represents the average power drop between V1 and V2 due to the large-scale fading. Furthermore, gn t ∈ C Nu ×1 is the n t th column of G, representing the channel vector between U and the n t th Tx of V1 , while hn t ∈ C Nr ×1 is the n t th column of the V2V channel matrix H ∈ C Nr ×Nt , indicating the complex fading envelope between V2 and the n t th Tx of V1 . Finally, w(·) denotes a complex additive white Gaussian noise (AWGN) vector with a power spectrum density of σ02 per entry. For the inter-vehicle channel, the path loss is considerable in (11.2), while it is neglected between the in-car user and the antenna array of V1 . In our system, the transmitter and both receivers are assumed to have perfect synchronization in both time and frequency. Full channel state information is assumed to be available at receivers (i.e. CSIR). In principle, both V2 and U first have to detect the signal destined for V2 , i.e. the activated Tx index n̂ t and the APM symbol χm̂ at each particular time instant. The corresponding optimum maximum likelihood (ML) detector is invoked at U and V2 according to 2  √     n̂ t , χm̂ = arg min yU − 1 − αgn t χm  , n t ,m  2      n̂ t , χm̂ = arg min yV − p0 (1 − α)hn t χm  . n t ,m (11.3) (11.4)   After eliminating the interference imposed by nˆt , χm̂ on yU , U becomes capable of performing another ML detection to acquire the desired signal γlˆ. 11 NOMA in Vehicular Communications 343 11.2.2 V2V Massive MIMO Channel Model In contrast to the conventional fixed-to-mobile cellular radio systems, in V2V systems, both the transmitter and receiver are in motion and both are equipped with low-elevation antennas, which will result in quite different propagation conditions. Hence, a non-isotropic scattering V2V stochastic model was proposed in [51] for characterizing a wide variety of V2V scenarios by adjusting relevant model parameters. In [41], a novel three-dimensional V2V geometry-based stochastic channel was proposed for accurately capturing the effect of vehicular traffic density on the channel. In this article, we consider a spatio-temporally correlated Rician channel model for characterizing our narrowband V2V massive MIMO channel, which has also been exploited in [43] and [52]. We describe the underlying V2V channel as a matrix of complex fading envelopes, i.e. H ∈ C Nt ×Nr , which can be expressed as H=  K H̄ + K +1  1 H̃, K +1 where K is the Rician factor, while H̄ is the fixed part related to the LoS component. Furthermore, H̃ represents the variable part, whose entries are correlated complex ˜ , we assume that Gaussian variables. Given H̃ = h p,q p,q E h̃ Rp,q h̃ Rp̂,q̂ = E h̃ Ip,q h̃ Ip̂,q̂ , E h̃ Rp,q h̃ Ip̂,q̂ = E h̃ Ip,q h̃ Rp̂,q̂ = 0, where p, p̂ ∈ {1, . . . , Nr } and q, q̂ ∈ {1, . . . , Nt }. Explicitly, for each h̃ p,q , the autocorrelations of the real and imaginary parts are identical and the cross-correlations between real and imaginary parts are equal to zero. Hence, the correlated channel matrix can be described by the widely used Kronecker correlation model [53], which is expressed as 1 1 H̃ =  r2 Ĥ t2 . Here,  t ∈ C Nt ×Nt and  r ∈ C Nr ×Nr are the correlation matrices at V1 and V2 , t {1, . . . , Nt }, and respectively, with the elements defined as [ t ]q,q̂ = σq, q̂ for q, q̂ ∈ r {1, . . . , Nr }. Furthermore, Ĥ is the independent Rayleigh [ r ] p, p̂ = σ p, p̂ for p, p̂ ∈ channel matrix whose entries are i.i.d complex Gaussian random variables, i.e. [Ĥ] p,q = ĥ p,q ∼ C N (0, 1). Specifically, the correlation matrices  t and  r can be determined according to a concrete model. Here, the exponential model of Loyka |q−q̂ | t and [54] is adopted, and the correlation matrix entries are formed as σq, q̂ = κt p− p̂ | | r , where κt and κr are the adjacent antenna correlation coefficients at σ p, p̂ = κr V1 and V2 , respectively. 344 Y. Chen et al. In order to mimic the influence of the V2V channel’s time-varying effects, we take the temporal correlation into consideration, which is defined as δ (τ ) = E Ĥ (t) Ĥ (t + τ ) , where τ is the sampling time. In [43], Jakes’ model is used for describing the temporal correlation expressed as δ (τ ) = J0 (2π f D τ ), where f D is the maximum Doppler frequency related to both the carrier frequency and the velocity of the terminal. For simplicity of analysis, in the following, we omit the index τ . Observe that δ = 1 indicates that the underlying V2V channel is quasi-static, while δ < 1 is related to a time-varying channel due to mobility. Naturally, both the spatial and temporal correlations would affect the performance of the receivers. 11.3 Capacity Analysis of the NOMA-SM System Recall that the proposed NOMA-SM transmission scheme relies on a pair of independent√ spaces: the √ classical signal-domain, pertaining to the radiated superimposed symbol αγl + 1 − αχm , and the Tx-domain, pertaining to the activated Tx index n t . More specifically, the message intended for V2 is conveyed by both of the two streams. While the message destined for U is only mapped to the classical signaldomain, superimposed with part of V2 ’s signal in the power domain. In what follows, we investigate the capacity of the collaboration-aided vehicle V2 and the in-car user U . Monte Carlo estimates are also provided for MI evaluation, followed by an illustrative example to augment the theoretical analysis. 11.3.1 Capacity Analysis of the Collaboration-Aided Vehicle In the NOMA protocol, the transmit power assigned by V1 to the distant user V2 has to be higher than that to the close-by user U . Then, the distant user directly detects its signal, since the interference induced by the close-by user is lower and can thus be treated as background noise. Considering that all Txs of V1 are activated with the same probability for NOMA-SM, the instantaneous capacity pertaining to the classical signal-domain of V2V transmission is given by sig C V = max I (χ ; yV |n t ) fχ   Nt E p h 2 +σ 2 1 log2 α Es p0 hi 2 +σ0 2 . = Nt i=1 s 0 i (11.5) 0 Observe that no practical modulation constellation is assumed, when performing these capacity calculations. Since the channel capacity relates to the highest rate in 11 NOMA in Vehicular Communications 345 bits per channel use at which information can be sent with arbitrarily low probability of error, in (11.5), we substitute χm by χ , which denotes a random input signal alphabet with a distribution of f χ . On the other hand, the MI conveyed by the spatialdomain Tx-constellations can be written as I (n t ; yV ) = Nt  Pr ( yV | hi ) 1  dyV , Pr ( yV | hi ) log2 Nt i=1 Pr (yV ) (11.6) where Pr (yV |hi ) denotes the probability density function (PDF) of the channel output yV received over the ith channel vector of H, given by Pr (yV |hi ) = π Nr   1 exp −yVH  i−1 yV , det ( i ) where  i = σ02 I + pE s hi hi H . As a result, the instantaneous capacity of V2 in the NOMA-SM system is formulated as sig C V = C V + I (n t ; yV ) . (11.7) sig Remark It is worth noting that in (11.5), C V is achievable where the optimum input distribution for χ is Gaussian. In fact, this optimum input distribution is also regarded as the optimum input distribution for a conventional SM system. This is a common assumption in the majority of SM capacity-related contributions [28, 55– 57], effectively simplifying the analysis. Nevertheless, a fundamental weakness of the Gaussian input assumption is that f χ affects both I (χ ; yV |n t ) and I (n t ; yV ). Clearly, the Gaussian input distribution maximizes I (χ ; yV |n t ), but it is unclear whether it maximizes I (n t ; yV ). In addition, the equiprobable activation of antennas is a widely accepted assumption for SM-enabled systems, albeit this activation regime cannot guarantee the optimal spatial design capable of achieving the capacity in the Tx-domain. Actually, Liu et al. in [55] studied the optimal antenna activation required for Tx-domain capacity maximization. Moreover, Basnayaka et al. [30] have demonstrated that the Gaussian input does not achieve the upper limit of the MI provided by an SM-aided system. As a further insight, although the MI conveyed by the Tx-domain cannot be formulated as an analytical expression, we are inspired to derive the capacity upper bound and to conceive the associated power allocation optimization schemes, which will be addressed in Sect. 11.5. 11.3.2 Capacity Analysis of the In-Car User In contrast to the receiver of V2 , the receiver of U can detect its own signal after removing the interference imposed by V2 , as seen in Fig. 11.2. To demonstrate the feasibility of this SIC procedure, we first deduce the maximum rate of which U can 346 Y. Chen et al. detect the message of V2 . Specifically, the maximum rate for U detecting the message related to the classical signal-domain of V2 is given by V,sig CU   Nt E s gi 2 + σ02 1  = log2 . Nt i=1 α E s gi 2 + σ02 (11.8) The MI associated with U detecting the information embedded in the Tx-constellation of V2 can be written as I (n t ; yU ) = Nt  Pr ( yU | gi ) 1  dyU , Pr ( yU | gi ) log2 Nt i=1 Pr (yU ) (11.9) where Pr (yU |gi ) denotes the PDF of the channel output yU received over the ith channel vector of G given by Pr (yU |gi ) = π Nu   1 exp −yUH i−1 yU , det (i ) where i = σ02 I + E s gi gi H . As a result, the instantaneous capacity for U detecting the signal of V2 can be expressed as V,sig CUV = CU + I (n t ; yU ) . (11.10) It may be readily seen that CUV > C V is always satisfied, since gi 2 > p0 hi 2 , guaranteeing the success of SIC. Hence, the capacity of U detecting its own desired signal is written as CU = max I ( γ ; yU | n t , χ , G) fγ   Nt (11.11) = N1t log2 1 + ασE2s gi 2 , i=1 0 where γ denotes the random input signal variable related to the desired message of U , with a distribution of f γ . The capacity for U detecting γ indeed becomes achievable when the channel’s input distribution f γ is Gaussian. 11.3.3 Mutual Information To appreciate the above theoretical analysis in terms of its relevance, next, we characterize the bandwidth efficiency of the proposed NOMA-SM. Assuming perfect knowledge of the instantaneous channel state information at both receivers, the MI achieved by V2 and U with the aid of practical APM constellations is evaluated by the classical Monte Carlo method. For the collaboration-aided vehicle V2 , the MI 11 NOMA in Vehicular Communications 347 between a discrete signal input (n t , χm ) and the received signal yV can be formulated as yV |n t ,χm ,H) I ( n t , χm ; yV | H) = En t ,χm ,yV log2 Pr( Pr( yV |H)  yV |χm ,hi ) = Nt1M × Pr ( yV | χm , hi ) log2 Pr(Pr(y dyV , V |H ) (11.12) where the conditional probability Pr ( yV | χm , hi ) is expressed as H  √ exp − yV − p0 (1 − α)hi χm   √ ×  i−1 yV − p0 (1 − α)hi χm , 1 Pr( yV | χm , hi ) = π Nr det( i) with  i = σ02 I + α p0 E s hi hiH . With regard to the in-car user U performing SIC first, the MI between the information input (n t , χm ) and the received signal yU is given by I ( n t , χm ; yU | G) = Nt1M ×  (11.13) yU |χm ,gi ) Pr ( yU | χm , gi ) log2 Pr(Pr(y dyU , U |G ) where the conditional probability Pr ( yU | χm , gi ) is expressed as 1 Pr ( yU | χm , gi ) = π Nu det( ×  Hi ) −1    √ √ exp − yU − 1−αgi χm i yU − 1−αgi χm , with i = σ02 I + α E s gi giH . Subsequently, the MI between the information input γl and the received signal yU after perfect SIC is expressed as   I γl ; ỹU | gn t = 1 Nt L  Pr ( ỹU | γl , gi ) log2 1 Nt L Pr( ỹU |γl ,gi ) dỹU , Pr( ỹU |γk ,g j ) (11.14) k, j √ where ỹU = yU − 1 − αgi χm with i ∈ {1, . . . , Nt } and m ∈ {1, . . . , M} denotes the received vector after SIC. The conditional probability Pr ( ỹU | γl , gi ) is given by    ỹU − √αgi γl 2 Pr( ỹU | γl , gi ) =  .  N exp − σ02 π σ02 u 1 348 Y. Chen et al. 11.3.4 An Illustration In this part, a simulation-based study of our theoretical expressions is provided with the aid of the MI attained by practical APM constellations. We set Nt = 64, Nr = Nu = 2 for our MIMO configurations in conjunction with α = 0.1, E s = 1 and p0 = 10−3 are given. The channel matrix H is generated according to Sect. 11.2.2, where K = 0.2, κt = κr = 0.5, and δ = 1 are used. Each entry of G is identically and independently generated according to a complex Gaussian distribution C N (0, 1). In our Monte Carlo evaluations, the 16PSK signal constellation is chosen as the APM for χm and γl ; hence, we have M = L = 16. The effective transmit signalto-noise ratio (SNR) at V1 is given by p0 E s /σ02 as the horizontal axis of Fig. 11.3. Notice that the transmit-SNR cannot be readily interpreted physically, because it relates the transmitter power to the noise power at the receiver, but its notion is convenient to use in NOMA-aided scenarios. Given the effective transmit-SNR at V1 as SNR = p0 E s /σ02 , the average receive-SNR at V2 can be computed as SNRrV2 = (1 − α) SNR . 1 + αSNR 14 12 bps/Hz 10 I(nt; y V) 8 I(nt; y U) C sig V 6 C V,sig U 4 m ; y V) I(nt, m ; y U) CV 2 0 -10 I(nt, CV U 0 10 20 30 40 SNR [dB] Fig. 11.3 Capacity and MI performance for Nt = 64, Nr = Nu = 2, M = L = 16, and α = 0.01. sig V,sig Specifically, C V , CU , I (n t ; yV ), and I (n t ; yU ) are obtained from (11.5), (11.8), (11.6), and (11.9), respectively. While I (n t , χm ; yV ) and I (n t , χm ; yU ) are generated from (11.12) and (11.13), respectively, after averaging over multiple channel realizations. Finally, C V and CUV are evaluated from (11.7) and (11.10), respectively 11 NOMA in Vehicular Communications 349 Similarly, the average receive-SNR at U for detecting the signal of V2 and that of itself is respectively expressed as (1−α)SNR , p0 +αSNR αSNR . p0 SNRrU,V2 = SNRrU = Hence, the effective transmit-SNR at V1 is unambiguously related to the SNRs at each receiver. Furthermore, we use SNR = p0 E s /σ02 in all of the subsequent performance analyses. The relevant results of Fig. 11.3 are discussed as follows. sig • The capacity of V2 gleaned from the signal-domain, that is C V obtained from (11.5), increases steadily up to a saturation point as the SNR increases. By contrast, V,sig the capacity for U detecting the signal-domain destined for V2 , i.e. CU obtained sig from (11.8), is higher than C V in the low and moderate SNR domain. Clearly, a successful detection of the signal-domain of V2 can be performed by U . • The MI I (n t ; yV ) generated using (11.6) increases with the SNR and saturates at 6 bps/Hz, since the input entropy of the Tx-domain space is log2 (Nt ). By contrast, the MI I (n t ; yU ) attained by (11.9) is as high as 6 bps/Hz across almost the entire SNR range, since the channel quality of U is much higher than that of V2 , implying that U can successfully detect the signal of V2 embedded in the Tx-domain. • The capacity of V2 , i.e. C V , grows steadily as the SNR increases up to its saturation at high SNRs, but it remains lower than CUV . Since C V is obtained by the summation V,sig sig of C V and I (n t ; yV ), and CUV equals to the sum of CU and I (n t ; yU ). Naturally, sig V,sig CUV > C V is satisfied, as CU and I (n t ; yU ) are higher than C V and I (n t ; yV ), respectively. Therefore, U can always perform successful SIC. • The MI curves I (n t , χm ; yV ) and I (n t , χm ; yU ) are generated from (11.12) and (11.13), respectively, after averaging over multiple channel realizations. It may be observed that the simulated curve I (n t , χm ; yV ) matches the analytical capacity C V quite closely upto an SNR of 5 dB, but beyond that I (n t , χm ; yV ) starts to drift away from C V . By contrast, the drift of I (n t , χm ; yU ) from CUV remains nearly unchanged. Both drifts are due to the fact that the MI attained with the aid of practical APM modulation is upper bounded by the capacity, namely by the maximum data rate related to the optimal input distribution. 11.4 Power Allocation Algorithms It has been demonstrated that the MI conveyed by the Tx-domain cannot be readily formulated as a closed-form expression, only by resorting to simulations. Thus, it is very hard to perform an optimal power allocation for NOMA-SM. To circumvent this problem, we first derive an upper bound of the NOMA-SM capacity. Then the power allocation, which is capable of maximizing the capacity bound is considered, leading to the optimal solution. 350 Y. Chen et al. 11.4.1 Problem Formulation Theoretically, the instantaneous capacity of V2 in the NOMA-SM system can be expressed as C V = max I (n t , χ ; yV ) = max h (yV ) − h (yV |n t , χ ) , fχ fχ (11.15) where h (·) denotes the differential entropy. The conditional differential entropy h (yV |n t , χ ) in (11.15) is explicitly given by h (yV |n t , χ ) = Nt    1  log2 det π e p0 α E s hi hiH + σ02 I . Nt i=1 To determine C V , we have to evaluate h (yV ), which requires the knowledge of the distribution of yV . It may be readily seen that the MI I (n t , χ ; yV ) is maximized if the vector variable yV has a Gaussian distribution. Thus, we assume that the received vector yV has a Gaussian distribution, which is a zero-mean vector having a covariance matrix presented as       E yV yVH = HEn t en t Eχ p0 (1 − α) χ χ ∗ enHt H H     + HEn t en t Eγ p0 αγ γ ∗ enHt H H + σ02 I   Nt 1  ei e H p0 (1 − α) E s H H =H Nt i=1 i   Nt 1  +H ei e H p0 α E s H H + σ02 I Nt n =1 i t p0 E s = HH H + σ02 I. Nt An upper bound of h (yV ) can be formulated as    p0 E s HH H + σ02 I . h (yV ) ≤ log2 det π e Nt Hence, we obtain an upper bound of C V which is written as 11 NOMA in Vehicular Communications 351    C V ≤ log2 det π e pN0 Et s HH H + σ02 I Nt    log2 det π e p0 α E s hi hiH + σ02 I − N1t i=1   Nr log2 pN0 Et s λ2j + σ02 = (11.16) j=1 − N1t Nt i=1   log2 p0 α E s hi 2 + σ02 = C VB1 , where λ j is the jth singular value of H with j ∈ {1, . . . , Nr }. Clearly, C VB1 has Nr DoFs, and it is the same as the capacity of an (Nt × Nr )-element spatially multiplexed MIMO system, subject to inter-user interference. On the other hand, the MI of the Tx-domain has a natural upper bound written as I (n t ; yV ) ≤ log2 (Nt ), which corresponds to the maximum MI that can be conveyed by the Tx-domain of the V2V transmission link. Now, another upper bound of C V may also be formulated as sig C V ≤ C V + log2 (Nt )   Nt (11.17) E p h 2 +σ 2 = N1t log2 α Es p0 hi 2 +σ0 2 + log2 (Nt ) = C VB2 . i=1 s 0 i 0 Before proceeding, we provide a numerical illustration in order to evaluate both of the upper bounds on the capacity of V2 . Figure 11.4 depicts C V and both upper bounds of the NOMA-SM system in conjunction with Nt = 64, Nr = 2, M = 16, and α = 0.1, which exhibit distinct approximations of C V within certain SNR regions. The upper bound C VB1 gives a tight bound of C V at low SNRs, indicating that the NOMA-SM capacity at V2 is almost the same as that of a spatially multiplexed MIMO system of the same configuration in the presence of inter-user interference. However, the MI embedded in the Tx-domain saturates as the SNR increases, which is due to the fact that Nt is finite. Hence, at high SNRs, C VB2 is much tighter. Based on the above observations, a refined upper bound on the capacity of V2 in the NOMA-SM system is represented as   C VB = min C VB1 , C VB2 . (11.18) Considering the QoS of the two receivers from a practical perspective, we define the minimum rate requirement of V2 and U as C̃ V and C̃U , respectively. The optimization problem constructed for maximizing the sum capacity with a power allocation factor of α can be formulated as P : max CU + C VB α 352 Y. Chen et al. 25 CB1 V 20 CB2 V CV bps/Hz 15 10 5 0 -10 0 10 20 30 40 50 SNR [dB] Fig. 11.4 Capacity and two upper bounds of the V2V transmission link with Nt = 64, Nr = 2, M = 16, and α = 0.01. Specifically, C V , C VB1 , and C VB2 are evaluated from (11.7), (11.16), and (11.17), respectively ⎧ ⎪ ⎨CU ≥ C̃U , (a) s.t. C VB ≥ C̃ V , (b) ⎪ ⎩ 0 < α < 21 . (c) (11.19) 11.4.2 The Proposed Power Allocation Algorithm To solve the proposed optimization problem, we first express the derivatives of CU , C VB1 , and C VB2 with respect to α as dCU dα B dC V 1 dα B dC V 2 dα = 1 Nt Nt i=1 = − N1t = − N1t E s gi 2 , α E s gi 2 +σ02 Nt n t =1 Nt n t =1 E s p0 hi 2 , α E s p0 hi 2 +σ02 (11.20) E s p0 hi 2 , α E s p0 hi 2 +σ02 respectively. Observe from (11.20) that CU is a monotonically increasing function of α, given its positive derivative, while both C VB1 and C VB2 are decreasing ones. Thus, when the constraint (c) of (11.19) is taken into account, there exist both minimum 11 NOMA in Vehicular Communications 353 and maximum capacities that V2 and U can achieve. Furthermore, to satisfy the constraint (a) and (b), we have the following conditions for C̃U and C̃ V , respectively   1 , 0 < C̃U < CU α = 2 C VB   1 α= < C̃ V < C VB (α = 0) . 2 Given the above conditions, we can rewrite the constraints of problem P in a compact form as     g −1 C̃U < α < f −1 C̃ V , where g −1 (·) and f −1 (·) indicate the inverse function of CU and C VB , respectively. To guarantee that the feasible set of problem P is non-empty, a further refined condition for setting C̃ V is given by     1 < C̃ V < C VB α = g −1 C̃U . C VB α = 2  2  2 since gn t  > p0 hn t  is always satisfied, the derivative of   Moreover, CU + C VB can be guaranteed to have a positive value. Accordingly, the objective function of problem P is a monotonically increasing function and can be maximized, when α reaches the upper bound of its feasible set. With C̃U and C̃ V being appropriately set, we find that the upper bound of α’s feasible set is related to the constraint (b) of (11.19), and the lower bound corresponds to the constraint (a) of (11.19). Thus, the optimal solution of problem P is   P = f −1 C̃ V . αopt (11.21) This optimal solution implies that the amount of power allocated to V2 is ‘just’ sufficient to meet the minimum rate requirement C̃ V , while the remaining power is used for U , aiming for maximizing its capacity. Nevertheless, we should notice that there may exist some practical considerations, which require us to give high priority to the V2V transmission link, such as those of safety applications, which have to be served reliably. By contrast, the transmissions for in-car users are typically related to infotainment applications, for example peer-to-peer video sharing and multimedia advertisements [58]. Hence, it may be desirable to maximize the data rate of the V2V link, while guaranteeing the minimum rate requirement of the in-car user. To this end, we develop an alternative optimization problem formulated as 354 Y. Chen et al. O : max C VB α ⎧ ⎪ ⎨CU ≥ C̃U , (a) s.t. C VB ≥ C̃ V , (b) ⎪ ⎩ 0 < α < 21 . (c) (11.22) Clearly, the objective function of (11.22) is a monotonically decreasing function of α, and it is maximized, when the constraint (a) is inactive. Therefore, the optimal solution of problem O can be written as   O = g −1 C̃U . αopt (11.23) So far, we have proposed a pair of power allocation schemes and analysed the solvability of the optimization problems considered. Explicitly, we provided an algorithm for finding the optimal solution of each problem, which are summarized in Table 11.1. The proposed algorithm essentially performs bounding through with the aid of a bisection procedure, yielding globally optimal solutions at linearly increasing computational complexity [59]. In specific, the minimum rate requirements of V2 and U are respectively set as C̃U = C̃ V = C U (α = 2 C VB (α = 1 2 ), 1 2 )+C VB 2 α = g −1 (C̃U ) (11.24) , for simplicity. Basically, both of the two power allocation optimization problems satisfy realistic practical considerations, and the suitable one can be flexibly selected based on the specific data priority of the distinct transmission links. 11.5 Simulations and Discussions In this section, simulation results are provided for evaluating the performance of the proposed NOMA-SM scheme. The system parameters are summarized as follows. The MIMO configurations for the NOMA-SM system are set as Nt = 64, Nr = Nu = 2. We fix p0 = 10−3 , or, equivalently, the path loss exponential is set to 3, and the distance between V1 and V2 is assumed to be 10 m, which is typical for urban environments, especially during rush hours. 11 NOMA in Vehicular Communications 355 Table 11.1 Power Allocation Algorithm Power Allocation Algorithm for Problem P and Problem O 1. Initialization     Set tolerance 0 < ε ≪ 1. Calculate CU α = 21 and set C̃U = CU α = 21 /2. 2. Determine the lower bound of α and find the optimal solution of problem O Set α L = 0 and αU = 21 . While α L − αU > ε U Set α = α L +α . Calculate CU (α). 2 If CU (α) − C̃U > 0 αU = α Else α L = α. End End      U Set C̃ V = C VB α = 21 + C VB α = α L +α /2. 2   O O U The optimal solution to the problem O is obtained as αopt and . Calculate CU αopt = α L +α 2   O C VB αopt . 3. Determine the upper bound of α and find the optimal solution of problem P U Set αmin = α L +α and αmax = 21 . 2 While αmax − αmin > ε max . Calculate C VB (α). Set α = αmin +α 2 If C VB (α) − C̃ V > 0 αmin = α Else αmax = α. End End   P P max The optimal solution of the problem P is obtained as αopt = αmin +α . Calculate CU αopt 2   P . and C VB αopt 11.5.1 BER Results and Discussions In this subsection, the BER performance of the NOMA-SM scheme is compared to NOMA relying on the popular VBLAST technique, termed NOMA-VBLAST. Specifically, we focus on the receiver performance of V2 . The effects of the Rician K -factor, adjacent antenna correlation coefficient, temporal correlation, and power allocation factor are all taken into consideration. The Rician K -factors are configured as K = 2.186 and K = 0.2 for low and high vehicular traffic density, respectively (see [51] for more details). More specifically, we consider a pair of references: NOMA-VBLAST applied with 16QAM and Nt = 2, and NOMA-VBLAST adopted QPSK and Nt = 4. The MIMO configuration of the references is the same as that of NOMA-SM except for Nt . Besides, QPSK is applied for NOMA-SM. Thus, the following BER comparisons are carried out for the same bandwidth efficiency of 8 bits per channel use (bpcu). The optimum ML detector described in (11.4) is 356 Y. Chen et al. 10 -1 -2 10 -3 10 -4 10 -5 10 -6 BER 10 NOMA-SM K=0.2 NOMA-SM K=2.168 NOMA-VBLAST K=0.2 16QAM NOMA-VBLAST K=2.168 16QAM NOMA-VBLAST K=0.2 QPSK NOMA-VBLAST K=2.168 QPSK 25 30 35 40 45 SNR [dB] Fig. 11.5 BER comparisons with different Rician K -factor when κt = κr = 0.2 and δ = 1 are given, and the power allocation factor is fixed at α = 0.001, as evaluated by the Monte Carlo simulation with 106 channel realizations 10 -1 NOMA-SM, t =0.2 NOMA-SM, t =0.8 10 NOMA-VBLAST, t =0.2,16QAM NOMA-VBLAST, t =0.8,16QAM -2 NOMA-VBLAST, t =0.2,QPSK NOMA-VBLAST, t =0.8,QPSK -3 10 -4 10 -5 10 -6 BER 10 25 30 35 40 45 SNR [dB] Fig. 11.6 BER comparisons with different adjacent antenna correlation coefficient at V1 , i.e. κt , when K = 0.2, κr = 0.5, and δ = 1 are given, and the power allocation factor is fixed at α = 0.001, as evaluated by the Monte Carlo simulation with 106 channel realizations employed at V2 in both schemes. All simulation results of this subsection are obtained through a Monte Carlo method. In Fig. 11.5, we show the BER performance for different Rician K -factor. It is observed that NOMA-SM outperforms the benchmark especially in the high SNR regime. Additionally, the increase of K imposes a more dominant degradation on 11 NOMA in Vehicular Communications 357 10-1 BER 10-2 NOMA-SM =1 NOMA-SM =0.9 NOMA-VBLAST NOMA-VBLAST NOMA-VBLAST NOMA-VBLAST =1 16QAM =0.9 16QAM =1 QPSK =0.9 QPSK 10-3 10-4 10-5 25 30 35 40 45 SNR [dB] Fig. 11.7 BER comparisons with different temporal correlation coefficient δ when K = 0.2 and κt = κr = 0.5 are given, and the power allocation factor is fixed at α = 0.001, as evaluated by the Monte Carlo simulation with 106 channel realizations -1 10 -2 10 -3 =0.01 NOMA-SM =0.001 NOMA-SM =0.0001 NOMA-VBLAST =0.01 16QAM NOMA-VBLAST =0.001 16QAM NOMA-VBLAST =0.0001 16QAM NOMA-VBLAST =0.01 QPSK NOMA-VBLAST =0.001 QPSK NOMA-VBLAST =0.0001 QPSK BER 10 NOMA-SM -4 10 -5 10 -6 10 25 30 35 40 45 SNR [dB] Fig. 11.8 BER comparisons with different power allocation factor α when K = 0.2, κt = κr = 0.5, and δ = 1 are given, as evaluated by the Monte Carlo simulation with 106 channel realizations 358 Y. Chen et al. both of the NOMA-VBLAST schemes, which rely more vitally on the presence of rich non-LoS scattering. This phenomenon can be explained as follows. The higher Rician factor K represents a stronger LoS component, which increases the spatial correlation among the adjacent channel paths. For NOMA-VBLAST schemes, the multiple-stream information is conveyed with the aid of multiple DoFs. By contrast, for NOMA-SM, although the more severe spatial correlation of the LoS scenario makes it difficult to determine the index of the activated Tx, the remaining information related to the APM signal-domain is transmitted over a single DoF; hence, it is less susceptible to spatial correlation. Figure 11.6 investigates the BER results associated with different adjacent Txcorrelation coefficients at V1 . Compared to κt = 0.8, κt = 0.2 represents an insignificant spatial correlation. Again, observe from Fig. 11.6 that NOMA-SM is less susceptible to spatial correlation. This phenomenon can be interpreted similarly to the trend of Fig. 11.5. Besides, for these two figures, we notice that NOMA-SM temporarily loses its advantage over NOMA-VBLAST adopted QPSK and Nt = 4 within moderate SNR regime. This observation results from the superiority that QPSK brings to NOMA-VBLAST compared to 16QAM. However, as the increment of SNR, NOMA-SM achieves its dominance in terms of higher diversity gain. Below, we investigate the impact of the V2V channel’s time-varying nature. Observe from Fig. 11.7 that compared to the performance of no time-varying effect associated with δ = 1, the BER has been substantially degraded in all schemes for δ = 0.9. Although a perfect channel estimation procedure is assumed for the receivers, the estimated channel coefficients used for ML detection becomes partially outdated due to the channel’s time-varying nature, hence resulting in a degraded BER performance. Nevertheless, the proposed NOMA-SM scheme maintains its advantage over the reference within the medium and high SNR regime, regardless of the grade of temporal correlation. Figure 11.8 shows the BER performance associated with different α values. For all schemes, the lower α values exhibit a better detection performance, since less power is allocated to U and hence V2 suffers from a lower inter-user interference. More importantly, we observe that NOMA-SM consistently outperforms the NOMA-VBLAST scheme applied with 16PSK and Nt = 2. For the cases of α = 0.001 and α = 0.0001, though NOMA-VBLAST adopted QPSK and Nt = 4 holds a dominance within the moderate SNR regime, NOMA-SM keeps improving its performance with the increase of SNR and outperform the references in terms of higher diversity gain. By jointly considering the above observations, we conclude that NOMA-SM constitutes a potent amalgam. 11.5.2 Capacity Results and Discussions Below, we evaluate the capacity of the NOMA-SM system associated with different power allocation strategies. All results presented in this subsection are obtained by averaging the instantaneous capacities over multiple channel realizations. In 11 NOMA in Vehicular Communications 359 particular, we fix K = 0.2, κt = κr = 0.5, and δ = 1 unless otherwise stated. For benchmarking, we use an OMA-SM system, where V1 transmits messages to V2 using SM in the first slot. Then, V1 sends messages through the previously activated antenna to U , without activating another antenna. This OMA-SM model constitutes a fair reference for the NOMA-SM system, since the signal intended for V2 is conveyed by both the APM signal- and Tx-domain, whereas the signal destined for U is only embedded in the classical signal-domain. The distinctive feature of OMA-SM is that data transmissions destined for V1 –V2 and V1 –U are operated in an orthogonal time division way within the classical APM signal-domain. Accordingly, the capacity upper bound for V2 and the capacity for U in the OMA-SM system are expressed as CVB = min CVB1 , CVB2 ,   Nt log2 1 + ασE2s gi 2 , CU = 2N1 t i=1 (11.25) 0 respectively, where CVB1 = 2N1 t Nt i=1  log2 1 + CVB2 = 21 log2 det I + (1−α)E s p0 hi 2 σ02 (1−α)E s p0 HH H σ02 Nt  + 21 log2 (Nt ), (11.26) . Let us first check the capacity associated with a fixed power allocation, that is α = 0.01. Figure 11.9 depicts the capacity of V2 and U , as well as the sum capacity versus SNR for both NOMA-SM and OMA-SM. Compared to OMA-SM, NOMASM provides substantial capacity gains both for the collaboration-aided vehicle V2 and for the in-car user U and accordingly obtains a significant sum capacity enhancement. Specifically, the capacity CU has been beneficially boosted by the proposed scheme, about twice as high as that of OMA-SM. Since the APM signal-domain of the proposed scheme is combined with a NOMA strategy, each user accesses the channel resources via power domain multiplexing. Subsequently, we investigate the efficiency of the proposed power allocation optimization. Specifically, the power allocation optimization denoted by Q is considered for OMA-SM, which is formulated as Q : max CU + CVB α⎧ ⎨ CU ≥ C˜U , s.t. CVB ≥ C˜VB , ⎩ 0 < α < 1. (11.27) For simplicity, the minimum rate requirements of V2 and U are set to C˜U = C U (α2 = 1) C B (α = 0) , which respectively correspond to the lower bound and upper and C˜B = V V 2 360 Y. Chen et al. 35 CU+CB in NOMA-SM V CB in NOMA-SM V 30 CU in NOMA-SM CU+CB in OMA-SM V 25 CB in OMA-SM V bps/Hz CU in OMA-SM 20 15 10 5 0 0 5 10 15 20 25 30 35 40 45 SNR [dB] Fig. 11.9 Capacity of V2 and U , or the sum capacity versus SNR for the NOMA-SM and OMA-SM scheme with a fixed power allocation factor, i.e. α = 0.01. Specifically, C VB and CU in NOMA-SM are evaluated from (11.18) and (11.11), while C VB and CU in OMA-SM are obtained from (11.25) bound of α’s feasible set. Then, a full-search algorithm is applied for OMA-SM within the feasible set. Figure 11.10 illustrates the capacity of V2 and U for NOMA-SM with optimization P or O, where the QoS of the collaboration-aided vehicle V2 and the in-car user U , i.e. C̃ VB and C̃U , are also plotted for reference. It can be observed that C VB always meets the requirement of C̃ VB with the aid of the optimization P, and CU associated with the optimization O exactly meets the QoS C̃U . This observation is in accordance with the foregoing analysis, which indicates that the optimization P intends to maximize CU , while maintaining the QoS C̃ VB for V2V transmission, whereas the optimization O aims for maximizing C VB while guaranteeing the minimum rate requirement C̃U for the in-car user. Thus, we find that the optimized CU of P is higher than that of O, whereas the optimized C VB of O outperforms that of P. Accordingly, the more appropriate optimization scheme can be readily selected based on the data priority of distinct transmission links. Figure 11.11 compares the results of the optimization Q to that of P and O. Let us contrast P and Q first. Clearly, both C VB and CU in NOMA-SM with optimization P have been remarkably improved, demonstrating that the NOMA strategy offers a bandwidth efficiency improvement. By considering the results of O and Q in Fig. 11.11, we find that CU of NOMA-SM associated with optimization O is tightly lower bounded by that of OMA-SM associated with optimization Q, and C VB with O provides a substantial gain, achieving nearly twice that of Q. 11 NOMA in Vehicular Communications 361 20 18 16 14 bps/Hz 12 10 8 CU in NOMA-SM with P CB in NOMA-SM with P V 6 CU in NOMA-SM with O CB in NOMA-SM with O V 4 QoS for CU 2 0 QoS for CB V 0 5 10 15 20 25 30 35 40 45 SNR [dB] Fig. 11.10 Capacity of V2 and U , or the respective QoS versus SNR for NOMA-SM with power allocation optimization P or O . Specifically, C VB and CU in NOMA-SM with P or O are evaluated with the aid of the algorithm in Table 11.1. The QoS for C VB and CU , i.e. C̃ V and C̃U , are set according to (11.24) 20 18 16 14 bps/Hz 12 10 8 CU in NOMA-SM with P 6 CB in NOMA-SM with P V 4 CU in NOMA-SM with O 2 CU in OMA-SM with Q CB in NOMA-SM with O V CB in OMA-SM with Q V 0 0 5 10 15 20 25 30 35 40 45 SNR [dB] Fig. 11.11 Capacity of V2 and U versus SNR for NOMA-SM with power allocation optimization P or O , and OMA-SM with power allocation optimization Q , respectively. Specifically, C VB and CU in NOMA-SM with P or O are evaluated with the aid of the algorithm in Table 11.1. While C VB and CU in OMA-SM with Q are obtained from a full-search algorithm 362 Y. Chen et al. 35 CU+CB in NOMA-SM with P V 30 CU+CB in NOMA-SM with O V CU+CB in OMA-SM with Q V bps/Hz 25 20 15 10 5 0 5 10 15 20 25 30 35 40 45 SNR [dB] Fig. 11.12 Sum capacity versus SNR for NOMA-SM with power allocation optimization P or O , and OMA-SM with power allocation optimization Q , respectively. Specifically, C VB and CU in NOMA-SM with P or O are evaluated with the aid of the algorithm in Table 11.1. While C VB and CU in OMA-SM with Q are obtained from a full-search algorithm Furthermore, it can be observed from Fig. 11.12 that the NOMA-SM systems achieve higher sum capacity than OMA-SM. Specifically, optimization P provides higher capacity gain than O, since P aims for maximizing the data rate of the incar user U , which experiences a much better channel than the collaboration-aided vehicle V2 . 11.6 Chapter Summary and Future Outlook In this chapter, we introduce NOMA and SM techniques into V2X scenarios in order to support high bandwidth efficiency and enhanced link reliability. The BER performance of the new NOMA-SM transmission strategy has been investigated with the impact of the Rician K -factor, spatial correlation of antenna array, timevarying effect of the V2V channel, and the power allocation factor being discussed. Compared to NOMA relying on VBLAST, NOMA-SM has been demonstrated to exhibit improved robustness against the spatial and temporal effects of the V2V channel. By analysing the capacity and deriving analytical upper bounds in closed form, a pair of power allocation optimization schemes have been formulated for NOMA-SM. The optimal solutions have also been shown to be achievable with the aid of the proposed power allocation algorithm. Our numerical results have verified that with the aid of an appropriate power allocation, NOMA-SM is capable of satisfying the QoS support of a low priority flow, whilst maximizing the throughput of the 11 NOMA in Vehicular Communications 363 high priority flow. In summary, NOMA-SM has been demonstrated to cooperatively improve the link reliability and bandwidth efficiency of V2V transmissions. Nonetheless, several open issues still need to be carefully addressed before NOMA can be practically exploited in vehicular environments. Here, we discuss two potential research topics in this field as examples. Parallel Interference Cancellation-Aided NOMA: There is a much broader range of V2X applications to be considered in VANETs, especially within the automated driving field, whose characteristics are more stringent, as captured by the ultrareliable low-latency constraints. The traditional NOMA schemes use the classic SIC technique, where a high received signal power difference is preferred. However, this condition cannot always be guaranteed, especially in a traffic jam, where all cars tend to have similar channel conditions. Furthermore, SIC receivers were reported to exhibit an error floor in high-order modulation modes due to error propagation across the cancellation stages [60]. By contrast, parallel interference cancellation (PIC) does not require any specific detection order and all users reconstruct the signals of all the other users in parallel. Then, they subtract the reconstructed signals from the composite signal. Hence, PIC outperforms SIC when the received signal powers for all users are similar. With the advent of a PIC receiver, NOMA is expected to possess enhanced transmission reliability as well as better applicability to highlyloaded vehicular scenarios. In a nutshell, the performance analysis of NOMA with PIC should be addressed in our further research. NOMA for Cognitive V2X: The large amount of data generated by the vehicles might impose excessive traffic demands on traditional cellular traffic. Hence, cognitive NOMA principles can be conceived for delay-tolerant vehicular communications services to opportunistically access the channels originally occupied by the cellular users. Given a dedicated spectral band, cellular users and vehicles can be regarded as primary users and secondary users, respectively. 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One kind of such systems is sparse code multiple access (SCMA), which is a promising code-domain non-orthogonal multiple access technique to address the challenges for the fifth-generation (5G) mobile networks [1–4]. Non-orthogonal multiple access has the potential to accommodate more users with limited resources, which provides many advantages over orthogonal multiple access including multiuser capacity, supporting overloaded transmission, enabling reliable and low latency grant-free transmission, enabling flexible service multiplexing, etc. Applications of non-orthogonal signaling for multi-user communications have been investigated several years ago, significant efforts were paid to the optimal signaling design and intensive multi-user detection techniques, to suppress the multiple access interference (MAI) for lowering probability of error or increasing capacity. Hoshyar and Guo suggest the low-density signature (LDS)-based multiple access [5], or sparsely spread code-division multiple access (CDMA) [6], which intentionally arranges each user to spread its data over a fraction of the chips, instead of all chips, to reduce both the MAI and the complexity of multi-user detection. Inspired by the overloading capability and the low-complexity feature of LDS, SCMA is developed by inheriting from LDS the sparse sequence structure, such that the message-passing algorithm (MPA) is available in multi-user detection to achieve near-optimal performance. In contrast to the LDS scheme, multi-dimensional signal constellations, instead of the spreading, are utilized in SCMA to combat the channel fading and MAI. As a result, Z. Ma (B) · J. Bao Southwest Jiaotong University, West Section, High-tech Zone, Chengdu, Sichuan, China e-mail: zma@swjtu.edu.cn J. Bao e-mail: jinchen_bao@my.swjtu.edu.cn © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_12 369 370 Z. Ma and J. Bao the larger coding gain and better spectrum efficiency are achievable for SCMA due to the improved codebooks, compared to LDS. As one of NOMA family, SCMA is capable of supporting overloaded access over the coding domain, hence increasing the overall rate and connectivity. By carefully designing the codebook and multi-dimensional modulation constellations, the coding and shaping gain can be obtained simultaneously. In an SCMA system, users occupy the same resource blocks in a low-density way, which allows affordable low multiuser joint detection complexity at receiver. The sparsity of signal guarantees a small collision even for a large number of concurrent users, and the spread-coding like codes design brings good coverage and anti-interference capability due to spreading gain as well. 12.1.1 System Model 12.1.1.1 Multiple Access Procedure An SCMA transmission system can be simply illuminated in Fig. 12.1. Suppose that there are J synchronous users multiplexing over K shared orthogonal resources, e.g., K time slots or orthogonal frequency division multiplexing (OFDM) tones, and each user employs one SCMA layer.1 The forward error control (FEC) coding scheme can be low-density parity-check (LDPC) codes or polar codes which have been adopted for 5G recently. Each SCMA modulator/encoder maps the coded bits to a K -dimensional complex codeword, and the resulted J codewords constitute an SCMA block, as is shown in Fig. 12.1 (J = 6, K = 4 in the figure). The multi-user codewords in each SCMA block are multiplexed over the air transmissions in uplink multiple access channel (MAC), or they are superimposed at the transmitter of the downlink broadcast channel (BC). Since each SCMA block occupies K resources for codeword transmitting, the resulted overloading factor is J/K . This multiple access process is similar to that of CDMA, where the spread signals in CDMA are replaced with the SCMA codewords. Multi-user detection is carried out at the receiver to recover the colliding codewords. For the uplink MAC, the received signal vector after the synchronous user multiplexing is expressed as y= J  j=1 diag(h j )x j + n (12.1)  t and h j = [h j [1], . . . , h j [K ]]t , are the where x j = x j [1], . . . , x j [K ] K -dimensional codeword and the corresponding channel gain for the jth user, respectively, and diag(h j ) denotes the diagonal matrix with h j [k] being the kth diagonal 1 In practical scenarios, each user employs one or multiple layers. 12 Sparse Code Multiple Access (SCMA) 371 Fig. 12.1 The system model for SCMA element. The K -vector n is the additive white Gaussian noise (AWGN) with zero mean and variance N0 per dimension. It is convenient to view the MAC model as an equivalent “MIMO” communication system, and the received vector in (12.1) becomes y = HX + n (12.2)   where H = diag(h1 ), diag(h2 ), . . . , diag(h J ) , is the equivalent “MIMO” channel  t matrix, and X = x1t , x2t , . . . , xtJ , is the combined multi-user codeword representing an SCMA block. For the downlink BC, the codewords from multiple users are superimposed before the transmission, so that they experience the same fading. In the case of absence of interference between K resources, the received signal vector is given by y = diag(h) J  j=1 x j + n = diag(h)X + n (12.3)  where a single receiver is considered here for simplicity, and X = Jj=1 x j , is the superimposed codeword of J users at the input of a BC, which also represents an SCMA codeword block. In the following, the upper case X always denotes the combined multi-user codeword of J users in the MAC model, or the superimposed codeword in the BC model. 12.1.1.2 SCMA Codebook Mapping Unlike the modulation used for 3G and 4G, the modulation and codebook mapping in SCMA are designed jointly in a multi-dimensional and sparsely spread way. An SCMA modulator/encoder maps the input bits to a K -dimensional sparse codeword, 372 Z. Ma and J. Bao Fig. 12.2 Illustration of SCMA codebooks and bits to codeword mapping which is selected from a layer-specific codebook of size M. The K -dimensional complex codewords of the codebook are sparse vectors with N < K nonzero entries, and all the codewords contain 0 in the same dimensions. Then, the codebook is sparse, and this is where the “sparse code multiple access” is named from. The codebooks are constructed by a mapping from an N -dimensional complex constellation with a mapping matrix. Denote the constellation for the jth layer/user with C j , which contains M j constellation points of length N j . The mapping matrix V j maps the N j -dimensional constellation points to SCMA codewords to form the codebook X j . To simplify our illumination and analysis, we assume that all layers have the same constellation size and length, i.e., M j = M, N j = N , ∀ j. In summary, the resulting codebook for the jth user contains M codewords, each codeword consists of K complex values from which only N are nonzero specified by the mapping matrix V j . An example of the codebook mapping is shown in Fig. 12.2, where a codebook set containing 6 codebooks for transmitting 6 SCMA layers is illustrated (J = 6). Each codebook contains 8 four-dimensional codewords (M = 8, K = 4), and two of the four entries in the codewords are nonzero (N = 2). Upon transmission, the codeword of each layer is selected based on the labeling of the bit sequence. 12.1.1.3 Factor Graph Representation The low-density structure of SCMA codewords can be efficiently characterized by a factor graph, which is analogous to that for LDPC codes. A binary column vector f j of length K is used to indicate the positions of zero (with digit 0) and nonzero (with digit 1) entries of the jth codebook. Then, a K × J sparse matrix F = [f1 , . . . , f J ], called factor graph matrix, can be used to indicate the relationships between the layers and resources. The rows of F indicate the resources and the columns indicate 12 Sparse Code Multiple Access (SCMA) 373 Fig. 12.3 Factor graph representation for SCMA the layers. The (k, j)th element of F, denoted as f k, j , is 1 if the jth layer contributes its data to the kth resource. Correspondingly, let the J variable nodes (VNs) and K function nodes (FNs) in the factor graph represent the layers and resources, respectively, and the jth VN is connected to the kth FN if and only if f k, j = 1. In the following, we denote φk = { j : 1 ≤ j ≤ J, f k, j = 1}, ϕ j = {k : 1 ≤ k ≤ K , f k, j = 1} (12.4) the index set of layers contributing to the kth resource, and the index set of resources occupied by the jth layer, respectively. For a regular factor graph matrix, |φ1 | = · · · = |φ K | and |ϕ1 | = · · · = |ϕ J |, and let d f = |φk | and dv = |ϕ j |. Example 1 Consider a 6-user SCMA transmission system with J = 6, K = 4, such a system permits a transmission overloading 150%, and the system model is depicted in Fig. 12.1. If we carefully design the factor graph matrix F to allow the users to collide over only one nonzero element, then a choice of F is given by ⎡ 1 ⎢0 F=⎢ ⎣1 0 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 ⎤ 0 1⎥ ⎥ 1⎦ 0 (12.5) In the sparse matrix settings, matrix (12.5) has d f = 3 and dv = 2, which means that each FN is connected to three VNs and each VN is connected to two FNs. The corresponding factor graph is shown in Fig. 12.3, and an example of a codebook (with size M = 4) is listed in Table 12.1. In summary, the main features of SCMA lie in: • Code-domain non-orthogonal multiplexing: SCMA allows superposition of multiple codewords from different users over several resources, which supports overloading. The superposition pattern on each resource is defined in codebooks. • Sparse spreading: SCMA uses sparse spreading to reduce inter-layer interference, so that more codewords collisions can be tolerated with low receiver complexity. 374 Z. Ma and J. Bao Table 12.1 An Example of SCMA Codebook (K = M = 4, N = 2, J = 6) SCMA SCMA Codebook for each layer codebook index ⎡ ⎤ 0.7851 −0.2243 0.2243 −0.7851 ⎢ ⎥ 0 0 0 0 ⎢ ⎥ Codebook 1 ⎢ ⎥ ⎣ −0.1815 − 0.1318i −0.6351 − 0.4615i 0.6351 + 0.4615i 0.1815 + 0.1318i ⎦ 0 0 0 0 ⎤ ⎡ 0 0 0 0 ⎢ −0.1815 − 0.1318i −0.6351 − 0.4615i 0.6351 + 0.4615i 0.1815 + 0.1318i ⎥ ⎥ ⎢ Codebook 2 ⎢ ⎥ ⎦ ⎣ 0 0 0 0 0.7851 −0.2243 0.2243 −0.7851 ⎡ ⎤ −0.6351 + 0.4615i 0.1815 − 0.1318i −0.1815 + 0.1318i 0.6351 − 0.4615i ⎢ 0.1392 − 0.1759i 0.4873 − 0.6156i −0.4873 + 0.6156i −0.1392 + 0.1759i ⎥ ⎢ ⎥ Codebook 3 ⎢ ⎥ ⎣ ⎦ 0 0 0 0 0 0 0 0 ⎤ ⎡ 0 0 0 0 ⎥ ⎢ 0 0 0 0 ⎥ ⎢ Codebook 4 ⎢ ⎥ ⎦ ⎣ 0.7851 −0.2243 0.2243 −0.7851 −0.0055 − 0.2242i −0.0193 − 0.7848i 0.0193 + 0.7848 0.0055 + 0.2242i ⎡ ⎤ −0.0055 − 0.2242i −0.0193 − 0.7848i 0.0193 + 0.7848 0.0055 + 0.2242i ⎢ ⎥ 0 0 0 0 ⎢ ⎥ Codebook 5 ⎢ ⎥ ⎣ ⎦ 0 0 0 0 −0.6351 + 0.4615i 0.1815 − 0.1318i −0.1815 + 0.1318i 0.6351 − 0.4615i ⎡ ⎤ 0 0 0 0 ⎢ ⎥ 0.7851 −0.2243 0.2243 −0.7851 ⎢ ⎥ Codebook 6 ⎢ ⎥ ⎣ 0.1392 − 0.1759i 0.4873 − 0.6156i −0.4873 + 0.6156i −0.1392 + 0.1759i ⎦ 0 0 0 0 • Multi-dimensional modulation: SCMA employs multi-dimensional constellations for better spectral efficiency. 12.1.2 Multi-user Detection This subsection discusses multi-user detection schemes for SCMA, including the optimal detection, the MPA receiver and other advanced receivers. 12 Sparse Code Multiple Access (SCMA) 12.1.2.1 375 Optimal/Quasi-optimal Multi-user Detection A. Optimal Multi-user Detection Assume that channel state is perfectly estimated at the receiver, given the received signal vector y, the joint optimum maximum a posteriori probability (MAP) detection, for multi-user codeword X and for the jth user’s codeword x j , can be written as  p(X|y) (12.6) X̂ = arg max p(X|y), and x̂ j = arg max x j ∈X j ,∀ j x j ∈X j xi ∈Xi ,∀i= j respectively. Using Bayes’ rule p(X|y) = where p(y|X) p(X) ∝ p(y|X)P(X) p(y) J p(X) = j=1 p(x j ), and p(y) =  p(y|X) p(X) x j ∈X j ,∀ j are the joint a prior probability2 for each user’s codeword, and the probability of the received signal vector, respectively. By assuming that the noise components over the K resources are identically independently distributed (i.i.d.), it holds K p(y|X) = p(y[k]|X) k=1 and considering only d f users actually collided over the kth resource, we have ⎞ ⎛ 2     1 1 p(y[k]|X) = h j [k]x j [k] ⎠ exp ⎝−  y[k] − π N0 N0 j∈φ (12.7) k Thus, the MAP decision for the jth user’s codeword is given by K x̂ j = arg max x j ∈X j  P(X) xi ∈Xi ,∀i= j k=1 p(y[k]|X), ∀ j (12.8) With the codeword probability for each user, it is straightforward to calculate the log-likelihood rate (LLR) for each coded bit, so that they can serve as the input for 2 Without feedback from the FEC decoder, p(x j ) = 1 M for all the users. 376 Z. Ma and J. Bao the FEC decoder. For the jth user, the LLR considering the mth bit b j,m is calculated by Pr{b j,m = 1|y} Λ(b j,m ) = log Pr{b j,m = 0|y}   K (12.9) x j ∈X1j,m xi ∈Xi ,∀i= j P(X) k=1 p(y[k]|X) = log   K x j ∈X0j,m xi ∈Xi ,∀i= j P(X) k=1 p(y[k]|X) where X1j,m and X0j,m are subsets of X j for which the mth bit of the jth user b j,m = 1 and b j,m = 0, respectively. Note that solving (12.8) is equivalent to solve the marginal product of functions (MPF) problem, which is of exponential complexity with bruteforce searching, and is prohibitive to employ when the number of users increases. B. MPA Detection As the SCMA encoding can be represented by a factor graph with sparse property, the low-complexity MPA can be used to solve the MPF problem with near-optimum performance. (t) Let Ik→ j be the extrinsic information to be passed from FN k to VN j at the tth (t) iteration, and I j→k be the extrinsic information to be passed from VN j to FN k. Given the a prior probability p(x j ), the probability that x j is transmitted by the jth user given the channel sample is updated as (t) I j→k (x j ) = p(x j ) (t) Il→ j (x j ) l∈ϕ j \k Then, for any x j ∈ X j , the probability of the received signal y[k] given that x j is transmitted by the jth user, marginalized over all possible codewords of the other users, is given by (t) Ik→ j (x j ) =  xi ∈Xi ,∀i∈φk \ j (t−1) Ii→k (xi ) p (y[k]|X) (12.10) i∈φk \ j After a number of iterations, the posterior probability of x j produced by the MPA is proportional to I j (x j ) = p(x j ) k∈ϕ j (T ) Ik→ j (x j ), x j ∈ X j , j = 1, . . . , J (12.11) where T is the number of iterations at the termination. Similar to that for MAP detection, the LLR of the mth bit of the jth user b j,m is calculated by  x j ∈X1j,m I j (x j ) Λ(b j,m ) = log  (12.12) x j ∈X0j,m I j (x j ) 12 Sparse Code Multiple Access (SCMA) 377 where X1j,m and X0j,m are the same as that in (12.9). The main complexity of MPA comes from the calculation of (12.10), the summation over xi adds up M |φk |−1 terms while M probabilities should be calculated in each iteration, which leads to a complexity order O(T K M d f ), and is far below that of the optimal MAP detection. In practical implementations, the exponential function in MPA algorithm may cause large dynamic ranges and very high storage burden, then the logarithmic domain MPA is preferred to avoid the exponential operations. For the log-MPA operation, the information exchanged between the FNs and VNs can be expressed as (t) (x j ) = log p(x j ) + I j→k (t) Ik→ j (x j ) = max xi ∈Xi ,∀i∈φk \ j  (t) Il→ j (x j ) l∈ϕ j \k ⎧ ⎨ ⎩ log p (y[k]|X) +  (t−1) Ii→k (xi ) i∈φk \ j ⎫ ⎬ ⎭  p  i where Jacobi’s logarithm formula log ≈ maxi pi is applied for a complexity i e reduction to a certain degree, which results in the max-log-MPA detection. The output LLR of the MPA detector is given by Λ(b j,m ) = max I j (x j ) − max I j (x j ) x j ∈X1j,m where I j (x j ) = log p(x j ) + 12.1.2.2 x j ∈X0j,m  (T ) Ik→ j (x j ) k∈ϕ j Other Advanced Detectors The MPA detector still has exponential complexity with respect to the codebook size (M) and the number of accessed users at each resource (d f ), which may become impractical for the implementation of very large codebook size (e.g., M ≥ 64) and very high overload (e.g., d f ≥ 8). Some other advanced detectors can harness the potential gain of SCMA while provide sufficient flexibility for a good trade-off between the performance and detection complexity [7, 8]. A. EPA Detector Expectation propagation algorithm (EPA) is an approximate Bayesian inference method in machine learning for estimating sophisticated posterior distributions with simple distributions through distribution projection, and it turns out to be an efficient iterative multi-user detector for SCMA as well as some other multiple access schemes [8]. It approximates the discrete message in MPA as continuous Gaussian message using the minimum Kullback–Leibler (KL) divergence criterion, and use the a posteriori probabilities fed back from the FEC decoder to compute the approx- 378 Z. Ma and J. Bao imate symbol belief and the approximate message, such that the message passing reduces to mean and variance parameters update. The detailed algorithm is given in [8]. With EPA, the complexity order of SCMA detection is reduced to linear complexity, i.e., it only scales linearly with the codebook size M and the average degree of the factor nodes d f , while simulation results show that the EPA detector shows nearly the same error performance as MPA for SCMA with receiver diversity. As a result, the computation burden of the SCMA receiver is significantly alleviated and is no longer a problem for implementation in real systems. B. SIC-MPA Detector Successive interference cancelation (SIC) receiver is a kind of multi-user receiver that treats all the other undecoded users as interference when decoding a target user, and can be implemented as either symbol level or codeword level. It works well when the received SNR among users are quite different from each other. However, the detecting performance deteriorates when the SNR difference is not obvious between users, in which case error propagation happens. To strike a good balance between link performance and implementation complexity, it is reasonable to combine SIC with an MPA (SIC-MPA) receiver. More specifically, MPA is applied to a limited number of users firstly, so that the number of colliding users over each resource does not exceed a given threshold value (e.g., ds users). Then, the successfully decoded users are removed by SIC and the procedure continues until all users are successfully decoded or no new user gets successfully decoded in MPA. In the case of ds = d f , full MPA is realized, and when ds = 1, it becomes a pure SIC receiver. Due to the fact that MPA is used for a very limited number of users instead of all the users, the decoding complexity is greatly reduced, which is of the order O(T K M ds ). 12.2 Performance Evaluation The error performance and capacity are excellent measures that indicating the goodness of a system, and more importantly, they serve as powerful tools for the practical system design. For SCMA, the multi-user codebook plays a key role in the system performance improvement, and it is necessary to establish performance criterion to guide the codebooks design. In this section, error performance and capacity analysis for uplink and downlink SCMA systems are provided, and independent Rayleigh fadings are assumed. 12.2.1 Average Error Probability The error probability, e.g., the average codeword error probability (ACEP), is one of the most important performance criteria, since it is most revealing about the nature 12 Sparse Code Multiple Access (SCMA) 379 of a system behavior. However, it is quite difficult to evaluate the exact ACEP for SCMA systems, since one needs to average over several fading statistics due to the multi-channel transmissions, and the integration involves a decision cell of a multidimensional signal point. As an alternative approach, it is convenient to resort to an upper bound or approximation on the ACEP [9]. In this subsection, we use union bound to evaluate the error performance of uplink and downlink SCMA under joint maximum likelihood (ML) multi-user detection. 12.2.1.1 PEP over Uplink MACs Consider the equivalent “MIMO” channel (12.2). Under the assumption of perfect channel estimation at the receiver, the joint ML detection of multi-user codewords is equivalent to the joint minimum distance decoding X̂ = arg min y − HX X The pairwise error probability (PEP), defined as the probability that received signal vector y is detected into Xb given that Xa is transmitted, is given by [10] ⎡ ⎛ P{Xa → Xb } = EH ⎣ Q ⎝ H(Xa − Xb ) 2N0 2 ⎞⎤ ⎠⎦ (12.13) ∞ 2 where, Q(x) = √12π x e−t /2 dt, is the well-known Gaussian function [11], and EH [·] denotes the mean. Let x j,a [k] and x j,b [k] be the kth entries of the jth user’s codewords x j,a and x j,b , corresponding to Xa and Xb , respectively. Due to the sparseness of the codewords, / φk . Now we define a distance for the MAC. x j,a [k] = x j,b [k] = 0 whenever j ∈ Definition 1 The kth dimension-wise distance, between the multi-user combined codewords Xa and Xb , for the uplink MAC is defined as λ2k = J  j=1 |x j,a [k] − x j,b [k]|2 =  j∈φk |δ j [k]|2 , ∀k (12.14) where δ j [k] = x j,a [k] − x j,b [k]. Assume that there are repeated values among the set {λ21 , . . . , λ2K }, such that they can be divided into V (1 ≤ V ≤ K ) groups, and each group contains the collection of a certain value λ̂2v . Let λ̂ = [λ̂21 , . . . , λ̂2V ]t , be the vector of V distinct elements among {λ21 , . . . , λ2K }, and r = [r1 , . . . , r V ]t , where rv is the number of elements in V {λ21 , . . . , λ2K } that equals to λ̂2v , such that v=1 rv = K . 380 Z. Ma and J. Bao Definition 2 Define the parameter V K Ar,λ = λ−2 k k=1 = λ̂v−2rv (12.15) v=1 as the reciprocal of the product of the dimension-wise distances. Definition 3 For positive integers l, v and vectors r and λ̂, define the parameter V l+1 Bv,l,r,λ̂ = (−1)  η∈Ωv,l j=1, j=v  ηj + rj − 1 ηj  1 λ̂2j − 1 −(r j +η j ) (12.16) λ̂2v where the vector η = [η1 , . . . , ηV ]t is created from the set Ωv,l of all nonnegative integer partitions of l − 1 (with ηv = 0). The set Ωv,l is defined as Ωv,l " ! V  t V η j = l − 1, ηv = 0, η j ≥ 0 ∀ j = η = [η1 , . . . , ηV ] ∈ Z ; j=1 Next, we provide the main result regarding the PEP. Theorem 1 For J users using K -dimensional codebooks in the uplink SCMA systems, the PEP between Xa and Xb is given by P{Xa → Xb } = Ar,λ̂ ×  rv V   λ̂2L v Bv,rv −L+1,r,λ̂ v=1 L=1 1 − µv 2 L  L−1  k=0 L −1+k k  1 + µv 2 where µv =  λ̂2v 4N0 + λ̂2v Proof Consider the metric in (12.13), H(Xa − Xb ) 2 = K  k=1 h[k]† (xa [k] − xb [k]) 2 k (12.17) 12 Sparse Code Multiple Access (SCMA) 381 where xa [k] = [x1,a [k], . . . , x J,a [k]]t , is the vector of the kth component for J users, and h[k]† = [h 1 [k], . . . , h J [k]], are the corresponding channel gains, and [·]† denotes conjugate transpose. Using the matrix decomposition, it holds that (xa [k] − xb [k])(xa [k] − xb [k])† = Uk Λk Uk† where Uk is unitary and Λk is a diagonal matrix, i.e., Λk = diag(λ̃2k,1 , . . . , λ̃2k,J ), with λ̃2k, j being the ordered singular values of the matrix (xa [k] − xb [k])(xa [k] − xb [k])† . Note that the matrix (xa [k] − xb [k])(xa [k] − xb [k])† is of rank 1 and the unique nonzero singular value in Λk is λ̃2k,1 = xa [k] − xb [k] 2 (a) =  j∈φk |x j,a [k] − x j,b [k]|2 where (a) is due to the sparseness of the codebooks. Obviously, the nonzero eigenvalue is equal to the dimension-wise distances defined in Definition 1, namely λ̃2k,1 = λ2k . Hence, h[k]† (xa [k] − xb [k]) 2 = h[k]† Uk Λk Uk† h[k] = h̃[k]† Λk h̃[k] = λ2k |h̃ 1 [k]|2 where we define, h̃[k]† = h[k]† Uk = [h̃ 1 [k], . . . , h̃ J [k]]. Thus, h̃[k] has the same distribution as h[k], since multiplying with unitary matrix Uk doesn’t change the amplitudes. Thus, the average PEP in (12.13) is equal to ⎡ ⎛  P{Xa → Xb } = Eh̃[1],...,h̃[K ] ⎣ Q ⎝ ⎞⎤ λ2k |h̃[k]|2 ⎠⎦ 2N0 K k=1 (12.18) where the index 1 is dropped here for h̃ 1 [k]2 . For i.i.d. Rayleigh fading, h̃[1], . . . , h̃[K ] are i.i.d. Gaussian random  K complex variables with zero mean and unit variance. Thus, k=1 λ2k |h̃[k]|2 is the sum of K exponential random variables with different means, or a linear combination of V independent χ 2 -distributed random variables with 2r1 , . . . , 2r V degrees of freedom, which follows Gamma or Generalized chi-squared distribution [12], with PDF given by f (x; r, λ̂) = Ar,λ̂ rv V   Bv,l,r,λ̂ v=1 l=1 (rv − l)! x rv −l e − x λ̂2 v 382 Z. Ma and J. Bao Then, the PEP can be obtained as P{Xa → Xb } = # ∞ 0 = Ar,λ̂ = Ar,λ̂ Q $ x 2N0  f (x; r, λ̂)dx # rv V   Bv,l,r,λ̂ v=1 l=1 rv V   (rv − l)!  0 ∞ Q $ x 2N0  x rv −l e rv −l+1 (1 − µv )λ̂2v Bv,l,r,λ̂ 2 v=1 l=1   rv −l   1 + µv k rv − l + k × k 2 k=0 − x λ̂2 v dx (12.19) where the last step follows from (13.4-15) in [11]. Substituting l with rv − L + 1, (12.17) is proved. This concludes the proof. The PEP is uniquely determined by the set of all λ2k s and the SNR, which is valid for any multi-dimensional codebooks and an arbitrary number of users. By reducing the number of users to 1, P{Xa → Xb } becomes the PEP between two multidimensional constellation points for a single-user transmission system. Therefore, the PEP of a joint multi-user detector is actually identical to that of the PEP of a single-user transmitting over a fading channel, where an equivalent K -dimensional constellation is employed such that the dimension-wise distances between the two constellation points are λ21 , . . . , λ2K . 12.2.1.2 PEP over Downlink BCs Consider the received signal vector of the downlink BC in (12.3), where X =  J j=1 x j is the superimposed codeword of multiple users at the transmitter. Obviously, the model is exactly the same with that in the single-user communications, where X is used as the K -dimensional transmitted codeword. The ML multi-user detection for the superimposed codeword X becomes X̂ = arg min y − diag(h)X X Similar to that in uplink SCMA, we define the distances for downlink BC model. Definition 4 Let Xa and Xb be two superimposed codewords, and x j,a [k] and x j,b [k] are the kth entries of the jth user’s codeword corresponding to Xa and Xb , respectively. The kth dimension-wise distance between Xa and Xb , for the downlink broadcast channel, is defined as 12 Sparse Code Multiple Access (SCMA) 383  2   J    2   δ j [k] , ∀k x j,a [k] − x j,b [k]  =  τk2 =  (12.20) j∈φk j=1 where δ j [k] = x j,a [k] − x j,b [k]. Theorem 2 The PEP of a Rayleigh broadcast channel is the same as that in (12.17), after the substitution of λ2k with τk2 . Proof Similar to that of the uplink case, the average PEP between Xa and Xb is equal to ⎞⎤ ⎡ ⎛ 2 diag(h) (Xa − Xb ) ⎠⎦ P{Xa → Xb } = Eh ⎣ Q ⎝ 2N0 ⎡ ⎛ ⎞⎤ (12.21) K 2 2 k=1 τk |h[k]| ⎠⎦ = Eh ⎣ Q ⎝ . 2N0 As h[1], . . . , h[K ] are independent Rayleigh distributed random variables, the integral has been solved in (12.18), and the PEP has the similar expression as that in the MAC case, after the substitution of λ2k with τk2 . This completes the proof. It should be noted that, while the PEP of a BC can be evaluated through the same expression as that in the MAC case, τk2 is different from the dimension-wise distance λ2k in MAC, due to the absence of cross components δ j [k] × δi [k], j = i, between different users. This is because in the MAC, the receiver distinguishes the multi-user signals by exploiting the differences among the channel coefficients, and only the amplitude of δ j [k] contributes to the PEP. However, in the broadcast channel case, since the receiver exploits the differences among the multiple users’ signals to perform the joint detection, both the amplitude and signs of δ j [k] will influence the result of PEP. 12.2.1.3 PEP over the AWGN Channel For the AWGN channel, where h j [k] is a constant for all j and k (assume that |h j [k]| = c), the expressions of the received signal vector in (12.1) for uplink channels and (12.3) for downlink channels are the same. Then, according to (12.21), it is straightforward to derive the PEP as ⎛ P{Xa → Xb } = Q ⎝ c2 ⎞ 2 τ k=1 k ⎠ 2N0 K ⎛& 2 ⎞ '    ' c2 K  ⎜( j∈φk δ j [k] ⎟ k=1 ⎟ ⎜ = Q⎝ ⎠ 2N0 (12.22) 384 Z. Ma and J. Bao where τk2 is the dimension-wise distance defined in (12.20), and δ j [k] = x j,a [k] − x j,b [k]. 12.2.1.4 Upper Bounds on PEP In the codebook design, sometimes it is sufficient and easier to optimize the performance through a bound or an approximation of PEP. The exact PEP in (12.17) is a little complicated for large K , due to the large number of enumerations in Ωv,l , when calculating Bv,rv −L+1,r,λ . An alternative way to evaluate (12.17) is to use an upper bound for the Q-function as [13] Q(x) ≤ N  2 ai e−bi x , for x > 0, i=1 where N , ai , bi are constants. Note that the upper bound in Sect. 12.2.1.4 tends to the exact value as N increases. For the multiple access and broadcast channels, since X = |ĥ[k]|2 is an exponen∞ 1 , for tial random variable with unit mean, holds that E X [et X ] = 0 et x e−x d x = 1−t t ≤ 1, and  * N  K 2 2 k=1 λk |h̃[k]| ai exp − P{Xa → Xb } ≤ Eh̃[1],...,h̃[K ] 2N0 i=1 *  + N K  bi λ2k |h̃[k]|2 ai = Eh̃[k] exp − 2N0 i=1 k=1 = N  i=1 bi + K ai 2N0 2 2N 0 + bi λk k=1 By choosing N = 1, a1 = b1 = 21 , we get the Chernoff bound with a scaling factor of 0.5 as K Pch {Xa → Xb } ≤ 4N0 1 2 k=1 4N0 + λ2k (12.23) In general, the Chernoff bound may be a little loose, but this does not affect the optimization criteria in the constellation design. It is obvious from (12.23) that a good direction is to design multi-dimensional multi-user codebooks, such that λ2k to span in as many dimensions as possible (maximizing the diversity) and to make the maximum PEP or maximum of Pch {Xa → Xb } as small as possible. If for any 12 Sparse Code Multiple Access (SCMA) 385 codeword pair Xa and Xb , all the λ2k are positive, then the maximal diversity order of K can be achieved. Due to the sparseness of the codebooks, the diversity is always less than K . A tight and simple bound (or approximation) is to choose N = 2, a1 = 1 , a2 = 41 , b1 = 21 , b2 = 23 , which is denoted as Pub {Xa → Xb }. 12 12.2.1.5 A Universal Bound of ACEP for Joint ML Detection of Multiple Signals A commonly used approach for the error performance analysis is the evaluation of the ACEP by using a union bound, assuming that the codewords are equiprobable transmitted. In general, the ACEP is dominated by the nearest neighbors of codewords, which result in a tight upper bound. However, it is quite difficult (if not impossible) to find the nearest neighbors in multi-user scenarios. To deal with this, we take into account all possible codewords that contribute to the ACEP. Let M1 , . . . , M J be the codebook size for J users, respectively. We define {X j } Jj=1  as the set of all Jj=1 M j possible combined codewords of J users, and let Xa , Xb ∈ {X j } Jj=1 be two different elements of {X j } Jj=1 . Here, the combined codeword Xa and Xb are a J K -dimensional vector for the MAC, or the sum of J K -dimensional codewords for the BC. Denote x j,a and x j,b the transmitted codewords of the jth user corresponding to Xa and Xb . Then, there are M j possible values for x j,a and x j,b . Note that x j,a and x j,b are K -dimensional vector with complex entries, i.e., SCMA codeword. Following the approach in [14] for multiple signals and [15] for MIMO channels, the ACEP for the jth user with joint ML detection of J users’ signals is upper bounded by P j (e) ≤  J 1 j=1 M j  Xa ⎛ ⎝  Xb ,x j,b =x j,a ⎞ P {Xa → Xb }⎠ (12.24) The ACEP of the system can  be obtained by taking the mean of all the single-user ACEPs, namely P(e) = 1J Jj=1 P j (e).  On the right-hand side of (12.24), the summation over Xa will add up Jj=1 M j  terms and the summation over Xb will add up (M1 − 1) Jj=2 M j terms. Thus, there  will be up to (M12 − M1 ) Jj=2 M 2j PEPs in (12.24), which is intractable for a large constellation size and number of users. However, we can simplify it by using the symmetry of the dimension-wise distances. For example, consider the ACEP for the first user P1 (e) here. The upper bound of P1 (e) can be decomposed into the summation of two parts as 386 Z. Ma and J. Bao 1 J j=1 + J Mj  Xa Xb ,x1,b =x1,a , [x2,b ,...,x j,b ]=[x2,a ,...,x j,a ] Mj   1 j=1 Xa  P {Xa → Xb } P {Xa → Xb } . Xb ,x1,b =x1,a , [x2,b ,...,x j,b ]=[x2,a ,...,x j,a ] (12.25) The first part in (12.25) is the union bound of the probability of the event that all users’ signals are correctly detected except for the first user, namely the ACEP for the first user with single-user  detection in the absence of interference. This part is a summation of (M1 − 1) Jj=1 M j PEPs, while only 21 M1 (M1 − 1) different PEP values should be calculated, due to the symmetry of the dimension-wise distance for the first user. The second part is the probability of the event that the errors happen for the first user  and for at least  one user among {2, . . . , J }, which is the summation of (M1 − 1)( Jj=2 M j − 1) Jj=1 M j PEPs, but only one-fourth of them should be considered. A further simplification can be achieved for the MAC by considering more decompositions of the second part. In general, SCMA codebooks of all users are constructed from a common mother constellation [16], with some layer-specific operations over this constellation to get their own layer’s codebook. These layer-specific operations do not change the fundamental properties of the mother constellation, such as the Euclidean distance. The layer operation losses their efficiency in the uplink multiple access fading channels, due to the distinctness of each user’s channel gain. If the factor graph matrix is regular as that in (12.5), every user will suffer from the same interference from other users. Then, the system results in the same performance for all users, while for other cases, the ACEP is asymmetric for each user. The MPA detection is believed to be an efficient approach for SCMA systems. Theoretically, the MPA detector is asymptotically equivalent to the optimal MAP detector [17, 18] (or ML conditioned on equal probably transmissions) for a sparsely spread system with long signatures. The analytical bounds, proposed in this subsection, work for ML detector as well as for the MPA detector. 12.2.1.6 Numerical Results and Simulations We consider an SCMA system illustrated in Example 1, and the four-dimensional four-ary codebooks are listed in Table 12.1. The ACEP of SCMA over AWGN and uplink Rayleigh fading channels for 2, 4, and 6 user cases are evaluated. For the Rayleigh fading channel, we give analytical results of the union bound on the ACEP, corresponding to exact PEP (denoted as Pjml ), the upper bound on PEP Pub , and the scaled Chernov bound Pch , respectively. Results for AWGN channels are shown in Fig. 12.4a. The analytical bound of a joint ML detector closely coincides with the simulation curves for large SNR. The bound is quite tight for values of ACEP below 10−3 , even for six users. Thus, this 12 Sparse Code Multiple Access (SCMA) 387 0 0 10 10 ML MPA Bound − Pjml Bound − Pub Bound − Pch −1 10 −1 10 −2 10 −2 ACEP ACEP 10 −3 10 −4 10 2 Users 4 Users −3 10 −4 6 Users 10 −4 −5 10 10 ML MPA Bound − Pjml −6 10 0 5 22.5 −5 10 10 15 20 0 23 23.5 5 24 10 15 SNR [dB] (a) SCMA over AWGN channel. 0 −1 −2 10 −4 MPA ML Bound − Pjml Bound − Pub Bound − Pch 10 −1 ACEP ACEP 30 10 0 MPA ML Bound − Pjml Bound − Pub Bound − Pch 10 −3 25 (b) SCMA with 2 users in Rayleigh fading. 10 10 20 SNR[dB] 10 −2 10 −3 10 −4 10 −4 10 −4 10 26 −5 10 5 26.5 10 27 27.5 28 15 20 25 30 SNR [dB] (c) SCMA with 4 users in Rayleigh fading. 10 −5 26 5 26.5 10 27 27.5 28 15 20 25 30 SNR [dB] (d) SCMA with 6 users in Rayleigh fading. Fig. 12.4 ACEP of uplink SCMA over AWGN and Rayleigh fading channels bound is sufficient for the analysis and design of a signal constellation in AWGN. Surprisingly, there are bends for the ACEP curves of an ML detector and analytical bound for the six user case. The performance turns better than expected within the SNR region from 12 to 18 dB, which is due to the sparse codebooks. This phenomenon happens if the distance profile of the multi-user codebooks is uneven. For example, a quite small distance exists while the others are very large. In general, the ACEP of a constellation in AWGN channels is proportional  to√the summation of Q-function of the distances d and SNR, i.e., P(SNR) ∝ d Q( dSNR), where d is the set of distances among the constellation points. If there is a large difference between two distance components, P(SNR) is not a convex function and a bend appears in the P(SNR) vs SNR curves in log–log scale at low SNR. The theoretical bound is still quite close to the actual ACEP within the bend region. It can be seen from Fig. 12.4a that there is nearly 0.4 dB gap between the performance of the MPA detector and the 388 Z. Ma and J. Bao ML detector at the SNR of 14 dB. The performance of the MPA detector is improved asymptotically and approaches that of ML detector at high SNRs. Figure 12.4b–d present the performance for 2, 4, and 6 users over Rayleigh fading channels, respectively. All the bounds are asymptotically tight as SNR increases. The analytical bound Pjml is quite tight for values of ACEP below 10−3 for all numbers of users, and the gap between Pjml and the exact ACEP is almost constant at high SNRs, when the number of users increases. Moreover, the bounds become looser at low SNRs as the number of users increases. The upper bound Pub shows superiority over all the other bounds, since it is much easier to calculate than Pjml while it has only a little difference. It should be noted that the scaled Pch is much looser compared to Pub . As expected, the MPA detector shows exactly the same performance as the ML detector for any number of users and any values of SNR over Rayleigh fading channels. 12.2.2 Capacity and Cutoff Rate This subsection discusses the sum rate analysis of SCMA systems. The channel capacity characterizes the limit information rate that can be reliably transmitted over a channel. It is well known that the sum rate of multi-channel transmissions is simply the sum of per channel rate, and in the uplink SCMA, the communications over each SCMA resource constitutes a multiple access process, then, the sum rate of uplink SCMA is ⎡ ⎛ ⎞⎤ K   |h j [k]|2 ⎠⎦ Eh 1 [1],...,h J [K ] ⎣log2 ⎝1 + ρ C= j∈φk k=1 d f d f −i (12.26) K e1/ρ   (−1)d f − j−i ρ i+ j = df Γ ( j, 1/ρ) ρ ln 2 i=1 j=0 j!(d f − i − j)! ∞ where ρ is the SNR, and Γ (a, x) = x t a−1 e−t dt, is the incomplete Gamma function. In the above sum rate evaluation, we assume that the users have the same transmitting power, and each SCMA resource carries the same number of users, i.e., d f = |φk |. To achieve any point on the sum rate curve, codebooks with Gaussian distributions and successive interference cancelation (SIC) receivers are generally required. In practical cases, it is more valuable to investigate the capacity restricted by specific codebooks, i.e., the discrete codebook-constrained capacity (DCCC). Consider the equivalent linear system of uplink SCMA in (12.2). Assuming that perfect channel knowledge is available at the receiver. The conditional probability density function (PDF) of the received signal vector is 12 Sparse Code Multiple Access (SCMA) 389  1 y − HX f (y|X, H) = exp − N0 (π N0 ) K 2 (12.27) The mutual information I (X; y) between the discrete input X and the continuous output y, or the DCCC, is given by [11]  , Xb f (y|Xb , H) I (X; y) = log2 M − Ey,Xa ,H log2 f (y|Xa , H) ⎤ ⎡  #  f (y|X , H) 1 b Xb = log2 M J − EH ⎣ J dy⎦ f (y|Xa , H) log M y X f (y|Xa , H) J a (12.28) where Xa , Xb ∈ {X j } Jj=1 are two combined codewords for J users. Obviously, it is quite difficult—if not impossible—to deal with the expression for the mutual information, and a closed-form solution is unattainable. In the following, we resort to the cutoff rate analysis. 12.2.2.1 Cutoff Rate of Uplink MAC The channel cutoff rate R0 , which is a lower bound on the channel capacity, is another commonly used metric characterizing the channel rate. The cutoff rate is more informative than the DCCC, since it provides a good estimate of the capacity as well as a tight upper bound on the error probability of an optimal detector. The cutoff rate can be defined by [11] ⎡ R0 = − log2 ⎣  Xa Xb ⎤ (12.29) p(Xa ) p(Xb )∆Xa ,Xb ⎦ where p(Xa ) = p(Xb ) = M1J , and ∆Xa ,Xb is the Bhatacharyya bound on the PEP between Xa and Xb , which is given by [11] ∆Xa ,Xb = EH ,# . , − 1 = EH e 4N0 0 1 − = EH e 4N0 p (y | Xa , H) p (y | Xb , H) dy H(Xa −Xb ) 2 H(Xa −Xb ) 2 # 1 - / H(Xa +Xb ) /2 1 − N1 /y− / 2 0 e dy (π N0 ) K / Note that ∆Xa ,Xa = 1, then the cutoff rate can be written as / - (12.30) 390 Z. Ma and J. Bao ⎞   1 ∆Xa ,Xb ⎠ R0 = log2 M J − log2 ⎝1 + J M X X =X ⎛ a b (12.31) a   It is observed that, the term M1J Xa Xb =Xa ∆Xa ,Xb , inside the bracket of (12.31), is the union-Bhatacharyya bound on the joint codeword error probability for multiple users. Therefore, optimizing the mean cutoff rate is equivalent to the optimization of the error probability, and cutoff rate can be used as a good performance criterion for the system design. For the uplink MAC, according to the analysis in Theorem 1, H(Xa − Xb ) 2 = K  k=1 λ2k |h̃[k]|2 where λ2k is the kth dimension-wise distance in the MAC defined in Definition 1, and |h̃[1]|, . . . , |h̃[K ]| are independent Rayleigh distributed random variables. Therefore, −1 K K  0 1 2 1 λ2 − λ |h̃[k]|2 = Eh̃[k] e 4N0 k ∆Xa ,Xb = 1+ k 4N0 k=1 k=1 and thus the cutoff rate for uplink MAC is given by ⎡ ⎤ −1   K  λ2k 1 ⎦ R0 = log M − log ⎣1 + J 1+ M X X ,b=a k=1 4N0 J a (12.32) b The average sum rate and cutoff rate for uplink SCMA in Rayleigh fading are depicted in Fig. 12.5, where the 4-ary codebook in Table 12.1 is adopted. The sum rates of SCMA with 2, 4 and 6 users are represented by the uppermost curves, which increase almost linearly with the SNR when SNR becomes very large. Due to the discrete codebooks, the DCCC and the cutoff rate are upper bounded by KJ log2 (M). However, significant rate improvement can be achieved by overloaded access for moderate to large SNRs. As it is observed, the cutoff rate establish a lower bound to the DCCC, and it asymptotically approaches the DCCC with increasing SNRs. 12.2.2.2 Cutoff Rate of Downlink BC Consider the downlink BC model in (12.3), for the jth user, the cutoff rate corresponding to the mutual information I (x j ; y) is given by [19] R0 = log2 M − log2   1   1+ ∆x j,a ,x j,b M x x =x j,a j,b j,a 12 Sparse Code Multiple Access (SCMA) Fig. 12.5 Capacity and cutoff rate of uplink SCMA in Rayleigh fading Shannon limits Mutual information Cutoff rate 2 users 4 users 6 users 3 Mutual Information [bits/s/Hz] 391 2.5 2 1.5 1 0.5 0 −10 −5 0 5 10 15 20 SNR [dB] and the Bhatacharyya parameter is ∆x j,a ,x j,b ,# 2     p y | x j,a , h p y | x j,b , h dy = Eh ⎡ ⎤ #    1 = Eh ⎣ J −1 p (y|Xb , h)dy⎦ p (y|Xa , h) M x ∈X ,∀i= j x ∈X ,∀i= j i,a i,b i i (12.33) In the integral of (12.33) a square root of the double sum of the products “ p (y|Xa , h) p (y|Xb , h)” is involved, which makes it excessively complex for a large number of users. Hence, we will attempt to obtain reasonable bounds for the cutoff rate. To deal with the expression, we first calculate ∆Xa ,Xb . According to the PEP analysis in (12.21) for downlink SCMA, the channel-dependent metric is equal to K  τk2 |h[k]|2 diag(h)(Xa − Xb ) 2 = k=1 where τk2 is the dimension-wise distance defined in Definition 2, and |h[k]| is the Rayleigh distributed random variables. Similar to that in the uplink case, the Bhatacharyya parameter considering the superimposed codewords Xa and Xb is given by 1 0 1 − diag(h)(Xa −Xb ) 2 ∆Xa ,Xb = Eh e 4N0 −1 K  1 0 1 K 2 τ2 τ |h[k]|2 − 1+ k = Eh e 4N0 k=1 k = 4N0 k=1 392 Z. Ma and J. Bao By applying Holder’s inequality,    p (y|Xa , h) p (y|Xb , h) ≥ xi,b ∈Xi ,∀i = j xi,a ∈Xi ,∀i = j  (xi,a ,xi,b )∈P i ,∀i = j . p (y|Xa , h) p (y|Xb , h) where Pi is the set of a point-to-point pairing of codewords (xi,a , xi,b ) for all i = j, which contains M elements.3 Then it holds that ⎡ ⎤ # .  1 ∆x j,a ,x j,b ≥ Eh ⎣ J −1 p (y|Xa , h) p (y|Xb , h)dy⎦ M (xi,a ,xi,b )∈P i ,∀i= j = 1 M J −1  ∆Xa ,Xb (xi,a ,xi,b )∈P i ,∀i= j Thus, an upper bound on the cutoff rate of downlink SCMA is R0 | upper = log2 M − log2 , 1   1+ J M x x =x j,a j,b  −1 τk2 1+ 4N0 k=1 K j,a  (xi,a ,xi,b )∈P i ,∀i= j (12.34) For the sake of deriving the lower bound of the cutoff rate, we may invoke the following simple inequality    p (y|Xb , h) xi,b ∈Xi ,∀i= j xi,a ∈Xi ,∀i= j ≤  p (y|Xa , h)  xi,a ∈Xi ,∀i= j xi,b ∈Xi ,∀i= j . p (y|Xa , h) p (y|Xb , h) we get that ∆x j,a ,x j,b ≤ 1 M J −1   ∆Xa ,Xb xi,a ∈Xi ,∀i= j xi,b ∈Xi ,∀i= j and a lower bound on the cutoff rate is obtained , R0 |lower = log2 M − log2 1 + 1  MJ x K   j,a x j,b  =x j,a xi,a ∈Xi ,∀i = j xi,b ∈Xi ,∀i = j k=1 , 1  = log2 M − log2 1 + J M K  Xa Xb ,x j,b  =x j,a k=1 3 There   1+ τk2 4N0  1+ τk2 4N0 −1 - −1 - (12.35) are M! possible pairing patterns for (xi,a , xi,b ), hence M! choices for Pi . The tightness of the bound is determined by the specific selection of the pairing patterns. A detailed seek for the appropriate pairing pattern can be found in [19]. 12 Sparse Code Multiple Access (SCMA) (a) Cutoff rate. 393 (b) ACEP. Fig. 12.6 Cutoff rate and ACEP of the worst user for downlink SCMA in Rayleigh fading As discussed in (12.24), the expression inside the square bracket of (12.35) is the union-Bhatacharyya bound on the ACEP for the jth user. With the derived upper and lower bound on R0 or ∆x j,a ,x j,b , the corresponding   bounds to the union-Bhatacharyya bound on ACEPs, i.e., M1 x j,a x j,b =x j,a ∆x j,a ,x j,b , can be obtained straightforwardly. We will now verify the cutoff rate bounds and the corresponding bounds for ACEPs by simulations. If R0 |upper is sufficiently tight, it may be regarded as satisfactory approximation of R0 . In order to emphasize the primary common characteristic between R0 |upper and R0 |lower , we can readily refer to R0 |upper as the approximated Chernov bound (Approxi. CB), and R0 |lower as relaxed Chernov bounds (Relaxed CB). The results for the cutoff rate of the worst user in downlink SCMA and the corresponding ACEPs are plotted in Fig. 12.6a and Fig. 12.6b, respectively. The curve of “Approxi. CB” and that of “Relaxed CB” merge with each other in the high-SNR region, while the “Approxi. CB” bound gets significantly close to the associated practical performance. Hence we may claim that R0 |upper indeed represents a satisfactory approximation of R0 . This conclusion may be verified by the associated simulation results shown in Fig. 12.6b, where the “Approxi. CB” over ACEP gives a better estimation of the practical ACEP than that of “Relaxed CB”, for both full-rank (with 4 users) and rank-deficient (with 6 users) SCMA systems. 12.3 Codebook Design As the performance of SCMA strongly depends on the multi-dimensional codebooks, codebook design constitutes one of the most important issues for SCMA, and it is what distinguishes SCMA from other non-orthogonal multiple access schemes. 394 Z. Ma and J. Bao Incorporating a sophisticated codebook design into SCMA has the potential of significantly improving the spectrum efficiency, and reducing the detection complexity. 12.3.1 General Design Rules The design of SCMA codebook is a joint optimization of the sparse mapping matrix and the multi-dimensional constellations. Assume that all layers have the same constellation size and length. An SCMA codebook can be represented by structure S(V , C ; J, M, N , K ), where V = {V j } Jj=1 , is the set of mapping matrices, and C = {C j } Jj=1 , is the set of signal constellations for J layers. Thus, the SCMA codebook designing is equivalent to solve the optimization problem [1] V + , C + = arg max Υ (S(V , C ; J, M, N , K )) V ,C (12.36) where the function Υ (·) is somehow the design criterion. Unfortunately, for a given criterion Υ (·) and such a multi-dimensional problem, the optimum solution cannot be found. In practice, a suboptimal multi-stage optimization approach is adopted, by optimizing the mapping matrices and constellations separately. The set of mapping matrices V is generally selected in order to meet the maximum overloading, while the design of J multi-dimensional constellations is simplified to the design of a mother constellation and multiple layer-specific operators. 12.3.1.1 Mapping Matrices The set of mapping matrices V should be pre-determined before the constellation design, since it determines the number of users/layers interfering at each resource node and complexity of the multi-user detection. As V can be characterized and uniquely determined by the factor graph matrix, the design of V can borrow the idea from the design of LDPC codes. However, here we introduce general rules for the designing: • V j ∈ B K ×N , and Vi = V j , ∀i = j [φ] [φ] • V j = I N , where V j is obtained by removing all-zero rows in V j Thus we may insert K − N all-zero row vectors into rows of I N to obtain the unique solution V + for problem (12.36). If we take Example 1 as the illumination, we have following properties and relations for SCMA encoding parameters • Choose the constellation length N = 2,  the codebook length K = 4  and K =6 • The maximum number of layers J = N 12 Sparse Code Multiple Access (SCMA) 395 • The number of multiplexed layers over each resource d f = JKN = 3 • Overloading factor λ = KJ = 1.5 • max(0, 2N − K ) ≤ l ≤ N − 1, where l is the number of the overlapping elements of any two distinct f j vectors. Thus 0 ≤ l ≤ 1 if K = 4 means that the codeword are either orthogonal or collide at only one overlap nonzero element over any two rows. The resulting factor graph matrix F is the same as (12.5) and the factor graph is shown in Fig. 12.3. 12.3.1.2 Multi-dimensional Constellations Having the mapping set V + , the optimization problem of an SCMA is reduced to   C + = arg max Υ S(V + , C ; J, M, N , K ) C (12.37) which is to find J different N -dimensional complex constellations, each contains M signal points. In general, the joint design of multiple multi-dimensional constellations is challenging, a further simplification of (12.37) can be conducted by dividing the problem into the design of a mother constellation and J layer-specific operators, and optimizing them separately. Without loss of generality, define C j ≡ Θ j (C), ∀ j, where Θ j (·) denotes a constellation operator. Thus the optimization problem in (12.37) becomes 4J 3 C+ , Θ +j j=1 = arg 5 8 6 7J max Υ S(V + , C ≡ Θ j (C) j=1 ; J, M, N , K ) J C,{Θ j } j=1 (12.38) A. Mother Constellation In general, a constellation with large minimum Euclidean distance achieves good performance when no collisions occur among users/layers over a tone. With increasing number of users/layers, the collisions are unavoidable and the multi-user interference will be introduced. To mitigate such interference, it is required to induce dependency among the nonzero elements of the codewords, such that the receiver can recover colliding codewords from other tones. In general, the mother constellation can be any form of a multi-dimensional constellation with a maximized minimum Euclidean distance. To control dimensional dependency and power variation without destroying the Euclidean distance profile, a unitary rotation can be applied to the mother constellation. For transmission over fading channels, the performance is dominated by the product distance of a constellation at high-SNR region. Thus the goal of designing a good mother constellation for SCMA is trying to optimize both the minimum Euclidean distance and product distance. Fortunately, the optimization of the product distance could be realized by unitary rotation as well. Thus the two types of distances can be optimized separately. In [20], using the Chernoff bounding 396 Z. Ma and J. Bao technique, it is shown that for Rayleigh fading channels, the error probability of a multi-dimensional signal set is essentially dominated by four factors. To improve performance is necessary to • minimize the average energy per constellation point; • maximize the modulation or signal-space diversity; • maximize the minimum product distance d p,min = min xa ,xb xa [k]=xb [k] |xa [k] − xb [k]| (12.39) between any two points xa and xb in the constellation; • minimize the product kissing number for the minimum product distance, i.e., the total number of points at the minimum product distance. For low rates, constellation design can be done by brute-force searching, however, this is not necessarily the case for higher rates and a larger number of users/layers due to the prohibitive searching complexity. Under this circumstance, the structured construction is required. Lattice constellation construction can be considered as a possible way to design good mother constellations. If we construct a constellation from the lattice Z2N with gray labeling, the construction could be done effectively by forming orthogonal QAM constellations on different complex planes. To maximize the minimum product distance of rotated lattice, the unitary rotations of QAM lattice constellations might be optimized as in [20, 21]. B. Constellation Operators After obtaining the mother constellation C+ , layer-specific operators should be designed to guarantee the unique decodability of the multi-layer signals at the receiver, and also lower the multi-user interference. The optimization problem for the operators can be formulated as   {Θ +j } Jj=1 = arg max Υ S(V + , C ≡ {Θ j (C+ )} Jj=1 ; J, M, N , K ) {Θ j } Jj=1 (12.40) Note that here, the design criterion Υ (·) are not necessarily the same as that in (12.38) for the joint design of mother constellation and constellation operators. The constellations for different SCMA layers might be constructed with different operators Θ j (·), and the constellation operators generally include complex conjugate, phase rotation and dimensional permutation. Generally speaking, if the different users have different power levels, the interfering codewords would be easily separated at receiver due to the power diversity. To do this, it is obliged to have a diverse average power level over the constellation dimensions when designing the mother constellations, which could be done by an appropriate rotation of the lattice constellation as discussed in [16]. Thus the task of optimization problem can be the permutation operators which enable the SCMA codebooks to capture as much power diversity as possible over the interfering users. The optimization for power variation 12 Sparse Code Multiple Access (SCMA) 397 over users can be designed to permute each codebook set to avoid interfering with the same dimensions of a mother constellation over a resource node. As discussed in Sect. 12.2.1.2, the constellation operators is unnecessary for the uplink SCMA in fading channels. On the one hand, in MAC, the fading itself takes the role of constellation operations, and the receiver exploiting the differences among the channel fadings to separate the multi-user signals. On the other hand, the constellation operators like phase rotation and complex conjugate don’t change λ2k in Definition 1, hence don’t change the error probability. However, it is important to design layerspecific operator for downlink SCMA, because all users experience the same channel condition and the destructive codeword collision can be avoided by careful design of Θ j (·) in the downlink. 12.3.1.3 Constellations for Lower Receiver Complexity This part introduces two kinds of multi-dimensional constellations for SCMA, that allow MPA receiving with reduced complexity. A. Shuffled Multi-dimensional Constellation The dependency among the complex dimensions of the mother constellation guarantees an efficient detection and diversity for fading channels. It is possible to construct a mother constellation such that the real part and imaginary part are independent with each other, while the complex dimensions are still dependent. One kind of approach is the shuffling [16], which enables the MPA to reduce the complexity from M d f to M d f /2 . The shuffling method rotates two independent N -dimensional real constellations to maximize the minimum product distances, with the same or different unitary rotations, then generates an N -dimensional complex mother constellation by concatenation of the two N -dimensional rotated real constellations. One of the two N -dimensional real constellations corresponds to the real part of the points of Fig. 12.7 Shuffling construction of the mother constellation [16] 398 Z. Ma and J. Bao Fig. 12.8 An example of shuffling construction of two-dimensional 16-ary SCMA constellation [16] the mother constellation, and the other one corresponds to the imaginary part. The construction is illustrated in Fig. 12.7. Example 2 The construction of a 16-point SCMA mother constellation applicable to codebooks with two nonzero position 5 (N√ =8 2) by shuffling is illuminated in Fig. 12.8. Its optimum rotation angle is tan−1 1+2 5 , which maximizes the minimum product distance. B. Low-Projected Multi-dimensional Constellation A key feature of SCMA codebooks is that the multi-dimensional constellation allows a few constellation points to collide over some of the dimensions, as they can still be separated through other components. An example is shown in Fig. 12.9, in which the constellation points corresponding to 01 and 10 collide over the first dimension, but are separated over the second tone, making the number of projection points equal to 3 instead of 4. By employing this low-projected constellation, the MPA receiver is able to reduce the number of probability calculations at the FNs during each iteration. d As a result, the complexity is reduced to O(M p f ), where M p ≤ M is the number of projection points. To do this, it is obliged to let the minimum “product distance”4 be zero during the mother constellations design by rotation. However, the zero minimum product 4 This is the relaxed product distance that takes the product of all the dimension-wise distance between two points into consideration. 12 Sparse Code Multiple Access (SCMA) 399 Fig. 12.9 An example of a 4-ary constellation with 3 projections per complex plane distance would cause the performance degradation at high SNR, thus the tradeoff between the performance and complexity should be considered for different scenarios. 12.3.2 Multi-user Codebooks Design for Uplink SCMA Systems In this subsection, we introduce a practical codebook design approach for uplink SCMA systems over Rayleigh fading channels. Instead of optimizing the mother constellation and constellation operators separately, we address the joint design of multi-user constellations for small constellation size and number of users [22]. 12.3.2.1 Design Criterion To address the design of good codebooks, we need to establish appropriate performance criteria for a given system, i.e., determine the Υ (·) in (12.37). It is straightforward to use the DCCC I (X, y) in (12.28), or the ACEP in (12.24), as the criterion, for increasing capacity or lowering probability of error. However, it is inefficient to use the DCCC or the ACEP as the cost function directly, since the evaluation of I (X, y) involves either Monte Carlo simulations or a large amount of numerical integration, and the calculation of the union bound on the ACEP is a little bit complicated. As an alternative metric, the cutoff rate also gives an approximated evaluation for the capacity as well as the error probability and allows us to optimize the codebooks at a target value of SNR. Therefore, we can formulate the criterion for the multi-user codebooks design, by making the cutoff rate as large as possible, or equivalently the union-Bhatacharyya bound on the ACEP as small as possible. According to the cutoff rate analysis in (12.32) for MAC, maximizing R0 is equivalent to choose the combined codewords such that 400 Z. Ma and J. Bao  −1 λ2 1+ k 4N0 ,b=a k=1 K {X+j } Jj=1 = arg min {X j } Jj=1 E s.t.  X   Xa Xb  2 (12.41) = Es where E s is the total power constraint on the multi-user codebooks. It is expected that the criterion is optimal in the sense of designing codebooks with large DCCC or small ACEP, since it involves all pairs of possible codewords for multiple users. 12.3.2.2 Signal-Space Diversity Scheme Recall that the problem of designing multi-dimensional constellations for fading channels has been solved by using signal-space diversity (SSD), which rotates the QAM constellations with a unitary matrix, constructed either from the algebraic number theory or by computer search [21, 23]. Therefore, here, we use the rotated constellation to build the multi-user codebooks. In particular, we obtain the rotation matrices through computer search over compact parameterizations of unitary matrices. Note, that the null dimensions of codebooks are discarded before the rotations. The N × N unitary matrix can be written as [23] R= N −1 N Tm,n (12.42) m=1 n=m+1 where Tm,n is a complex Givens matrix given by [24] ⎡ Tm,n ⎤ .. ⎢ . ⎢ cos θm,n ⎢ ⎢ .. =⎢ . ⎢ iηm,n ⎢ −e sin θm,n ⎣ . . . e−iηm,n sin θm,n .. .. . . ... cos θm,n .. . ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ which changes the identity matrix by replacing its (m, m)th, (n, n)th, (m, n)th and (n, m)th elements with cos θm,n , cos θm,n , e−iηm,n sin θm,n and −eiηm,n sin θm,n , respectively. The angles satisfy θm,n ∈ [−π, π ] and ηm,n ∈ [−π/2, π/2]. So, the search for R is reduced to the search for a sequence of phase of θ = {θ1,2 , . . . , θ1,N , θ2,3 , . . . , θ2,N , . . . , θ N −1,N }, η = {η1,2 , . . . , η1,N , η2,3 , . . . , η2,N , . . . , η N −1,N }. (12.43) It seems that optimizing the constellations using the above criterion is intractable even for a moderate number of users and codebook size, since we need to search 12 Sparse Code Multiple Access (SCMA) 401 N (N − 1)J angles, and the summation in the right-hand side of (12.41) will add up M J (M J − 1) terms. However, searching results for two-dimensional constellations with a small number of users show that the rotation matrices are the same for all codebooks, and are independent of the number of users. Therefore, we simplify the optimization process by searching over a single rotation matrix, and reducing the number of accessed users, even though this is suboptimal. Furthermore, we can use the approach developed in [25], where all the entries of the rotation matrix6are7 equal in magnitude. Therefore, by expanding the product in (12.42), we get θ = π4 7 6 and θ = π4 , 0.6155, π4 for N = 2 and N = 3, respectively. Exhaustive search is computationally feasible, provided, that each user occupies a moderate number of resources such that N ≤ 3. In the signal-space diversity scheme, the constellations are restricted to lattice constellations such that the rotated QAMs are suggested. In practice, the rotation can be done over any multi-dimensional constellations to improve the cutoff rate, e.g., the rotated spherical codebook [26] and rotations over the product of other low-dimensional constellations. 12.3.2.3 Simulations and Discussions Consider the SCMA system in Example 1, which supports an overloading factor 150%. Simulation results of packet error rate (PER) for uncoded SCMA and DCCC are provided, which are performed over i.i.d Rayleigh fading channels for 4-ary and 16-ary codebooks. Four kinds of codebooks including the codebooks through SSD scheme discussed in this subsection (named as G4,4 /G16,16 ), the codebook from [16] (named as C4,4 /C16,16 ), spherical codebook [26], star-QAM-based codebooks [27], are employed, and we also provide the results of the star-QAM-based codebooks after optimization using the criterion (12.41), for which we extend α to complex numbers and get β = 1, α = −i and α = 0.8 − 0.8i for 4-ary and 16-ary codebooks, respectively.5 Figure 12.10 plots the DCCC of 4-ary and 16-ary codebooks for four users, together with the theoretical limit rates of i.i.d Gaussian inputs. As it is evident, the SSD scheme outperforms all the other codebooks in the high rate region for both 4-ary and 16-ary codebooks, while the mutual information gain is more clear for 4-ary case. While the rate of the star-QAM scheme is small, a significant gain is achieved after optimization with the criterion in (12.41), and it becomes as good as the SSD scheme for 4-ary codebook. Figure 12.11 compares the PER performance of different codebooks for uplink SCMA with six users, where two antennas are employed for receive diversity, and the MPA detector is used with six iterations all the time. As it is observed, the SSD scheme has a gain about 0.8 dB over C4,4 and 0.6 dB over C16,16 , and a gain about 0.5 dB and 0.3 dB over the spherical codebook for 4-ary and 16-ary cases, respectively. 5 The star-QAM-based codebook targets on downlink channels, while its performance deteriorates in the uplink and for large constellation size. 402 Z. Ma and J. Bao 4 Shannon limit on 4 users C4,4 2 Shannon limit on 4 users C 16,16 Star−QAM Star−QAM with optimization Spherical codebook SSD − G 1.8 Mutual Information [bits/s/Hz] Mutual Information [bits/s/Hz] 2.2 4,4 1.6 1.4 1.2 1 1.7 0.8 0.6 1.62 4.8 0.4 0 2 4 Star−QAM Star−QAM with optimization Spherical codebook SSD − G16,16 3.5 3 2.5 2 3 1.5 5.8 6 8 6 1 10 2 3 4 5 6 7 8 7 9 10 Eb/N0 [dB] Eb/N0 [dB] (a) 4-ary SCMA codebooks. (b) 16-ary SCMA codebooks. Fig. 12.10 Mutual information of uplink SCMA with 4 users 0 10 0 10 −1 10 −1 PER PER 10 −2 10 −2 10 C 4,4 −3 10 10 C Star−QAM Star−QAM with optimization Spherical codebook SSD − G4,4 12 14 16,16 −3 16 18 E /N [dB] b Star−QAM with optimization Spherical codebook SSD − G16,16 0 (a) 4-ary SCMA codebooks. 20 10 10 12 14 16 18 20 E /N [dB] b 0 (b) 16-ary SCMA codebooks. Fig. 12.11 PER of uplink SCMA systems over Rayleigh fading channels Without optimization, the star-QAM scheme yields the worst error performance. However, it performs much better after optimization, which coincides with the result of mutual information in Fig. 12.10. 12 Sparse Code Multiple Access (SCMA) 403 12.3.3 Low-Projected Multi-dimensional Constellations Design As is discussed above, by employing the multi-dimensional constellations with low projections, the MPA receiver is able to utilize the constellation structure to reduce the receiver complexity. This subsection introduces an approach of constructing low-projected multi-dimensional constellations for uplink coded SCMA. In particular, constellation optimization for bit-interleaved convolutional coded SCMA with iterative multi-user detection is considered. 12.3.3.1 Transfer Characteristics of Turbo-MPA Detector Extrinsic information transfer (EXIT) characteristics are investigated to find the effect of multi-user constellations on the performance of the MPA detector, and give us insights on the constellation optimization criteria. For each user, the EXIT chart analysis computes the average mutual information (AMI) between the extrinsic LLR (L e ), or the a priori LLR (L a ), and each coded bit. Thus, the extrinsic AMI is calculated as [28] 1 Idet,e = 1 − . 2π σe2 # +∞ −∞  2 + − l − σe2 /2 log2 (1 + e−l )dl exp 2σe2 * 1 1 1 or 2 users 1 or 2 users 0.8 0.6 0.6 6 users I I det,e det,e 0.8 6 users 0.4 4 users 0.4 4 users 8−ary constellation, Eb/N0 = 6 dB, Rc = 0.5 8−ary constellation, Eb/N0 = 6 dB, Rc = 0.5 0.2 0.2 Labeling 1 Labeling 2 Labeling 3 Constellation 1: G 8,8 Constellation 2: LDS − 8PSK 0 0 0.2 0.4 0.6 0.8 1 Idet,a (a) EXIT charts for different constellations. 0 0 0.2 0.4 0.6 0.8 1 Idet,a (b) EXIT charts for different labelings. Fig. 12.12 Impacts of constellations and labelings on the detector’s transfer characteristics (E b /N0 = 6 dB, MPA detector with 3 iterations) 404 Z. Ma and J. Bao where σe2 is the variance of the extrinsic LLR. It is worth noting that due to the multi-user interference, the a priori AMI (Idet,a ) and extrinsic AMI (Idet,e ) for each user will be influenced by the other users, and hence, a J -dimensional EXIT chart is necessary to characterize the transfer function. Here, the AMI is averaged over all the users such that the EXIT curves can be depicted on a one-dimensional complex plane. Now, we investigate the transfer characteristics of the turbo-MPA detector for uplink SCMA over i.i.d. Rayleigh fading, and a factor graph matrix as in (12.5) is considered. Figure 12.12 presents the detector’s transfer characteristics of twodimensional 8-ary constellations for different number of users (J ) at E b /N0 = 6 dB, where the MPA detector performs 3 iterations. Note that when J = 2, the signals from the two users are orthogonal with each other, so that they have the same AMI as that in the single-user case. The impact of different constellations on the detector’s transfer characteristics is shown in Fig. 12.12a, where the detector’s EXIT curves of two different 8-ary constellations6 with the same labelings are provided. Obviously, constellation 1 outperforms constellation 2, and the superiority of constellation 1 over constellation 2 is independent with the number of users. This implies that the effect of the constellation on the single-user system agrees with its multi-user counterpart, even though the EXIT curves become steeper as the number of users increases. In Fig. 12.12b, the results for the same 8-ary constellation with different mappings/labelings are demonstrated. It is observed that different labelings result in transfer characteristics curves of different slopes, for all the number of users cases, and the labeling with a steeper EXIT curve in the single-user case shows the larger slope in its multi-user counterpart. The conclusion implies that the influence of constellations and labelings on the single-user system is consistent with that on the multi-user case. More precisely, a constellation or a labeling that is good for single-user systems will be beneficial to the multi-user systems. Therefore, we suggest to simplify the complicated multi-user constellations optimization in SCMA to the suboptimal single-user system design. It is expected that the constellation designing criteria for the single-user system is efficient for multi-user cases. 12.3.3.2 Design Criteria of Multi-dimensional Constellations A. Links Between EXIT Charts and Constellation Design The EXIT chart is a good tool to guide the system design. For iteratively decoded systems, given an outer convolutional code, the constellation should be designed to form a tunnel between the transfer curves of the detector and the decoder, and the 6 The constellation 1 is constructed by rotation over the product of a binary phase-shift keying (BPSK) and a quadrature phase-shift keying (QPSK) constellation with Gray labelings, using the approach in Sect. 12.3.2.2 (G8,8 ), and constellation 2 is the repetition over an 8PSK constellation with Gray labeling, i.e., the LDS scheme [29]. 12 Sparse Code Multiple Access (SCMA) 405 starting point of the detector curve and the intersection point between the detector curve and the decoder curve should be as high as possible, to guarantee a low threshold as well as a low error floor. At a given value of SNR, the transfer characteristics of the detector are affected by both the constellation itself and the labeling, as shown in Fig. 12.12. In terms of the constellation, it is known that the area under the detector’s EXIT curve is approximately equal to the DCCC per number of bits of a constellation point [30]. Based on this property, once a constellation with a larger DCCC is constructed, a larger area is obtained and then it has the potential of providing a wider EXIT tunnel, or equivalently, it would be easier to let the detector’s curve to be above the decoder’s curve. In terms of the labeling, for a given constellation, the detector curves corresponding to distinct labelings are rotations with each other, since the labeling does not change the DCCC and hence the area below the detector curve. A good labeling rotates the detector curve such that a large AMI is produced when Idet,a = 1, which provides an error floor that reaches the BER range of practical interest, and at the same time, to make the tunnel between the transfer curves of the detector and the decoder still open. Based on these facts, we divide the constellation optimization framework into two steps. First, try to design the multi-dimensional constellation by maximizing the DCCC; Second, optimize the labeling by EXIT curve-fitting. In the following, we introduce two figures of merits for the constellation and the labeling. B. Constellation Figure of Merit As discussed above, the cutoff rate, corresponding to the DCCC, is a good criterion that allows us to optimize the constellation at a target value of SNR. Considering the received signal y = diag(h)x + n, the cutoff rate constrained by an M-ary K dimensional signal set C in i.i.d. Rayleigh fading, is given by [25] ⎡ 1   ΨCFM (C) = log2 M − log ⎣1 + M x ∈C x ∈C,x =x a b b a ⎤  −1 δk2 ⎦ 1+ 4N 0 k=1 K (12.44) where δk = |xa [k] − xb [k]|, is the dimension-wise distance between any two distinct K -dimensional symbols xa and xb . We take the quantity ΨCFM (C) as the SNRdependent constellation figure of merit, which is a function of SNR and the constellation C, or the set of all pairwise distances between the constellation points. It involves all pairs of multi-dimensional symbols, and is independent of the labeling or any channel codes. C. Labeling Figure of Merit The constellation labeling is a crucial design parameter to achieve a high coding gain over the iterations for iteratively decoded bit-interleaved coded modulation (BICM) systems. To obtain an optimization criterion for the labelings, we resort to the error performance of multi-dimensional constellations under ideal interleaving. Let x̃(i) = [x̃(i) [1], . . . , x̃(i) [K ]]t , be the symbol having the same label with that of x 406 Z. Ma and J. Bao except at the ith bit position. The effect of labeling µ on the performance of BICM with iterative decoding (BICM-ID) systems employing multi-dimensional signal constellation can be characterized by [31]  1 m K  1   1 2 −1 1+ ΨLFM (µ) = δ m M i=1 b=0 4N0 k b k=1 (12.45) x∈Ci where δk = |x[k] − x̃(i) [k]|, and Cib is the subset of C that consisting of symbols whose label has the value b in the ith bit position. The SNR-dependent object function ΨLFM (µ) is able to characterize the influence of both the constellation C and the labeling µ to the bit error rate (BER) performance of BICM-ID systems. With this criterion, one can optimize the bit labeling when fixing the signal constellation, or optimize the constellation for a given labeling, or optimize them jointly. Since optimizing the labeling µ by decreasing ΨLFM (µ) improves the BER performance, we take ΨLFM1 (µ) as the labeling figure of merit to guide the labeling design for a given multi-dimensional constellation. 12.3.3.3 Design Multi-dimensional Constellations The multi-dimensional constellation with the same projections over each dimension can be viewed as a multi-modulation scheme [32], where the data bits are modulated into multiple one-dimensional symbols that are chosen from a one-dimensional complex constellation A, called subconstellations in the following. The difference among the modulations for each dimension is that they have different labelings. This implies that the multi-dimensional constellation can be constructed by permutations of the one-dimensional subconstellation A, dimensionally. Therefore, the problem is to design an M-ary subconstellation A with M p distinct signal points, and the specific mapping or permutation for each dimension. In the following, we propose a multi-stage optimization, and the K -dimensional constellation is constructed by three steps: (a) Determine the desired number of projection points M p such that M p ≤ M, choose a one-dimensional M-ary subconstellation A with M p projections; (b) Based on the one-dimensional subconstellation A, construct a K -dimensional constellation C using permutations; (c) Design a labeling for the K -dimensional constellation C. A. Design One-Dimensional Subconstellation A Different from the traditional constellation design, the M-ary constellation with M p projections imply that there are M − M p signal points that overlap with others. We first choose an M p -ary constellation A p without overlappings, then allocate the M p signal points with M labels to obtain A. The choice of A p is various, any onedimensional complex constellation, e.g., quadrature amplitude modulation (QAM) 12 Sparse Code Multiple Access (SCMA) 407 or phase-shift keying (PSK), is available. Here, we construct A p using the amplitude phase-shift keying (APSK) constellation, since it is able to provide good DCCC compared to other conventional modulations [33, 34]. An M p -APSK constellation is composed of L concentric rings, each with uniformly spaced PSK points. The M p -APSK constellation can be expressed as [33] 6 7 A p = r1 e jθ1 P(m 1 ), . . . , r L e jθL P(m L ) where P(m l ) is an m l -ary PSK constellation with unit average energy, and rl , θl are the radius and phase offset of the lth ring, respectively. Let m= [m 1 , . . . , m L ]t , be L m l . To guarantee the vector of the number of points over each ring so that M p = l=1 a good distance profile, it is preferred to locate fewer constellation points on the inner rings than that on the outer rings. Then, for a set of ordered radius r1 < · · · < r L , it is suggested that m 1 ≤ · · · ≤ m L . Following the general APSK design procedure proposed in [34], the M p -APSK constellation can be constructed as • Select the number of rings L and the number of constellation points on each ring L m l , such that M p = l=1 ml ; • Determine the radius of each ring rl , & *  l−1 + ' '  m 1 l mi + rl = (− ln 1 − ; M p i=1 2 • Set θl as 0 or π/m l . Given the designed M p -APSK constellation A p , we allocate the M-ary constellation with the M p signal points. The problem can be formulated as how to put M numbers, 0, 1, . . . , M − 1, into M p sets, where each set represents a signal point in A p . The allocation strategy is preferred to follow the rules: • The numbers that are allocated to a set should be less than or equal to M p , and greater than or equal to 1, such that the overlapped points can be separated through other dimensions; • The numbers in each set should be as less as possible, such that the resulted multidimensional constellation has a good distance profile; • Symmetry of the constellation A is preferred so that it has a zero mean; • The sets with low power levels may be allocated with more numbers, such that the constellation has a small average energy. Note that in some cases, the allocation yields a constellation A with nonzero mean, then we shift A toward the origin, such that the mean of all signals is zero and therefore more energy-efficient. Now, we give an example to illustrate the allocations. For a given 9-APSK A p that is constructed with 3 rings and m = [1, 3, 5]t , a 16-ary subconstellation A can be obtained by allocating 16 numbers into 9 sets. Some possible allocations are given 408 Z. Ma and J. Bao in Fig. 12.13. Among the four strategies A, B, C, and D, while the strategy A is the most energy-efficient, it shows the worst performance when used to construct multi-dimensional constellations. This is because too many points overlap with each other, leading to a very poor distance profile for the multi-dimensional constellation. Numerical results show that the strategies B and C are equally efficient, and the strategy D is the best one, since the largest number of overlappings is only two. B. Construct K -dimensional Constellation C Given the designed M-ary subconstellation A, denote πk (A) a column vector of the kth permutation of the signals in A, and let π1 (A) = A. The K -dimensional constellation through the permutation construction can be expressed with a K × M matrix as C = [π1 (A), . . . , π K (A)]t where each column of the matrix corresponds to a K -dimensional symbol. Then, constructing a K -dimensional constellation requires to find K − 1 permutations π2 , . . . , π K , such that the constellation figure of merit ΨCFM (C) in (12.44) is maximized. We focus on two-dimensional constellations (K = 2), by maximizing the constellation figure of merit, the unique permutation function π is selected as  −1 |xa [k] − xb [k]|2 1+ 4N0 k=1 K π = arg min π̂   xa ∈C xb ∈C,xb =xa There are M! different choices for the permutations, for small constellation size where M ≤ 8, the optimum solution can be solved by exhaustive search with a reasonable complexity. However, it becomes intractable for high order constellations. Note that this problem is similar to the labeling map of a constellation in bit-interleaved coded modulation with iterative decoding (BICM-ID) systems, which can be efficiently solved by using the binary switching algorithm (BSA) [35], or iteratively searching inside a randomly selected list, and a local optimum permutation can be found for a given cost function. As for a larger dimension where K > 2, the search for K − 1 permutations is challenging. A suboptimal solution can be used by successively optimizing the multidimensional constellation from lower dimensions to higher dimensions, such that only one permutation is needed to be checked in every round. C. Labeling the K -dimensional Constellation When a multi-dimensional constellation is found, we should choose an appropriate labeling for the constellation. In terms of EXIT chart, optimizing the labeling is to adjust the slope of the detector’s curve. Our approach is to obtain a set of labelings with various slopes in their EXIT curves, firstly. Then, to choose a labeling from the set such that the detector EXIT curve matches with the decoder curve of a given convolutional code. 12 Sparse Code Multiple Access (SCMA) (a) (b) 10 51 11 2 6 9 7 12 7 2 4 13 3 (c) 12 6 1 14 0 8 15 4 409 0 8 9 2 5 15 3 10 15 9 11 7 12 11 6 4 1 0 4 14 3 14 13 1 8 7 10 13 (d) 6 11 5 8 2 13 5 0 3 12 15 10 14 9 Fig. 12.13 Examples of 16-ary subconstellation A based on 9-APSK with m = [1, 3, 5]t The labeling figure of merit ΨLFM1 (µ) in (12.45) represents the ultimate performance with perfectly known a priori AMI, and in EXIT charts, it corresponds to the maximum achievable value of Idet,e with Idet,a = 1, denoted as I ∗ , and I ∗ becomes larger with decreasing ΨLFM (µ). It is observed in Fig. 12.12a, b that for single-user systems, the detector’s EXIT curves corresponding to distinct labelings can be approximated to straight lines, with a common intersection around the point with Idet,a = 0.5. Then, the slope of the EXIT curve corresponding to a labeling can thus be determined by I ∗ , approximately. A labeling with a larger I ∗ can have a steeper transfer curve. Therefore, we can use ΨLFM (µ) to approximately control the slope of the EXIT curves, and the detector’s curve becomes steeper with decreasing ΨLFM (µ). Denote the set of labelings as Ω. For constellations with small sizes (M ≤ 8), Ω is chosen to be the set of all possible labelings with distinct ΨLFM (µ). For higher order constellations, the BSA can be used once again to obtain Ω. Begin with a given original labeling, by minimizing ΨLFM (µ) using the BSA, new labelings with increasing slopes may be obtained during the search, we output these labelings and store them into Ω. Similarly, new labelings with decreasing slopes may be obtained by maximizing ΨLFM (µ) with the same original labeling. Then, the labelings in Ω are sorted with increasing values of ΨLFM (µ). The set Ω can also be obtained by iteratively searching inside a randomly selected list. Now, we choose a labeling from Ω, with the aid of EXIT chart analysis. At an appropriate SNR, the following two conditions have to be fulfilled for the labeling: (a) the slope of either the single-user or multi-user detector EXIT curve should be as steep as possible, to achieve a low BER error floor; (b) the tunnel between the decoder and the multi-user detector curves should be open, or the intersection point between them should be as high as possible, to guarantee a low threshold. As an illustrative example, Fig. 12.14 shows the choices of labelings for a twodimensional 8-ary constellation (with 3 projections) and a 16-ary constellation (with 9 projections) for SCMA. The detector curves of several labelings, with distinct ΨLFM (µ), as well as the decoder curve are provided, where a half-rate four-state non-recursive convolutional code with generator [5, 7]8 is employed as the outer channel code, and Idec,a (Idec,e ) denotes the AMI between the a priori (extrinsic) LLR and the transmitted coded bit at the input (output) of the convolutional decoder. 410 Z. Ma and J. Bao 1 1 user, Eb/N0 = 7.5 dB 0.8 1 user, Eb/N0 = 6 dB Idet,e/Idec,a 0.6 I /I det,e dec,a 0.8 6 users, Eb/N0 = 6 dB 0.6 6 users, Eb/N0 = 7.5 dB 0.4 0.4 Labeling with minimum Ψ (µ) Labeling with minimum Ψ LFM( µ ) LFM Labeling with maximum Ψ LFM( µ ) Labeling with maximum Ψ LFM( µ ) Proposed labeling CC [5, 7] 0.2 0 0.2 0.4 I 0.6 Proposed labeling 0.2 0.8 1 CC [5, 7] 0 0.2 I /I 0.6 0.8 1 /I det,a dec,e det,a dec,e (a) Labeling for an 8-ary constellation with 3 projections. 0.4 (b) Labeling for a 16-ary constellation with 9 projections. Fig. 12.14 Examples of labelings for two-dimensional constellations As it is observed in Fig. 12.14, the labeling with the maximum ΨLFM (µ) yields a relatively flat slope in the 6 users case, and the one with the minimum ΨLFM (µ) closes the tunnel between the decoder and the multi-user detector curves. In contrast, the proposed labeling, which shows a very steep slope in the EXIT curve while still keeps the tunnel open, achieves a good trade-off between the threshold and the BER performance. With the proposed approach, it is possible to construct constellations with any projections. Figure 12.15a and 12.15b show the examples of the designed twodimensional 8-ary constellations with various projections, and a four-dimensional 64-ary constellation with 16 projections, respectively. 12.3.3.4 Simulations and Discussions In the following, simulations are conducted to evaluate the performance of lowprojected constellation for an uplink convolutional coded SCMA system. The detailed simulation configuration is given in Table 12.2. The SCMA system follows a factor graph matrix as (12.5), which supports an effective system loading of 75% (J Rc /K ). For the sake of simplicity, let A M,M p denote the APSK-based M-ary constellation with M p projections. The constellations in [16, 36, 37] are named as C M,M p (the SSD scheme in Sect. 12.3.2 is denoted as G M,M p ), which are used as the benchmark. The simulated BER performances of 4-ary, 8-ary and 16-ary codebooks are depicted in Figs. 12.16 and 12.17, respectively. It is obvious that the A M,M p code- 12 Sparse Code Multiple Access (SCMA) 3 6 4 A83 2 7 3 5 2 4 2 5 0 A85 7 2 6 7 7 0 5 3 2 1 7 A87 4 0 5 2 0 1 7 6 6 3 A88 4 2 5 7 37 46 0 50 20 9 53 42 61 6 17 39 62 4 60 36 16 1 49 48 10 22 6 0 3 0 5 54 58 20 44 5 (a) Two-dimensiona l8-ary constellations with various projections. 0 33 41 40 7 36 31 23 28 59 42 62 24 57 32 46 56 27 5 44 58 30 51 55 22 63 14 15 7 63 23 53 25 45 26 40 12 24 50 55 54 22 62 31 1 32 34 52 33 48 13 45 12 21 19 3 42 5 31 47 13 57 19 17 31 24 3 29 43 51 41 49 8 18 19 59 11 27 28 56 9 35 30 14 59 32 27 20 26 18 0 43 35 52 60 58 23 63 34 6 21 16 51 46 24 47 11 29 7 2 29 15 53 61 50 26 56 48 2 30 21 38 7 6 8 34 1 35 54 17 3 52 4 4 2 19 14 11 39 5 3 1 6 4 5 0 18 16 25 10 2 43 47 37 3 1 4 A86 1 3 2 1 5 2 0 5 7 4 6 3 6 1 6 6 7 4 5 0 0 38 15 2 5 3 6 3 0 6 4 1 2 1 1 7 3 4 A84 4 7 0 1 411 9 1 40 13 8 0 49 21 18 9 28 16 32 15 33 29 23 51 42 20 22 13 60 43 39 57 11 27 25 34 4 61 48 63 59 58 46 12 47 36 44 38 62 50 36 8 45 39 41 61 38 28 57 37 10 44 49 12 60 33 25 4 55 17 14 1 53 26 5 37 10 3 35 30 7 56 41 2 55 52 40 6 45 54 (b) A four-dimensional 64-ary constellation with 16 projections. Fig. 12.15 Examples of APSK-based low-projected SCMA constellations Table 12.2 Simulation parameters Parameters Channel model Target spectral efficiency Number of users FEC coding Interleaving Codebooks Receiver Values Uplink Rayleigh fading channel 1.5, 2.25, 3 bits/resource 6 1/2-rate convolutional code with generator [5, 7]8 Random interleaver, interleave length: 1024 bits C43 /A43 , C44 /A44 /G44 , C83 /A83 , C84 /A84 , C85 /A85 , A88 /G88 C169 /A169 , C1616 /G1616 /A1616 Turbo-MPA (3 MPA iterations + 6 BICM iterations) books outperform others in the large SNR region, for almost all the simulation cases. In Fig. 12.16a, the BER floor of A4,3 is lower than C4,3 when SNR is less than 8 dB, and A4,4 shows much better performance than C4,4 , and has a gain about 0.25 dB over G4,4 . Note that A4,4 outperforms G4,4 in the whole SNR region. This is because A4,4 has the same labeling with G4,4 but the larger DCCC. For the 8-ary codebooks shown in Fig. 12.16b, the error floors of the A M,M p codebooks happen at the BER level below 10−5 , which are much lower than the other codebooks. Thus, much smaller values of SNR are required to achieve a BER value of 10−6 . Similar results are also obtained for 16-ary codebooks, which are shown in Fig. 12.17. The gain of the A M,M p codebooks over C M,M p is smaller than that in the 8-ary codebooks case. The BER curve of C16,16 degrades earlier than the others, but it arrives the error 412 Z. Ma and J. Bao 0 0 10 10 C4,4/A4,4/G4,4 A8,3/C8,3 C4,3/A4,3 −1 10 A8,4/C8,4 A88 −1 10 A /C A /G 8,5 A85 8,8 8,5 8,8 A83 −2 −2 10 10 A84 BER BER C44 −3 10 C43 −3 10 C85 −4 −4 10 C83 10 C84 A43 G88 G44 −5 10 −5 10 A44 −6 −6 10 1 2 3 4 5 6 7 8 10 9 3.5 4.5 E /N [dB] b 5.5 6.5 7.5 8.5 9.5 E /N [dB] 0 b (a) 4-ary codebooks. 0 (b) 8-ary codebooks. Fig. 12.16 BER performance of uplink coded SCMA for 4-ary and 8-ary codebooks Fig. 12.17 BER performance of uplink coded SCMA for 16-ary codebooks 0 10 −1 10 C1616 C169 A169 −2 10 G1616 BER A1616 −3 10 −4 10 −5 10 A16,9/C16,9 −6 10 5 A16,16/C16,16/G16,16 6 7 8 9 10 Eb/N0 [dB] floor quickly. The codebook G16,16 shows a very large threshold, and A16,16 achieves a good trade-off. Even though the BER threshold of A16,9 is larger than C16,9 , A16,9 shows almost the same BER performance with C16,9 at the SNR around 9.5 dB. To achieve the BER performance of 10−6 , equal or less SNR are required for the A M,M p codebooks. 12 Sparse Code Multiple Access (SCMA) 413 12.4 SCMA for 5G Radio Transmission 12.4.1 Application Scenarios for 5G Networks In addition to achieving higher transmission rates, faster access, supporting of larger user density and better user experience in enhanced mobile broadband (eMBB), the 5G air interface connects to new vertical industries and new devices, creating new application scenarios such as massive machine type communications (mMTC) and ultra reliable low latency communications (URLLC) services, by supporting massive number of devices and enabling mission-critical transmissions with ultra high reliability and ultra-low latency requirement, respectively. This presents new challenges and considerations for the radio multiple access to be fully scalable to support these diverse service requirements. The current orthogonal multiple access might not be able to fulfill some of the requirements, such as services for dense MTC devices deployments, and SCMA can be considered as a promising candidate to meet the 5G performance requirements. In particular, SCMA is proposed for 5G to achieve the following benefits: • for eMBB: larger capacity region by non-orthogonal multiplexing; robustness to fading and interference with code-domain design; robust link adaptation with relaxed CSI accuracy. • for URLLC: higher reliability through diversity gain achieved by multidimensional constellations, and robustness to collision by carefully design the codebooks; latency reduction and more transmission opportunities by enabling grant-free access; Non-Orthogonal Multiplexing of mixed traffic types. • for mMTC: higher connection density with high overloading; reduction of signaling overhead and power consumption by enabling grant-free access. Moreover, it is also possible to extend SCMA application to unlicensed spectrum and V2X systems, since the non-orthogonal transmissions can help to increase the system efficiency and deal with the interference. The link-level performance evaluation for some uplink SCMA scenarios is provided in [38], which compares SCMA with orthogonal frequency division multiple access (OFDMA) in typical scenarios and investigate the robustness of SCMA to overloading and codebook collision. Results show that SCMA achieves significant gain over orthogonal multiple access with good codebook design, and the gain increases as the supported number of users and target spectrum efficiency increases. Moreover, high overloading with stable performance is feasible with SCMA design, which enables robust overloaded transmission, and the performance loss with codebook collision is negligible with SCMA design, which enables robust grant-free transmission. 414 Z. Ma and J. Bao 12.4.2 Challenges and Future Works While SCMA is able to greatly enhance the system capability for 5G networks, some further issues on design and implementation of SCMA remain to be resolved, which can be listed as follows: • Reduced complexity receiver design: Even though MPA or EPA receiver is able to significantly reduce the complexity of SCMA, the complexity of MPA is still very high and iterative multi-user receiver is usually required, which brings several challenging issues for practical implementation: – It limits the capability of SCMA to support massive connectivity; – The iterative multi-user detection brings a large processing delay; – The complexity makes it difficult for SCMA to employ constellations with large sizes, hence limits the transmission rate. Sophisticated multi-user detection schemes should be developed to address the high complexity. • Theoretical analysis: Further theoretical analysis of SCMA is needed to get more insights on the practical system design. For example, the capacity or error performance with randomly codebook allocations. Also, interference cancelation may be incorporated into the MPA detection for lower complexity, then it is desirable to determine the performance and capacity under practical detectors. • Codebook design: The codebook design is complicated, especially for highdimensional codebooks and that with large size. Advanced multi-dimensional constellation construction is necessary, and the joint design of factor graph matrix and constellations is to be developed, for further performance improvement. Moreover, the design for the scenario that all the overloaded users have different transmission rates (codebook sizes) is to be investigated, to improve the link adaptation. • Other issues: System scalability of supporting various loading, SCMA in both uplink and downlink transmissions, supporting of other techniques such as MIMO, resource/codebook allocation, channel estimation for uplink SCMA, etc. References 1. H. Nikopour, H. Baligh, Sparse code multiple access, in IEEE 24th International Symposium on Personal Indoor and Mobile Radio Communications (PIMRC’13) (2013), pp. 332–336 2. S. Zhang, X. Xu, L. Lu, Y. Wu, G. He, Y. Chen, Sparse code multiple access: an energy efficient uplink approach for 5G wireless systems, in IEEE Global Communications Conference (GLOBECOM’14) (2014), pp. 4782–4787 3. Y. Wu, S. Zhang, Y. Chen, Iterative multiuser receiver in sparse code multiple access systems, in Proceedings of IEEE International Conference on Communications (ICC’15) (2015), pp. 2918–2923 4. J. Zhang, L. Lu, Y. Sun et. al., PoC of SCMA-Based Uplink Grant-Free Transmission in UCNC for 5G. IEEE J. Sel. Areas Commun. 35, 1353–1362 (2017) 12 Sparse Code Multiple Access (SCMA) 415 5. R. Hoshyar, F.P. Wathan, R. Tafazolli, Novel low-density signature for synchronous CDMA systems over AWGN channel. IEEE Trans. Signal Process. 56, 1616–1626 (2008) 6. D. Guo, C. Wang, Multiuser detection of sparsely spread CDMA. IEEE J. Sel. Areas Commun. 26, 421–431 (2008) 7. R1-166098: Discussion on feasibility of advanced MU-detector. Huawei, HiSilicon, 3GPP TSG RAN WG1 Meeting #86 (2016) 8. X. Meng, Y. Wu, Y. Chen, M. Cheng M, Low complexity receiver for uplink SCMA system via expectation propagation, in Proceedings of Wireless Communications and Networking Conference (WCNC’ 17), (2017), pp. 1–5 9. J. Bao, Z. Ma, G.K. 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Chen, Spherical codes for SCMA codebook, in Proceedings of IEEE 83th Conference on Vehicular Technology (VTC Spring’16) (2016), pp. 1–5 27. L. Yu, X. Lei, P. Fan, D. Chen, An optimized design of SCMA codebook based on star-QAM signaling constellations, in Proceedings of International Conference on Wireless Communications & Signal Processing (WCSP’15) (2015), pp. 1–5 28. S.T. Brink, Convergence behavior of iteratively decoded parallel concatenated codes. IEEE Trans. Commun. 49, 1727–1737 (2001) 416 Z. Ma and J. Bao 29. J.V.D. Beek, B.M. Popović, Multiple access with low-density signatures, in Proceedings of IEEE Conference on Global Communications (GLOBECOM) (2009) 30. A. Ashikhmin, G. Kramer, S.T. Brink, Extrinsic information transfer functions: model and erasure channel properties. IEEE Trans. Inf. Theory. 50, 2657–2673 (2004) 31. N.H. Tran, H.H. Nguyen, Design and performance of BICM-ID systems with hypercube constellations. IEEE Trans. Wirel. Commun. 5, 1169–1179 (2006) 32. A. Seyedi, Multi-QAM modulation: a low-complexity full rate diversity scheme, in Proceedings of IEEE International Conference on Communications (ICC) (2006), pp. 1470–1475 33. C.M. Thomas, M.Y. Weidner, S.H. Durrani, Digital amplitude-phase keying with M-ary alphabets. IEEE Trans. Commun. 22, 168–180 (1974) 34. Q. Xie, Z. Yang, J. Song, L. Hanzo, EXIT-chart-matching-aided near-capacity coded modulation design and a BICM-ID design example for both gaussian and rayleigh channels. IEEE Trans. Veh. Tech. 62, 1216–1227 (2013) 35. F. Schreckenbach, N. Görtz, J. Hagenauer, G. Bauch, Optimization of symbol mappings for bit-interleaved coded modulation with iterative decoding. IEEE Commun. Lett. 7, 593–595 (2003) 36. M.T. Boroujeni, A. Bayesteh, H. Nikopour, M. Baligh, System and method for generating codebooks with small projections per complex dimension and utilization thereof, U.S. Patent 0,049,999, 18 Feb 2016 37. A. Bayesteh, H. Nikopour, M. Taherzadeh, H. Baligh, J. Ma, Low complexity techniques for SCMA detection, in Proceedings of IEEE Globecom Workshops (2015), pp. 1–6 38. R1-164037: LLS results for uplink multiple access. Huawei, HiSilicon, 3GPP TSG RAN WG1 Meeting #85 (2016) Chapter 13 Interleave Division Multiple Access (IDMA) Yang Hu and Li Ping 13.1 Overview The capacity of a multiple access channel was studied in [1, 2]. It can generally be achieved by random coding together with other techniques, e.g., power control, linear precoding, and dirty paper coding [3–6]. Random coding does not involve orthogonality among users so it is inherently non-orthogonal. The sub-optimality of orthogonal multiple access (OMA) was investigated in [7, 8]. The gain of nonorthogonal multiple access (NOMA) over OMA was assessed in [9, 10] for both single-input single-output (SISO) and multiple-input multiple-output (MIMO) systems. Recently, NOMA has been promoted for improving system fairness [11, 12]. However, many practical systems still belong to OMA category. This is mainly due to complexity concerns. OMA can work with low-cost single-user detection (SUD), while NOMA may require more complex multi-user detection (MUD). Thus, these two options entail different trade-off between cost and performance. Historically, the third-generation (3G) direct-sequence code-division multiple access (DS-CDMA) system is non-orthogonal. Normally, only SUD is used in DSCDMA to avoid high complexity, which is sub-optimal. The spreading operation in DS-CDMA reduces rate, so DS-CDMA is not convenient for high-rate applications. The fourth-generation (4G) orthogonal frequency division multiple access (OFDMA) system returns to OMA. OFDMA allows flexibility for resource allocation over time and frequency, which can bring about noticeable gain. Y. Hu (B) · L. Ping Department of Electronic Engineering, City University of Hong Kong, Hong Kong, SAR, China e-mail: yhu228-c@my.cityu.edu.hk L. Ping e-mail: eeliping@cityu.edu.hk © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_13 417 Y. Hu and L. Ping average sum rate (bits/channel use) 418 3 2.5 2 SNRsum = 10 dB 1.5 1 0.5 0 SNRsum= 0 dB 0 4 8 12 16 20 24 28 number of users K Fig. 13.1 Achievable sum-rate of NOMA and OMA under equal energy constraint. Note that the two strategies have exactly the same performance. SNRsum is the sum signal-to-noise ratio (SNR) of all users. Complex channels with both slow- and fast-fading factors are considered. Path loss is based on a hexagon cell with a normalized side length = 1. The minimum normalized distance between users and the base station is 35/289, corresponding to an unnormalized distance of 35 m for an LTE cell with radius 289 m. Path loss factor = 3.76 and lognormal fading deviation = 8 dB. The channel samples are normalized such that the average power gain = 1 Recently, NOMA has been discussed widely for the fifth-generation (5G) [13– 16]. A natural question is whether the possible gain of NOMA can justify its higher receiver cost. We examine this question using achievable sum-rate below. In Fig. 13.1, we compare the achievable sum-rate of NOMA and OMA with equal energy constraint per frame per user in a multi-user SISO system. We can see NOMA improves sum-rate when the number of users K increases, but OMA can achieve the same gain through resource allocation. NOMA has no advantage here since OMA is capacity achieving in this case. The situation is slightly different if the energy per frame per user can also be freely optimized under the sum energy constraint. For example, consider maximizing sum-log-rate under the proportional fairness criterion [3]. Figure 13.2 illustrates the related numerical results for K = 2. The sum-rate curves in Fig. 13.2 show that NOMA is only slightly better than OMA with resource allocation (about 8% gain at SNRsum = 10 dB). The advantage of NOMA over OMA seen in Figs. 13.1 and 13.2 is disappointing compared with many results in the literature. This is mainly for two reasons. First, comparisons with OMA without resource allocation are not fair as resource allocation has already been widely used in LTE. Second, a practical signal-to-noise ratio (SNR) range should be used for comparison. A standard way for this purpose is using the following approximation of the signal-to-noise-plus-interference ratio (SINR) in a cellular system [3]: Psum , (13.1) SINRsum = βPsum + σ 2 Fig. 13.2 Achievable sum-rate of NOMA and OMA under sum energy constraint with proportional fairness, K = 2. Other system settings are the same as those in Fig. 13.1 average sum rate (bits/channel use) 13 Interleave Division Multiple Access (IDMA) 419 3 2.5 NOMA 2 OMA with resource allocation 1.5 1 OMA without resource allocation 0.5 0 0 2 4 6 8 10 12 14 16 18 20 SNRsum (dB) where Psum is the sum received powers of all users in a cell, β a cross-cell interference factor and σ 2 the noise power. As a rule of thumb, a typical value is β = 0.6. Then, we have SINRsum ≤ 2.2 dB. Treating interference as noise and allowing a certain range of β, we may consider 0–10 dB as a typical range for SNRsum , as used in Figs. 13.1 and 13.2. Clearly, the gain of NOMA is marginal over such a practical SNR range. Furthermore, NOMA performance may deteriorate seriously if a practical forward error control (FEC) code is used. To see this, consider a successive interference cancelation (SIC) process with K users in a descendant order of user index k. We employ an FEC code that can achieve (almost) error-free decoding at SNR = Γ . Assume that, when we decode for user k, the signals of all users with indexes k ′ > k have been successfully decoded and subtracted from the received signal. Let qk be the received power of user k. Then, user k can achieve error-free decoding provided that qk ≥ Γ. (13.2a) SNRk =  2 ′ k <k qk ′ + σ For sum-power minimization, qk can be calculated using the following recursion (with q0 = 0):    2 qk = Γ × qk ′ + σ . (13.2b) k ′ <k Ideally, the minimum value for Γ can be found from Shannon capacity R = log2 (1 + Γ ). For a practical code, however, a larger Γ is required to ensure (almost) error-free decoding. This overhead accumulates during SIC, which can amount to a considerable loss. Specifically, we consider a practical rate-1/6 channel coding as an example. It achieves bit error rate (BER) ≈ 10−5 (approximately error-free) at about SNR = −3.5 dB with a relatively short block length. There is a 2.35 dB gap compared with SNR = −5.85 dB calculated from the Shannon formula for Gaussian signaling. Such 420 30 required sum-SNR (dB) Fig. 13.3 Accumulated loss in the SIC process Y. Hu and L. Ping 2.35dB 25 20 15 8dB Rate-1/2 7.4dB 10 5 Rate-1/6 0 SIC Capacity -5 -10 2 4 6 8 10 12 number of users K loss accumulates in the SIC process as shown in Fig. 13.3. The accumulated loss is roughly 7.4 dB for 12 users. The problem is more serious for higher rate, as seen from Fig. 13.3 for rate-1/2 coding. Assume the same initial single-user gap of 2.35 dB. The accumulated gap for six users is 8 dB. Nevertheless, NOMA is still useful. In many situations, it is difficult to establish orthogonality due to the lack of centralized control or accurate channel state information (CSI). Then, we may have to resort to NOMA. In particular, as we will show below, NOMA based on interleave-division multiple access (IDMA) [17–22] offers robust and flexible solutions in such environments. IDMA can also recover a considerable portion of the accumulated loss suffered by SIC as shown in Fig. 13.3. We will use numerical results to verify these claims. Some of the software used in this chapter are available at: http://www.ee.cityu.edu.hk/%7Eliping/Research/Simulationpackage/. 13.2 Basic Principles of IDMA Following the advent of turbo and low-density parity-check (LDPC) codes [23– 25], iterative detection techniques were developed in late 1990s for equalization in multipath channels [26] and MUD in DS-CDMA systems [27–29]. It was shown in [30] that two independently interleaved code sequences can be separated by iterative detection. This inspired the IDMA scheme in which users are solely separated by interleavers [17]. Intuitively, a randomly interleaved code results in a different code. Thus we may also say that different users in IDMA are separated by different codes. This follows the basic principle of CDMA, except now the set of codes are generated by a master code followed by different interleavers. We therefore can regard IDMA as a special case of CDMA. However, IDMA is fundamentally different from DSCDMA. The former does not rely on spreading for user separation and so can avoid the rate loss suffered by the latter. In the following, we will start with a graphic model originally presented in [31]. We will show that the primary motivation behind IDMA is to break short cycles, 13 Interleave Division Multiple Access (IDMA) 421 since the latter is detrimental for message-passing detection. The same principles have been successfully used in turbo and LDPC codes. 13.2.1 IDMA Transmitter Principles Throughout this chapter, we will assume an underlying OFDMA layer that removes inter-symbol interference (ISI). We will focus on uplink multiple access techniques built on this OFDMA layer. Let K be the user number and ck = {ck (j), j = 1, 2, . . . , J } a length-J codeword generated by users k. The transmitted symbols {xk (j), j = 1, 2, . . . , J } are generated from {ck (j)} after certain operations, such as spreading, scrambling, interleaving, and modulations. At the receiver, the received signals {y(j)} are given by y(j) = K  k=1 √ hk ek xk (j) + η(j), j = 1, 2, . . . , J , (13.3) where hk is the channel coefficient for user k, ek the transmitted power of user k and η(j) an additive white Gaussian noise (AWGN) sample with variance σ 2 per dimension. Figure 13.4 illustrates the factor graph representation [32] for (13.3) with K = 2, J = 8, and the same LDPC coding for all users. A circle in Fig. 13.4 represents a variable and a square a constraint. Three types of constraints are involved, namely a square marked with “+” for linear additions in (13.3), a white square for LDPC coding, and a square marked with “×” for modulation. Optimal detection for the system in (13.3) typically requires prohibitively high complexity. Low-cost message-passing detection, similar to that used for LDPC codes, can be applied instead, as Fig. 13.4 is sparse when K ≪ J . However, short cylices constitute a problem. To see this, let us call a circle involving m coded (1) Channel y(2) y(3) y(4) y(5) y(6) y(7) y(8) y(1) Received signals + (2) + (3) (4) (5) (6) (7) (8) + + + + + + h1 h2 Transmitted signals x1(1) x1(2) x1(3) x1(4) x1(5) x1(6) x1(7) x1(8) x2(1) x2(2) x2(3) x2(4) x2(5) x2(6) x2(7) x2(8) c1(1) c1(2) c1(3) c1(4) c1(5) c1(6) c1(7) c1(8) c2(1) c2(2) c2(3) c2(4) c2(5) c2(6) c2(7) c2(8) user1 user2 Modulation & Scrambling LDPC coding Fig. 13.4 Factor graph of a NOMA LDPC-coded system where K = 2 and J = 8 422 Y. Hu and L. Ping (a) Channel y(2) y(3) y(4) y(5) y(6) y(7) y(8) y(1) Received signals (1) + (2) + (3) (4) (5) (6) (7) (8) + + + + + + h1 h2 Transmitted signals x1(1) x1(2) x1(3) x1(4) x1(5) x1(6) x1(7) x1(8) Modulation x2(1) x2(2) x2(3) x2(4) x2(5) x2(6) x2(7) x2(8) Interleaving c1(1) c1(2) c1(3) c1(4) c1(5) c1(6) c1(7) c1(8) c2(1) c2(2) c2(3) c2(4) c2(5) c2(6) c2(7) c2(8) user1 user2 LDPC coding (b) y(1) y(2) y(3) y(4) y(5) y(6) y(7) y(8) Received signals (2) (1) + Channel + (3) (4) (5) (6) (7) (8) + + + + + + h1 h2 Transmitted signals x1(1) x1(2) x1(3) x1(4) x1(5) x1(6) x1(7) x1(8) Modulation x2(1) x2(2) x2(3) x2(4) x2(5) x2(6) x2(7) x2(8) c1(2) c1(4) c1(5) c1(7) c1(8) c1(1) c1(6) c1(3) c2(1) c2(5) c2(4) c2(3) c2(6) c2(8) c2(2) c2(7) Interleaved LDPC coding user1 user2 Fig. 13.5 a Factor graph of a two-user IDMA system. b An equivalence form of (a) with re-shuffled {ck (j)}. Note that in (a) the interleavers for LDPC coding are the same for both users, while in (b) they are different bits as a size-m cycle. An example of a size-4 cycle is shown by the black circles {c1 (1), c1 (3), c2 (1), c2 (3)} in Fig. 13.4. There are a large number of such size-4 cycles in Fig. 13.4. Correlation may build up along these short cycles during messagepassing detection, which is detrimental to performance [33]. Figure 13.4 can also be used to represent a DS-CDMA system. For example, a spreading operation involving binary sequence of ±1 can be regarded as a repetition code plus a scrambling operation of sign changes. Repetition coding can be merged with FEC coding. Scrambling can be incorporated in function of the modulation nodes in Fig. 13.4. Note that scrambling does not change the topology of the graph. The problem of short cycles remains the same with or without scrambling. The same conclusion applies to other modulation techniques. Inspired by the success of turbo and LDPC codes [23, 25], the IDMA scheme proposed in [31] employs user-specific interleaving to reduce short cycles in a statistical sense. This is illustrated by the shuffled edge connections between {ck (j)} and {xk (j)} in Fig. 13.5a. For example, compared with Fig. 13.4, {c1 (1), c1 (3), c2 (1), c2 (3)} no longer form a size-4 cycle after interleaving in Fig. 13.5a. This is beneficial for 13 Interleave Division Multiple Access (IDMA) 423 Fig. 13.6 An MA node. Here, an inbound message LLRDEC (xk (j)) is from an LDPC decoder. An outbound message LLRESE (xk (j)) is generated from the MA node. ESE stands for “elementary signal estimation” and DEC for decoder y(j) (j) + LLRESE(xk(j)) LLRDEC(xk(j)) x1(j) ... xk(j) ... xK(j) message-passing detection. Similar principles can also be applied to systems involving convolutional or turbo coding. Two interleavers are used by each user in Fig. 13.5a, one for LDPC encoding and one for multiple access. The former is the same for all users, and the latter is user-specific. We can combine these two interleavers by re-shuffling {ck (j)} for each user, resulting in Fig. 13.5b. The latter interpretation of IDMA is based on [34]. The advantage of Fig. 13.5b is its simpler implementation. If an interleaver is random, its shifted version can be approximately regarded as another independent random interleaver. Thus, Fig. 13.5b can actually be realized by using a shifted version of an LDPC encoder involving a common underlying interleaver. The users, in this case, are separated by the amount of shift after the common encoder structure. Such shifting can be realized with very little cost. We may name such a scheme as code-shift division multiple access (CsDMA).1 This concept was first discussed in [35]. Note that the use of interleavers in Fig. 13.5 does not incur any rate loss. This is a noticeable advantage of IDMA for high-rate applications. 13.2.2 Operations on a Multiple Access Node We now consider message-passing detection based on Fig. 13.5. For convenience, we refer to a square marked with “+” in Fig. 13.5 as a multiple access (MA) node. We will only discuss the operations for such an MA node shown in Fig. 13.6. The operations for other nodes follow the standard treatments for an LDPC code [25]. Denote by DEC k the decoder for user k. We define two messages: an inbound message LLRDEC (xk (j)) and an outbound message LLRESE (xk (j)) that are, respectively log-likelihood ratios (LLRs), generated by DEC k and elementary signal estimation 1 The underlying code in CsDMA should be properly interleaved. An LDPC code naturally meets this requirement. Without interleaving, however, the correlation among the consecutive bits in a convolutional code may cause a problem in CsDMA. This problem can be easily avoided by shuffling the coded sequence. 424 Y. Hu and L. Ping (ESE) operations at the jth MA node in Fig. 13.6. The discussions for LLRDEC (xk (j)) follow the standard LDPC decoding principles [25] and so will be omitted. We will focus on LLRESE (xk (j)) below since its computation is not part of a standard LDPC decoder. For simplicity, let us first assume binary phase-shift keying (BPSK) modulation xk (j) = ±1 for all k. We define LLRESE (xk (j)) by the following LLR for xk (j): LLRESE (xk (j)) = log Pr (y(j)|xk (j) = +1) , Pr (y(j)|xk (j) = −1) (13.4) where y(j) and xk (j) are defined in (13.3). Assume that η(j) in (13.3) is Gaussian with mean μ(j) = E(η(j)) and variance v = Var(η(j)). (For simplicity, we will assume that v is not a function of j. We will explain the rationale for this assumption later.) For a single-user system with K = 1 in (13.3), the conditional probabilities in (13.4) are given by   √ 2 y(j)−(μ(j)±h1 e1 )) ( exp − 2v Pr (y(j)|x1 (j) = ±1) = , √ 2π v so √ y(j) − μ(j) . LLRESE (x1 (j)) = 2h1 e1 v (13.5) (13.6) For K > 1, the problem is much more complicated. We need to consider all possible combinations of {xk (j)}. The exact result is the maximum likelihood (ML) estimator [36] below:  i  i (j) (j) Pr X∼k Pr y(j)|xk (j) = +1, X∼k  i  LLRESE (xk (j)) = log  , i i Pr y(j)|xk (j) = −1, X∼k (j) Pr X∼k (j)  i (13.7) i (j) is one among all 2K−1 possibilities of the set {x1 (j), x2 (j), . . . , where X∼k ′ xk−1 (j), xk+1 (j), . .i . , xK (j)} (since xk ′ (j) ∈ {−1, +1}, ∀k ). In i (13.7), Pr y(j)|xk (j) = ±1, X∼k (j) can be computed similarly to (13.5) and Pr X∼k (j) can be computed from messages {LLRDEC (xk (j))}. Following [25], we define LLRDEC (xk (j)) by an LLR: Pr (xk (j) = +1) . (13.8) LLRDEC (xk (j)) = log Pr (xk (j) = −1) We can obtain Pr(xk (j) = ±1) by solving (13.8) together with Pr(xk (j) = +1) + Pr(xk (j) = −1) = 1. Then,  i Pr X∼k (j) = Pr (xk ′ (j)). k ′ =1,k ′ =k (13.9) 13 Interleave Division Multiple Access (IDMA) 425 The complexity of ML is O(2K ) for BPSK, which increases exponentially with K. For a higher order modulation with an M -point constellation, the complexity of ML is O(M K ). This can be a serious problem in practice. Gaussian approximation (GA) is a low-cost alternative. We rewrite (13.3) as √ y(j) = hk ek xk (j) + ζk (j), (13.10a) where √ ζk (j) = y(j) − hk ek xk (j) = K  k ′ =1,k ′ =k √ hk ′ ek ′ xk ′ (j) + η(j) (13.10b) is the distortion (including interference-plus-noise) with respect to user k. From the central limit theorem, we apply GA to ζk (j) in (13.10b) and assume ζk (j) ∼ N (μk (j), Var(ζk (j))). Now we can treat (13.10a) as a single-user system. For simplicity, we assume a real channel. (We will discuss a complex channel in Sect. 13.5.2.) Then, we have   √ 2 y(j)−(μk (j)±hk ek )) ( exp − 2Var(ζk (j)) Pr (y(j)|xk (j) = ±1) = . (13.11) 2π Var(ζk (j)) Substituting (13.11) into (13.4) and evaluating μk (j) via (13.10b), we have the following ESE operations for the jth MA node in Fig. 13.6. (We will discuss the generation of Var(ζk (j)) in (13.12c) later in (13.13).) ESE operations (i) E(xk (j)) = Pr(xk (j) = +1) − Pr(xk (j) = −1), (ii) μk (j) = K  k ′ =1 √ √ hk ′ ek ′ E(xk ′ (j)) − hk ek E(xk (j)), √ y(j) − μk (j) . (iii) LLRESE (xk (j)) = 2hk ek Var(ζk (j)) (13.12a) (13.12b) (13.12c) The following are some details related to the ESE operations. • Initially, there is no decoder feedback, and we can set E(xk (j)) = 0 in (13.12a) for ∀k, j. • GA is approximate. However, we observed a very good performance based on GA. • The summation in (13.12b) can be shared by all users. The cost per information bit per user is independent of the number of users K. • The principles of GA for higher-order modulations (such as quadrature phaseshift keying (QPSK)) and complex channels can be derived similarly. Related discussions will be shown in Sect. 13.5.2. 426 Y. Hu and L. Ping We now discuss the evaluation of Var(ζk (j)) involved in (13.12c). From (13.10b), Var(ζk (j)) = K  k ′ =1 |hk ′ |2 ek ′ Var(xk ′ (j)) − |hk |2 ek Var(xk (j)) + σ 2 . (13.13a) We observed that the system performance is not sensitive to Var(ζk (j)). Therefore, we take the following approximation Var(ζk (j)) ≈ K  k ′ =1 |hk ′ |2 ek ′ vk ′ − |hk |2 ek vk + σ 2 (13.13b) based on the following assumption Var(xk (j)) = 1 − (E(xk (j)))2 ≈ vk , ∀j. (13.13c) To evaluate vk in (13.13c), we can simply compute a few samples of Var(xk (j)) and take their average. In practice, computation for each Var(xk (j)) can be implemented using a look-up table. We observed that the required number of samples is small, so the related cost is negligible. From the above discussions, the total cost for the ESE operations is (approximately) four additions and two multiplications per chip per iteration. (Some √ √ operations, such as hk ek and 2hk ek /Var(ζk (j)), can be precalculated and need not be repeated for every j. The related cost is negligible.) 13.2.3 Overall IDMA Receiver Now return to an overall IDMA system in Fig. 13.5. We divide the receiver into an ESE module and K DEC modules. The operations in the ESE module is based on (13.12). Consider two types of schedules below. • Serial schedule: In each iteration, operations are carried out as follows: ESE for user 1, DEC for user 1, ESE for user 2, DEC for user 2, . . . As an example, ESE for user 1 means running (13.12) for every j with k fixed to 1. The LLRs generated in (13.12) are then fed to DEC for user 1. Then, {μ1 (j)} and v1 are updated based on the DEC outputs and the process continues to user 2. • Parallel schedule: In each iteration, operations are carried out as follows: ESE for all users in parallel, then DEC for all users. In the above, in each iteration, (13.12) is run through every pair of j and k. After the ESE operations, the outputs are fed to the K local DECs. Then, the K DECs are run in parallel. Afterward, {μk (j)} and vk are updated simultaneously for all k. 13 Interleave Division Multiple Access (IDMA) Fig. 13.7 Performance of IDMA with GA in AWGN channels 427 0 10 -1 Scramling, 4 users, 1 inner itearion, 30 outer itearions 10 -2 single user BER 10 -3 10 IDMA,4 users, 5 inner itearions, 20 outer itearions -4 10 IDMA, 4 users, 1 inner itearion, 10 and 20 outer itearions -5 10 0 0.5 1 1.5 2 2.5 3 Eb/N0 (dB) Note that in the parallel schedule, the value of the summation in (13.12b), that is generated using the results of the previous iteration, remains the same for all users in one iteration. In the serial schedule, this summation is updated user-by-user. We noticed that serial scheduling converges slightly faster than the parallel one. A question arises whether it is helpful if multiple inner DEC iterations are carried out between two consecutive ESE iterations. For example, with the serial schedule, we can run multiple LDPC decoding iterations for one user before going to the next user. We observed that such inner iterations are generally unnecessary. For fixed overall cost, better performance is achieved without inner iterations. However, slipping some inner or outer iterations may lead to reduced complexity. Intuitively, we can treat an IDMA system in Fig. 13.5 as a generalized code system on a graph (where an MA node is just for a special type of constraint). The inneriteration method means more iterations on some parts of the graph for LDPC coding constraints. Such uneven message-passing process does not help in general. For comparison of overall cost, let us consider a K-user TDMA system in which LDPC decoding is individually carried out for each user. The only difference between SUD for such a TDMA system and MUD for IDMA is the ESE operations. As we have seen above, the cost of the extra ESE operations is quite moderate. Thus, an IDMA receiver involving (13.12) has only moderately higher complexity than SUD for corresponding TDMA systems with the same number of users. Figure 13.7 shows an example of a four-user IDMA system in AWGN channels. A rate-1/2 LTE turbo code with 1200 information bits per user is used, followed by a rate-1/8 repetition code and QPSK modulation. We can see that iterative detection nearly converges with about ten outer iterations between ESE and DECs with one inner-iteration. Here, one inner-iteration means running each component decoder once in a turbo decoder. We can also run multiple iterations in each turbo decoder within each outer iteration. The result of five inner iterations is shown in Fig. 13.7. We can see that multiple inner iterations can only offer marginal improvement on performance, even though at considerably higher overall cost. 428 Y. Hu and L. Ping Figure 13.7 also shows the result with user-specific scrambling by spreading each coded bit with a random binary sequence of +1 and −1 before modulation. No user-specific interleaving is used in this case. It is seen that interleaving offers better performance than scrambling. This is due to the short cycle problem as explained earlier. 13.2.4 Performance Evaluation Through SNR Evolution We now outline an SNR evolution technique [17] for tracking IDMA performance. Similar techniques have been successfully applied to turbo and LDPC codes [25] and more recently to AMP algorithms [37]. This analysis method also provides the basis of power allocation for IDMA performance optimization in the next section. Figure 13.8a is a so-called protograph [38] representation of Fig. 13.5, in which each circle represents a vector and each edge represents a vector connection. Relatively thick lines are used in Fig. 13.8a to distinguish it from Fig. 13.5. The messages LLRDEC,k and LLRESE,k in Fig. 13.8a are LLR sequences generated by the coding and MA constraints, respectively. We will use the following SNR-variance relationship to characterize the behavior √ of the system in Fig. 13.8a. Recall (13.10a): yk (j) = hk ek xk (j) + ζk (j). We define the average SNR for user k as  √ E |hk ek xk (j)|2 |hk |2 ek = . SNRk ≡ E(Var(ζk (j))) E(Var(ζk (j))) (13.14) From (13.13), we have E(Var(ζk (j))) = k ′ =1,k ′ =k (a) |hk ′ |2 ek ′ vk ′ + σ 2 . (13.15) (b) Received signals Received signals y MA constraints LLRESE,1 y ESE MA constraints + LLRDEC,1 Transmitted signals K  LLRESE,2 LLRDEC,2 x1 x2 c1 c2 Modulation ESE + SNR1 = 1(v2) v1 = (SNR1) Transmitted signals x1 Modulation SNR2 = 2(v1) v2 = (SNR2) x2 DEC DEC Coded bits LDPC constraints Coded bits c1 c2 LDPC constraints Fig. 13.8 a Protograph representation of Fig. 13.5. b Evolution characterization for (a) 13 Interleave Division Multiple Access (IDMA) 429 Since LLRDEC,k is the output of DEC k with input LLRESE,k characterized by SNRk , we can write vk as a function vk = ψ(SNRk ). Here for simplicity, we assume that all users employ the same LDPC code and therefore have the same ψ(·). (Note that the re-shuffling operation in Fig. 13.5b has no effect on ψ(·).) Combining this with (13.14) and (13.15), we have the following recursion to characterize an iterative IDMA detector: SNR(t) k = K |hk |2 ek (t) 2 2 k ′ =1,k ′ =k |hk ′ | ek ′ vk ′ + σ vk(t) =ψ SNRk(t−1) (t) (t) ≡ φk v1(t) , . . . vk−1 , vk+1 , . . . vK(t) , , (13.16a) (13.16b) where t is an iteration index. The initialization is vk(0) = 1, ∀k, implying no information from DECs. In general, there is no closed form expression for ψ(·), but it can be obtained by simulating a single-user APP decoder in an AWGN channel with specified SNRs. Using {SNRk(T ) } in the final iteration in (13.16), we can estimate the BER by a function (13.17) BERk = g SNRk(T ) , where g(·) can be obtained through simulation of DECs [17]. 13.2.5 Superposition Coded Modulation (SCM) The above IDMA scheme involves multiple signal streams from different users. We may simply allocate these signal streams to a single-user. Such scheme is referred to as superposition coded modulation (SCM) [39, 40]. We define a standard QPSK constellation as SQPSK = {00 → (+1, +1), 01 → (+1, −1), 10 → (−1, +1), 11 → (−1, −1)}. Figure 13.9a is a 16-ary scheme formed by superimposing SQPSK and a scaled version of SQPSK with a scaling factor of 2 and a 45◦ phase shift [39]. Figure 13.9b is a 64-ary scheme formed by superimposing SQPSK and two scaled versions of SQPSK with, respectively, scaling factors of 1.18 and 1.10 plus 60◦ and 120◦ phase shifts. From the central limit theorem, the SCM signaling is more Gaussian-like when the number of streams is large. This can offer the so-called shaping gain as analyzed in [41, 42]. It has been proved that, among all possible signaling methods, an SCM constellation achieves the minimum mean squared error (MMSE) bound; that is, it minimizes the function ψ(·) in (13.16b) for a fixed underlying binary decoder. The details are discussed in [39]. 430 Y. Hu and L. Ping 1000 0000 00 (a) 10 1100 0100 1010 0010 1001 0001 00 + 11 10 01 1101 0101 1110 0110 1011 0011 01 11 1111 0111 (b) + + Fig. 13.9 a A 16-ary SCM signaling by superimposing two streams of QPSK constellations. b A 64-ary SCM signaling by superimposing three streams of QPSK constellations 13.3 Power Control for IDMA At a relatively low sum-rate, such as less than 1 in a complex channel, IDMA with a GA receiver works well with equal received power. At a higher sum-rate, unequal power control is required. The situation is similar to power control for SIC in (13.2), except that iterative detection makes the problem more complicated. 13.3.1 Transmitted and Received Power Minimization Denoting the received powers by qk = |hk |2 ek . (13.18) We combine (13.16a) and (13.16b) into a compact form as SNR(t) k = K k ′ =1,k ′ =k qk + σ2 qk ′ ψ SNR(t−1) k′ , ∀k. (13.19)   Let ek and qk be, respectively, sum transmitted power and sum received power. We can minimize them, respectively. The latter is simpler since the channel gains {|hk |2 } are not involved. The following Remark establishes a connection between these two problems [7, 9, 43]. 13 Interleave Division Multiple Access (IDMA) 431  ∗ Remark 1 Assume for qk . Then, {ek∗ = qk∗ /|hk |2 } is a  that {qk } is a minimizer ∗ 2 minimizer for ek provided that {qk } and {|hk | } have the same order, i.e., qk∗ ≤ qk∗′ if |hk |2 ≤ |hk ′ |2 .  Based on Remark 1, we can first find the minimizer {qk∗ } for qk . We then re∗ 2 ek label {qk } such that it has the same order as {|hk | }. Then, the minimizer for can be obtained as {ek∗ = qk∗ /|hk |2 }. Incidentally, Remark 1 implies that a user with a higher channel gain should be assigned a higher transmitted power and vice versa. Next, we focus on minimizing {qk }. 13.3.2 Feasible Profile We now impose an SNR requirement Γ after T iterations. This SNR requirement can be equivalently translated into a BER requirement through (13.17). We write the received power optimization problem as follows. minimize  (13.20a) qk , subject to SNR(t) k = K k ′ =1,k ′ =k SNRk(T ) ≥ Γ, ∀k. qk + σ2 qk ′ ψ SNR(t−1) k′ , ∀k, (13.20b) (13.20c) The problem in (13.20) is non-convex. We will outline two searching techniques for this problem. For convenience, we will call {qk } a feasible profile if it ensures the constraints in (13.20b) and (13.20c). Incidentally, it is interesting to compare (13.2a) and (13.20b). In (13.20b), represents the residual interference from user k ′ after soft canqk ′ ψ SNR(t−1) k′ celation. Such terms disappear in (13.2a) for decoded users due to the error-free assumption and hard cancelation. 13.3.3 Greedy Search We first set T = 1, i.e., only one iteration. Assume that approximate error-free decoding can be achieved at a sufficiently large SNR in the single-user case. We can construct an initial feasible profile Q = {qk } according to (13.2). The sum-power for such a Q is typically large. We next consider a general T . Starting from the above initial Q, we search for a minimum value for each qk individually to achieve (13.20c), while keeping other elements in Q unchanged. This involves a one-dimensional search, so its complexity 432 Y. Hu and L. Ping is affordable. Let the search result be qk∗ . We then update qk ← qk − ǫ(qk − qk∗ ) in Q, where ǫ is a damping factor (e.g., ǫ = 0.5). We repeat the above process for all k iteratively. We observed reasonably good performance of this simple method for a relatively small K. 13.3.4 Approximate Linear Programming Method Inspired by the linear program technique for LDPC code design [25], we can use the approximate technique below for a large K. The key idea is to transform the problem of finding the power for each user into finding the number of users on different given power levels, which makes the problem convex [44, 45]. Let us quantize the received power into M + 1 discrete values: {q(m), m = 0, 1, . . . , M } with q(m − 1) < q(m). The received powers of all users are selected from {q(m)}. We partition K users into M + 1 groups according to their power levels. Let λ(m) be the number of users assigned with power level q(m) and z(m) be the total power of these λ(m) users. As such,  λ(m) = K,  λ(m)q(m) = m (13.21a) z(m) = λ(m)q(m) (13.21b) and the sum received power  k qk = m  z(m). (13.21c) m Denote by SNR(m) the SNR for the users in the mth group with power q(m). Define I=  m z(m)ψ(SNR(m)) + σ 2 , (13.22) which is the total interference power (including noise) after soft cancelation. When K is large, (13.20b) can be approximated as SNR(m)(t) = I (t) q(m) q(m)  ≈ (t) , (t−1) I − q(m)ψ SNR(m) (13.23) where I (t) denotes the value of I at the tth iteration. Using (13.22) and (13.23), we have the update rule    q(m) I (t) = (13.24) z(m)ψ (t−1) + σ 2 . I m 13 Interleave Division Multiple Access (IDMA) Fig. 13.10 IDMA with EPC and UPC in AWGN channels. Rate-1/3 turbo coding followed by rate-1/2 repetition coding is used for each user. Information length of each user is 1200. QPSK modulation. Sum-rate = 1, 2, and 4 for K = 3, 6, and 12, respectively 433 0 10 K=12 EPC UPC -1 K=6 10 -4 10 -5 10 -2 0 Capacity for K = 6 10 Capacity for K = 12 K=3 -3 Capacity for K = 3 BER -2 10 2 4 6 8 10 K=12 12 14 16 18 SNR sum (dB) Equation (13.24) characterizes the evolution of the total interference at each iteration. If iterative detection converges, I (t) should be lower than I (t−1) . Equivalently, we can write the convergence condition as  z(m)ψ m  q(m) I  + σ 2 ≤ (1 − δ)I , Imin ≤ I ≤ Imax . (13.25) where 0 < δ < 1 is a decay factor that controls the convergence speed. Imax and Imin specify the total interference at the beginning and end of the iterative detection. In summary, we re-formulate the optimization problem as, minimize subject to   m (13.26a) z(m), z(m)ψ  z(m) ≥ 0, ∀m. q(m) I  + σ 2 ≤ (1 − δ)I , Imin ≤ I ≤ Imax , (13.26b) (13.26c) The above optimization problem is linear with respect to {z(m)}. Hence, it can be resolved by linear programming. More details can be found in [44, 45]. Figure 13.10 is an example to illustrate the necessity of unequal power control (UPC). We can see that UPC improves the system performance for all the three cases of K = 3, 6, and 12 compared with equal power control (EPC). Particularly, when K = 12, the IDMA system does not work at all with EPC, but works well with UPC. The gap between IDMA with UPC and capacity is about 3.7 dB for K = 12. Compared with the 12-user example in Fig. 13.3, we can see that IDMA recovers about 3.7 dB relative to the loss incurred by SIC. This is impressive, but there is definitely room for improvement. The gap towards capacity can potentially be further narrowed using more sophisticated techniques, such as curve-matching-based degree 434 Y. Hu and L. Ping sequence optimization [46, 47], spatial coupling [48–57], or more sophisticated modulations [58]. A criterion called overloading, which represents the user capacity in a NOMA system, is used to assess the system performance in the recent literature. Figure 13.10 demonstrates that IDMA can offer very high overloading with centralized power control. In the next section, we will discuss IDMA techniques to achieve high throughput without centralized power control. 13.4 Random Access via IDMA 13.4.1 Limitations of Conventional Systems A conventional uplink system with centralized control involves a connection setup procedure before data transmission. The overhead incurred by this procedure is not serious for services with long-lasting connections since it can be amortized across the connection duration. One of the main tasks envisaged for the next 5G cellular systems is to support machine-type communication (MTC) which is characterized by short and sporadic data communication. In this case, the cost of establishing centralized control can be substantial. Random access is decentralized, in which each user makes an individual decision to transmit data packets. This avoids the overhead of connection setup, but the packets from different users may collide. In a conventional random-access scheme, such as ALOHA [59], colliding packets are discarded, which reduces throughput. For this reason, such techniques cannot satisfy the demand of high spectral efficiency in 5G cellular systems. In a fading channel, the received powers of different users may form a feasible profile defined in Sect. 13.3.2, even without centralized control. This is captured in the multi-packet reception (MPR) model [60, 61]. Most existing MPR techniques rely on channel fading to form feasible profiles. Such a passive approach achieves limited throughput gain. In what follows, we will discuss an active approach based on the power control technique. The main idea is to optimize the probability of forming a feasible profile through decentralized power control at the transmitters. 13.4.2 Random IDMA with Decentralized Power Control 13.4.2.1 Problem Formulation Recall from Sect. 13.3.2, a feasible received power profile {qk } can be formed by centralized control. Without centralized control, however, it is difficult to guarantee 13 Interleave Division Multiple Access (IDMA) 435 this. A randomized power control (RPC) technique [62–67] is discussed below to handle this difficulty. The principles of RPC are as follows. Let {Q(l) , l = 1, 2, . . . , L} (L is the maximum level index) be a set of pre-defined power levels and {P (l) , l = 1, 2, . . . , L} a set of related probabilities. Upon a packet arrival, each user randomly draws a power level Q(l) with probability P (l) and uses it to transmit. Different users act individually and so their transmissions may collide. However, as long as their received powers {qk } form a feasible profile, their signals can still be recovered. Our aim is to optimize the probability that {qk } form a feasible profile. Mathematically, {qk } defined above are the realizations of an underlying random variable. The random variable is characterized by a probability mass distribution {P (l) }. Each qk is independently drawn by a user from the support {Q(l) }. Once the distribution is given, there is no need for centralized control. 13.4.2.2 Type-2 Collisions The problem formulated above turns out to be difficult when K is large. So far, we have no general solution. We will discuss a sub-optimal technique below. We say that a collision is of type-M if it involves M active users. Let Q(0) = 0 be an element in {Q(l) }. In RPC, a user will not transmit if its selected power is Q(0) . The related P (0) is equivalent to the back-off probability in 802.11 Wi-fi systems. Intuitively, collisions are dominated by type-2 ones when P (0) is sufficiently large. Therefore, we will focus on type-2 collisions. We consider a type-2 collision involving user i and user j. We define the union of all possible feasible profiles of the received power pair {qi , qj } as a feasible region. The collision is resolvable if {qi , qj } falls in this region. Figure 13.11a shows an example of the feasible region for SIC with ideal coding and decoding. The area marked by “A” in Fig. 13.11a is formed by all possible {qi , qj } that meets the following conditions (see (13.2)): qj ≥ Γ, qi + σ 2 qi SNRi = 2 ≥ Γ. σ SNRj = (13.27a) (13.27b) The area marked by “B” in Fig. 13.11a is formed similarly by changing decoding order. The value of Γ here is determined by the Shannon capacity R = log2 (1 + Γ ). Any power pair in the feasible region is resolvable by SIC.2 Figure 13.11b is an example of an IDMA system involving two LDPC-coded users with coding rate 0.5 per user. The receiver can achieve BER ≤ 10−5 in the feasible region, which is regarded as approximately error-free. The border of this feasible region is obtained using simulation. 2 Figure 13.11a is for R < 1. If R ≥ 1, the feasible region is divided into two disjoint sub-regions A and B symmetric to the 45◦ line qi = qj [63]. 436 Y. Hu and L. Ping (a) (b) qj Q(2) (Q(3), Q(4)) (Q(2), Q(3)) qj ... A B ... ... Q (3) B (Q(4), Q(3)) (Q(3), Q(2)) (Q(1), Q(2)) Q (4) Q(2) qj = ϕ (qi) (Q(3), Q(4)) qi = Q(1) Q(4) Q(3) qj = ϕ (qi) qi = Q(1) ... A 0 (Q(2), Q(1)) (Q(4), Q(3)) (Q(1), Q(2)) Q (1) Q (2) Q (3)Q (4) Q(1) qj = Q(1) ... qi 0 ... (Q(2), Q(3)) (Q(3), Q(2)) qi = ϕ (qj) qi = ϕ (qj) Q(1) ... qj = Q(1) (Q(2), Q(1)) Q (1) Q (2) Q (3) Q (4) ... qi Fig. 13.11 Feasible regions for a a two-user ideally coded system with SIC, and b a two-user LDPC-coded IDMA system Each of the two feasible regions in Fig. 13.11 is bounded by four curves qi = Q(1) , qj = φ(qi ), qj = Q(1) and qi = φ(qj ). Here, the function φ(·) is determined by (13.27a) taking mark of equality (for Fig. 13.11a) or by simulation (for Fig. 13.11b). Q(1) is the minimum power for successful single-user transmission. We construct the set {Q(l) } as follows: ⎧ 0, l = 0, ⎪ ⎨ (1) (l) l = 1, (13.28) Q = Q , ⎪ ⎩ (l−1) φ(Q ), l > 1. For {qk } randomly selected from {Q(l) }, we have the following situations: Case 1: All {qk } are zeros. In this case, throughput is zero. Case 2: Only one element in {qk } is nonzero. The transmission of the only active user will be successful. Case 3: Exactly two elements in {qk }, say qi and qj , are nonzero. This is a type-2 collision. It can be shown that {qi , qj } falls in the feasible region (so collision is resolvable) provided that qi = qj . Case 4: More than two elements in {qk } are nonzero. For simplicity, such events are regarded as unresolvable, which is a pessimistic assumption. Based on the above cases, we can find an optimized probability set {P (l) } for a K user system. For convenience, we assume that the packets of all users arrive independently, following the Bernoulli process with parameter λ. The system throughput is then given by (13.29) T = T1 + T2 . In (13.29), T1 is the throughput related to transmissions without collision: 13 Interleave Division Multiple Access (IDMA) T1 = K  k=1 437 CKk λk (1 − λ)(K−k) Ck1 (1 − P (0) )(P (0) )k−1 (13.30) =Kλ(1 − P (0) )(1 − λ + λP (0) )K−1 , where CKk λk (1 − λ)(K−k) is the probability of k users among total K users having packets to transmit and Ck1 (1 − P (0) )(P (0) )k−1 the probability of only one user among these k users transmitting with nonzero power. Also in (13.29), T2 is the throughout related to type-2 collisions. From case 3 above, a type-2 collision is unresolvable only if the two transmitting users are using the same received powers. Therefore, we have   K   (0) 2 (l) 2 k k (K−k) 2 (P ) (P (0) )k−2 CK λ (1 − λ) Ck (1 − P ) − T2 =2 l>0 k=2 2 (0) K−2 =K(K − 1)λ (1 − λ + λP ) where (1 − P (0) )2 −   (0) 2 (1 − P ) −  l>0 (l) 2 (P )  (13.31) , (P (l) )2 is the probability that two active users transmit with l>0 different received powers. We further consider an average transmitted power constraint q̄ for each user. In AWGN channels with unit channel power gain, the constraint is given by  l≥0 Q(l) P (l) ≤ q̄. (13.32) Under such power constraint, we can search for {P (l) } that maximize the throughput T in (13.29). It can be verified that the problem is convex if P (0) is fixed. The treatments for fading channels are somewhat more complicated. The details can be found in [63]. Figure 13.12 shows two examples, one for an ideally coded system and the other for an LDPC-coded IDMA system [63]. Conventional ALOHA is included as a reference. We can see that the RPC-based scheme can offer noticeably throughput gain compared with ALOHA. It is proved in [63] that the {Q(l) } in (13.28) forms an optimal support for the decentralized power control when K = 2. It is sub-optimal for K > 2, but it can still provide excellent performance gain, as seen in Fig. 13.12. As a short summary, we can treat collisions as NOMA cases. A conventional scheme, such as ALOHA, treats such NOMA cases as failures. The discussions in this section aim at optimizing the probability of successful detection in such NOMA cases. 438 Y. Hu and L. Ping (b) 1.5 1 RPC 0.5 ALOHA 0 2 4 6 8 system throughput system throughput (a) 1.5 RPC 1 0.5 ALOHA 02 10 12 14 16 18 20 number of active users K 4 6 8 10 12 14 16 18 20 number of active users K Fig. 13.12 Performance comparison of RPC and ALOHA in a an ideally coded system and b an LDPC-coded IDMA system. Rayleigh fading channel with averaged power gain 1 for all users. Same power constraint for RPC and ALOHA 13.5 IDMA in MIMO Systems 13.5.1 Multi-User Gain in MIMO Systems Fig. 13.13 Achievable sum-rate of ZF under perfect CSI. The number of antennas at the base station is 64. Single antenna assumed is for each user. Other system parameters are the same as those in Fig. 13.1 average sum rate (bits/channel use) MIMO is a wireless technology employing multiple transmit and receive antennas [68–74]. Multi-user gain refers to the advantage of allowing a large number of users to transmit simultaneously over the same time and same frequency in MIMO [10, 75]. This is illustrated in Fig. 13.13 by the potential sum-rate capacity gain for a singlecell system. Perfect CSI is assumed in Fig. 13.13. The curves apply to both up- and downlinks following the duality principle [3, 76, 77]. We can see that multi-user gain is very attractive. The potential gain is in the order of tens of times. Diversifying power over more users, i.e., increasing K, is a very effective way to increase sum-rate. 60 50 Capacity 40 OMA via ZF 30 20 10 0 0 8 16 24 32 40 number of users K 48 56 64 13 Interleave Division Multiple Access (IDMA) 439 Figure 13.13 also includes the performance of zero-forcing (ZF) with proper power allocation. ZF is an OMA technique [75]. Different users are divided into different orthogonal subspaces in ZF, which avoids interference among users. It is seen from Fig. 13.13 that, with accurate CSI, ZF can offer very good multi-user gain. Since the gap between ZF and capacity is small, any further gain by NOMA is limited. In this case, OMA via ZF can be preferred for its low-cost SUD receiver. However, in practice, we usually do not have reliable CSI to establish spatial orthogonality initially. ZF performance deteriorates seriously when CSI is not accurate. In the following, we will see that NOMA via IDMA offers a solution to the problem. 13.5.2 Iterative Maximum Ratio Combining (I-MRC) We first extend the GA-based detection technique in Sect. 13.2.2 to MIMO. Consider a multi-user uplink system model with NBS antennas at the base station. For simplicity, we assume a single antenna at each user. The IDMA principle discussed in Sect. 13.2.1 can be directly used here. We assume perfect CSI first and will return to the CSI estimation problem later. The received signal at time j is written as y(j) = K  k=1 √ hk ek xk (j) + η(j), (13.33) where y(j) is an NBS × 1 signal vector received at base station antennas, hk an NBS × 1 complex channel coefficient vector, ek the transmitted power of user k, xk (j) a symbol transmitted from user k, and η(j) an NBS × 1 vector of complex AWGN with mean 0 and variance σ 2 = N0 /2 per dimension. Maximum ratio combining (MRC) is a common strategy for MIMO systems. An MRC estimator is defined in a symbol-by-symbol form as x̂k (j) = hH k y(j). (13.34) x̂k (j) = λk xk (j) + ξk (j), (13.35a) Substituting (13.33) into (13.34), √ √ where λk ≡ hk 2 ek = hH k hk ek is a scalar and ξk (j) ≡ K  k ′ =1,k ′ =k √ H hH k hk ′ ek ′ xk ′ (j) + hk η(j) (13.35b) 440 Y. Hu and L. Ping is an interference (from {xk ′ (j), k ′ = k} to xk (j)) plus noise term. MRC does not involve matrix inversion and so has low cost. However, interference is a problem for MRC, especially when K is large. Iterative GA technique can alleviate this problem. Similar as that in Sect. 13.2.2, we approximate ξk (j) in (13.35b) by a Gaussian random variable. We assume that the real and imaginary parts of xk (j) carry two bits of information in the QPSK modulation. Similar to (13.12), the real part of xk (j) can be estimated as LLRESE (Re(xk (j))) =   2λk   Re x̂k −E ξk (j) . Var Re ξk (j) (13.36) The mean and variance in (13.36) are updated as  E ξk (j) = hk H   Var Re ξk (j) = K  k ′ =1  K  k ′ =1 hk ′ √  √ ek ′ E(xk ′ (j)) − hk ek E(xk (j)) ,   √ Var Re hH k hk ′ ek ′ xk ′ (j) (13.37a)   √ − Var Re hk 2 ek xk (j) + hk 2 σ 2 . (13.37b) Some detailed computation techniques for (13.37) can be found in [75]. The imaginary part of xk (j) can be estimated similarly. We call the above process iterative MRC (I-MRC). Figure 13.14 illustrates the effectiveness of I-MRC [75]. We consider three different settings: (i) K = 1 and sum-rate Rsum = 5 with five signal streams (each stream with rate 1) assigned to the sole user using the SCM principle discussed in Sect. 13.2.5, (ii) K = 8 and Rsum = 16 with two streams per user, and (iii) K = 8 and Rsum = 24 with three streams per user. For K = 1, all the signal streams see the same channel so there is no spatial diversity among them, which results in poor performance. Increasing K from 1 to 8 results in drastically enhanced rate or reduced power or both in Fig. 13.14. Figure 13.14 is a compelling evidence for multi-user gain: allowing more concurrent transmitting users is more efficient than increasing single-user rate. 13.5.3 Data-Aided Channel Estimation (DACE) We now consider the CSI acquisition problem. Many factors may affect CSI accuracy in MIMO. In particular, the correlation among the pilots used by different users can lead to the pilot contamination problem [78]. 13 Interleave Division Multiple Access (IDMA) 441 0 10 -1 10 BER -2 10 -3 10 8 users, Rsum = 16 Single user, Rsum = 5 8 users, Rsum = 24 -4 10 -5 10 -4 -3 -2 -1 0 1 2 3 4 SNRsum (dB) Fig. 13.14 Multi-user gain for K = 8 with I-MRC. Rayleigh fading. NBS = 64. Equal transmitted power is assumed for different users. Power control is used for the streams assigned to the same user. The power allocation levels are obtained through heuristic search. Rate-1/2 turbo coding and information length = 1200 for each stream. QPSK modulation. A codeword is transmitted over ten resource blocks. Each resource block contains 180 symbols experiencing the same fading conditions IDMA with data-aided channel estimation (DACE) [17, 79–83] technique can be used to improve CSI accuracy. The basic principle of DACE is as follows. Recall that a key difference between pilot and data is that the former is known at the receiver, while the latter is not. Therefore, if a data symbol is known, it can be used as a pilot. Furthermore, partial information of a data symbol, such as its mathematical mean, can also be used to refine the channel estimates. Such partial information is readily available in an IDMA receiver (as given in (13.8)). DACE can be used jointly with I-MRC, which involves iterations of the following two operations [75]: (a) using both pilots and partially decoded data information to refine CSI, and (b) using improved channel estimates to refine data estimation by I-MRC. The advantages of DACE are twofolds: (i) With DACE, the estimated data is gradually used for channel estimation. Pilot energy can be greatly reduced since only very coarse CSI is required initially. (ii) Data sequences are typically much longer than pilots and correlation is low among them. Therefore, DACE is robust against the pilot contamination problem. Such problem is typically caused by the correlation among the pilot sequences re-used in neighboring cells. Without DACE, longer pilot sequences will be required to reduce such correlation. Thus, DACE also reduces the time overhead related to pilots. I-MRC and DACE can be naturally combined in an overall iterative process. After MRC and decoding operations in each iteration, partially detected data are used to refine channel estimates that are in turn used for MRC and decoding in the next iteration. This is referred to as I-MRC-DACE. Figure 13.15 compares BER 442 Y. Hu and L. Ping (a) 100 (b) 10 0 -1 -1 ZF 10 10 -2 BER BER -2 10 β = 0.9 -3 10 β = 0.6 β=0 -4 10 -4 I-MRC-DACE -3 10 β = 0 β = 0.6 β = 0.8 β = 0.8 -4 β = 0.9 10 -5 10 10 -5 -2 0 2 4 SNRsum (dB) 6 8 10 10 -4 -2 0 2 4 6 8 10 SNRsum (dB) Fig. 13.15 Performance comparison of a ZF and b I-MRC-DACE with different β values. Rayleigh fading. NBS = 64 and K = 16. Rate-1/3 turbo coding and information length = 1312 for each user. QPSK modulation. Each codeword is divided into 12 sections, and each section is transmitted over a resource block (including 16 pilot symbols). Different users in a cell are assigned different orthogonal pilots. These pilots are repeated for users in different cells. The pilot and data symbols have the same average power performance for ZF and I-MRC-DACE, in which β is the cross-cell interference factor defined in (13.1). A larger β indicates a more serious pilot contamination problem due to more severe interference among the pilots. From Fig. 13.15, we can see that I-MRC-DACE noticeably outperforms ZF. The difference becomes very significant when β is large (e.g., β ≥ 0.6). IDMA is a natural choice for I-MRC-DACE since it is beneficial for iterative detection. Note that Fig. 13.5 can also be used to characterize IDMA in MIMO systems, if each scalar y(j) in Fig. 13.5 is replaced by its vector counterpart y(j) in (13.33). The discussions on short cycles in Sect. 13.2.1 are still applicable after such replacement. IDMA also allows a superimposed pilot scheme that can reduce the power overhead and rate loss. The related discussions can be found in [83–86]. 13.6 Prospective Applications of IDMA in 5G Systems Various approaches have been proposed recently for 5G radio link under LTE, including IDMA [87], RSMA [88], IGMA [89], PDMA [90] and SCMA [91, 92]. In the following, we will show that these schemes all share, explicitly or implicitly, the basic principle of IDMA. We first represent these different schemes using a unified protograph framework. Assume that N resource blocks (RBs) defined in LTE are available for transmission. We label the observations from these RBs by {y(1) , y(2) , . . . , y(N ) }. Figure 13.16 shows a scheme in which each user transmits on all available RBs as illustrated for two system settings: (a) three users over two RBs, and (b) six users 13 Interleave Division Multiple Access (IDMA) 443 (b) (a) | RB1 | RB2 y(1) | | RB2 | RB3 | RB4 y(3) (2) + + + | y(2) (1) (2) (1) RB1 y(1) y(2) | y(4) (3) (4) + + + x1 x2 x3 x1 x2 x3 x4 x5 x6 c1 c2 c3 c1 c2 c3 c4 c5 c6 Fig. 13.16 Protograph representations of DS-CDMA and IDMA with a three users over two RBs, and b six users over four RBs over four RBs. This can be realized by transmitting replicas of each xk over multiple RBs. Alternatively, we may use a low-rate code to generate each xk . Each xk can be segmented into several blocks, with each block transmitted over an RB. The latter approach can potentially provide higher coding gain [93]. We may also use different modulations for the bits on different RBs, as for SCMA [92]. Incidentally, both DS-CDMA and IDMA can be represented using the protographs in Fig. 13.16. They are distinguished by the absence or presence of user-specific interleaving within each RB. RSMA [88] is a DS-CDMA scheme. However, userspecific interleaving is stated as an option for RSMA. If this option is used, it is equivalent to IDMA. The advantage of this option can be seen in Fig. 13.7. Alternatively, each user can transmit over only some of the available RBs. This is referred to as sparse coding in [91]. Figure 13.17 shows an example for sparse coding. IGMA, PDMA, and SCMA all involve such treatment. Note that symbol-level interleaving as in Fig. 13.5a is not explicitly seen in Figs. 13.16 and 13.17. If such underlying interleaving is not used, size-4 cycles can be a problem in Fig. 13.16. Sparse coding in Fig. 13.17 avoids this problem. Clearly, sparse coding leads to user-specific edge connections between users and RBs. It has the same effect as symbol-level interleaving; they both reduce short cycles. With sparse coding, each user does not fully occupy all RBs. This may cause problem for decentralized grant-free [13] or random-access applications, where each user determines its activity individually. In these cases, the number of active users, denoted by Kactive , is a random variable. When Kactive is small, sparse coding may lead to inefficient use of the available RBs and so low power efficiency. This implies poor scalability of user numbers. On the other hand, an IDMA system in Fig. 13.16 based on symbol-level interleaving does not have this problem, since all available RBs are fully used for any value of Kactive . 444 Y. Hu and L. Ping Fig. 13.17 Protograph representation of sparse coding | RB1 | y(1) RB2 | y(2) (1) RB3 (2) RB4 | y(4) (3) (4) + + | y(3) + + x1 x2 x3 x4 x5 x6 c1 c2 c3 c4 c5 c6 Fig. 13.18 IDMA with GA versus SCMA with ML. Ten iterations for both schemes 0 10 BLER SCMA with ML IDMA with GA -1 10 -2 10 -4 -2 0 2 4 6 8 single-user SNR (dB) Also, multi-user gain in MIMO is determined by the number of users concurrently transmitting in each RB. Therefore, sparse coding may not be an efficient option in MIMO (especially in massive MIMO). Figure 13.18 compares IDMA and SCMA in quasi-static Rayleigh fading channels. The channels remain unchanged within each transmission. Both schemes are with six users, two receiver antennas, sum-rate = 3 and equal transmitted power per user. A rate-1/2 LTE turbo code is used followed the following transmitter structures: • IDMA is with a length-2 spreading and QPSK modulation. • SCMA is based on Fig. 13.17 with the 16-point modulation in [92]. We can see from Fig. 13.18 that the two schemes have similar performance. However, SCMA in Fig. 13.18 is based on ML, while IDMA based on GA. The latter has much lower complexity. 13 Interleave Division Multiple Access (IDMA) 445 13.7 Summary We have shown that the real attractiveness of NOMA is in systems without centralized control or without accurate CSI. It is difficult or too costly to establish orthogonality in such channels, so we have to resort to NOMA. Iterative processing holds the key; interference can be gradually resolved and CSI can be gradually refined through iterative processing. IDMA is a simple implementation technique for NOMA. The features of IMDA can be seen from its sparse graphic representation. The interleaved edge connections in IDMA facilitate iterative processing at the receiver. We have demonstrated that IDMA can offer significant performance gain in random access and MIMO systems. IDMA also offers lower detection complexity compare with other alternatives. References 1. R. Ahlswede, Multi-way communication channels, in Second International Symposium on Information Theory (1971), pp. 103–135 2. 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This chapter provides a whole picture of PDMA, including the origination and principle, pattern design, receiver algorithms, performance evaluation, extension design, applications, challenges and trends. 14.1 Origination and Principles of PDMA As described above, PDMA is originated from SAMA which takes successive interference cancellation (SIC) detection in the receiver. To solve the error propagation problem of SIC, transmitter and receiver joint design is a good solution [8]. Therefore, in this section, based on the explanation of error propagation problem in SIC S. Chen (B) · S. Sun · S. Kang · B. Ren China Academy of Telecommunication Technologies, No. 40, Xueyuan Road, Haidian District, Beijing, China e-mail: chensz@datanggroup.cn S. Sun e-mail: sunshaohui@catt.cn S. Kang e-mail: kangshaoli@catt.cn B. Ren e-mail: renbin@catt.cn © Springer International Publishing AG, part of Springer Nature 2019 M. Vaezi et al. (eds.), Multiple Access Techniques for 5G Wireless Networks and Beyond, https://doi.org/10.1007/978-3-319-92090-0_14 451 452 S. Chen et al. and the idea of transmitter receiver joint design, PDMA definition and framework are described, and also PDMA transmitting and receiving schemes are explained. 14.1.1 Error Propagation Problem in SIC According to theoretical results of multi-user channel [9], superposition coding at a transmitter and SIC at a receiver, are able to achieve capacity boundary of multiple access channels (MAC) or degraded broadcast channels (BC) when transmitter and receiver are working together. From theoretical perspective it is rational to use SIC to achieve channel capacity, since the packet error rate tends to be zero with the increased code length as long as a user’s transmission rate is below the channel capacity. However, in a real system, detection error is inevitable due to various nonideal conditions, such as, limited code length, channel fading, and glitches, etc. For SIC receiver, if a former user’s packet is detected erroneously, it is very unlikely that the following user’s packet could be detected correctly. This is the so-called error propagation problem. Since multiple users are detected one by one in serial order, the detection order of all users is usually arranged according to their signal strength. That is, the signal of the first detected user is the strongest, the signal of the second detected user is weaker, and so on. For the first detected user, it is recovered directly from the original receiving. While for the following detected users, they are recovered respectively from related cancellation receiving which should cancel those former detected users from the original receiving by user reconstruction. If a user is not correctly detected, its reconstruction is impossible to be correct. In addition, the accuracy of reconstruction also impacts on the performance of following users. For example, based on distorted channel estimation, the reconstructed signal will also be distorted. Even though the user’s packet is detected correctly, it still has an adverse effect on the following users’ detection. Error propagation is a crushing blow for multi-user detection and it will deteriorate the performance of SIC-based multi-user system. In general, two approaches can be considered to alleviate the error propagation problem. The first is to enhance the reliability of those early-decoded users, the second is to adopt more advanced and complex detection algorithm than SIC. These approaches relate to joint design between transmitter and receiver, which is the origination of PDMA technology. 14.1.2 Transmitter and Receiver Joint Design One approach to alleviate error propagation problem is to enhance the reliability of those early-decoded users, either by selecting users with good channel condition or by designing transmission parameters such that the early-decoded users have higher reliability and better channel condition. 14 Pattern Division Multiple Access (PDMA) 453 Analytical results of multiple-input and multiple-output (MIMO) detection from [10, 11] show that the ith detected layers of SIC receiver could achieve diversity order (14.1) Ndiv (i) = N R − N T + i where N R is the receiving antenna number, N T is number of data layers. The diversity order increases with the detection order. PDMA design is inspired by above result [1–8]. Multi-user channel can be viewed as a virtual MIMO channel and the above result could be generalized to multi-user non-orthogonal transmission. For non-orthogonal transmission employing SIC receiver, diversity order of each user varies with the order of detection. The first detected user has the lowest diversity order, and the last detected user has the highest diversity order. In a fading channel, diversity order affects transmission reliability significantly. Increasing the diversity order typically leads to more reliable transmission. With SIC receiver, the first detected user actually determines the overall detection performance, but unfortunately its diversity order is the lowest. To optimize system performance, it is desirable to have identical pro-detection diversity order for each user. Diversity could be obtained from transmission or reception, or from both. Assuming that transmission diversity order of the ith detected user is DT (i), the diversity order after SIC receiver can be expressed as Ndiv (i) = DT (i) − K + i (14.2) where K is the number of users. By joint design from transmitter and receiver, PDMA deliberately selects DT (i) so that the diversity order after SIC receiver is as close as possible. The definition of transmission diversity means that multiple copies of a signal are transmitted from independent resources to avoid transmission error due to deep fading on one resource. The resources could be time, frequency, code, spatial or power resource. PDMA maps transmitted data onto a group of resources according to PDMA pattern to realize disparate transmission diversity order. A PDMA pattern defines the mapping from transmitted data to a resource group. A resource group can consist of time resource, frequency resource, code resource, spatial resource, power resource or any combination of these resources. The number of mapped resources in a group determines the order of transmission diversity. Data of multiple users can be multiplexed onto the same resource group with different PDMA patterns. In this way, non-orthogonal transmission is realized. By assigning PDMA pattern with different diversity order, disparate transmission diversity orders among users could be achieved. Another approach to alleviate error propagation problem is to adopt more advanced and complex detection algorithm such as maximum likelihood (ML) or maximum a posterior (MAP). It is anticipated that PDMA with advanced detection algorithm can alleviate error propagation effect to a substantial degree. However, ML or MAP algorithm incur tremendous detection complexity and it is hard to implement. 454 S. Chen et al. Fig. 14.1 PDMA pattern for 6 users on 4 REs Fortunately, the detection complexity could be reduced significantly by making the PDMA pattern sparse. That is, data are only mapped to a small part of the resources in the resource group. This draws on the idea of sparse coding in low density parity check (LDPC) coding. Sparsity makes it possible to use low complexity belief propagation (BP) algorithm to approach the MAP detection. In addition, the convergence of BP algorithm could be speeded up by disparate transmission diversity of PDMA. In summary, PDMA uses PDMA pattern to define sparse mapping from data to a group of resources. PDMA pattern could be represented by a binary vector. The dimension of the vector equals to number of resources in a group. Each element in the vector corresponds to a resource in a resource group. A “1” means that data shall be mapped to the corresponding resource. Actually, number of “1” in the PDMA pattern is defined as its transmission diversity order. Figure 14.1 shows an example of resource mapping according to PDMA pattern. Six users are multiplexed on four resource elements (REs). A PDMA pattern is assigned to a user. User1’s data are mapped to all four REs in the group, and user2’s data are mapped to the first three REs, etc. The order of transmission diversity of the six users is 4, 3, 2, 2, 1, and 1 respectively. 14.1.3 PDMA Definition and Framework PDMA is proposed as a novel NOMA scheme based on code pattern. Joint optimization of transmitting and receiving is considered with SIC amenable pattern design at the transmitter side and SIC-based detection at the receiver side. PDMA pattern is designed to offer different orders of transmission diversity, so that disparate diversity order between multiple users could be introduced to alleviate the error propagation problem of SIC receiver. PDMA pattern is also required to be sparse to facilitate advanced detection algorithm such as BP. Iterations between BP and channel decoding could further boost system performance. PDMA pattern can be also extended to include power scaling and phase shifting to harvest additional constellation shaping gain. PDMA can design pattern for a specific user in time, frequency and space resources. Figure 14.2 shows the technical framework of the PDMA uplink application, and Fig. 14.3 shows that of the PDMA downlink application. As shown 14 Pattern Division Multiple Access (PDMA) 455 Fig. 14.2 The technical framework of the PDMA uplink application Fig. 14.3 The technical framework of the PDMA downlink application in Figs. 14.2 and 14.3, the PDMA technical framework includes two parts: the transmitter and the receiver, which reflects that the PDMA technology considers the joint design of the transmitter and the receiver based on the optimization point of view for multi-user communication system. At transmitter side, users are distinguished by the non-orthogonal characteristic pattern based on the multiple signals domain (including time, frequency, code, power and the space domain, etc.). At the receiver side, general SIC type sub-optimal multiuser detection algorithms can be realized based on the features of the user pattern. 456 S. Chen et al. 14.1.4 PDMA Transmitting and Receiving Scheme With above framework of PDMA, taking orthogonal frequency division multiplexing (OFDM) waveform as a baseline, the transmitting and receiving schemes of a PDMA based system are further explained. Figure 14.4 shows the uplink process of the PDMA based system. At the transmitting end, the system completes transmitting signal processing by multiple user data forward error correction channel coding, PDMA code modulation, PDMA subcarrier resource mapping and OFDM modulation. At the receiving end, the base station performs the opposite process, namely the system gets the transmitting data of each terminal through the OFDM demodulation and general SIC type detection like belief propagation iterative detection and decoding (BP-IDD). In the process of the PDMA modulation and coding, the symbol level mapping and spread spectrum in the frequency domain are achieved at the same time. The receiver adopts the BP-IDD algorithm which is essentially a joint iterative processing of the Turbo decoder and BP detecting. Figure 14.5 shows the downlink process of the PDMA based system. At the transmitting end of the downlink PDMA system, the base station performs data forward error correction channel coding for multiple user like PDMA code modulation, PDMA subcarrier resource mapping and OFDM modulation. At the receiving end of the downlink PDMA system, each user performs the opposite process, including Fig. 14.4 Illustration of the transmitting and receiving scheme of the PDMA based uplink system 14 Pattern Division Multiple Access (PDMA) 457 Fig. 14.5 Illustration of the transmitting and receiving schemes of the PDMA based downlink system OFDM demodulation and general SIC type detection like BP-IDD. In this process, the downlink PDMA coding is conducted on modulation symbol level and finishes mapping on symbol level and realizes spread spectrum in frequency domain. The receiver adopts the BP-IDD algorithm which is essentially the Turbo decoding and a joint iterative processing of BP detecting. 14.2 Pattern Design of PDMA The PDMA pattern defines the rule of mapping data to the radio resource, which can be defined as a binary vector. Each binary vector represents the PDMA pattern of one user equipment (UE). The dimension of the vector equals to the number of resources in a group. Those patterns with UEs sharing the same set of resources are arranged together to constitute the PDMA pattern matrix. Overload factor (OF) is defined as the ratio of the number of columns to the number of rows in a PDMA pattern matrix. It reflects the excessive number of UEs multiplexed on the same resources of PDMA relative to that of orthogonal multiple access (OMA) scheme. Taking resource number N = 4 and user number K = 8 as an example, the OF is then α = K /N = 200%, which means that PDMA supports two times the number of UEs compared with that of OMA. Properties of PDMA pattern matrix such as dimension and level of sparsity contribute to both receiver complexity and system performance. 458 S. Chen et al. 14.2.1 Basic Pattern Matrix Without loss of generality, both transmitter and receiver are assumed using single antenna, K UEs map onto N REs in the domains of time and frequency, in which each UE has a unique PDMA pattern. The PDMA received signal on the resource group composed by N REs at the base station (BS) is expressed as: y= K  diag(hk )g k xk + n = H x + n (14.3) k=1 where y denotes a vector composed by received signal on the N resources, with length N ; x = [x1 , x2 , . . . , x K ]T represents modulated signal vector transmitted by K UEs, with length K , and xk are the modulated signals of kth UE; n indicates the Gaussian ,K ] noise vector with length N , where n ∼ C N (0, N0 I N ×N ); H = H C H • G [N P D M A and H C H = [h1 , h2 , . . . , h K ] are the PDMA equivalent channel response matrix and original channel response matrix of K UEs multiplexed on N REs, respectively and both have dimensions of N × K , h K is the uplink channel response of the kth UE with length N, diag(hk ) represents a diagonal matrix with elements from hk , the (n, k) elements of H C H is the channel response from the kth UE to the BS on the nth ,K ] RE, and • denotes element-wise product of two matrices; G [N P D M A denotes a PDMA pattern matrix with the dimensions of N × K , where g k is the PDMA pattern used ,K ] by the kth UE, corresponding to the kth column of G [N P DM A. Given a certain overload factor, there are a number of pattern matrices available, as long as resource number N and user number K are selected properly. For example, overload factor of 150% could be achieved by a 2 × 3 pattern matrix, i.e., 3 users are multiplexed on 2 REs. The pattern matrix is:   110 = G [2,3] P DM A 101 and another design for 150% overload is 4 × 6 pattern matrix: ⎡ ⎤ 101110 ⎢1 1 0 1 0 1⎥ ⎢ ⎥ G [4,6] P DM A = ⎣ 1 1 1 0 1 0 ⎦ 011001 Though both pattern matrices having the same overload factor, G [4,6] P D M A can achieve better performance while the cost of detection complexity is higher comparing to that of G [2,3] P DM A. PDMA pattern matrices with different dimensions are able to achieve a given overload factor. With a higher dimension, detection complexity is also higher, and better performance is expected. Given an overload factor, the dimension of pattern 14 Pattern Division Multiple Access (PDMA) 459 matrix shall be selected to reach a tradeoff between detection complexity and system performance. If N is the size of resource group (row number of PDMA pattern matrix), there are 2 N − 1 possible binary vectors for a pattern matrix. Assuming K is the column number determined based on overload factor, we can thus choose K patterns out from 2 N − 1 candidates to construct PDMA pattern matrix. Selection of patterns also gives impacts on performance and complexity: (1) A pattern with heavier weight (number of “1” elements in the pattern) provides higher diversity order. More reliable data transmission can be anticipated, and detection complexity is also increased. If the system can conduct complex computation, patterns with heavy weight will be preferable; otherwise, light weight patterns have to be selected, aiming at sparse PDMA pattern matrix. (2) According to the design principle of PDMA, it is desirable to have different diversity orders in the pattern matrix to alleviate error propagation problem of SIC receiver or fasten convergence of BP receiver. Thus the selected patterns shall have as many different diversity orders as possible. (3) For patterns with identical diversity order, smaller inner product between the patterns leads to less interference against each other. Small inner product means that the two patterns have less “1” elements in common positions. That is, the number of REs shared by the two patterns is low. Data of two users are multiplexed on only few REs. For example, if two patterns have inner product of 0, the two patterns actually map data onto a different set of REs, hence there is no interference between the two patterns. For a given diversity order, the selected patterns shall minimize the maximum inner product between any two patterns. Of course this rule is also applied to patterns with different diversity order. The design of pattern matrix shall take overload factor, diversity order and detection complexity into account. A good pattern matrix can reach good trade-off among these aspects. 14.2.2 Pattern Optimization Method Taking PDMA pattern matrix G [2,3] P D M A as an example, data of 3 users are mapped onto two REs. The transmission signal on these REs can be expressed as:  v1 v2  ⎡ ⎤  x 1 1 0 ⎣ 1⎦ x2 = 101 x3  (14.4) where v j is the transmission signal on the jth RE, and xk is the modulation symbol of the k th user. Unlike orthogo