Receiver Design for High Spectral Efficiency Communication Systems in Beyond 5G 9789811980893, 9789811980909

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Weijie Yuan Nan Wu Jingming Kuang

Receiver Design for High Spectral Efficiency Communication Systems in Beyond 5G

Receiver Design for High Spectral Efficiency Communication Systems in Beyond 5G

Weijie Yuan · Nan Wu · Jingming Kuang

Receiver Design for High Spectral Efficiency Communication Systems in Beyond 5G

Weijie Yuan Department of Electronic and Electrical Engineering Southern University of Science and Technology Shenzhen, Guangdong, China

Nan Wu School of Information and Electronics Beijing Institute of Technology Beijing, China

Jingming Kuang Beijing Institute of Technology Beijing, China

ISBN 978-981-19-8089-3 ISBN 978-981-19-8090-9 (eBook) https://doi.org/10.1007/978-981-19-8090-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword

This research monograph addresses one the most pressing issues of wireless communications, namely the avoidance of the impending ‘spectrum crunch’. In the early era of wireless communications only the police and military as well as the rich and famous have had access to mobile telephony, which resulted in rather limited teletraffic. Hence having high spectral efficiency was much less critical than in the era at the time of writing, when low-cost commercial devices are readily available to the general public. The first digital mobile radio system supporting international roaming was the Global System of Mobile communications known as GSM, which was designed and standardized during the late 1980s by the European Union and spread to more than 150 countries across the globe. In this early era it adopted the then extremely radical design principle of using a partial response modulation scheme referred to as Gaussian Minimum Shift Keying (GMSK). The benefit of this partial response scheme is that each modulating symbol’s effect is spread across multiple symbol intervals, which hence facilitates having very smooth, slowly evolving modulated waveforms. When combined with signaling relying on Gaussian shaping of the waveform, it results in the most compact spectral-domain representation that may be attained. This is however achieved at the cost of consciously introducing conrolled intersymbol interference, which necessitates the employment of channel equalization even in the absence of any channel-induced Intersymbol Interference (ISI) or Multiuser Interference (MUI). By contrast, the Third-Generation (3G) wireless systems opted for the employment of Code Division Multiple Access (CDMA) relying on so-called Orthogonal Variable Spreading Factor (OVSF) codes and a total bandwidth of 5 MHz, which resulted in frequency-selected fading in typical cellular scenarios and hence suffered from the deleterious effects of dispersion that required sophisticated receiver techniques. This realization triggered a spate of research activities on the conception of more bandwidth-efficient non-orthogonal CDMA and a whole host of other non-orthogonal signaling solutions. This timely research-oriented book follows a similar line of investigation in the emerging 6G era. The rationale is that with the advent of Moore’s Law a whole v

vi

Foreword

suite of sophisticated receiver techniques are available for mitigating the deleterious effects of ISI and MUI. Hence the benefits of Faster-than-Nyquist (FTN) signaling are quantified in the presence of radically new receivers. Furthermore, the merits of Sparse Code Multiple Access (SCMA) are discussed, which is also eminently suitable for grant-free access in support of transmitting short messages from IoT nodes for example, with the objective of avoiding the excessive handshaking delays of IoT systems. Wishing you intellectual stimulation and fun, while consulting the book. October 2022

Lajos Hanzo Chair of Telecommunications University of Southampton Southampton, UK

Preface

With the rapid development of modern society, people have an increasing demand for higher data rates. Due to the limited available bandwidth, how to improve the spectral efficiency becomes a key issue in the beyond 5G mobile communication system. Recent researches show that by introducing non-orthogonalties, the nonorthogonal system is capable of transmitting more information and supporting more users within the same resource elements, leading to improved spectral efficiency. In general, the high spectral efficiency communication system can be achieved by using Faster-Than-Nyquist (FTN) signaling which breaks the Nyquist criterion to transmit more data symbols in the same time period. Another solution is to multiplex more users over the orthogonal resource elements to provide wireless connectivity to more users, which is known as Non-orthogonal Multiple Access (NOMA). Among several NOMA technologies, the code domain NOMA, e.g., Sparse Code Multiple Access (SCMA) could achieve further shaping gain. Nevertheless, FTN signaling will result in intersymbol interference (ISI) while NOMA system will introduce interference between different users using the same resource elements, leading to challenges on receiver design. This book focuses on the receiver design for high spectral efficiency communication systems in beyond 5G era. We summarize the main innovations contained in this book as follows: Chapter 2 presents an energy minimization-based SCMA decoding algorithm for uplink system. Relying on the optimization theory and variational free energy (VFE) framework, the a posteriori distribution of each transmitted data symbol is derived. Then, the convergence behavior of the proposed algorithm is further analyzed, which proves the convergence of the a posteriori variance of the data symbol. Chapter 3 develops a low-complexity receiver relying on a reconstructed stretched factor graph for downlink MIMO-SCMA system. To address the convergence issue of the current MPA-baed receiver on the loopy factor graph, a convergence guaranteed message passing receiver is proposed by convexifying the Bethe free energy. Finally, cooperative detection schemes that allow the downlink users to share information are proposed to achieve diversity gain. In Chap. 4, we solve the problem of FTN signaling detection over frequency selective channels with unknown channel knowledge. Instead of the commonly used vii

viii

Preface

composite ISI channel model, we intentionally separate the known ISI induced by FTN and the unknown ISI introduced by channel fading, which results in better detection performance and reduced complexity. In Chap. 5, FTN transmission over doubly selective channels is considered. Two low-complexity receiver designs, namely Gaussian message passing (GMP)-based and variational inference-based methods are proposed. In particular, both mean field and Bethe approximations are considered and compared in terms of detection performance and complexity. The GMP-based scheme is then extended to an imperfect channel case. Chapter 6 studies a novel non-orthogonal communication system by integrating FTN signaling into SCMA system, which leads to an even higher spectral efficiency. To eliminate the ISI and inter-user interference (IUI), a low-complexity receiver is designed based on the factor graph framework for joint channel estimation and SCMA decoding. Then, considering the grant-free transmission, the factor graph is modified by including the user activity detection. Chapter 7 of this book investigates a general FTN-NOMA system with random access in practical dynamic environments. Joint user activity tracking, channel estimation, and data detection are done by novel expectation maximization (EM)-MPA receiver iteratively. All messages and beliefs of unknown variables are determined in parametric forms, resulting in low-complexity signal reception. Finally, Chap. 8 summarizes this book by emphasizing the innovative results. Then, we provide several future research directions for further investigation. Our hope is that this book would provide a good reference for the researchers in the area of iterative receiver design for supporting new communication systems in the era of beyond 5G. This book can also be used as a textbook for postgraduate courses. The authors would like to express their sincere thanks to the people who have helped the accomplishment of this book: Lajos Hanzo, Yonghui Li, Hua Wang, Xiaojing Huang, Andrew Zhang, Qinghua Guo, Jinhong Yuan, Derrick Wing Kwan Ng, Chengwen Xing, and Chaoxing Yan. The authors also want to thank Hongjia, Zhongjie, Xiaoqi, Jun, Buyi, Xinyuan, Xiang, and Kecheng for helping edit the figures in this book. In the end, the author Weijie Yuan would thank Ms. Jasmine Zhang for her company and support. Shenzhen, China August 2022

Weijie Yuan Nan Wu Jingming Kuang

Contents

1 Introduction of High Spectral Efficiency Communication Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Faster-Than-Nyquist Signaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Sparse Code Multiple Access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 7 10

2 Uplink Multi-user Detection for SCMA System . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Low-Complexity Receiver Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Bayesian Inference-Based Approach . . . . . . . . . . . . . . . . . . . . 2.3.2 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 14 15 15 19 20 24 26

3 Downlink Multi-user Detection for MIMO-SCMA System . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 MIMO-SCMA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Probabilistic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Stretched Factor Graph and Low-Complexity Message Passing Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Factor Graph Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Message Passing on Stretched Factor Graph . . . . . . . . . . . . . 3.3.3 Algorithm Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Convergence-Guaranteed Message Passing Receiver . . . . . . . . . . . . . 3.4.1 VFE and Belief Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Convergence-Guaranteed BP-EP Receiver . . . . . . . . . . . . . . . 3.4.3 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Distributed Cooperative Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Belief Consensus-Based Method . . . . . . . . . . . . . . . . . . . . . . .

29 29 30 31 32 32 32 37 37 37 40 44 45 46 ix

x

Contents

3.5.2 Bregman ADMM-Based Method . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Algorithm Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 50 51 56

4 FTN Data Detection and Channel Estimation over Frequency Selective Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 FTN Signaling Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Low-Complexity Message Passing Receiver Design . . . . . . . . . . . . . 4.3.1 Forney-Style Factor Graph Representation . . . . . . . . . . . . . . . 4.3.2 Gaussian Message Passing Algorithm . . . . . . . . . . . . . . . . . . . 4.3.3 Computation of Extrinsic LLR . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 60 62 63 65 69 70 71 75

5 Receiver Design for FTN Signaling over Doubly Selective Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 FDE-MMSE Based Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 VFE-Based Receiver Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 VFE-Based Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Complexity Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 GMP-Based Receiver Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 GMP-Based Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Imperfect Channel Information . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 78 79 79 80 83 84 85 86 88 89 91

6 Receiver Design for FTN-SCMA Communication System . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Receiver Design for FTN-SCMA Systems . . . . . . . . . . . . . . . . . . . . . 6.3.1 Approximation of Colored Noise . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Probabilistic Model and Factor Graph Representation . . . . . 6.3.3 Message Passing Receiver Design . . . . . . . . . . . . . . . . . . . . . . 6.4 User Activity Detection in Grant-Free System . . . . . . . . . . . . . . . . . . 6.4.1 Probability-Based Active User Detection . . . . . . . . . . . . . . . . 6.4.2 Message Passing Based Active User Detection Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 93 94 96 96 96 98 101 101 103 107 113

Contents

xi

7 Receiver Design for FTN-NOMA System with Random Access . . . . . 7.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Faster Than Nyquist Signaling . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Non-orthogonal Multiple Access . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 FTN-NOMA System with Random Access . . . . . . . . . . . . . . 7.2 Factor Graph Representation of FTN-NOMA System . . . . . . . . . . . . 7.2.1 Probabilistic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Factor Graph Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 EM-MPA Receiver for FTN-NOMA System . . . . . . . . . . . . . . . . . . . 7.3.1 Multiuser Detection and Decoding Part . . . . . . . . . . . . . . . . . . 7.3.2 Summary of the Proposed Receiver . . . . . . . . . . . . . . . . . . . . . 7.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 116 116 117 117 118 119 122 123 123 128 129 134

8 Current Achievements and The Road Ahead . . . . . . . . . . . . . . . . . . . . . . 8.1 Summary of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Road Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Receiver Design for FTN Signaling in Nonlinear Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Receiver Design for Orthogonal Time Frequency Space (OTFS) Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Receiver Design for Integrated Sensing and Communication (ISAC) System . . . . . . . . . . . . . . . . . . . .

135 135 137 137 138 140

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Acronyms

1G 2G 3G 4G 5G 6G ADMM AMP AR ARMA AWGN BCJR BER BP BS CDMA CS CSI dB DD DFT DSC EM EP EXIT FBMC FDE FDMA FFT FTN GMP

The First Generation The Second Generation The Third Generation The Fourth Generation The Fifth Generation The Sixth Generation Alternating Direction Method of Multipliers Approximate Message Passing Auto Regressive Auto Regressive Moving Average Additive White Gaussian Noise Bahl-Cocke-Jelinek-Raviv Bit Error Rate Belief Propagation Base Station Code Division Multiple Access Compressive Sensing Channel State Information Decibel Delay Doppler Discrete Fourier Transform Doubly Selective Channel Expectation Maximization Expectation Propagation Extrinsic Information Transfer Filter Bank Multi Carrier Frequency Domain Equalization Frequency Shaped Sliding Mode Control Fast Fourier Transform Faster Than Nyquist Gaussian Message Passing xiii

xiv

GSM HPA ICI IDFT IDMA ISAC ISI IUI KLD LDPC LDS LEOS LLR LMMSE LS MAP MCA MF MIMO MMSE MPA MPSK MSE MUSA NOMA OFDM OFDMA OMA OTFS PDMA PMF p-NOMA QAM RRC SCM SCMA SIC SPA TDE TDMA TF UAV VFE VMP

Acronyms

Global System for Mobile Communication High Power Amplifier Inter Carrier Interference Inverse Discrete Fourier Transform Interleave Division Multiple Access Integrated Sensing and Communication Inter Symbol Interference Inter User Interference Kullback-Leibler Divergence Low Density Check Code Low Density Signature Low-Earth-Orbit Satellites Log-Likelihood Ratio Linear Minimum Mean Squared Error Least Square Maximum A Posteriori Mobile Communications on board Aircraft Mean-Field Multiple Input Multiple Output Minimum Mean Squared Error Message Passing Algorithm M-ary Phase Shift Keying Mean Squared Error Multi-User Shared Access Non-Orthogonal Multiple Access Orthogonal Frequency Division Multiplexing Orthogonal Frequency Division Multiple Access Orthogonal Multiple Access Orthogonal Time Frequency Space Pattern Division Multiple Access Probability Mass Function power domain NOMA Quadrature Amplitude Modulation Root Raised Cosine Single-Carrier Modulation Sparse Code Multiple Access Successive Interference Cancellation Sum Product Algorithm Time Domain Equalization Time Division Multiple Access Time Frequency Unmanned Aerial Vehicles Variational Free Energy Variational Message Passing

Notations

a A {an } A:,i N (x) ∝ diag(·) G(m, v) O(·) |·| (·)T (·)∗ (·)H (·)−1 In E C BK CK R K ×K ∗

A vector a A matrix A A sequence of parameters, i.e., a1 , a2 ,...an The i-th column of matrix A The set of factor (variable) nodes connected to variable (factor) node x Equality up to a constant normalization factor Convert a vector to a diagonal matrix A Gaussian distribution with mean m and variance v The order of complexity Modulus of a variable or cardinality of a set Transpose operator Conjugate operator Hermitian operator Inverse operator Identity matrix of dimension n × n Expectation operator Denotes a constant A K -dimensional binary space A K -dimensional complex space A K × K -dimensional complex space Convolution operator

xv

Chapter 1

Introduction of High Spectral Efficiency Communication Systems

The first chapter of this book commences with the evolution of mobile communication systems and the key problems for future wireless communication. Then the technologies of faster-than-Nyquist (FTN) signaling and non-orthogonal multiple access (NOMA) are briefly reviewed.

1.1 Background Since the 1970s, mobile communication technology has evolved rapidly, which profoundly changes the work and lifestyle of human beings and promotes economic development [1]. In 1978, Bell Labs designed a cellular mobile communication network using analog technology and frequency division multiple access (FDMA) [2]. This communication network, also known as the first generation mobile communication system (1G), effectively solved the capacity requirements at that time. After that, in order to overcome the problem of low call quality in the 1G era, the second generation mobile communication system (2G) based on digital voice communication was proposed [3]. The new technology employed in 2G is time division multiple access (TDMA) [4]. With the increasing demand for high-speed data transmission, the third generation mobile communication system (3G) using code division multiple access (CDMA) technology [5] came into being. Relying on more bandwidth and a high data rate, 3G can provide users with more wireless applications. Nevertheless, it still has some limitations in mobile multimedia services [6]. The fourth generation mobile communication system (4G) using orthogonal frequency division multiple access (OFDMA) [7] and multiple input multiple output (MIMO) [8] can address the signal coverage problem in 3G. At present, various new mobile Internet-based services have emerged that greatly improved the informationization of society. In

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Yuan et al., Receiver Design for High Spectral Efficiency Communication Systems in Beyond 5G, https://doi.org/10.1007/978-981-19-8090-9_1

1

2

1 Introduction of High Spectral Efficiency Communication Systems

Fig. 1.1 The evolution of mobile communications

Fig. 1.1 (From Ref. [7]), the evolution of mobile communications from 1G to 5G is illustrated. The history of mobile communication over the last four decades suggests that the research on new mobile communication systems will never stop. On one hand, the rapid development of integrated circuit and chip design has greatly enhanced the performance of mobile devices, which enables the implementation of many complex technologies such as low-density partial check (LDPC) code [9]. On the other hand, the demand for higher quality wireless services continues growing [10, 11]. In the last decade, the explosive growth of smartphones led to an exponential increase in the amount of mobile data [12]. At the same time, the applications including the Internet of things, smart home, and virtual reality require more accessed devices, higher coverage, lower transmission delay, and smaller communication overhead [13–16]. According to the tutorial paper [17], the future mobile networks in beyond 5G (B5G) and 6G era should satisfy the following performance indicators: (1) transmission rate increased by 10–100 times; (2) capacity enhanced by 1000 times of current network capacity; (3) end-to-end delay reduced to 1/10 of the current one [18]. Figure 1.2 shows the variation in transmission rate as the mobile communication systems evolve. Undoubtedly, the standardization of 5G and the future development of the B5G system will further change our lives. In order to achieve the above objectives, academia and industry have conducted research from three different points of view. The first one is to use new technology to further improve the spectral efficiency with limited bandwidth [19–21]. The second one is finding new available spectrum resources based on mm-Wave, terahertz (THz), and even visible light communications [22–25]. The third one is to increase the throughput by using communications between users [26–28]. Higher spectral efficiency can be achieved by designing new modulation formats and new multiple access systems. There exist several new modulation methods, e.g., filter bank multi-carrier (FBMC), FTN signaling, and single-carrier modulation (SCM) that can improve spectral efficiency [29–33]. By introducing intentional inter symbol interference (ISI), FTN signaling can transmit more data using the remaining bandwidth in the 4G system [34, 35]. For the multiple access technologies, B5G wireless systems will consider the use of non-orthogonal multiple access (NOMA) to support more users using the same resources [36, 37]. Existing NOMA technologies include

1.2 Faster-Than-Nyquist Signaling

3

Transmission rate >100Mbps

5G 10x increase 10Mbps

4G 5x increase

2Mbps

3G 10x increase 25Kbps

2G

Times

Fig. 1.2 The variation of data rate from 1G to 5G

power domain NOMA (p-NOMA) [38], interleave division multiple access (IDMA) [39, 40], multiuser shared access (MUSA) [41], pattern division multiple access (PDMA) [42] and sparse code multiple access (SCMA) [43–45]. Amongst them, SCMA has received extensive attention because of its extra shaping gain. Moreover, the environment will introduce various interference, such as the multi-path effect in the indoor environment and time selective fading on high-speed trains. How to tackle these challenges while realizing high spectral efficiency is one of the key goals of B5G. In this book, the low complexity reception methods for high spectral efficiency communication systems will be studied.

1.2 Faster-Than-Nyquist Signaling According to the Nyquist theorem, the maximum symbol rate should be twice the channel bandwidth. When the symbol rate equals the Nyquist rate, a sample of the matched filter output only depends on one transmitted symbol impulse. That is to say, the received sample contains all the necessary information for symbol decision and symbol-by-symbol detection is used to obtain all transmitted information. Although the Nyquist criterion ensures ISI-free transmission, it wastes certain spectrum resources to keep the orthogonality of waveforms [32]. In order to take advantage of these spare resources, more data symbols can be transmitted in the same time period, which is the basic principle of FTN signaling. In 1970, the concept of FTN was first introduced [46]. Then Mazo detailedly analyzed FTN signaling and proved that FTN signaling can transmit 25% more information bits than Nyquist signaling

4

1 Introduction of High Spectral Efficiency Communication Systems

Fig. 1.3 Comparison of a Nyquist signaling and b FTN signaling

in additive white Gaussian noise (AWGN) channel while preserving the bit error rate (BER) performance [31]. Assuming that cn is the transmitted symbol at time n and q(t) is a shaping pulse, the Nyquist signaling can be expressed as s(t) =



cn q(t − nT0 ).

(1.1)

n

Considering the same shaping pulse q(t), the FTN signaling model reads, s(t) =



cn q(t − nτ T0 ).

(1.2)

n

It can be seen when τ = 1, (1.2) and (1.1) are identical. In FTN signaling, setting τ < 1 can transmit more information with the same bandwidth and time period, therefore τ is named as the packing factor. Figure 1.3 illustrates the Nyquist signaling waveform and FTN signaling waveform with τ = 0.8. It can be observed that in the same time period, FTN signaling can transmit more sinc pulses. It can be also observed that when sampling with symbol interval τ T0 , the samples contain information corresponding to other symbols, which means that the interference between transmitted symbols should be canceled when detecting FTN signals. For the signal defined by (1.2), the error probability when performing detection is determined by the minimum distance of signals [47]. Considering two signals si (t) 2 is and s j (t) given by (1.2), the minimum Euclidean distance dmin  2 dmin

=

1 Eb



∞

−∞

|si (t) − s j (t)|2 dt, i = j,

(1.3)

where E b is the signal energy per bit. Assuming that the signal transmits through an AWGN channel, the symbol error rate of optimal detection is given by function

1.2 Faster-Than-Nyquist Signaling

5

 2 d Eb Q( min ), where the Q-function is the tail distribution function of the standard N0 normal distribution, formulated as 1 Q(x) = √ 2π



∞ x

 2 u du. exp − 2

(1.4)

For binary symbol Nyquist signaling, the minimum  Euclidean distance of signal Eb is always 2. The corresponding symbol error rate Q( 2min ) is referred to as the N0 matched filter bound [48]. For FTN signaling using sinc pulse, the minimum distance will not drop immediately for τ < 1. In fact for binary symbols, although ISI is 2 = 2 still holds. Hence, induced by FTN signaling, when τ is larger than 0.802, dmin τ = 0.802 is known as the Mazo limit [31]. When the packing factor is above the Mazo limit, it is capable of increasing the symbol rate while not affecting the BER performance. Subsequently, research on root raised cosine (RRC) shaping pulse showed that with roll-off factor α = 0.3, Mazo limit can be reduced to 0.703, which shows the data rate can be as 1.43 times high as Nyquist signaling [32]. According to [49], the Mazo limit applies to non-binary transmissions, nonlinear pulses, and even nonlinear modulations. Based on these works, it is seen that FTN signaling is very attractive in future mobile communications. However, due to the non-orthogonality of the transmitted waveform, FTN signaling introduces severe ISI. It can be seen from the signal model (1.2) that each symbol is interfered by several neighboring transmitted symbols. Consequently, designing receivers for FTN signaling in order to recover original transmitted data is very important. Next, a basic discrete time FTN signaling model is considered. The transmitted signal s(t) passes through an AWGN channel and is received at the receiver side. After matched filtering, the received signal r (t) can be expressed as r (t) =

∞ 

cn g(t − nτ T0 ) + γ (t),

(1.5)

n−∞

where g(t) is the convolution of shaping filter and matched filter and γ (t) is the random noise process. After sampling using symbol period τ T0 , the received samples {rn } are obtained. The objective of FTN signaling receiver is to recover sequence {cn } from the received signal samples {rn }. Since the shaping pulse is non-orthogonal with respect to symbol period τ T0 , the noise samples at the output of the matched filter have correlations [32]. Usually, a post filter is employed to whiten the noise [50] and then signal detection is performed. Assuming that the discrete time sequence obtained from g(t) is denoted by {gn }, FTN signaling can be regarded as the trellis code of {cn } by {gn }. Detection of {cn } is equivalent to decoding the noisy trellis coded sequence. Viterbi algorithm [51] can be employed to find the most likely sequence {cn } by comparing the received signal and the shortest path. Viterbi algorithm provides the maximum likelihood estimate of the transmitted symbol. When the symbols are with

6

1 Introduction of High Spectral Efficiency Communication Systems

equal probability, the maximum likelihood-based receiver is optimal. However, when the probabilities of symbols are unequal, the maximum a posteriori estimator should be employed, given by cˆn = arg max cn

p(cn |{rn }) . p({rn })

(1.6)

Considering the symbol by symbol MAP estimator, Bahl et al. proposed Bahl– Cocke–Jelinek–Raviv (BCJR) algorithm for symbol detection [52], which is also based on a trellis diagram. By computing the likelihood probability of {cn }, the transmitted sequence is detected iteratively. If a large value of τ is chosen, the FTN induced ISI is slight, then using Viterbi algorithm or BCJR algorithm is effective in symbol detection. However, when ISI becomes severer, the increased number of trellis states makes the receiver complexity prohibitively high. To reduce the complexity of trellis-based receivers, three approaches were proposed, i.e., channel shortening [53], reduced search [54], and reduced trellis [55]. Channel shortening aims for equalizing the FTN-induced ISI to an ISI channel with fewer taps in order to reduce the complexity. Reduced search method only considers searching the optimal path in part of the trellis diagram. For example, the M-algorithm only considers M paths in the diagram [56]. Reduced trellis method can reduce the total number of trellis states by reducing the number of states of a symbol [55]. For FTN signaling, an M-algorithm-based BCJR receiver was proposed in [57] to eliminate the ISI introduced by FTN signaling. Although only M-paths are considered, the complexity of the receiver in [57] still increases exponentially. Based on the frequency domain equalization (FDE) method for conventional frequency selective fading channels, an FDE-based minimum mean squared error (MMSE) algorithm was proposed in [58]. In FDE, the time domain signal is transformed to the frequency domain through fast Fourier transform (FFT), then ISI becomes frequency channel coefficients and FTN symbols are detected with linear complexity. Nevertheless, FDE has to insert cyclic prefix symbols in the transmitted sequence to convert linear convolution into circular convolution, which decreases the spectral efficiency. In time domain equalization, the authors in [59] employed autoregressive (AR) process to model the colored noise and proposed a linear MMSE FTN signaling receiver. Current research on receiver design for FTN signaling mainly focuses on the AWGN channel. In practical environments, the channel suffers from multipath effect and Doppler spread, which lead to frequency and time selective fading. The interference induced by fading channels makes the FTN signaling receiver more complex. Moreover, when the channel information is unknown, how to achieve good detection performance of FTN symbols is still under investigation.

1.3 Sparse Code Multiple Access

7

1.3 Sparse Code Multiple Access In conventional orthogonal multiple access (OMA) technology, each user will be assigned to certain orthogonal radio resource elements, such as orthogonal frequency division multiplexing (OFDM) sub-carriers and MIMO antennas. So the number of users cannot exceed the total number of orthogonal resource elements. At the receiver side, the signals corresponding to different users can be obtained through simple single user detection. When the channel condition of a user is poor, OMA has to set a high priority to transmit this user’s data in order to satisfy fairness. This results in both the waste of spectrum and the decline of system throughput. Theoretically, OMA cannot achieve the sum-rate capacity of multiuser systems. To this end, non-orthogonal resource allocation is proposed in a multiple access system, which is the NOMA technology. By introducing interference between users, one resource element in NOMA can support more users to transmit information, and the increment of the information rate makes the improvement of spectral efficiency. The multiuser capacity in AWGN channel is analyzed in [60], which demonstrated that NOMA can reach the capacity bound. In multipath fading channels, NOMA achieves the optimal capacity when the channel information is only known at the receiver side, while OMA is strictly sub-optimal. Moreover, since more users are supported by NOMA, the requirement of massive connectivity in 5G can be satisfied [36]. The NOMA technologies can be categorized into power domain methods and code domain methods [36]. The power domain NOMA (p-NOMA) maximizes the system gain by assigning different power levels to users according to their channel conditions. This enables us to distinguish different users at the receiver side, then use successive interference cancellation (SIC) to detect the signals. In code domain NOMA technologies, the users will be assigned different codewords to achieve multiplexing gain. Compared to p-NOMA, code domain NOMA technologies can achieve spreading gain and higher sum rate [61]. SCMA technology can be regarded as the extension of the low-density signature (LDS) method [43, 62]. Different from LDS technology, SCMA maps bit streams of different users to SCMA codewords directly, which makes joint codebook and constellation optimization possible. Figure 1.4 illustrates the SCMA encoding process of a 6-user, 4-resource SCMA system. Each user maps its bits to an SCMA codeword chosen from a predefined SCMA codebook and then multiplexed over 4 resource elements. Multiuser detection is performed at the receiver side to determine the bit sequences of users. In the SCMA system, a binary vector fk is employed to indicate the resource elements occupied by user k. The jth element in fk is defined as  f k, j =

0 xk, j = 0 1 xk, j = 0.

(1.7)

By stacking fk , we have the indicator matrix F = [f1 , . . . , f K ]. In F, the non-zero entries in the jth row denote the conflicting users over the jth antenna while the

8

1 Introduction of High Spectral Efficiency Communication Systems User 1

User 2

User 3

User 4

User 5

User 6

The bitstreams are mapped to codewords (0, 0)

Multiplexing

(1, 0)

(0,1)

(1,1)

(1,1)

(0, 0)

Multiuser Detection

Fig. 1.4 SCMA encoding process

non-zero entries in the kth column indicate the resources occupied by user k. The indicator matrix corresponding to the SCMA system shown in Fig. 1.4 is given as follows, ⎡ ⎤ 111000 ⎢1 0 0 1 1 0⎥ ⎥ (1.8) F=⎢ ⎣0 1 0 1 0 1⎦. 001011 It is observed from F that when designing the codebook, the 0 valued positions of different users’ codewords are different, which efficiently avoids packet collisions. Besides, only partial resource elements are assigned to one user, which ensures one resource element will support a few users and the sparsity property makes the complexity of the SCMA system still controllable [63]. In the overloaded SCMA system, the received signal contains the interference introduced by other users and this should be taken into account when designing receivers. For the indicator matrix F in (1.8), a factor graph can be used to show the relationship between the users and resource elements [64], as shown in Fig. 1.5. The factor graph contains factor vertices E and variable vertices U , where a variable vertex represents one user’s transmitted symbols and a factor vertex represents the function relationship of the source elements and transmitted symbols. If and only if a user occupies a resource element, the variable vertex and factor vertex are connected by an edge. Thanks to the sparsity of SCMA codewords, message passing algorithm (MPA), also known as belief propagation (BP) can be used on a factor graph to derive the MAP estimate of the transmitted symbol. On the factor graph, there are two kinds of messages, namely, the message from variable vertex x to factor vertex f and the message from factor vertex f to variable vertex x, denoted by μ f →x (x) and μx→ f (x), respectively. According to the BP update rules [64], μ f →x (x) and μx→ f (x) are given by

1.3 Sparse Code Multiple Access

9

Fig. 1.5 The factor graph representation of SCMA system

 μ f →x (x) ∝ μx→ f (x) ∝



f (x)





μx  → f (x )dx ,

(1.9)



x ∈N ( f )\{x}



μ f  →x (x),

(1.10)

f  ∈\{ f }

where N (x) and N ( f ) denote the sets of all factors connected to x and all variables in the function f , respectively. And the belief (approximate marginal) of variable x can be expressed as b(x) ∝



μ f →x (x).

(1.11)

f ∈N (x)

By passing messages on the factor graph, the marginal distribution of SCMA symbol is determined iteratively. From (1.9), it can be observed that the message calculation requires integration over all other interfered symbols. Therefore the complexity of the conventional BP algorithm increases exponentially with the number of interfered symbols. To reduce the complexity of conventional MPA receiver, several modified BP message passing receivers are proposed for SCMA system. In [65], the authors proposed a shuffled message passing algorithm to accelerate the convergence. Reference [66] presented a fixed low complexity detector for uplink SCMA system based partial marginalization. In [67], a Monte Carlo Markov Chain (MCMC) based SCMA decoder was proposed which features low complexity when the codebook size is large. In [68], the authors compute the messages in the log-domain and then the multiplication operations become simple summations. In a word, many researchers have proposed low-complexity receiver designs for SCMA systems. However, there are still a few points that should be addressed. The first problem is the symbol detection in complex environment. Existing works consider a very simple channel model and assume that the channel state information is perfectly known, which ignores complex channel conditions in practice. The second

10

1 Introduction of High Spectral Efficiency Communication Systems

problem is the convergence of BP algorithm. Although [69] showed that a loopy BP is still efficient in a factor graph with cycles, the convergence problem may result in performance loss. The third one is the combination of NOMA and non-orthogonal waveform. SCMA technology and FTN signaling can improve the spectral efficiency from different degrees. Naturally, a combination of both technologies is expected to achieve ever higher spectral efficiency.

1.4 The Objectives In summary, the main aims of this book are as follows: i. Conventional MPA receivers for SCMA systems may suffer from convergence problem since BP does not guarantee convergence on loopy graphs. Therefore, the convergence analysis of the SCMA receiver is necessary. In addition, the interference becomes more severe in complex environments, which results in a factor graph with more short cycles. To tackle the problem that MPA fails to converge, it is necessary to design a convergence-guaranteed message passing algorithm for this scenario. ii. In downlink multiuser system, cooperation can be enabled amongst users to exchange necessary information, which provides further diversity gain. The basic idea to share the measurements among all users is not realistic due to high power consumption. Therefore, developing new cooperative detection schemes with a low cost is necessary. iii. Existing FTN signaling receivers work under the assumption of known channel state information. When the multipath channel is unknown, how to tackle the combined ISI induced by fading channels and FTN signaling and then estimate the channel coefficients is very important. It is highly demanded to design new reception algorithms that jointly estimate channel taps and detect FTN data symbols. iv. The FDE-based receivers can tackle the ISI introduced by FTN signaling. However, in high mobility environments, due to the time-variant channel, existing FDE-based algorithms experience prohibitively high complexity. Therefore, it is necessary to design novel low complexity FDE-based algorithm to eliminate the interference caused by time selectivity and improve the robustness of FTN signaling in high mobility environments. v. A combination of SCMA and FTN technologies is expected to further improve the spectral efficiency of communication systems at the cost of high complexity receivers. Moreover, in uplink system, communication overhead at the base station can be reduced by detecting the users’ activities. Therefore, a novel low complexity receiver aiming for detecting active users, decoding, and channel estimation should be designed. vi. Additionally, considering the FTN-NOMA system working in dynamically fluctuating environments where the user activity, as well as channel state vary over

1.4 The Objectives

11

time, user activity tracking, channel estimation, and information decoding are all required. Therefore, how to model the dynamic states of users and design low-complexity receivers for joint channel estimation, user activity tracking, and decoding need investigation.

Chapter 2

Uplink Multi-user Detection for SCMA System

2.1 Introduction In this chapter, we will investigate the receiver design problem for Uplink SCMA System. With the aid of appropriate sparse codebook design, SCMA achieves an improved performance. However, due to the non-orthogonal resource allocation of the SCMA system, the optimal maximum a posteriori (MAP) detectors impose a high complexity. By exploiting the sparsity of the codewords, several factor graph and message passing algorithm (MPA) [64] based receivers have been developed [65, 68, 70, 71]. Nevertheless, the rank-deficient SCMA system results in a factor graph having short cycles, for which the MPA may not be able to converge. Therefore, it is important to investigate the convergence of iterative SCMA receivers. In the following of this chapter, we will first introduce the system model for SCMA uplink. Then, a low-complexity receiver based on Bayesian inference [72] is proposed. Instead of applying conventional probabilistic factorization, we formulate the joint a posteriori distribution of data symbols as the product of several local clique potentials. The employment of clique potentials is capable of reducing the number of short cycles on graphical models [73]. In particular, we construct the variational free energy (VFE) by adopting the Bethe approximation [74]. By minimizing the corresponding free energy given probability constraints, we determine the closed forms of the marginals of data symbols. The overall complexity of the receiver developed in this chapter only increases linearly with the number of uplink users. Moreover, we further analyze the convergence behavior of the proposed iterative scheme. It is proved that the a posteriori variance can guarantee its convergence by running iterations. As for the a posteriori mean, we derive the necessary and sufficient conditions for ensuring its convergence.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Yuan et al., Receiver Design for High Spectral Efficiency Communication Systems in Beyond 5G, https://doi.org/10.1007/978-981-19-8090-9_2

13

14

2 Uplink Multi-user Detection for SCMA System

2.2 System Model In this section, we will show the SCMA uplink model with K users, which is given in Fig. 2.1. In existing orthogonal multiple access (OMA) used in current 4G and 5G standards, each user will occupy an orthogonal resource element for information transmission. The number of orthogonal resource elements J is in general larger than the number of users K . Nevertheless, as discussed in Chap. 1, the available resource elements are not able to meet the rapid increase of the number of network users. This motivates us to find some new schemes to allocate more users using the same resource blocks. In particular, in the SCMA system considered in this book, we set K > J , indicating the SCMA system is a rank-deficient one. To evaluate the capability of supporting users, the normalized overloaded factor is defined as λ = KJ > 1. In the SCMA uplink, a user will transmit a bit stream bk , ∀1 ≤ k ≤ K , which is directly mapped to an SCMA codeword of dimension J , denoted by xk . The operation follows the mapping φ : Blog2 M → χ , where |χ | = M. Without loss of generality, the transmitted SCMA codeword of user k is denoted by xk = [xk,1 , . . . , xk,J ]. Each uplink user will have a predefined codebook, whose sparsity is characterized by the variable D < J , denoting the number of nonzero entries in xk . For each symbol of xk , it transmits through a channel h k, j and is received at the base station. Assuming that the uplink users and the base station have been perfectly synchronized [75], the received samples corresponding to the kth user can be expressed as yk = diag(hk )xk + nk .

(2.1)

Accordingly, the received signals containing the information of all K users are given by y=

K 

diag(hk )xk + n,

k=1

where n is the Gaussian noise having a power spectral density of N0 .

Fig. 2.1 Block diagram of the SCMA system

(2.2)

2.3 Low-Complexity Receiver Design

15

2.3 Low-Complexity Receiver Design Obviously, the received signal contains the information from all K users. A key issue is to extract the transmitted information of individual users k from the received signal y. In this section, we will propose a low-complexity receiver for an uplink SCMA system based on Bayesian inference.

2.3.1 Bayesian Inference-Based Approach We commence from the optimal maximum a posteriori detection, whose goal is to find the estimate of every data symbol, i.e. xˆk, j , which is equivalent to obtain the marginal a posteriori probability p(xk, j |y). Nevertheless, direct marginalization of p(x|y) will result in an exponentially increased complexity. Therefore, we will resort to the probabilistic factorization approach. According to the Bayesian theorem, we can write the a posteriori distribution of the transmitted data symbols as p(x|y) ∝ p(x) p(y|x),

(2.3)

In (2.3), p(x) is the joint a priori distribution. In general, the transmitted symbols corresponding to different users are independent. Therefore, p(x) can be factorized as p(x) ∝

K 

G(xk ; m0xk , V0x,k ),

(2.4)

k=1

where the mean vector and the covariance matrix are given by m0xk = [m 0xk,1 , . . . , m 0xk,J ]T , and

V0x,k

=

diag([vx0k,1 , . . . , vx0k,J ]),

(2.5) (2.6)

respectively, which are determined by exploiting the extrinsic information output from the channel decoder. Moreover, following the zero mean Gaussian assumption of the noise samples n, the joint likelihood function p(y|x) can be factorized as, 

 K  1 2 p(y|x) ∝ exp − y − diag(hk )xk  . N0 k=1

(2.7)

In addition, by further expanding the local likelihood function in (2.7), we can derive the following factorization expression,

16

2 Uplink Multi-user Detection for SCMA System

p(x|y) ∝

J  

j

φk

j

φi,k ,

(2.8)

(i,k)

j=1 k j



j

where φk and φi,k are the so-called potential cliques [76], having the expressions of  j φk

= p(xk, j ) exp − 

j φi,k

=

j φk,i

= exp −

2 h 2k, j xk, j − 2h k, j y j x k, j

N0 2h 2k, j xi, j xk, j N0

 ,

(2.9)

 .

(2.10)

Motivated by the Bayesian inference framework, we aim for finding a trail distribution b(x) is sufficiently close to the original distribution p(x|y) and is easy to be marginalized. The similarity between the trail and the original distributions is characterized by the VFE, which is defined by [77]  F = FH +

b(x) ln

b(x) dx, p(x|y)

(2.11)

 where F H = − ln Z is termed as Helmholtz free energy [77] with Z = p(x|y)dx being the normalization factor. Considering that the trail distribution is sufficiently simple, i.e., can be fully factorized, we have the expression of bMF (x) =  j b (x k, j k k, j ), which is the well-known mean-field (MF) approximation [76, 78]. From MF approximation, we can see that the variables in x are all independent and j bk (xk, j ) denotes the marginal distribution (‘belief’) of the variable xk, j . Substituting the expression of bMF (x) into the free energy F , we arrive at the MF j free energy FMF . By setting the derivative of FMF with respect to bk (xk, j ) to zero, we j can readily compute the MF approximation of the belief bk (xk, j ). Although based on MF approximation, the marginal belief can be easily obtained, the assumption of independent variables in x will inevitably degrade the performance. Therefore, an approximation scheme with a higher accuracy is desirable. By considering the conditional dependencies between two transmitted data symbols, we adopt the Bethe approximation as follows b(x) =

 j

j

k

j

bk (xk, j )

j  bi,k (xk, j , xi, j ) j

(i,k)

j

bk (xk, j )bi (xi, j )

,

(2.12)

where bi,k (xk, j , xi, j ) denotes the joint belief of two interfered symbols over the same resource element. By substituting the expression of (2.12) into the expression of VFE, (2.11) can be written as

2.3 Low-Complexity Receiver Design

FBethe = C +

 

17 j

j

bi,k (xk, j , xi, j ) ln

bi,k (xk, j , xi, j ) j

φi,k

j  j bk (xk, j ) bk (xk, j ) ln + (K − 1) dxk, j , j φk k

dxk, j dxi, j

(i,k)

j

(2.13)

j

where C is a constant. Considering that the marginals bk (xk, j ) and the joint beliefs j bi,k (xk, j , xi, j ) are probabilistic density functions, they are subject to the normalization constraint, i.e.,  j (2.14) bk (xk, j )dxk, j = 1,  j bi,k (xk, j , xi, j )dxi, j dxk, j = 1. (2.15) Moreover, the following marginalization constraint should hold for the joint belief  j

j

bi,k (xk, j , xi, j )dxi, j = bk (xk, j ).

(2.16)

With the above constraints, the minimization of the VFE can be determined using the classic method of multipliers. The Lagrangian can be constructed as L=F + 

  j



 j λk 1 − bk (xk, j )dxk, j +

i



λi,k (xk, j )

(2.17)

k (xk, j , xi, j )dxi, j bi,k

−

j bk (xk, j )

dxk, j ,

(i,k)

where λk and λk,i (xk, j ) denote the multipliers corresponding to the normalization and the marginalization constraints. j To compute the marginal distribution bk (xk, j ), we set the partial derivative of L j j with respect to bk (xk, j ) and bi,k (xk, j , xi, j ) to zero, which is equivalent to the variation of the functional L to zero, i.e., δ(L) = 0. Consequently, we have the stationary points of the functional, formulated as  j j j k (xk, j , xi, j ) ∝ φi,k bi (xi, j )bk (xk, j ) exp −λk,i (xk, j ) − λi,k (xi, j ) , bi,k ⎛ ⎞  j j λi,k (xk, j )⎠ . bk (xk, j ) ∝ φk exp ⎝ (i,k)

Substituting (2.19) into (2.18) yields,

(2.18) (2.19)

18

2 Uplink Multi-user Detection for SCMA System

⎛



bi,k (xk, j , xi, j ) ∝ φk φi φi,k exp ⎝ j

j

j

j



λk,l (xk, j ) +

(k,l),l=i

⎞ λi,k (xi, j )⎠ .

(i,m),m=k

For ease of exposition, we define an auxiliary belief in the following. ⎛ bi\k (xi, j ) ∝ φi exp ⎝ j

j



⎞ λi,m (xi, j )⎠ .

(2.20)

(i,m),m=k j

By performing integration of bi,k (xk, j , xi, j ) over variable xi, j and comparing it to the expression of (2.19), we can obtain the multiplier λk,i (xk, j ) as 

j j φi,k bi\k (xi, j )dxi, j

λk,i (xk, j ) = ln

.

(2.21)

j

Assuming that bi\k (xi, j ) obeys Gaussian distribution, whose mean and variance are j

j

denoted as m i\k and vi\k . Then the multiplier λk,i (xk, j ) can be rewritten as a quadratic polynomial, following j

λk,i (xk, j ) =

h 4k, j vi\k N02

j

2 xk, j

j

−2

h 2k, j m i\k N0

xk, j + C

j

2 = −βk,i xk, j + 2γk,i x k, j + C.

(2.22)

Thus, from (2.19) and (2.22), we can calculate the mean and variance for the marginal j bk (xk, j ), which are ⎛

m xk, j

and vxk, j

⎞  j y 2h k, j j = vxk, j ⎝ 0 + + γk,i ⎠ , vxk, j N0 (i,k) ⎞−1 ⎛ h 2k, j  k 1 =⎝ 0 + + βk,i ⎠ , vxk, j N0 (i,k) m 0xk, j

(2.23)

(2.24)

respectively. j Note that from the above derivations, the updates of parameters m xk, j , m i\k , vxk, j , j

and vi\k also rely on the information from other variables. Hence they are updated j

in an iterative fashion. Again, for simplicity, we define ρk = m 0xk, j vx0k, j

+

2h k, j y j N0

1 vx0k, j

+

h 2k, j N0

j

and k =

since they do not change during the iterations. To further improve the

detection performance in a densely connected network [79], we introduce a damping j factor 0 < α ≤ 1. Let us denote the belief at the lth iteration as bk (l), then after

2.3 Low-Complexity Receiver Design

19

damping, the damped belief is given by j j j b˜k (l) = (bk (l))α (bk (l − 1))(1−α) .

(2.25)

We can see the damped belief is the combination of the belief obtained in the current iteration and the one we get in the last iteration. Obviously, α = 1 corresponds to the original non-damping belief. With damping factor α, the mean and variance are modified as

αm xk, j (l) (1 − α)m xk, j (l − 1) + , (2.26) m˜ xk, j (l) = v˜ xk, j (l) vxk, j (l) vxk, j (l − 1)

−1 1−α α v˜ xk, j (l) = + . (2.27) vxk, j (l) vxk, j (l − 1) Then, based on the Gaussian belief b˜k (xk, j ), we are able to calculate the loglikelihood ratios (LLRs) fed to the channel decoder. After performing decoding, the decoder will output LLRs, which are converted to the a priori probability of data symbol and fed to the detector for commencing the next turbo iteration. We summarize the proposed algorithm in in Algorithm 2.1. j

Algorithm 2.1 Uplink SCMA Receiver Design 1: Initialization: 2: The a priori distributions of users’ transmitted data symbols are set as zero mean Gaussian distribution with an infinite variance; 3: for iter=1:NIter (number of iterations) do 4: For all users ∀k j j 5: Determine βk,i and γk,i according to (2.22); j

j

Compute bi\k (xi, j ) and bk (xk, j ) according to (2.20) and (2.23), (2.24); Update the mean and variance of the damped belief according to (2.26), (2.27); j Calculate LLR L ak based on b˜k (xk, j ) and feed them to the channel decoder; Perform standard BP channel decoding; Calculate m 0xk, j and vx0k, j based on the output extrinsic information from the channel decoder ; 11: end for 6: 7: 8: 9: 10:

2.3.2 Computational Complexity Designing low-complexity receivers is always essential for modern communication systems. As for the proposed algorithm, its complexity is dominated by the integration step in (2.21). Taking the data symbol xk, j as an example. It is interfered with by a total of (D − 1) symbols. Given Gaussian distributions for the beliefs of data

20

2 Uplink Multi-user Detection for SCMA System

Table 2.1 Complexity comparison Algorithm name

Computational complexity O(K |χ | D ) O(K |χ | D ) O[K |χ |(D − 1)] O[K |χ |(D − 1)]

Conventional MPA Max-log based MPA Modified MPA [71] The proposed algorithm

symbols, calculating the belief of xk, j only involves simple addition and multiplication operations. Therefore, As a result, a complexity on the order of O[(D − 1)]) is required. Considering all J resource elements, the order of complexity of the proposed algorithm is O[J (D − 1)]. In contrast, the original MPA receiver which relies on the maximum a posteriori criterion, imposes a complexity order of O(|χ | D ), which increases exponentially with the number of interfered symbols. A reducedcomplexity SCMA detector proposed in [68] utilizes the max-log message passing, which significantly reduces the number of operations. Nevertheless, it still has the order of complexity of |χ | D . Compared to those classic schemes, the complexity of the proposed algorithm only increases linearly with the number of users. For brevity, we summarize the complexity for different SCMA detectors in Table 2.1.

2.4 Convergence Analysis For the proposed iterative detection scheme, it is important to analyze its convergence behavior. In this section, we aim for discussing and deriving the conditions to guarantee the convergence of the proposed algorithm. Taking the lth iteration as an j j example. Denoting m i\k (l) and vi\k (l) as the mean and variance of (2.20) at the lth iteration, which are updated based on the parameters obtained in the l − 1th iteration, following ⎛ vi\k (l) = ⎝ρi + j

⎞−1



j

βi,m (l − 1)⎠ j

(i,m),m=k

⎛

m i\k (l) = vi\k (l) ⎝i + j

j

j



,

(2.28) ⎞

γi,m (l − 1)⎠ . j

(2.29)

(i,m),m=k

The convergence of the proposed algorithm depends on the convergence of the mean and variance of the belief, which is given by the following two propositions. Statement 2.1 The variance v˜ xk, j of the belief is guaranteed to converge, satisfying

2.4 Convergence Analysis

21

v˜ xk, j (l) ≤ v˜ xk, j (l − 1). Proof Observe from (2.22) and (2.28), the updating of parameter β kj,i (l) can be expressed as ⎛



β K ,i (l) = −a ⎝ρiJ + j

⎞−1 βi,m (l − 1)⎠ j

,

(2.30)

(i,m),m=k j

where we use the shorthand notation a = h 4k, j /N02 . If βi,k ≤ 0, we can derive the following inequality,   j j β (l) − β (l − 1) (i,m),m=k i,m i,m j j   βk,i (l + 1) − βk,i (l) =    j j j j ρi + (i,m),m=k βi,m (l − 1) ρi + (i,m),m=k βi,m (l)    j a j βi,m (l) − βi,m (l − 1) . (2.31) ≥ j (ρi )2 (i,m),m=k a



By stacking all β values sharing the same index j for the resource element to form the vector β j , the vector-form for the above inequality is formulated as β j (l + 1) − β j (l) ≥ ≥

a j (ρi )2 l

 A β j (l) − β j (1 − 1)

a

j (ρi )2l

 Al β j (1) − β j (0) ,

(2.32)

where A is the multi-user adjacent matrix with Aik = 1 if and only if user i and k interfere with each other. At the initial stage, the symbol variance is infinity, which j j indicates that vxk, j (1) ≤ vxk, j (0) holds. Therefore, we have βk,i (1) ≥ βk,i (0) and furj ther β j (l + 1) − β j (l) ≥ 0. We see that βk,i (l) is monotonically increasing during the iterations. According to (2.24), the variance of the belief will decrease with more iterations, i.e., vxk, j (l + 1) < vxk, j (l). We prove that the variance vxk, j will guarantee its convergence for the proposed iterative algorithm. Considering our damping scheme above, it can be seen that v˜ xk, j (l + 1) < v˜ xk, j (l) is equivalent to α 1−α α 1−α + ≥ + . vxk, j (l) vxk, j (l + 1) vxk, j (l − 1) vxk, j (l)

(2.33)

Obviously, with the convergence-guaranteed variance vxk, j , we have vx α (l) ≥ k, j

and vx 1−α ≥ (l+1) k, j to converge.

1−α . vxk, j (l)

α vxk, j (l−1)

Hence, the variance of the damped belief is also guaranteed 

22

2 Uplink Multi-user Detection for SCMA System

Next, we move our focus on the convergence of the a posteriori mean m˜ xk, j . With the increase of the iteration index l, the variation of the parameter m˜ xk, j is expected to decrease in two iterations, i.e., |m˜ xk, j (l + 1) − m˜ xk, j (l)| ≤ |m˜ xk, j (l) − m˜ xk, j (l − 1)|.

(2.34)

As proved in Proposition 3.1, the variance is guaranteed to converge. Therefore, we assume that after a sufficiently large number of iterations, the parameters related to j variance, e.g., vxk, j and vi\k have already converged to fixed values v∗ and v¯ ∗ for all k, respectively. Therefore, it is easily seen that v˜ xk, j will converge to v∗ . The original a posteriori mean can then be expressed as ⎛ m xk, j (l) = v∗ ⎝k + j



⎞ γk,i (l)⎠ . j

(2.35)

(i,k)

Given that the a posteriori mean m˜ xk, j (l) = αm xk, j (l) + (1 − α)m xk, j (l − 1), we have m˜ xk, j (l + 1) − m˜ xk, j (l) = α(m xk, j (l + 1) − m xk, j (l)) + (1 − α)(m xk, j (l) − m xk, j (l − 1))      j j j j = v∗ α γk,i (l + 1) − γk,i (l) + (1 − α) γk,i (l) − γk,i (l − 1) . (i,k)

(2.36) j

Equation (2.36) indicates that the convergence of m˜ xk, j depends on the variation γk,i in the iterative process. j To further discuss the parameter γk,i , we substitute (2.29) into (2.22) and yields, ⎛ γi,m (l) = −b ⎝i + j

j



⎞ γi,m (l − 1)⎠ , j

(2.37)

(i,m),m=k

where b is the shorthand notation for j

h 2k, j v˜ ∗ . N0

j

Since i is fixed during the iterations,

the variation of γk,i in two consecutive iterations is given by j

j

γi,m (l + 1) − γi,m (l) = b



  j j γi,m (l − 1) − γi,m (l) .

(2.38)

(i,m),m=k

Again, we stack all γ with the same resource index j as a vector and (2.38) can be written in a matrix form as  γ j (l + 1) − γ j (l) = bA γ j (l) − γ j (l − 1) .

(2.39)

2.4 Convergence Analysis

23

Statement 2.2 The sufficient and necessary condition to guarantee the convergence of the a posteriori mean m˜ xk, j is that the largest absolute value of adjacent matrix A’s eigenvalues (spectral radius) satisfies ρ(A) < b1 . Proof We commence our discussion from the necessary condition. According to (2.39), it is readily to express γ j (l + 1) − γ j (l) as  γ j (l + 1) − γ j (l) = bl Al γ j (1) − γ j (0) .

(2.40)

When l is approaching infinity we derive the limits of both sides of (2.40), which is formulated as   lim γ j (l + 1) − γ j (l) = lim (bA)l γ j (1) − γ j (0) .

l→∞

l→∞

(2.41)

Given the convergence of m˜ xk, j ,      lim α γ j (l + 1) − γ j (l) + (1 − α) γ j (l) − γ j (l − 1) l→∞   = lim α(bA)l + (1 − α)(bA)l−1 · γ j (1) − γ j (0) = 0. l→∞

(2.42)

As γ j (1) − γ j (0) = 0 holds, the above equation requires that liml→∞ (bA)l = 0. Assuming that A has an eigenvalue of λ and an associated eigenvector of ν, we have   ν lim (bA)l = lim (bA)l ν = lim (bλ)l ν l→∞

l→∞

l→∞

= ν lim (bλ)l . l→∞

(2.43)

Since the eigenvector is not zero, we must have liml→∞ (bλ)l = 0, which is equivalent to |bλ| < 1. Therefore, for any eigenvalue λ of matrix A, its absolute value satisfies |λ| < b1 . This gives the necessary condition of Statement 2.2. As for the sufficient condition, according to the basic matrix theorem, we can write γ j (l + 1) − γ j (l) as  γ j (l + 1) − γ j (l) = bA γ j (l) − γ j (l − 1)  ≤ b · ρ(A)γ j (l) − γ j (l − 1),

(2.44)

where ρ(A) denotes the spectral radius of matrix A. If ρ(A) < 1/b, we have γ j (l + 1) − γ j (l) < γ j (l) − γ j (l − 1).

(2.45)

As shown in (2.36), the convergence of γ will guarantee the convergence of m˜ xk, j , which gives the sufficient condition. 

24

2 Uplink Multi-user Detection for SCMA System

So far, we have analyzed the convergence behavior of the proposed iterative algorithm. In particular, the a posteriori variance is guaranteed to converge while the convergence of the a posteriori mean depends on the spectral radius of the adjacent matrix.

2.5 Simulation Results In this section, we will validate the proposed algorithm through numerical simulations. In the simulations, we consider two different SCMA uplink systems, i.e., K = 6 and J = 4 with overloaded factor λ = 150%, and K = 12 and J = 6 with overloaded factor λ = 200%. The codebooks for the above two systems can be found in [80, 81], respectively. The information bits are encoded using an LDPC code with a rate of 5/7. The channel is flat Rayleigh fading and we assume that the channel state information (CSI) is perfectly known. For turbo iteration, the maximum number of iterations is set to NIter = 10. We first compare the bit error rate (BER) performance of our proposed algorithm and some existing methods in the literature. Figures 2.2 and 2.3 show the performance corresponding to the SCMA systems with λ = 150% and λ = 200%, respectively. As for the damping factor, there have been different schemes for searching for the optimal value. In our simulations, we use the α = 0.3 given in [79] for simplicity. From Figs. 2.2 and 2.3, we can observe that the proposed algorithm is capable of approaching the optimal MPA receiver with negligible performance loss. However, compared to the exponentially increased complexity of the MPA receiver, the complexity of our proposed algorithm only increases linearly with the number of users.

Fig. 2.2 BER performance of different algorithms (λ = 150%)

2.5 Simulation Results

25

Fig. 2.3 BER performance of different algorithms (λ = 200%)

Fig. 2.4 BER performance versus the number of iterations (λ = 150%)

The variational inference method relying on the MF approximation has very low complexity, but its assumption of independent data symbols will lead to significant performance degradation. The modified MPA receiver proposed in [71] is comparable to the proposed algorithm with a relatively low overloaded factor of λ = 150%. However, when the overloaded factor becomes large, i.e., λ = 200%, a performance loss can be observed. This is mainly because more short cycles will appear on the factor graph, which will severely affect the convergence of the MPA. Next, we evaluate the convergence of the proposed algorithm in Fig. 2.4, where the BER performance parameterized by different values of E b /N0 versus the number of iterations is illustrated. Obviously, increasing the number of iterations will lead

26

2 Uplink Multi-user Detection for SCMA System

Fig. 2.5 EXIT chart between the SCMA detector and channel decoder (λ = 150%)

to a lower BER. After running a few iterations, the BER performance will converge and the performance gain becomes marginal, especially for the E b /N0 = 3 dB case. This motivates us to reduce the number of iterations when the value of E b /N0 = 3 is small. To further show the convergence behavior, we depict the extrinsic information transfer (EXIT) chart in Fig. 2.5 for representing the convergence of the mutual information between the channel decoder and the SCMA uplink detector. We use I A,dec and I E,dec to denote the mutual information between the information bits and the LLRs fed to the decoder, and the mutual information between the information bits and the output LLRs of the decoder, respectively. Similarly, the mutual information between the transmitted information bits and the LLRs related to the SCMA detector can be defined as I A,det and I E,det . In Fig. 2.5, we can see that an open tunnel is attained at E b /N0 = 4d B, which indicates that the proposed algorithm will converge.

2.6 Conclusions In this chapter, we develop a Bayesian inference-based approach for low-complexity uplink SCMA symbol detection. In particular, we commence our receiver design by factorizing the joint a posteriori distribution by the product of several local functions. Then the Bethe approximation is used for constructing the trial distribution, which aims for approximating the original a posteriori distribution. By constructing the Lagrangian and the corresponding constrained optimization problem, the marginals of data symbols can be obtained. Relying on the Gaussian representation of the beliefs, the updating of parameters only involve simple operations, which leads to a complexity that only increases linearly with the number of users. We further prove the convergence of the a posteriori variance and derive the convergence conditions

2.6 Conclusions

27

of the a posteriori mean for the proposed algorithm. Simulation results verified the effectiveness of our proposed algorithm for two different SCMA systems with overloaded factors of λ = 150% and λ = 200%. It can be seen that the performance of the proposed algorithm can closely attach to that of the optimal MPA receiver.

Chapter 3

Downlink Multi-user Detection for MIMO-SCMA System

3.1 Introduction Chapter 2 presents the low complexity receiver design for uplink SCMA system. In this chapter, we will focus on the downlink SCMA system and its receiver design issue. In general, by exploiting spatial diversity, the the multiple-input multipleoutput (MIMO) systems can be combined with SCMA to further improve the spectral efficiency. Due to the inter-user interference and possible inter-antenna interference, the optimal detection for MIMO-SCMA system is not feasible due to very high computational complexity [82]. In [83], the data symbols transmitted by a MIMONOMA system are approximated as Gaussian variables and two low-complexity MPA-based detection schemes were developed. What’s more, the commonly adopted wide-band communication systems will also experience frequency selective fading and the received signal suffers from inter-symbol interference (ISI) as well [76, 79]. It is well acknowledged that running the classic MPA, a.k.a., belief propagation (BP), on a loop-free factor graph yields the exact marginals of the variables [84]. However, for MIMO-SCMA system over frequency selective channels, MPA may suffer from convergence issue due to the loopy factor graph representations. This can be explained by the theorem of variational free energy (VFE). The BP message passing rules are equivalent to minimizing the VFE under Bethe independence constraints [77]. When the factor graph has cycles, the VFE is non-convex and the resultant MPA receiver fails to converge. Therefore, how to guarantee the convergence of the widely adopted MPA detection is of great importance and needs further investigation. Moreover, by revisiting the inter-user interference in downlink systems, we observe that the users also receive the information corresponding to other users. Therefore, there is a chance to achieve further diversity gain by sharing information among the users. The direct way is to transmit all received samples of one user to the others. However, this decentralized mechanism is not realistic. On the one hand, two downlink users may be far away from each other and direct information transmission consumes huge power. On the other hand, a complex scheduling scheme is required to avoid packet collision and to ensure all received samples are collected. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Yuan et al., Receiver Design for High Spectral Efficiency Communication Systems in Beyond 5G, https://doi.org/10.1007/978-981-19-8090-9_3

29

30

3 Downlink Multi-user Detection for MIMO-SCMA System

To this end, the distributed processing scheme based on cooperation between users was proposed for in-network information fusion, which relies on only local computations and communications with neighboring users. For tracking problems, the distributed processing scheme has been widely used [85, 86]. However, for communications, only a few papers considered the distributed processing strategy and did not investigate its application in fading channels [87–89]. In this chapter, we aim for deriving a convergent version of the classic MPA receiver for MIMO-SCMA systems. Moreover, by enabling low-cost communications between neighboring users, we develop the cooperative detection schemes to enhance communication performance. We first introduce the general downlink MIMO-SCMA model with frequency selective fading channels. To reduce the complexity of the MPA receiver, we construct a stretched factor graph by intentionally adding auxiliary variables for factorizing the joint a posteriori distribution. Moreover, to address the convergence issues of MPA receiver in loopy factor graph, we convexify the Bethe free energy and proposed a convergence-guaranteed BP-EP receiver. As a further step, we observe that the global messages on the factor graph can be expressed as the product of several user-dependent local messages. By exploiting this property, we develop a cooperative detection scheme in a distributed way. A consensus-based scheme is proposed to avoid double counting of local messages. Since all messages are derived in Gaussian forms, only means and variances of the local messages are exchanged between users and updated. Considering that nonideal inter-user links with noise will affect the convergence speed of the original consensus-based method, we proposed an alternative direction method of multipliers (ADMM)-based algorithm [90] which minimizes the Kullback–Leibler divergence [91] between the global message and the product of local messages in each iteration. Finally, we demonstrate the performance of the proposed convergence-guaranteed MPA receiver and of the cooperative detection method for downlink MIMO-SCMA systems via numerical simulations.

3.2 MIMO-SCMA Model Throughout this chapter, a MIMO-SCMA system with an J -antenna base station and K single-antenna downlink users is considered. As discussed in Chap. 2, the SCMA encoder maps every log2 M bits to an SCMA codeword of dimension J . Let us denote n n T , . . . , xk,J ] , which the transmitted codeword of user k at time instant n by xkn = [xk,1 is multiplexed over over J antennas and transmitted. The block diagram for the considered MIMO-SCMA system is depicted in Fig. 3.1. Let the transmitted symbol at the jth antenna and the nth time instant be s nj , which is given by s nj =

K  k=1

n xk, j.

(3.1)

3.2 MIMO-SCMA Model

31

Fig. 3.1 System model for downlink MIMO-SCMA

We assume that the signal transmits through multipath channels with L paths and is acquired by different downlink users. In particular, the received signal at time instant n and user k can be expressed as ykn =

L−1 J  

h lj,k s n−l + ωkn , j

(3.2)

j=1 l=0

where h lj,k is the channel coefficient of the lth path between the jth transmit antenna and the kth user, and ωkn is the corresponding additive white Gaussian noise (AWGN) sample with zero mean and power spectral density N0 .

3.2.1 Probabilistic Model Assume that a total number of N codewords are transmitted to the users. The transmitted SCMA codewords and received signal samples of the kth user are denoted by Xk and yk . Based on the received signal yk at the kth user, the symbol detection can be carried out by performing the maximum a posteriori (MAP) detection, expressing as ˆ k = arg max p(Xk |yk ) X Xk  = arg max p(X|yk )dX\Xk . Xk

(3.3)

32

3 Downlink Multi-user Detection for MIMO-SCMA System

Following Bayesian rules, p(X|yk ) reads p(X|yk ) ∝ p(X) p(yk |X),

(3.4)

where p(X) and p(yk |X) denote the joint a priori distribution the joint likelihood function, respectively. In general, the transmitted data symbols are independent. Therefore, the joint a priori distribution can be fully factorized by p(X) =  n n j,k,n p(x k, j ), where the a priori distribution p(x k, j ) can be calculated based on the LLRs output from the channel decoder. The optimal MAP receiver (3.3) experiences a computational complexity that increases exponentially with the number of users, antennas, and channel paths. To reduce the receiver complexity, we will propose low-complexity message passing receivers for the considered MIMO-SCMA system.

3.3 Stretched Factor Graph and Low-Complexity Message Passing Receiver 3.3.1 Factor Graph Representation With the assumption of independent noise samples at different time instants, we can factorize p(yk |X) as

p(yk |X) ∝

 n

⎛ 2 ⎞ n J L−1 l K n−l yk − j=1 l=0 h j,k k=1 xk, j ⎟ ⎜ exp ⎝− ⎠. 2N0 



(3.5)



f nk

Therefore, the joint a posteriori distribution is factorized as ⎛ 2 ⎞ n J L−1 l K n−l − j=1 l=0 h j,k k=1 xk, j  ⎟ ⎜ yk n p(xk, p(X|yk ) ∝ ⎠ , (3.6) j ) exp ⎝− 2N 0 n j,k which can be represented by a factor graph, as shown in Fig. 3.2.

3.3.2 Message Passing on Stretched Factor Graph The standard message passing rules are derived in (1.9)–(1.11). In Fig. 3.2, there are J K L variables connecting to a single factor vertex. According to (1.9), calculation

3.3 Stretched Factor Graph and Low-Complexity Message Passing Receiver

33

Fig. 3.2 Factor graph representation of the factorization in (3.6), where f kn denotes the local likelihood function corresponding to the received sample ykn . For ease of exposition, we only illustrate part of the factor graph including the variable vertices connected to factor vertex f kn

n n of the message μ fkn →xk,n j (xk, j ) requires integrating (J K L − 1) variables over f k , 2 which introduce a complexity order of O(N (J K L) ) corresponding to the standard BP message passing receiver for MIMO-SCMA systems based on the factor graph illustrated in Fig. 3.2. For complexity reduction purpose, we introduce auxiliary variables to reduce the number of to-be-updated messages, leading to a significantly lower receiver complexity. By using the following notation,

n rk, j

=

L−1 

h lk, j s n−l j ,

(3.7)

l=0

the likelihood function (3.5) is rewritten as p(yk |X) ∝

 n

 exp −

|ykn −

j

2N0

r nj,k |2

 ψkn



φ nj

 ,

(3.8)

j

L−1 l n−l

K n n n with the shorthand notations ψkn = δ(rk, l=0 h k, j s j ) and φ j = δ(s j − k=1 j − n xk, j ) being the equality constraints. Based on the factorization in (3.8), we are capable of constructing a novel ‘stretched factor graph’ with auxiliary variables, as illustrated in Fig. 3.3. Following the message updating rules (1.9) and (1.10), we can obtain the messages on factor graph as: n • Message μxk,n j →φ nj (xk, j ) (Expectation Propagation):

34

3 Downlink Multi-user Detection for MIMO-SCMA System

Fig. 3.3 Stretched version of factor graph in Fig. 3.2

n We first focus on the incoming message μxk,n j →φ nj (xk, j ), which can be regarded as n the a priori distribution p(xk, j ) of symbols, given by

n p(xk, j) =

M 

n pi δ(xk, j − χi ),

(3.9)

i=1

where χi and pi are the ith constellation point and associated probability, respectively. The probability pi is determined based on the output LLRs from the channel n decoder. The discrete distribution p(xk, j ) can be approximated by a Gaussian one through direct moment matching. Nevertheless, it leads to inevitable performance degradation. To tackle this problem, we resort to expectation propagation (EP) which matches the moments of belief by exploiting the information from detector [92], which is expected to improve the performance. With the assumption of Gaussian repn n n n n resentation of the message μφ nj →xk,n j (xk, j ), i.e., G(m φ j →xk, j , vφ j →xk, j ), we can obtain n as the mean and variance of the belief of xk, j m xk,n j = vxk,n j =

1 2π vφ nj →xk,n j 1 2π v f →xk,n j − |m xk,n j |2 .

·

M 

 χi pi exp −

i=1

·

M  i=1

(m φ nj →xk,n j − χi )2

 |χi | pi exp − 2



vφ nj →xk,n j (m φ nj →xk,n j − χi )2

,

(3.10)



vφ nj →xk,n j (3.11)

3.3 Stretched Factor Graph and Low-Complexity Message Passing Receiver

35

n Consequently, the message μxk,n j →φ nj (xk, j ) can be obtained in Gaussian, whose mean and variance are   m xk,n j m φ nj →xk,n j n n n n − m xk, j →φ j = vxk, j →φ j , (3.12) vxk,n j vφ nj →xk,n j vxk,n j vφ nj →xk,n j . (3.13) vxk,n j →φ nj = vφ nj →xk,n j − vxk,n j

• Messages related to φ nj and ψk,n j (Belief propagation): n Having the messages μxk,n j →φ nj (xk, j ), ∀k in hand, with the Gaussian assumption of n n μs nj →φ nj (s j ) as G(m s nj →φ nj , vs nj →φ nj ), we obtain the message μφ nj →xk,n j (xk, j ) as follows,  n μφ nj →xk,n j (xk, j) ∝

δ(s nj −

K 

n xk, j)



μx n

k ,j



k =k

k=1

n n n n →φ nj (x k  , j )μs nj →φ nj (s j )ds j dx k  , j

∝ G(m φ nj →xk,n j , vφ nj →xk,n j ),

(3.14)

where the mean m φ nj →xk,n j and variance vφ nj →xk,n j are given by, m φ nj →xk,n j = m s nj →φ nj −



m x n

k ,j

k  =k

vφ nj →xk,n j = vs nj →φ nj +

 

k =k

vx n

k ,j

→φ nj ,

(3.15)

→φ nj .

(3.16)

In a similar way, the message from φ nj to s nj can be easily obtained in a Gaussian form, whose mean and variance are  m xk,n j →φ nj , (3.17) m φ nj →s nj = k

vφ nj →s nj =



vxk,n j →φ nj .

(3.18)

k=k

As for the ψk, j -related messages, their means and variances can be obtained as

m ψk,n j →s n−l j

⎛ ⎞ L−1   1 ⎝ = l h lk, j m s n−l  →ψ n ⎠ , m rk,n j →ψk,n j − j k, j h k, j  

vψk,n j →s n−l = j and

vrk,n j →ψk,n j +

L−1

(3.19)

l =0,l =l

l  =0,l  =l



|h lk, j |2 vs n−l  →ψ n

|h lk, j |2

j

k, j

,

(3.20)

36

3 Downlink Multi-user Detection for MIMO-SCMA System

m ψk,n j →rk,n j =

L−1 

h lk, j m s n−l n , j →ψk, j

(3.21)

h lk, j vs n−l n , j →ψk, j

(3.22)

l=0

vψk,n j →rk,n j =

L−1  l=0

with m s nj →ψk,n j and vs nj →ψk,n j being  m s nj →ψk,n j = vs nj →ψk,n j

m φ nj →s nj vφ nj →s nj



vs nj →ψk,n j = vφ−1n →s n + j

L−1 

j

+

L−1 m n+l  ψk, j →s nj

vψk,n+lj →s nj

l=1

 ,

(3.23)

−1

vψ−1n+l →s n

.

j

k, j

l=1

(3.24)

n • Messages related to rk, j (Belief Propagation): n Finally, let us calculate the message μ fkn →rk,n j (rk, j ). As we have already obtained n the message μψk,n j →rk,n j , the outgoing message from f kn to variable rk, j is expressed as 

n 2   |ykn − j rk, j| n n n μr n  → fkn (rk, )drk, , (3.25) μ fkn →rk,n j (rk, j ) ∝ exp − j j k, j 2N0  j = j

which can be written as a Gaussian distribution with mean and variance  m r n  → fkn , m fkn →rk,n j = ykn − 

j = j

and v fkn →rk,n j = N0 +

 j  = j

(3.26)

k, j

vr n

k, j



→ f kn ,

(3.27)

respectively. It can be observed that in the original factor graph based on the factorization in n (3.6), when calculating the message xk, j , integration over all the other (N L J − 1) variables are required. In contrast, by adopting auxiliary variables, the stretched factor graph representation based on (3.8) can reduce the number of integration to (J + K + L − 1), which is much lower than J K L − 1, especially for large antenna size and massive access applications.

3.4 Convergence-Guaranteed Message Passing Receiver

37

3.3.3 Algorithm Summary In the above, we have proposed a BP-MP message passing receiver relying on a novel stretched factor graph, which is constructed by introducing several auxiliary variables. We can observe the resultant factor graph is with loops, indicating the message updating will involve other variables. Therefore, all messages are updated in an n iterative fashion. At the initial stage, the messages μgk,n j →xk,n j (xk, j ), ∀k, j, n can be set as Gaussian distributions with zero means, since there is no a priori information concerning the transmitted data symbols. Then, in each iteration, the means and variances of all messages are computed according to (3.12)–(3.27). Finally, the extrinsic n information of bits can be determined based on the message message μφ nj →xk,n j (xk, j ), which are fed to the channel decoder. The channel decoder will update the LLRs of bits after channel decoding and starts the next turbo iteration.

3.4 Convergence-Guaranteed Message Passing Receiver Although the stretched factor graph helps to reduce the complexity of the message passing receiver, the loopy factor graph, e.g., Fig. 3.3 still results in the convergence issue of the iterative receiver. Several works have made efforts to improve the convergence, e.g., the tree reweighted BP and damping message method. In this section, we focus on the generalized VFE framework [84] for designing an iterative message passing receiver with guaranteed convergence. First, we briefly introduce how to derive the standard message passing rules based on free energy. Then, we show that by convexifying the Bethe free energy, a convergence-guaranteed message passing receiver can be determined even for the loopy factor graph.

3.4.1 VFE and Belief Propagation Without loss of generality, we consider a joint distribution p(x) of random variables x = [x1 , . . . , xi , . . .], which can be factorized as the product of non-negative functions as  f a (xa ), (3.28) p(x) = a

where f a is the the ath local function of variables xa . We use N(a) as the set containing the indices i if xi is an element of xa . The factorization of (3.28) can be represented by a factor graph, which is loopy if same variables appear

in different local functions f a . To calculate the marginal distribution p(xi ) = x\xi p(x),

38

3 Downlink Multi-user Detection for MIMO-SCMA System

we have to perform the summation (integration) over all variables except xi . The VFE framework is adopted to efficiently find the approximate marginals (beliefs) of variables. For the variational approach, a positive function b(x) (trail distribution) is used for approximating the target distribution p(x). The similarity of two distributions can be characterized by the VFE, which is defined as the Kullback–Leibler divergence between two distributions [91], i.e.,  b(x) dx F = D[b(x)| p(x)] = b(x) ln p(x)   = b(x) ln b(x)dx − b(x) ln p(x)dx, (3.29)    −H (b)

where H (b) can be regarded as the entropy of b(x). Our aim is to find a distribution b(x) which minimizes the VFE and can be marginalized readily. Considering the conditional dependencies of variables in x, we adopt the Bethe approximation, given by 

a ba (xa ) . b (xi )|N(i)|−1 i i

b(x) = 

(3.30)

where ba (xa ) and bi (xi ) denote the joint belief of variables xa and the belief of xi , respectively. Substituting (3.30) into (3.29) yields the Bethe free energy as 

FB = −

ba (xa ) ln f a (xa )dxa +

a



+

ba (xa ) ln ba (xa )dxa

a

 (1 − |N(i)|)



bi (xi ) ln bi (xi )dxi .

(3.31)

i

 As for the beliefs, they obey the normalization constraints ba (xa )dxa = 1, and bi (xi )dxi = 1 as well as the marginalization constraint bi (xi ) = ba (xa )dxa \xi . Letting the associated Lagrangian multipliers be βi , βa and βai (xi ), the constrained Lagrangian is expressed as L B FB +



 βi

i

+

   i

f a ∈N(i)

    βa bi (xi )dxi − 1 + ba (xa )dxa − 1  βai (xi ) bi (xi ) −



a

 ba (xa )dxa \xi dxi .

(3.32)

3.4 Convergence-Guaranteed Message Passing Receiver

39

Calculating the partial derivatives of L B with respect to the multipliers and setting them to zero yields the normalization and marginalization constraints. Setting the partial derivative with respect to ba (xa ) to zero gives ln ba (xa ) = ln f a (xa ) +



βai (xi ) + βa − 1.

(3.33)

i∈N(a)

Similarly, by computing the partial derivative of L B with respect to bi (xi ), we obtain (|N(i)| − 1) ln bi (xi ) = 1 − βi +



βai (xi ).

(3.34)

f a ∈N(i)

Setting exp (βai (xi )) = (3.34) reads

 f a  ∈N(i)\ f a

ba (xa ) ∝ f a (xa ) bi (xi ) ∝

μ fa →i (xi ) and substituting it into (3.33) and





i∈N(a) f a  ∈N(i)\ f a



μ fa →i (xi ),

μ fa →i (xi ).

(3.35) (3.36)

f a ∈N(i)

Following the marginalization constraint, we have  bi (xi ) ∝ ∝



ba (xa )dxa \xi f a (xa )





i  ∈N(a)\i f a  ∈N(i  )\ f a

μ f  →i  (xi  )dxi  a

 f a  ∈N(i)\ f a

μ fa →i (xi ). (3.37)

By comparing (3.37) with (3.36), we arrive at the expression of μ fa →xi (xi ), which is formulated as    μ fa →xi (xi ) ∝ f a (xa ) μ fa →xi  (xi  )dxa \xi . (3.38) i  ∈N(a)\i f a  ∈N(i  )\ f a

It is observed that (3.38) is the message from factor vertex f a to variable vertex xi , as defined (1.9). Following the definition bi (xi ) = μxi → fa (xi )μ fa →xi (xi ), we finally have the message passing rule in (1.10), i.e., μxi → fa (xi ) =

bi (xi ) ∝ μ fa →xi (xi )

 f a  ∈N(i)\ f a

μ fa →xi (xi ).

(3.39)

40

3 Downlink Multi-user Detection for MIMO-SCMA System

3.4.2 Convergence-Guaranteed BP-EP Receiver We briefly introduce the derivations of message passing rules from the VFE perspective. The standard BP updating rules (1.9) and (1.10) are in fact equivalent to minimizing the constrained Bethe free energy. Based on the free energy framework, the convergence issue of the standard BP can be interpreted by the fact that the Bethe free energy is non-convex and has several local minima, which motivates us to derive a novel convergence guaranteed version of the classic MPA. Using the shorthand notations  Hi (b) = −



and Ha (b) = −

bi (xi ) ln bi (xi )dxi ,

(3.40)

ba (xa ) ln ba (xa )dxa ,

(3.41)

the entropy H (b) is rewritten as H (b) =



(1 − N(i)) Hi (b) +



Ha (b).

(3.42)

a

i

The pioneering work by Yedidia [84] provides a generalized form of Bethe approximation-based entropy, which is written as the linear combination of Hi (b) and Ha (b), H˜ (b) =



ci Hi (b) +

i



ca Ha (b),

(3.43)

a

where ci and ca denote the counting numbers, satisfying ci = 1 − fa ∈N(i) ca . Obviously, when ca = 1, H˜ (b) = H (b). With the aid of counting numbers, the optimization problem in (3.32) is rewritten as min −

 a

ba (xa ) ln f a (xa )dxa − 

s.t bi (xi ) = ba (xa )dxa \xi ,  ba (xa )dxa = 1,  bi (xi )dxi = 1.

 a

ca Ha (b) −



ci Hi (b)

i

(C1) (C2) (C3)

To solve the optimization problem in (C1), we construct the Lagrangian and derive the message passing rules. Using the method of multipliers, the corresponding beliefs are expressed as

3.4 Convergence-Guaranteed Message Passing Receiver



41



βai (xi ) ca i∈N(a)    βai (xi ) bi (xi ) ∝ . exp − ci f ∈N(i) 1

ba (xa ) ∝ f a (xa ) ca



exp

(3.44) (3.45)

a

For ease of exposition, we use shorthand notations τi = (1 − ci )/|N(i)|, μxi → fa (xi ) = exp( βaic(xa i ) ) and μ fa →xi (xi ) = biτi (xi ) exp(−βai (xi )), and obtain the following equations exp(βai (xi )) = μcxai → fa (xi ) exp(−βai (xi )) = μ fa →xi (xi ) ·

(3.46) bi−τi (xi ).

(3.47)

By substituting (3.46) and (3.47) into (3.44) and (3.45), we have the beliefs 1

ba (xa ) ∝ f a (xa ) ca



μxi → fa (xi )

(3.48)

μ fa →xi (xi ).

(3.49)

i∈N(a)

c +τi |N(i)|

bi i



(xi ) ∝

f a ∈N(i)

Following the standard message passing rules, we define two auxiliary messages as  μ˜ fa →xi (xi ) ∝ μ˜ xi → fa (xi ) ∝

1

f aca (xa )

 i  ∈N(a)\i

 f a  ∈N(i)\ f a

μxi  → fa (xi  )dxi 

μ fa →i (xi ).

(3.50) (3.51)

After straightforward manipulations, the message bi (xi ) is given by bi (xi ) = μ˜ xi → fa (xi )μ fa →xi (xi )  = ba (xa )dxa \xi = μ˜ fa →xi (xi )μxi → fa (xi ).

(3.52)

Comparing (3.52) with μcxai → fa (xi )μ fa →xi (xi ) = biτi (xi ) yields τi

τi −1

μxi → fa (xi ) = μ˜ xcai → fa (xi )μ faca→xi (xi ).

(3.53)

Based on (3.53), we have the message from factor vertex f a to variable vertex xi , expressed as

42

3 Downlink Multi-user Detection for MIMO-SCMA System τi −ca c −τ +1

ca c −τ +1

i μ fa →xi (xi ) = μ˜ xai →i fa (xi )μ˜ faa →x (xi ). i

(3.54)

Substituting (3.54) into (3.53) gives the expression of μxi → fa (xi ) based on the auxiliary messages, τi −1 c −τ +1

1 c −τ +1

i μxi → fa (xi ) = μ˜ xai →i fa (xi )μ˜ faa →x (xi ). i

(3.55)

Finally, we arrive at the generalized message passing rules, with the definition of τi , given by  γ  γ −1 μ fa →xi (xi ) = μ˜ fa →xi (xi ) ai μ˜ xi → fa (xi ) ia ,  γ −1  γ μxi → fa (xi ) = μ˜ fa →xi (xi ) ai μ˜ xi → fa (xi ) ia ,

(3.56) (3.57)

and γia = |N(i)|/ where γai = |N(i)|ca /(|N(i)|ca + ci + |N(i)| − 1) (|N(i)|ca + ci + |N(i)| − 1). In particular when we set ca = 1 and ci = 1 − |N(i)|, the generalized message passing rules are identical to the standard BP. As discussed above, the classic MPA fails to converge mainly because of the non-convex free energy. With the construction of the generalized Bethe free energy, we can convexify the free energy by choosing appropriate counting numbers. A well-known example is the tree re-weighted BP [93], which uses the edge appearance probability of the spanning tree as counting number ca , yielding a convexified free energy. Nevertheless, tree re-weighted BP is not suitable for most cases as spanning trees can only represent a few convex free energies. To this end, we consider the following proposition, which gives the general conditions for convex free energies. Proposition 3.1 The generalized Bethe free energy is convex if the counting numbers cia , cii , and caa satisfy cia ≥ 0, cii ≥ 0, caa ≥ 0  cia , ca = caa +

(3.58) (3.59)

i∈N(a)



ci = cii −

cia .

(3.60)

f a ∈N(i)

Proof To prove the above proposition, we first substitute (3.59) and (3.60) into (3.31), yielding FB = −



ba (xa ) ln f a (xa )dxa −

a

−

 i

cii Hi (b) −

 i, f a ∈N(i)



caa Ha (b)

a

cia (Ha (b) − Hb (b)).

(3.61)

3.4 Convergence-Guaranteed Message Passing Receiver

43

The convexity of FB depends on the second-order partial derivative. For the first term on the RHS of (3.61), its partial derivative with respect to the belief is 0. Therefore, we focus on the convexity of the following functional Fconv = −



caa Ha (b) −

a

 i



cii Hi (b) −

cia (Ha (b) − Hi (b)).

(3.62)

i, f a ∈N(i)

By calculating the second-order partial derivatives, we obtain 1 ∂ 2 Hi (b) , =− ∂bi (xi )2 bi (xi ) ∂ 2 Ha (b) 1 =− . ∂ba (xa )2 ba (xa )

(3.63) (3.64)

Since the beliefs are with positive values, the first two

terms on the RHS of Fconv are also convex. Hence, Fconv is convex if and only if − i, fa ∈N(i) cia (Ha (b) − Hi (b)) is convex. Now, the problem becomes the analysis of the convexity of Ha (b) − Hi (b). Note  that bi (xi ) = ba (xa )dxa \xi = ba (xi ). Then, Fai = Hi (b) − Ha (b) is rewritten as  Fai =

 ba (xa ) ln ba (xa )dxa −

ba (xi ) ln bi (xi )dxi .

(3.65)

To prove the convexity of Fai is equivalent to showing that the Hessian matrix is positive definite, i.e., 

 b˜

∂ 2 Fai ∂ 2 Fai ∂ba (xa )2 ∂ba (xa )∂bi (xi ) ∂ 2 Fai ∂ 2 Fai ∂bi (xi )∂ba (xa ) ∂bi (xi )2

 b˜ T dxa ≥ 0,

(3.66)

where b˜ = [b˜a (xa ), b˜i (xi )] denotes arbitrary beliefs. The components of the Hessian matrix are obtained by calculating the second-order partial derivatives, expressed as 1 ∂ 2 Fai , = ∂ba (xa )2 ba (xa ) ∂ 2 Fai ba (xi ) =− , ∂bi (xi )2 (bi (xi ))2 ∂ 2 Fai 1 ∂ 2 Fai = =− . ∂ba (xa )∂bi (xi ) ∂bi (xi )∂ba (xa ) bi (xi ) Substituting the components (3.67)–(3.69), we have

(3.67) (3.68) (3.69)

44

3 Downlink Multi-user Detection for MIMO-SCMA System

   ˜ (ba (xa ))2 2b˜a (xa )b˜i (xi ) b˜a (xi )(b˜i (xi ))2 − + dxa ba (xa ) (bi (xi ))2 b˜i (xi )  2  (b˜a (xa )) b˜i (xi ) ˜ − = ba (xa ) dxa ≥ 0, ba (xa ) bi (xi )

(3.70)

indicating that Fai is convex. So far, we prove that the conditions depicted in Proposition 3.1 guarantee the convexity of the generalized Bethe free energy.  From Proposition 3.1, we see that several groups of counting numbers can be found for convexifying the generalized free energy. Since our goal is to derive the convergence-guaranteed version of the original MPA, intuitively the convexified free energy should be close to the original Bethe free energy. As for Bethe approximation, the counting numbers are defined as da = 1 and di = 1 − |N(i)|. A straightforward solution is to minimize the 2 norm c − d 2 as

min



cii ,caa ,cia

a

s.t ci = 1 −

⎛ ⎝caa + 



⎞2 cia − 1⎠

(3.71)

i∈N(a)

ca , (3.59), (3.60).

f a ∈N(i)

Standard solvers can be readily used to solve the optimization problem of (3.71). In the above, we have derived the convergence-guaranteed MPA based on the convexified Bethe free energy by choosing counting numbers. Following the convergence-guaranteed message passing updating rules, we first calculate the auxclosed iliary message μ˜ fa →xi (xi ), which is obtained in Gaussian  γ form. Using the property that (ea )b = eab for real numbers a and b, μ˜ fa →xi (xi ) ai is still a Gaussian distribution with the same mean of μ˜ fa →xi (xi ) and variance divided by γai . Therefore, Gaussian messages can still be derived based on convergence-guaranteed message passing rules with ci and ca .

3.4.3 Complexity Analysis In this subsection, we briefly discuss the complexity for our proposed algorithms. The complexity of the proposed message passing receiver based on the stretched version of factor graphs have been analyzed already. As for the convergenceguaranteed message passing receiver, its complexity also depends on the number of integration operations, which is O(N (J + K + L − 1)2 ). Note that to determine the counting numbers, we need to solve the optimization problem of (3.71), which increases the number of operations. Nevertheless, given the channel information, the quadratic programming can be solved offline before data detection. The pro-

3.5 Distributed Cooperative Detection

45

Table 3.1 Computational complexities of different receivers Receivers Computational complexity O(N · 2 J K L−1 ) O(N (J K L − 1)2 ) O(N (J + K + L − 1)2 ) O(N (J + K + L − 1)2 )

MPA BP-EP (Original Factor Graph) BP-EP (Stretched Factor Graph) Convergence-guaranteed BP-EP

posed convergence-guaranteed message passing receiver has a complexity order of O(N (J + K + L − 1)2 ). In Table 3.1, the computational complexity for different receivers is summarized.

3.5 Distributed Cooperative Detection From the system model, it can be observed that the term s j appears in the received signals of all users, which means it is possible to achieve further diversity gain by collecting the signals from all users [94]. Obviously, the basic idea is to share all users’ measurements in the network. Nevertheless, this mechanism is not realistic due to two reasons: (1) sharing the measurements means that every user will collect the measurements of all users, which will consume a tremendous amount of communication overhead; (2) a scheduling of packets transmission is required to ensure all measurements are collected, which is not always available. Hence, we consider an in-network distributed cooperative detection scheme. The cooperative network for the MIMO-SCMA system enables the users to communicate with each other and share information. Thanks to the factor graph representation, we can illustrate the relationship between a transmitted symbol s nj and received signals at different users graphically, as shown in Fig. 3.4. For user information detection, we adopted the stretched factor graph and the convergence-guaranteed message passing receiver proposed in previous sections. Define μk (s nj ) as the message to variable vertex s nj based on the received signal of user k, which can be written as μk (s nj ) =

L−1 

μψkn+l →s nj (s nj ).

(3.72)

l=0

After obtaining the messages μk (s nj ), ∀k, we can calculate the message to factor n vertex φ nj as well as the extrinsic message of xk, j following the message passing rules. Obviously, if there exists a central unit in the cooperative network that can collect all received signals, the message μs nj →φ nj (s nj ) is simply determined as the product of μk (s nj ), ∀k. Nevertheless, the high power cost is impractical. In contrast, the distributed processing which relies on only local computation and in-neighbor

46

3 Downlink Multi-user Detection for MIMO-SCMA System

Fig. 3.4 Factor graph representation for cooperative detection

communications is more attractive. By exchanging information between neighboring downlink users, all users can fully take use of the information corresponding to s nj . Next, we will develop two distributed methods for cooperative detection in our considered MIMO-SCMA system.

3.5.1 Belief Consensus-Based Method Denoting the neighboring set of user k as Sk , containing the users that can communicate with user k. Our goal is to obtain the product of μk (s nj ), ∀k, i.e., μs nj →φ nj (s nj ) (the ‘global message’) distributively, which can be done by the prominent belief consensus method. The standard belief consensus iteration is given as follows p+1 ρk (s nj )

=

p ρk (s nj )





p

ρi (s nj ) p

i∈Sk

ρk (s nj )

η ,

(3.73)

where local message μk (s nj ) is obtained based on the received signal at user k, the superscript p is the pth consensus iteration, and η denotes the update rate. Before starting consensus iteration, the local belief is initialized as ρk0 (s nj ) = μk (s nj ). For standard belief consensus, the same update rate η is adopted for all users, which may lead to performance degradation, especially in asymmetric graphs. To tackle this problem, the metropolis weight [95] can be used, given by

3.5 Distributed Cooperative Detection p+1

ρk

47

(s nj ) = ρk (s nj )ηkk p



η

p

ρi (s nj ) ki ,

(3.74)

i∈Sk

where the update rate ηik is defined as  ηik = ηki =

1/ max(|S

k |, |Si |), for i = k 1 − i  ∈Sk ηi  k , for i = k.

(3.75)

Since all messages are determined in Gaussian, instead of exchanging the distributions, the users can exchange only the corresponding means and variances. Consequently, (3.74) is rewritten as p+1

θk

p

= ηkk θ k +



p

ηki θ i ,

(3.76)

i∈Sk p

p

p

p

where θ k = [m k→s n /vk→s n , 1/vk→s n ]T denotes the parameters to be shared to other j j j users. Note that two users, e.g., i and k may fail to communicate even if they are within the communications range in a consensus iteration. In this circumstance, an additional variable θ¯ ik is introduced to store the parameter in the previous consensus iteration. p When link failure happens, θ¯ ik can be used as θ k . It was shown in [86] that all users are N guaranteed to reach consensus on the global message, i.e., ρk p (s nj ) = μs nj →φ nj (s nj )1/K , ∀k, after running a few consensus iterations if the topology graph is connected. Due to the existence of white noise when users are exchanging parameters, the variance of θ k grows unbounded with the increase of consensus iteration. A vanishing parameter is usually used to reduce the impact of noise, leading to the updating rule of p+1

θk

p

= θk + αp



 p p p ηki θ i + ωki − θ k .

(3.77)

i∈Sk

In (3.77), ωki and α p are the additive noise and the vanishing parameter, respectively. The vanishing parameter α p is set to be monotonically decreasing with the increase of p, which significantly affect the convergence speed of the consensus method. To this end, we will develop an optimization-based consensus method for distributed cooperative detection.

3.5.2 Bregman ADMM-Based Method Motivated by the fact that the product of all local beliefs after running the consensus method should be as close as possible to the global message, we can formulate an optimization problem that minimizes the Kullback–Leibler divergence between the local belief and the global message under the constraint ρk (s nj ) = ρi (s nj ),

48

3 Downlink Multi-user Detection for MIMO-SCMA System

min D[μs nj →φ nj (s nj )|



ρ

ρk (s nj )]

(3.78)

k

s.t ρk (s nj ) = ρi (s nj ), ∀ k, i ∈ Sk . where D[ p|q] follows the expression of (3.29). Considering that ρk (s nj ) is a Gaussian distribution and can be fully characterized by the parameter θ k , the constraint can be replaced by θ k = θ i . For decoupling purpose, a set of additional variables π ki are adopted for each inter-user link, and (3.78) can be rewritten as min D[μs nj →φ nj (s nj )| θ



ρk (s nj )]

(3.79)

k

s.t θ k = π k , θ i = π k , ∀k, i ∈ Sk . The alternative direction method of multipliers (ADMM) could be used to solve the optimization problem in (3.79) subject to equality constraints. ADMM iteratively updates variables by solving the augmented Lagrangian of (3.79), which is expressed as  L(θ , π , λ) = D

μs nj →φ nj (s nj )|



 ρk (s nj )

+

  λTkk (θ k − π k ) + λTki (θ i − π k )

k

k i∈Sk

 c θ k − π k 22 + θ i − π k 22 , + 2

(3.80)

k i∈Sk

where λ is the associated Lagrangian multipliers and c > 0 denotes a penalty coefficient. Because of the nonlinearity, the commonly adopted quadratic penalty term may result in a high complexity in our considered problem. To this end, we employ the Bregman divergence as the penalty term to form a generalized ADMM. For formulating the Bregman divergence, a Bregman function ε is considered, which is continuously differentiable and strictly convex. The Bregman divergence is then defined as Bε (x, y) = ε(x) − ε(y) − x − y, ∇ε (y) ,

(3.81)

for two arbitrary variables x and y. In (3.81), ∇ε (y) denotes the gradient of ε and · denotes the inner product. Based on Bregman divergence, the following augmented Lagrangian is formulated, L Br eg (θ, π , λ) =D[μs nj →φ nj (s nj )| +c

 k i∈Sk



ρk (s nj )] +

k

Bε (θ i , π k ).



 T λTki (θ k − π k ) + λik (π k − θ i )

k i∈Sk

(3.82)

3.5 Distributed Cooperative Detection

49

We can see that when choosing Bregman function ε = · , the penalty term in (3.82) is quadratic. Similar to the original ADMM, the Lagrangian L Br eg is minimized with respect to one set of variables while fixing the others. In particular, at the ( p + 1)th iteration, the Bregman ADMM update the variables as θ p+1 = arg min L Br eg (θ , π p , λ p ),

(3.83)

π p+1 = arg min L Br eg (θ p+1 , π , λ p ),

(3.84)

θ

π

p+1

λki

p

p+1

= λki + c(θ i

p+1

− π ki ).

(3.85)

For efficient computations, we aim for choosing an appropriate Bregman function to formulate the Bregman divergence. In the considered problem, since the passed messages are Gaussian distributions, we can adopt the log partition function as Bregman function. Then, the Bregman divergence between two variables is equivalent to the KLD between two Gaussian distributions, i.e., Bε (a, b) = D[ f (x|a)| f (x|b)], where a and b follows the expression of θ k to characterize the Gaussian distribution. Then (3.83)–(3.85) are obtained as p+1 θk

=

p+1

=

πk

p+1

λki

θ 0k +



i∈Sk ∪k

1 + c(|Sk | + 1)  

p+1 p − λki i∈Sk ∪k cθ i c(|Sk | + 1) p

p+1

= λki + c(θ k p

p

p

λki + cπ i

p+1

− πi

,

,

).

(3.86)

(3.87) (3.88)

p

By sharing the parameters θ k and π k , all users in the network finally arrive at the message μs nj →φ nj (s nj ) in a distributed manner. For the purpose of accelerating convergence speed, the penalty parameter c is set to be varying for different users in each iteration, which are given in [96], p+1

ck

⎧ p ⎨ ck · (1 + τ ) if  Tk 2 > κ ιTk 2 p = ck · (1 + τ )−1 if ιTk 2 > κ  Tk 2 ⎩ p ck otherwise,

(3.89) p

where  Tk 2 and ιTk 2 are the primal and dual residuals, defined as  Tk 2 = θ k −

p p−1 p p p θ¯ k 2 , ιTk 2 = θ¯k − θ¯ i 2 , θ¯k = |S1k | i∈Sk θ i . Typical values of κ and τ are suggested as constant κ = 10 and τ = 1. Proposition 3.2 All local parameters will reach consensus on the global parameter after a few iterations following (3.86)–(3.88) Proof For any user k, the second order partial derivative of the functional D[μs nj →φ nj  (s nj )| k ρk (s nj )] with respect to ρk (s nj ) satisfies

50

3 Downlink Multi-user Detection for MIMO-SCMA System

Algorithm 3.1 Distributed Cooperative Detection for MIMO-SCMA system 1: Each user calculates message μk→s nj (s nj ), ∀k, j, n based on its received signal. 2: Initialize ρk0 (s nj ) as μk→s nj (s nj ) 3: for p=1:Pmax do p 4: For Belief Consensus-based method: Each user broadcasts the parameters θk , ∀k to its neighboring users; p For Bregman ADMM-based method: Each user broadcasts the parameters θk , ∀k to its neighboring users; 5: Belief Consensus: Each user updates its local parameters using (3.76) Bregman ADMM: Each user updates its local parameters according to (3.86)–(3.88) ; 6: end for 7: Calculate the message μs nj →φ nj (s nj ) at all users; 8: Computes other messages on factor graph with μs nj →φ nj (s nj ).

∂ 2 D[μs nj →φ nj (s nj )|



∂ρk (s nj )2

k

ρk (s nj )]

=

1 > 0, ρk (s nj )

(3.90)

which shows the objective in (3.79) is convex. Moreover, since the Bregman function is strictly convex, the Bregman penalty term is also convex, indicating that the optimization problem is convex and the convergence is guaranteed.  N

After several ADMM iterations, the local belief ρk p (s nj ), ∀k will converge to the global message μs nj →φ nj (s nj ). In the presence of noisy inter-user links, the parameter updating in (3.86)–(3.88) can be seen as stochastic gradient updates, whose variances are still bounded [97]. Compared to the belief consensus-based method, it can be seen that the Bregman ADMM-based algorithm needs to transmit an additional variable, which doubles the communication overhead.

3.5.3 Algorithm Summary For the distributed cooperative detection, the goal is to obtain the product of all messages μk→s nj (s nj ), ∀k, j, n based on users’ local received signals in a distributed fashion. The algorithm is commenced by initializing the local belief ρk0 (s nj ) as μk→s nj (s nj ). According to the belief consensus-based method and Bregman ADMMbased method, all users can update their local beliefs and finally reach a consensus on the global message. Due to the Gaussian representations of the messages, only a few parameters are transmitted for both schemes, having a complexity order of O(N ). The complexity for both schemes only linearly increases with the number of users, making them attractive in practical applications. For clarity, we summarize the proposed distributed cooperative detection methods as follows.

3.6 Simulation Results

51

3.6 Simulation Results This section illustrates the performance of the proposed multiuser receiver for MIMO-SCMA system and cooperative detection scheme. Performance variations of the proposed receiver and state-of-the-art methods are first considered. Then, for the proposed cooperative detection scheme, the impact of the number of consensus iterations and noisy links is investigated. In the simulations, we consider a MIMOSCMA system with J = 4 antennas, K = 6 users, D = 2 nonzero entries in each codeword, M = 4 and therefore the overloading factor is ρ = 150%. The SCMA codebook is designed according to [43] with the indicator matrix F defined as ⎡

1 ⎢1 F=⎢ ⎣0 0

0 1 1 0

1 0 1 0

0 0 1 1

1 0 0 1

⎤ 0 1⎥ ⎥. 0⎦ 1

(3.91)

A 5/7-rate LDPC code is employed with variable and check node degree distributions v(X ) = 0.0005 + 0.2852X + 0.2857X 2 + 0.4286X 3 and c(X ) = 0.0017X 9 + 0.9983X 10 , respectively. Quadrature phase shifting key (QPSK) is utilized as the modulation scheme. We assume a frequency selective channel with L = 10 paths, and the channel gain for each path is independently generated according to the distribution

l . h lk, j ∼ G(0, q l ), ∀k, j, where the normalized power delay profile is q l = exp(−0.1l) q The simulation results are averaged from 1000 independent Monte Carlo trails. Figure 3.5 depicts the bit error rate (BER) performance of the proposed stretched factor graph-based message passing receiver (denoted as ‘Stretch-BP-EP’). For comparison purpose, the performances for the MPA receiver [98], Gaussian approximated BP (denoted as ‘GaussAppro-BP’) algorithm and a combined MMSE-PMMPA algorithm are included. The GaussAppro-BP algorithm is also running on the proposed stretched factor graph while the extrinsic information of data symbols is approximated by Gaussian via direct moment matching, instead of adopting EP. The combined MMSE-PM-MPA receiver first performs the MMSE-based MIMO equalization and then decodes the SCMA codewords relying on PM-MPA [66]. A K = 4 orthogonal multiple access case is considered as the performance benchmark. It can be seen from Fig. 3.5 that the MMSE-PM-MPA method suffers from significant performance loss since MMSE detector only outputs hard information for the PM-MPA-based SCMA detector. The proposed Stretch-BP-EP algorithm slightly outperforms GaussAppro-BP and performs close to the classic MPA receiver. However, the complexity of the proposed algorithm is reduced significantly compared to the MPA receiver, which has an exponentially increased complexity. Moreover, the proposed MIMO-SCMA system has a similar BER performance with the orthogonal multiple access case while supporting 50% more users. In Fig. 3.6, we compare the BER performance of the Stretch-BP-EP method and the proposed convergence-guaranteed message passing receiver (denoted as ‘ConvBP-EP’) at different values of E b /N0 . Both algorithms are observed to have a perfor-

52

3 Downlink Multi-user Detection for MIMO-SCMA System 100

10-1

BER

10-2

10-3

10-4

10-5

6.2dB 7.0dB 7.8dB 8.6dB

10-6 1

2

3

4

5

6

7

8

9

10

Number of Iterations

Fig. 3.5 BER performance comparison for MIMO-SCMA system

mance improvement as the number of iterations increases in the first few iterations. By comparing Fig. 3.6a with Fig. 3.6b, we observe that the Conv-BP-EP method converges faster than the Stretch-BP-EP algorithm. This is because that the StretchBP-EP algorithm may converge to the local minima while Conv-BP-EP is guaranteed to converge to the global minimum, verifying the effectiveness of the Conv-BP-EP method. Next, let us evaluate the performance of the proposed distributed cooperative detection schemes. We consider a scenario of 40 × 40 m2 unit square, where six users are uniformly distributed. The communications range is set to d = 25 m such that two users can communicate and exchange information if their distance is smaller than d. The inter-user communications links between neighboring users are modeled additional white Gaussian noise channel and the power spectral density is the same for all links. The vanishing parameter for belief consensus is set to α p = 1p . In Fig. 3.7, the BER performances of the proposed two distributed cooperative detection schemes under noise-free links are illustrated. As a benchmark, the BER performance for the centralized scheme is also plotted. For fair comparison, we assume that for the centralized scheme, only received signals from connected users are collected. The averaged BER performance of all users based on their local received signal as in Fig. 3.6b is also included. It is observed that the proposed cooperative detection scheme significantly improves BER performance, which reveals that

3.6 Simulation Results

53

10 0

10 -1

BER

10 -2

10 -3

10 -4

6.2dB

10 -5

7.0dB 7.8dB 8.6dB

10 -6

1

2

3

4

5

6

7

8

9

10

8

9

10

Number of Iterations

(a) Stretch-BP-EP

BER

10

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

6.2dB 7.0dB 7.8dB 8.6dB 1

2

3

4

5

6

7

Number of Iterations

(b) Conv-BP-EP Fig. 3.6 Impact of the number of iterations on BER performance

54

3 Downlink Multi-user Detection for MIMO-SCMA System

Fig. 3.7 BER performance of the proposed distributed cooperative detection schemes with p = 5 and p = 10

diversity gain can be exploited by information sharing between users. By comparing the belief consensus-based method and the Bregman ADMM-based method, we see that both algorithms perform almost the same under noise-free inter-user links for p = 5 and p = 10. After 10 iterations, both methods achieve a similar performance to the centralized scheme. The communication range is critical to users’ power consumption. We compare the BER performance of Bregman ADMM-based algorithm with different communication ranges, i.e., d = 4 m, d = 10 m, d = 25 m, d = 32 m and d = 40 m. Obviously, increasing d will result in a better BER performance since more neighboring users are available, as shown in Fig. 3.8. However, the performance gain becomes marginal when d is sufficiently large. This motivates us to strike a trade-off between the power cost and BER performance in practical applications since the power consumption will increase quadratically with d. We further extend the performance evaluation of our proposed distributed cooperative detection methods to a more practical scenario with noisy inter-user links. In Fig. 3.9, we plot the BER performance of the proposed distributed algorithms versus E b /N0 , where the SNR corresponding to the inter-user links is set to be 10 dB. The number of consensus iterations is p = 10. Due to the existence of additional noise, both distributed algorithms are not capable of approaching the centralized scheme in 10 iterations. Compared to the belief consensus-based algorithm, the Bregman ADMM-based method is shown to have better performance. As a further step, we analyze the convergence of two distributed schemes, in Fig. 3.10. We depict the mean

3.6 Simulation Results 10

55

0

-1

10

-2

10

-3

BER

10

10

d=2m d=6m d=10m d=14m d=20m

-4

3

4

5

6

7

8

9

E b / N0 (dB)

Fig. 3.8 Impact of communications range on the BER performance 100

10-1

BER

10-2

10-3

10-4

10-5

Centralized Processing Bregman ADMM Belief Consensus

10-6 3

3.5

4

4.5

5

5.5

Eb /N0 (dB)

Fig. 3.9 Impact of noisy inter-user links on the BER performance

6

6.5

7

56

3 Downlink Multi-user Detection for MIMO-SCMA System

Belief consensus 5dB Belief ADMM 5dB Belief consensus 10dB Belief ADMM 10dB Belief consensus 20dB Belief ADMM 20dB

101

MSE

100

10-1

10-2

10-3 1

2

3

4

5

6

7

8

9

10

Number of iteration

Fig. 3.10 Impact of the number of consensus iterations

squared error (MSE) of local parameter θ k versus the number of iterations. The MSE is given by MSE[θ k ] =

K 

θ k − θ¯ 22 ,

(3.92)

k=1

θ

where θ¯ = Kk k . Three cases with different SNRs of the inter-user links are considered, i.e., SNR = {5, 10, 20}dB. Obviously, a higher SNR leads to a better MSE performance. The belief consensus-based algorithm converges slower than the Bregman ADMM-based method because of the adoption of the vanishing factor α p = 1p . It is interesting to see the performance gap between two algorithms becomes even larger in higher SNR region. This is because the Bregman ADMM-based method can benefit from small noise variance, showing that Bregman ADMM-based distributed detection scheme is more efficient in scenarios with noisy links.

3.7 Conclusions In this chapter, we develop the low-complexity message passing receiver for MIMOSCMA system. Since the direct factorization of the joint a posteriori distribution leads to huge complexity in on-graph message updating, we introduce auxiliary variables and construct a stretched version of the factor graph for complexity reduction. Con-

3.7 Conclusions

57

sidering the convergence issue of the standard message passing receiver on the loopy factor graph, we convexify the Bethe free energy by introducing appropriate counting numbers. The convergence-guaranteed message passing receiver is then derived based on the minimization of the generalized Bethe free energy. Moreover, to exploit the diversity gain in downlink communications, we enable the cooperation between users and propose two distributed detection schemes, i.e., the belief consensus-based algorithm and the Bregman ADMM-based method. The proposed iterative receivers are evaluated by Monte Carlo simulations and compared with the state-of-the-art schemes. The proposed Stretch-BP-EP receiver performs close to the MMSE-based receiver with significantly reduced complexity. The proposed Conv-BP-EP receiver can efficiently enhance the convergence of the MPA and therefore improve the error performance. Compared with the orthogonal multiple access counterpart, MIMO-SCMA system with the proposed receivers can support 50% more users with negligible BER performance loss. In cooperative networks, it is verified that BER performance can be further improved by using the proposed distributed cooperative detection schemes. Particularly with noisy inter-user links, the Bregman ADMM-based method is superior to the belief consensus-based algorithm.

Chapter 4

FTN Data Detection and Channel Estimation over Frequency Selective Channels

4.1 Introduction It is widely acknowledged that FTN signaling can improve the spectral efficiency of its Nyquist counterpart, making it attractive in the era of beyond 5G wireless communications. However, the theoretically infinite ISI induced by FTN signaling is very challenging for receiver design. This issue becomes more severe in frequency selective channels, where the multipath propagation effect also results in ISI. In a naive way, one can design the optimal receiver based on a composite ISI channel by combining the interference from both FTN signaling and fading channels. Nevertheless, the fading channel information may be unknown or not perfectly known on the one hand. To obtain an accurate channel estimate, a large number of pilot symbols have to be used [99–101], which is contrary to our goal of spectral efficiency improvement. On the other hand, the ISI induced by FTN signaling is inherently known to the receiver. Therefore, combining it with unknown channel taps is not optimal from the receiver design perspective. Considering the channel coefficients are not known, it is necessary to design a receiver that jointly detects data symbols and estimates the ISI channels based on the received signal. Early contributions [102–104] have shown that a joint channel estimation and detection scheme not only improves the channel estimation accuracy but also leads to a reduced number of pilots adopted. In this chapter, we will develop a low complexity receiver for FTN signaling in a frequency selective channel, where both symbol detection and channel estimation are considered. In particular, we focus on the heuristic iterative approaches on probabilistic graphical models [92, 105]. Based on the signaling model, we construct a Forney-style factor graph, on which MPA is executed to infer the unknown variables. Due to the fact that both FTN symbols and channel coefficients are unknown, the classic message passing rules will fail to work. To this end, we reconstruct the factor graph and employ variational message passing (VMP). As a result, all messages can be written in Gaussian closed form, which leads to a much lower complexity than the optimal MAP receiver.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Yuan et al., Receiver Design for High Spectral Efficiency Communication Systems in Beyond 5G, https://doi.org/10.1007/978-981-19-8090-9_4

59

60

4 FTN Data Detection and Channel Estimation over Frequency …

Fig. 4.1 System model for considered FTN signaling system

4.2 FTN Signaling Model Without loss of generality, we consider a coded FTN system, as shown in Fig. 4.1. The information bit sequence b is encoded and mapped to the transmitted symbols x = [x0 , . . . , x N −1 ]T at the transmitter side. Then the data symbols are converted to continuous time signal using shaping filter g(t), i.e., s(t) =



g(t − iτ T0 )xi ,

(4.1)

i

where 0 ≤ τ ≤ 1 is the so-called packing ratio and T0 is the symbol interval. In particular, when τ = 1, s(t) becomes the classic Nyquist signaling. We can see that by choosing a smaller τ , a higher data rate can be achieved at the cost of severer ISI. In theory, the length of FTN-induced ISI is infinite [106]. According to [33], we can use a sufficiently large value of tap length L FTN = 2L f + 1 to approximate the original model, yielding s(t) =

Lf 

g(t − iτ T0 )xi .

(4.2)

i=−L f

The transmitted signal passes through a frequency selective with L nyq taps, expressed as h˜ = [h˜ L nyq −1 , . . . , h˜ 0 ]T . Due to the smaller symbol period of FTN signaling, the equivalent number of channel taps becomes L = L nyq τ  and the coefficient ˜ According to [107], the discrete for each tap can be obtained via interpolation over h. Fourier transform (DFT) matrix can be adopted and exploited for obtaining the channel taps for FTN signaling. Assuming D and D F are two DFT matrices with dimensions L nyq and L, respectively, the equivalent channel h = [h L−1 , . . . , h l , . . . , h 0 ]T is given by

4.2 FTN Signaling Model

61

 h = DHF

 Dh˜ . 0

(4.3)

For FTN signaling, since different channel taps are correlated, leading to a nondiagonal covariance matrix of h, expressed as  Vh = DHF

 DVh˜ DH 0 DF , 0 0

(4.4)

˜ where Vh˜ is a diagonal matrix denoting the covariance matrix of h. After transmitting through the fading channel, the received signal can be represented as y(t) =

Lf L−1  

h l g(t − (i + l)τ T0 )xi + ξ(t),

(4.5)

l=0 i=−L f

where ξ(t) is the additional white Gaussian noise (AWGN) with power spectral density (PSD) of N0 . The received signal y(t) passes through the matched filter g ∗ (t) and is sampled with interval τ T0 . After matched filtering, the kth sample is given by rk =

Lf L−1  

h l q(kτ T0 − (i + l)τ T0 )xi + ξ(kτ T0 )

l=0 i=−L f

=

Lf L−1  

h l qk−l−i xi + ξk ,

(4.6)

l=0 i=−L f

where  qm−n = ξk =



g(t − mτ T0 )g ∗ (t − nτ T0 )dt,

(4.7)

ξ(t)g ∗ (t − kτ T0 )dt.

(4.8)

Obviously, the filtered noise term {ξk } is colored, whose auto-correlation function (ACF) is expressed as E[ξm ξn ] = N0 qm−n .

(4.9)

With the expression (4.6), we arrive at the vector form received signal of r = [r0 , . . . , r N −1 ]T as r = HQx + ξ ,

(4.10)

62

4 FTN Data Detection and Channel Estimation over Frequency …

where ξ = [ξ0 , . . . , ξ N −1 ]T denotes the noise sample vector, H and Q are the channel matrix and FTN ISI matrix, formulating as ⎡

h0 h1 .. .

0

⎤

⎢ ⎥ h0 ⎢ ⎥ ⎢ ⎥ . .. ⎢ ⎥ ⎥ H=⎢ ⎢ h L−1 h L−2 · · · h 0 ⎥ ⎢ ⎥ ⎢ ⎥ .. .. ⎣ . . ⎦ 0 h L−1 · · · h 1 h 0

(4.11)

and ⎡

q0 q1 ⎢ .. ⎢ . q0 ⎢ ⎢ q−L f · · · Q=⎢ ⎢ q−L f ⎢ ⎢ ⎣ 0

⎤ · · · qL f 0 ⎥ .. ⎥ . ⎥ ⎥ q0 · · · q L f ⎥. · · · q0 · · · q L f ⎥ ⎥ .. ⎥ .. . . ⎦ q−L f · · · q0

(4.12)

Based on (4.9), the autocorrelation matrix of ξ is N0 Q.

4.3 Low-Complexity Message Passing Receiver Design This section presents the message passing-based iterative receiver for joint symbol detection and channel estimation for FTN system. All messages are determined in Gaussian form, leading to a low-complexity receiver. Considering a coded system, ‘turbo’ equalization is performed by exchanging LLR between the detector and channel decoder. For simplicity, we adopt the standard BP decoding algorithm and the output LLR from the channel decoder is given by L 0 (cn,m ) = ln

p(cn,m = 0) , p(cn,m = 1)

(4.13)

where the mth code bit in the nth sub-sequence is denoted by cn,m . Due to the non-orthogonality of the matched filter, the noise sample ξk is colored. While a whitening filter can be applied to decorrelate the noise, the implementation complexity is increased [108]. To address this issue, here we employ an autoregressive (AR) model to approximate the colored noise [109], formulated as

4.3 Low-Complexity Message Passing Receiver Design

ξk =

P 

a j ξk− j + wk = aT ξ k−1 + wk ,

63

(4.14)

j=1

where a = [a1 , . . . , a P ]T is the coefficients for the Pth order AR model and wk is zero-mean white Gaussian noise term with variance σw2 . The vector ξ k−1 = [ξk−1 , . . . , ξk−P ]T contains the noise samples that are correlated with ξk . By solving the following Yule-Walker equation, the coefficients a can be obtained N 0 qk =

N0 Pj=1 a j q− j + σw2 k = 0

N0 Pj=1 a j qk− j othertwise.

(4.15)

Note that based on (4.10) and (4.14), we can adopt the sophisticated minimum mean square error (MMSE) algorithm for estimating the channel coefficients and detecting data symbols. However, the inverse operation in the MMSE algorithm requires a very high complexity, which avoids its application in practice. To this end, we develop the factor graph approach, which will be detailed in what follows.

4.3.1 Forney-Style Factor Graph Representation The Forney-style factor graph relies on the state evolution equations of the unknown variables and the observation equations [110]. We first rewrite (4.6) as sk = qT xk ,

(4.16)

r k = h sk + ξk ,

(4.17)

T

where the vectors x and s are expressed as xk = [xk−L f , . . . , xk , . . . , xk+L f ]T and sk = [sk−L+1 , . . . , sk ]T , respectively, which follow xk = Gxk−1 + f xk+L f , sk = Gs sk−1 +

fsT sk ,

(4.18) (4.19)

   02L f I2L f 0 L−1 I L−1 T T , f=[0 and fs = [0TL−1 , 1]T . , 1] , G = s T 2L f 0 02L 0 0TL−1 f Similarly, (4.14) can be rewritten as 

where the G=

ξ k = Aξ k−1 + fw wk , ξk =  with fw = [0TP−1 , 1]T and A =

0

fwT ξ k ,

aT

0 P−1 I P−1

 .

(4.20) (4.21)

64

4 FTN Data Detection and Channel Estimation over Frequency …

Fig. 4.2 Factor graph representation for joint channel estimation and decoding for FTN system. The sub-graphs 1–4 correspond to symbol detection, channel equalization, channel estimation, and colored noise process, respectively

Based on the above linear equations (4.16)–(4.21), the Forney-style factor graph is illustrated in Fig. 4.2. A factor node in Fig. 4.2 represents a state transition or observation function and each edge represents an unknown variable. The function nodes representing equality constraint allow different functions to share the same variables. Furthermore, a multiplier node denote the relationship of r = hT s.

4.3 Low-Complexity Message Passing Receiver Design

65

4.3.2 Gaussian Message Passing Algorithm The Gaussian message passing (GMP) is an efficient tool on Forney-style factor graph for solving linear equations. Usually, we can use the mean m and the covariance matrix V to parameterize the Gaussian message. For the case that the covariance matrix V is singular, the weight matrix W = V−1 and the transformed mean Wm can be used for characterizing messages. For ease of exposition, we summarize the basic update rules of GMP as follows [110],

y  x =

z 

− → − → − → Wz = Wx + Wy − →− − → → − → → Wz→ m z = Wx− m x + Wy− my ← − − → ← − Wx = Wy + Wz ← − ← − → → ← − − − W m =W − m +W ← m x

x

y

y

y  x +

z

z 

− → − → − → Vz = Vx + Vy − → → → mz = − mx + − my ← − ← − − → Vx = Vz + Vy ← − =← − −− → m m m x

z

x

A

y

y 

− → → my = A− mx − → − → T Vy = A VxA ← − ← − W x = AT W y A ← − − ← − − m = AT W ← m W ← x

x

y

y

z

66

4 FTN Data Detection and Channel Estimation over Frequency …

→ − The notations − · and ← · denote the message passing along and opposite the arrow direction, respectively. Following the GMP rules, we can derive the Gaussian messages on four sub-graphs.  Messages Updating for Sub-graph 1 (symbol detection): − → → − → − → → − → m xk−1 , we can derive V x˜ k−1 and W x˜ k−1 − m x˜ k−1 Having obtained W xk−1 and W xk−1 − as ← − → − − → −1 T V x˜ k−1 = G W x + W xk−1 G , k−1   − → − − → → ← − −  , W x˜ k−1 → m x˜ k−1 = G W xk−1 − m xk−1 + W x ← m x k−1

k−1

(4.22) (4.23)

← − ← − −  are given by where W x and W x ← m x k−1

k−1

k−1

qqT ← − W x = , k−1 Vsk−1 ← − −  = qm sk−1 . W x ← m xk−1 k−1 Vsk−1

(4.24) (4.25)

Similarly, we can derive the backward message as ← ← − − ← − −1 V xk = W x + W x , k k  ← − ← − ← − ← ← − = V W ← − −    m xk xk x m x + Wx m x . k

k

k

k

(4.26) (4.27)

Now, we are capable of determining the parameters for xk+L f , given by −  → ← − T ← m xk − − m x˜ k−1 , m xk+L f = f − → ← −  ← − V xk+L f = f T V x˜ k−1 + V xk f.

(4.28) (4.29)

In order to convert the output LLR L 0 (cn,m ) to Gaussian messages, we resort to EP scheme [111]. Based on the LLRs, we can obtain the probability pk,i associated with the ith constellation χi for the kth symbol. Therefore, the distribution of xk can be written as  − → μ (xk ) = pk,i δ(xk − χi ), (4.30) χi ∈A

where A is the set of constellations. The EP algorithm matches the first two order moments of the belief of xk , which are expressed as

4.3 Low-Complexity Message Passing Receiver Design

67

  − )2  m 1 (χi − ← xk m˜ xk = χi pk,i exp − , ← − ← − V xk 2π V xk χi ∈A   − )2  m 1 (χi − ← xk 2 ˜ Vxk = |χi | pk,i exp − − |m xk |2 . ← − ← − V xk 2π V xk χi ∈A

(4.31)

(4.32)

− → → The mean − m xk and variance V xk of incoming message are obtained consequently. Then, the parameters of the message passed to Sub-graph 2 are given as follows −  ← → → − − −  ← − −1 − → − → W xk → m x˜ k−1 + f − m x , m xk+L f + W x ← m sk =qT W xk + W x k k k − −1 → ← − − → V sk =qT W xk + W x q, k

(4.33) (4.34)

−1 − − → → − → where W xk = V x˜ k−1 + f V xk f T .  Messages Updating on Sub-graph 2 (channel equalization): Similar to the derivation of messages related to xk , the backward messages associated with sk are given by − − →  ← − = f T ← m sk s m sk − m s˜k , − → ← −  ← − V sk = fsT V s˜k + V sk fs ,

(4.35) (4.36)

where the parameters corresponding sk and s˜k have similar expressions as (4.22)– − → → (4.27). Moreover, the messages − m s and V s are given by k

k

→ → − → − ← − −  − → m sk + W s ← m s , m s = V s W sk − k k k k − − → → ← − −1 V s = W sk + W s , k

(4.37) (4.38)

k

which are used for computing the messages on Sub-graph 3.  Messages Updating on Sub-graph 3 (channel estimation): − → − → → Having the messages − m s , V s , and V ξk , the message from the multiplier node k k to hk can be written as     ← − −(s )− →  μ (hk ) ∝ δ(rk − hkT sk )← μ k μ (r k ) dsk dr k       (rk − hkT sk )2 − →−1  −   − → → H ∝ exp −(sk − m s ) V s (sk − m s ) exp − dsk − → k k k V ξk ⎛ ⎞ − → − → H − →  m s m s  r m sk k k k ⎠. ∝ exp ⎝−hkH − hk + 2hkH − (4.39) → − → → → H− V ξk + hk V s hk V ξk + hkH V s hk k

k

68

4 FTN Data Detection and Channel Estimation over Frequency …

Fig. 4.3 “Soft” node reconstruction. The factor f k can  be expressed as f k ∝  exp −(rk − hkT sk )2 /Vξk

−(s ) can be calculated in the same way. However, we can observe And the message ← μ k that it is infeasible to write (4.39) in a Gaussian form. To this end, we resort to VMP [112] to derive Gaussian messages. In particular, the message from factor vertex f to variable vertex x using VMP is formulated as ⎛  ⎝ ln f (x) μ f →x (x) ∝ exp



⎞ μx  → f (x )dx ⎠ . 



(4.40)



x ∈S( f )\{x}

Based on VMP rules, (4.39) can be rewritten as ← −(h ) ∝ exp μ k





ln δ(rk −

    hkT sk )b(sk )b(rk ) dsk





drk .

(4.41)



Note that the VMP rules involve the beliefs of sk and rk . A new problem emerges that the logarithm of the delta function in (4.41) is not mathematically defined. To tackle this issue, a “soft” node reconstruction scheme in [113] is adopted by grouping the multiplier node and noisy measurement, providing a new function node f k , as depicted in Fig. 4.3. Based on the “soft” node f k , the message can be obtained as      (rk − hkT sk )2  H −1   ) V  (s − m  ) exp −(s − m dsk k k sk sk − → sk V ξk ⎛ ⎞ Vs + ms msH  r m sk k k k k ⎠. ∝ exp ⎝−hkH hk + 2hkH − (4.42) − → → V ξk V ξk

← −(h ) ∝ exp μ k





−

→ −(s ) and − μ (sk ), we have the parameters ms and Vs Given Gaussian messages ← μ k k k as follows, −  → − ← −−1 ← → − , (4.43) ms = Vs V −1  m s + V  m s sk sk k k k k − → ← −−1 −1 Vs = V −1 . (4.44)  + V  s s 

k



k

k

4.3 Low-Complexity Message Passing Receiver Design

69

Since VMP updating utilize the beliefs of variables, (4.43) and (4.44) can be seen as the a posteriori mean and variance of sk . Then, the mean and variance of message ← −(h ) read μ k −1 ← − − →  V hk = V ξk Vs + ms msH , k k k −1  ← − = V  + m  mH m ms rk . hk s s s k

k

k

k

(4.45) (4.46)

← − −  as Similarly, we have the expressions of V s and ← m s k

k

−1 ← − − →  V s = V ξk Vhk + mhk mhHk , k   −1 ← −  = V + m mH m mhk rk , hk hk hk s k

(4.47) (4.48)

where mhk and Vhk are computed similar to (4.43) and (4.44).  Messages Updating on Sub-graph 4 (colored noise estimation): Notice that the means of the color noise samples are still zero, indicating that → − = m ξk = ← m all messages related to ξk on Sub-graph 4 are with zero means, i.e., − ξk E[ξk ] = 0, ∀k. Therefore, we focus on the variance calculation. In particular, the − → variance V ξk can be obtained as − − → →  − → V ξk = fwT V ξ  fw = V ξ  k

k

P,P

,

(4.49)

− → where V ξ  has the form of k

 −1 −1 − → − − → 2 T T←  Vξ = V ξ˜ k + σw fw fw + A W ξ˜ k+1 A . k

(4.50)

4.3.3 Computation of Extrinsic LLR Finally, the detected data symbols are converted to extrinsic LLR, which are fed for − − and variance ← channel decoding. Based on the mean ← m V xk of symbol xk , the LLR xk is calculated by p(cn,m = 0|r) − L 0 (cn,m ) p(cn,m = 1|r)

d =0 p(r|cn = di ) p(cn = di ) = ln i,m − L 0 (cn,m ), di,m =0 p(r|cn = di ) p(cn = di )

L e (cn,m ) = ln

(4.51)

70

4 FTN Data Detection and Channel Estimation over Frequency …

Algorithm 4.1 GMP-based Iterative Receiver for FTN Signaling over Unknown Frequency Selective Channels

− → → 1: The incoming messages are initialized as − m 0xk = 0 and V x0k = +∞. The a priori of h is obtained by using coarse channel estimation with few pilots and can be expressed as   − → → → p(h) ∝ exp (h − − m 0h )H W 0h (h − − m 0h ) ,

2: for Iter=1:I do 3: Compute messages from Sub-graph 1 to Sub-graph 2 based on (4.33) and (4.34); 4: Compute messages on Sub-graph 2 based on (4.35)–(4.38); 5: Compute messages on Sub-graph 3 based on (4.49) and (4.50); 6: Compute messages on the “soft” node using (4.45)–(4.48); 7: Calculate the extrinsic LLR according (4.52) and feed them for decoding; 8: Perform standard BP decoding; 9: Compute incoming messages using EP algorithm; 10: end for

where di is the coded bit sequence corresponding to the constellation symbol χi . Alternatively, L e (cn,m ) can be represented in a more concise form, which was derived in [114] and reads

L e (cn,m ) = ln

χi ∈A0m

  − )2  (χi −← m xk exp − ← p(cn,m  = si,m  ) − Vx k

m  =m

  , − )2 

(χ −← m xk   exp − i ← p(c = s ) − n,m i,m

χi ∈A1m

Vx k

(4.52)

m  =m

where A0m and A1m are the subsets of A, having values values 0 and 1 in position m. After channel decoding, the output LLRs are converted to Gaussian messages again for next turbo iteration. We summarize the proposed GMP-based receiver for joint channel estimation and FTN symbol detection in Algorithm 4.1.

4.3.4 Complexity Analysis It can be observed that the GMP-based algorithm relies on basic matrix manipulations, whose complexity mainly depends on the inversion operations in (4.22), (4.26), (4.38), (4.50), (4.44), and (4.45). For a non-sparse K -dimensional matrix, calculating its inverse requires a complexity of O(K 3 ). Therefore, the total complexity of the proposed receiver is O(N (L 3FTN + L 3 + P 3 )), where L FTN , L, and P are the length of FTN-induced ISI, the number of channel taps, and the order of AR model, respectively. Compared to the classic MMSE algorithm having cubically increased complexity, the complexity of our proposed method only increases linearly with block length N .

4.4 Simulation Results

71

4.4 Simulation Results Throughout the simulations, we adopt a 5/7-rate LDPC code with variable and check node degree distributions v(X ) = 0.0005 + 0.2852X + 0.2857X 2 + 0.4286X 3 and c(X ) = 0.0017X 9 + 0.9983X 10 [115], respectively. The code bits are interleaved and mapped to QPSK symbols. The shaping filter is an RRC one with a roll-off factor of 0.4. The symbol interval for Nyquist signaling is T = 0.2 µs. The frequency selective fading channel has L = 10 taps and the coefficients {h˜ l } are generated

l . Through DFT according to the distribution h˜ l ∼ G(0, q l ), where q l = exp(−0.05l) q interpolation, the channel taps under FTN transmission can be obtained. Unless otherwise specified, the FTN-induced ISI tap length is L FTN = 11 and the packing factor is τ = 0.7. The number of iterations is set to I = 10. We first compare the BER performance of the proposed algorithm with different packing factors τ . As a reference scheme, the performance of Nyquist signaling over the same channel is illustrated. We can observe that when the packing factor τ ≥ 0.7, the FTN system relying on our proposed receiver achieves a similar BER performance to the Nyquist system while improving up to 40% transmission rate given the same bandwidth. The BER performance loss is smaller than 0.2 dB even for τ = 0.6, where the spectral efficiency is increased by more than 65%. To evaluate the impact of the roll-off factor, we depict the BER performance with a roll-off factor 0.05 in Fig. 4.4b. We can see that for the case with τ = 0.6, the performance gap becomes 0.5 dB, due to severer ISI resulted by a smaller roll-off factor. Nevertheless, the proposed receiver still works well for τ = 0.8 case, where the 25% improvement of the transmission rate gain is achieved by employing FTN signaling. According to the complexity analysis, we see that L FTN will affect the total computational complexity. Although choosing a smaller number of L FTN can reduce the complexity, the underestimation of ISI induced by FTN will lead to performance loss. Figure 4.5a presents the BER performances with L FTN = {5, 11, 41}, where the roll-off factor is 0.4. Significant performance degradation can be observed for L FTN = 5. By increasing L FTN , the BER performance can be improved. However, the gain becomes marginal by further increasing L FTN when L FTN ≥ 11. Therefore, for the considered FTN system, L FTN = 11 is a sufficiently large number for characterizing the FTN-induced ISI. In addition, we illustrate the BER performance with a smaller packing factor of τ = 0.5 in Fig. 4.5b, where the ISI induced by FTN is obviously stronger. In this case, L FTN = 11 is not sufficient for approximating the theoretically infinite ISI, and 0.3 dB performance loss is observed. Therefore, it is very important to choose an appropriate value of L FTN in practical receiver design under system constraints. Figure 4.6 compares the BER performance of the proposed GMP-based receiver and state-of-the-art methods, i.e., the MMSE scheme [102] and the variational inference (VI) method [104]. To validate the effectiveness of the channel estimation, an ideal case with perfect channel information is also depicted. The MMSE detection performs over the composite channel by combining ISI from both FTN and fading channels, leading to a significantly increased number of channel taps. The VI method

72

4 FTN Data Detection and Channel Estimation over Frequency …

BER

10

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

=0.5 =0.6 =0.7 =0.8 Nyquist signaling

5.5

6

6.5

7

7.5

Eb /N0 (dB)

(a) Roll-off factor 0.4 10 0

10 -1

BER

10 -2

10 -3 =0.6

10

=0.7

-4

=0.8 Nyquist signaling

10 -5

10 -6 5.8

6

6.2

6.4

6.6

6.8

7

Eb /N0 (dB)

(b) Roll-off factor 0.05 Fig. 4.4 BER performance of the proposed algorithm with different packing factor τ

4.4 Simulation Results

73

100 L TN=5 F

10

LFTN=11

-1

LFTN=41

BER

10-2

10-3

10-4

10-5

10-6 6

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

Eb /N0 (dB)

100

10-1

BER

10-2

10-3

L TN=5 F

L TN=11 F

10

LFTN=41

-4

L TN=81 F

10-5

6.2

6.4

6.6

6.8

Eb /N0 (dB)

Fig. 4.5 BER performance with different L FTN

7

7.2

7

74

4 FTN Data Detection and Channel Estimation over Frequency … 10

-1

-2

10

-3

10

-4

10

-5

10

-6

BER

10

Perfect CSI Proposed BP-EP-VMP Proposed BP-VMP MMSE method VI method

5

5.5

6

6.5

7

7.5

8

8.5

Eb /N0 (dB)

Fig. 4.6 Performance comparison of different algorithms

is not able to exploit the conditional dependencies of data symbols at the receiver side and therefore suffers from performance loss. Moreover, the performance for direct moment matching scheme without using EP is plotted, where 0.2 dB performance loss exists compared to the EP-based approximation.

Fig. 4.7 MSE of channel estimation

4.5 Conclusions

75

The channel estimation accuracy based on the proposed joint channel estimation and FTN symbol detection algorithm is evaluated in Fig. 4.7 using the metric of mean square error (MSE). Some classic channel estimation algorithms, including least square (LS) [100] and expectation maximization (EM) method [103] are also included. Given L = 10, the number channel taps for τ = 0.7 FTN signaling is 14. We see that by exploiting the data symbols for channel estimation, the MSE is much lower than the LS scheme relying on only limited number of pilots. Moreover, the proposed algorithm outperforms the EM-based method, since the soft information is exchanged between the channel estimation and equalization.

4.5 Conclusions This chapter developed a low-complexity receiver for joint channel estimation and FTN data detection in frequency selective fading channels. We intentionally separate the ISI imposed by FTN signaling and by fading channel, which benefits from the known FTN-induced ISI. Considering the colored noise samples at the receiver side, we resort to the AR model for characterizing the correlation between different samples. Based on the linear state models, we construct a Forney-style factor graph, on which all messages are derived in Gaussian form. Therefore, the message updating only involves simple calculations of means and variances. In particular, to solve the message updating on the multiplier node, we use VMP rules and reconstruct a “soft” node for keeping Gaussian messages. As a result, the complexity of the proposed GMP-based receiver only increases linearly with the block length N , which is significantly lower than the MMSE algorithm. Simulation results show that by using the proposed receiver, the FTN system is capable of increasing 40% transmission rate in frequency selective channels with negligible performance loss.

Chapter 5

Receiver Design for FTN Signaling over Doubly Selective Channels

5.1 Introduction In Chap. 4, we study the low-complexity receiver design for FTN signaling over frequency selective channels. In the beyond 5G wireless communications, reliable data transmission in high-mobility environments is vital. However, the Doppler shift introduced by the movement of transceivers leads to frequent variation of the channel coefficients. In such circumstances, the channel is also time selective in addition to frequency selectivity, which is the doubly selective channels (DSCs). In general, frequency domain equalization (FDE) can efficiently detect the symbols in a low complexity fashion by transforming the time domain frequency selective channel into independent sub-channels in the frequency domain. Nevertheless, due to the time-varying channel coefficients, inter-carrier interference (ICI) exists in the frequency domain, making the low-complexity receiver design challenging. Some early contributions aimed for developing low-complexity receiver under DSCs based on techniques of channel shortening [116, 117], basis expansion [118], and compressive sensing [119, 120]. For slow time-varying DSCs, the authors proposed a linear MMSE (LMMSE)-based method that has linearly increased complexity [121]. All previous works focused on Nyquist signaling. As for FTN signaling, we can still adopt the FDE algorithm [58] and use MMSE to mitigate the ICI. However, its complexity will be very high and avoids its application in high-mobility scenarios. Therefore, a low-complexity receiver for FTN signaling over DSCs needs further investigation. In this chapter, we investigate the FDE-based FTN receiver for DSCs. In particular, we first present the frequency domain FTN signaling model and discuss the application of the FDE-MMSE algorithm [58] in DSCs. Then, we propose two different algorithms for solving the FTN symbol detection problem, i.e., the VFE-based algorithm and the GMP-based algorithm. For the former scheme, we adopt (mean field) MF [122] and Bethe [123] methods for approximating the a posteriori distri-

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Yuan et al., Receiver Design for High Spectral Efficiency Communication Systems in Beyond 5G, https://doi.org/10.1007/978-981-19-8090-9_5

77

78

5 Receiver Design for FTN Signaling over Doubly …

bution and lead to two receiver structures. For the GMP-based algorithm, we further extend the case to imperfect channel information and derive a robust message passing receiver for FTN detection in DSCs.

5.2 System Model The system model for FTN signaling over DSCs is similar to the one in Fig. 4.1, where the frequency selective channel is replaced by DSC. The information bits b are coded and mapped to data symbols x = [x0 , . . . , x N −1 ]T . For FDE, 2M cyclic prefix (CP) symbols are inserted at the transmitter side. The transmitted symbol vector pass through a shaping filter g(t) with period T0 = τ T and 0 < τ ≤ 1 denotes the packing factor. The transmission rate is increased by 1/τ by employing FTN signaling. Then, the DSC is assumed to have L paths and the channel response at the kth time is [h k,0 , . . . , h k,L−1 ]. Again, the signal is acquired at the receiver side, given by y(t) =

L−1   l=0

h l,i g(t − (i + l)xi τ T ) + ξ(t).

(5.1)

i

Then y(t) is matched filtered and sampled with period τ T . After CP removal, the kth received sample is given by [58, 124] rk =

L−1  l=0

h k,l

M 

qi xk−l−i + ξk ,

(5.2)

i=−M

with qk−m and ξk defined in (4.7) and (4.8), respectively. Since g(t) is non-orthogonal with period τ T , the colored noise ξk has the ACF given in (4.9). Similar to (4.10), the received signal vector r  [r0 , . . . , r N −1 ]T can be represented in a matrix form as r = HQx + ξ ,

(5.3)

where Q is an N × N matrix having the kth row being [0k , q−M , . . . , q M , 0 N −k−2M ]. Since the channel coefficients are time-varying, the kth row of H is given by [0k , q−M , . . . , q M , 0 N −k−2M ]. As for the colored noise vector ξ  [ξ0 , . . . , ξ N −1 ]T , its autocorrelation matrix is expressed as E[ξ ξ H ] = N0 Q. To adopt the FDE, the received signal r is transformed to frequency domain, following r f = FHQx + ξ f ,

(5.4)

5.3 VFE-Based Receiver Design

79

where F is the N × N normalized DFT matrix whose element on the mth row and nth column is Fm,n = √1N exp(−2π jmn). With the insertion of CP, the matrix Q is circulant and follows the eigenvalue decomposition Q = FH F, where  is a diagonal matrix with the ith element being the ith eigenvalue of Q. Finally, we can rewrite (5.4) as r f = Cx f + ξ f ,

(5.5)

where x f = Fx = [x f,0 , . . . , x f,N −1 ]T is the frequency domain symbol and C = FHFH can be regarded as the equivalent frequency domain channel matrix. The autocorrelation matrix of the frequency domain noise ξ f = Fξ is E[ξ f ξ f H ] = E[Fξ ξ H FH ] = N0 FQFH = N0 ,

(5.6)

which indicates that the noise samples are decorrelated in the frequency domain.

5.2.1 FDE-MMSE Based Algorithm Relying on the above model, the FDE-MMSE algorithm [58] can be used to estimate x f , i.e., xˆ f = Wr f ,

(5.7) −1

where W =  C (C C + N0 ) . H

H

H

H

(5.8)

By adopting inverse DFT (IDFT), we have the estimate of the FTN data symbols as xˆ = FH xˆ f . Note that the matrix inversion operation in (5.8) requires a complexity order of O(N 3 ). To maintain the same complexity of the frequency selective channel case in [58], we can approximate C by its diagonal entries and yield diagonal matrix W. However, the detection performance will be significantly affected. To reliably detect the FTN symbols in DSCs with low complexity, we develop two different algorithms for receiver design, which will be discussed in the following sections.

5.3 VFE-Based Receiver Design From the Bayesian detection perspective, we focus on the a posteriori distribution of x f , which reads

80

5 Receiver Design for FTN Signaling over Doubly …

p(x f |r f ) ∝ p(r f |x f ) p(x f ),

(5.9)

where the likelihood function p(r f |x f ) can be written as p(r f |x f ) ∝ exp(−(r f − Cx f )H (N0 )−1 (r f − Cx f )).

(5.10)

The a priori distribution p(x f ) can be modeled as a Gaussian distribution with mean vector mx0 f = Fmx0 and covariance matrix Vx0 f = FVx0 FH , where mx0 = [m 0x0 , . . . , m 0x N −1 ]T and Vx0 = diag{Vx00 , . . . , Vx0N −1 } are the mean and covariance matrix for FTN symbols x, respectively. Note that a diagonal matrix Vx0 will become non-diagonal after domain transformation. To this end, we adopt the approximation [125] for simplifying the calculations, where Vx0 is approximated by  N −1 Vx0

 aI, with a =

i=0

Vxi0

N

.

(5.11)

Consequently, after transforming to the frequency domain, the covariance matrix is still diagonal, i.e., Vx0 f = aI. Now, we are able to factorize (5.9) as the product of several local potentials, following p(x f |r f ) ∝

 i

p(x f,i ) exp(−R{dii }|x f,i |2 + R{di x ∗f,i })    ϕii (x f,i )

 i, j

exp(−R{di j x ∗f,i x f, j }),    ϕi j (x f,i ,x f, j )

(5.12) CH r

:,i f −1 with the shorthand notations di j = CH :,i (N0 ) C:, j and di = N0 λi . We further let ϕi j = ϕ ji hereafter since two frequency domain symbols are mutually interfered.

5.3.1 VFE-Based Method With the a posteriori distribution (5.12), the MAP detection of x f,i focuses on the marginal distribution p(x f,i |r f ), which can be obtained by directly marginalizing p(x f |r f ) but involves prohibitively high complexity. As discussed in Chap. 3, the VFE framework helps to find a distribution q(x f ) which is easy to be marginalized to approximate the actual distribution p(x f |r f ). The VFE [84] between the distributions q(x f ) and p(x f |r f ) is formulated as F = with a constant C0 .

q(x f ) ln

q(x f ) dx f + C0 , p(x f |r f )

(5.13)

5.3 VFE-Based Receiver Design

81

We first consider the MF

approximation, where all data symbols are assumed to be independent, i.e., q(x f ) = i q(x f,i ) is first considered Substituting trail distribution q(x f ) into (5.13) yields the VFE as FM F = −

q(x f ) ln p(x f |r f )dx f +



q(x f,i ) ln q(x f,i )dx f,i .

(5.14)

i

 To minimize F M F , we take into account the normalization constraint q(x f,i )dx f,i = 1 and construct the Lagrangian as 

LM F = FM F +

μi (x f,i ) 1 − q(x f,i )dx f,i ,

(5.15)

i

where μi (x f,i ) is the associated Lagrangian multiplier. By setting the partial derivative of Lagrangian with respect to q(x f,i ) to zero, we have the expression of q(x f,i ) q(x f,i ) ∝ ϕii (x f,i )



N −1 

exp

ln ϕi j (x f,i , x f, j )q(x f, j )dx f, j .

(5.16)

j=0, j=i

Assuming that the marginal distributions of all frequency domain symbols are Gaussian q(x f, j ) ∝ CN(x f, j , m x f, j , vx f, j ), q(x f,i ) is also Gaussian with mean m x f,i and variance vx f,i . The extrinsic information is then extracted from the q(x f,i ) by subtracting the a priori information, which are given by m ex f,i

=

vxe f,i (m x f,i /vx f,i

−

m 0x f,i /vx0 f,i )

vxe f,i = (1/vx f,i − 1/vx0 f,i )−1 =

=

di −

N −1

j=0, j=i

R{dii }

1 . R{dii }

di j m x f, j

,

(5.17) (5.18)

We can see that the calculation of extrinsic parameters corresponding to x f,i does not depend on the variances of other symbols. That is to say, the variances of other symbols are ignored in MF approximation, which results in performance loss. To improve the performance of MF approximation, we consider the Bethe approximation used in Sect. 3.7. Here, we consider a pairwise form of the Bethe approximation, given by q(x f ) =

 i

q(x f,i )

 q(x f,i , x f, j ) . q(x f,i )q(x f, j ) i, j

(5.19)

In a similar way, we substituting the Bethe approximation (5.19) into (5.13) and yield the VFE as

82

5 Receiver Design for FTN Signaling over Doubly …

FB = − +

q(x f ) ln p(x f |r f )dx f + 



q(x f,i ) ln q(x f,i )dx f,i

i

q(x f,i , x f, j ) ln q(x f,i , x f, j )dx f,i dx f, j

i, j, j=i

−



q(x f,i , x f, j ) ln q(x f,i )q(x f, j )dx f,i dx f, j

i, j, j=i

=



q(x f,i , x f, j ) dx f,i dx f, j ϕi j (x f,i , x f, j ) i, j, j=i    + (N − 1) q(x f,i ) ln q(x f,i ) − ln ϕii (x f,i ) . q(x f,i , x f, j ) ln

(5.20)

i

For Bethe approximation, the minimization is subject to marginalization con straint q(x f,i , x f, j )dx f, j = q(x f,i ), in addition to normalization constraint. Defining μi (x f,i ) and μi j (x f,i ) as the multipliers, the corresponding Lagrangian is formulated as

 μi j (x f,i ) q(x f,i ) − q(x f,i , x f, j )dx f, j LB = FB + i, j, j=i

+



μi (x f,i ) 1 − q(x f,i )dx f,i .

(5.21)

i

Again, by setting the partial derivatives ∇q(x f,i ) L B and ∇q(x f,i ,x f, j ) L B to zero, we obtain  μi j (x f,i ), (5.22) ln q(x f,i ) = μi (x f,i ) + ln ϕii (x f,i ) + j, j=i

ln q(x f,i , x f, j ) = ϕi j (x f,i , x f, j ) + ln q(x f,i )q(x f, j ) − μi j (x f,i ) − μ ji (x f, j ). (5.23) Substituting (5.22) into (5.23) and taking exponential of both sides provides q(x f,i , x f, j ) = C1 ϕii (x f,i )ϕ j j (x f, j )ϕi j (x f,i , x f, j )



exp(μik (x f,i )) exp(μ jk (x f, j ))

k,k =i,k = j

(5.24) with constant C1 . Performing integration over x f, j and comparing it with (5.22) yield exp(μi j (x f,i )) ∝

ϕi j (x f,i , x f, j )

 k,k=i k= j

exp(μ jk (x f, j ))dx f, j .

5.3 VFE-Based Receiver Design

83

Let us define q˜\i (x f, j ) = k,k=i,k= j exp(μ jk (x f, j )) and substitute it into (5.22), we finally arrive at the marginal distribution q(x f,i ) ∝ ϕii (x f,i )



N −1 

ϕi j (x f,i , x f, j )q˜\i (x f, j )dx f, j ,

(5.25)

j=0, j=i

 where q˜\i (x f, j ) = q(x f, j )/ ϕi j (x f,i , x f, j )q˜\ j (x f,i )dx f,i . With the Gaussian assumption of q˜\i (x f, j ) ∝ CN(x f, j , m˜ \i,x f, j , v˜ \i,x f, j ), we derive the extrinsic information based on Bethe-approximation m ex f,i vxe f,i

=

di −

N −1

j=0, j=i

di j m˜ \i,x f, j

R{dii } + v˜ \i,x f, j |di j |2 1 = . R{dii } + v˜ \i,x f, j |di j |2

,

(5.26) (5.27)

For both MF and Bethe approximations, the extrinsic information can be converted to LLR and fed to channel decoder.

5.3.2 Complexity Reduction The complexity of FDE algorithm mainly depends on two parts, the DFT processing complexity and symbol detection complexity. The complexity of DFT and IDFT is fixed with O(N log N ) based on faster Fourier transformation (FFT). For the symbol detection part, the complexity is O(N ) for (5.18), (5.27) and O(N 2 ) for (5.17), (5.26). Therefore, since the messages are computed once per block, the detection complexity for the MF-based algorithm is O(N 2 ). As for Bethe approximation, an additional complexity of O(N 2 ) is required for calculating m˜ \i,x f, j and v˜ \i,x f, j for different frequency domain symbols. However, this does not affect the complexity order, which still remains O(N 2 ). In fact, we can observe that q˜\i (x f, j ) only differs in one term compared to q(x f, j ). Hence, it is capable of exploiting q(x f, j ) to approximate q˜\i (x f, j ), which enjoys the same computational complexity as MF-based approach. As a further step, we could take into account only R frequency-domain interfered symbols to reduce the complexity, which can be chosen based on the modulus of the entries in C:,i . As a result, the complexity can be further reduced to O(N R) for the VFE-based receiver relying on both MF and Bethe approximation, which linearly increases with the block length N .

84

5 Receiver Design for FTN Signaling over Doubly … 10 -1

10 -2

10 -3

BER

BPSK 10

8QAM

-4

MMSE equalizer ref ( =1) MMSE equalizer ref ( =0.8) Time domain-based ref ( =0.8) Bethe approximation ( =1) Bethe approximation ( =0.8) app-Bethe approximation ( =0.8) Mean Filed approximation ( =0.8)

10 -5

10 -6

5

5.5

6

6.5

7

7.5

8

SNR (dB)

Fig. 5.1 BER performance for different algorithms in DSCs

5.3.3 Simulation Results We consider a coded FTN system with an LDPC code of rate R = 1/2. The roll-off factor for the RRC shaping filter is α = 0.5. The Nyquist symbol duration is T = 0.25 µs and the packing factor is τ = 0.8. We assume that the number of channel taps is L = 10 and the number of CP symbols is 2M = 20. In high-mobility scenarios, the channel coefficients vary symbol by symbol. For each symbol interval k, the coefficients h k,l are generated from distribution CN(h k,l , 0, q l ) with the power delay profile is q l = exp(−0.1l). We first compare the BER performance of the proposed algorithm and the FDEMMSE scheme in Fig. 5.1. The performance of a time domain equalization (TDE) method extended from [59] is also included as a reference. The number of interfered symbols for both MF and Bethe approximations is set to R = 10. From Fig. 5.1, we can observe that the FTN system based on the proposed receiver can increase 25% transmission rate and approach the performance of the Nyquist system. The proposed Bethe approximation-based VFE receiver performs close to the FDE-MMSE algorithm and the TDE algorithm while significantly reducing the complexity. Due to the underestimation of variances of interfered symbols, the MF approximation-based method has a small performance loss. Moreover, the app-Bethe method which use the marginal q(x f, j ) instead of q˜\i (x f, j ) can strike a good trade-off between detection performance and complexity. In Fig. 5.2, the BER performances of the proposed VFE receiver based on MF and Bethe approximations with different values of R are plotted. Obviously, increasing R

5.4 GMP-Based Receiver Design

85

10 0

10 -1

BER

10 -2

10 -3

10 -4

10 -5

Bethe approximation (R=N) Bethe approximation (R=20) Bethe approximation (R=10) Bethe approximation (R=5) Mean Filed approximation (R=N) Mean Filed approximation (R=20) Mean Filed approximation (R=10) Mean Filed approximation (R=5)

10 -6 5.5

6

6.5

7

SNR (dB)

Fig. 5.2 BER performance for different values of R

improves the detection performance at the cost of higher complexity. Nevertheless, when R > 10, the performance gain by further increasing R is negligible. Therefore, setting R = 10 is sufficient for achieving good performance in piratical applications.

5.4 GMP-Based Receiver Design Generally, it is of high complexity to mitigate the interference in C. Nevertheless, as shown in [117], a “Q-tap” approximation can be used by considering the interference from only Q − 1 adjacent frequency domain symbols. Therefore, we only keep Q non-zero elements in the kth row of C, i.e., Ck,: = [0, . . . , 0, Ck,k−Q+1 , . . . , Ck,k , 0, . . . , 0]. And the received sample r f,k is rewritten as r f,k =

Q−1 

Ck,k− j λk− j x f,k− j + ξ f,k .

(5.28)

j=0

To apply the GMP algorithm, we aim for constructing the Forney-style factor graph, which relies on the following linear state space model, sk = Ask−1 + bk x f,k , r f,k =

ckT sk

+ ξ f,k ,

(5.29) (5.30)

86

5 Receiver Design for FTN Signaling over Doubly …

Fig. 5.3 Forney-style factor graph of GMP-based receiver for FTN signaling in DSC

where sk  [λk−Q+1 x f,k−Q+1 , . . . , λk x f,k ]T , ck  [Ck,k−Q−1 , . . . , Ck,k ]T , A =  T 0 Q−1 I Q−1 , and bk = [0 Q−1 , λk ]T . Based on (5.29) and (5.30), we illustrate the 0Q Forney-style factor graph for the considered FTN system in Fig. 5.3. Again, we use − → − · to denote the message passed along the arrow direction and ← · to denote the message passed along the opposite arrow direction.

5.4.1 GMP-Based Method With the Gaussian assumption of frequency domain symbols, all messages are characterized by their means and covariance matrices. The parameters from the channel − → − → − → → → → decoder are given by − m x f =[− m x f,0 , . . . , − m x f,N −1 ]T , V x f = diag{ V x f,0 , . . . , V x f,N −1 } − = while the extrinsic parameters fed to the channel decoder are defined by ← m xf ← − ← − ← − − ,...,← − T [← m m ] and V = diag{ V , . . . , V }. As discussed in the x f,0 x f,N −1 xf x f,0 x f,N −1 previous section, the incoming messages are the a priori mean and covariance matrix, which are calculated based on the parameters corresponding to time-domain symbols. For complexity reduction purpose, we also adopt the approximation method in [125], i.e., − → − → V x ≈ aI, with a = tr[ V x ]/N .

(5.31)

− → Then the covariance matrix of the frequency domain incoming message is V x f = aI. Next, we move our focus on the calculation of the extrinsic parameters. Assuming

5.4 GMP-Based Receiver Design

87

− → → that − m s and V s have been already obtained, we can derive the parameters k−2 k−2 associated with sk−1 as − → − → − → V sk−1 = V s + b|λk−1 |2 V x f,k−1 bT , k−2 − → → → m sk−1 = − m x f,k−1 . m s + bλk−1 − k−2

(5.32) (5.33)



As for the message regarding sk−1 , it is more convenient to consider the transformed mean and covariance matrix, which are determined as − → V s

k−1

− → − W s → m s k−1

k−1

← − − → −1 T = A W s˙k−1 + W sk−1 A ,  − → → ← − − m sk−1 + W s˙k−1 ← m = A W sk−1 − s˙k−1 ,

(5.34) (5.35)

where H ck−1 ck−1 ← − , W s˙k−1 = N0 λk−1 ck−1r f,k−1 ← − − W s˙k−1 ← m . s˙k−1 = N0 λk−1

(5.36) (5.37)

− − and ← Moreover, the backward parameters ← m V sk are given by sk ← ← − − ← − −1 V sk = W s˙k + W s¨k ,  − − − ← ← − −  ← − =← m m s˙k + W s¨k ← m s¨k . V sk W s˙k ← sk

(5.38) (5.39)

Having the messages (5.34)–(5.39), we obtain the extrinsic parameters ← − = m x f,k

  − −− → bTk ← m m s sk k−1

λk

,

− → ← −  bTk V s + V sk bk ← − k−1 . V x f,k = λ2k

(5.40) (5.41)

− − and ← V x f are determined using (5.40) and Then, the vector-form messages ← m xf (5.41). By performing inverse DFT, the time domain extrinsic information is derived, which is fed for channel decoding.

88

5 Receiver Design for FTN Signaling over Doubly …

Fig. 5.4 Factor graph modification for imperfect channel information

5.4.2 Imperfect Channel Information The proposed GMP-based receiver relies on the knowledge of channel information C, which may not be perfectly known in practice. In this case, we can model the ¯ as actual channel matrix C ¯ = ρC + , C

(5.42)

where ρ and  denote the skew coefficient and channel uncertainty, respectively. The (i, j)th element of  has zero mean and variance {φi, j }, which describes the accuracy of channel estimation of h i, j . Obviously, using C for FTN symbol detection is not robust to channel uncertainty. To this end, we extend our proposed GMP-based FTN receiver to take the channel uncertainty into consideration. Substituting the revised channel model in (5.42) into (5.30) yields r f,k = (ρck + δ k )T sk + ξ f,k ,

(5.43)

where δ k  [k,k−Q−1 , . . . , k,k ]T . Consequently, the dashed-line boxed part of the factor graph in Fig. 5.3 is modified, as illustrated in Fig. 5.4. Again we introduce a multiplier node  to represent the function δ(y f,k − c¯ kT s˙k ). Let us denote the vector ¯ k,k−Q+1 , . . . , C ¯ k,k ]T and matrix k = diag{φk,k−Q+1 , . . . , φk,k }. Based on c¯ k = [C the classic sum-product algorithm, the message from  to s˙k is given by ← − = μ s˙k



− − ← ¯k δ(y f,k − c¯ kT s˙k )← μ y f,k μ c¯ k d y f,k d c −

1

(r

−¯cT s˙ )2

−1

e N0 λk f,k k k e−(¯ck −ρck ) k (¯ck −ρck ) d¯ck ,

ρ 2 ck ckH ρck r f,k , s˙k − 2˙skH ∝ exp s˙kH N 0 λk + s N 0 λk + s

∝

H

(5.44)

where s = s˙kH k s˙k . Considering that the PSK symbols are energy normalized − in (5.44) can be written in Gaussian |xk |2 = |xk | = 1. Therefore, the message ← μ s˙k

5.4 GMP-Based Receiver Design

89

after some tedious but straightforward manipulations. Then, the mean and variance of (5.44) are given by ρck r f,k , N0 λk + tr[k ] ρ 2 ck ckH ← − . W s˙k = N0 λk + tr[k ]

← − − m s˙k = W s˙k ←

(5.45) (5.46)

The other messages on the factor graph have already been derived in the previous subsection and are not given here for brevity. The complexity of the proposed GMP-based receiver is dominated by message calculations. Same as [125], the complexity is O(Q 3 ) for a Q-length symbol vector. Therefore the detection complexity is O(Q 2 ) per FTN symbol. Considering the complexity of DFT and IDFT, The total computational complexity for the proposed algorithm is O(N log N + N Q 2 ). Since Q is much smaller than N , the complexity of the proposed algorithm is significantly lower than that of the FDE-MMSE detector in DSCs.

5.4.3 Simulation Results In the simulations, we adopt a convolutional code of rate-1/2 and BPSK symbol mapping. The shaping filter and symbol duration are set identically to Sect. 5.3.3. The number of channel taps is set to L = 30. The coefficients h k,l are independently generated according to the distribution h k,l ∼ G(0, q l ) with normalized power delay profile q l = exp(−0.1l). The considered number of interfered symbols is Q = 5, unless otherwise specified. The parameters of incoming message are initialized as − →(0) − → T m (0) x = 0 and V x = I N . Figure 5.5 illustrates the BER performance of the proposed algorithm under perfect channel information. For comparison purpose, the performances for the MMSE detector, the TDE method extended from [59], and the MF-based method are depicted. The performance of the approximate MMSE detector using only diagonal entries of C is also included. We see that the proposed algorithm outperforms the approximate MMSE detector and the MF-based method by fully exploiting the correlation of the received samples. Moreover, it can be observed that with τ = 0.6, the FTN signaling can transmit 66% more symbols than Nyquist signaling in DSCs using the same time-frequency resource, with only 0.4 dB performance degradation. Next, To validate the robustness of the proposed receiver with imperfect channel information, we depict the BER performance versus E b /N0 in Fig. 5.6. Two different levels of channel uncertainties are considered, having φi j = 0.003 and φi j = 0.01. The skew coefficient is ρ = 0.99. The performance of Nyquist signaling with roll-off factor α = 0.2 is plotted as a reference. It is interesting to see given the same spectral efficiency, the FTN signaling achieves a better BER performance than Nyquist

90

5 Receiver Design for FTN Signaling over Doubly … 10 0

10 -1

BER

10 -2

10 -3

TDE MMSE equalizer =1 TDE MMSE equlaizer =0.8 Time domain-based method =0.8 Proposed algorithm =1 Proposed algorithm =0.8 Proposed algorithm =0.6 Variational-based method =0.8 Approximated MMSE =0.8

10 -4

10 -5

10 -6

1

2

3

4

5

6

7

8

9

10

E b/N0 (dB)

Fig. 5.5 BER performance of different algorithms -1

10

-2

10

-3

Perfect CSI Robust receiver

BER

10

Nyquist 10

Robust receiver

-4

Nyquist

ij

VI method 10

ij

Original method -6

4

5

ij

=0.01

=0.01, =0.2

MMSE method 10

=0.003

=0.01, =0.2

Original method

-5

ij

=0.003, =0.2 ij

6

ij

=0.003

=0.01 ij ij

=0.01 7

8

Eb /N0 (dB)

Fig. 5.6 BER performance of the proposed robust detection algorithm

9

10

5.5 Conclusions

91

signaling. Compared to the algorithm that ignores channel uncertainty, the proposed algorithm achieves significant performance improvement in the high SNR region. While in the low SNR region, both algorithms have similar performance. This is because the noise dominates at low E b /N0 .

5.5 Conclusions This chapter developed low-complexity FDE algorithms for FTN signaling transmitted over DSCs. We first develop a VFE-based receiver, which relies on the MF and Bethe approximations. By minimizing the VFE under certain constraints, the extrinsic mean and variance corresponding to MF- and Bethe-based methods are derived, enabling low-complexity calculations of the parameters. Motivated by the GMP algorithm in Chap. 4, we extend the GMP-based receiver to DSCs. Based on a reformulated Forney-style factor graph, we derive all messages in Gaussian closedform. By considering a limited number of interfered frequency domain data symbols, the detection complexity can be further reduced. Our analysis show that our proposed low-complexity FTN detection schemes enjoy a complexity that only increases linearly with the block length N . Through simulation results, we see that relying on the designed receivers, the FTN system in DSCs can approach the Nyquist counterpart while increasing the transmission rate significantly. Moreover, our proposed lowcomplexity receiver can achieve similar performance to the high-complexity MMSE detector. Finally, we show that when the channel information is imperfect, our modified GMP-based receiver can reliably detect the FTN symbol by considering the channel uncertainty.

Chapter 6

Receiver Design for FTN-SCMA Communication System

6.1 Introduction In Chaps. 2 and 3, we discuss the receiver design for SCMA system while Chaps. 4 and 5 investigate the receiver design for FTN signaling. With the rapid evolution of wireless applications, higher spectral efficiency is always expected. Motivated by the fact that SCMA systems and FTN signaling introduce non-orthogonality from different perspectives, we can naturally combine both techniques to achieve even higher spectral efficiency. In the new non-orthogonal communication system, the transmission rate of each user can be further increased using FTN signaling. Nevertheless, the new system will be affected by both ISI and inter-user interference (IUI), imposing challenges on receiver design. To tackle the above problem, we study the low-complexity receiver design for combined non-orthogonal waveform and non-orthogonal multiple access systems. Without loss of generality, we take the uplink system for an example. The proposed receiver algorithm can be easily extended to downlink system. In particular, we first formulate and factorize the joint a posteriori distribution of data symbols, channel taps, and colored noise samples. Then, the factorization can be represented by a factor graph. To reduce the complexity of on-graph message calculations, we resort to the EP method to restrict the discrete messages output from the channel decoder to Gaussian distributions. Compared to direct moment matching, the EP method aims for minimizing a specified relative entropy and can exploit the extrinsic information when performing Gaussian approximation, which enhances the detection performance. Moreover, a similar problem in Chap. 4 emerges that the product of channel coefficient and data symbol makes the derivation of Gaussian messages infeasible. Therefore, we reform a modified factor node, on which VMP is adopted for obtaining Gaussian messages. Since the proposed FTN-SCMA receiver only relies on updating of means and variances of different variables, the complexity scales linearly with the number of users.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Yuan et al., Receiver Design for High Spectral Efficiency Communication Systems in Beyond 5G, https://doi.org/10.1007/978-981-19-8090-9_6

93

94

6 Receiver Design for FTN-SCMA Communication System

Moreover, we extend our proposed receiver to a grant-free application case. The grant-free transmission scheme mainly aims for reducing the high communication overhead induced by the request-grant procedure in the classic grant-based transmission schemes. For the classic scheme, a user first sends a request to the BS and the BS allocates an orthogonal resource element to the user for uplink information transmission [126]. For the massive connectivity application with a large amount of users, the communication overhead and transmission latency will be very high [36]. On the contrary, in grant-free transmission scheme, the active users directly send signals to the BS without sending requests. The BS has to identify the active users and decode their transmitted information. Because only a small percentage of users are active at the same time [127], the sparsity of active users can be exploited for multiuser detection based on the compressive sensing (CS) technique [128]. Later, several joint channel estimation and user activity detection algorithms were developed. The two-stage algorithm in [129] proposed a CS-based active user detection algorithm followed by channel estimation and symbol detection. A fully joint method based on approximate message passing (AMP) and expectation maximization (EM) was proposed in [130]. Relying on the factor graph framework, [131, 132] modeled the user activity as the hyper-parameter of the channel coefficient. However, the resultant factor graph will contain more short loops, which leads to possible convergence issues. Instead of using a hyper-parameter, we introduce a binary variable to represent the active/inactive users, which clearly shows the user state. By modifying the factor graph, we develop a message passing algorithm to determine the distribution of the binary variable. As EP is adopted for approximating the binary variable to Gaussian, the proposed receiver still enjoys low complexity.

6.2 System Model We consider an uplink SCMA system with K users and J orthogonal resource elements. For brevity, the codeword of user k at time instant n has the same notation in n n T , . . . , xk,J ] . After SCMA encoding, each user adopts FTN Chap. 3, i.e., xkn = [xk,1 signaling for improving the transmission rate through a shaping filter q(t) with symbol period T = τ T0 , where τ is the FTN packing factor. After shaping filter, the transmitted signal of user k over the jth resource element is given by sk, j (t) =



n xk, j q(t − nτ T0 ).

(6.1)

n

We assume that the uplink channel associated with user k is given by hk = [h k,1 , . . . , h k,J ]T . Consequently, we illustrate the transmitter block diagram in Fig. 6.1. With the assumption of perfect synchronization between users and BS, the received signal at BS is written as,

6.2 System Model b1

Channel Encoder 1

95 c1

SCMA Encoder 1

X1

shaping Filter (t)

h1

…

n

Channel Encoder k

ck

bK

Channel Encoder K

cK

SCMA Encoder k

Xk

shaping Filter (t)

hk

SCMA Encoder K

XK

shaping Filter (t)

hK

y

Iterative Receiver

…

bk

Fig. 6.1 Transmitter block diagram

SISO Equalizer 1

Channel Encoder k

SISO Equalizer k

Channel Encoder K

SISO Equalizer K

Channel Decoder 1

bˆ1

Channel Decoder k

bˆk

Channel Decoder K

bˆK

…

Channel Encoder 1

y

Matched Filter

r

Multiuser Detector

*

q ( t)

…

Fig. 6.2 Receiver block diagram

y(t) =

K 

hk  s j (t) + xi(t),

(6.2)

k=1

where s j (t) = [s1, j (t), . . . , s K , j (t)]T denotes the transmitted signals of all users over the jth resource element and xi(t) is the noise process with power spectral density N0 . As shown by Fig. 6.2, the received signals pass through the matched filter q ∗ (−t). With the notation of g(t) = q(t) ∗ q ∗ (−t), the filtered signal can be expressed as r(t) =

K 

hk 



n xk, j g(t − nτ T0 ) + ξ (t).

(6.3)

n

k=1

With sampling rate 1/τ T0 , the vector of received samples at the nth time instant is rn =

K  k=1

where the jth entry in s˜kn is given by

hk  s˜kn + ξ n ,

(6.4)

96

6 Receiver Design for FTN-SCMA Communication System L 

n s˜k, j =

n−i gi xk, j ,

(6.5)

i=−L



q(t − nτ T0 )q ∗ (t − iτ T0 )dt.

and gn−i =

(6.6)

Note that in (6.5), we use a sufficiently large number of L to characterize the ISI taps induced by FTN signaling. The noise samples ξ n is expressed as  ξn =

n(t)q ∗ (t − nτ T0 ).

(6.7)

Since we are employing FTN signaling, the noise is colored and the ACF of noise sample ξ nj has been discussed in previous chapters. In the next section, we will propose a factor graph approach to perform channel estimation and detection.

6.3 Receiver Design for FTN-SCMA Systems 6.3.1 Approximation of Colored Noise Similar to Chaps. 4 and 5, the colored noise can be approximated by a P-order AR model as ξ nj

=

P 

n− p

apξ j

+ δ nj .

(6.8)

p=1

6.3.2 Probabilistic Model and Factor Graph Representation Given all the received samples of all N time instants, we aim for determining the n n marginal distribution of xk, j based on r. To calculate the marginal of x k, j in a lowcomplexity way, we develop a factor graph approach based on the joint distribution p(X, h, ξ |r), where X, h and ξ denote the transmitted symbols, channel coefficients, and colored noise, respectively. According to Bayesian theorem, p(X, h, ξ |r) reads p(X, h, ξ |r) ∝ p(X) p(h) p(ξ ) p(r|X, h, ξ ). Assuming that all transmitted symbols and channel coefficients are independent, the a priori distribution p(X) p(h) is given by p(X) p(h) =

 k, j

 p(h k, j )

 n

 n p(xk, j)

,

(6.9)

6.3 Receiver Design for FTN-SCMA Systems

97

Fig. 6.3 Factor graph representation of the jth resource element

n where p(xk, j ) and p(h k, j ) are obtained based on the output LLR of channel decoder and pilot-based coarse channel estimation, respectively. The distribution p(ξ ) is factorized according to the AR model (6.8),

p(ξ ) ∝

 j

n

 exp −

ξ nj −



P

n− p p=1 a p ξ j 2σδ2

 ψ nj

,

(6.10)

Conditioned on ξ nj , the received samples at different time n and different resource element r nj are independent. Consequently, p(r|X, h, ξ ) can be factorized as1 p(r|X, h, ξ ) ∝

 j,n

 δ

r nj

−

K   k=1

n h k, j s˜k, j

 f jn





−

ξ nj

·δ



n s˜k, j

−

L  i=−L



n φk, j

. (6.11)

n−i gi xk, j

According to (6.9)–(6.11), we fully factorize p(X, h, ξ |r), which can be represented by the factor graph in Fig. 6.3, where the factor node pk, j denotes the function p(h k, j ). For ease of exposition, we split the factor graph into four parts, i.e. 1. decoding part, 2. equalization part, 3. channel estimation part, and 4. colored noise part.

1

As discussed in Chap. 3, we introduce auxiliary variables to reduce the receiver complexity [133].

98

6 Receiver Design for FTN-SCMA Communication System

6.3.3 Message Passing Receiver Design In this section, we derive the messages on Fig. 6.3. We first consider the decoding part. As for channel decoding, we adopt the optimal BCJR decoding algorithm [52]. After decoding, the output LLR is expressed as L a (cn,m ) =, where the subscripts n and m are the bit index and constellation index, respectively. The LLRs are then converted to n the probability constellation points, which gives the a priori distribution of p(xk, j) = M n i=1 pi δ(x k, j − χi ). For complexity reduction purpose, we employ the EP algorithm n to approximate the incoming message p(xk, j ) by Gaussian distribution. Following the EP updating rules discussed in previous chapters, the mean and variance of the n Gaussian approximation of p(xk, j ) are determined as  vx0k,n j

=

1 vxk,n j 

m 0xk,n j = vx0k,n j

−

1

−1 ,

vxe n

k, j

m xk,n j vxk,n j

−

m ex n

k, j

vxe n

(6.12) ,

(6.13)

k, j

where m ex n and vxe n are the mean and variance of the outgoing (‘extrinsic’) message k, j k, j after equalization. Based on m 0x n and vx0n , we can calculate the messages in the equalization part. k, j k, j Provided that the message μs˜k,n j →φk,n j = μ f jn →˜sk,n j has already been determined as a Gaussian distribution μs˜k,n j →φk,n j = G(m s˜k,n j →φk,n j , vs˜k,n j →φk,n j ),

(6.14)

the message μφk,n j →xk,n+lj is also Gaussian with mean m φk,n j →xk,n+lj = m s˜k,n j →φk,n j −

L  i=−L ,i=l

gi m xk,n+ij →φk,n j ,

(6.15)

gi2 vxk,n+ij →φk,n j .

(6.16)

and variance vφk,n j →xk,n+lj = vs˜k,n j →φk,n j +

L  i=−L ,i=l

n+l Following standard message passing rules, the message μxk,n j →φk,n j to φk, j with different l has to be calculated for 2L + 1 times. Nevertheless, we can calculate message n μxk,n j →φk,n j based on the fact of μxk,n j →φk,n j · μφk,n j →xk,n j = bG (xk, j ) in a low-complexity way as

6.3 Receiver Design for FTN-SCMA Systems

 vxk,n j →φk,n j =

1 vxk,n j

99

−

m xk,n j →φk,n j = vxk,n j →φk,n j

1

−1

, vφk,n j →xk,n j  m xk,n j m φk,n j →xk,n j − . vxk,n j vφk,n j →xk,n j

(6.17) (6.18)

Having messages μφk,n+lj →xk,n j , ∀l = −L L in hand, the outgoing message is characterized by mean and variance of  vxek,n j =

−1

L 

,

1/vφk,n+lj →xk,n j

l=−L



m exk,n j = vxek,n j

L m n+l n  φk, j →xk, j

l=−L

vφk,n+lj →xk,n j

(6.19)

.

(6.20)

Then, the extrinsic LLRs can be calculated and fed for decoding the information bits of users. Then, we consider the message derivations in the colored noise part. As the first order moment remains zero, we only focus on the variance vψ nj →ξ nj , which has the form of vψ nj →ξ nj = σδ2 +

P  p (a p )2 vξ n− →ψ nj . j

(6.21)

p=1

In a causal system, the sample at time instant n depends on noise samples in previous instants. Consequently, the variances of μξ nj → f jn and μψ nj →ξ nj are identical, i.e. vξ nj to f jn = vψ nj →ξ nj . Next, we move our focus to the channel estimation part. The message μh k, j → f jn can be easily obtained using (1.10) as μh k, j → f jn = p(h k, j )

 

n =n

μ f n →h , j

(6.22)

k, j

where the a priori distribution of channel coefficient p(h k, j ) is usually modeled as a Gaussian variable with mean m 0h k, j and variance vh0k, j . Assuming that the message μ f n →h is obtained in Gaussian form as μ f n →h = (m f n →h , v f n →h ), j

k, j

j

the mean and variance of μh k, j → f jn are formulated as

k, j

j

k, j

j

k, j

100

6 Receiver Design for FTN-SCMA Communication System

⎛ m h k, j → f jn = vh k, j → f jn ⎝ ⎛ vh k, j → f jn = ⎝

1 + vh0k, j

m 0h k, j vh0k, j

 

n =n

+

 m f jn →h k, j 

n =n

v f n →h j

⎞−1

1 ⎠ v f n →h j

⎞ ⎠,

(6.23)

k, j

.

(6.24)

k, j

The parameters of the belief b(h k, j ) are obtained by modifying the summation oper ation over all n . And the estimate of channel is given by hˆ k, j = arg maxh k, j b(h k, j ) = m h k, j . So far, we have derived the messages in Gaussian form for four parts of the factor graph, which however relies on the Gaussian assumption of all messages from f jn to connected variable nodes. Following (1.9), the message μ f jn →˜sk,n j can be obtained as  μ f jn →˜sk,n j ∝

 exp −

|r nj − v

ξ nj → f jn

K

n 2 n k=1 [m h k, j → f j s˜k, j ]|  n 2 n + kk=1 |˜sk, j | vh k, j → f j



 k  =k

μs˜n

k j

skn j . → f jn d˜

(6.25)

We can see that μ f jn →˜sk,n j is not able to be written in Gaussian form since the variable n s˜k, j appears in both numerator and denominator of the exponent term. To this end, we employ the VMP rule (4.40) again to calculate the message (6.25), whose mean and variance can be obtained as, m f jn →˜sk,n j =

(r nj −

K

k  =1,k=k

m h k  j → f jn m s˜n

k j

→ f jn )m h k, j → f jn

|m h k, j → f jn |2 + vh k, j → f jn

,

(6.26)

and variance v f jn →˜sk,n j =

vξ nj → f jn |m h k, j → f jn |2 + vh k, j → f jn

.

(6.27)

The message from f jn to h k, j can be calculated using VMP in the same way, yielding the mean and variance as  (r nj − kK =1,k=k m h k  j → f jn m s˜n → f jn )m s˜k,n j → f jn k j , (6.28) m f jn →h k, j = |m s˜k,n j → f jn |2 + vs˜k,n j → f jn vξ nj → f jn , (6.29) and v f jn →h k, j = |m s˜k,n j → f jn |2 + vs˜k,n j → f jn respectively.

6.4 User Activity Detection in Grant-Free System

101

6.4 User Activity Detection in Grant-Free System The grant-free transmission scheme does not require the authors to receive a grant before sending uplink information to BS. In this section, we use a binary variable to capture the user activity and develop two factor graph-based user activity detection algorithms. We define the activity of user k by variable ηk = {0, 1}, where ηk = 0 represents inactive user while ηk = 0 indicates the user is active. Following this definition, we can rewrite the nth received at the jth resource element as r nj =

K 

n n h k, j ηk s˜k, j + ξj .

(6.30)

k=1

In general, ηk can be modeled as a Bernoulli distributed variable obeying η

p 0 (ηk ) = p1 k (1 − p1 )1−ηk ,

(6.31)

where p1 denotes the a priori probability that user k is active, which is determined based on empirical data.

6.4.1 Probability-Based Active User Detection The activity of user k can be characterized by the probability of being active, denoted by γk . Therefore, we aim for calculating the probability γk based on the received samples. For this reason, we can revise the original factor graph in Fig. 6.4 by including ηk , The modified sub-graph part is illustrated in Fig. 6.4. Note that in Fig. 6.4, an auxiliary variable ηk, j is associated with factor node h k, j to extract the information from the received sample at the jth resource element. Based on the modified factor graph, we are capable of deriving the MPA to determine γk, j , which represents the probability of ηk, j = 1.

Fig. 6.4 Modified factor graph model

102

6 Receiver Design for FTN-SCMA Communication System

The mean m f jn →h k, j and variance v f jn →h k, j of message μ f jn →h k, j can be obtained based on (6.28) and (6.29). Consequently, we have the mean and variance of the intrinsic message of ηk, j h k, j as follows  m f jn →h k, j − → → v ηk, j h k, j , m ηk, j →h k, j = − v f jn →h k, j n  −1  1 − → v ηk, j →h k, j = . v f jn →h k, j n

(6.32)

(6.33)

By integrating h k, j from the joint distribution, we obtain the distribution of ηk, j , given by 

 → (h k, j − m 0h k, j )2 (ηk, j h k, j − − m ηk, j h k, j )2 p(ηk, j ) ∝ exp − exp − dh k, j − → vh0k, j v ηk, j h k, j  → (ηk, j m 0h k, j − − m ηk, j h k, j )2 ∝ exp − 2 0 . (6.34) → η v +− v 

k, j h k, j

ηk, j h k, j

Then the probability γk, j can be calculated as p(ηk, j = 1) p(ηk, j = 0) + p(ηk, j = 1) 1 . = p(ηk, j =0) 1 + p(ηk, j =1)

γk, j =

(6.35)

Having the probability of a user being active based on the received sample on the jth resource element, ∀ j, the probability γk is readily obtained as  j

γk =  j

γk, j p1 +



γk, j p1 j (1

− γk, j )(1 − p1 )

.

(6.36)

Finally, to determine the user activity, a pre-defined threshold can be used. If γk is higher than the threshold, we conclude that user k is active and vice versa. The message from h k, j to f jn can be obtained as follows μh k, j → f jn = μ pk, j →h k, j

 

n =n

μ f n →h . j

k, j

(6.37)

Calculating the message μ pk, j →h k, j involves the combined variable ηk, j h k, j , which is formulated as

6.4 User Activity Detection in Grant-Free System

⎛ μ pk, j →h k, j ∝ ⎝γk, j e

−

(h k, j −m 0h

k, j vh0 k, j

103

)2

+ (1 − γk, j )e

−

[m 0h

k, j vh0 k, j

]2

⎞ ⎠.

(6.38)

Apparently, μ pk, j →h k, j is a Gaussian mixture distribution, indicating that μh k, j → f jn is also a Gaussian mixture distribution. To be compatible with the message passing receiver, we approximate μh k, j → f jn by Gaussian distribution by matching its first two order moments as m h k, j → f jn = Eμhk, j → f n [h k, j ], j

vh k, j → f jn = Eμhk, j → f n [h 2k, j ] − m 2h k, j → f jn . j

(6.39) (6.40)

We can observe that our proposed user activity detection algorithm can be readily extended based on the receiver design in Sect. 6.3 with a small amount of modifications. The details of the proposed probability-based active user detection scheme are summarized in the following table. Algorithm 6.1 Probability-based User Activity Detection Algorithm 1: 2: 3: 4: 5:

Run the message passing algorithm in Sect. 6.3; Compute intrinsic message to h k, j based on (6.32) and (6.33); Calculate the probability γk, j via (6.35); Determine γk based on (6.36) and perform threshold-based decision; Approximate μh k, j → f jn by Gaussian distribution and continue running message passing receiver.

6.4.2 Message Passing Based Active User Detection Algorithm In the probability-based user activity detection algorithm, the probability γk, j has to be calculated separately, leading to an increased receiver complexity. Moreover, the message derivations are not intuitive from the perspective of probabilistic graphical model. To this end, we develop another message passing-based user activity detection method in this subsection. To be concise with the message passing receiver in Sect. 6.3, we add a new variable node ηk to the original factor graph. Based on the signal model (6.30), a Dirac delta n n function representing s¯k, j = ηk s˜k, j is introduced. Consequently, we rewrite the joint likelihood function (6.11) as follows,

104

6 Receiver Design for FTN-SCMA Communication System

Fig. 6.5 Modified factor graph structure. The product node ×nk, j represents the n − η s˜ n ) constraint δ(¯sk, k k, j j

p(r|X, h, ξ , ξ ) ∝

 j,n

 δ

r nj

−

K   k=1



n h k, j s¯k, j







 n δ(¯sk, j

− ξ nj

n − ηk s˜k, j)δ

n s˜k, j

−

i=−L





f jn

L 

n−i gi xk, j

.

n φk, j

(6.41) Based on the new factorization, the modified factor graph is given in Fig. 6.5. For purpose of deriving Gaussian messages, here we adopt EP again to approximate the message μηk →×nk, j from binary variable ηk to the product vertex ×nk, j by Gaussian. Provided the mean m ηk → pηk and variance vηk → pηk of the message μηk → pηk , the mean and variance of b(ηk ) is m ηk =



p1 exp(−

1−2m ηk → pηk vηk → pηk

1−2m η → p p1 exp(− vη →kp ηk ηk k

)  , )−1 +1

(6.42)

vηk = m ηk − m 2ηk .

(6.43)

It is interesting to see that in (6.42), the absolute value of the exponent term dominates the value of m ηk . When ηk is approaching 0 or 1, the variance vηk decreases. This can be explained by the fact that the belief ηk becomes ‘concentrated’ during the iterations. Having m ηk and vηk , the mean and variance of μηk →×nk, j via EP can be readily obtained as  μηk →×nk, j ∼ G

m ηk vηk − m ×nk, j →ηk v×nk, j →ηk vηk − v×nk, j →ηk

,

vηk v×nk, j →ηk vηk − v×nk, j →ηk

.

(6.44)

To calculate the message μs¯k,n j → f jn , we alternatively calculate the distribution of n n n n n n s¯k, j = ηk s˜k, j , where ηk and s˜k, j obey distributions μηk →×k, j and μs˜k, j →×k, j , respec-

6.4 User Activity Detection in Grant-Free System

105

tively. Given Gaussian messages μηk →×nk, j and μs˜k,n j →×nk, j , the probability density n function (pdf) of s¯k, j is given by  n f (¯sk, j)

=

n n n n f (˜sk, sk, sk, j ) f (ηk )δ(¯ j − s˜k, j ηk )d˜ j dηk  n   s¯k, j 1 f (ηk )dηk = f |ηk | ηk ⎛ s¯n ⎞  k, j n n )2 n )2 ( − m − m (η s ˜ →× k ηk →×k, j 1 η k, j k, j ⎠ dηk . (6.45) exp ⎝− k ∝ − |ηk | vs˜k,n j →×nk, j vηk →×nk, j

Nevertheless, we can not derive an analytical expression for the integral in (6.45). As our goal is to derive messages in Gaussian closed-from, we focus more on the mean and variance of μs¯k,n j → f jn instead of the distribution itself. According to Mellin transform [134], for two independent variables x and y, the nth-order of their product x y is given by E[(x y)n ] = E(x n )E(y n ).

(6.46)

Therefore, we can derive the first two order moments of μs¯k,n j → f jn as n n n n n E[¯sk, sk, j ] = E[˜ j ]E[ηk ] = m ηk →×k, j m φk, j →˜sk, j ,

(6.47)

n 2 n 2 2 sk, E[(¯sk, j ) ] = E[(˜ j ) ]E[ηk ]

=

(m 2ηk →×nk, j

+

(6.48)

vηk →×nk, j )(m 2φk,n j →˜sk,n j

+ vφk,n j →˜sk,n j ).

Based on the above moments, the mean and variance of message μs¯k,n j → f jn are expressed as m s¯k,n j → f jn = m ηk →×nk, j m φk,n j →˜sk,n j , m s¯k,n j → f jn =

n 2 2 n E[(¯sk, sk, j ] j ) ] − E [¯

(6.49) =

vηk →×nk, j m 2φ n →˜s n k, j k, j

+ (m 2ηk →×n k, j

+ vηk →×nk, j )vφk,n j →˜sk,n j .

(6.50) The mean and variance for μs¯k,n j →×nk, j are calculated in a similar way as (6.26) and (6.27). Next, we focus on the message from the product node to its connected variable nodes. The problem occurs again that formulating a Gaussian message for ηk is intractable. To overcome this challenge, a new ‘soft’ node is constructed by grouping the message μm s¯n →×nk, j as well as the delta constraint. Thus, we formulate the joint k, j n pdf of s˜k, j and ηk as  p(m s˜k,n j , ηk ) ∝ exp −

n 2 (m s¯k,n j →×nk, j − ηk s˜k, j)

vs¯k,n j →×nk, j

μηk →×nk, j μs˜k,n j →×nk, j .

(6.51)

106

6 Receiver Design for FTN-SCMA Communication System

n Motivated by the variational inference framework, we adopt b(ηk )b(˜sk, j ) to approxn imate the pdf (6.51). The KLD between b(ηk )b(˜sk, j ) and p(m s˜k,n j , ηk ) is given as

 n KLD(ηk , s˜k, j) =

n b(ηk )b(˜sk, j ) ln

n p(˜sk, j , ηk )



 =−

n b(ηk )b(˜sk, j)

b(ηk )



n n n sk, sk, p(˜sk, j , ηk )b(˜ j )d˜ j

ln

 +

n dηk d˜sk, j

dηk

b(ηk ) ln b(ηk )dηk + C,

(6.52)

with a constant C. We aim for minimizing the KLD. Note that  b(ηk ) = exp

 n n , η )b(˜ s ) . ln p(˜sk, k j k, j

(6.53)

Substituting (6.51) into (6.53) reads  m 2n + vs˜k,n j m s¯k,n j →×nk, j m s˜k,n j b(ηk ) 2 s˜k, j ∝ exp −ηk + 2ηk , μηk →×nk, j vs¯k,n j →×nk, j vs¯k,n j →×nk, j

(6.54)

where −1 vs˜k,n j = (vs−1 ˜ n →×n + v×n k, j

n sk, k, j →˜ j

k, j

)−1 ,

(6.55)

−1 n n and m s˜k,n j = vs˜k,n j (m s˜k,n j →×nk, j vs−1 ˜ n →×n + m ×k, j →˜sk, j v×n k, j

n sk, k, j →˜ j

k, j

).

(6.56)

Therefore, the mean and variance of message μ×nk, j →ηk can be calculated as m ×nk, j →ηk = and v×nk, j →ηk

m s¯k,n j →×nk, j m s˜k,n j

m 2s˜n + vs˜k,n j k, j vs¯k,n j →×nk, j = 2 , m s˜n + vs˜k,n j

,

(6.57) (6.58)

k, j

respectively. In a similar way, we determine the message μ×k, j n →˜sk,n j in Gaussian as  μ×k, j n →˜sk,n j ∝ G

m ηk →×nk, j m ηk m 2ηk + vηk

,

vηk →×nk, j m 2ηk + vηk

,

(6.59)

where m ηk and vηk are the a posteriori mean and variance of ηk . Finally, the message μηk → pηk is determined in Gaussian following some straightforward manipulations. To obtain the value of ηk , we resort to the standard MAP estimator. Since b(ηk ) is a Gaussian distribution, the MAP estimate is its mean (6.42). Again, a pre-defined

6.5 Simulation Results

107

threshold is compared with m ηk , based on which we can tell the state of user k. The details of the proposed message passing-based user activity detection algorithm II are summarized in the table below Algorithm 6.2. Algorithm 6.2 Message Passing-based User Activity Detection Algorithm 1: 2: 3: 4: 5: 6:

Run the message passing algorithm in Sect. 6.3; Derive Gaussian message from ηk to the product node via EP based on (6.42) and (6.44); Compute the mean m s¯k,n j → f jn and variance vs¯k,n j → f jn using (6.49) and (6.50); n and η following (6.57)–(6.59); Calculate messages from the product mode to s˜k, k j Compute message μηk → pηk and estimate ηk according to (6.42); Continue executing message passing receiver.

6.5 Simulation Results In the simulations, we consider an SCMA system supporting K = 6 users relying on J = 4 orthogonal resource elements, whose indicator matrix is given by ⎡

1 ⎢1 F=⎢ ⎣0 0

1 1 1 0

0 1 0 1

0 0 1 0

1 0 0 1

⎤ 0 0⎥ ⎥. 1⎦ 1

(6.60)

The SCMA codebook is defined according to [80] with size M = 4. For each user k, a sequence of information bits is encoded with LPDC of rate 5/7 and mapped to SCMA codewords. The total number of transmitted symbols for each user is N = 2048. We assume that the same RRC shaping filter with roll-off factor α = 0.5 and packing factor τ = 0.8 is employed for different resource elements at the transmitter side. The length of ISI taps induced by FTN signaling is set as 2L + 1 = 21. We consider a Rayleigh fading channel model whose coefficient is We first compare the detection performance of the proposed algorithm with the state-of-the-art methods, i.e., SPA, MPA-Gauss, and MMSE-MPA methods Fig. 6.6. The SPA receiver follows the standard sum-product rules and experiences an exponentially increased complexity. The ‘MPA-Gauss’ method refers to the method that n approximates the a priori distribution p(xk, j ) by Gaussian through direct moment matching. The ‘MMSE-MPA’ method first performs MMSE for channel equalization, followed by an MPA-based SCMA decoder. We observe that our proposed receiver performs similarly to the classic SPA receiver while outperforming the other two algorithms. Compared with the ‘MPA-Gauss’ method, using the EP algorithm in our proposed receiver can exploit the extrinsic information and lead to a more accurate approximate distribution. Due to the error propagation, the ‘MMSE-MPA’ method is observed to experience a certain performance loss. Moreover, the performance of

108

6 Receiver Design for FTN-SCMA Communication System 10-1

10-2

BER

10-3

10-4

MMSE-MPA MPA-Gauss The Proposed Algorithm Conventional MPA Orthogonal System

10-5

10-6 6

6.2

6.4

6.6

6.8

7

7.2

7.4

7.6

7.8

8

Eb /N0 (dB)

Fig. 6.6 BER performance for different algorithms

a Nyquist signaling-based OMA system is plotted as a reference. Compared to the conventional orthogonal communication system, the proposed FTN-SCMA system relying on the proposed receiver can support 50% more users and improve 25% transmission rate at the cost of around 0.2 dB performance loss. That is to say, the spectral efficiency can be improved by 87.5% based on the developed new non-orthogonal communication system. In Fig. 6.7, the BER versus E b /N0 of the proposed algorithm with different packing factor τ is depicted, where τ = 1 represents the Nyquist signaling. It is seen that the developed receiver for the FTN-SCMA system can approach the performance of the Nyquist signaling system given τ ≥ 0.8. With the decrease of the packing factor, the transmission rate is improved. However, the interference becomes more severe and the performance gap to the Nyquist system becomes higher. In addition to the interference, another reason is that the number of ISI taps L is not sufficient for modeling the FTN-induced ISI. To further analyze this issue, we plot the BER performance for various values of L in Fig. 6.8 while the parameter τ = 0.7. With the increase of L, the performance gap between the FTN system and Nyquist system becomes smaller, showing that we can adopt a smaller τ for further improving transmission rate, at the cost of higher receiver complexity. Nevertheless, the Mazo limit indicates that the packing factor should be higher than a certain bound in order to guarantee information transmission. To evaluate the convergence of the proposed iterative receiver, we show the BER performance versus the number of iterations in Fig. 6.9. For different values of E b /N0 , the proposed algorithm can converge after running the proposed algorithm for a few

6.5 Simulation Results

109

10-1

10-2

BER

10-3

10-4

10-5

10-6 6.0

= 0.8 = 0.7 = 0.6 = 1(Nyquist) 6.5

7.0

7.5

8.0

Eb/N0 (dB)

Fig. 6.7 BER performance of the proposed algorithm with different packing factors 10 -1

10 -2

BER

10 -3

10 -4

10 -5

10 -6 6.0

L=5 L = 10 L = 20 Nyquist 6.5

7.0

7.5

8.0

Eb/N0 (dB)

Fig. 6.8 BER performance of the proposed algorithm with different FTN-induced ISI tap

110

6 Receiver Design for FTN-SCMA Communication System -1

10

-2

10

-3

10

-4

10

-5

10

-6

BER

10

6.8dB 7.2dB 7.6dB 8.0dB 0

2

4

6

8

10

Number of iterations

Fig. 6.9 BER performance of the proposed algorithm versus the number of iterations

iterations. Moreover, the convergence speed of the proposed algorithm slows down in high SNR region. The channel estimation performance based on the proposed algorithm is illustrated in Fig. 6.10 in terms of the normalized mean squared error (NMSE) of the estimated channel coefficients. The NMSE is defined as K NMSEh =

ˆ 2 k=1 hk − hk  , K 2 k=1 hk 

(6.61)

where hˆ k is the estimated channel coefficients. The NMSEs based on LS channel estimation relying on 5 pilots and all pilots are illustrated in Fig. 6.10 for benchmarks. We can see from Fig. 6.10 that the proposed algorithm can accurately estimate the channel coefficients, achieving a similar performance to the full pilot-based LS channel estimation. The LS channel estimation with 5 pilot symbols suffers from a significant estimation error. Moreover, a joint channel estimation and symbol detection algorithm based on expectation maximization (EM) is considered for comparison, Since EM provides the point estimate and neglects the uncertainties when calculating variables, it performs worse than the proposed algorithm. Considering the grant-free transmission, we illustrate and evaluate the proposed two user activity detection algorithms through BER performance in Fig. 6.11. The a priori probability of user being active is p1 = 0.3. To verify the effectiveness of the proposed algorithm, the message passing receiver in Sect. 6.3 with known knowledge of user activity (denoted by ‘MPA-Known’), the message passing receiver that

6.5 Simulation Results

111

100

NMSE

10-1

10-2

10-3

The Proposed Algorithm EM Algorithm Only Pilots based Algorithm 10-4 6.0

6.5

7.0

7.5

8.0

Eb/N0 (dB)

Fig. 6.10 NMSEs for different algorithms

assumes all users are active (‘Approx-known’), and a two-stage CS-MPA algorithm [135] are used for reference. The two-stage method detects user activity first based on the CS technique and then performs multiuser detection. Obviously, the ‘Approxknown’ suffers significant performance degradation. The two-stage CS-MPA algorithm can not provide soft information regarding the user activity also experiences performance loss. The proposed two user activity detection algorithms can achieve a similar performance to the ideal case with known user activity. Between the two proposed algorithms, the message passing-based scheme has lower complexity than the probability-based one, making it more attractive in practice. Finally, we consider the channel estimation result in the grant-free FTN-SCMA system. Figure 6.12 depicts the channel estimation NMSE of the proposed algorithm parameterized by the a priori probability of p1 . We can see that the increase of p1 results in the degradation of the NMSE performance. This is because a higher value of p1 indicates more users are active simultaneously in the FTN-SCMA system. Therefore, both IUI and ISI become more severe. Moreover, the NMSE performance of the MPA-Known algorithm is illustrated as performance bound. The proposed algorithm can attain the bound when p1 is small. A performance gap will emerge when we increase p1 . However, these channel coefficients can still be accurately estimated based on the proposed algorithm.

112

6 Receiver Design for FTN-SCMA Communication System -1

10

-2

10

-3

BER

10

-4

10

MPA-Known Activity Dection Algorithm I CS-MPA Approx-Known Activity Dection Algorithm II

-5

10

-6

10

6.0

6.5

7.0

7.5

8.0

Eb/N0 (dB)

Fig. 6.11 BER performance of different algorithms with grant-free transmission

-2

NMSE

10

-3

10

-4

10

6.0

MPA-Known p1=0.3 The Proposed Algorithm p1=0.3 MPA-Known p1=0.1 The Proposed Algorithm p1=0.1 AMPA-Known p1=0.6 The Proposed Algorithm p1=0.6 6.5

7.0

7.5

Eb/N0 (dB)

Fig. 6.12 Channel estimation NMSE with different active probability p1

8.0

6.6 Conclusions

113

6.6 Conclusions In this chapter, we develop a novel non-orthogonal communication system by combining the sophisticated techniques of FTN signaling and SCMA for further increasing the spectral efficiency. We adopt the AR model to decorrelate the colored noise samples at the receiver side. Then the joint a posteriori distribution was factorized and represented by a factor graph. A low-complexity message passing receiver is then proposed to jointly estimate the channel coefficients and SCMA decoding. Moreover, we extend our proposed iterative receiver to grant-free transmission case and proposed two user activity detection methods, i.e., the probability-based and the message passing-based methods. Numerical results show that the new non-orthogonal communication system with the proposed receiver can improve more than 80% spectral efficiency than the orthogonal counterpart.

Chapter 7

Receiver Design for FTN-NOMA System with Random Access

In Chap. 6, we have developed a new communication system by combining nonorthogonal waveform and NOMA. The grant-free transmission scheme, also known as random access was further investigated relying on the proposed factor graphbased receiver framework. The random access scheme was originally developed for supporting machine-type communications (MTC) in the beyond 5G era, where a massive number of devices need wireless connectivity. It is shown that only a few fractions of users are active in networks even in busy hours [136]. Therefore, we can exploit the sparsity of active users to reduce detection complexity as well as communication latency. Moreover, in practical MTC applications, the user activity may be time-varying. Some sleeping users may become active in the next time instant and some active users turn to be inactive after information transmission [137]. In such dynamic scenarios, the user activity has to be tracked in real-time at BS. In addition to the user activity, the channel state fluctuates over time. Therefore, tracking channel state as well user state on top of symbol detection is highly desirable for dynamic environments. This chapter considers a practical application case of the FTN-NOMA system developed in Chap. 6. In particular, we deal with the low-complexity receiver design issue with dynamically varying user activity and channel coefficient. As before, we define a binary variable to represent the user state, instead of using a precision parameter. By exploiting the temporal correlations between the user states and channel states in two consecutive time instants, we build the transition models for both variables as a Markov process. By approximating the colored noise samples by the autoregressive moving average (ARMA) model, we are capable of factorizing the joint a posteriori distribution. To represent the factorization, a factor graph is constructed and we can execute MPA for determining the beliefs of unknown variables. For user activity detection, we derive an expectation maximization (EM) algorithm, which can be fully embedded in the message passing receiver, benefiting from the flexibility of the factor graph framework. Upon Gaussian approximation of data symbols and channel coefficients, the messages and beliefs of variables are determined in

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Yuan et al., Receiver Design for High Spectral Efficiency Communication Systems in Beyond 5G, https://doi.org/10.1007/978-981-19-8090-9_7

115

116

7 Receiver Design for FTN-NOMA System with Random Access

a low-complexity way. Overall, the user activity detection, channel estimation, and symbol detection are performed iteratively based on the proposed EM-MPA algorithm. Relying on the closed form solutions for the messages on the factor graph and M-step in the EM process, the receiver complexity only increases linearly with the numbers of active users and FTN-induced ISI taps. Simulation results verify the effectiveness of the proposed iterative receiver for a general FTN-NOMA system with random access in dynamic environments. While the data rate can be increased by 80% compared to the orthogonal system, the performance loss is negligible.

7.1 System Model We consider an uplink scenario where a massive number of users require wireless connectivity. A simple LDS-based NOMA with random access scheme is adopted to support more users than available orthogonal resource elements. The FTN signaling is utilized to further improve the user transmission rate. In what follows, we detailed introduce the considered non-orthogonal communication system.

7.1.1 Faster Than Nyquist Signaling Considering we are transmitting a sequence of data symbols {xn }, the FTN signaling based on a shaping filter of q(t) is given by s(t) =



x n q(t − nτ T0 ),

(7.1)

n

where 0 < τ ≤ 1 is the packing factor and T0 is the Nyquist symbol interval. The modulated signal s(t) then transmits over channel h(t) and is acquired at the receiver. By adopting a matched filter q ∗ (−t) followed by sampling, the nth received sample is expressed as r n = hn



gi xn−i + ξ n ,

(7.2)

i

where h n is the channel coefficient at time n. The term gm−l and ξ n are the FTNinduced ISI tap and colored noise samples, whose expressions are given in (6.6) and (6.7). The colored noise samples at different instants are correlated with correlation function E(ξ l ξ m ) = N0 gl−m , where N0 is the PSD of additional white Gaussian noise.

7.1 System Model

117

7.1.2 Non-orthogonal Multiple Access Similar to Chap. 6, K users are multiplexed over J orthogonal resource elements, where J < K and ρ = KJ is the normalized overloaded factor. The bit sequence of the kth user is mapped to a sequence of transmitted symbols, which are then spread onto an LDS sequence xnk of dimension J . The LDS sequences corresponding to different users are pre-designed, which avoids the packet collision by differentiating the non-zero entry positions in xnk . The nth received sample on the jth resource element at BS can be written as

y nj =

K 

n n h nk, j xk, j + wj,

(7.3)

k=1

where h nk, j denotes the channel coefficient and wnj denotes the noise sample.

7.1.3 FTN-NOMA System with Random Access The considered FTN-NOMA system model is illustrated in Fig. 7.1. To combine the FTN signaling and NOMA system, the transmitted sequences corresponding to different users are passing through the same shaping filter q(t) with symbol period τ T0 , yielding the transmitted signal of user k as sk (t) = [sk,1 (t), . . . , sk,J (t)]T .

(7.4)

As shown in Fig. 7.1, after passing through the time-variant channel hk (t) = [h k,1 (t), . . . , h k,J (t)]T , the received signal on all J resource elements is expressed as follows,

Fig. 7.1 Block diagram of the considered FTN-NOMA system. L kA and L kE denote the extrinsic LRR and output LLR, respectively

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7 Receiver Design for FTN-NOMA System with Random Access

y(t) =

K 

diag{hk (t)} · sk (t) + ξt ,

(7.5)

k=1

where y(t) and wt are of dimension J , whose jth entries denote the received signal and noise process on the jth orthogonal resource element, respectively. Following the standard FTN signaling process, the received signal is matched filtered and sampled, yielding the discrete time sample as r nj =

K 

h nk, j

k=1

L 

n−i n gi xk, j + ξj ,

(7.6)

i=−L

where L is the number of FTN-induced ISI taps. Moreover, as we are considering a grant-free transmission scheme, K+ and K− are used to denote the sets containing all active users and inactive users, respectively. Hence, we can rewrite (7.6) by including only active users as r nj =

 +

k∈K

L 

h nk, j

n−i n gi xk, j + ξj .

(7.7)

i=−L

To label the user activity, we use a binary variable λnk = {0, 1} to represent the user k’s state at time instant k. The state λnk = 1 corresponds to a user being active and vice versa. Finally, we arrive at a concise form of received sample r nj , formulated as r nj =

K  k=1

λnk h nk, j

L 

n−i n gi xk, j + ξj .

(7.8)

i=−L

To design a low-complexity receiver for joint user state tracking, channel estimation, and symbol detection, the LDS sequence and the sparsity of active users are expected to be fully exploited. In what follows, we will propose a receiver design based on the sophisticated factor graph framework.

7.2 Factor Graph Representation of FTN-NOMA System From the optimal detection perspective, we aim for inferring the channel estimates and transmitted bits from the received samples. To tackle the correlated colored noise samples, we employ the ARMA model. For the noise sample ξ nj , the ARMA model of AR order n b and MA order n a is expressed as,

7.2 Factor Graph Representation of FTN-NOMA System

ξ nj =  nj +

na 

ai  n−i − j

i=1

nb 

119

bi ξ n−i j ,

(7.9)

i=1

where  nj is a zero mean Gaussian variable with variance σ2 , ai and bi are the MA coefficient and AR coefficient, respectively. We further set enj

=

na 

ai  n−i j , with a0 = 1,

(7.10)

i=0

for simplifying the expression. Note that enj is still Gaussian distributed with zero  mean and variance σe2 = σ2 ai2 . Then the ACF of colored noise samples is expressed as  N0 gn =

n b for n > 0, i=1 nbb i gn−i bi gn−i for n = 0. σe2 + i=1

(7.11)

Provided known coefficients {gn } based on the knowledge of shaping filter q(t), the AR coefficients {bi } and variance σe2 are estimated by through solving the linear equations in (7.11). With the parameters {bi } and σe2 , all colored noise samples can be written as combinations of other noise samples. n n n n n By stacking the variables xk, j , h k, j , r j , λk , and ξ j into vectors x, h, r, λ, and ξ , we have the joint a posteriori distribution regarding the transmitted symbols, channel coefficients, user activities, and noise samples conditioned on the received samples, i.e., p(x, h, λ, ξ |r). For any unknown variable, we focus on its marginal a posteriori distribution, which can be efficiently derived based on the factor graph framework by leveraging the conditional independencies of variables. The factor graph representation relies on the probabilistic model of the considered system, which will be studied in the following subsection.

7.2.1 Probabilistic Model According to Bayes theorem, the joint distribution p(x, h, λ, ξ |r) can be factorized as p(x, h, λ, ξ |r) ∝ p(x) · p(h) · p(λ) · p(ξ ) · p(r|x, h, λ, ξ ).

(7.12)

Since the transmitted symbols with different indices k and n are independent, the a priori distribution p(x) is fully factorized as p(x) =

 k, j,n

n p(xk, j ),

(7.13)

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7 Receiver Design for FTN-NOMA System with Random Access

n where p(xk, j ) is determined based on the output LLR from the channel decoder. Next, we focus on p(h). According to Bello’s pioneering paper [138], the temporal correlation of time-varying channel coefficients is characterized by f D τ T0 , where f D denotes the Doppler shift. Considering that the adjacent samples have the highest correlation, [139], the channel coefficient h nk, j can be modeled by a Gauss-Markov process as follows n h nk, j = αh n−1 k, j + ε ,

(7.14)

where the coefficient α has the form of α = E[h nk, j (h nk, j )∗ ] = J0 (2π f D τ T0 ),

(7.15)

with J0 (x) being the zero-order Bessel function of the first kind, and εn is a Gaussian distributed variable with zero mean and variance 1 − |α|2 . Therefore, the distribution p(h) is factorized as p(h) =



p(h 0k, j )



p(h nk, j |h n−1 k, j ).

(7.16)

n

k, j

In (7.16), we assume a Gaussian prior for p(h 0k, j ) with mean m h 0k, j and variance vh 0k, j , which are obtained using pilot-based coarse channel estimation. In particular, the transition probability p(h nk, j |h n−1 k, j ) is determined based on (7.14), having the n−1 2 expression of G(αh k, j , 1 − |α |). In a dynamic application case, the user activity λ evolves according to a Markov chain, i.e. the activity of user k at the current time instant depends on its previous state. Hence, we can write the distribution p(λ) as p(λ) =

K 

p(λ0k ) ·



p(λnk |λn−1 k ).

(7.17)

n

k=1

The state transition function p(λnk |λn−1 k ) relies on the state of user k at the previous = 0, a birth probability pbnk is given when user k turns instant. In particular, for λn−1 k to be active at instant n and probability for user k to remain inactive state if 1 − pbnk . In the same way, we define a mortality probability pmn k to describe a previous active user being inactive at instant n, while 1 − pmn k indicates that user k is still active. Based on above assumptions, p(λnk |λn−1 k ) is given by  p(λnk |λn−1 k )

=

( pbnk )1−λk (1 − pmn k )λk , λnk = 1 n−1 n−1 (1 − pbnk )1−λk ( pmn k )λk , λnk = 0 n−1

n−1

(7.18)

Now the question is how to model the birth probability pbnk pbnk and mortality probability pmn k . The Poisson distribution is usually employed to describe the possibility

7.2 Factor Graph Representation of FTN-NOMA System

121

of event occurrence in n a fixed time duration. Assume that there are N + users that change states from inactive to active at the current instant, the variable N + obeys the Poisson distribution of +

p(N + ) =

 N − e , N +!

(7.19)

where  is the average number of occurring events. Therefore, the birth probability of user k, ∀k is set as pbnk = pb = /K . As for the mortality probability pmn k , the exact model is quite challenging and requires a large amount of data collected. Motivated by the fact that an active user will remain active in the next few instants with high probability, we adopt a fair setting that pmn k = 0.5. the ARMA model (7.9) The factorization of p(ξ ) can be easily performed according to the ARMA model (7.9), which reads   b p(ξ 0j ) p(ξ nj |ξ n−n , . . . , ξ n−1 ), (7.20) p(ξ ) = j j n

j

b where p(ξ 0j ) ∝ G(0, σe2 ) and p(ξ nj |ξ n−n , . . . , ξ n−1 ) is given by j j

 b p(ξ nj |ξ n−n , . . . , ξ n−1 ) ∝ exp − j j

(ξ nj − bT ξ nj )2 σe2

 ,

(7.21)

b T , . . . , ξ n−n ] , where b = [b1 , . . . , bn b ]T contains the AR coefficients and ξ nj = [ξ n−1 j j whose state transition can be written in a matrix form as

= B1 ξ nj + b1 ξ nj , ξ n+1 j

(7.22)

0Tn b −1 0 and b1 = [1, 0Tn b −1 ]T . In b −1 0n b −1 Finally, we consider the joint likelihood function p(r|x, h, λ, ξ ). According to (7.8), the likehood function can be written

where B1 =

p(r|x, h, λ, ξ ) =

 j,n

 δ

r nj

−

K  k=1

λnk h nk, j

L 

 n−i gi xk, j

−

ξ nj

.

(7.23)

i=−L

Motivated by the stretched factor graph model 3 for complexity reduction,  L in Chap. n−i n T n ˜ k, j , where g and x˜ nk, j we introduce an auxiliary variables sk, = g x j i=−L i k, j = g x ˜ n−L have the expressions of [g−L , . . . , g L ]T and [˜xn+L k, j , . . . , x k, j ], respectively. As a result,the joint likelihood function p(r|x, h, λ, ξ ) can be factorized as

122

7 Receiver Design for FTN-NOMA System with Random Access

p(r|x, h, λ, ξ ) =



 δ

r nj

−

j,n

K 

 n λnk h nk, j sk, j

−

ξ nj

k=1 n T n ˜ k, j ), · δ(sk, j −g x



where

x˜ nk, j

=

B2 x˜ n−1 k, j

+

n+L b2 xk, j

0T 0 with B2 = 2L I2L 02L

(7.24)

and b2 = [1, 0T2L ]T .

7.2.2 Factor Graph Representation Based on the above factorizations give in (7.13)–(7.24) and the shorthand notations n−1 n n n−n b ,..., φkn , ψk,n j and ξ nj denote the function p(λnk |λn−1 k ), p(h k, j |h k, j ) and p(ξ j |ξ j n−1 ξ j ). we can represent the factorization of p(x, h, λ, ξ |r) by a factor graph, which is illustrated in Fig. 7.2. For ease of exposition, the multiuser detection and decoding part is depicted in Fig. 7.3. Equality factor nodes = are further introduced on the factor graphs to make sure that each edge represents a unique variable.

Fig. 7.2 Factor graph representation for joint distribution (7.12)

7.3 EM-MPA Receiver for FTN-NOMA System

123

Fig. 7.3 Sub-graph for multiuser detection and decoding part

7.3 EM-MPA Receiver for FTN-NOMA System In this section, we solve the joint user activity tracking, channel estimation, and data detection problem by the proposed EM-MPA method. To simplify the message → → expressions, we use − μ (x) and − μ (x) to denote the messages flowing along and in opposite the arrow direction. The standard MPA rules can be found in (1.9)–(1.11).

7.3.1 Multiuser Detection and Decoding Part We focus on the message passing on the sub-graph in Fig. 7.3. Based on the output LLR L ek from the channel decoder, it is capable of calculating the corresponding a n priori distributions of user k’s transmitted symbols, i.e., p(xk, j ), which is the intrinsic − → n n message μ (xk, j ). Again, as xk, j is discretely distributed, we resort to the EP algo→ n rithm to obtain Gaussian form message − μ (xk, j ), which is done by approximating n n ˜ k, the original belief b(xk, j ) to a Gaussian distribution b(x j ) firstly and dividing it by ← − n the extrinsic message μ (xk, j ), given by n ˜ k, b(x j) → → − → n μ (xk, v xk,n j ) = G(− m xk,n j , − j) = ← − n μ (xk, j )   −n vn ← m xk,n j ← m v−xk,n j − ← v−xk,n j vxk,n j xk, j xk, j ,← ∝G . ← v−xk,n j − vxk,n j v−xk,n j − vxk,n j

(7.25)

→ → → v xk,n j , the parameters for − μ (˜xn−L After determining − m xk,n j and − k, j ) can be readily updated as

124

7 Receiver Design for FTN-NOMA System with Random Access

− → → → m x˜ k,n−Lj =b2 − m xk,n j + B2 − m x˙˜ k,n−L−1 , j − → − → → V x˜ k,n−Lj =b2 − v xk,n j bT2 + B2 V x˙˜ k,n−L−1 BT2 . j

(7.26) (7.27)

The message simply passes through the equality node, providing the mean and → covariance matrix of message − μ (˙xn−L k, j ), based on which the message output → n−L from the detection and decoding part can be derived in Gaussian as − μ (sk, j )∝ − → → T− T G(g m x˙ k,n−Lj , g V x˙ k,n−Lj g). As for the intrinsic message fed to the channel decoder, its mean and variance can be obtained based on the knowledge of x˜ nk, j , given by  − n−L − B − → ← − n = bT ← m m ˙˜ n−L−1 , xk, j 2 2 m x˜ k, x j k, j ← − − → ← − T T v xk,n j = b2 V x˜ k,n−Lj − B2 V x˙˜ k,n−L−1 B 2 b2 , j

(7.28) (7.29)

− − n−L and ← where ← m V x˜ k,n−Lj are given by x˜ k, j ← − ← − − ← − n−L = ← − ← − ← n−L n−L m ˙ n−L + g w n−L m n−L W , m V ˙ sk, j sk, j x˜ k, j x˜ k, j x˜ k, j x˜ k, j  ← − ← − − n−L gT −1 , V x˜ k,n−Lj = W x˙˜ k,n−Lj + g← w sk, j

(7.30) (7.31)

← − ← − − = 1/← where W = V −1 and ← w v− are the weight (matrix) of a variable. To commence the message derivations, we first focus on the colored noise. Given the assumption of a causal system, only messages forwarding along the arrow direction are considered. Since the means of noise parameters are zero, the variances of corresponding messages are given by − → − → v ξ nj = bT V ξ nj b, − → − → → V ξ n+1 = B1 V ξ nj B1T + b1 − v ξ nj b1T . j

(7.32) (7.33)

 → − rn as a Gaussian distribution Given the variance − v ξ nj , we can determine ← μ k− j  ← − ← − G m rk,n j , v rk,n j with ← − n = rn −  − → m m rkn , j , rk, j j

(7.34)

− ← → → v−rk,n j = − v ξ nj + v rkn , j ,

(7.35)

k =k

k =k

 → where the messages − μ rkn , j , ∀k = k will be discussed later in this section.

7.3 EM-MPA Receiver for FTN-NOMA System

125

We then move to the calculations  of messages regarding channel estimation. − → in the Gaussian closed-form, the message Assuming that we have obtained μ h˙ n−1 k, j  − → μ h˜ n can be calculated as k, j

  → → − → μ h˜ nk, j ∝ G − m h˜ nk, j , − v h˜ nk, j   → → ∝ α− m h˙ k,n−1j , 1 + |α|2 − v h˙ k,n−1j − 1 .

(7.36)

  → → μ h˙ nk, j are expressed as Based on − μ h˜ nk, j , the mean and variance of message − − → ← −n  m h˜ nk,1 m h k,1 +← , − → − v h nk,1 m h˜ nk,1 − → v−h nk,1 v h˜ nk,1 + ← = − . → v h˜ nk,1 ← v−h nk,1

− → → m h˙ nk,1 = − v h˙ nk,1

(7.37)

− → v h˙ nk,1

(7.38)

 − hn Note that when deriving (7.37) and (7.38), we assume that ← μ k, j is Gaussian ← − ← − with mean m h nk, j and variance v h nk, j . 

→ can be seen as the belief of user As for the user activity, the message − μ λ˙ n−1 k k’s state at the n − 1th time instant, expressed as λkn−1  1−λkn−1

 − → − → − → n−1 n−1 μ λ˙ n−1 = p · 1 − p , λ˙ k λ˙ k k

(7.39)

→ where − p λ˙ kn−1 denotes the the probability of λn−1 = 1. It is noted that the probability k

n−1  − → − → p λ˙ [n−1] can fully characterize the message μ λ˙ k . Therefore, we can pass the k probability alternative to passing the message to have

 simplified expressions. Then → the message flowing along the arrow direction − p λnk is given by 

→ → − → p λ˙ nk . p λ˙ nk + pbnk 1 − − p λnk = 1 − pmn k −

(7.40)

→ Finally, the probability − p λ˙ nk is determined by multiplying the messages forwarded to the equality node, which is formulated as − → p λ˙ nk =

− → − p λnk ← p λ˙ nk,1 − → − n 1− pλ −← p k

λ˙ nk,1

,

(7.41)

→ The probability − p λnk, j for λnk, j with other indices j to the multiplier nodes are derived in a similar way.

126

7 Receiver Design for FTN-NOMA System with Random Access

Finally, we move our focus to the multiplier node × , which imposes challenges on deriving the backward messages in Gaussian closed-form. As the conventional MPA rules fail to provide Gaussian messages, we invoke the EM algorithm to tackle this problem. The EM algorithm was originally proposed for obtaining the maximum likelihood (ML) estimate in the presence of latent variables [140, 141]. The E-step and M-step are performed iteratively, where the E-step determines the expected log-likelihood function based on the distribution of latent variables and the M-step maximizes the expected log-likelihood function to give the estimates of parameters. Without loss  n n n of generality, we take the joint a posterioiri distribution p λk, j , sk, j , h nk, j | rk, j corresponding to the kth user, jth resource element, and nth time instant. We aim for estimating the user activity λnk, j , which is regarded as the unknown n n n variable. Correspondingly, rk, j is the incomplete data and h k, j , sk, j are the latent variables. To incorporate the joint distribution into EM algorithm, we adopt the expected log augmented density, which is given by      n n n n n n n n Q λnk, j ∝ b sk, j b h k, j ln p λk, j , h k, j , sk, j rk, j dh k, j dsk, j  =−



← − n − λn s n h n m rk, j k, j k, j k, j ← v− n rk, j

2

   − → n n n n n · b sk, j b h k, j dh s, j dh k, j + ln μ λk, j + C,

(7.42) where the constant C is irrelevant to λnk, j . Then in M-step, the estimate λˆ nk, j is obtained by maximizing (7.42), which is equivalent to finding the solution of the following equation  ∂q λnk, j ∂λnk, j

= 0.

(7.43)

It should be noticed that the log density of (7.42) only considers the received sample on the jth element, while λnk, j has the same value for different j. Therefore, the Mstep is performed after collecting the information from all J resource elements. To thisend, each multiplier node corresponding to resource element ∀ j feed the message ← − λn μ k, j back to the equality node. Afterwards, we can obtain the probability of  λ˙ nk . Based on the expected log augmented density q λnk, j from (7.42), we have the backward message    n exp q λ k, j ← − λn  ∝  . μ k, j − → n μ λk, j

(7.44)

7.3 EM-MPA Receiver for FTN-NOMA System

127

 − λn is also fully Similar to the forward messages, the backward message ← μ k, j ← − n described by the normalized probability p , which is expressed as λk, j

← − p λnk, j

 → 1−− p λnk, j q(1) . = → 1 − pλnk, j q(1) + − p λnk, j q(0)

(7.45)

The probability is passed on the factor graph instead of the actual message. Having − − − the probability ← p λnk, j , it is readily to determine the probability ← p λnk, j−1 , then ← p λnk, j−2 , ← − − → − → until p λnk,1 . Finally, we obtain the probability p λ˙ nk . By comparing p λ˙ nk to a preset threshold, we can detect the activity of user k. n The calculation (7.42) depends on the assumption of available beliefs b(sk, j ) and n n b(h k, j ). To obtain the beliefs, we adopt the EM algorithm again. Now sk, j is the parameter of interest and h nk, j is still seen as latent variable. Based on the concept of n EM, the belief b(sk, j ) is given by

n  − → n  b sk, j ∝ μ sk, j · exp

 b

h nk, j



  n n n n n ˆ ln p rk, j , sk, j | h k, j , λk, j dh k, j . (7.46)

Here, we use the estimate λˆ nk, j obtained from the M-step (7.43), given by λˆ nk, j

 ← − n m n m n +← → m p λnk, j v−rk,n j 1 − 2− rk, j sk, j h k, j    . =     n 2  n 2 m sk, j  + vsk,n j m h k, j  + vh nk, j

(7.47)

   − → n ← − n n Based on the fact that b sk, j = μ sk, j · μ sk, j , the second term on the RHS   − s n . Given Gaussian form of belief b h n ∝ of (7.46) is actually the message ← μ k, j k, j   − s n as G m h nk, j , vh nk, j , we can derive the mean and variance of message ← μ k, j ← −n n ← − n = m rk, j m h k, j , m  2 sk, j  n  m h k, j  + vh nk, j ← v−rk,n j ← . v−sk,n j =    n 2 n m + v  h k, j  h k, j

(7.48)

(7.49)

n n Consequently, the belief of sk, j can be determined. To obtain the belief b(h k, j ), we n n set h k, j as the unknown variable and sk, j becomes the latent variable. Following the   − h n and belief b h n . same EM updating rules, we arrive at both message ← μ k, j

k, j

128

7 Receiver Design for FTN-NOMA System with Random Access

  n n Based on the obtained beliefs b sk, j and b h k, j , the expected log augmented  density Q λnk, j is updated and executes the next EM iteration. Given the belief of channel coefficient, the estimate of h nk, j is expressed as   hˆ nk, j = Eh b h nk, j = m h nk, j .

(7.50)

 → n Finally, we consider the message − μ rk, j . According to the independent assumpn n n n tion of variables λk, j , sk, j , and h k, j , the first two order moments of rk, j are equal to the product of the same order moments of these three variables. Then, the mean and  − → n variance of μ rk, j are expressed as − → (7.51) m rk,n j = λˆ nk, j m h nk, j m h ns, j ,  2  2  2 2            − → v rk,n j = λˆ nk, j 1 − λˆ nk, j m h nk, j  m sk,n j  + m h nk, j  vsk,n j + m sk,n j vh nk, j + vh nk, j vsk,n j  .

(7.52)

7.3.2 Summary of the Proposed Receiver In the above, we have derived the beliefs and corresponding messages of all variables on the factor graph in closed form. For portraying the proposed EM-MPA receiver, we depict the block diagram in Fig. 7.4. The soft information is exchanged between the channel decoder and multiuser detector. The output LLRs from the channel decoder are converted to discrete distributions of transmitted symbols and then approximated by Gaussian distributions via the EP method. Three basic blocks, i.e., channel estimation, detection, and user activity detection are performed by passing messages to each other. The whole iterative process is done by applying the EM concept.

Fig. 7.4 Block diagram of the proposed EM-MPA receiver

7.4 Simulation Results

129

At each time instant, the proposed algorithm is commenced by extracting the a priori information of user activity, channel state, and noise from the state evolution model. Then, the EM-MPA algorithm is adopted for estimating the unknown variables and calculating the extrinsic message fed to the decoder. In a ‘turbo’ fashion, the decoded information is output and fed back to the detector for refining the estimation of the channel coefficients and user states. As for the complexity, it is dominated by the integration operation in each ‘turbo’ iteration. For the optimal detection scheme, the complexity scales exponentially with the numbers of users and FTN-induced ISI taps. On the contrary, the proposed joint channel estimation, detection, and user activity tracking algorithm  FTN for the NOMA system with random access has a complexity order of O 2|K+ |L , which only increases linearly with the numbers of users and ISI taps. This is because in the proposed algorithm, all messages are written in parametric forms, which fully exploit the sparsity of user activity distribution. Furthermore, for the purpose of supporting more users in MTC applications, the LDS sequences can be designed to control the receiver complexity.

7.4 Simulation Results The simulation results are discussed in this section. In particular, an LDPC code of 1/2-rate is adopted, which is specifically designed for FTN signaling based on the criteria in [142]. QPSK is used for symbol mapping and the length of the transmitted symbol sequence is 1080. The number of users is K = 180 and the number of orthogonal resource elements is J = 120. Therefore, the overloaded factor ρ = 150%. To spread the users’ transmitted symbols over all J resource elements, the LDS scheme in [98] is adopted. To employ the FTN signaling, a standard RRC filter with packing factor τ = 0.8 and roll-off factor α = 0.4 is used. The one-side length of the FTNinduced ISI taps is set to L = 10. The channel is Rayleigh fading, whose coefficients are generated according to Jake’s model. The parameter  = 20 is adopted unless otherwise specified, showing that 11% of all users are active. We further assume that the user activity does not change during one transmission block. For user activity detection, a threshold of 0.5 is used for decision. In Fig. 7.5, the BER performance of the proposed algorithm is compared with those of the reference algorithms. We first assume the user activities are known to the BS. The Genie-aided method which has the knowledge of perfect channel information is used as the performance bound. Compared to the MMSE-PIC scheme which performs channel estimation using the MMSE method and detects data symbols by parallel interference cancellation (PIC), the proposed algorithm is shown to have better performance due to the iterative information exchange between the channel estimation and multiuser detection blocks. Moreover, with much lower complexity, our proposed algorithm suffers from a very slight performance loss compared to the optimal detection based on classic MPA rules. We can also observe that the GAEM-MPA method which approximates non-Gaussian messages by Gaussian ones

130

7 Receiver Design for FTN-NOMA System with Random Access 10 0

10 -1

BER

10 -2

10 -3

10 -4

10

-5

10 -6 4.5

Conventianal-MPA Genie-aided GA-EM-MPA MMSE-PIC EM-MPA

5

5.5

6

Eb /N0 (dB)

Fig. 7.5 BER performance for different algorithms with known user activity

via direct moment matching degrades the performance, which shows the benefits of using the EP algorithm to exploit the extrinsic information gleaned from the detector. Then, we consider the grant-free transmission scenario where the user activity needs to be tracked. The BER performance of the proposed algorithm, the MPAAPP, the Genie-aided, and the LS-AMP-MPA methods are depicted in Fig. 7.6. The LS-AMP-MPA receiver adopts the LS-AMP scheme for detecting the active users and then performs MPA-based multiuser detection. Since the two-stage LS-AMPMPA method can not provide the uncertainties of the user activities to multiuser detection, a considerable performance degradation is observed. Moreover, it can be seen that when neglecting the user activity detection, the performance of the MPAAPP method will suffer significant performance loss. Compared to the Genie-aided scheme with perfectly known user activity, there exists only a negligible performance gap for the proposed algorithm, showing the effectiveness of the proposed algorithm. Obviously, increasing the overloaded factor ρ or reducing the packing factor τ will lead to the improvement of spectral efficiency but affect the BER performance. To this end, we introduce a new performance metric, i.e., equivalent spectral efficiency, defined as η=

ρ (1 − BER) , τ

(7.53)

to evaluate the communication performance. In Fig. 7.7, we plot the equivalent spectral efficiency for different packing factors τ . We can observe that since more data symbols can be transmitted in the same interval, the equivalent spectral efficiency

7.4 Simulation Results

131

10 0

10 -1

BER

10 -2

10 -3

10 -4

4.5

MPA-APP Genie-aided LS-AMP-MPA EM-MPA

5

5.5

6

Eb /N0 (dB)

Fig. 7.6 BER performance for different algorithms with unknown user activity

Equivalent Spectral Efficienci

1.4

1.3

1.2

1.1

1 FTN(tau=0.6) FTN(tau=0.7) FTN(tau=0.8) Nyquist(tau=1)

0.9

0.8 4.5

5

5.5

6

Eb /N0 (dB)

Fig. 7.7 BER performance for different algorithms with unknown user activity

is increased, indicating more information can be carried given fixed time-frequency resources. However, it is seen that the gain becomes marginal when decreasing τ . This is because the ISI induced by FTN signaling becomes more severe which degrades the BER performance.

132

7 Receiver Design for FTN-NOMA System with Random Access 1.8

Equivalent Spectral Efficienci

1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 4.5

=100% =125% =150% =200%

5

5.5

6

Eb /N0 (dB)

Fig. 7.8 BER performance for different algorithms with unknown user activity

In Fig. 7.8, we illustrates the equivalent spectral efficiency for different numbers of users, i.e., K = 120, 160, 180, and 240, with overloaded factors ρ = 100%, 125%, 150%, and 200%. Similar to increasing τ , the increase of ρ also leads to higher equivalent spectral efficiency, showing the capability of supporting more users based on the proposed FTN-NOMA system. From both Figs. 7.7 and 7.8, we can see that the system throughput can be improved at the cost of higher complexity and degraded BER performance. Therefore, in practical system design, these key factors can be jointly optimized to fulfill specific applications. The NMSEs based on the proposed receiver design and based on the sparse Bayesian learning (SBL) method are illustrated in Fig. 7.9. The initial coarse channel estimation result based using 5 pilots is included for comparison. By utilizing the data symbols for aiding channel estimation, the proposed algorithm has significantly improved the estimation performance compared to the initial estimation result. The SBL scheme adopting the whole transmission block as pilots has the best performance. Nevertheless, our proposed algorithm is shown to have a 0.3 dB performance loss, which verifies the capability of achieving accurate channel estimation. Finally, we investigate user activity detection in dynamic environments. In Fig. 7.10, the BER performance versus the average number of active users  is plotted, where the E b /N0 is fixed at 6 dB. The performance of the EM-MPA receiver relying on known user activity is depicted as a reference. We observe that the performance of the proposed algorithm is approaching the ideal case with the increase of , indicating that the user activity detection becomes more accurate with a higher value . However, at the same time, more active users in the FTN-NOMA system will result in more severe IUI and affect the BER performance.

7.4 Simulation Results

133

10 0

NMSE

10 -1

10 -2

10 -3

EM-MPA Initial estimation Full-pilot SBL

4.5

5

5.5

6

Eb /N0 (dB)

Fig. 7.9 NMSE performance for channel estimation 10 -2 EM-MPA EM-MPA-Ideal

BER

10 -3

10 -4

20

30

40

50

Fig. 7.10 BER performance with different 

60

70

80

90

100

134

7 Receiver Design for FTN-NOMA System with Random Access

7.5 Conclusions In this chapter, we extend the non-orthogonal communication system in Chapter 6 to dynamic environments, where both the user activity and channel state are varying over time. We start the low-complexity receiver design problem by analyzing the probabilistic model. Relying on an ARMA model for approximating the colored noise samples, we fully factorize the joint a posteriori distribution and represent it by a factor graph. Then an iterative EM-MPA algorithm is proposed, which jointly estimates the channel, detects the user activity, and decodes information bits. Since all beliefs and messages of unknown variables are determined in parametric forms using the proposed algorithm, the complexity increase linearly with the number of users and ISI taps. Through simulation results, we verify the effectiveness of the proposed algorithm in the considered FTN-NOMA system with random access.

Chapter 8

Current Achievements and The Road Ahead

8.1 Summary of This Book In this book, we focus on the receiver design for high spectral efficiency communication system in the beyond 5G era. The concepts of FTN signaling and NOMA system are introduced. As both FTN signaling and NOMA technology will introduce nonorthogonalities, the received samples are impacted by inter-user interference and inter-symbol interference, which imposes challenges on low-complexity receiver design. The first Chapter has overviewed the evolution of wireless communication systems from 1G to 5G and briefly introduced FTN signaling and the SCMA system. The following two chapters present low-complexity receiver designs for both uplink and downlink SCMA systems. Then, in Chaps. 4 and 5, the receiver design issue for FTN signaling over frequency selective channels and doubly selective channels are investigated, respectively. In Chap. 6, we introduce a new non-orthogonal communication system by combining both FTN signaling and the NOMA system, which is expected to further improve the spectral efficiency. Finally, Chap. 7 discusses the application of the FTN-NOMA system in a dynamic environment with grant-free transmission. A joint channel estimation, user activity tracking, and decoding algorithm is proposed with low complexity. In summary, the innovative research results are listed as below: i. We design an energy minimization-based receiver for uplink SCMA system and the convergence behavior of the proposed receiver is analyzed. Relying on the concept of clique potentials, the joint a posteriori distribution is factorized as the product of several clique potentials. Then, an SCMA symbol detection algorithm is proposed based on the VFE framework. Next, the convergence of the proposed algorithm is analyzed, which shows that the variance is guaranteed to convergence while the mean converges under certain conditions. ii. A convergence guaranteed message passing receiver is designed for downlink MIMO-SCMA system. By allowing information exchange between neighboring

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Yuan et al., Receiver Design for High Spectral Efficiency Communication Systems in Beyond 5G, https://doi.org/10.1007/978-981-19-8090-9_8

135

136

iii.

iv.

v.

vi.

8 Current Achievements and The Road Ahead

downlink users, two cooperative detection schemes are developed. To reduce the receiver complexity, a stretched version of the factor graph is constructed by introducing auxiliary variables. Considering the convergence issue of MPA, a convergence guaranteed version of MPA is derived by convexifying the Bethe free energy. To realize the cooperative detection with noisy inter-user links, the belief consensus-based and ADMM-based methods are proposed. A low-complexity iterative receiver based on MPA is developed for FTN transmission over unknown ISI channels, which jointly detects FTN symbols and estimates channel coefficients. By intentionally separating the ISI caused by FTN signaling and fading channels, we fully exploit the known FTN-induced ISI, which helps to reduce the receiver complexity. A P-order AR process is used to model the colored noise without the need of whitening filtering. EP algorithm is utilized to approximate the non-Gaussian a priori distributions of data symbols to Gaussian forms. Two FDE algorithms, i.e., GMP-based and VFE-based receivers are designed to solve the FTN symbol detection problem over doubly selective channels. An MF approximation-based receiver is first proposed, which has the problem of variance underestimation. To this end, a Bethe approximation-based receiver is developed, which can exploit the conditional dependencies of data symbols. Then, by formulating the state space model based on the received samples, a Forney-style factor graph is constructed and a GMP-based receiver is proposed. Considering uncertain CSI, a modified GMP-based receiver is proposed, which is shown to be robust to channel uncertainty. A novel non-orthogonal communication system is introduced, which relies on the combination of non-orthogonal FTN waveform and NOMA techniques. A low-complexity joint channel estimation and multiuser detection method is developed for the considered system. The receiver is later extended to a grantfree transmission scenario that requires to detect user activity. This is done by a modified factor graph representation based on the probability associated with user activity. Furthermore, a unified message passing receiver is proposed that solves the problem of joint active user detection, channel estimation, and SCMA decoding. An application case of FTN-NOMA system in dynamically fluctuating environments is considered, where both the user activity and channel state are timevarying. Relying on an ARMA model for approximating the colored noise samples, we fully factorize the joint a posteriori distribution and represent it by a factor graph. Then an iterative EM-MPA algorithm is proposed, which jointly estimates the channel, detects the user activity, and decodes information bits.

8.2 The Road Ahead

137

8.2 The Road Ahead The receiver design problem is a classic topic in wireless communications but is always a new topic due to the rapid development of new communication systems. In the following, we discuss several promising research directions related to receiver design for high spectral efficiency communication systems.

8.2.1 Receiver Design for FTN Signaling in Nonlinear Channels In digital communication systems, high power amplifiers (HPA) are usually adopted to overcome propagation path loss, especially in the scenarios of satellite communications [143]. Since the HPAs are with limited linear regions, it may happen frequently that the transmitted signal is clipped and distorted by nonlinearity effect, resulting in BER performance loss. To tackle the nonlinear distortion, the digital pre-distortion technique can be used at the transmitter side, which is however not practical for beyond 5G mobile systems. To this end, several works have studied the receiver design under nonlinear channels. The AM/AM-AM/PM can well model a memoryless or quasi-memoryless channel. Nonetheless, the channel memory effect is unavoidable. Moreover, as the main scope of this book, we are also expecting high spectral efficiency in nonlinear channels. For example, with the application of FTN signaling, a higher data transmission rate is achieved at the cost of introducing ISI. To model a nonlinear channel with a strong memory, we resort to the Volterra model, which has a more general expression and can be applied to various nonlinear cases. Since the even-order components fall out of the frequency band, the Volterra model of memory length L for a nonlinear system with input sequence {xn } is given by rn = r1,n + r3,n + r5,n + · · · ,

(8.1)

where the ith order component is expressed as ri,n =

L−1 L−1   l1 =0 l2 =l1

L−1 

···

L−1 

l i+1 =l i−1 l i+3 =0 2

2

2

···

L−1  l1 =li−1

∗ h li1 ...li s˜n−l1 s˜n−l2 · · · s˜n−l i+1 s˜n−l i+3 · · · s˜n−l , i 2

2

(8.2) with h li1 ...li being the ith order Volterra coefficient. Generally, it is sufficiently in practice to consider the third-order Volterra series for modeling a nonlinear channel. By adopting the shaping filter and matched filter associated with FTN signaling, we finally arrive at

138

8 Current Achievements and The Road Ahead L−1  k=0

rn =

L−1  k=0

h k sn−k +

L−1  L−1  L−1 

∗ h 3klm sn−k sn−l sn−m + ξn ,

(8.3)

k=0 l=0 m=0

where ξn is the colored noise sample and sn is the equivalent sampled symbol output from the shaping filter, which is given by sn =

Lf 

qn− j x j ,

(8.4)

j=−L f

where x j and qn− j are the original data symbol and the FTN-induced ISI tap. Based on the received signal model (8.2), it is a challenge to efficiently detect the data symbols in the presence of strong nonlinearity. Particle filtering method is an efficient tool to deal with nonlinear systems and several works have been proposed based on particle filtering to solve the symbol detection problem in nonlinear channels [144, 145]. Nevertheless, the performance highly depends on the number of particles. In order to achieve good detection performance, a large number of particles are required, which imposes prohibitively high complexity. For reducing the complexity, one solution is to perform channel shortening which shortens the channel memory length from the perspective of minimal information loss, which is still difficult for nonlinear channels [146]. Alternatively, we may find a linear channel for approximating the original nonlinear one. Then the low-complexity receiver designs discussed in this book can be easily employed. The key issue that needs to be addressed is how to find such linear channels, which demands further investigation.

8.2.2 Receiver Design for Orthogonal Time Frequency Space (OTFS) Modulation With the commercialization of 5G wireless systems globally, research on the next generation communication system has been taken on the way. The sixth-generation (6G) wireless communications are expected to provide connectivity for not only high data rate as well as reliability, but also seamless coverage at any place on earth, which is not satisfactorily supported by currently deployed terrestrial wireless systems. To fulfill this requirement, several emerging applications such as low-earth-orbit satellites (LEOS), mobile communications on board Aircraft (MCA), and unmanned aerial vehicles (UAV) communications scenarios will be involved. A vital issue in the above use cases is reliable information transmission in high-mobility environments. For example, in MCA and LEOS scenarios, the relative speed of communication devices will be even higher than 1000 km/h. The Doppler shift induced by highspeed movements of transceivers will make the conventional orthogonal frequency division multiplexing (OFDM) technology fail to work. In particular, due to the fast time-varying of channel fluctuations, it is not possible to acquire channel information

8.2 The Road Ahead

139

in a low overhead way. Moreover, the detection complexity will be high because of severe ICI. To this end, quite a few researches are focusing on how to achieve ultrareliable communications in high-mobility environments. The orthogonal time-frequency space (OTFS) technique, which was proposed in 2017, has been recognized as a key enabler for supporting future high-mobility communications [147, 148]. Instead of the commonly used time-frequency (TF) domain, OTFS technology modulates data symbols in the delay Doppler (DD) domain, which provides strong resilience against high Doppler shifts in highly dynamic complex environments. Mover, each data symbol in DD domain will experience the whole TF channel, which gives the chance of achieving full TF diversity [149]. As a new modulation waveform, receiver design for OTFS has drawn numerous attention in the last few years. In particular, the DD domain input-output relationship is given by y [k, l] =

N −1 M−1       x k, l  hw k − k, l − l  ,

(8.5)

k  =0 l  =0

where k and l denote the Doppler index and delay index of DD grid, respectively, h w [k, l] is the DD domain effective channel. Or in an equivalent vector form, the input-output relationship is expressed as       yDD = F N ⊗ FHM V (I N ⊗ F M ) Ht I N ⊗ FHM U FHN ⊗ F M xDD + F N w, (8.6) t where F M ∈ C M×M and F N ∈ C N ×N denote the DFT matrices. The matrix H denotes the time domain channel, formulated as t = H

P 

h i lτi kνi +κνi ∈ C N (M+MCP )×N (M+MCP ) ,

(8.7)

i=1

where ⎡

⎤ 0 ··· ··· 0 1 ⎢ .. ⎥ ⎢1 . 0 ··· 0⎥ ⎢ ⎥ ⎥ =⎢ ⎢0 1 0 ··· 0⎥ ⎢. . . . ⎥ ⎣ .. . . . . . . 0 ⎦ 0 ··· 0 1 0 ⎡ j2π 0 e NM 1 ⎢ e j2π N M ⎢ and  = ⎢ .. ⎣

(8.8)

⎤ ⎥ ⎥ ⎥. ⎦

. e j2π

N M−1 NM

(8.9)

140

8 Current Achievements and The Road Ahead

For receiver design in OTFS system, the MPA proposed in [150] is commonly adopted for OTFS detection followed by DD channel estimation [151, 152]. Some variations including a hybrid (MAP) and parallel interference cancellation detection in [153], a variational Bayes detection [154], a unitary approximate message passing (UAMP)-based OTFS detector [155], and a cross time-DD domain detector [156]. As for the high spectral efficiency communication system, although some pioneering papers have considered the adoption of NOMA in OTFS system [157, 158], there are still some fundamental issues that are not clearly studied. For example, how to multiplex multiple users in the DD domain. Due to the two-dimensional convolution relationship between the effective channel and transmitted symbols, the special interference pattern should be taken into account for receiver design in high spectral efficiency OTFS system.

8.2.3 Receiver Design for Integrated Sensing and Communication (ISAC) System It has been widely agreed that the sensing capability is expected in 6G communication networks, in addition to the classic functionality of communications. In the past decades, researches on communications and sensing are on two separate ways. With the fast development of wireless technology, the mmWave frequency band has been used for communications, which was occupied by the radar sensing system. Therefore, the trend of integrating both functionaries using the same spectrum resources and hardware architectures has become very popular since 2020, which is the ISAC technology. By reusing the same resources, the system throughput and efficiency are improved relying on ISAC [159]. By exploiting the sensing capability, it is capable of determining the locations and speeds of the communication users. Then, the number of pilots can be reduced to improve the spectral efficiency. For example, the works of [160–163] discussed a novel sensing-aided predictive beam alignment scheme for mmWave communication systems. As shown in Fig. 8.1, the frame structures for communications-only beam alignment and ISAC-based beam alignment are compared. In particular, the communications-only scheme utilizes a few pilots for estimating the angular parameters. Then the estimated angles are fed back through the uplink channel for transmit beamforming. On the contrary, the ISAC-based algorithm does not rely on dedicated pilots for beam direction estimation. In fact, the whole downlink block is known at the transmitter side. Then, based on the sensing echoes, the angular parameters can be extracted at the transmitter side. On the one hand, the spectral efficiency is undoubtedly improved. On the other hand, as no uplink feedback is required, the communication latency is reduced. Moreover, since the whole block is used as pilots, which provides a higher SNR gain and results in better estimation performance. Although ISAC enables an inherently high spectral efficiency communication system, receiver design is also a critical issue. The communications receiver aims for

8.2 The Road Ahead

Communications-only

ISAC

141

Parameter Estimation and Feedback

Communication Data Transmission

ISAC Signal Transmission

Parameter Estimation and Feedback

Communication Data Transmission

ISAC Signal Transmission

Fig. 8.1 Frame structures for Communications-only and ISAC-based beam alignment

extracting the transmitted information from the ISAC signal while the radar sensing receiver will infer the sensing parameters, i.e., delay, Doppler, and angle from the sensing echoes (mono-static) or reflected signals (bi-static). Efficient Interference mitigation technique is required to cancel the interference between two sub-systems. In addition, exploiting the mutual information between communications and sensing will help to further improve the spectral efficiency as well as overall performance.

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