Sustainable Wireless Communications 9789811904479, 9789811904486, 9811904472

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Jianping An · Kai Yang · Xiaozheng Gao · Neng Ye

Sustainable Wireless Communications

Sustainable Wireless Communications

Jianping An · Kai Yang · Xiaozheng Gao · Neng Ye

Sustainable Wireless Communications

Jianping An Beijing Institute of Technology Beijing, China

Kai Yang Beijing Institute of Technology Beijing, China

Xiaozheng Gao Beijing Institute of Technology Beijing, China

Neng Ye Beijing Institute of Technology Beijing, China

ISBN 978-981-19-0447-9 ISBN 978-981-19-0448-6 (eBook) https://doi.org/10.1007/978-981-19-0448-6 Jointly published with Beijing Institute of Technology Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Beijing Institute of Technology Press. © Beijing Institute of Technology Press 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 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 publishers, 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 publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain 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

Preface

With more and more intelligent devices connected to the wireless communication networks, the performance of wireless communication networks needs to be largerly improved, and various promising technologies needs to be investigated to achieve sustainable wireless communications. To this end, this book discusses the architecture of future wireless communications networks, reliable communications between different nodes, and energy-efficient resource allocations for achieving sustainable wireless communications. In particular, for the architecture of future wireless communication networks, we discuss the sustainable ultra-dense heterogeneous networks and the sustainablity issues of non-orthogonal multiple access. For the reliable communications between different nodes, we investigate the space-time network coding with maximal-ratio combining and antenna selection, and develop a compressive sensing-based dynamic estimation algorithm for a unified laser telemetry, tracking, and command system. For the energy-efficient resource allocation, we investigate the resource allocation for heterogeneous orthogonal frequency division multiple access (OFDMA) networks and the power control scheme for device-to-device communications. Also, based on channel distribution information, we further investigate the user scheduling and power control for multi-cell OFDMA networks, and base station association and beamforming for multi-cell multiuser systems. We believe that the results in this book can provide useful insights into the design of future wireless communication networks and achieving sustainable wireless communications. This book is intended for graduate students, researchers, and engineers in the field of wireless communications. We hope that the contents in this book can provide useful guidance to the readers. Beijing, China

Jianping An Kai Yang Xiaozheng Gao Neng Ye

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Future Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Reliable Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Energy-Efficient Resource Allocation . . . . . . . . . . . . . . . . . . . . . . . 1.5 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 3 4 6

2

Sustainable Ultra-Dense Heterogeneous Networks . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Multi-Tier Heterogeneous Networks . . . . . . . . . . . . . . . . . 2.2.2 Downlink and Uplink Decoupling . . . . . . . . . . . . . . . . . . . 2.2.3 Cloud Radio Access Network . . . . . . . . . . . . . . . . . . . . . . . 2.3 Random Access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Challenges of UDHN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Interference Management . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Mobility Management and Mobile Association . . . . . . . . 2.4.3 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 8 11 11 12 16 16 17 17 18 18

3

Non-Orthogonal Multiple Access: Achieving Sustainable Future Radio Access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Achievable Power Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Energy Efficiency Improvement in NOMA . . . . . . . . . . . . . . . . . . . 3.3.1 Energy-Aware Resource Allocation . . . . . . . . . . . . . . . . . . 3.3.2 Grant-Free Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 A Hybrid NOMA Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Future Directions about NOMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Multi-Domain MA Signatures . . . . . . . . . . . . . . . . . . . . . .

21 21 22 24 24 25 27 29 30 vii

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Contents

3.5.2 3.5.3

SE/EE Performance Boost with Full-Duplex . . . . . . . . . . Ultra Long Battery Life Cycle with Backscatter Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30 30 31 31

4

Space-Time Network Coding with TAS/MRC . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Performance with Perfect Feedback . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Exact Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Asymptotic Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Performance with Delayed Feedback . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Exact Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Asymptotic Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 34 38 38 41 43 43 46 48 51 55

5

Space-Time Network Coding with Antenna Selection . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Performance with Outdated CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Symbol Error Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Performance with Spatial Correlation . . . . . . . . . . . . . . . . . . . . . . . 5.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 58 60 61 66 67 69 73 75

6

Compressive Sensing-Based Dynamic Estimation in Unified Laser TTC System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Dynamic Estimation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Pre-Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Dynamic Estimation by CS-sDPT-sFrFT . . . . . . . . . . . . . 6.3.3 Pilot Delete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Clock-Data Recovery and Demodulation . . . . . . . . . . . . . 6.4 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Estimation Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 78 80 81 82 86 87 87 87 88 89 93 94

Contents

7

ix

Energy-Efficient Resource Allocation in Heterogeneous OFDMA Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Optimal Resource Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Outer Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Inner Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Computational Complexity Analysis . . . . . . . . . . . . . . . . . 7.5 Sub-Optimal Resource Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 RB Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Transmit Power Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 95 96 98 101 101 101 104 104 104 106 107 112 116

8

Energy-Efficient Power Control for D2D Communications . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Total EE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Individual EE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Optimal Power Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Total EE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Individual EE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Sub-Optimal Power Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Total EE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Individual EE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 120 122 122 124 126 126 129 131 131 131 133 134 134 138 141

9

Energy-Efficient User Scheduling and Power Control for Multi-Cell OFDMA Networks Based on CDI . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . 9.3 Joint User Scheduling and Power Control . . . . . . . . . . . . . . . . . . . . 9.4 Decentralized Power Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Convergence and Complexity of Algorithms . . . . . . . . . . 9.5.2 Performance Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 146 149 157 164 164 166 168 171

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Contents

10 Energy-Efficient Base Station Association and Beamforming for Multi-Cell Multiuser Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . 10.3 Joint BS Association and Beamforming . . . . . . . . . . . . . . . . . . . . . 10.4 Decentralized Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Passive Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Millimeter-Wave and Terahertz Communications . . . . . . 11.2.3 Space-Air-Ground Integrated Information Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Joint Radar and Communication . . . . . . . . . . . . . . . . . . . . 11.2.5 Artificial Intelligence-Enabled Wireless Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 175 176 178 187 193 196 199 201 201 204 204 205 205 206 206

Acronyms

1G 2G 3G 3GPP 4G 5G 6G ACK ADC ADMM AI AS AWGN BB BBU BC BLER BP BPSK BS CAPEX CDF CDI CD-NOMA CDR CoMP CP-OFDM CPRI C-RAN CS CSI

The first generation The second generation The third generation Third-Generation Partnership Project The fourth generation The fifth generation The sixth generation Acknowledge Analog-to-digital converter Alternating direction method of multipliers Artificial intelligence Antenna selection Additive white Gaussian noise Branch-and-bound Baseband unit Broadcast channel Block error ratio Basis pursuit Binary phase-shift keying Base station Capital expenditure Cumulative distribution function Channel distribution information Code-domain non-orthogonal multiple access Clock-data recovery Coordinated multipoint Cyclic prefix orthogonal frequency division multiplexing Common public radio interface Cloud radio access network Compressive sensing Channel state information xi

xii

D2D D.C. DF DFT DPT DRL DUDe EE eNB ESE-PIC FD FDD FFR FFT FP FRA FrFT GFP GONORA HPPP HTC IA ICI ICIC IDFT IDMA ID-NOMA IE IGMA IMDD i.i.d. i.n.i.d. IoT IRS IS JRC KKT LHS LP LTE MA MAC MBB MeNB MGF

Acronyms

Device-to-device Difference of two concave functions Decode-and-forward Discrete Fourier transform Discrete polynomial-phase transform Deep reinforcement learning Downlink and uplink decoupling Energy efficiency Evolved Node B Elementary signal estimator parallel interference cancellation Full-duplex Frequency division duplexing Fractional frequency reuse Fast Fourier transform Fractional programming Future radio access Fractional Fourier transform Generalized fractional programming Generalized orthogonal/non-orthogonal random access Homogeneous Poisson point process Human-type communication Interference alignment Inter-cell interference Inter-cell interference coordination Inverse discrete Fourier transform Interleave division multiple access Interleave-domain non-orthogonal multiple access Interference exploitation Interleave grid multiple access Intensity modulation direct detection Independent and identically distributed Independent but not necessarily identically distributed Internet of Things Intelligent reflecting surface Interference shaping Joint radar and communication Karush-Kuhn-Tucker Left-hand side Linear programming Long-term evolution Multiple access Multiple access channel Mobile broadband Macro evolved Node B Moment-generating function

Acronyms

MIMO MINLFP MINLP MISO mmWave MPA MRC MTC MUSA NCO NLP NOMA OBSAI OFDM OFDMA OMA OMP OP OPEX PDF PDMA PD-NOMA PHR PHR-AL PRB PRP PUCCH QoS RB RF RHS RMSE RRC RRH RU SCA SC-FDMA SCMA sDPT SE SeNB SER sFFT sFrFT SIC

xiii

Multiple-input multiple-output Mixed-integer nonlinear fractional programming Mixed-integer nonlinear programming Multi-input single-ouput Millimeter wave Message passing algorithm Maximal-ratio combining Machine-type communication Multi-user shared access Numerical controlled oscillator Nonlinear programming Non-orthogonal multiple access Open base station architecture initiative Orthogonal frequency division multiplexing Orthogonal frequency division multiple access Orthogonal multiple access Orthogonal matching pursuit Outage probability Operating expenditure Probability density function Pattern division multiple access Power-domain non-orthogonal multiple access Powell-Hestenes-Rockafellar Powell-Hestenes-Rockafellar augmented Lagrangian Physical resource block Physical resource pool Physical uplink control channel Quality of service Resource block Radio frequency Right-hand side Root-mean-square error Radio resource control Remote radio head Resource unit Successive convex approximation Single-carrier frequency division multiple access Sparse code multiple access Sparse discrete polynomial-phase transform Spectral efficiency Small cell evolved Node B Symbol error rate Sparse fast Fourier transform Sparse fractional Fourier transform Successive interference cancellation

xiv

SINR SNR SOC SOCP SRS STNC SWIPT TAS TAS/MRC TDD TD-LTE TDMA THz TPC TTC UDHN UE

Acronyms

Signal-to-interference-plus-noise ratio Signal-to-noise ratio Second-order cone Second-order cone programming Sounding reference signal Space-time network coding Simultaneous wireless information and power transfer Transmit antenna selection Transmit antenna selection and maximal-ratio combining Time division duplexing Time division long-term evolution Time division multiple access Terahertz Transmit power control Telemetry, tracking, and command Ultra-dense heterogeneous networks User equipment

Chapter 1

Introduction

This chapter first introduces the necessity of achieving sustainable communications. Then, three aspects to achieve sustainable communications, including future network architecture, reliable communications, and energy-efficient resource allocation are discussed. Finally, the organization of this book is presented.

1.1 Background Fueled by the proliferation of mobile devices and applications, mobile data traffic has increased exponentially in the past decade, and this trend will certainly continue and accelerate even further in the future [1]. It has been reported that the mobile device density (the device number per unit area) and traffic density (the traffic volume per unit area) will rise to several million per square kilometer and tens of terabits per second per square kilometer in local hotspot areas, respectively [2]. Future wireless communication networks are expected to provide an unprecedented quality of service (QoS) to our networked society, and these explosions of mobile devices and traffic would impose new challenges to future cellular networks. To meet the increasing demands of wireless communication networks and achieve sustainable wireless communications, various promising technologies should be further investigated. Specifically, firstly, considering the various transmission demands of devices, traditional homogeneous network cannot effectively provide the wireless services. Therefore, developing ultra-dense heterogeneous networks (UDHN) architecture is essential to improve the spectral efficiency (SE) and energy efficiency (EE) [3, 4]. Also, for the future radio access (FRA) technology, orthogonal frequency division multiple access (OFDMA) and single-carrier frequency division multiple access (SC-FDMA) cannot achieve the optimal capacities of the broadcast channel (BC) and multiple access channel (MAC), respectively [5], and non-orthogonal © Beijing Institute of Technology Press 2023 J. An et al., Sustainable Wireless Communications, https://doi.org/10.1007/978-981-19-0448-6_1

1

2

1 Introduction

multiple access (NOMA) has been introduced to fulfill the requirements of FRA [6]. Secondly, the reliability of wireless communications needs to be further considered. An example is that the synchronization between different nodes is imperfect [7, 8]. Therefore, developing the wireless communication technologies with imperfect synchronization is important to the reliability of wireless communications. Furthermore, with the explosive increase of wireless data exchange, the energy consumption of wireless communications needs to be paid attention to [9]. Otherwise, it would bring many economic and ecological issues. We would like to highlight that the energy consumption can be decreased by resource allocation, and developing energy-efficient resource allocation schemes for wireless networks is of special necessity. Motivated by the above observations, in this book, we will investigate the architecture of future wireless networks, the reliable communications between different nodes, and the energy-efficient resource allocation schemes for achieving sustainable wireless communications, which will provide useful guidance for the design of wireless networks.

1.2 Future Network Architecture To meet the rapid increase of data traffic, the UDHN, with a very high network densification, has been acknowledged as a key mechanism in the future wireless communication networks [1, 2, 10, 11]. Note that the network densification includes both the mobile device densification and base station (BS) densification, and the density of BSs may exceed that of mobile devices. According to the trend in cellular networks evolution, future networks will be a heterogeneous one consisting of macrocells along with a large number of small cells, device-to-device (D2D) pairs, and machine-type communication (MTC) based tiers [12]. Each tier has a different size with a different transmit power level. Typically, macrocell is covered by a high power node with a large radius to provide wide area coverage for the remote and rural areas, whereas the main purpose of small cells, which encompass microcells, picocells, and femtocells, is to enhance network capacity in hot spots with very dense services through offloading traffic load from the macrocells and increasing the frequency reuse factor. Compared with microcell and picocell, which are deployed by operator, femtocell is typically in the mode of plug and play, and any consumer can install a femtocell at home. Due to the private property, consumer needs to declare the femtocell access control protocol, namely, open access control, where any nearby users can access the femtocell, or closed access control, where only the pre-registered users are allowed to access the femtocell. For the future radio access, NOMA has been introduced to satisfy the requirements of FRA [6]. The core concept behind NOMA is to superimpose multi-user signals on the same radio resource through the use of dedicated multiple access (MA) signatures to manage inter-user interference. The design of MA signatures is usually performed in the power, code, or interleave domains [13]. Moreover, the advanced multi-user detector is employed at the receiver to separate different signals by exploiting the

1.2 Future Network Architecture

3

structures of MA signatures. With elaborate designs, NOMA outperforms OFDMA and SC-FDMA in both SE and connectivity. Theoretically, NOMA can achieve the entire capacity regions of Gaussian BC and MAC [13].

1.3 Reliable Communications With the communication devices deployed in more and more scenarios, the reliability of data transmission should be paid attention to. In some scenarios, it is of special necessity to guarantee the reliablity of data transmission. We would like to highlight that it is very difficult to align all the signals from multiple sources at multiple destinations [7, 8], and high-accuracy synchronization requires complicated control mechanisms and extra control messages, which results in high system complexities and overheads [8]. When synchronization is imperfect, the performance of wireless communications can be severely degraded. Space-time network coding (STNC) can be employed as a possible solution to overcome the problem caused by imperfect synchronization. In particular, STNC combines information from different sources at each relay node and transmits the combined signal in dedicated time slots, which jointly exploit the benefits of both network coding and space-time coding [14]. Also, for the high-speed laser telemetry, tracking, and command (TTC) systems, it is difficult to achieve precise temporal synchronization because of the high dynamic resulting from the movement of space platforms [15, 16]. Thus, it is necessary to preferentially estimate dynamic parameters including velocity and acceleration, and then employ the estimated results to facilitate temporal synchronization, demodulation, and ranging in laser TTC systems.

1.4 Energy-Efficient Resource Allocation During the past decades, tremendous efforts have been made to enhance thebreak system capacity and SE to deal with the increasing demands for better QoS and explosive growth of high-data-rate applications. Actually, the data transmission in wireless communications has consumed a large amount of energy, and reducing the energy consumption in wireless communications is essential [9]. The increase of the energy consumption of wireless communications does not only cause much more energy costs, but also increase carbon dioxide emission, which aggravates the global warming and sea level rise. Therefore, the energy consumption of wireless communications becomes a non-negligible problem from both the economic and environmental perspectives [17]. In particular, if the average energy consumption of transmitting per unit amount of data is not reduced significantly, considering the exponential increase of the data traffic, the total energy consumption of the data traffic would increase dramatically. Therefore, reducing the energy consumption of data exchange is essential to achieve sustainable wireless communications.

4

1 Introduction

EE, which is usually defined as the ratio of the achieved system throughput to the total power consumption, is an important metric to quantify energy conversion efficiency in wireless communication networks [18]. Unfortunately, EE and SE do not always coincide. Improving SE directly leads to an increase in energy consumptions in many situations [19–21]. Considering the steadily rising energy costs and environmental concerns, EE is attracting more and more attention [9, 19]. Resource allocation, including but not limited to power control, subcarrier assignment, and base station association, is an effective method to improve the EE, and energy-efficient resource allocation is important to achieve sustainable wireless communications.

1.5 Organization To achieve sustainable wireless communications, this book discusses the architecture of future wireless networks in Chaps. 2 and 3, reliable communications between different nodes in Chaps. 4–6, and energy-efficient resource allocation schemes in Chaps. 7–10. The rest of this book is specified as follows. Chapter 2 discusses the sustainable UDHN. To meet the exponentially increased demands of wireless communication services, network densification is acknowledged as an important way in future cellular networks, and UDHN is a promising technique in the future. We present a potential network architecture for UDHN and develop a generalized orthogonal/non-orthogonal random access scheme to improve the network efficiency while reducing the signaling overhead. Chapter 3 discusses the sustainability issues of NOMA. NOMA has been considered as a highly promising FRA technology to improve the SE and massive connectivity. With the growing concern on the EE, we investigate the theoretical power regions of NOMA to show the minimum transmission power with fixed data rate requirement. Also, we investigate a hybrid NOMA strategy which reaps the joint benefits of resource allocation and grant-free transmission to simultaneously achieve high throughput, large connectivity, and low energy cost. Chapter 4 investigates the STNC with transmit antenna selection and maximalratio combining (TAS/MRC). It is difficult or impossible to implement perfect synchronization in practical multi-node wireless networks due to the fact that it is very challenging to align all the signals from multiple sources at multiple destinations. Therefore, the STNC was employed to deal with the imperfect synchronization issues. We integrate cascaded TAS/MRC into STNC as a solution to preserve full transmit and receive diversity with low computational complexity and reduced feedback overhead, and derive new closed-form exact and asymptotic expressions for the outage probability (OP) and symbol error rate (SER) with perfect feedback and delayed feedback, respectively. Chapter 5 investigates the STNC with antenna selection (AS). It is noticed that TAS/MRC requires multiple radio frequency (RF) chains, which inevitably increases the complexities and costs of the transceivers. Therefore,we integrate AS into STNC to circumvent this drawback since AS preserves the diversity gain while significantly

1.5 Organization

5

reducing hardware complexity. Also, a new closed-form SER expressions for STNC with AS to quantify the effect of outdated channel state information (CSI) on the system performance is derived, and a new compact expression for the asymptotic SER over flat spatially correlated Rayleigh fading channels is derived to examine the performance degradation of STNC with AS due to spatial correlation. Chapter 6 investigates a compressive sensing (CS) based dynamic estimation in a unified laser TTC system. Due to the high dynamic resulting from the movement of space platforms, the deployment of unified laser TTC system is hindered by the imprecise temporal synchronization. Therefore, we develop a CS based dynamic estimation algorithm to address the dynamic estimation problem in spaceborne intensity modulation direct detection (IMDD) based laser TTC systems. Analytical and simulation results show that the developed algorithm exhibits a better performance than conventional CS algorithms in terms of computational complexity, suggesting its attractiveness for spaceborne IMDD-based laser TTC systems. Chapter 7 investigates energy-efficient resource allocation in heterogeneous OFDMA networks. Deploying small cells jointly with macro cells to form heterogeneous networks can achieve improved EE and throughput owing to the fact that eNBs are brought closer to users. Therefore, we investigate energy-efficient resource allocation in heterogeneous OFDMA networks, where both optimal and suboptimal resource allocation schemes are developed. Chapter 8 investigates energy-efficient power control for D2D communications. D2D communications have been expected as a key component for future cellular networks. It is noticed that improving the resource efficiency or frequency reuse factor by allowing multiple D2D pairs to share the same resource is possible, especially when the short distance characteristics of D2D communications are considered. As such, we investigate the energy-efficient optimal and low-complexity suboptimal power control schemes for multiple D2D pairs underlaying cellular networks, where total EE and individual EE are maximized, respectively. Chapter 9 investigates energy-efficient user scheduling and power control for multi-cell OFDMA networks based on channel distribution information (CDI). Designing user scheduling and power control for multi-cell OFDMA networks can improve the network EE. It is noted that obtaining the perfect CSI is very challenging owing to the channel variation over time, the channel estimation error, and the feedback delay. Therefore, we employ the CDI, which remains unchanged for a relatively long period of time, to design energy-efficient user scheduling and power control scheme for multi-cell OFDMA networks. Chapter 10 investigates energy-efficient BS association and beamforming for multi-cell multiuser systems. Frequency reuse would severely deteriorate the performance of the cell-edge users due to the increased interference, and coordinated beamforming is a practical approach for interference management. Also, the conventional greedy scheme, where each user is associated with the BS providing the highest signal, is no longer effective. In addition, CDI is much easier to be obtained than CSI. Therefore, we develop energy-efficient BS association and beamforming scheme for multi-cell multiuser systems based CDI.

6

1 Introduction

Chapter 11 summarizes this book and discusses the future directions to achieve sustainable wireless communications.

References 1. J. Liu, W. Xiao, Advanced carrier aggregation techniques for multi-carrier ultra-dense networks. IEEE Commun. Mag. 54(7), 61–67 (2016). 2. IMT-2020, 5G network technology architecture. White Paper, May 2015. 3. W. Wang, G. Shen, Energy efficiency of heterogeneous cellular network, in IEEE VTC-Fall IEEE(2010), pp. 1–5. 4. Y. Li, H. Celebi, M. Daneshmand, C. Wang, W. Zhao, Energy-efficient femtocell networks: challenges and opportunities. IEEE Wireless Commun. 20(6), 99–105 (2013). 5. Y. Yuan, Z. Yuan, G. Yu, C. Hwang, P. Liao, A. Li, K. Takeda, Non-orthogonal transmission technology in LTE evolution. IEEE Commun. Mag. 54(7), 68–74 (2016). 6. Z. Ding, Y. Liu, J. Choi, Q. Sun, M. Elkashlan, C.-L. I, H.V. Poor, Application of non-orthogonal multiple access in LTE and 5G networks. IEEE Commun. Mag. 55(2), 185–191 (2017). 7. H. Zhang, N.B. Mehta, A. F. Molisch, J. Zhang, H. Dai, Asynchronous interference mitigation in cooperative base station systems. IEEE Trans. Wireless Commun. 7(1), 155–165 (2008). 8. X. Li, T. Jiang, S. Cui, J. An, Q. Zhang, Cooperative communications based on rateless network coding in distributed MIMO systems. IEEE Wireless Commun. 17(3), 60–67 (2010). 9. J. Wu, S. Rangan, H. Zhang, Green Communications: Theoretical Fundamentals, Algorithms, and Applications. CRC Press, 2012. 10. N. Bhushan, J. Li, D. Malladi, R. Gilmore, D. Brenner, A. Damnjanovic, R.T. Sukhavasi, C. Patel, S. Geirhofer, Network densification: the dominant theme for wireless evolution into 5G. IEEE Commun. Mag. 52(2), 82–89 (2014). 11. S. F. Yunas, M. Valkama, J. Niemela, Spectral and energy efficiency of ultra-dense networks under different deployment strategies. IEEE Commun. Mag. 53(1), 90–100 (2015). 12. M. Agiwal, A. Roy, N. Saxena, Next generation 5G wireless networks: a comprehensive survey. IEEE Commun. Surveys Tuts. 18(3), 1617–1655 (2016). 13. 3GPP TR 38.812, Study on non-orthogonal multiple access (NOMA) for NR. May 2017. 14. H.-Q. Lai, K.J.R. Liu, Space-time network coding. IEEE Trans. Signal Process. 59(4), 1706– 1718 (2011). 15. H. Someya, I. Oowada, H. Okumura, T. Kida, A. Uchida, Synchronization of bandwidthenhanced chaos in semiconductor lasers with optical feedback and injection. Opt. Express 17(22), 19536–19543 (2009). 16. D.O. Caplan, Laser communication transmitter and receiver design. J. Opt. Fiber Commun. Rep. 4(4-5), 225–362 (2007). 17. S. Buzzi, C.-L. I, T.E. Klein, H.V. Poor, C. Yang, A. Zappone, A survey of energy-efficient techniques for 5G networks and challenges ahead. IEEE J. Sel. Areas Commun. 34(4), 697–709 (2016). 18. D. Feng, C. Jiang, G. Lim, L.J. Cimini, G. Feng, G.Y. Li, A survey of energy-efficient wireless communications. IEEE Commun. Surveys Tuts. 15(1), 167–178 (2013). 19. Y. Chen, S. Zhang, S. Xu, G.Y. Li, Fundamental trade-offs on green wireless networks. IEEE Commun. Mag. 49(6), 30–37 (2011). 20. C. Xiong, G.Y. Li, S. Zhang, Y. Chen, S. Xu, Energy- and spectral-efficiency tradeoff in downlink OFDMA networks. IEEE Trans. Wireless Commun. 10(11), 3874–3886 (2011). 21. C. He, B. Sheng, P. Zhu, X. You, G.Y. Li, Energy- and spectral-efficiency tradeoff for distributed antenna systems with proportional fairness. IEEE J. Sel. Areas Commun. 31(5), 894–902 (2013).

Chapter 2

Sustainable Ultra-Dense Heterogeneous Networks

In this chapter, we discuss the sustainable ultra-dense heterogeneous networks (UDHN). Section 2.1 introduces the motivation of developing sustainable UDHN. Section 2.2 presents one potential network architecture for UDHN, and Sect. 2.3 investigates the random access problem in UDHN, where we focus on the machinetype communication (MTC) devices and develop a random access scheme for MTC devices to improve the network efficiency while reducing the signaling overhead. Section 2.4 analyzes the challenges of UDHN, and Sect. 2.5 concludes this chapter.

2.1 Introduction To tackle the 1000x mobile traffic challenge and to accommodate the massive devices, UDHN has been identified as a key mechanism to handle orders of magnitude increase in the data traffic volume [1–4]. The UDHN refers to the idea of densifying the cellular networks with a very high network densification, where the density of base stations (BSs) may exceed that of mobile devices. As a result, the UDHN would potentially allow for orders of magnitude improvement in both spectral efficiency (SE) and energy efficiency (EE) [4]. The more traffic generated, the more BSs will be needed to serve the devices. The evolution of the cellular networks can be viewed as the process of network densification to some extent. In the first generation (1G) networks, the cell radius is around 10 mi, and cell splitting may occur, where the radio coverage of one cell is partitioned into two or more new smaller cells to mitigate path-loss and to support more devices. In the second generation (2G) networks, the typical radius of macrocell varies from a couple of hundred meters to several kilometres. Besides cell splitting, small cells, which are low-powered radio access nodes that have a range of tens of meters to 1 or 2 km, are introduced in 2G networks to offload traffic from macrocells. In the third generation (3G) and the fourth generation (4G) networks, small cells are © Beijing Institute of Technology Press 2023 J. An et al., Sustainable Wireless Communications, https://doi.org/10.1007/978-981-19-0448-6_2

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2 Sustainable Ultra-Dense Heterogeneous Networks

further popular and viewed as a vital element for boosting network throughput and balancing traffic load. Deploying more and more small cells to serve a certain geographical area leads to smaller and smaller cell coverage areas, and thereby achieving cell splitting gains. With small cells being deployed, homogeneous cellular networks become heterogeneous because of difference in transmit power and coverage area of BSs. In the fifth generation (5G) networks, the density of small cells is further increased compared with that in 4G networks to guarantee seamless coverage and high data rate. As a result, the UDHN is emerging as one of the core characteristics of 5G cellular networks [5]. However, the performance of the UDHN does not increase monotonously with BS density, and it is limited by the inter-cell interference and the fronthaul and backhaul network capacity [5, 6]. The full benefits of network densification can be realized only if it is complemented by advanced transceivers capable of interference cancellation and backhaul densification [3].

2.2 Network Architecture The current long-term evolution (LTE) heterogeneous networks are mainly designed for mobile broadband scenario. In addition, the communications between different cells are limited by the existing Un and Uu interface, which may not satisfy the demand of growing traffic. To meet the explosive growth in traffic volumes and unprecedented increase in connected mobile devices, dramatic changes in the design of network architecture are required. As a result, we develop a potential network architecture for the UDHN, as shown in Fig. 2.1, where macrocells, microcells, picocells, and femtocells are connected via the cloud radio access network (C-RAN) structure to facilitate the collaboration among cells. The developed UDHN can be viewed as an evolution of the current LTE heterogeneous networks, and is compatible with the current LTE architecture.

2.2.1 Multi-Tier Heterogeneous Networks Due to the ultra-densities of BSs and users, a tractable model for the UDHN is desired to facilitate the network performance analysis compared to complex systemlevel simulations. Fortunately, we can model multi-tier UDHN based on stochastic geometry, where we model the locations of BSs of the same tier as a homogeneous Poisson point process (HPPP) of density λ. Different tier is with different λ to characterize its density property. It is easy to theoretically analyze the performance of multi-tier UDHN based on HPPP model by using Palm measure. However, the locations of the macro BSs are usually dominated by the network planning, and are not random in general. To model the location of macro BS more appropriate, we can resort to Matern hard core Poisson point process, whose repulsive nature makes it a suitable candidate to model macro BSs in cellular networks. Furthermore, we can

2.2 Network Architecture

9

Fig. 2.1 The developed potential network architecture for ultra-dense heterogeneous networks

use Poisson cluster process to characterize the clustering property of small cells, as small cells are usually clustered around highly populated areas.

2.2.1.1

Device-to-Device Communications

To take advantage of the physical proximity of communicating devices, such as increasing resource utilization, improving cellular coverage, and offloading traffic load from BSs, device-to-device (D2D) communications are developed as a key component for future cellular networks, as shown in Fig. 2.1. Two D2D user equipments (UEs) can communicate with each other using the licensed radio resources in three modes, i.e., cellular mode, overlay mode, and underlay mode, to coordinate the cellular and D2D links and to manage the interference while remaining control under the network. Compared with the underlay mode, the overlay mode is with the advantage in facilitating the challenging interference management. In underlay mode, resource sharing between D2D pair and cellular UE incurs interference, which is usually viewed as an obstacle to cellular networks. To realize the direct communications between two D2D UEs, the transceiver architecture of current UE shall be changed, since LTE adopts orthogonal frequency division multiplexing (OFDM) and single-carrier frequency division multiple access (SC-FDMA) in the downlink and uplink, respectively. As a result, we can choose either OFDM or SC-FDMA for D2D communications. One simple solution is to add inverse discrete Fourier transform (IDFT) module in the UE receive chain for receiving SC-FMDA signal or bypass the discrete Fourier transform (DFT) module in the UE transmit chain for transmitting OFDM signal. The D2D communication ranges with different transceiver architectures are shown in Table 2.1, where we observe

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2 Sustainable Ultra-Dense Heterogeneous Networks

Table 2.1 D2D communication ranges (meters): OFDM versus SC-FDMA Modulation scheme QPSK 16QAM 64QAM Code rate 1/2 3/4 1/2 3/4 1/2 2/3 Indoor scenario Outdoor scenario

3/4

5/6

SC-FDMA

64.7

58.8

54.6

48.9

48.7

44.2

42

39.8

OFDM SC-FDMA

56.4 378

50.6 283.5

49.5 231.1

43.9 173.3

45 171.8

40.5 136.5

38.4 121.6

36.2 108.4

OFDM

252.6

189.5

178.9

134.1

142.1

112.8

100.6

89.6

that the range improvement of SC-FDMA-based architecture is about 10∼15% and 20∼50% for indoor and outdoor scenarios, respectively, compared with OFDMbased architecture. This is due to the fact that the peak-to-average power ratio of OFDM is higher than that of SC-FDMA, which means that the power back-off of SC-FDMA is lower than that of OFDM.

2.2.1.2

Machine-Type Communications

Compared with human-type communication (HTC), one of the key characteristics of MTC is that it involves a potentially huge number of power-limited devices, where each connected device transmits low volume of non-delay sensitive data with a certain period. Since the current cellular networks are designed for human-type applications, the operator should accommodate their networks to support MTCs. It is noticed that handling a huge amount of MTC devices causes problems in connection establishment and radio resource allocation [7], and significant congestion problem will be risen due to simultaneous signaling or data transmissions from massive MTC devices. Before establishing a radio resource control (RRC) connection for an MTC device, a random access procedure shall be initialized for the MTC device. With a large number of MTC devices performing the random access procedure simultaneously, the succession probability of access for each device would degrade seriously. In order to mitigate the congestion in a random access procedure, a simple solution is to assign new dedicated resource for the access of MTC devices based on the traffic conditions and the number of MTC devices. In this chapter, we will develop a random access scheme for massive MTC devices to mitigate the congestion and to improve the network performance later.

2.2 Network Architecture

11

2.2.2 Downlink and Uplink Decoupling Compared with the traditional coupled downlink-uplink association strategy in current cellular networks, decoupling of downlink and uplink is inspired to balance the traffic load between downlink and uplink in multi-tier UDHN, where each user may be associated with different BSs in the downlink and uplink directions and the downlink and uplink communication sessions are treated as two separated entities. From the perspective of optimization, the region of coupled association is a subset of that of decoupled association, which indicates that a well designed decoupled association strategy principally outperforms coupled one in terms of network throughput, SE, and EE [8]. Downlink and uplink decoupling (DUDe) can be viewed as an evolution of the biased coupled association, where the small cell range is extended through the use of a positive cell selection offset to the largest received downlink power. Compared with biased coupled association, the gains of DUDe are substantial. A simple access rule of DUDe is that users connect to the nearest BS in uplink and access to the BS with the strongest received downlink signal in downlink. This is much more beneficial for MTC applications, whose uplink traffic load is usually much higher than downlink traffic load. Compared with coupled association, DUDe results in new challenges from the perspective of network design, where the logical and physical channels are much easier to design and operate for coupled association [8]. Furthermore, decoupling the downlink and uplink requires a strict synchronization and signalling connectivity between the two association BSs, which can be easily implemented with C-RAN. In C-RAN, two BSs can exchange information directly to perform call admission, handover, and resource assignment, which could facilitate the DUDe. As the C-RAN implements the protocol stack in a centralized fashion, it is preferred to employ a centralized fashion for DUDe in our developed UDHN with C-RAN architecture.

2.2.3 Cloud Radio Access Network Since ultra-dense networks meet new challenges of densely deploying small cells due to the significant capital expenditure (CAPEX) and operating expenditure (OPEX), C-RAN is viewed as a promising paradigm for UDHN, as shown in Fig. 2.1. C-RAN integrates cloud computing into radio access networks to virtualize the functionalities of BSs, and it may be viewed as an evolution of the distributed BS. In distributed BS, the radio function unit, i.e., remote radio head (RRH), is separated from the digital function unit, i.e., baseband unit (BBU). In C-RAN, BBUs are brought together to construct a BBU pool, and RRHs are deployed in a distributed manner to provide wide coverage and high data rate [9, 10]. A number of RRHs are connected a BBU pool via high bandwidth and low latency links, e.g., optical fiber or microwave connections, using the open base station architecture initiative (OBSAI) or common public radio

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interface (CPRI) standard [11]. As the BBU pool is responsible for all the baseband signal processing, C-RAN could facilitate the collaboration among cells, such as coordinated multipoint (CoMP) and interference mitigation. In Fig. 2.1, all the macrocells, microcells, picocells, and femtocells can be served by RRHs, which transmit radio frequency (RF) signals to users in the downlink and forward the baseband signals to BBU pool in the uplink. As most of the functions of the protocol stack are conducted in BBU pool, RRHs could be relatively simple and cheap with physical layer functions only. Compared to stand-alone BS, which conducts the entire protocol stack and has its own backhaul, battery, air condition, and so on, RRH is a pure RF unit, and can be easily deployed to extend the coverage and to improve the capacity in a cost-efficient manner. Compared to traditional network architecture with stand-alone BSs, the baseband processing resources are shared among RRHs through cloud computing in BBU pool, and fewer BBUs are needed in C-RAN to cover the same geographic area. It is reported that one BBU pool could serve up to one thousand RRHs to cover a geographic area with a radius of tens of kilometers. Due to the cloud computing property of C-RAN, it could also effectively solve the challenge of traffic load fluctuation of each cell caused by the user mobility, and save the processing resources and power at idle times. Because of the dense deployment of inexpensive RRHs and processing resources sharing among cells, C-RAN architecture could significantly reduce the CAPEX and OPEX compared to current network architecture with stand-alone BSs and distributed BSs. Furthermore, due to the scalability of C-RAN, it is easy to add new BBUs in the BBU pool to improve the processing ability and to deploy new RRHs to enhance the network coverage, thereby facilitating system roll out and network maintenance. However, C-RAN architecture brings huge overhead on the fronthaul links between RRHs and BBU pool, and the capacity of fronthaul has a crucial influence on the performance of C-RAN. The traffic load on the fronthaul is proportional to the aggregate volume of all the users, and it is 50 times higher than that on the backhaul. For example, the CPRI data rate per carrier of time division long-term evolution (TD-LTE) is 10 Gbps. Hence, the fronthaul capacity is a bottleneck of C-RAN. To tackle this problem, fronthaul compression is necessary, where RRHs and BBU pool compress the baseband signals in uplink and downlink respectively. With the continued user densification in UDHN, the impact of the constrained fronthaul will become more and more serious, and joint design from both the wireless and transport perspectives is required for the advanced fronthaul. With constrained fronthaul, fronthaul-aware resource allocation schemes are also required to alleviate the fronthaul constraints.

2.3 Random Access In UDHN, the fact that the density of MTC devices will be much higher than that of UEs makes the random access for MTC devices very challenging. To support a massive number of MTC devices, we develop a generalized orthogonal/non-orthogonal

2.3 Random Access

13

Fig. 2.2 Markov Chain model for the GONORA scheme

random access (GONORA) scheme for UDHN, which is a grant-less transmission scheme. Let physical resource pool (PRP) denote the basic transmission block in GONORA, where a PRP consists of ω resource units (RUs). The procedure of GONORA can be modelled as a Markov chain, as shown in Fig. 2.2. An MTC device with a new packet to transmit generates a random discrete-time backoff interval before transmitting to minimize the congestion, where the backoff interval is uniformly chosen in the range (0, Wv ), 0 ≤ v ≤ V, with subscript v denoting the repetition times of the new packet. Here, we adopt an exponential backoff scheme [12], where the content window Wv = 2v W0 with W0 being the initial content window size. When the backoff time counter reaches zero, the MTC device chooses Nv RUs from the next coming PRP to transmit part of the new packet based on the size of Nv . The value of Nv is related with the size of the packet, the repetition times, and the payload of PRP. If it is successfully transmitted, the MTC device returns to idle state to wait for transmitting the rest of the new packet or to wait for the next packet. If the MTC device does not receive an acknowledge (ACK) in time, it will prepare for the next repetition. After reaching the maximum allowed repetition times, the MTC device will give up repetition and return to the idle state no matter whether the packet is successfully transmitted or not.

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In this Markov chain, the one-step transition probabilities are ⎧ Pr ( v, k| v, k + 1) = 1, ⎪ ⎪ ⎨ Pr ( v, k| Sv ) = 1 Wv , Pr ( Sv+1 | Nv ) = pve , ⎪ ⎪ ⎩ Pr ( idle| Nv ) = pvs ,

k ∈ (0, Wv − 2) , v ∈ (0, V) k ∈ (0, Wv − 1) , v ∈ (0, V) v ∈ (0, V − 1) v ∈ (0, V)

(2.1)

where the success and error probabilities pvs and pve of the v-th repetition are dominated by the channel state and congestion among MTC devices. To simplify the choosing of Nv , we can choose each RU from one PRP with a probability p with Nv following Bernoulli distribution and E [Nv ] = pω. Obviously, p is related with the traffic volume and the size of the PRP, and the value of p can be derived as p = M

βγ

m=1

(2.2)

αm λm τ

where M is the number of MTC devices associated with the same cell, αm is the average package size of MTC device m, λm is the Poisson parameter, τ is the time duration of each PRP, β is the average capacity of one PRP without signal superposing, and γ denotes the average resource reuse factor. Here, the package arrival of each MTC device is assumed to follow Poisson distribution and, consequently, the M αm λm τ . average aggregate traffic load during one PRP is m=1 The block error ratio (BLER) and normalized throughput of GONORA versus overload factor with different RRH numbers are shown in Figs. 2.3 and 2.4, respectively. Here, the overload factor is defined as the ratio of the MTC device number and the RU number per PRP, and the normalized throughput is defined as the average number of UEs that successfully transmit their packets on one RU. In Fig. 2.3, it is shown that the BLER increases with the increase of overload factor due to the

Fig. 2.3 BLER of GONORA with different RRHs

10

0

1 RRH 3 Centralized RRHs 2 Distributed RRHs 3 Distributed RRHs

10 -1

-2

BLER

10

10 -3

10 -4

10 -5 0.8

1

1.2

1.4

1.6

1.8

2

Overload Factor

2.2

2.4

2.6

2.8

2.3 Random Access

15

Fig. 2.4 Normalized throughput of GONORA with different RRHs

2.8

Offered traffic 3 Distributed RRHs 2 Distributed RRHs 3 Centralized RRHs 1 RRH

2.6

Normalized Throughput

2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Overload Factor

Fig. 2.5 BLER of GONORA with different p

10

0

BLER

10 -1

10

-2

p=0.14 p=0.12 p=0.10 p=0.08 p=0.06 p=0.04 p=0.02

10 -3

10

-4

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Overload Factor

congestion of MTC devices. In Fig. 2.4, we observe that the normalized throughput approaches the offered traffic with low overload factor, which is due to the fact that the packets are quasi orthogonal in such a scenario. With the increase of overload factor, the orthogonality among the transmitted packets is destroyed, which results in that more RRHs are required to mitigate the congestion. In Fig. 2.5, the BLER of GONORA versus overload factor with different RU selecting probability p is plotted. We observe that the BLER decreases with p when overload factor is small. Obviously, the higher the value of p, the larger the repetition times per packet, and hence the higher the received signal-to-interference-plus-noise ratio (SINR). With the increase of overload factor, the congestion becomes serious, which leads to higher BLER.

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2.4 Challenges of UDHN The challenges of the UDHN are from deploying and operating the cellular networks to satisfy the unprecedented mobile device increase and the explosive traffic load growth with limited radio resources. Hence, the greatest challenge of the UDHN is the interference management. Furthermore, due to the mobility of massive mobile devices, mobility management, mobile association, and channel estimation are also key challenges of the UDHN.

2.4.1 Interference Management Due to the decrescent distance between neighbouring cells caused by continued network densification, the co-channel interference will become more and more serious and may severely deteriorate the network performance, especially the quality of service (QoS) of cell-edge users. Hence, effective interference management is crucial for guaranteeing the user experience [13]. To address the co-channel interference, inter-cell interference coordination (ICIC) has been introduced in LTE networks, where the neighbouring BSs allocate different radio resources to their users in some way to mitigate inter-cell interference (ICI) based on the received interference status of their neighbours from X2 interface. A straightforward way of allocating different resources to cell-edge users is adopting fractional frequency reuse (FFR) strategy, where different frequency reuse factors are applied in the cell center and cell edge regions to mitigate ICI. It is noticed that we can also adopt the concept of FFR to mitigate the co-channel interference among geographically overlapped cells by assigning different frequency resources to different cells in overlapped area. Another effective ICIC strategy in LTE is CoMP, which turns the ICIs into useful signals, especially for cell-edge users, through joint processing or coordinated scheduling/coordinated beamforming. In UDHN, besides FFR and CoMP, advanced interference management strategies are needed to tackle the more serious interference status. Recently, two new interference management strategies have emerged: interference shaping (IS) and interference exploitation (IE) [13]. The concept of IS is to linearly combine the interference signals in a certain way to eliminate the aggregated interference effect at receivers, and the representative IS technique is interference alignment (IA), which aligns multiple interfering signals such that the received signal can be projected into the null space of the interfering signals to decode the desired signal with no interference. IE exploits the interference as side information to improve the throughput, such as network coding and index coding. The abovementioned four interference management strategies are network-side approaches. To further enhance the interference management, user-side approach is another option to alleviate the interference issues in UDHN. In user-side approach, advanced receiver is required to take use of the interference signals structure, and the

2.4 Challenges of UDHN

17

desired signals can be derived by subtracting the reconstructed interfering signals from the received signals. Joint network-side and user-side interference managements could significantly mitigate the interference.

2.4.2 Mobility Management and Mobile Association The continued network densification and increased heterogeneity in the UDHN also pose challenges for the mobility management [14]. A natural way for mobility management in the UDHN is to decouple the user and control planes, where the user plane protocol stack is conducted through small cell to improve the throughput and the control plane protocol stack is preferred to be conducted by macrocell to reduce the handover frequency. After adopting C-RAN architecture, the mobility management will become more easier with the decoupling of user and control planes, as all necessary information is in the same BBU pool and is easy to be passed from source RRH to target RRH to prepare the handover procedure. Furthermore, the advanced cloud processing could also reduce the intra-BBU pool handover delay. Another way to ease the mobility management in the UDHN is to restrict highly mobile users to macrocells at lower frequencies, thereby forcing them to tolerate lower data rates while minimizing the frequency of the handover. Most of the existing mobile association schemes are developed based on the downlink channel state information (CSI). In the UDHN, different types of small cells with different transmit powers are densely deployed, where multiple access points are available in both uplink and downlink. Hence, mobile association, which adaptively selects the uplink and downlink access points to guarantee the QoS while balancing the traffic load in different cells, becomes a challenging and fundamental issue. One natural way for mobile association is to select the serving cell with the highest instantaneous receiving power. To reduce the probability of handover, we can also select the serving cell based on long-term CSI. With C-RAN, the mobile association can be performed in the BBU pool to ease the collaboration among cells to guarantee the QoS from the perspective of the UEs and balance the traffic load from the perspective of the networks.

2.4.3 Channel Estimation CSI is essential not only for resource scheduling and demodulation, but also for coordination among cells in UDHN. And the use of wide channel bandwidths and massive antennas poses significant challenges for the channel estimation. For example, CoMP relies on the availability of CSIs strictly, and fast CSI feedback is required to tackle the channel changes. To alleviate the CSI estimation and signalling overhead, instead of allowing coordination among all the cells, the cells can be clustered

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so that only certain cells in the same cluster join the CoMP. In IA algorithm, global and instantaneous CSI at transmitter is also crucial for aligning and removing the interference. One simple method of reducing the CSI overhead is to use the channel stochastics to replace the instantaneous CSI, e.g., using the channel autocorrelation function instead of CSI itself in blind IA, at the cost of performance degradation. The second method is to adopt the channel reciprocal properties to estimate the uplink (downlink) channel based on the downlink (uplink) signals to avoid the reference signals and to reduce the feedback overhead. The second method is attractive for multiple-input multiple-output (MIMO) channel, especially for massive MIMO channel. However, the channel reciprocality only exists in time division duplexing systems, and the channel reciprocal is also unavailable for DUDe.

2.5 Conclusions Network architecture dominates the performance of the networks. This chapter provides a potential network architecture to satisfy the 1000x traffic increase. The future cellular networks will be an ultra-dense multi-tier heterogeneous networks along with D2D pairs and MTC devices. To accommodate different tier cells, DUDe and C-RAN will be introduced. We have developed the GONORA scheme for massive MTC devices and analyzed the challenges of the UDHN.

References 1. J. Liu, W. Xiao, Advanced carrier aggregation techniques for multi-carrier ultra-dense networks. IEEE Commun. Mag. 54(7), 61–67 (2016). 2. IMT-2020, 5G network technology architecture. White Paper, May 2015. 3. N. Bhushan, J. Li, D. Malladi, R. Gilmore, D. Brenner, A. Damnjanovic, R.T. Sukhavasi, C. Patel, S. Geirhofer, Network densification: the dominant theme for wireless evolution into 5G. IEEE Commun. Mag. 52(2), 82–89 (2014). 4. S. F. Yunas, M. Valkama, J. Niemela, Spectral and energy efficiency of ultra-dense networks under different deployment strategies. IEEE Commun. Mag. 53(1), 90–100 (2015). 5. X. Ge, S. Tu, G. Mao, C.-X. Wang, T. Han, 5G ultra-dense cellular networks. IEEE Wireless Commun. 23(1), 72–79 (2016). 6. H. Zhang, Y. Dong, J. Cheng, M.J. Hossain, V.C.M. Leung, Fronthauling for 5G LTE-U ultra dense cloud small cell networks. IEEE Wireless Commun. 23(6), 48–53 (2016). 7. H. Shariatmadari, R. Ratasuk, S. Iraji, A. Laya, T. Taleb, R. Jantti, A. Ghosh, Machine-type communications: current status and future perspectives toward 5G systems. IEEE Commun. Mag. 53(9), 10–17 (2015). 8. F. Boccardi, J. Andrews, H. Elshaer, M. Dohler, S. Parkvall, P. Popovski, S. Singh, Why to decouple the uplink and downlink in cellular networks and how to do it. IEEE Commun. Mag. 54(3), 110–117 (2016). 9. J. Wu, Green wireless communications: from concept to reality. IEEE Wireless Commun. 19(4), 4–5 (2012).

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10. C. Fan, Y.J. Zhang, X. Yuan, Advances and challenges toward a scalable cloud radio access network. IEEE Commun. Mag. 54(6), 29–35 (2016). 11. M. Peng, Y. Sun, X. Li, Z. Mao, C. Wang, Recent advances in cloud radio access networks: system architectures, key techniques, and open issues. IEEE Commun. Surveys Tuts. 18(3), 2282–2308 (2016). 12. G. Bianchi, Performance analysis of the IEEE 802.11 distributed coordination function. IEEE J. Sel. Areas Commun. 18(3), 535–547 (2000). 13. N. Lee, R.W.H. Jr, Advanced interference management technique: potentials and limitations. IEEE Wireless Commun. 23(3), 30–38 (2016). 14. J.G. Andrews, S. Buzzi, W. Choi, S.V. Hanly, A. Lozano, A.C.K. Soong, J.C. Zhang, What will 5G be? IEEE J. Sel. Areas Commun. 32(6), 1065–1082 (2014).

Chapter 3

Non-Orthogonal Multiple Access: Achieving Sustainable Future Radio Access

This chapter investigates the non-orthogonal multiple access (NOMA) to achieve sustainable future radio access (FRA). In Sect. 3.1, we introduce the motivation of investigating NOMA. In Sect. 3.2, we compare the achievable power region of NOMA with that of orthogonal multiple access (OMA) and demonstrate the energy efficiency (EE) performance advantage of NOMA over OMA. In Sect. 3.3, we investigate the roles played by energy-aware resource allocation and grant-free transmission to further enhance the EE performance of NOMA. In Sect. 3.4, we examine a hybrid NOMA strategy to harness the joint benefits of resource allocation and grant-free transmission in different application scenarios of FRA, and in Sect. 3.5, we identify and discuss future research directions for further boosting the EE performance of NOMA, followed by conclusions in Sect. 3.6.

3.1 Introduction The future cellular networks are expected to provide an unprecedented quality of service to our networked society, which has imposed significant challenges to the design and implementation of FRA [1]. In long-term evolution (LTE) standards, orthogonal frequency division multiple access (OFDMA) and single-carrier frequency division multiple access (SC-FDMA) are adopted for downlink and uplink transmissions, respectively. However, OFDMA and SC-FDMA cannot achieve the optimal capacities of the broadcast channel (BC) and multiple access channel (MAC), respectively [2]. Against this background, NOMA has been introduced to fulfill the requirements of FRA [3]. The current NOMA studies have mainly focused on its superior spectral efficiency (SE) performance and its ability to connect massive users [2, 3]. With the rise of the worldwide environmental awareness [4], the energy consumption has © Beijing Institute of Technology Press 2023 J. An et al., Sustainable Wireless Communications, https://doi.org/10.1007/978-981-19-0448-6_3

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become a major concern, especially in the Internet of Things (IoT) scenario [1]. Motivated by this, in this chapter we investigate NOMA from the EE perspective, providing pivotal insights into the development of green NOMA transmission.

3.2 Achievable Power Region Generally speaking, the state-of-the-art NOMA schemes are categorized into three types: power-domain multiplexing, code-domain multiplexing, and interleave-domain multiplexing. Specifically, power-domain NOMA (PD-NOMA) exploits the near-far effect in the multi-user system [3]. Code-domain NOMA (CDNOMA), including sparse code multiple access (SCMA), pattern division multiple access (PDMA), and multi-user shared access (MUSA), relies on low-correlated spreading signatures. Interleave-domain NOMA (ID-NOMA), including interleave division multiple access (IDMA) and interleave grid multiple access (IGMA), exploits the structure of different interleaving patterns of users [1, 2]. It is noted that these NOMA schemes are mainly based on the centralized scheduling.1 Therefore, they are classified as scheduling-based NOMA. Scheduling-based NOMA achieves higher SE and larger overloading than OMA [2], thus appearing attractive and suitable for mobile broadband (MBB) transmissions. In addition to this advantage, NOMA surpasses OMA in terms of EE, which will be elaborated in the following example. Consider a NOMA system consisting of one base station (BS) and two users, i.e., user 1 and user 2. The capacity regions of the NOMA system, defined as the closures of all achievable rate tuples under certain power constraints, have been derived to reflect the SE performance in both uplink and downlink [2, 3]. Reciprocally, we can define the power regions as the closure of the power tuples where the data rate requirements of users can be simultaneously satisfied to illustrate the EE performance of NOMA. Denoting the uplink transmit power pair of users 1 and 2 as (P1 , P2 ) and the downlink transmit power pair for users 1 and 2 as (P1 , P2 ), the uplink and downlink power regions are shown in Fig. 3.1. Obviously, the power regions are bounded in the negative direction but unbounded in the positive direction. Also, the power pair which is closest to the coordinate origin is the minimum power consumption to meet the capacity requirement. In Fig. 3.1 we compare the power regions of NOMA and OMA in the uplink and downlink. In the uplink depicted in Fig. 3.1a, the gray shaded area illustrates that the peak-power-constrained OMA is strictly suboptimal compared to NOMA in terms of power consumption. Moreover, at certain operation point, i.e., “◦”, the averagepower-constrained OMA can achieve the same EE as NOMA, at the cost of a higher peak transmit power. Furthermore, the tangency point between the dashed line and the boundary of the achievable power region of the average-power-constrained OMA, 1

It is noticed that SCMA could also work in grant-free transmission mode. When SCMA was first proposed, centralized scheduling was assumed in the design of SCMA codebooks to avoid the hard collisions among users.

3.2 Achievable Power Region

(a) Uplink

23

(b) Downlink

Fig. 3.1 The power regions of NOMA and OMA. In the uplink, both the peak-power-constrained and average-power-constrained OMAs are considered, where the instantaneous transmit power and the average transmit power are limited, respectively

illustrated by “∗”, indicates the minimum overall power consumption of OMA. In addition, it is clearly observed that the maximum EE is achieved by NOMA at the operation point A. Similarly, in the downlink depicted in Fig. 3.1b, the operation point illustrated by “•” indicates that the maximum EE is achieved by NOMA. Also, we observe from Fig. 3.1b that the minimum power consumption of NOMA is strictly lower than that of OMA. Particularly, the gap between the dashed and solid lines shows the power-saving gain of NOMA over OMA in the downlink, which implies that NOMA is inherently energy efficient. To intuitively explain why NOMA achieves a better EE performance than OMA, we present the relationship between SE/EE and power in Fig. 3.2. We first see from Fig. 3.2 that the SE logarithmically increases with power. Based on this, the x-axis can be divided into three regions: Power-limited region, NOMA region, and bandwidthlimited region. It is found that the cell-edge and cell-center users are usually located in the power-limited and bandwidth-limited regions, respectively. Second, we see from Fig. 3.2 that the EE first increases and then decreases as power increases, which is due to the fact that the throughput increases with power logarithmically while the power consumption increases linearly. We clarify that in the power-limited and bandwidth-limited regions, the mismatch between power and bandwidth constrains the performance of cell-center and cell-edge users in OMA. However, this mismatch is mitigated in the NOMA region. In uplink NOMA, more bandwidth is allocated to the cell-center user. In downlink NOMA, more power is allocated to the cell-edge user and more bandwidth is allocated to the cell-center user. Hence, NOMA becomes more energy efficient than OMA. To further improve the EE performance of NOMA, we resort to energy-aware resource allocation and grant-free transmission, which will be elaborated in the following section.

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Fig. 3.2 The illustration of the relationship between SE/EE and power

3.3 Energy Efficiency Improvement in NOMA 3.3.1 Energy-Aware Resource Allocation Resource allocation has been recognized as an effective method to enhance the SE performance of NOMA. Also, this method has great potentials to improve the EE performance of NOMA. In this subsection, we discuss the energy-aware resource allocation for NOMA in three typical deployment scenarios, namely, single-cell NOMA, network-level NOMA, and NOMA with wireless power transfer.

3.3.1.1

Single-Cell NOMA

Energy-efficient resource allocation in single-cell NOMA can be generally formulated as maxs∈S M(s)/P(s) [4], where s is the transmission strategy including radio resource and multiple access (MA) signature allocation, M(s) is the achieved utility, and P(s) is the total power consumption. It is noted that the problem formulation in the uplink of NOMA is slightly different from that of downlink, since the individual power constraint is required at each user in the uplink, while the total power constraint is required at the BS in the downlink [5]. The problem of the optimal energy-efficient resource allocation is generally NPhard. Since multiple users share the same radio resources, the EE optimization problem in NOMA is even more complex than that in OMA [6]. To address this, we can decompose the optimization problem into two subproblems, i.e., user scheduling and MA signature allocation, and then solve them sequentially. Notably, the determination of M(s) in the user scheduling stage is strongly correlated with the MA signature allocation outcome. Thus, the iterations between user scheduling and MA signature allocation are needed to produce better solutions.

3.3 Energy Efficiency Improvement in NOMA

3.3.1.2

25

Network-Level NOMA

With the popularity of network densification, inter-cell interference (ICI) constitutes a major obstacle to the enhancement of the network-level throughput and EE performance. Due to the benefit of interference cancellation, NOMA has been incorporated in network-level scenarios [7, 8] where a large number of users/cells share the same resources. In network-level NOMA, cell clustering and user association are essential to provide a stable and efficient access service. Moreover, interference-aware MA signatures, e.g., ultra-sparse spreading sequences designed for SCMA or PDMA and small power levels designed for PD-NOMA, need to be carefully assigned to celledge users for ICI mitigation and EE improvement [9]. When the dual connectivity is enabled, traffic offloading needs to be considered to optimize the traffic burden in different radio links. Furthermore, in network-level NOMA with the cloud radio access structure, one can benefit from the cooperation among neighboring cells. In such cooperation, the joint transmission and detection can be employed to enhance the EE performance [2, 9, 10].

3.3.1.3

NOMA with Wireless Power Transfer

Simultaneous wireless information and power transfer (SWIPT) is a cutting-edge technique which can prolong the battery life of wireless devices through harvesting energy from radio waves. When the SWIPT technique is deployed, the resources for data transmission are reduced since some resources are utilized for energy harvesting. SWIPT has been integrated into NOMA to enhance the SE, EE, and access opportunities [11–13]. After this integration, user scheduling is one of the primary design challenges [11–13]. For example, the users with a low latency requirement need to be scheduled to perform NOMA transmission, while the delay-tolerant users may perform energy harvesting to prolong the battery life. Also, the tradeoff between energy harvesting and data transmission needs to be considered. When the cooperative communications are considered, the power allocation design needs to be further optimized to accommodate the energy storage constraint [12, 13].

3.3.2 Grant-Free Transmission Scheduling-based NOMA relies on the grant signaling from the BS, which leads to large signaling overhead and low EE, especially when facing the challenges of sporadic traffic and small payloads in massive machine-type communications (MTCs). As such, grant-free NOMA has been introduced to exploit the statistical multiplexing principle in massive MTC [1, 2], and it is also a promising technology for ultrareliable low-latency communications. As per the rules of grant-free NOMA, a user becomes active and performs uplink instant transmission once it has data in buffer.

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Following such rules, each user transmits its signals according to an MA signature which is randomly selected from an MA signature pool. At the receiver, blind detection or preamble aided detection is performed to simultaneously detect user activity information, MA signatures, and data packets. We clarify that some scheduling-based NOMA, such as SCMA, PDMA, and MUSA, can be easily extended to the grant-free mode by allowing users to randomly select MA signatures [2, 10].

3.3.2.1

Collision Management

The collision in grant-free NOMA is detrimental to the reliability of data transmission. Fortunately, the collided users can be distinguished if they choose different MA signatures. This indicates that the design of the MA signature pool is vital in the collision management of grant-free NOMA. With a larger signature pool, the probability of hard collision where multiple users select the same MA signature is lower. However, a larger signature pool also increases the mutual correlation among signatures, which in return deteriorates the detection accuracy. Therefore, a good tradeoff needs to be achieved between the pool size and mutual correlation for producing the optimal system-level throughput. Preamble transmission is another approach to mitigate the collision in grant-free NOMA. A preamble, which indicates the activity information and the selected MA signature, can be used to remind the receiver about possible collision and thus aid data detection. When the cyclic prefix orthogonal frequency division multiplexing (CPOFDM) is adopted, the preamble can also act as the time synchronization sequence. In an extreme case where several collided users choose the same MA signature, we can resort to the advanced multi-user detector, such as successive interference cancelation (SIC), elementary signal estimator parallel interference cancellation (ESE-PIC), and message passing algorithm (MPA), to recover the received data streams by exploiting the characteristics of user channels [10].

3.3.2.2

Energy Consumption

We now investigate the energy consumption in grant-free NOMA. For a given transmission strategy s, the energy consumption of users with sporadic traffic is formulated as η = α(s)Pac (s) + (1 − α(s)) Pina (s), where α(s) is the duty cycle, and Pac (s) and Pina (s) represent the power consumptions in active and inactive modes, respectively. Notably, Pac (s) consists of the power consumptions of data transmission, signaling interaction, and standby state, mathematically given by Pac (s) = Pdata (s) + Psignaling (s) + Pstandby (s)  Pina (s). The values of α(s), Pac (s), and Pina (s) may vary if different transmission strategies are adopted. Compared with the conventional transmission strategies where users are always connected, i.e., α(s) = 1, grant-free access not only utilizes the sparsity of the traffic but cuts down the signaling overhead to reduce the power consumption. In grant-free NOMA, users remain inactive in most of time, and the registration and grant sig-

3.3 Energy Efficiency Improvement in NOMA

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naling are canceled. After successful transmission, the user gets back to the inactive mode once the acknowledgment is received. Otherwise, the user performs retransmission in either the grant-free or scheduling-based fashion. Hence, the duty cycle is extremely small, i.e., α(s) → 0. In addition, the simplified signaling procedure further decreases Pac (s). Therefore, the energy consumption in grant-free NOMA is very low.

3.4 A Hybrid NOMA Strategy NOMA with centralized resource scheduling is particularly suitable for the MBB scenario since it achieves high SE, EE, and reliability. Differently, grant-free NOMA meets the targets of the massive MTC scenario since it enables uplink instant transmission to profoundly reduce the signaling overhead and energy consumption. Motivated by this, we discuss a hybrid NOMA strategy which takes the advantage of grantfree NOMA in terms of connectivity and signaling procedure while maintaining the high reliability, SE, and EE of scheduling-based NOMA, to satisfy the requirements of various scenarios. The core idea behind the hybrid NOMA strategy is that a user can either be a scheduling-based NOMA user or a grant-free NOMA user, according to its business characteristics, the instantaneous channel conditions, and the network congestion. Usually, machine-type users with small data packets and a low latency requirement can adopt the grant-free access for initial transmission, and legacy users with streaming media data can employ the scheduling-based transmission, as shown in Fig. 3.3. We clarify that the BS is able to configure not only the NOMA scheme for each user but the volume of radio resources allocated for each NOMA scheme, according to the network load. For example, if a high collision probability is expected in grant-free NOMA, e.g., by estimating the interference level according to the previous observations, the BS may assign more radio resources for grant-free access, or even enforce

Fig. 3.3 An illustration of the hybrid NOMA strategy

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some grant-free NOMA users to conduct scheduling-based access. With the flexible switching between different NOMA schemes, users can make a full use of the channel and high reliability can still be achieved even in a highly congested channel, since an appropriate amount of radio resources is elaborately scheduled to different NOMA users. To further boost the SE and EE performance, the hybrid NOMA strategy allows spectrum sharing between scheduling-based and grant-free transmissions. As shown in Fig. 3.3, the legacy users, which conduct scheduling-based access, are normally scheduled with a large amount of radio resources due to the use of large data packets. Differently, the machine-type users, which conduct grant-free access, perform autonomous transmission with limited radio resources. Since the legacy users and the machine-type users coexist with each other, the hybrid NOMA strategy has the great potential to increase connectivities while maintaining a high throughput. We note that the legacy users and the machine-type users may interfere with each other. To reduce such interference, the legacy users use additional redundancy to combat the interference, while the machine-type users adjust their transmit power as per the received power of the downlink reference signal to ensure that their received powers are at the level of thermal noise. To randomize the interference, the machine-type users may randomly use spreading-based MA signatures, occupy radio resources in a sparse manner, and adopt frequency hopping. Moreover, interference cancellation techniques can be employed at the BS. Since the received signals of schedulingbased transmission are often stronger than those of grant-free transmission, they can be detected and canceled firstly. Then the remaining signals can be recovered by exploiting the distinct structures of MA signatures. Finally, although hybrid NOMA can potentially provide a performance gain, the elaborate design of resource allocation and transmission/reception algorithms is still required to realize this gain. To demonstrate the advantage of the hybrid NOMA strategy over the existing OMA and NOMA schemes, we compare their throughput and EE performance in a massive MTC scenario in Fig. 3.4. From Fig. 3.4, we observe that the hybrid NOMA strategy outperforms OMA, grant-free NOMA, and scheduling-based NOMA in terms of both throughput and EE, which is in accordance with our previous claims. The detailed observations made from Fig. 3.4 are summarized as follows. In Fig. 3.4a, the throughput of OMA first increases with the number of machinetype users, and then remains unchanged. This can be explained by the fact that the number of served machine-type users is limited by the radio resource. Due to the same reason, the throughput of scheduling-based NOMA enters a plateau regime with sufficiently large machine-type users. Nevertheless, scheduling-based NOMA achieves a higher throughput than OMA due to the radio resource sharing among users. The throughput of grant-free NOMA first increases and then decreases when the number of machine-type users becomes higher, since the collision becomes increasingly serious in this case. Thus, we conclude that the overloading factor needs to be controlled within a reasonable regime to support grant-free NOMA, by adopting either semi-static methods, e.g., allocating more radio resources in network planning, or dynamic methods, e.g., transmitting access barring signaling to limit the number of active users. Finally, the hybrid NOMA strategy achieves the highest through-

3.4 A Hybrid NOMA Strategy

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Fig. 3.4 Comparisons between NOMA and OMA. The total bandwidth is 10 MHz and the bandwidth allocated for MTC is 2 physical resource blocks (PRBs). Non-full buffer small packet traffic model specified in 3GPP TR 45.820 is adopted, the maximum transmission power of users is 23 dBm, the open loop power control is used, Pdata (s) + Psignaling (s) = 23 dBm, and Pstandby (s) = 20 dBm

put. This is due to the fact that the hybrid NOMA employs the flexible switching between grant-free and scheduling-based transmissions to avoid severe collision and the cliff-like performance drop experienced by grant-free NOMA, and multiplexes machine-type users with legacy users to fully utilize the entire bandwidth. In Fig. 3.4b, the EE performance of all schemes decreases when the number of machine-type users increases, which is due to the increase in the mutual interference among users. With a small number of active users, grant-free NOMA achieves the highest EE, which is due to the low collision among users and low signaling overhead. In this case, the hybrid NOMA achieves a slightly worse EE performance than grantfree NOMA, due to the signaling cost of partial scheduling, but still achieves a significantly higher EE performance than scheduling-based NOMA and OMA. With the increase in the number of machine-type users, the reliability of grant-free NOMA is deteriorated due to the significantly increased collision. This leads to the fact that grant-free NOMA underperforms hybrid NOMA and even scheduling-based NOMA in terms of EE.

3.5 Future Directions about NOMA Despite that NOMA has shown to exhibit a superior performance than OMA in various scenarios, numerous research efforts are still needed to fully unlock the great potentials of NOMA for futuristic wireless applications. In this section, we present and discuss some future directions of NOMA to shed a light on the multiple access schemes for FRA.

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3.5.1 Multi-Domain MA Signatures Different from OFDMA, MA signatures can be involved into FRA. Also, the involved MA signatures may vary in different application scenarios, which implies that the MA signatures can be treated as virtual resources of the wireless network. Therefore, one important and challenging task of FRA is to design and allocate these virtual resources to meet the targets of FRA. It is noted that the existing NOMA technologies usually construct the MA signatures in a single domain for the sake of simplicity, e.g., the MA signatures of PD-NOMA are constructed in the power domain while the MA signatures of SCMA are constructed in the code domain. To break the limits of the existing NOMA technologies, it is worthwhile to explore the multi-user detection through multi-domain MA signatures by reaping the benefits offered by additional dimensions. For example, CD-NOMA can be jointly designed with PDNOMA. With multi-domain MA signatures, two types of receiving structure can be employed to exploit the features in multiple domains, i.e., successive receiving and joint receiving, where the features of different domains are successively and jointly exploited, respectively. In addition, the joint allocation of physical radio and virtual resources is a challenging problem in FRA and needs further exploration.

3.5.2 SE/EE Performance Boost with Full-Duplex According to the multi-user information theory, NOMA has harnessed the utmost theoretical potentials of MAC and BC. Therefore, the only way to further boost the SE/EE performance of NOMA is to introduce new technical advancement into it. One promising introduction is to integrate full-duplex (FD) techniques into NOMA [14], since the co-frequency co-time FD transceiver theoretically increases the achievable capacity twice with the deployment of ideal self-interference cancellation. When FD is used, users are able to transmit and receive signals simultaneously and thus, the flexibility of the cellular network deployment is enhanced. It is noted that the integration of NOMA and FD may lead to unexpected inter-user and inter-cell interference. To address this, advanced signal processing and resource allocation algorithms need to be developed for interference mitigation.

3.5.3 Ultra Long Battery Life Cycle with Backscatter Communications A high EE is of paramount significance for machine-type communications, since machine-type devices are typically battery-operated and the IoT business may ask for a battery life cycle as long as several decades [2]. To further improve the EE performance of NOMA in IoT scenarios, one promising strategy is to integrate the

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31

cutting-edge technologies with NOMA, e.g., scattering-based passive transmission (or equivalently, backscatter communications) [15]. In this technology, the active radio frequency amplifier module can be removed from machine-type devices and thus, an ultra-low power consumption is achieved. We note that advanced multiple user detection algorithms are required in backscatter communications since the backscattering is performed simultaneously among machine-type devices and multiple scattered signals overlap at the receiver.

3.6 Conclusions Motivated by the growing concerns on green communications in FRA, we discussed the sustainability of NOMA in this chapter. We first illustrated the achievable power regions, which demonstrated that NOMA is more energy efficient than OMA. Moreover, we investigated the role of scheduling-based resource allocation and grant-free transmission in improving the EE performance of NOMA. Furthermore, we examined a hybrid NOMA strategy by effectively integrating resource allocation and grant-free transmission within NOMA, which was shown to provide better EE and throughput than using them separately. Finally, we discussed some pressing challenges and identified future research directions of NOMA.

References 1. 3GPP TR 38.812, Study on non-orthogonal multiple access (NOMA) for NR. May 2017. 2. Y. Yuan, Z. Yuan, G. Yu, C. Hwang, P. Liao, A. Li, K. Takeda, Non-orthogonal transmission technology in LTE evolution. IEEE Commun. Mag. 54(7), 68–74 (2016). 3. Z. Ding, Y. Liu, J. Choi, Q. Sun, M. Elkashlan, C.-L. I, H.V. Poor, Application of non-orthogonal multiple access in LTE and 5G networks. IEEE Commun. Mag. 55(2), 185–191 (2017). 4. J. Wu, S. Rangan, H. Zhang, Green Communications: Theoretical Fundamentals, Algorithms and Applications. CRC Press, 2012. 5. X. Sun, C. Shen, Y. Xu, S.M. Al-Basit, Z. Ding, N. Yang, Z. Zhong, Joint beamforming and power allocation design in downlink non-orthogonal multiple access systems, in IEEE Globecom Workshops IEEE(2016), pp. 1–6. 6. Y. Zhang, H.M. Wang, T.X. Zheng, Q. Yang, Energy-efficient transmission design in nonorthogonal multiple access. IEEE Trans. Veh. Tech. 66(3), 2852–2857 (2017). 7. S Han, L. Chih, Z. Xu, Q. Sun, Energy efficiency and spectrum efficiency co-design: from NOMA to network NOMA. IEEE COMSOC MMTC E-Letter 9(5), 21–24 (2014). 8. Y. Wu, L.P. Qian, Energy-efficient NOMA-enabled traffic offloading via dual-connectivity in small-cell networks. IEEE Commun. Lett. 21(7), 1605–1608 (2017). 9. X. Dai, Z. Zhang, B. Bai, S. Chen, S. Sun, Pattern division multiple access: a new multiple access technology for 5G. IEEE Wireless Commun. 25(2), 54–60 (2018). 10. J. Zhang, L. Lu, Y. Sun, Y. Chen, J. Liang, J. Liu, H. Yang, S. Xing, Y. Wu, J. Ma, I. Berberana, F. Murias, F.J.L. Hernando, PoC of SCMA-based uplink grant-free transmission in UCNC for 5G. IEEE J. Sel. Areas Commun. 35(6), 1353–1362 (2017).

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3 Non-Orthogonal Multiple Access: Achieving Sustainable Future Radio Access

11. P.D. Diamantoulakis, K.N. Pappi, Z. Ding, G.K. Karagiannidis, Wireless-powered communications with non-orthogonal multiple access. IEEE Trans. Wireless Commun. 15(12), 8422–8436 (2016). 12. Z. Yang, Z. Ding, P. Fan, N. Al-Dhahir, The impact of power allocation on cooperative nonorthogonal multiple access networks with SWIPT. IEEE Trans. Wireless Commun. 16(7), 4332–4343 (2017). 13. Y. Liu, Z. Ding, M. Elkashlan, H.V. Poor, Cooperative non-orthogonal multiple access with simultaneous wireless information and power transfer. IEEE J. Sel. Areas Commun. 34(4), 938–953 (2016). 14. D. Korpi, J. Tamminen, M. Turunen, T. Huusari, Y. Choi, L. Anttila, S. Talwar, M. Valkama, Full-duplex mobile device: pushing the limits. IEEE Commun. Mag. 54(9), 80–87 (2016). 15. B. Kellogg, V. Talla, J. Smith, Passive Wi-Fi: bringing low power to Wi-Fi transmissions. Getmobile Mobile Computing & Communications 20(3), 38–41 (2017).

Chapter 4

Space-Time Network Coding with TAS/MRC

In this chapter, we investigate the space-time network coding (STNC) with transmit antenna selection and maximal-ratio combining (TAS/MRC). Section 4.1 introduces the motivation of investigating the STNC with TAS/MRC. Section 4.2 presents the system model. In Sect. 4.3, the exact and asymptotic performance of the STNC with TAS/MRC in cooperative multiple-input multiple-output (MIMO) networks with perfect feedback is analyzed. In Sect. 4.4, the impact of delayed feedback on the performance of the STNC with TAS/MRC is quantified. Section 4.5 presents the simulation results, and Sect. 4.6 concludes this chapter.

4.1 Introduction Cooperative communications have recently attracted considerable attention in emerging wireless applications due to their advantages of network coverage extension and capacity expansion [1–3]. The inherent concept of cooperative communications is to provide spatial diversity by employing relay nodes to forward signals from the source to the destination. Against this background, various cooperative diversity schemes have been proposed and analyzed [4, 5]. We note that the assumption of perfect synchronization may be difficult or impossible in practical multi-node wireless networks as it is very challenging to align all the signals from multiple sources at multiple destinations [6, 7]. High-accuracy synchronization requires complicated control mechanisms and extra control messages, which leads to high system complexities and overheads [7]. When synchronization is imperfect, the performance of cooperative communications can be severely degraded. In order to overcome the problem caused by imperfect synchronization, the STNC scheme was proposed in [8], which uses time division multiple access (TDMA) to deal with the imperfect synchronization issues. Importantly, this scheme achieves the full diversity order. STNC combines information from different sources at each © Beijing Institute of Technology Press 2023 J. An et al., Sustainable Wireless Communications, https://doi.org/10.1007/978-981-19-0448-6_4

33

34

4 Space-Time Network Coding with TAS/MRC

relay node and transmits the combined signal in dedicated time slots, which jointly exploits the benefits of both network coding and space-time coding. Motivated by the above observations, we examine the performance of STNC in the cooperative MIMO network, where closed-form exact and asymptotic expressions for the outage probability (OP) and symbol error rate (SER) with perfect feedback and delayed feedback are presented, respectively. In the network, U users communicate with a common destination D with the assistance of R relays and all the nodes are equipped with multiple antennas. We consider the independent but not necessarily identically distributed (i.n.i.d.) Rayleigh fading channels and include the direct links between the users and destination. Particularly, we focus on decode-and-forward (DF) relaying protocol due to its application in the 3rd Generation Partnership Project (3GPP) long-term evolution (LTE) and IEEE 802.16m [9, 10] such that the relays need to decode the received signal, reencode it, and forward it to the destination. For user-destination (u–D) and relay-destination (r –D) links, we adopt TAS/MRC [11, 12], where a single transmit antenna that maximizes the output signal-to-noise ratio (SNR) is selected and all the receive antennas are combined using maximalratio combining (MRC). As such, the transmitter can be implemented with a single front-end and an analog switch, and the receiver only needs to feed back the index of the selected transmit antenna.

4.2 System Model We consider a cooperative MIMO network where U users transmit their own information to a common destination D with the aid of R relays. In this MIMO network, user u, 1 ≤ u ≤ U , relay r , 1 ≤ r ≤ R, and destination D are equipped with Nu , Nr , and ND antennas, respectively. The channel coefficient between the jth antenna νμ −α ), of transmitter μ and the ith antenna of receiver ν is defined as h i j ∼ CN(0, dμν μ ∈ {u, r }, ν ∈ {r, D}, μ = ν, where dμν and α denote the distance between μ and ν and the path loss exponent, respectively. The transmission of STNC takes place over U + R time slots, which are divided into two consecutive phases [8]. In the first phase, the U users take turns to broadcast their symbols to the relays and destination in the first U time slots, i.e., user u broadcasts its symbol in time slot u while other users staying in silence. In the second phase, R relays combine the overheard symbols from multiple users during the first phase to a single symbol and then take turns to transmit it to destination in the last R time slots, i.e., relay r transmits its symbol in time slot U + r . We outline the STNC with TAS/MRC in the cooperative MIMO network as follows. In the first phase, TAS/MRC is applied between each user and the destination D. The optimal antenna amongst the Nu antennas at user u is selected to maximize the instantaneous SNR of the signal from user u to the destination D. Therefore,the index  of the optimal transmit antenna, n ∗u , is determined as n ∗u = arg max1≤n u ≤Nu  hn u D  F , where hn u D is the channel vector between the n u th antenna at user u and the multiple antennas at the destination D, and  ·  F denotes the Frobenius norm. Since the

4.2 System Model

35

multiple antennas at one node do not appear colocated and the MIMO links are close to independent and identically distributed (i.i.d.) Rayleigh fading in the richscattering scenario [13], we assume that hn u D has i.i.d. Rayleigh entries. The signals received at the destination D and relay r from user u in time slot u are  yuD (t) = hn ∗u D P0u xu su (t) + nuD (t)

(4.1)

and yur (t) = hn ∗u r



P0u xu su (t) + nur (t)

(4.2)

respectively, where hn ∗u D and hn ∗u r denote the Rayleigh channel vectors between the n ∗u th antenna at user u and multiple antennas at the destination D and between the n ∗u th antenna at user u and multiple antennas at relay r respectively, nuD (t) and nur (t) are the additive white Gaussian noise (AWGN) vectors with zero mean and variances of N0 I ND and N0 I Nr respectively, In , n ∈ {ND , Nr }, denotes the identity matrix of size n, P0u is the transmit power of xu , and xu and su (t) denote the symbol with unit energy transmitted by user u and the related spreading code, respectively. The cross correlation between su (t) and su  (t) is defined as ρuu  = su (t) , su  (t) , where T  f (t) , g (t) = T1 0 f (t) g ∗ (t) dt is the inner product between f (t) and g (t) with the symbol interval T , and g ∗ (t) is the complex conjugate of g (t). Moreover, we assume that ρuu = su (t)2F = 1. After combining the received signal replicas from different antennas using MRC and applying matched-filtering, the received signals at the destination D and relay r are given by      u = wuD yuD (t) , su (t) =  hn ∗u D  F P0u xu + n uuD yuD

(4.3)

     u = wur yur (t) , su (t) =  hn ∗u r  F P0u xu + n uur yur

(4.4)

and

respectively, where weight vectors wuD = h†n ∗u D /hn ∗u D  F and wur = h†n ∗u r /hn ∗u r  F , (·)† denotes conjugate transpose, and n uuD and n uur are AWGN with zero mean and variance of N0 . In the second phase, TAS/MRC is applied between each relay and the destination D. The index ofthe optimal transmit antenna at relay r , n r∗ , is determined as n r∗ =    arg max1≤nr ≤Nr hnr D F , where hnr D is the channel vector between the n r th antenna at relay r and the multiple antennas at the destination D with i.i.d. Rayleigh entries. Each relay linearly combines the overheard symbols during the first phase to a single encoded signal and then forwards it to the destination D in the second phase. The signal received at the destination D from relay r in time slot U + r is

36

4 Space-Time Network Coding with TAS/MRC

yr D (t) = hnr∗ D

U 

βr u



Pr u xu su (t) + nr D (t)

(4.5)

u=1

where hnr∗ D denotes the Rayleigh channel vector between the n r∗ th antenna at relay r and multiple antennas at the destination D, nr D (t) is the AWGN vector with zero mean and variance of N0 I ND , Pr u is the transmit power of xu at relay r , and βr u denotes the detection state of relay r on xu . If relay r correctly decodes xu , βr u = 1; otherwise, βr u = 0. The relay detection state can possibly be done by examining the included cyclic-redundancy-check digits or the received SNR levels. Here, we assume that the destination D knows the detection states of the relays, which can be obtained through the indicators sent by the relays. By combining the received signal replicas from multiple antennas with MRC and applying matched-filtering, the received signal at the destination D from user u  through relay r is U         yruD = wr D yr D (t) , su  (t) =  hnr∗ D  F βr u Pr u xu ρuu  + n ruD

(4.6)

u=1 

where weight vector wr D = h†nr∗ D /hnr∗ D  F , and n ruD is AWGN with zero mean and variance of N0 . Rewriting (4.6) into matrix form yields   ˜yr D =  hnr∗ D  F RPr x + n˜ r D

(4.7)

where ⎡

1 ρ21 ⎢ ρ12 1 ⎢ R=⎢ . .. ⎣ .. . ρ1U ρ2U

··· ··· .. .

⎤ ρU 1 ρU 2 ⎥ ⎥ .. ⎥ . ⎦

··· 1

T    √ √ √ with ˜yr D = yr1D , yr2D , · · · , yrUD , Pr = diag βr 1 Pr 1 , βr 2 Pr 2 , . . . , βrU PrU , T  x = [x1 , x2 , . . . , xU ]T , n˜ r D = n r1D , n r2D , . . . , n rUD is the AWGN vector with zero mean and variance matrix of N0 R, and [·]T denotes transpose. Here, the correlation amongst the entries of n˜ r D is from the matched-filtering operation. Assuming that R is invertible with the inverse matrix R−1 [14], the soft symbol of xu from relay r is derived by multiplying the both sides of (4.7) with R−1 , which yields    y˜ruD =  hnr∗ D  F βr u Pr u xu + n˜ ruD

(4.8)

where n˜ ruD is the AWGN with zero mean and variance of N0 θu with θu being the uth diagonal element of matrix R−1 associated with symbol xu .

4.2 System Model

37

By combining the information on xu from user u and R relays with MRC, the instantaneous end-to-end SNR of xu at the destination D is written as ϒu = ϒuD +

R 

βr u ϒr D

(4.9)

r =1

  2 2 where ϒuD =  hn ∗u D  F P0u /N0 and ϒr D =  hnr∗ D  F Pr u / (N0 θu ). Similarly, the  2 instantaneous SNR of xu at relay r can be given by ϒur =  hn ∗u r  F P0u /N0 . By defining the active relay set for user u as Du = {r : βr u = 1, r = 1, 2, . . . , R}, (4.9) is rewritten as  ϒr D . (4.10) ϒu = ϒuD + r ∈Du

When all the relays fail to decode the symbol xu , i.e., Du = φ, it is obvious that the STNC with TAS/MRC reduces to the conventional single hop TAS/MRC MIMO scheme. We note that the optimal transmit antenna for the destination D corresponds to a random transmit antenna for relay r , which is due to the fact that the optimal antenna at user u is entirely determined by the channel state between user u and the destination D and that the channels from user u to the destination D and to relay r are mutually independent. Therefore, the output SNR at r after MRC combing, ϒur , is the sum of the SNR at each antenna of r . As such, the probability density function (PDF) of ϒur is the convolution of exponential PDFs, and is given by [15] γ

f ϒur (γ ) =

γ Nr −1 e− γ¯ur  (Nr ) γ¯urNr

(4.11)

  where γ¯ur = E h ri ju 2F P0u /N0 is the average per-antenna SNR between user u and relay r , and E[x] denotes the expectation of x. Since hn ∗u D is the channel vector between the optimal transmit antenna n ∗u at user u and ND receive antennas at the destination D, the PDF of ϒuD is f ϒuD (γ ) =

   Nu −1  Nu − 1 g e−ξ0 γ γ φg +ND −1 Nu  ×  (ND ) g =0 γ¯uD g0 (−1)g0 γ¯uD

(4.12)

0

  2 where γ¯uD = E h iDu  j F P0u /N0 is the average per-antenna SNR between user u and the destination D, and g , φg , and ξ0 are defined in Appendix 1. Proof The proof is presented in Appendix 1.



Following the similar procedures of deriving the PDF of ϒuD , we can derive the PDF of ϒr D as

38

4 Space-Time Network Coding with TAS/MRC

f ϒr D (γ ) =

   N r −1   Nr − 1 r e−ξr,0 γ γ φr +ND −1 Nr ×  (ND ) h =0 γ¯r D h r,0 (−1)hr,0 γ¯r D

(4.13)

r,0

 ND −1 hr,i−1  hr,i−1  1 hr,i −hr,i+1   ND −1 , φr = i=1 where r = i=1 h r,i , h r,ND = 0, h r,i =0 h r,i i!     2 ξr,0 = h r,0 + 1 /γ¯r D , and γ¯r D = E h iDrj  F Pr u /(N0 θu ) is the average per-antenna SNR between relay r and the destination D. In (4.12) and (4.13), the MIMO links between the same transmitter/receiver pair are assumed i.i.d. Rayleigh distributed. If they are i.n.i.d. Rayleigh distributed, we have f ϒμD (γ ) =

Nμ  j=1

αj

ND 

βi j e

−

γ Dμ γ¯i j

i=1

 m

m0

1

Nμ 

×



j  =1, j  = j

−α j 

ND 

βi  j  γ¯iDμ  j e

−

γ Dμ γ¯   i j

m j 

i  =1

(4.14) where μ ∈ {u, r }, m is the set of nonnegative integers {m 0 , m 1 , . . . , m j−1 ,  Nμ m j+1 , . . . , m Nμ } such that j  =0, j  = j m j  = Nμ − 1 with 0 ≤ m 0 ≤ Nμ − 1 and Dμ m j  ∈ {0, 1}, 1 ≤ j  ≤ Nμ , and α j , βi j , and γ¯i j are defined in Appendix 2. Proof The proof is presented in Appendix 2.



The PDF of ϒμD given by (4.14) is the summation of exponential functions. Therefore, (4.14) is in the same form as (4.12) and (4.13). As such, it is easy to generate our theoretical analysis in the following sections to the i.n.i.d. Rayleigh fading case.

4.3 Performance with Perfect Feedback In this section, we analyze the performance of STNC in the cooperative network without feedback delays. We first present the closed-form expressions for the exact OP and SER. We then present the asymptotic expressions for the OP and SER to provide the useful insights into the behavior of STNC with TAS/MRC in the high SNR regime.

4.3.1 Exact Performance 4.3.1.1

Outage Probability

The OP is an important quality-of-service measure as it characterizes the probability that the instantaneous end-to-end SNR falls below a predetermined threshold ϒth . Here, we define ϒth = 2R − 1, where R is the target transmission rate of the users.

4.3 Performance with Perfect Feedback

39

From (4.10), the OP of STNC associated with xu can be given by Pout,u =

R 

Pr ( ϒu < ϒth | Du ) Pr (Du )

(4.15)

|Du |=0

  where |Du | is the size of Du . As there are ϑ = |DRu | possible choices of the active relay set Du for a given |Du |, (4.15) is rewritten as Pout,u =

ϑ R  

    Pr ϒu < ϒth | Du,v Pr Du,v

(4.16)

|Du |=0 v=1

where Du,v isthe vth  possible choice of |Du | relays from the R relays, and the probability Pr Du,v is       1 − Ps,ur Pr Du,v = Ps,ur r ∈Du,v

(4.17)

r ∈D / u,v

where Ps,ur is the SER of detecting xu at relay r . We will derive Ps,ur later in this subsection. In the the conditional probability  following, we proceed  to evaluate  Pr ϒu < ϒth | Du,v and the probability Pr Du,v . Without loss of generality, we now re-number the indices of the active relays belonging to Du,v as {1, 2, . . . , }, where  = |Du |. As such, the end-to-end SNR of xu conditioned on active relay set Du,v , ϒ u|Du,v , is ϒ u|Du,v = ϒuD +

 

ϒr D .

(4.18)

r =1

Given that ϒ u|Du,v is the summation of  + 1 independent largest order statistics of Gamma distributed variables, we present the PDF of ϒ u|Du,v in the following lemma. Lemma 4.1 The PDF of ϒ u|Du,v is derived as f ϒ u|Du,v (γ ) =

N u −1 N 1 −1  g0 =0 h 1,0 =0

N −1

···



h ,0 =0

⎞  φr +ND p−1 e−ξr,0 γ  cu γ p−1 e−ξ0 γ  c γ r ⎠ ⎝ +  ( p)  ( p) ⎛

φg +ND p=1

r =1

p=1

(4.19) where , cu , and cr are defined in Appendix 3. Proof The proof is presented in Appendix 3.



Integrating (4.19) from 0 to γ with the aid of the definite integral of the exponential function, which is given by [16, Eq. (3.351.1)], we obtain the corresponding cumulative distribution function (CDF) of ϒ u|Du,v , as shown in (4.20).

40

4 Space-Time Network Coding with TAS/MRC

Fϒ u|Du,v (γ ) =

N −1

N u −1 N 1 −1  g0 =0 h 1,0 =0

⎛

×⎝

φg +ND



···

 cu

γ 0

p=1

=1− ⎛ ×⎝



h ,0 =0

N u −1 N 1 −1 

p−1 −ξ0 x

x e  ( p)

dx

r =1

p=1

γ 0

p−1 −ξr,0 x

x e  ( p)

⎞ dx

⎠

N −1

···

g0 =0 h 1,0 =0





h ,0 =0

φg +ND p−1

  cu γ k e−ξ0 γ p−k

p=1 k=0

+

 φr +ND   cr

k!ξ0

+

 φr +ND p−1    cr γ k e−ξr,0 γ p−k

r =1

p=1 k=0

k!ξr,0

⎞ ⎠.

(4.20)

  Based on (4.20), it is easy to derive the conditional probability Pr ϒu < ϒth | Du,v as   Pr ϒu < ϒth | Du,v = Fϒ u|Du,v (ϒth )

(4.21)

where Fϒ u|Du,v (ϒth ) is the CDF of ϒ u|Du,v evaluated at γ = ϒth .   We now derive the SER Ps,ur to calculate the Pr Du,v according to (4.17). The closed-form expression for SER could be given directly in terms of the CDF of the received SNR, F (γ ), as [17] a Ps = 2

#

b π

$∞ 0

F (γ ) −bγ √ e dγ , γ

(4.22)

where the parameters a and b are up to a specific used modulation scheme, which encompasses a variety of modulations such as binary phase-shift keying (BPSK) (a = b = 1), M-ary phase-shift keying 2, b = sin2 (π/M)), and M-ary quadrature √ (a = %√ M, b = 3/(2(M − 1))) [15]. Integrating amplitude modulation (a = 4( M − 1) (4.11) with the aid of [16, Eq. (3.351.1)] and then substituting the outcome into (4.22) with the aid of identity [16, Eq. (3.371)] yield Ps,ur

√ Nr −1  −i− 21 a b  a (2i − 1)!! 1 = − +b . 2 2 i=0 i!2i γ¯uri γ¯ur

(4.23)

Substituting (4.23) into (4.17), the probability of the active relay set Du,v is obtained. Therefore, we can derive the exact closed-form expression for the OP of STNC with TAS/MRC by substituting (4.17) and (4.21) into (4.16).

4.3 Performance with Perfect Feedback

4.3.1.2

41

Symbol Error Rate

Similar to (4.16), the SER of STNC associated with xu is given by Ps,u =

ϑ R  

  P s,u|Du,v Pr Du,v

(4.24)

|Du |=0 v=1

where P s,u|Du,v denotes the SER of detecting xu at the destination D conditioned on active relay set Du,v . Substituting the CDF of end-to-end SNR associated with xu , (4.20), into (4.22) with the aid of identity [16, Eq. (3.371)], we derive the exact SER of xu conditioned on Du,v , as shown in (4.25). P s,u|Du,v

√ Nu −1 N1 −1 N −1   a a b  = − ···  2 2 g =0 h =0 h ,0 =0 0 1,0 ⎛ ⎞ φg +ND p−1  φr +ND p−1 k− p   cu (2k − 1)!!ξ k− p    cr (2k − 1)!!ξr,0 0 ⎠ ×⎝ + 1  k+ 21 . k (ξ + b)k+ 2 k ξ k!2 0 k!2 + b p=1 k=0 r =1 p=1 k=0 r,0 (4.25)

Substituting (4.17) and (4.25) into (4.24) yields the exact closed-form SER expression for STNC with TAS/MRC. The SER expression in (4.24) is valid for a variety of modulations and applies to arbitrary numbers of users and relays and arbitrary antenna numbers at the nodes. It is noticed that our result encompasses the SER expression for single-antenna STNC in [8] as a special case.

4.3.2 Asymptotic Performance In this subsection, we derive the asymptotic OP and SER expressions in the high SNR regime to characterize the behavior of STNC with TAS/MRC.

4.3.2.1

Outage Probability

In the high SNR regime, the probability that relay r decodes the symbol xu correctly approaches one, i.e., lim βr u = 1. In such a scenario, γ¯ur →∞

ϒu∞

=

∞ ϒuD

+

R  r =1

ϒr∞D .

(4.26)

42

4 Space-Time Network Coding with TAS/MRC

In order to obtain the CDF of ϒu∞ , the first order expansions of the PDFs of ϒuD and ϒr D are needed. Applying the Taylor series expansion of the exponential function in ∞ (γ ) and f ∞ (γ ) (4.12) and (4.13) and retaining the first order term, we obtain f ϒuD ϒr D as ∞ (γ ) ≈ f ϒuD

Nu γ ND Nu −1 ND Nu  (ND ) ( (ND + 1)) Nu −1 γ¯uD

(4.27)

and f ϒr∞D (γ ) ≈

Nr γ ND Nr −1  (ND ) ( (ND + 1)) Nr −1 γ¯rNDD Nr

(4.28)

respectively. Performing the Laplace transforms of (4.27) and (4.28) with the aid of ∞ and [16, Eq. (3.351.3)] to calculate the moment generating functions (MGFs) of ϒuD ∞ ∞ ϒr D and multiplying the MGFs together, we have the MGF of ϒu as Mϒu∞ (s) ≈ (s γ¯uD )−G d

(4.29)

where  Nu  (ND Nu ) Nr  (ND Nr ) κrND Nr × Nu −1  (ND ) ( (ND + 1))  (ND ) ( (ND + 1)) Nr −1 r =1 R

=

   with κr = γ¯uD /γ¯r D and G d = Nu + rR=1 Nr ND . Performing the inverse Laplace transform of (4.29) and integrating the outcome produce the CDF of ϒu∞ as Fϒu∞ (γ ) ≈

 Gd  (G d + 1) γ¯uD

γ Gd .

(4.30)

By evaluating Fϒu∞ (γ ) at γ = ϒth , the OP of the end-to-end SNR associated with xu in the high SNR regime is given by ∞ = Fϒu∞ (ϒth ) . Pout,u

4.3.2.2

(4.31)

Symbol Error Rate

We now derive the asymptotic SER based on Fϒu∞ (γ ). Substituting (4.30) into (4.22) and calculating the resultant integral, we derive the asymptotic SER as ∞ ≈ (G a γ¯uD )−G d Ps,u

(4.32)

4.3 Performance with Perfect Feedback

43

− G1  d a(2G d −1)!! where the array gain G a = b (G . Based on (4.32), we demonstrate G d +1 +1)2 d that STNC with  TAS/MRC achieves the full diversity order of G d =  R Nu + r =1 Nr ND in the cooperative MIMO network and the system performance is significantly improved through integrating multiple antennas in the nodes. The contributions of the u–D and r –D links to the diversity order are Nu ND and u (ND Nu ) and Nr ND respectively, and their contributions to the array gain are (N N)((N +1)) Nu −1 D

N N

D

Nr (ND Nr )κr D r (ND )((ND +1)) Nr −1

respectively. In the special case where all the nodes are equipped with a single antenna with Nu = Nr = ND = 1, the diversity order reduces to R + 1, which agrees with the result given in [8, 18].

4.4 Performance with Delayed Feedback In the presence of perfect feedback, the optimal transmit antenna is selected based on accurate channel state information (CSI). However, the feedback processes usually lead to delay, which results in that the transmit antenna selection (TAS) is performed based on outdated CSI. In this section, we examine the detrimental impact of delayed feedback on the performance of STNC with TAS/MRC.

4.4.1 Exact Performance Due to the delay between the instants of channel estimation and data transmission, we assume that the transmit antennas of user u and relay r are selected based on the outdated CSI with τu and τr time delays, respectively.  Dμ Dμ  To model the relationship between h i j (t) and h i j t − τμ , μ ∈ {u, r }, we employ the time-varying channel feedback   error model to express the channel coefDμ Dμ  ficient as h i j (t) = ρμ h i j t − τμ + 1 − |ρμ |2 eμ (t) [19, 20], where eμ (t) ∼   Dμ −α and ρμ is the normalized correlation coefficient between h i j (t) and CN 0, dμD     t − τμ . For Clarke’s fading spectrum, ρμ = J0 2π f μ τμ , where f μ is the h iDμ j Doppler frequency and J0 (·) is the zeroth-order Bessel function of the first kind [16, Eq. (8.402)]. Define ϒ˜ μD , μ ∈ {u, r }, as the outdated SNR, and the following lemma gives the PDF of ϒ˜ μD . Lemma 4.2 The PDFs of ϒ˜ uD and ϒ˜ r D are f ϒ˜ uD (γ˜ ) =

φg N u −1  g0 =0 k=0

and

˜ 0,k γ˜ ND +k−1 −ξ˜ γ˜  e 0  (ND + k)

(4.33)

44

4 Space-Time Network Coding with TAS/MRC

f ϒ˜ r D (γ˜ ) =

φr N r −1   h r,0

˜ r0 ,kr γ˜ ND +kr −1 −ξ˜ γ˜  e r,0  (ND + kr ) =0 k =0

(4.34)

r

˜ 0,k ,  ˜ r0 ,kr , ξ˜0 , and ξ˜r,0 are defined in Appendix 4. respectively, where  

Proof The proof is presented in Appendix 4.

In the presence of perfect feedback, i.e., ρμ = 1, ϒ˜ μD = ϒμD . In such a case, it is easy to demonstrate that (4.33) and (4.34) are equal to (4.12) and (4.13), respectively, by setting k = φg in (4.33) and kr = φr in (4.34). For the case of fully delayed feedback, i.e., ρμ = 0, ϒ˜ μD reduces to a gamma distributed variable with shape parameter ND and scale parameter γ¯μD , and we demonstrate that (4.33) and (4.34) reduce to the same form as that in (4.11) by setting k = 0 and kr = 0 in (4.33) and (4.34), respectively.

4.4.1.1

Outage Probability

Following the procedure outlined in Sect. 4.3.1.1, the end-to-end SNR of xu conditioned on active relay set Du,v with delayed feedback is ϒ˜ u|Du,v = ϒ˜ uD +

 

ϒ˜ r D .

(4.35)

r =1

Based on the PDFs of ϒ˜ uD and ϒ˜ r D , the PDF of ϒ˜ u|Du,v is presented in the following lemma. Lemma 4.3 The PDF of ϒ˜ u|Du,v is derived as f ϒ˜ u|Du,v (γ˜ ) =

φg N1 −1 N u −1  

φ1 

g0 =0 k=0 h 1,0 =0 k1 =0

⎛

×⎝

N D +k p=1

N −1

···

φ  

˜ 

h ,0 =0 k =0

⎞  ND +kr ˜   c˜r γ˜ p−1 e−ξ˜r,0 γ˜ c˜u γ˜ p−1 e−ξ0 γ˜ ⎠ +  ( p)  ( p) r =1 p=1

where c˜u = (−1) ND +k− p

  ˜ u r =1 



ND +kr +ir −1 ir



  ND +kr +ir ξ˜r,0 − ξ˜0

(4.36)

4.4 Performance with Delayed Feedback

c˜r =(−1) ND +kr − p

 ˜r 



45

ND +k+i 0 −1 i0



 



ND +kr  +ir  −1 ir 



  ND +k+i0 ×   ND +kr  +ir  r  =1,r  =r ξ˜r  ,0 − ξ˜r,0 ξ˜0 − ξ˜r,0

   ˜ u is the set of nonnegative integers i 1 , i 2 , . . . , i  , such that j=1 i j = ND + with    ˜ r is the set of nonnegative integers i 0 , i 1 , . . . , ir −1 , ir +1 , . . . , i  , such that k − p,    ˜ ˜ ˜ r =1 r0 ,kr . j=0, j=r i j = ND + kr − p, and  = 0,k Proof Following the similar procedures outlined in Appendix 3, we derive the MGF of ϒ˜ u|Du,v by multiplying the MGFs of ϒ˜ uD and ϒ˜ r D . Here, the MGFs of ϒ˜ uD and ϒ˜ r D can be obtained based on the PDFs of ϒ˜ uD and ϒ˜ r D . Then performing the inverse Laplace transform with some algebraic manipulations, we derive the PDF of ϒ˜ u|Du,v as (4.36).  Integrating (4.36) with the aid of [16, Eq. (3.351.1)], we obtain the CDF of ϒ˜ u|Du,v as Fϒ˜ u|Du,v (γ˜ ) =1 −

φg N1 −1 N u −1  

φ1 

N −1

···

g0 =0 k=0 h 1,0 =0 k1 =0

⎛ ×⎝

p−1 N D +k  p=1 q=0

φ  

˜ 

h ,0 =0 k =0

⎞  ND +kr p−1 ˜  c˜r γ˜ q e−ξ˜r,0 γ˜ c˜u γ˜ q e−ξ0 γ˜   ⎠. + p−q p−q q!ξ˜0 q!ξ˜r,0 r =1

(4.37)

p=1 q=0

In the presence of perfect feedback, i.e., ρu = ρr = 1 and ϒ˜ u = ϒu , it is easy to demonstrate that (4.37) is equal to (4.20) by setting k = φg and kr = φr in (4.37). As we mentioned previously that the optimal transmit antenna of user u for the destination D corresponds to a random transmit antenna for relay r in the first phase, the delayed feedback has no impact on the SNR of xu at relay r . Hence, substituting the conditional CDF Pr(ϒ˜ u < ϒth |Du,v ) = Fϒ˜ u|Du,v (ϒth ) and (4.17) into (4.16), we can derive the exact closed-form expression for the OP of STNC with TAS/MRC in the presence of delayed feedback.

4.4.1.2

Symbol Error Rate

In the presence of delayed feedback, the SER of STNC associated with xu is given by P˜s,u =

ϑ R  

  P˜ s,u|Du,v Pr Du,v

(4.38)

|Du |=0 v=1

where P˜ s,u|Du,v denotes the SER of detecting xu at the destination D conditioned on active relay set Du,v in the presence of delayed feedback. Substituting (4.37) into

46

4 Space-Time Network Coding with TAS/MRC

(4.22) and applying the identity [16, Eq. (3.371)] to solve the resultant integral, we derive the exact closed-form expression for P˜ s,u|Du,v as P˜ s,u|Du,v

√ Nu −1 φg N1 −1 φ1 N −1 φ      a a b  ˜  = − ··· 2 2 g =0 k=0 h =0 k =0 h ,0 =0 k =0 0 1,0 1 ⎞ ⎛  ND +kr p−1 p−1 q− p N q− p D +k  ˜    ˜ c˜r (2q − 1)!!ξr,0 ⎟ c˜u (2q − 1)!!ξ0 ⎜ ×⎝   q+ 21 + q+ 21 ⎠ . p=1 q=0 q!2q ξ˜ + b r =1 p=1 q=0 q!2q ξ˜ + b 0 r,0 (4.39)

For the extreme case of perfect feedback with ρu = ρr = 1, it is easy to verify that (4.39) is equivalent to (4.25) by setting k = φg and kr = φr in (4.39). Substituting (4.17) and (4.39) into (4.38) yields the exact closed-form SER expression P˜s,u with delayed feedback.

4.4.2 Asymptotic Performance In this subsection, we derive the asymptotic OP and SER expressions in the high SNR regime to quantify the detrimental impact of delayed feedback and offer useful insights into the behavior of STNC with TAS/MRC.

4.4.2.1

Outage Probability

In the presence of delayed feedback, the end-to-end SNR of xu in the high SNR regime is given by ∞ ϒ˜ u∞ = ϒ˜ uD +

R 

ϒ˜ r∞D

(4.40)

r =1 ∞ and ϒ˜ r∞D are the received SNRs of xu in the high SNR regime through the where ϒ˜ uD u–D and r –D links respectively, whose asymptotic PDFs can be derived by applying the Taylor series expansion of the exponential function in (4.33) and (4.34) and discarding the high order items as follows

f ϒ˜ uD ˜) ≈ ∞ (γ

N u −1  g0 =0

Nu − 1 g0



  φ Nu g (−1)g0  ND + φg 1 − ρu2 g ND −1 ×     N +φ ND γ˜ ( (ND ))2 1 + 1 − ρu2 g0 D g γ¯uD (4.41)

4.4 Performance with Delayed Feedback

f ϒ˜ r∞D (γ˜ ) ≈

N r −1  h r,0 =0



Nr − 1 h r,0



47

φ  Nr r (−1)hr,0  (ND + φr ) 1 − ρr2 r ND −1 × γ˜ .     N +φ ( (ND ))2 1 + 1 − ρr2 h r,0 D r γ¯rNDD (4.42)

Performing the Laplace transforms of (4.41) and (4.42) with the aid of [16, Eq. (3.351.3)] and multiplying the resulted expressions together, we have the MGF of ϒ˜ u∞ as ˜ γ¯uD )−(R+1)ND Mϒ˜ u∞ (s) ≈ (s

(4.43)

where   φ g  Nu − 1 Nu g (−1)g0  ND + φg 1 − ρu2 ˜ =    N +φ   g0  (ND ) 1 + 1 − ρu2 g0 D g g0 =0 φ   R N r −1    Nr r (−1)hr,0  (ND + φr ) 1 − ρr2 r κrND Nr − 1 × × .   N +φ   h r,0  (ND ) 1 + 1 − ρr2 h r,0 D r r =1 h r,0 =0 N u −1 

Performing the inverse Laplace transform of (4.43) yields the PDF of ϒ˜ u∞ , from which we derive the asymptotic CDF of ϒ˜ u∞ in the high SNR regime as ˜  γ˜ (R+1)ND . (R+1)ND  ((R + 1) ND + 1) γ¯uD

Fϒ˜ u∞ (γ˜ ) ≈

(4.44)

Through evaluating Fϒ˜ u∞ (γ˜ ) at γ˜ = ϒth , the asymptotic OP of the end-to-end SNR associated with xu under delayed feedback conditions is given by ∞ = Fϒ˜ u∞ (ϒth ) . P˜out,u

4.4.2.2

(4.45)

Symbol Error Rate

Based on Fϒ˜ u∞ (γ˜ ), we now derive the asymptotic SER of STNC with TAS/MRC in the high SNR regime. Substituting (4.44) into (4.22) and calculating the resultant integral, we derive the asymptotic SER as −G˜ d  ∞ ≈ G˜ a γ¯uD P˜s,u where the diversity − ˜1  Gd ˜ (2G˜ d −1)!! a b .  (G˜ d +1)2G˜ d +1

order

G˜ d = (R + 1) ND

(4.46) and

array

gain

G˜ a =

48

4 Space-Time Network Coding with TAS/MRC

Comparing (4.46) with (4.32), it is evident that delayed feedback has a severely detrimental effect  on the SER and degrades the diversity order from  R Nu + r =1 Nr ND to (R + 1) ND , which indicates that the contribution of the multiple antennas at the transmit end to the diversity vanishes due to the delayed feedback. In the extreme case of fully delayed feedback, the optimal transmit antenna selected based on the fully outdated CSI actually corresponds to a random transmit antenna. The significant impact of delayed feedback implies that a high feedback rate may be required in practice in order to attain the full benefits of TAS.

4.5 Numerical Results In this section, numerical and simulation results are presented to show the validity of our analysis and the impacts of the network parameters on the OP and SER of STNC with TAS/MRC. We assume that the coordinations of the destination D, relay r , and user u are (0, 0), ((d + r d) cos(r ψ), (d + r d) sin(r ψ)), and (cos(uψ  ), sin(uψ  )) respectively, which implies that the distances from the destination D to relay r and to user u are d + r d and 1 respectively. In the simulations, d = 0.4, d = 0.1, and ψ = ψ  = π/18. The cross correlations between different spread codes are set to be zero, and the value of the path-loss exponent α is 3.5 [21]. We also assume equal transmit power at each node. Figures 4.1 and 4.2 plot the OP and SER of STNC versus transmit SNR Pu /N0 for different antenna configurations in the presence of perfect feedback, respectively, where Pu = P0u + rR=1 Pr u . The arrows in the figures are used to link each curve with corresponding network parameters. It is observed that the simulation results perfectly match with exact theoretical results and exact curves approach asymptotic

Fig. 4.1 Outage probability of STNC for a threshold ϒth = 10 dB in the presence of perfect feedback (ND = 1)

0

10

Exact OP Simulation Asymptotic OP

−1

Outage Probability

10

−2

10

−3

10

−4

10

R=1, Nu=2, Nr=1 (Gd=3) R=1, N =1, N =2 (G =3) u r d R=2, N =2, N =1 (G =4) u r d R=3, N =2, N =1 (G =5) u r d R=2, Nu=2, Nr=2 (Gd=6)

−5

10

−6

10

0

5

15

10

P /N (dB) u

0

20

25

4.5 Numerical Results

49

Fig. 4.2 Symbol error rate of BPSK modulation of STNC in the presence of perfect feedback (ND = 1)

−1

10

Exact SER Simulation Asymptotic SER

−2

Symbol Error Rate

10

−3

10

−4

10

R=1, N =2, N =1 (G =3) u r d R=1, N =1, N =2 (G =3) u r d R=2, Nu=2, Nr=1 (Gd=4) R=3, N =2, N =1 (G =5) u r d R=2, N =2, N =2 (G =6)

−5

10

−6

10

u

r

d

−7

10

−5

0

5

10

15

P /N (dB) u

Fig. 4.3 Outage probability of STNC for a threshold ϒth = 10 dB in the presence of delayed feedback (R = 2, Nu = Nr = 2, ND = 1)

0

0

10

Exact OP Simulation Asymptotic OP

−1

Outage Probability

10

−2

10

−3

10

−4

10

ρu=ρr=0 (Gd=3) ρ =ρ =0.5 (G =3) u r d ρ =ρ =0.9 (G =3) u r d ρu=ρr=1 (Gd=6)

−5

10

−6

10

0

5

10

15

20

25

P /N (dB) u

0

curves in the high SNR regime, which validate the accuracy of our theoretical analysis in Sect. 4.3. In thepresence of perfect  feedback, STNC with TAS/MRC provides full  diversity order of Nu + rR=1 Nr ND , as indicated by (4.32). Notably, the diversity order increases when the antenna number and relay number increase, which in turns leads to a pronounced SNR gain. For the single relay case in Figs. 4.1 and 4.2, it is shown that the antenna configuration of Nu = 1 and Nr = 2 exhibits a better performance than that of Nu = 2 and Nr = 1, although both of them have the same diversity order. This is due to the fact that the relay is involved in both transmission phases.

50

4 Space-Time Network Coding with TAS/MRC

Fig. 4.4 Symbol error rate of BPSK modulation of STNC in the presence of delayed feedback (R = 2, Nu = Nr = 2, ND = 1)

−1

10

Exact SER Simulation Asymptotic SER

−2

Symbol Error Rate

10

−3

10

−4

10

−5

10

−6

10

ρ =ρ =0 (G =3) u r d ρ =ρ =0.5 (G =3) u r d ρ =ρ =0.9 (G =3) u r d ρu=ρr=1 (Gd=6)

−7

10

−5

0

5

10

15

20

P /N (dB) u

0

10

−1

10

16QAM

−2

10

Symbol Error Rate

Fig. 4.5 Symbol error rates of QPSK and 16QAM modulations of STNC in the presence of delayed feedback (R = 2, Nu = Nr = ND = 2)

0

−3

10

QPSK

−4

10

−5

10

ρ =ρ =0 u

r

ρu=ρr=0.5 −6

10

ρu=ρr=0.9 ρ =ρ =1

−7

10 −10

u

−5

r

0

5

10

15

20

P /N (dB) u

0

Figure 4.3 plots the OP of STNC with different delay correlation coefficients versus transmit SNR Pu /N0 . Figures 4.4 and 4.5 plot the SERs of STNC with different modulations and different delay correlation coefficients versus transmit SNR Pu /N0 . The simulation results are marked by “•” in Fig. 4.5. The precise agreement between the theoretical and the simulation results in Figs. 4.3 and 4.4 verifies our theoretical analysis in Sect. 4.4. In Fig. 4.5, the theoretical and simulation curves are consistent in the medium and high SNR regime; whereas there exists a very minor gap between the theoretical and the simulation curves in the low SNR regime, especially for 16QAM. This is due to the fact that the closed-form SER expression (4.22) is an approximation for higher degree of modulation, e.g., QPSK and 16QAM. We find that the performance of the STNC improves as the delay correlation coefficients ρu and ρr increase. It is shown that, in the presence of delayed feedback, the asymptotic

4.5 Numerical Results

51

curves in Figs. 4.3 and 4.4 keep the same slope of −3, which means that the delayed feedback reduces the full diversity order to (R + 1) ND regardless of the values of ρu and ρr . This indicates that the diversity advantage from transmit end vanishes and only the one from receive end remains due to the delayed feedback. In Fig. 4.5, it is shown that fully delayed feedback incurs an SNR loss of approximately 3 dB compared to perfect feedback at an SER of 10−6 .

4.6 Conclusions In this chapter, we have analyzed the performance of STNC in cooperative MIMO network in terms of OP and SER, where multi-antenna diversity is guaranteed via TAS/MRC. We have presented closed-form expressions of the exact and asymptotic OP and SER for both perfect and delayed feedback. In the presence of perfect feedback, confirmed that STNC with TAS/MRC preserves the full diversity order   we have R of Nu + r =1 Nr ND . In the presence of delayed feedback, we have quantified the detrimental effect of delayed feedback on the OP and SER of STNC. It is shown that the diversity advantage from the transmit end vanishes and the diversity order is degraded to (R + 1) ND due to delayed feedback.

Appendix 1 Proof of Eq. (4.12) We denote γu as the instantaneous SNR of the channel from a random transmit antenna at user u to the ND antennas at the destination D. The PDF and CDF of γu , f γu (γ ) and Fγu (γ ), are given by [15] f γu (γ ) =

γ ND −1 e

− γ¯ γ

uD

ND  (ND ) γ¯uD

(4.47)

and Fγu (γ ) = 1 − e

− γ¯ γ

N D −1

uD

i=0

respectively. Based on (4.47) and (4.48), we have

γi i i!γ¯uD

(4.48)

52

4 Space-Time Network Coding with TAS/MRC

f ϒuD (γ ) = =

Nu γ ND −1 e

− γ¯ γ



uD

1−e

ND  (ND ) γ¯uD

Nu γ

γ u −1  ND −1 − γ¯uD N

e

ND  (ND ) γ¯uD

g0 =0

N D −1

− γ¯ γ

γi i i!γ¯uD

uD

i=0

 Nu −1

 N −1 g0  D  g0 γ Nu − 1 γi g0 − γ¯uD × . (−1) e i g0 i!γ¯uD i=0 (4.49)

According to [22, Eq. (9)], we have  N −1 g0   D  γ φg γi = g i γ¯uD i!γ¯uD i=0

(4.50)

 ND −1 gi−1  gi−1  1 gi −gi+1   ND −1 , φg = i=1 gi , and g ND = 0. Subwhere g = i=1 gi =0 gi i! stituting (4.50) into (4.49) yields (4.12), where ξ0 = (g0 + 1) /γ¯uD .

Appendix 2 Proof of Eq. (4.14) Dμ

Using γi j , μ ∈ {u, r }, 1 ≤ i ≤ ND , and 1 ≤ j ≤ Nμ , to denote the i.n.i.d. distributed SNR between the jth antenna of transmitter μ and the ith antenna of destination D, we have 1

f γ Dμ (γ ) =

Dμ

γ¯i j

ij

e

−

γ Dμ γ¯i j

(4.51)

  Dμ Dμ Dμ where γ¯i j = E γi j . Based on (4.51), the MGF of γi j is Mγ Dμ (s) = ij

Dμ

1/(s γ¯i j + 1). If we MRC combine the ND signal replicas from the jth antenna Dμ of transmitter μ at the destination D, the total SNR, γμ , is the summation of γi j , 1 ≤ i ≤ ND . As such, the MGF of γμ is Mγμ (s) =

where α j =

 ND

ND 

1

i=1

s γ¯i j + 1

Dμ

1

Dμ . Expanding

i=1 γ¯ ij

= αj

ND 

 s+

i=1

 i=1 s +

−1

1

(4.52)

Dμ

γ¯i j

−1

 ND

1

in poles and residuals with

Dμ

γ¯i j

the aid of partial fraction decomposition [16, Eq. (2.102)] and performing inverse Laplace transform on (4.52), we obtain f γμ (γ ) = α j

ND ND   i=1 i  =1,i  =i



1 γ¯iDμ j

−

1 γ¯iDμ j

−1 e

−

γ Dμ γ¯i j

.

(4.53)

4.6 Conclusions

Defining βi j = 1 − αj

 ND i=1

53



 ND

i  =1,i  =i

Dμ

βi j γ¯i j e

f ϒμD (γ ) =

Nμ 

αj

j=1

γ − Dμ γ¯i j

ND 

1 Dμ

γ¯i  j

−

−1

, the CDF of γμ is given by Fγμ (γ ) =

1 Dμ

γ¯i j

. Hence, the PDF of ϒμD is given by

βi j e

−

γ Dμ γ¯i j

Nμ 

×

 1 − α j

j  =1, j  = j

i=1

ND 

Dμ βi  j  γ¯i  j  e

−

γ Dμ γ¯   i j

 . (4.54)

i  =1

Based on (4.54), (4.14) can be derived directly by expanding the product term.

Appendix 3 Proof of Eq. (4.14) Here, we first derive the MGF of ϒ u|Du,v , and then obtain the PDF of ϒ u|Du,v based on the MGF. From (4.12) and (4.13), the MGFs of ϒuD and ϒr D are obtained by performing the Laplace transform with the aid of the definite integral of the exponential function [16, Eq. (3.351.3)] as MϒuD (s) =

N u −1

0 (ξ0 + s)−(φg +ND )

(4.55)

 −(φr +ND ) r,0 ξr,0 + s

(4.56)

g0 =0

and Mϒr D (s) =

N r −1  h r,0 =0

respectively,

where

Nr (φr +ND ) r φ +N (ND )(−1)hr,0 γ¯r Dr D

0 =



Nu −1 g0



Nu  (φg +ND ) g (ND )(−1)

g0

φg +N γ¯uD D

,

and

r,0 =



Nr −1 h r,0



.

Since the MGF of the sum of multiple independent random variables is equal to the product of the MGFs of the random variables [15], the MGF of ϒ u|Du,v is Mϒ u|Du,v (s) =

N u −1 g0 =0

0 (ξ0 + s)−(φg +ND ) ×

  r =1

⎛ ⎝

N r −1 

⎞ −(φr +ND )  ⎠. r,0 ξr,0 + s

h r,0 =0

(4.57) Expanding (4.57) in poles and residuals with the aid of partial fraction decomposition [16, Eq. (2.102)] and Faa di Bruno Formula and performing inverse Laplace transform yield (4.19), where

54

4 Space-Time Network Coding with TAS/MRC

cu = (−1)φg +ND − p

  u r =1

cr =(−1)φr +ND − p

 r



φg +ND +i 0 −1 i0



φr +ND +ir −1 ir



 φ +N +i ξr,0 − ξ0 r D r



 



φr  +ND +ir  −1 ir 



 φ +N +i ×  φ  +N +i  ξ0 − ξr,0 g D 0 r  =1,r  =r ξr  ,0 − ξr,0 r D r

   with  0 r =1 r,0 , u is the set of nonnegative integers i 1 , i 2 , . . . , i  , such =  that r is the set of nonnegative integers j=1 i j = φg + ND −p, and    i 0 , i 1 , . . . , ir −1 , ir +1 , . . . , i  , such that j=0, j=r i j = φr + ND − p.

Appendix 4 Proof of Lemma 4.2 Denoting γ˜u as the outdated SNR relative to the original SNR γu , the joint PDF of γ˜u and γu is [23] γ˜ +γ    −    ND2−1  2 2 ρu2 γ˜ γ e (1−ρu )γ¯uD 1 ND +1 γ˜ γ  × I ND −1    f γ˜u ,γu (γ˜ , γ ) = γ¯uD ρu2  (ND ) 1 − ρu2 1 − ρu2 γ¯uD (4.58)

where In (·) stands for the nth-order modified Bessel function of the first kind [16, Eq. (8.406.1)]. According to the order statistics, the time-delayed SNR after TAS, ϒ˜ uD , is the induced order statistics of the original ordered SNR after TAS, ϒuD [24]. Therefore, the PDF of ϒ˜ uD is derived as $∞ f ϒ˜ uD (γ˜ ) = 0

f γ˜u ,γu (γ˜ , γ ) f ϒuD (γ ) dγ = Nu f γu (γ )

$∞

  N −1 f γ˜u ,γu (γ˜ , γ ) Fγu (γ ) u dγ .

0

(4.59) Substituting (4.58) and the CDF of γu , (4.48), into (4.59) and performing some algebraic manipulations with the aid of [16, Eq. (6.643.2)] yield    ND  − ND +φg  2 Nu g  ND + φg γ˜ 2 −1 1 f ϒ˜ (γ˜ ) = × + g0 uD   ND 1 − ρu2 2 g0 =0 ( (ND ))2 (−1)g0 ρu2 γ¯uD     ⎛  ⎞ −1 2 1+ 1−ρu2 g0 γ˜ −ρu2 γ˜ −  2   2  ρu2 γ˜ 1 − ρu2 ⎜ ⎟ 2 1−ρu γ¯uD 1+ 1−ρu g0     ⎠ ×e × M  ND N −1 ⎝ − 2 +φg , D2 2 γ¯uD 1 + 1 − ρu g0 N u −1 

Nu − 1 g0



(4.60)

4.6 Conclusions

55

where Ma,b (·) is the Whittaker function [16, Eq. (9.220.2)]. With the aid of the finite series expansion of Whittaker function [20], we derive the PDF of ϒ˜ uD given in (4.33), where ˜ 0,k = 



Nu − 1 g0



φg k



  φ −k  Nu g (−1)g0  ND + φg ρu2k 1 − ρu2 g ×    N +φ +k ND +k  1 + 1 − ρu2 g0 D g  (ND ) γ¯uD

and ξ˜0 =

g0 +1 . Following the similar procedures of deriving the PDF of (1+(1−ρu2 )g0 )γ¯uD ϒ˜ uD , we derive the PDF of ϒ˜ r D , as shown in (4.34), where

˜ r0 ,kr =  and ξ˜r,0 =



Nr − 1 h r,0 1+h r,0



(1+(1−ρr2 )hr,0 )γ¯r D

φr kr



 φ −k Nr r (−1)hr,0  (ND + φr ) ρr2kr 1 − ρr2 r r ×     N +φ +k  (ND ) γ¯rNDD +kr 1 + 1 − ρr2 h r,0 D r r

.

References 1. J.N. Laneman, D.N.C. Tse, G.W. Wornell, Cooperative diversity in wireless networks: efficient protocols and outage behavior. IEEE Trans. Inf. Theory 50(12), 3062–3080 (2004). 2. A. Nosratinia, T.E. Hunter, A. Hedayat, Cooperative communication in wireless networks. IEEE Commun. Mag. 42(10), 74–80 (2004). 3. Q. Li, R.Q. Hu, Y. Qian, G. Wu, Cooperative communications for wireless networks: techniques and applications in LTE-advanced systems. IEEE Wireless Commun. 19(2), 22–29 (2012). 4. J.N. Laneman, G.W. Wornell, Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks. IEEE Trans. Inf. Theory 49(10), 2415–2425 (2003). 5. M. Xiao, M. Skoglund, Multiple-user cooperative communications based on linear network coding. IEEE Trans. Commun. 58(12), 3345–3351 (2010). 6. H. Zhang, N.B. Mehta, A.F. Molisch, J. Zhang, H. Dai, Asynchronous interference mitigation in cooperative base station systems. IEEE Trans. Wireless Commun. 7(1), 155–165 (2008). 7. X. Li, T. Jiang, S. Cui, J. An, Q. Zhang, Cooperative communications based on rateless network coding in distributed MIMO systems. IEEE Wireless Commun. 17(3), 60–67 (2010). 8. H.-Q. Lai, K.J.R. Liu, Space-time network coding. IEEE Trans. Signal Process. 59(4), 1706– 1718 (2011). 9. K. Loa, C.-C. Wu, S.-T. Sheu, Y. Yuan, M. Chion, D. Huo, L. Xu, IMT-advanced relay standards. IEEE Commun. Mag. 48(8), 40–48 (2010). 10. C. Hoymann, W. Chen, J. Montojo, A. Golitschek, C. Koutsimanis, X. Shen, Relaying operation in 3GPP LTE: challenges and solutions. IEEE Commun. Mag. 50(2), 156–162 (2012). 11. Z. Chen, J. Yuan, B. Vucetic, Analysis of transmit antenna selection/maximal-ratio combining in Rayleigh fading channels. IEEE Trans. Veh. Technol. 54(4), 1312–1321 (2005). 12. J.P. Pena-Martin, J.M. Romero-Jerez, C. Tellez-Labao, Performance of TAS/MRC wireless systems under Hoyt fading channels. IEEE Trans. Wireless Commun. 12(7), 3350–3359 (2013). 13. M. Jankiraman, Space-Time Codes and MIMO Systems. Artech House, 2004. 14. R. K. Mallik, The uniform correlation matrix and its application to diversity. IEEE Trans. Wireless Commun. 6(5), 1619–1625 (2007). 15. M.K. Simon, M.-S. Alouini, Digital Communication over Fading Channels: A Unified Approach to Performance Analysis. Wiley, 2000.

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16. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products. 7th ed. Academic, 2007. 17. M.R. McKay, A.J. Grant, I.B. Collings, Performance analysis of MIMO-MRC in doublecorrelated Rayleigh environments. IEEE Trans. Commun. 55(3), 497–507 (2007). 18. K. Xiong, P. Fan, T. Li, K.B. Letaief, Outage probability of space-time network coding over Rayleigh fading channels. IEEE Trans. Veh. Technol. 63(4), 1965–1970 (2014). 19. S. Zhou, G.B. Giannakis, Adaptive modulation for multiantenna transmissions with channel mean feedback. IEEE Trans. Wireless Commun. 3(5), 1626–1636 (2004). 20. N. Yang, M. Elkashlan, P.L. Yeoh, J. Yuan, Multiuser MIMO relay networks in Nakagami-m fading channels. IEEE Trans. Commun. 60(11), 3298–3310 (2012). 21. D. Tse, P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2005. 22. S. Choi, Y.-C. Ko, Performance of selection MIMO systems with generalized selection criterion over Nakagami-m fading channels. IEICE Trans. Commun. E89-B(12), 3467–3470 (2006). 23. J.L. Vicario, C. Anton-Haro, Analytical assessment of multi-user vs. spatial diversity trade-offs with delayed channel state information. IEEE Commun. Lett. 10(8), 588–590 (2006). 24. H.A. David, Order Statistics, 2nd ed. Wiley, 1981.

Chapter 5

Space-Time Network Coding with Antenna Selection

In this chapter, we investigate the space-time network coding (STNC) with antenna selection (AS). Section 5.1 introduces the motivation of investigating the STNC with AS. Section 5.2 presents the system model. The effects of outdated channel state information (CSI) and spatial correlation on the performance of the STNC with AS are quantified in Sects. 5.3 and 5.4, respectively. Section 5.5 presents the simulation results, and Sect. 5.6 concludes this chapter.

5.1 Introduction As discussed in Chap. 4, integrating cooperation between different nodes usually requires perfect timing and/or frequency synchronization [1, 2], which may be difficult or impossible in practical multi-node systems as it is very challenging to align all the signals especially when the nodes are spatially distributed. Since multiple-input multiple-output (MIMO) technology can significantly increase communication reliability through the use of spatial diversity [3], we integrated transmit antenna selection and maximal-ratio combining (TAS/MRC) into STNC in Chap. 4. It is noticed that TAS/MRC requires multiple radio frequency chains, which inevitably increases the complexities and costs of the transceivers [4, 5]. Motivated by this, we integrate AS into STNC here to circumvent this drawback since AS preserves the diversity gain while significantly reducing hardware complexity [4]. With AS, the transceiver can be easily implemented with a single front-end and an analog switch. In this chapter, we consider STNC with AS in the cooperative MIMO network over flat Rayleigh fading channels, where U users communicate with a common destination D with the assistance of R decode-and-forward (DF) relays and all nodes are equipped with multiple antennas. We adopt AS over user-destination (u-D) and relaydestination (r -D) links, where the optimal transmit/receive antenna pair is selected to © Beijing Institute of Technology Press 2023 J. An et al., Sustainable Wireless Communications, https://doi.org/10.1007/978-981-19-0448-6_5

57

58

5 Space-Time Network Coding with Antenna Selection

provide the highest received signal-to-noise ratio (SNR) based on the CSI estimated at the receiver. For the user-relay (u-r ) link, only the receive antenna at the relay is selected to maximize the received SNR as the transmit antenna of the user is determined by the CSI of u-D link. In particular, the impacts on the performance of STNC with AS due to outdated CSI and spatial correlation are examined, respectively.

5.2 System Model Consider a cooperative MIMO network composed of U users sending information to a common destination D with the aid of R DF relays over flat Rayleigh fading channels. We use Hνμ to denote the channel coefficient matrix between transmitter νμ −α ), is the channel μ and receiver ν, and the ( j, i)th element of Hνμ , h ji ∼ CN(0, dνμ coefficient between the ith antenna of transmitter μ and the jth antenna of receiver ν. Here, (μ, ν) ∈ {(u, r ) , (u, D) , (r, D)}, and dνμ and α denote the distance between μ and ν and the path loss exponent, respectively. The transmission of STNC takes place over U + R time slots, which are partitioned into two consecutive phases [6]. We now outline the STNC with AS in the cooperative MIMO network as follows. In the first phase, U users take turns to broadcast their signals to the relays and the destination D in the first U time slots, i.e., user u broadcasts its signal in time slot u. The optimal transmit antenna amongst the Nu antennas at user u and the optimal receive antenna amongst the ND antennas at the destination D are selected to maximize the instantaneous received SNR at destination D. Therefore, the indices of the optimal transmit and receive in the sense of SNR are determined as  antennas   Du  ∗ ∗ {i , j }=arg max1≤i≤Nu ,1≤ j≤ND h ji  , where  ·  F denotes the Frobenius norm. F With optimal transmit/receive antenna pair {i ∗ , j ∗ }, the signals received at the destination D and relay r from user u are yDu = h Du j ∗i ∗



P0u xu + n Du

(5.1)

P0u xu + n r u

(5.2)

and yr u = h rku∗ i ∗



respectively, where P0u is the transmit power of xu , xu denotes the signal with unit energy transmitted from user u, n Du and n r u are the additive  white Gaussian  noise  (AWGN) with zero mean and variance of σ 2 , and h rku∗ i ∗ , h rku∗ i ∗  F = max h rkiu∗  F , 1≤k≤Nr

is the channel coefficient between the i ∗ th antenna at user u and optimal receive antenna at relay r . In the second phase, R relays combine the overheard signals from multiple users during the first phase to a single symbol by assigning a unique spreading code to each signal, and then take turns to transmit the combined symbol to the destination D in the last R time slots, i.e., relay r transmits its symbol in time slot U + r . The indices of

5.2 System Model

59

the optimal transmit antenna at relay r and optimal receive antenna at the destination    D in the sense of SNR are determined as {k  , j  }=arg max1≤k≤Nr ,1≤ j≤ND h Dr jk F , and the signal received at the destination D from relay r is yDr (t) =

h Dr j k

U 

βr u



Pr u xu su (t) + n Dr (t)

(5.3)

u=1

where βr u denotes the detection state of relay r on xu , Pr u is the transmit power of xu at relay r , su (t) denotes the spreading code assigned to xu , and n Dr (t) is AWGN with zero mean and variance of σ 2 . With spreading codes, the bandwidth requirement of STNC is TTcs times of that of traditional time division multiple access (TDMA) scheme, where Ts and Tc are the symbol and spreading code chip periods, respectively. If relay r correctly decodes xu , βr u = 1; otherwise, βr u = 0. The relay detection state can possibly be done by examining the included cyclic-redundancy-check digits or the received SNR levels. Here, we assume that the destination D knows the detection states of the relays, which can be obtained through the indicators sent by the relays. Through matched-filtering, the received signal at the destination D from user u  through relay r is 

u yDr = yDr (t) , su  (t) = h Dr j k

U 

βr u





Pr u xu ρuu  + n uDr

(5.4)

u=1 

where n uDr is AWGN with zero mean and variance of σ 2 , ρuu  = su (t) , su  (t) is 

the cross correlation between spreading codes su (t) and su  (t) with  f (t) , g (t) =  1 T f (t) g ∗ (t) dt being the inner product between f (t) and g (t) with the symbol T 0 interval T , and g ∗ (t) is the complex conjugate of g (t). Here, we assume that ρuu = su (t)2F = 1 and define the cross correlation matrix as ⎡

1 ρ21 ⎢ ρ12 1 ⎢ Rρ = ⎢ . .. ⎣ .. . ρ1U ρ2U

··· ··· .. .

⎤ ρU 1 ρU 2 ⎥ ⎥ .. ⎥ . . ⎦

··· 1

Based on Rρ , we rewrite (5.4) into matrix form as yDr = h Dr j  k  Rρ Pr x + nDr

(5.5)

√ √ √ U T 1 2 , yDr , . . . , yDr ] , Pr = diag{βr 1 Pr 1 , βr 2 Pr 2 , . . . , βrU PrU }, where yDr = [yDr x = [x1 , x2 , . . . , xU ]T , nDr = [n 1Dr , n 2Dr , . . . , n UDr ]T is the AWGN vector with zero mean and variance matrix of σ 2 Rρ , and [·]T denotes transpose. The correlation amongst the entries of nDr is from the matched-filtering operation of inner product between different spreading codes.

60

5 Space-Time Network Coding with Antenna Selection

Assuming Rρ being invertible with the inverse matrix Rρ−1 [7] and multiplying the both sides of (5.5) with Rρ−1 yield the soft signal of xu from relay r as u = h Dr y˜Dr j  k  βr u



Pr u xu + n˜ uDr

(5.6)

where n˜ uDr is the AWGN with zero mean and variance of σ 2 θu with θu being the uth diagonal element of matrix Rρ−1 associated with signal xu . With orthogonal spreading codes, i.e., ρuu  = 0, ∀u = u  , Rρ is an identity matrix of size U . In such a case, θu = 1, 1 ≤ u ≤ U . Combining the information on xu from user u and R relays via maximal-ratio combining (MRC) yields the end-to-end SNR of xu at the destination D as ϒu = ϒuD +



ϒr D

(5.7)

r ∈Du

2   2 2   2  and Du = ϒr D = h Dr where ϒuD = h Du ∗ ∗ j  k  F Pr u / σ θu , j i  P0u /σ , F {r : βr u = 1, r = 1, 2, . . . , R} is the active relay set associated with user u. When all the relays fail to decode the signal xu , i.e., Du = φ, it is obvious that STNC with AS reduces to the conventional single-hop AS scheme.

5.3 Performance with Outdated CSI Due to channel fluctuations, the CSI employed in AS process may differ from the exact CSI in data transmission instant, in other words, the employed CSI is outdated [8]. In this section, we examine the detrimental impact of outdated CSI on the performance of STNC with AS. Here, we assume that the transmit/receive antenna pair of r -D link1 and the receive antenna of u-r link are selected based on the outdated CSI with τr and τur time delays between the instants of channel estimation and data transmission, respectively. Dr To model the relationship between h Dr jk (t) and h jk (t − τr ), we employ the timevarying channel feedback error model to express the channel coefficient as h Dr jk (t) 

−α Dr 2 = ρr h jk (t − τr ) + 1 − |ρr | er (t) [9–11], where er (t) ∼ CN 0, dDr and ρr is Dr the normalized delay correlation coefficient between h Dr jk (t) and h jk (t − τr ). For Clarke’s fading spectrum, ρr = J0 (2π fr τr ), where fr is the Doppler frequency and J0 (·) is the zeroth-order Bessel function of the first  kind [12, Eq. (8.402)]. Similarly, for the u-r link, h rkiu (t) = ρur h rkiu (t − τur ) + 1 − |ρur |2 eur (t), where and ρur = J0 (2π f ur τur ) is the normalized delay correlation eur (t) ∼ CN 0, dr−α u coefficient between h rkiu (t) and h rkiu (t − τur ) with f ur being the Doppler frequency.

1

Here, the r -D link with r = 0 represents the u-D link.

5.3 Performance with Outdated CSI

61

Defining ϒ˜ r D and ϒ˜ ur as the actual received SNRs of r -D and u-r links when AS is performed based on outdated CSI, the probability density functions (PDFs) of ϒ˜ r D and ϒ˜ ur are given by [13] (m+1)γ˜ −  2 N ND −1 Nr ND − 1 (−1)m e (m(1−ρr )+1)γ¯r D Nr ND r

f ϒ˜ r D (γ˜ ) = γ¯r D m=0 m m 1 − ρr2 + 1

(5.8)

and

f ϒ˜ ur

(m+1)γ˜ −  2 Nr −1 Nr − 1 (−1)m e (m(1−ρur )+1)γ¯ur Nr 

γ ˜ = ( ) 2 +1 γ¯ur m=0 m m 1 − ρur

(5.9)

    2 Dr 2 2 2 P  /σ , γ ¯ = E h  respectively, where γ¯0D = E h Du 0u r D ji F jk F Pr u /(σ θu ), 1 ≤  ru 2  r ≤ R, γ¯ur = E h ki  F P0u /σ 2 , and E[x] denotes the expectation of x. In the presence of perfect CSI, by substituting ρr = 1 and ρur = 1 into (5.8) and (5.9) respectively, the PDFs of ϒuD , ϒr D , and ϒur can be expressed using a generic expression given by f ϒ (γ ) =

 N −1  m+1 N − 1 N  (−1)m e− γ¯ γ γ¯ m=0 m

(5.10)

where (ϒ ,N , γ¯ ) ∈ {(ϒuD , Nu ND , γ¯uD ) , (ϒr D , Nr ND , γ¯r D ) , (ϒur , Nr , γ¯ur )}, 2 and ϒur = h rku∗ i ∗  F ·P0u /σ 2 is the instantaneous received SNR of xu at relay r . For the extreme case of fully outdated CSI, i.e., ρr = ρur = 0, ϒ˜ r D and ϒ˜ ur reduce to exponentially distributed variables, which is in line with the fact that the transmit/receive antenna is actually selected randomly.

5.3.1 Symbol Error Rate 5.3.1.1

Exact Symbol Error Rate

In the presence of outdated CSI, the symbol error rate (SER) of STNC with AS associated with xu is given by P˜s,u =

ϑ˜ R  

|D˜ u |=0 v=1

  ˜ u,v P˜ s,u|D˜ u,v Pr D

(5.11)

62

5 Space-Time Network Coding with Antenna Selection

˜ u is the active relay set associated with xu in the outdated CSI scenario, D ˜ u,v where D   R ˜  ˜ ˜ is the vth, 1 ≤ v ≤ ϑ with ϑ = |D˜ u | , possible choice of Du  active relays from the ˜ u,v . After R relays, and P˜ s,u|D˜ u,v denotes the SER of detecting xu conditioned on D ˜ re-numbering the indices of relays belonging to Du,v , the corresponding received ˜ u,v , ϒ˜ u|D˜ , is SNR of xu conditioned on D u,v ϒ˜ u|D˜ u,v =

p˜ 

ϒ˜ r D

(5.12)

r =0

    ˜  ˜ u,v is ˜ 0D = ϒ˜ uD . In (5.11), the probability Pr D where p˜ = D u  and ϒ       ˜ u,v = Pr D 1 − P˜s,ur P˜s,ur ˜ u,v r ∈D

(5.13)

˜ u,v r∈ /D

CSI where P˜s,ur is the SER of detecting xu at relay r in the presence of outdated  with  ˜ u,v to give τur time delay. In the following, we proceed to derive P˜ s,u|D˜ u,v and Pr D the SER for the outdated CSI scenario based on the PDF expressions (5.8) and (5.9). From (5.8) and (5.12), the PDF of ϒ˜ u|D˜ u,v is presented in the following lemma. Lemma 5.1 The PDF of ϒ˜ u|D˜ u,v is derived as f ϒ˜ u|D˜

u,v

˜ (γ˜ ) = 

N0 ND −1

N p˜ ND −1

···

m 0 =0



m p˜ =0

ϕ˜

p˜ 

˜

μ˜ r e−ξr γ˜

(5.14)

r =0

where p˜  Nr N D ˜ =  γ¯r D r =0 p˜ 

ϕ˜ =(−1)r =0

mr

p˜  r =0

Nr ND −1 mr m r 1 − ρr2 + 1

mr + 1

ξ˜r = m r 1 − ρr2 + 1 γ¯r D μ˜ r =

p˜ 

 −1 ξ˜r  − ξ˜r .

r  =0,r  =r

Proof The proof is presented in Appendix 1.



5.3 Performance with Outdated CSI

63

Integrating (5.14) with the aid of [12, Eq. (3.351.1)], we obtain the cumulative distribution function (CDF) of ϒ˜ u|D˜ u,v as Fϒ˜ u|D˜

u,v

˜ (γ˜ ) = 1 − 

N0 ND −1

N p˜ ND −1

···

m 0 =0



m p˜ =0

ϕ˜

p˜ ˜  μ˜ r e−ξr γ˜ r =0

ξ˜r

.

(5.15)

Based on the CDF of the received SNR, the closed-form expression for SER is given by [14] a Ps = 2



b π

∞ 0

F (γ ) −bγ √ e dγ γ

(5.16)

where the parameters a and b are up to a specific used modulation scheme, which encompasses a variety of modulations such as binary phase-shift√keying (BPSK) √ M, (a = b = 1) and M-ary quadrature amplitude modulation (a = 4( M − 1) b = 3/(2(M − 1))) [15]. Substituting (5.15) into (5.16) and applying the identity [12, Eq. (3.371)] to solve ˜ u,v , the resultant integral, the exact SER of xu at the destination D conditioned on D ˜ P s,u|D˜ u,v , is given by √ N p˜ ND −1 p˜ ND −1 − 21   ˜ N0 a b μ˜ r  a ˜ ξ˜r + b ··· ϕ˜ . P s,u|D˜ u,v = − 2 2 ξ˜ m =0 m =0 r =0 r

(5.17)

p˜

0

For the extreme case of perfect CSI with ρr = 1, it is easy to verify that (5.17) is reduced as P s,u|Du,v

√ N p ND −1 p N ND −1   a b 0 μr a 1 = − ··· ϕ (ξr + b)− 2 2 2 ξ m =0 m =0 r =0 r 0

(5.18)

p

p

p p  = r =0 Nγr¯rNDD , ϕ = (−1) r =0 m r r =0 Nr NmDr −1 , ξr = mγ¯rr+1 , μr = D −1 r  =0,r  =r (ξr  − ξr ) , p = |Du,v |, and Du,v denotes the active relay set in the perfect CSI scenario.   ˜ u,v , we now derive the SER of u-r link, P˜s,ur , in In order to calculate Pr D the presence of time delay τur between the instants of CSI estimation and data transmission. Based on (5.9), the CDF of ϒ˜ ur is given by

where p

Fϒ˜ ur (γ˜ ) = 1 − Nr

N r −1  m=0

 (m+1)γ˜ Nr − 1 (−1)m − (m(1−ρ 2 ur )+1)γ¯ur e . m+1 m

(5.19)

64

5 Space-Time Network Coding with Antenna Selection

Substituting (5.19) into (5.16) with the aid of identity [12, Eq. (3.371)], we have P˜s,ur

− 21  √  Nr −1  Nr − 1 (−1)m m+1 a a bNr 

= − +b . 2 + 1 γ¯ 2 2 m + 1 m 1 − ρur m ur m=0 (5.20)

By substituting ρur = 1 into (5.20), the SER of u-r link in the presence of perfect CSI, Ps,ur , is given by Ps,ur

√ Nr    − 21 Nr m a b a m−1 = − +b . (−1) 2 2 m=1 m γ¯ur

(5.21)

˜ Substituting (5.20) into (5.13), the probability  of Du,v in the presence of outdated  ˜ u,v , is obtained. Substituting (5.13) and CSI with time delay τur over u-r link, Pr D (5.17) into (5.11) yields the exact closed-form SER expression P˜s,u in the outdated CSI scenario. In the extreme case of fully outdated CSI with ρr = ρur = 0, we can conclude that the STNC with AS reduces to the single-antenna STNC, which agrees with the fact that the optimal transmit/receive antenna in the sense of SNR selected based on the fully outdated CSI actually corresponds to random transmit/receive antenna. For the perfect CSI scenario, we can derive the closed-form SER expression based on (5.18) and (5.21). Obviously, the SER of STNC with AS in the outdated CSI scenario encompasses that of STNC with AS in the perfect CSI scenario as a special case with ρr = 1 and ρur = 1.

5.3.1.2

Asymptotic Symbol Error Rate

Here, we derive the asymptotic SER expression to characterize the diversity order and array gain to offer useful insights into the behavior of STNC with AS in the high SNR regime for both perfect and outdated CSI scenarios. In the high SNR regime, the probability that relay r decodes the signal xu correctly approaches one, i.e., lim βr u = 1. As such, the received SNR of xu in the presence γ¯ur →∞

of perfect CSI is given by ϒu∞ =

R 

ϒr∞D

r =0 ∞ ∞ where ϒ0D = ϒuD . The asymptotic CDF of ϒu∞ is given by

(5.22)

5.3 Performance with Outdated CSI

65

Fϒu∞ (γ ) ≈

 Gd  (G d + 1) γ¯uD

R Nr N D where  =   (Nr ND + 1), r =0 κr  R Nu + r =1 Nr ND .

γ Gd

κr = γ¯uD /γ¯r D ,

(5.23) κ0 = 1,

and

Gd =



Proof The proof is presented in Appendix 2.

Based on (5.16) and (5.23), the asymptotic SER of STNC with AS in the high SNR regime can be easily derived by substituting (5.23) into (5.16) and calculating the resultant integral as ∞ ≈ (G a γ¯uD )−G d Ps,u

(5.24)

− G1  d a(2G d −1)!! where the array gain G a = b (G . G +1 d +1)2 d wedemonstrate that STNC with AS achieves the full diversity order From (5.24), R of Nu + r =1 Nr ND in the cooperative MIMO network, where the contributions of the u-D and r -D links to the diversity order are Nu ND and Nr ND , respectively. In the special case where all the nodes are equipped with a single antenna with Nu = Nr = ND = 1, the diversity order reduces to R + 1, which is consistent with the result given in [6, 16]. In the presence of outdated CSI, the received SNR of xu in the high SNR regime is given by ϒ˜ u∞ =

R 

ϒ˜ r∞D

(5.25)

r =0

where ϒ˜ r∞D is the received SNR of xu in the high SNR regime from relay r , and ∞ ∞ ϒ˜ 0D = ϒ˜ uD . The asymptotic CDF of ϒ˜ u∞ is given by Fϒ˜ u∞ (γ˜ ) ≈

˜  γ˜ R+1 R+1  (R + 2) γ¯uD

(5.26)

where ˜ = 

R  r =0

N

   Nr ND − 1  Nr ND κr (−1)m r

. mr m r 1 − ρr2 + 1 m =0 r ND −1

r

Proof The proof is presented in Appendix 3.



Based on Fϒ˜ u∞ (γ˜ ), we now derive the asymptotic SER by substituting (5.26) into (5.16) and calculating the resultant integral as

66

5 Space-Time Network Coding with Antenna Selection

−G˜ d  ∞ ≈ G˜ a γ¯uD P˜s,u where the diversity order G˜ d = R + 1 and array gain G˜ a = b

(5.27) 

˜ (2G˜ d −1)!! a  (G˜ d +1)2G˜ d +1

− ˜1

Gd

.

Comparing (5.27) with (5.24), it is evident that outdated CSI has a severely detri  mental effect on the SER and degrades the full diversity order of Nu + rR=1 Nr ND to R + 1, which indicates that the contribution of the multiple antennas to the diversity vanishes due to outdated CSI.

5.3.2 Capacity 5.3.2.1

Outage Probability

The outage probablity (OP) is an important quality-of-service measure as it characterizes the probability that the instantaneous capacity falls below a predetermined threshold Rth , which corresponds to the SNR threshold ϒth = 2 Rth − 1. With outdated CSI, from (5.7), the OP associated with user u is given by P˜out,u = Pr (Cu < Rth ) =

ϑ˜ R  

|D˜ u |=0 v=1

  ˜ u,v P˜ out,u|D˜ u,v Pr D

(5.28)

  ˜ u,v is given by (5.13), and P˜ out,u|D˜ is where Cu is the capacity of user u, Pr D u,v ˜ the OP of Cu conditioned on Du,v , which can be derived from (5.15) directly as P˜ out,u|D˜ u,v = Fϒ˜ u|D˜

u,v

Rth

2 −1

(5.29)

R

2 th − 1 is the CDF of ϒ˜ u|D˜ u,v evaluated at γ˜ = 2 Rth − 1. Substitutwhere Fϒ˜ u|D˜ u,v ing (5.13) and (5.29) into (5.28) yields the OP associated with user u. For the perfect CSI scenario, the OP can be obtained by substituting ρr = 1 and ρur = 1 into (5.28).

5.3.2.2

Ergodic Capacity

In the presence of outdated CSI, the ergodic capacity of user u is C˜ u =

ϑ˜ R  

|D˜ u |=0 v=1

  ˜ u,v C˜ u|D˜ u,v Pr D

(5.30)

5.3 Performance with Outdated CSI

67

˜ u,v , and where C˜ u|D˜ u,v denotes the ergodic capacity of user u conditioned on D   ˜ u,v is given by (5.13). Based on (5.14), the ergodic capacity of user u condiPr D ˜ u,v is mathematically formulated as tioned on D C˜ u|D˜ u,v =

∞ log2 (1 + γ˜ ) f ϒ˜ u|D˜

u,v

(γ˜ ) d γ˜

0

˜ = − log2 (e)

N0 ND −1

N p˜ ND −1

···

m 0 =0



m p˜ =0

ϕ˜

p˜  μ˜ r r =0

ξ˜r

  ˜ × eξr Ei −ξ˜r

(5.31)

x t where Ei (x) = −∞ et dt, x < 0, is the exponential integral function. We note that (5.31) can be obtained directly by using the definite integral of exponential function, given in [12, Eq. (4.337.2)]. Substituting (5.13) and (5.31) into (5.30), we obtain the ergodic capacity of user u in the presence of outdated CSI directly. For the perfect CSI scenario, the ergodic capacity of user u can be easily derived by substituting ρr = 1 and ρur = 1 into (5.30).

5.4 Performance with Spatial Correlation Correlated fading occurs in many practical scenarios due to the limited antenna separation or the lack of local scatters [17]. In this section, we examine the impact of spatial correlation on the performance of STNC with AS. According to the common Kronecker structure, the spatially correlated channel cor , 0 ≤ r ≤ R, between relay r and the destination D can be decomposed matrix HDr as [14, 18] 1

1

cor HDr = RD2 HDr Rr2

(5.32)

where RD and Rr denote the spatial correlation matrices at the destination D and relay r , respectively. Here, user u is assumed as the zeroth relay for the sake of simplicity. The ( p, q)th elements of RD and Rr , R Dp,q and R rp,q , satisfy [19]  R Dp,q =

Dp qD , p = q 1, p=q

(5.33)

p = q rp qr , 1, p=q

(5.34)

and  R rp,q =

68

5 Space-Time Network Coding with Antenna Selection

respectively, where 0 ≤ Dp < 1, 1 ≤ p ≤ ND , 0 ≤ qr < 1, and 1 ≤ q ≤ Nr . Consequently, RD and Rr can be parameterized by column vectors D = T T   D D 1 , 2 , . . . , DND and r = 1r , 2r , . . . , rNr , respectively. Based on the Kronecker product of the transmit and receive spatial correlation matrices, we have [14, 20] cor 1 = vec (HDr ) R 2 vec HDr

(5.35)

where vec (·) denotes the matrix vectorization operation, R = Rr ⊗ RD , and ⊗ denotes the Kronecker product. Based on the mixed-product property of Kronecker product, R is rewritten as R =  T

(5.36)

where the column vector  = r ⊗ D with the ith element i = mr nD , 1 ≤ i ≤ Nr ND , m = i/ND , n = i − (i/ND  − 1) ND , and x is the ceiling function of x. And the ( p, q)th element of R, R p,q , satisfies  R p,q =

 p q , p = q . 1, p=q

(5.37)

With AS, we would like to select the transmit/receive antenna pair that corresponds to the highest channel coefficient out of the ND × Nr correlated channel coefficients between relay r and destination D. With the selected transmit/receive antenna pair, the single polynomial expansion of the asymptotic CDF of ϒr∞D,cor in the high SNR regime is given by [13, 21]  Fϒr∞D,cor (γ ) ≈ ωr

γ γ¯r D

 Nr N D (5.38)

where ωr =

N N r D  i=1

−1 N N r D  i 1 +1 . 1 − i 1 − i i=1

Taking the first derivative of (5.38) yields the asymptotic PDF of ϒr∞D,cor in the high SNR regime. Based on the PDF of ϒr∞D,cor , we follow the similar procedure specified in Sect. 5.3.1.2 to derive the asymptotic SER of STNC with AS over flat spatially correlated Rayleigh fading channels as

−G d ∞ ≈ G acor γ¯uD Ps,u,cor

(5.39)

5.4 Performance with Spatial Correlation

69

− G1   d a(2G d −1)!! where array gain G acor = b (G , and  = rR=0 ωr . It is easy to demonG d +1 +1)2 d strate that (5.39) is equal to (5.24) by setting i = 0 in (5.39). we observe that STNC with AS achieves the full diversity order of  From(5.39),  R Nu + r =1 Nr ND over flat spatially correlated Rayleigh fading channels regardless of the values of spatial correlation coefficients and the impact of spatial correlation lies in the array gain. It is expected that the higher the value of spatial correlation coefficient, the lower the array gain, and therefore the higher the SER. We now characterize the performance gap between spatially correlated and uncorrelated scenarios. Specifically, it is determined by the ratio of the average transmit SNR in the correlated scenario and the average transmit SNR in the uncorrelated scenario for the same SER. We derive this ratio as 1

∞ =  Gd . ϒgap

(5.40)

It is indicated from (5.40) that for the same SER, the uncorrelated scenario is superior ∞ dB. to the correlated scenario by an SNR gap of 10 log ϒgap

5.5 Numerical Results In this section, the numerical and simulation results are presented to verify the accuracy of our analysis and examine the impacts of the network parameters on the performance of STNC with AS. We assume that the coordination of the destination D, relay r , and user u are (0, 0), ((d + r d) cos(r ψ), (d + r d) sin(r ψ)), and (cos(uψ  ), sin(uψ  )), respectively. In the simulations, d = 0.4, d = 0.1, and ψ = ψ  = π/18. The cross correlations between different spread codes are set to be zero, and the value of the path-loss exponent α is 3.5 [22]. We assume equal transmit power at each node. Figures 5.1 and 5.2 plot the SER of STNC with AS versus transmit SNR  Pu /N0 for perfect CSI and outdated CSI scenarios, respectively, where Pu = P0u + rR=1 Pr u . We observe an excellent agreement between the calculation results using analytical expressions and simulation results marked by “◦”. Moreover, the exact curves approach asymptotic curves in the high SNR regime. This validates the accuracy of our theoretical analysis in Sect. 5.3. In Fig. 5.1, the curve with tuple {2, 1, 1, 1} denotes the SER of single-antenna STNC. Figure 5.1 shows that increasing the antenna number could improve the system performance in terms of both diversity order and SER. With perfect CSI, STNC with AS provides full diversity order of   Nu + rR=1 Nr ND , as indicated by (5.24), which implies that it is preferred to distribute more antennas to destination to improve the network performance. The asymptotic curves in Figs. 5.1 and 5.2 have different slopes. In Fig. 5.1, the larger the value of antenna/relay number, the steeper the slope of asymptotic curve, which in turns indicates a significant diversity gain. Figure 5.2 shows that the diversity

70

5 Space-Time Network Coding with Antenna Selection

Fig. 5.1 SER of BPSK modulation of STNC with AS for perfect CSI scenario with different network configurations. (Tuple {a, b, c, d} indicates that R = a, Nu = b, Nr = c, and ND = d, respectively.)

Fig. 5.2 SER of BPSK modulation of STNC with AS for outdated CSI scenario with R = 2, Nu = Nr = 2, and ND = 1

gain vanishes due to the outdated CSI. With the decrease of the delay correlation coefficient, the SER of STNC with AS increases. It is noticed that the exact and asymptotic SER curves of STNC with AS for ρr = ρur = 0 in Fig. 5.2 are identical to that for Nu = Nr = ND = 1 in Fig. 5.1. This can be explained by the fact that, for extremely outdated CSI of ρr = ρur = 0, the selected antennas are actually random antennas, but not the optimal antennas that maximize the received SNR, and the diversity gain offered by multiple antennas vanishes. Figure 5.3 gives the SER of STNC with varying delay correlation coefficients. It further demonstrates the detrimental effect of outdated CSI on the multi-antenna diversity gain. We observe that the multi-antenna diversity gain is trivial with the delay correlation coefficient ρ being lower than 0.5 and it almost vanishes with ρ being lower than 0.2. With the same relay number and transmit SNR, STNC with AS associated with different antenna configurations shows the same SER when ρ = 0.

5.5 Numerical Results

71

Fig. 5.3 Detrimental impact of outdated CSI on the SER with variable delay correlation coefficient ρ. (ρr = ρur = ρ)

Fig. 5.4 OP of capacity for outdated CSI scenario with R = 2 and Nu = Nr = 2

It is shown that the SER increases dramatically as the delay correlation coefficient decreases from 1 to 0.8. Thus, it is reasonable to increase the delay correlation coefficient, e.g., decreasing the time delay between the instants of channel estimation and data transmission, to decrease the SER and increase the system performance. The OPs of capacities with different thresholds are presented in Fig. 5.4. The perfect match between the theoretical and simulation results demonstrates the correctness of our theoretical analysis in Sect. 5.3.2. In Fig. 5.4, the curve with ρr = ρur = 0 corresponds to single-antenna STNC due to the fact that fully outdated CSI vanishes the multi-antenna diversity gain completely. It is shown that increasing the antenna number and delay correlation coefficient improves the capacity significantly. The effect of spatial correlation on the SER of STNC with AS is shown in Figs. 5.5 and 5.6. In Fig. 5.5, the solid curves denote the simulation results. In Fig. 5.6, the curves are plotted based on the simulation results. We can observe that the spatial

72

5 Space-Time Network Coding with Antenna Selection

Fig. 5.5 SER of BPSK modulation of STNC with AS for spatial correlation scenario with R = 1, Nu = Nr = 2, and ND = 1

Fig. 5.6 Detrimental impact of spatial correlation on the SER with variable spatial correlation coefficient 2

correlation deteriorates the SER. The asymptotic curves are parallel with the same slope of −4 in Fig. 5.5, but are shifted to the right as 2 increases from 0 to 0.9. This verifies the correctness of our analysis on system diversity order and implies that the spatial correlation has no impact on the achievable diversity order but has impact on the array gain. We observe that the increase in spatial correlation coefficients deteriorates the SER, particularly for large correlation coefficients. Specifically, the effect of correlation is almost negligible when 2 < 0.5, but becomes profound when 2 > 0.5, which is in agreement with previous results on the effect of spatial correlation [23].

5.6 Conclusions

73

5.6 Conclusions In this chapter, we have analyzed the performance of STNC with AS in cooperative MIMO network over flat Rayleigh fading channels. We have confirmed that STNC   with AS preserves the full diversity order of Nu + rR=1 Nr ND . It is shown that the outdated CSI vanishes the diversity gain offered by multiple antennas and the diversity order degrades to R + 1 from full diversity, while spatial correlation has no impact on the diversity order. The significant impact of outdated CSI implies that a high channel estimate rate may be required in practice in order to attain the full benefits of AS.

Appendix 1 Proof of Lemma 5.1 Based on (5.8), the moment generating function (MGF) of ϒ˜ r D is derived by performing the Laplace transform with the aid of the definite integral of the exponential function [12, Eq. (3.351.3)] as ∞ Mϒ˜ r D (s) =

f ϒ˜ r D (γ˜ ) exp (−s γ˜ ) d γ˜ 0

=

Nr ND −1  m r =0

 −1 mr  ˜ Nr ND − 1 (−1) Nr ND ξr + s

. mr m r 1 − ρr2 + 1 γ¯r D

(5.41)

Since the MGF of the sum of multiple independent random variables is equal to the product of the MGFs of the random variables [15], the MGF of ϒ˜ u|D˜ u,v is Mϒ˜ u|D˜

u,v

˜ (s) = 

N0 ND −1

N p˜ ND −1

···

m 0 =0



m p˜ =0

p˜  −1  ξ˜r + s ϕ˜

(5.42)

r =0

where N0 = Nu . Expanding (5.42) in poles and residuals with the aid of partial fraction decomposition [12, Eq. (2.102)] yields Mϒ˜ u|D˜

u,v

˜ (s) = 

N0 ND −1 m 0 =0

N p˜ ND −1

···



m p˜ =0

ϕ˜

p˜ 

 −1 μ˜ r ξ˜r + s .

(5.43)

r =0

The desired result (5.14) can now be derived directly by performing inverse Laplace transform on (5.43).

74

5 Space-Time Network Coding with Antenna Selection

Appendix 2 Proof of Eq. (5.23) By applying the Taylor series expansion of the exponential function in the PDF of ϒr D in (5.10) and retaining the first order term, the first order expansion of the PDF of ϒr D , f ϒr∞D (γ ), is derived as f ϒr∞D (γ ) ≈

Nr N D γ¯rNDr ND

γ Nr ND −1 .

(5.44)

Performing the Laplace transform of (5.44) with the aid of [12, Eq. (3.351.3)] yields ∞ Mϒr∞D (s) =

f ϒr∞D (γ ) exp (−sγ )dγ ≈ 0

Nr N D γ¯rNDr ND

 (Nr ND ) s −Nr ND .

(5.45)

As ϒr∞D is independent of each other, based on (5.22) and (5.45), we have the MGF of ϒu∞ as Mϒu∞ (s) ≈ (s γ¯uD )−G d .

(5.46)

Performing the inverse Laplace transform of (5.46) and integrating the outcome produce the CDF of ϒu∞ , as shown in (5.23).

Appendix 3 Proof of Eq. (5.26) Applying the Taylor series expansion of the exponential function in (5.8) and discarding the high order items yield f ϒ˜ r∞D (γ˜ ) ≈

 N ND −1 Nr N D − 1 Nr ND r (−1)m

. γ¯r D m=0 m m 1 − ρr2 + 1

(5.47)

Performing the Laplace transform of (5.47), 0 ≤ r ≤ R, with the aid of [12, Eq. (3.351.3)], we have the MGF of ϒ˜ r∞D as ∞ Mϒ˜ r∞D (s) =

f ϒ˜ r∞D (γ˜ ) exp (−s γ˜ ) d γ˜ 0

≈

 N ND −1 Nr N D − 1 Nr ND r (−1)m

. γ¯r D m=0 m m 1 − ρr2 + 1 s

(5.48)

5.6 Conclusions

75

Based on (5.25) and (5.48), we have the MGF of ϒ˜ u∞ as ˜ γ¯uD )−(R+1) . Mϒ˜ u∞ (s) ≈ (s

(5.49)

Performing the inverse Laplace transform of (5.49) yields the PDF of ϒ˜ u∞ , from which we can derive the asymptotic CDF of ϒ˜ u∞ , as shown in (5.26).

References 1. H. Zhang, N.B. Mehta, A.F. Molisch, J. Zhang, H. Dai, Asynchronous interference mitigation in cooperative base station systems. IEEE Trans. Wireless Commun. 7(1), 155–165 (2008). 2. X. Li, T. Jiang, S. Cui, J. An, Q. Zhang, Cooperative communications based on rateless network coding in distributed MIMO systems. IEEE Wireless Commun. 17(3), 60–67 (2010). 3. E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, H.V. Poor, MIMO Wireless Communications. Cambridge University Press, 2007. 4. S. Sanayei, A. Nosratinia, Antenna selection in MIMO systems. IEEE Commun. Mag. 42(10), 68–73 (2004). 5. N.B. Mehta, S. Kashyap, A.F. Molisch, Antenna selection in LTE: from motivation to specification. IEEE Commun. Mag. 5(10), 144–150 (2012). 6. H.-Q. Lai, K.J.R. Liu, Space-time network coding. IEEE Trans. Signal Process. 59(4), 1706– 1718 (2011). 7. R.K. Mallik, The uniform correlation matrix and its application to diversity, IEEE Trans. Wireless Commun. 6(5), 1619–1625 (2007). 8. V. Kristem, N.B. Mehta, A.F. Molisch, Optimal receive antenna selection in time-varying fading channels with practical training constraints. IEEE Trans. Commun. 58(7), 2023–2034 (2010). 9. S. Zhou, G.B. Giannakis, Adaptive modulation for multiantenna transmissions with channel mean feedback. IEEE Trans. Wireless Commun. 3(5), 1626–1636 (2004). 10. N. Yang, M. Elkashlan, P.L. Yeoh, J. Yuan, Multiuser MIMO relay networks in Nakagami-m fading channels. IEEE Trans. Commun. 60(11), 3298–3310 (2012). 11. Y. Huang, F. Al-Qahtani, C. Zhong, Q. Wu, J. Wang, H. Alnuweiri, Performance analysis of multiuser multiple antenna relaying networks with co-channel interference and feedback delay. IEEE Trans. Commun. 62(1), 59–73 (2014). 12. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products. 7th ed. Academic, 2007. 13. G. Amarasuriya, C. Tellambura, M. Ardakani, Two-way amplify-and-forward multiple-input multiple-output relay networks with antenna selection. IEEE J. Sel. Areas Commun. 30(8), 1513–1529 (2012). 14. M.R. McKay, A.J. Grant, I.B. Collings, Performance analysis of MIMO-MRC in doublecorrelated Rayleigh environments. IEEE Trans. Commun. 55(3), 497–507 (2007). 15. M.K. Simon, M.-S. Alouini, Digital Communication over Fading Channels: A Unified Approach to Performance Analysis. Wiley, 2000. 16. K. Xiong, P. Fan, T. Li, K.B. Letaief, Outage probability of space-time network coding over Rayleigh fading channels. IEEE Trans. Veh. Technol. 63(4), 1965–1970 (2014). 17. R. Janaswamy, Radiowave Propagation and Smart Antennas for Wireless Communications. Kluwer Academic Publishers, 2000. 18. C.-N. Chuah, D.N.C. Tse, J.M. Kahn, R.A. Valenzuela, Capacity scaling in MIMO wireless systems under correlated fading. IEEE Trans. Inf. Theory 48(3), 637–650 (2002). 19. X. Zhang, N.C. Beaulieu, Performance analysis of generalized selection combining in generalized correlated Nakagami-m fading. IEEE Trans. Commun. 54(11), 2103–2112 (2006). 20. D.-S. Shiu, G.J. Foschini, M.J. Gans, J.M. Kahn, Fading correlation and its effect on the capacity of multielement antenna systems. IEEE Trans. Commun. 48(3), 502–513 (2000).

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21. K. Yang, N. Yang, C. Xing, J. Wu, Relay antenna selection in MIMO two-way relay networks over Nakagami-m fading channels. IEEE Trans. Veh. Technol. 63(5), 2349–2362 (2014). 22. D. Tse, P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2005. 23. M. Chiani, M.Z. Win, A. Zanella, On the capacity of spatially correlated MIMO Rayleighfading channels. IEEE Trans. Inf. Theory 49(10), 2363–2371 (2003).

Chapter 6

Compressive Sensing-Based Dynamic Estimation in Unified Laser TTC System

In this chapter, we investigate the compressive sensing (CS) based dynamic estimation algorithm in a unified laser telemetry, tracking, and command (TTC) system. Section 6.1 introduces the motivation of designing the CS-based dynamic estimation algorithm for the unified laser TTC system. Section 6.2 presents the system model. Section 6.3 details the CS-based dynamic estimation algorithm, and Sect. 6.4 analyses the performance and the complexity of developed algorithm. Section 6.5 presents the simulation results, and Sect. 6.6 concludes this chapter.

6.1 Introduction Unified laser TTC system is increasingly utilized in space exploration because it can take advantage of the ultra-wideband laser signal, which is beneficial for high speed command communication and high precision tracking and ranging [1, 2]. In unified laser TTC system, it is necessary to preferentially estimate dynamic parameters including velocity and acceleration, and then employ the estimated results to facilitate temporal synchronization, demodulation, and ranging. To address the dynamic estimation problem, a large body of works have been developed, among which digital transformation-domain algorithms are commonly utilized. Specifically, the discrete fractional Fourier transform (FrFT), discrete polynomial-phase transform (DPT), and keystone transform were adopted for dynamic estimation in [3–5], respectively. Nevertheless, their computational complexity is usually too high to meet the low power consumption requirement on space platforms. To reduce the computational complexity, a sparse algorithm based on the fast Fourier transform (FFT) namely sparse FFT (sFFT) was developed and supplemented by [6, 7], respectively. The sparse algorithm was extended to fractional frequency domain in [8], which inspires us to apply the sparse fractional transformation-domain © Beijing Institute of Technology Press 2023 J. An et al., Sustainable Wireless Communications, https://doi.org/10.1007/978-981-19-0448-6_6

77

78

6 Compressive Sensing-Based Dynamic Estimation in Unified Laser TTC System

Fig. 6.1 The model of unified laser TTC system, where the IMDD-based laser communications serve for the data links between spaceborne platform and ground station and among spaceborne platforms, respectively

algorithm to estimate velocity and acceleration. With lower complexity, the sparse algorithms are more suitable for dynamic estimation on low power consumption platforms than the conventional transformation algorithms. However, these sparse digital transformation-domain algorithms require extremely high sample rate in the scenario with high data rate, which is inapplicable in spaceborne systems. To reduce the sample rate, the CS algorithms are introduced into high data rate systems. Since the works in [9–11], CS algorithm has attracted immense research interest in high speed signal processing due to the fact that it allows a sparse signal to be acquired at a very low sample rate [12]. Motivated from the above observations, combining sparse transformation-domain algorithms and CS algorithms, this chapter develops a CS-based dynamic estimation algorithm via sparse DPT (sDPT) and sparse FrFT (sFrFT), which is of good accuracy, low sample rate, and low complexity. This algorithm can be applied to high dynamic and high data rate systems with the demands of low power consumption and small size.

6.2 System Model The model of the unified laser TTC system is presented in Fig. 6.1, where the intensity modulation direct detection (IMDD) based laser communications serve for the data links between spaceborne platform and ground station, and among spaceborne platforms. The tasks of TTC can be acheived with the aid of laser data links. In IMDD-based laser TTC systems, the laser signal is received by photodetection and is transformed to a baseband electrical signal, which is expressed as    s (t) = r (t − τ0 − τ (t)) + w (t) = r t − τ0 − a1 t + a2 t 2 + w (t) .

(6.1)

In (6.1), r (t) and τ0 denote the transmission signal and propagation delay, respectively, τ (t) denotes the dynamics containing two components related to velocity

6.2 System Model

79

(a1 ) and acceleration (a2 ), respectively, and w (t) is the zero-mean additive white Gaussian noise with variance σ 2 . The nature of dynamic estimation is to estimate a1 and a2 , which can be obtained with the aid of pilot. The transmission signal can be expanded as r (t) = rs (t) + r p (t), where rs (t) and r p (t) denote useful signal and pilot, respectively. Here, rs (t) can be expressed as rs (t) = As gd (t)

N s −1 

c (m) δ (t − mTs − τ (t))

m=0

= As

Ts N s −1  

     gd t˙ − t c (m) δ t˙ − mTs − τ t˙ d t˙.

(6.2)

m=0 0

In (6.2), As and Ts denote the amplitude and symbol duration of rs (t), respectively, gd (t) denotes the waveform of rs (t) in a signal interval, which is assumed to be square wave in this chapter, Ns denotes the total number of symbols, and c (m) ∈ {−1, 1} represents bipolar symbol in the m-th interval. The periodic square wave is selected as pilot, which is expressed as follows according to the Fourier series: r p (t) = A p

∞  k=1

  1 2π sin (2k − 1) t , 2k − 1 Tp

(6.3)

where A p and T p denote the amplitude and symbol duration of r p (t), respectively. Letting T p = Ts /2 to mitigate the mutual interference between rs (t) and r p (t) in frequency domain, and assuming A p = β As = β (the real constant, β, represents power efficiency), we have r p (t) = β

∞  k=1

  Ak 4π sin (2k − 1) t , 2k − 1 Ts

(6.4)

where Ak denotes the amplitude of the k-th harmonic, which decreases rapidly with the increase of k in a low-pass system. For brevity, the harmonics above 3rd order are omitted. The time-frequency expression of r (t) is presented in Fig. 6.2, which shows that r (t) ∈ {−1 − β, −1 + β, 1 − β, 1 + β} has only four different types of amplitude, and is applied to intensity modulation laser TTC systems. Moreover, the peaks of r p (t)-spectrum appear in the zero-positions of rs (t)-spectrum, and the mutual interference between rs (t) and r p (t) is zero. In addition, the transmission signal is of excellent sparsity in frequency domain, which makes CS algorithm possible.

80

6 Compressive Sensing-Based Dynamic Estimation in Unified Laser TTC System

Fig. 6.2 The time-frequency expression of r (t), where the first subfigure presents r (t) in time domain, and the second subfigure below presents r (t) in frequency domain

6.3 Dynamic Estimation Algorithm As r p (t) is the sum of sinusoidal functions, according to the maximum likelihood principle, the pilot-based parameters estimation problem can be expressed as follows:



Ns Ts

 2



s (t) e− j2π (a1 t+a2 t ) dt

aˆ 1 , aˆ 2 = argmax



a1 , a2 0 s.t. a1 > 0, a2 > 0.

(6.5)

For mathematical conciseness, we adopt the virtual sampling model to describe the procedure of CS. In particular, assuming the received signal is virtually sampled at f NSR = 8/Ts , the received signal takes the form as s (n) = r (n − τ0 − τ (n)) + w (n) = rs (n − τ0 − τ (n)) + r p (n − τ0 − τ (n)) + w (n) ,

(6.6)

where n = t · f NSR , and here we assume τ0 = 0 for brevity. After virtual sampling, we acquire NNSR entries. Then we present how to perform the dynamic estimation in four steps as shown in Fig. 6.3. We conduct the pre-process in the first step, and obtain aˆ 1 and aˆ 2 based on CS algorithm via sDPT and sFrFT (CS-sDPT-sFrFT) in the second step. The pilot,

6.3 Dynamic Estimation Algorithm

81

Fig. 6.3 Diagram of developed CS-sDPT-sFrFT based dynamic estimation scheme

i.e., r p (t), is deleted from s (t), and only the useful signal, i.e., rˆs (t), is reserved in the third step. Finally, we output the estimated results to facilitate the fourth step including clock-data recovery (CDR) and demodulation.

6.3.1 Pre-Process To prepare form   we  derive  the complex   of s (n) as follows. First,  data for estimation, letting s n − 21 , rs n − 21 , r p n − 21 , and w n − 21 denote s (n), rs (n), r p (n), and w (n) with TNSR -delay, respectively, we have 2        3 1 1 4π TNSR β Ak rp n − = sin (2k − 1) n − 2 2k − 1 2 Ts k=1   3  β(−1)k−1 Ak 4π TNSR cos (2k − 1) = n , (2k − 1) Ts k=1

(6.7)

where TNSR = 1/ f NSR . Then we get the complex form of s (n) by   1 = rsc (n) + r pc (n) + wc (n) , sc (n) = s (n) + js n − 2

(6.8)

  where rsc (n) = rs (n) + j · rs n − 21 can be regarded as pseudo-random sequence,   wc (n) = w (n) + j · w n − 21 , and   3  1 (−1)k−1 j·(−1)k−1 (2k−1) 4π Ts nTNSR . (6.9) r pc (n) = r p (n) + j · r p n − =β· e 2 2k − 1 k=1 We substitute sc (n) into CS-sDPT-sFrFT procedure and focus on the signal processing of the developed dynamic estimation algorithm in the following subsection.

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6 Compressive Sensing-Based Dynamic Estimation in Unified Laser TTC System

6.3.2 Dynamic Estimation by CS-sDPT-sFrFT In this part, we introduce the developed CS-sDPT-sFrFT dynamic estimation algorithm, which can be decomposed into two subalgorithms namely CS algorithm via sDPT (CS-sDPT) and CS algorithm via sFrFT (CS-sFrFT). The procedure of CSsDPT-sFrFT algorithm is presented in Fig. 6.4, where CS via sFFT (CS-sFFT) is the main module during CS-sDPT and CS-sFrFT procedures. We now elaborate the procedure of CS-sDPT as follows. First, sc (n) is conjugate multiplied by itself with ξ -delay, as follows x (n) = sc (n) · sc∗ (n − ξ )



= [r (n − τ (n)) + wc (n)] · r ∗ (n − τ (n) − ξ ) + wc∗ (n − ξ )

(6.10)

= x p (n) +λ¯ (n) +  (n) , where x (n) is dominated by pilot component, x p (n) = r pc (n − τ (n)) · r ∗pc (n − τ (n) − ξ ), and ∗  (n) = r pc (n − τ (n)) rsc (n − τ (n) − ξ ) + rsc (n − τ (n)) r ∗pc (n − τ (n) − ξ ) ∗ + rsc (n − τ (n)) rsc (n − τ (n) − ξ )

and

∗ λ¯ (n) = r ∗pc (n − τ (n) − ξ ) + rsc (n − τ (n) − ξ ) + wc∗ (n − ξ )

+ r pc (n − τ (n)) + rsc (n − τ (n)) + wc (n) · wc∗ (n − ξ ) · wc (n) represent signal and noise components, respectively. In addition, x p (n) can be further expanded by

Fig. 6.4 Proceudre of estimating a1 and a2 based on CS-sDPT-sFrFT algorithm

6.3 Dynamic Estimation Algorithm

x p (n) =

83

3  β 2 (−1)k−1 k=1

·

2k − 1

e− j(−1)

3  β 2 (−1)k−1

2k − 1

k=1

k−1

k−1

e j(−1)

(2k−1)

(2k−1)

4π TNSR Ts

4π TNSR Ts

(n−τ (n)−ξ )

(n−τ (n))

(6.11)

= β 2 e j2π (−2a2 ξ nTNSR −φ ) + (n) , 2

where β 2 e j2π (−2a2 ξ nTNSR −φ ) represents the principal component of x p (n), φ denotes a constant phase and is unrelated to the dynamic estimation, and (n) denotes the disturbance terms, which is separated from the principal component in spectrum. Accordingly, we have 2

x (n) = β 2 e j2π (−2a2 ξ nTNSR −φ ) + (n) +  (n) +λ¯ (n) . 2

(6.12)

Then, CS-sFFT is employed to obtain the spectrum of x (n), which is denoted by yˆ (n). This procedure is summerized in  Algorithm 6.1 with the following notations.   , and B = O C, R, NNSR

 NNSR  log2 ( NNSR )

denote complex matrices, real matrices,

data size after sampling, and data size after subsampling, respectively. x ∈ C NNSR  and L = log2 (NNSR ) denote the vector form of x (n) and the total number of  cycles, respectively. F ∈ C B×B denotes the Fourier matrix. S ∈ R NNSR ×NNSR and  S ∈ R B×NNSR with   , 1, k ≡ j · NNSR /NNSR S ( j, k) = 0, otherwise, 

and S ( j, k) =

1, (k − j) mod 0, otherwise,

 NNSR B

≡ 0,

denote the subsample matrices in time and frequency domains, respectively. Furthermore, let g (n) denote the window function with length ω, whose spectrum is given by ⎧ 

⎨ 1 − 1 c , 1 + 1 c , k ∈ −ε N  , ε N  NSR NSR , N N NSR NSR  G (k) ∈

1   ⎩ 0, NNSR , k∈ / −εNNSR , εNNSR . c Here ε ∈ (0, 1), ε ∈ (0, 1), and c is a positive integer. Therefore, we have the window   matrix, W ∈ C NNSR ×NNSR with  g ( j) , j ≡ k, W ( j, k) = 0, otherwise.

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6 Compressive Sensing-Based Dynamic Estimation in Unified Laser TTC System

Next, we derive the estimated acceleration from yˆ (n) as aˆ 2, CS−sDPT = −

2ξ



2 NNSR

2  NNSR TNSR



argmax yˆ (n) .

(6.13)

Newton method can be utilized to further improve the accuracy of aˆ 2, CS−sDPT , whose procedure can be found in [13], and is omitted for brevity. The output of  Newton method is denoted by aˆ 2, CS−sDPT . After the CS-sDPT procedure, we conduct CS-sFrFT to further achieve the dynamic estimation, which contains three stages including compensation, CS-sFFT, and Newton method. First, we compensate the rotation angle in sc (n) through compensation procedure as scomp (n, as ) = sc (n) · vcomp (n, as ) , 

where vcomp (n, as ) = and

(6.14)

1 NNSR

2 2 2 e j2πas n (U +TNSR )

  as ∈ aˆ 2, ˆ 2, CS−sDPT − 3σCS−sDPT , a CS−sDPT + 3σCS−sDPT

denote the compensation value and rotation angle of the s-th search, respectively, σCS−sDPT is the standard deviation of aˆ 2, CS−sDPT , and U denotes the sample interval of the CS-sFrFT output. Then, we substitute scomp (n, as ) into CS-sFFT Algorithm and obtain the spectrum scomp (n, as ), which is denoted by yˆCS−sFrFT (n, as ). This procedure is similar to that in Algorithm 6.1 and is omitted for brevity. Next, we repeat the above procedure to traverse the as -search, and derive aˆ 1 and aˆ 2 by 

aˆ 2 = αˆ m , f NSR aˆ 1 = nˆ mNNSR ,

(6.15)



 where nˆ m , aˆ m = arg max yˆCS−sFrFT (n, as ) . Newton method can be also utilized n, as

to further improve the accuracy of aˆ 1 , whose procedure is also omitted for brevity. After aˆ 1 and aˆ 2 are obtained, the pilot, r p (t), needs to be deleted from s (t), which is necessary for the subsequent procedures. The pilot delete method is presented in next subsection.

6.3 Dynamic Estimation Algorithm

85

Algorithm 6.1 Obtain the Spectrum based on CS-sFFT Input: x, S , S, F, W, and G (k). Output: yˆ (n) 1: for i = 1; i ≤ L; i + + do 2: Letting σi denote the hash factor in the i-th cycle, the permutation matrix of the i-th cycle is   Pσi ∈ R NNSR ×NNSR with   1, (σi · k) mod NNSR ≡ j, Pσi ( j, k) = 0, otherwise. 3:

B×NNSR with Define the sensing matrix of CS-sFFT in the i-th cycle, (i) sFFT ∈ C 

(i)

sFFT = F · S · W · Pσi · S . 4:

(i)

The compressed measurement, yCS−sFFT ∈ C B is (i)

(i)

yCS−sFFT, = sFFT · x. 5:

Define the hash function of the i-th cycle,    h σi (n) = σi · n B/NNSR .

6:

Define the offset function of the i-th cycle,  oσi (n) = σi · n − h σi (n) · B/NNSR .

7: 8:

(i) , and Let i be the support set of the l coordinates of the maximum magnitudes in yCS−sFFT,

 

the preimage set of i is Ii = n ∈ 1, NNSR h σi (n) ∈ i . The inverse-map result of yCS−sFFT, i is ⎧ (i)    ⎨ yCS−sFFT h σi (k)e− jπoσi kω/NNSR   (i) , k ∈ Ii , yˆimap (k) = G oσi (k) ⎩  0, k ∈ [1, NNSR ] ∩ I¯i .

9: end for 10: reconstruction 11: Letting vn be the occurrence times of coordinate n in the sets and only retaining the coordinate whose occurrence times are larger than L/2, we have I = {n ∈ I1 ∪ · · · ∪ I L |vn > L/2 } . 12:

(r ) For each coordinate in I , we obtain corresponding yˆimap (n) with n ∈ I and r = 1, 2, · · · L.

13:

Output the median of yˆimap (n),

(r )

 yˆ (n) = 14: end reconstruction

  (r ) median yˆimap (n) , n ∈ I , 0, n ∈ / I .

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6 Compressive Sensing-Based Dynamic Estimation in Unified Laser TTC System

6.3.3 Pilot Delete This step contains 3 substeps including pilot location, pilot regeneration, and pilot cancellation, which mean estimating τ0 , generating the replica of received pilot, and removing the recevied pilot with the assist of regenerated pilot, respectively. First, we focus on the pilot location. Since the received signal is with high data rate, conventional pilot location method in all-digital domain encounters the difficulty of insufficient sample rate. Therefore, we conduct this procedure in the analog domain, which can be implemented in the scenario with high data rate. Assuming the frame header, h (t), is included in rs (t), we can apply it to locate the received signal and obtain τ0 with the aid of analog correlator, as shown in Fig. 6.5, where h (t − τ ) denotes the local replica of h (t) with τ -delay, and it can be generated by numerical controlled oscillator (NCO). The procedure of τ0 estimation through analog correlator can be expressed as ⎞ ⎛NT s s s (t) · h (t − τ ) dt ⎠ , τˆ0 = arg max (Rx (τ )) = arg max ⎝ τ

τ

(6.16)

0

where τˆ0 denotes the estimated result of τ0 . Then, we utilize τˆ0 , aˆ 1 , and aˆ 2 to regenerate the estimated pilot with the assist of NCO, which is denoted by rˆ p (t) =

3 ! k=1

Ak 2k−1

   2 sin (2k − 1) 4π − a ˆ t − a ˆ t t − τ ˆ . 0 1 2 Ts

(6.17)

Finally, we remove the rececived pilot, and obtain the estimated useful signal component by rˆs (t) = s(t) − rˆ p (t) .

Fig. 6.5 Diagram of pilot location with the assist of analog correlator

(6.18)

6.3 Dynamic Estimation Algorithm

87

6.3.4 Clock-Data Recovery and Demodulation This step contains CDR and demodulation, which are employed to realize bit synchronization and information recovery, respectively. CDR can be implemented by phase locked loop with the assist of τˆ0 , aˆ 1 , and aˆ 2 , and outputs the recovery clock to faciliate demodulation and information recovery from rˆs (t).

6.4 Performance Analysis To further present the advantages of the developed algorithm, we theoretically analyze its performance in terms of estimation accuracy and computational complexity in this section.

6.4.1 Estimation Accuracy For the reason that aˆ 2 and aˆ 1 are derived from CS-sDPT and CS-sFrFT, respectively, we discuss the performance of these two algorithms as follows. As the procedures of CS-sDPT and CS-sFrFT are respectively the same as those of DPT and FrFT except for the implementation of Fourier transform, we preferentially focus on the performances of aˆ 2 and aˆ 1 based on DPT and FrFT, respectively, and then analyze the difference between CS-sFFT and FFT. First, we present the performance of aˆ 2 based on DPT. Assuming Ak = 0 when k > 1, the accuracy of aˆ 2 based on DPT with the aid of Newton method is given by [14] σaˆ22,DPT ≈

6 , 2  NNSR NNSR (2π ) SNRDPT TNSR

where SNRDPT =

2

(6.19)

 β 4 NNSR   1 + 2β 2 + 2 1 + β 2 σ 2 + σ 4

denotes the signal-to-noise ratio (SNR) after DPT. Note that the accuracy of aˆ 2 can be further improved by as -search in FrFT. Letting the estimation error before FrFT and

the search range of as are xaˆ 2 , DPT and Ras ∈ aˆ 2, DPT − 3σaˆ 2,DPT , aˆ 2, DPT + 3σaˆ 2,DPT , respectively, and assuming the search times are sufficient, the estimation error after as -search is

88

6 Compressive Sensing-Based Dynamic Estimation in Unified Laser TTC System

xaˆ 2 , FrFT

⎧ ⎨ xaˆ 2 , DPT + 3σaˆ 2 , DPT , xaˆ 2 , DPT < −3σaˆ 2 , DPT , = 0, xaˆ 2 ∈ −3σaˆ 2 , DPT , 3σaˆ 2 , DPT , ⎩ xaˆ 2 , DPT − 3σaˆ 2 , DPT , xaˆ 2 , DPT > 3σaˆ 2 , DPT .

(6.20)

The expectation and variance of xaˆ 2 , FrFT are presented in ⎧    ⎪ ⎨ E x aˆ 2 , FrFT = 0,       3 ⎪ σaˆ22 , DPT ≈ 0.027σaˆ22 , DPT , ⎩ var x aˆ 2 , FrFT = 10 1 − erf √ 2 



where p xaˆ 2 , DPT =

1 # 2πσaˆ2

e

−

x2 aˆ 2 , DPT 2σ 2 aˆ 2 , DPT

and erf (x) =

√2 π

x 0

(6.21)

e−t dt. If the search 2

range becomes as ∈ aˆ 2, DPT − 2σaˆ 2,DPT , aˆ 2, DPT + 2σaˆ 2,DPT , the variance turns into 0.35σaˆ22 , DPT accordingly. Then, we discuss the accuracy of aˆ 1 by FrFT and Newton method, which is given by 2 , DPT

σaˆ21 ≈

6 , 2  NNSR NNSR (2π )2 SNRFrFT TNSR

(6.22)

β2 N 

where SNRFrFT = σ 2NSR denotes the SNR after FrFT.  Finally, the performance of CS-sFFT is discussed as follows. Letting yˆ− (n) = /  , the estimation error probability based on CS-sFFT is [6] yˆ (n) |k ∈ 

2 P yˆ (n) − y (n) ≥

ε l

$ $ $ yˆ− (n)$2 + 3 2

1 N

NSR

2c

$  $   $ yˆ (n)$2 < O l . εB 1

(6.23)

CS-sFFT performs similarly to FFT with a sufficient small error probability if l B. Consequently, the accuracies of aˆ 2 and aˆ 1 based on CS-sFFT are approximately equal to those based on FFT, respectively, which demonstrates the accordance between the developed CS-sDPT-sFrFT algorithm and non-compressed algorithm.

6.4.2 Complexity Considering the fact that the computational complexity of reconstruction algorithm is an important metric, we analyze the complexity of developed CS-sDPT-sFrFT algorithm in this subsection. The CS-sDPT-sFrFT dynamic estiamtion algorithm mainly includes the CS-sFFT algorithm and the Newton method, the complexities of which are discussed as  denote data size after reconstruction, the complexities of follows. Letting NNSR iterations are CS-sFFT and Newton method with liter -times

6.4 Performance Analysis

89

Complexity

Fig. 6.6 The complexity comparison of the developed CS-sDPT-sFrFT algorithm and conventional OMP and BP algorithms

10

25

10

20

CS-sDPT-sFrFT OMP reconstruction algorithm BP reconstruction algorithm

1015 1010

10

5

10

0

10

3

10

4

10

5

10

6

10

7

Data size after reconstruction

 #           log2 NNSR and O liter NNSR , respectively. Therefore, O L NNSR log2 NNSR the total computional complexity of CS-sDPT-sFrFT algorithm is  #          O L NNSR log2 NNSR log2 NNSR +liter NNSR . In comparision, the complexities of conventional dynamic estimation methods with orthogonal matching pursuit (OMP) and basis pursuit (BP) reconstruction  algorithms       3     are O L NNSR log2 NNSR + liter NNSR and O L NNSR + liter NNSR , respectively, under the same condition. Assuming liter = 0 and L = 1 for brevity, the complexity comparison of these three algorithms are presented in Fig. 6.6, which indicates that the developed CS-sDPT-sFrFT algorithm outperforms the other two algorithms in terms of complexity.

6.5 Simulation Results In this section, we evaluate the performance of CS-sDPT-sFrFT algorithm with simulation experiments, where the main parameters are listed in Table 6.1. From (6.21) and (6.22), we find that the performance is related to three parameters including β, data size, and SNR. To illustrate the impact of these three parameters on the rootmean-square error (RMSE), we measure the RMSEs of a1 and a2 versus β, data size, and SNR, respectively. To further verify the performance, we compare the developed CS-sDPT-sFrFT algorithm with conventional OMP and BP algorithms. Note that the theoretical results of non-compressed transformation algorithm derived from (6.21)

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6 Compressive Sensing-Based Dynamic Estimation in Unified Laser TTC System

Table 6.1 The main parameters Parameters Modulation Demodulation Data rate The maximum velocity The maximum acceleration Compressive ratio

Values AM Direct detection 5 Gbps 7 km/s 300 m/s2 1/8

and (6.22) are presented by the dotted lines in all the figures, and that the results of aˆ 1 and aˆ 2 are presented with m/s and m/s 2 , respectively. The RMSEs of aˆ 1 and aˆ 2 versus β are shown in Figs. 6.7 and 6.8, respectively, where SNR = 14 dB with the data size being 4096 and 16384, respectively. From these figures, we find a minor gap between the results of CS-sDPT-sFrFT algorithm and theoretical results. This can be explained by (6.23) that the error probability of sparse algorithm can be sufficiently small when the spectrum is sparse enough. However, the received signal is of non-ideal sparsity, resulting in the minor gap. Except this issue, the developed CS-based dynamic estimation algorithm performs almost the same as non-compressed algorithm. Also, the RMSEs decrease with the increased β when β < 4, and remain almost unchanged after β ≥ 4, indicating that the developed algorithm can accurately estimate a1 and a2 without high pilot power. Actually, the performance usually meets the requirement even if the pilot power is less than the signal power, i.e., β < 1, which means that the developed dynamic estimation algorithm can be implemented with affordable overhead. Moreover, increasing data size from 4096 to 16384 results in an RMSEs advantage of one order of magnitude in terms of aˆ 1 and aˆ 2 , respectively. This is due to the fact that the performance is positively related to the data size. To further investigate the relationship between RMSE and data size, we conduct the simulation of CS-sDPT-sFrFT on estimating a1 and a2 versus data size as follows. The RMSEs of aˆ 1 and aˆ 2 versus data size are presented in Figs. 6.9 and 6.10, respectively, where SNR = 14 dB with β = 1, 4, or 32. It is observed that the developed CS-sDPT-sFrFT algorithm performs similarly to the non-compressed algorithm, which shows that the CS-sDPT-sFrFT algorithm reduces the computational complexity without significant performance loss. Also, although the RMSEs of aˆ 1 and aˆ 2 decrease with an increased data size, the change rate of RMSEs gradually degrades as the data size increases from 2048 to 32768. This means that it is difficult to benefit from the increased data size when the data size is sufficiently large. In addition, the complexity of the algorithm increases with an increased data size from Fig. 6.6. Therefore, data size should be carefully chosen to seek a good tradeoff between performance and complexity. Considering the general demand of laser TTC  = 4096 is chosen in this chapter, and the RMSEs system on space platform, NNSR of aˆ 1 and aˆ 2 get the orders of 0.1m/s and 1m/s2 , respecively, where SNR = 14 dB

6.5 Simulation Results

91

Fig. 6.7 The RMSEs of CS-sDPT-sFrFT algorithm on estimating a1 versus β when SNR = 14 dB with  = 4096 or 16384 NNSR

Fig. 6.8 The RMSEs of CS-sDPT-sFrFT algorithm on estimating a2 versus β when SNR = 14 dB with  NNSR = 4096 or 16384

and β = 4. The results are accurate enough for general TTC system. In addition, the results also reveal that the configration of β = 1 or β = 4 exhibits a similar performance with that of β = 32, which further demonstrates that the accurate estimation can be acheived without high pilot power. Finally, we compare the performance of CS-sDPT-sFrFT algorithm with conventional OMP and BP reconstruction algorithms. The RMSEs versus SNR are shown  = 4096 and β = 4. In both figin Figs. 6.11 and 6.12, respectively, where NNSR ures, we observe a prominent improvement in performance as the SNR increases, demonstrating the effectiveness of developed dynamic estimation algorithm based on CS-sDPT-sFrFT. Combining Figs. 6.11 and 6.12, the developed CS-sDPT-sFrFT

92

6 Compressive Sensing-Based Dynamic Estimation in Unified Laser TTC System

Fig. 6.9 The performance of CS-sDPT-sFrFT algorithm on estimating a1 versus data size when SNR = 14 dB with β = 1, 4, or 32

Fig. 6.10 The performance of CS-sDPT-sFrFT algorithm on estimating a2 versus data size when SNR = 14 dB with β = 1, 4, or 32

algorithm acheives accurate estimation of velovity and acceleration in the low SNR regime. For example at SNR = 8 dB, the RMSEs of aˆ 1 and aˆ 2 are the orders of 0.1m/s and of 1m/s2 , respectively. In addition, the developed CS-sDPT-sFrFT algorithm performs similarly and keeps the same slope with non-sparse algorithm and conventional OMP and BP algorithms. Moreover, as CS-sDPT-sFrFT algorithm outperforms conventional OMP and BP algorithms in terms of complexity from Fig. 6.6, it is a better choice than conventional reconstruction algorithms on resource constrained platforms.

6.6 Conclusions

93

Fig. 6.11 The comparison of the developed CS-sDPT-sFrFT algorithm and conventional OMP and BP algorithms in terms of the accuracy of aˆ 1 versus SNR when the data size  NNSR = 4096 and β = 4

Fig. 6.12 The comparison of the developed CS-sDPT-sFrFT algorithm and conventional OMP and BP algorithms in terms of the accuracy of aˆ 2 versus SNR  when NNSR = 4096 and β=4

6.6 Conclusions In this chapter, we have developed a CS-based dynamic estimation algorithm namely CS-sDPT-sFrFT for non-coherent laser TTC system on space platforms. The performance and complexity of the developed algorithm has been analyzed in comparison with the conventional CS-based dynamic estimation algorithms. The analytical and simulation results demonstrate that the developed algorithm strikes a excellent balance between good accuracy and low complexity. The results indicate that the developed CS-sDPT-sFrFT algorithm can be applied to spaceborn laser TTC systems.

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6 Compressive Sensing-Based Dynamic Estimation in Unified Laser TTC System

References 1. Q.-L. Xing, J.-T. Li, J. Tang, X. Gao, Conception of a unified laser TTC system. J. Spacecraft TT&C Technol. 28(2), 36–44 (2009). 2. M.A. Khalighi, M. Uysal, Survey on free space optical Communication: a communication theory perspective. IEEE Commun. Surveys Tuts. 16(4), 2231–2258 (2014). 3. W.-C. Du, X.-Q. Gao, G.-H. Wang, Using FRFT to estimate target radial acceleration, in Int. Conf. Wavelet Anal. Pattern Recognit. (2007), pp. 442–447. 4. R.F. Brcich, A.M. Zoubir, The use of the DPT in passive acoustic aircraft flight parameter estimation, in IEEE Region 10 Annu. Conf. Speech Image Technol. Comput. Telecommun. IEEE(1997), pp. 819–822. 5. F. Pignol, F. Colone, T. Martelli, Lagrange-polynomial-interpolation-based Keystone Transform for a passive radar. IEEE Trans. Aerosp. Electron. Syst. 54(3), 1151–1167 (2018). 6. H. Hassanieh, P. Indyk, D. Katabi, E. Price, Simple and practical algorithm for sparse Fourier transform, in 23rd Annu.ACM-SIAM Symp. Discrete Algorithms (2012), pp. 1183–1194. 7. A.C. Gilbert, P. Indyk, M. Iwen, L. Schmidt, Recent developments in the sparse Fourier transform: a compressed Fourier transform for big data. IEEE Signal Process. Mag. 31(5), 91–100 (2014). 8. S. Liu, T. Shan, R. Tao, Y.D. Zhang, G. Zhang, F. Zhang, Y. Wang, Sparse discrete fractional Fourier transform and its applications. IEEE Trans. Signal Process. 62(24), 6582–6595 (2014). 9. D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 4(52), 1289–1306 (2006). 10. E.J. Candes, M.B. Wakin, An introduction to compressive sampling. IEEE Signal Process. Mag. 25(2), 21–30 (2008). 11. E.J. Candes, T. Tao, Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006). 12. S. Wang, S.-N. An, X.-Q. Miao, Y.-Y. Ma, S.-X. Luo, Compressed sensing assisted joint channel estimation and detection for DS-CDMA uplink. IEEE Commun. Lett. 19(10), 1730– 1733 (2015). 13. T.J. Abatzoglou, A fast maximum likelihood algorithm for frequency estimation of a sinusoid based on Newton’s method. IEEE Trans. Acoust., Speech, Signal Process. 33(1), 77–89 (1985). 14. S. Peleg, B. Porat, B. Friedlander, The discrete polynomial transform (DPT), its properties and applications, in Conf. Rec. 25th Asilomar Conf. Signals, Syst. Comput. (2002), pp. 116–120.

Chapter 7

Energy-Efficient Resource Allocation in Heterogeneous OFDMA Networks

In this chapter, we investigate the energy-efficient resource allocation in heterogeneous orthogonal frequency division multiple access (OFDMA) networks. Section 7.1 introduces the motivation of developing energy-efficient resource allocation for heterogeneous OFDMA networks. Section 7.2 presents the system model. Section 7.3 formulates the energy-efficient resource allocation problem as a mixed-integer nonlinear fractional programing (MINLFP) problem and transforms it into an equivalent concave mixed-integer nonlinear programing (MINLP) problem in a parametric subtractive form. Section 7.4 derives the global optimal solution. Section 7.5 decomposes the resource allocation into two sub-problems, namely resource block (RB) allocation and transmit power control (TPC), and develops the sub-optimal scheme. Section 7.6 presents the simulation results, and Sect. 7.7 concludes this chapter.

7.1 Introduction To deal with the ever-increasing data demand in wireless communications, various efforts have been made to improve the system capacity. However, only focusing on improving the capacity of communication systems would cause economic and environmental issues [1, 2]. To find a trade-off between the system capacity and total power consumption, energy efficiency (EE) metric, which is defined as the system throughput for unit-energy consumption, is used to quantify energy conversion efficiency [3]. It has been shown that EE is strictly quasi-concave with respect to spectral efficiency (SE) in single-cell OFDMA networks under the constraint of peruser quality of service (QoS) [4]. As the throughput logarithmically increases with the transmit power, SE and EE do not always coincide [5]. Recent research results have shown that deploying small cells jointly with macro cells to form heterogeneous networks can achieve improved EE as well as increased throughput since the distances between evolved Node B (eNB) and users become © Beijing Institute of Technology Press 2023 J. An et al., Sustainable Wireless Communications, https://doi.org/10.1007/978-981-19-0448-6_7

95

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7 Energy-Efficient Resource Allocation in Heterogeneous OFDMA Networks

Fig. 7.1 Illustration of system model

smaller [6, 7]. Therefore, in this chapter, we address the EE maximizing resource allocation in downlink of heterogeneous OFDMA networks, where the users can receive signals from either the macro eNB (MeNB) or small cell eNB (SeNB). This is a non-concave MINLFP problem. As mixed-integer programming is NP-hard, the MINLFP problem is challenging to be solved. Through exploiting the fractional programming (FP) and changing the variables, we first transform the non-concave MINLFP problem into an equivalent subtractive-form problem, which is proved to be a concave MINLP problem, and then develop optimal and sub-optimal resource allocation schemes.

7.2 System Model We consider a downlink heterogeneous OFDMA network consisting of one MeNB, N SeNBs, and M users, as shown in Fig. 7.1. Let N = {0, 1, 2, . . . , N }, M = {1, 2, . . . , M}, and K = {1, 2, . . . , K } denote the sets of eNBs, users, and RBs, respectively. Here, eNB 0 denotes the MeNB, and eNB n, 1 ≤ n ≤ N , denotes the nth SeNB. We assume that each eNB could assign one RB to at most one user and intelligent interference coordination or mitigation techniques are employed such that the interference between different users that are assigned with the same RB by different eNBs is negligible [8, 9]. Assuming that all eNBs share user data and combining the received signals on the same RB from different eNBs using maximal ratio combining [10], the achievable throughput of user m is given by Rm = B

K  k=1

 log2 1 +

 2  N  δn,m,k pn,m,k h n,m,k  n=0

N0 B

(7.1)

7.2 System Model

97

where B is the bandwidth of RB, δn,m,k ∈ {1, 0} is the binary RB allocation variable to indicate whether eNB n assigns RB k to user m (δn,m,k = 1) or not (δn,m,k = 0), pn,m,k denotes the transmit power of eNB n to user m on RB k, h n,m,k is the channel coefficient between eNB n and user m on RB k, and N0 is the power spectral density of the additive white Gaussian noise. Throughout this chapter, we assume perfect channel state information (CSI), which can be estimated through accurate channel measurements [11]. In the case of difficult to get the perfect CSI, the CSI can be also estimated according to the statistical information of channel gains, which yields a chance constrained program [12] that emphasizes on long-term EE of the heterogeneous networks. In this chapter, we consider instantaneous energy-efficient resource allocation with perfect CSI. It is noticed that the results obtained by our developed scheme serves as an upper bound on the achievable EE with channel estimation errors [13].     In the following sections of this chapter, we use ψ = δn,m,k and ϕ = pn,m,k , ∀n ∈ N, ∀m ∈ M, and ∀k ∈ K, to denote the feasible RB allocation variable set and transmit power set, respectively. Based on (7.1), the total network throughput is given by R=

M 

Rm .

(7.2)

m=1

For the downlink transmissions, the total power consumption at transmitter consists of the power consumption of radio frequency power amplifiers and that of other circuits incurred by signal processing and active circuit blocks [4]. We assume that the circuit power consumption is related to the network architecture, e.g., in cloud radio access network [14], and it can be divided into two parts: static part and throughput-dependent dynamic part which is proportional to the total throughput [4, 15], namely Pc = Ps + ξ R

(7.3)

where Ps is the static term, and ξ is a constant denoting dynamic power consumption per unit throughput. As such, the total power consumption is given by P = ς Pt + Ps + ξ R

(7.4)

N M K where Pt = n=0 m=1 k=1 δn,m,k pn,m,k is the total transmit power, and ς is the reciprocal of drain efficiency of power amplifier. Based on (7.2) and (7.4), the EE is defined as ηE E =

R . ς Pt + Ps + ξ R

From (7.5), it is shown that the maximum EE, ηE E , is bounded by

(7.5)

98

7 Energy-Efficient Resource Allocation in Heterogeneous OFDMA Networks

0 < ηE E
0, the continuous relaxation of P2 is biconcave [20], but not concave. In such case, one common method to solve the continuous relaxation of P2 is alternatively updating relaxed ψ and ϕ by fixing one of them and solving the corresponding concave optimization problem [20]. As this method optimizes relaxed ψ and ϕ independently through alternate concave optimization, it does not guarantee the global optimality. In P2, we observe that there is a consistent one-to-one mapping between each element of ψ and each element of ϕ, i.e., between δn,m,k and pn,m,k , in the objective function, which is only concerned with the cross product of δn,m,k and pn,m,k , i.e., δn,m,k pn,m,k . As such, we can linearize the cross product term by replacing δn,m,k pn,m,k with a new variable xn,m,k , and P2 is now reformulated as [21]

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7 Energy-Efficient Resource Allocation in Heterogeneous OFDMA Networks



P3 : max

X,ψ,ϕ

s.t.



 R˜ − λ ς P˜t + Ps + ξ R˜

M  K 

(7.12a)

xn,m,k ≤ pnmax , ∀n ∈ N

(7.12b)

R˜ m ≥ Rmth , ∀m ∈ M   xn,m,k ≤ min pn,m,k , pnmax δn,m,k +  xn,m,k ≥ pn,m,k + pnmax δn,m,k − pnmax

(7.12c)

m=1 k=1

(7.12d) (7.12e)

(7.7b) where R˜ m = B

K 

 log2

k=1

 2  N  xn,m,k h n,m,k  1+ N0 B n=0

R˜ =

M 

R˜ m

(7.13)

(7.14)

m=1

P˜t =

N  M  K 

xn,m,k

(7.15)

n=0 m=1 k=1

  with X = xn,m,k and (a)+ = max{0, a}. The constraints (7.12d) and (7.12e) of P3 imply that xn,m,k = pn,m,k with δn,m,k = 1 and xn,m,k = 0 with δn,m,k = 0, which means xn,m,k = pn,m,k δn,m,k with δn,m,k ∈ {0, 1}. Thus, it is easy to conclude that P3 is equivalent to P2. As P˜t and R˜ are linear and concave in X respectively, the objective function of P3 can be deemed as the difference of one concave function and one linear function due to the fact of 1 − λξ > 0, which can be concluded from Theorem 7.1. Hence, this objective function is concave, and P3 is a concave MINLP problem. It is noticed that P3 is equivalent to the spectral-efficient resource allocation problem if we fix λ = 0 in (7.12a), which means that our investigated energyefficient resource allocation problem in this chapter encompasses the spectralefficient resource allocation problem as a special case. From P1 and P3, the initial non-concave MINLFP problem is now transformed into an equivalent concave MINLP problem, whose optimal solution will be elaborated in the subsequent section.

7.4 Optimal Resource Allocation

101

Algorithm 7.1 Outer Loop based on Dinkelbach Method 1: Set tolerance , and initialize i = 0 and λi = 0. 2: Solve the concave MINLP P3 with λi to obtain the optimal solution X, ψ, and ϕ, i.e., Xi , ψi , and ϕi . 3: while |F (λi ) | > do 4: i ← i + 1 ˜  5: λi ← ˜ R ˜ with Xi−1 ς Pt +Ps +ξ R

6: Solve P3 with λi to obtain Xi , ψi , and ϕi . 7: end while

7.4 Optimal Resource Allocation In Sect. 7.3, the resource allocation problem is reformulated as an equivalent concave MINLP problem by introducing new variables λ and X. In this section, we will present a two-loop resource allocation scheme to give the global optimal solution without resorting to exhaustive enumeration. In the outer loop, the optimal λ is derived based on Dinkelbach method. In the inner loop, the optimal X, ψ, and ϕ with given λ are derived based on BB method.

7.4.1 Outer Loop Based on the application of Newton’s method, the Dinkelbach method is used to determine the optimal value of λ with given X iteratively, as shown in Algorithm 7.1, which produces an increasing sequence of λ and converges to the global optimal value at a superlinear convergence rate [22]. In each iteration, the optimal X, i.e., Xi , is obtained with given λi , which is derived  . If the resultant |F (λi ) | is sufficiently small, i.e., |F (λi ) | ≤ , the based on Xi−1 iteration is stopped and the determined Xi is optimal. Otherwise, we calculate new λi and start the next iteration. It is obvious that the key step in Algorithm 7.1 lies in solving the concave MINLP P3 with given λi to obtain the optimal Xi , ψi , and ϕi in each iteration. In the following, we will elaborate how to solve P3 with given λi .

7.4.2 Inner Loop One way to solve P3 with given λ is by exhaustive search, which means we need to solve 2(N +1)M K concave nonlinear programming (NLP) problems and then choose the maximum one of these optimal values, which is obviously unacceptable. Here, we will use a BB method to solve this problem [19]. The BB method consists of a systematic search of continuous solutions in which the integer variables are succes-

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7 Energy-Efficient Resource Allocation in Heterogeneous OFDMA Networks

Algorithm 7.2 Inner Loop based on Branch and Bound Method 1: Set tolerance , and initialize global lower bound L = −∞, global upper bound U = ∞, node set S ← φ, depth set D ← φ, lower bound set L ← φ, and upper bound set U ← φ. 2: Add the continuous relaxation of P3, denoted as node s, into S, and add its related depth ds = 0, lower bound ls = −∞, and upper bound u s = ∞ into D, L, and U, respectively. Solve s based ∗ ∗ ∗ on PHR-AL method to obtain optimal solutions xn,m,k , δn,m,k , and pn,m,k . ∗ 3: if δn,m,k ∈ {0, 1}, ∀n ∈ N, ∀m ∈ M, ∀k ∈ K then 4: Stop. 5: end if 6: while U − L >  do 7: Sort the nodes in S in descending order of optimal objective value, and renumber the corresponding elements in sets D, L, and U according to S sorting results respectively. 8: i ← 1 9: while dsi = (N + 1)M K do 10: i ←i +1 11: end while 12: {n ∗ , m ∗ , k ∗ } ← arg min |δn,m,k − 0.5|, where δn,m,k is from si . n∈N,m∈M,k∈K

13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33:

Substitute δn ∗ ,m ∗ ,k ∗ = 0 and δn ∗ ,m ∗ ,k ∗ = 1 into si to form two new nodes s|S|+1 and s|S|+2 , respectively. Remove si , dsi , lsi , and u si from sets S, D, L, and U, respectively. for j ∈ {1, 2} do if s|S|+ j is infeasible then Continue. else ∗ ∗ ∗ Solve s|S|+ j based on PHR-AL method to obtain q ∗ , xn,m,k , δn,m,k , and pn,m,k . ∗ if q < L then Continue. else Add s|S|+ j , u s|S|+ j = q ∗ , and ds|S|+ j = dsi + 1 into S, U, and D, respectively. ∗ ∗ Round δn,m,k and substitute the rounded δn,m,k into s|S|+ j to produce s˜|S|+ j . if s˜|S|+ j is feasible then Solve s˜|S|+ j to obtain its objective value q˜ ∗ , and add ls|S|+ j = q˜ ∗ into L. else Add ls|S|+ j = −∞ into L. end if end if end if end for U ← max {u s }, L ← max{ls } u s ∈U

ls ∈L

34: Remove s, ds , ls , and u s from sets S, D, L, and U, respectively, if u s < L, ∀s ∈ S. 35: end while

sively forced to take on integral values [23]. The detailed procedures of the developed BB method are summarized in Algorithm 7.2. With BB method, the first step of solving P3 with given λ is to solve its continuous relaxation, i.e., the problem defined by relaxing the integrality constraints on the variable set ψ with 0 ≤ δn,m,k ≤ 1, ∀n ∈ N, ∀m ∈ M, ∀k ∈ K, which is represented as node s from a mathematical point of view in Algorithm 7.2. Since P3 is a concave MINLP problem, its continuous relaxation, s, is a concave NLP problem with

7.4 Optimal Resource Allocation

103

continuous variables, and therefore easily solved. Hence, we can use Lagrangian method to solve s. Here, the Powell-Hestenes-Rockafellar augmented Lagrangian (PHR-AL) method is adopted, as shown in Appendix 2. Algorithm 7.2 starts by finding the global optimal solution to s. If the obtained ∗ ∈ {0, 1}, ∀n ∈ N, ∀m ∈ solution of each relaxed binary variable is integral, i.e., δn,m,k M, ∀k ∈ K, it is optimal. Thus, we have the optimal solution to P3 with given λ, ∗ ∗ ∗ ∗ ∗ ∗ , δn,m,k , and pn,m,k , where xn,m,k = δn,m,k pn,m,k . which is explicitly indicated by xn,m,k In such a case, the BB method is terminated, as shown from Lines 2 to 5. Otherwise, we need to form two new sub-problems (nodes) by enforcing one selected continuous relaxed variable from one selected node to be 0 and 1 respectively, which is called the branching process in BB method, as shown from Lines 7 to 14. The efficiency of BB method usually depends on the strategy used for selecting the branching nodes and branching variables. From Lines 7 to 12, the node, which currently has the highest optimal objective value, is selected for branching to minimize the total amount of computation [23]. The “while” loop from Lines 9 to 11 is due to the fact that a node at depth of (N + 1)M K , which corresponds to a sub-problem in which all the (N + 1)M K binary variables δn,m,k ’s have fixed binary values, cannot be branched anymore. In Line 12, we select the most fractional binary variable, which is farthest from its nearest integer value. This corresponds to getting the greatest improvement of the objective value when branching is performed so that more nodes can be fathomed at an early stage [23]. Obviously, the earlier the nodes are fathomed, the lower the computational complexity. After branching selected node on selected variable, two new nodes are generated, as shown in Line 13. For each new node, we first check whether it is feasible with certain fixed binary variables, as shown in Lines 16 and 17. If it is feasible, we can solve it based on PHR-AL method, which is outlined in Appendix 2, to obtain its optimal objective value q, q ∗ , and the related optimal ∗ ∗ ∗ , δn,m,k , and pn,m,k , as shown in Line 19. Given the optimal objective solutions xn,m,k ∗ value q of one node, we can fathom it if q ∗ is lower than the global lower bound, as shown in Lines 20 and 21, since this node cannot yield a better solution to P3 with given λ than the node that has already reached the global lower bound. From Lines 23 to 29, the related upper and lower bounds of each feasible new node are obtained, which is called the bounding process in BB method. Here, the global lower bound L is nondecreasing, and the global upper bound U is nonincreasing [19]. After updating the global upper and lower bounds based on the upper and lower bounds of the nodes belonging to S, as shown in Line 33, we further fathom the nodes in S if their upper bounds are lower than the new global lower bound L, as shown in Line 34. Finally, the BB method stops when the difference between the global upper bound U and global lower bound L is no larger than the required tolerance  [19], as shown in Line 6. In the worst case, we end up solving the concave sub-problems at most 2(N +1)M K times, i.e., performing an exhaustive search. Actually, some generated new nodes can be fathomed immediately due to the exclusive RB allocation policy of constraint (7.7b), which would decrease the computational complexity of BB method significantly.

104

7 Energy-Efficient Resource Allocation in Heterogeneous OFDMA Networks

7.4.3 Computational Complexity Analysis As the outer loop iteration converges at a superlinear convergence rate, the computational complexity of the two-loop resource allocation scheme is dominated by the inner loop, whose complexity is mainly determined by solving at most 2(N +1)M K concave nonlinear continuous relaxations based on PHR-AL method. However, due to the fathom operation in BB method, as shown in Algorithm 7.2, it is noticed that the actual iteration times of the inner loop are usually much lower than 2(N +1)M K . In PHR-AL method, we transform the constrained optimization problem as an unconstrained problem, which is solved through Newton’s method. Since we have (N + 1) M K variables in the unconstrained problem, as shown in (7.24) of Appendix 2, the Hessian matrix involved in Newton’s method has a dimension of (N + 1) M K × (N + 1) M K . As such, the computational complexity of the PHRAL method, which mainly lies in the computation of Newton step that involves  matrix inversion, is O ((N + 1) M K )3 [24]. Therefore, we conclude that the computational complexity of the developed global optimal resource allocation scheme is   O 2(N +1)M K ((N + 1) M K )3 in the worst case.

7.5 Sub-Optimal Resource Allocation The global optimal resource allocation scheme is presented in Sect. 7.4, and its computational complexity increases exponentially with the problem size [19]. In this section, we decompose the energy-efficient resource allocation problem into two sub-problems consisting of RB allocation and TPC by determining ψ and ϕ successively to obtain the sub-optimal solution with lower computational complexity. It is noticed that the problem decomposition may make P3 infeasible due to the fact that the RB allocation results may not belong to the domain of P3.

7.5.1 RB Allocation In the following, Theorem 7.2 illustrates the principle of RB allocation for the downlink of heterogeneous OFDMA networks from the EE perspective. Theorem 7.2 The 2 EE, η E E , is a strictly monotonically increasing function of channel gain h n,m,k  . With no constraint on the maximum transmit power, δn,m,k ∈ ψ shall satisfy the following restriction for the downlink of heterogeneous OFDMA networks from the EE perspective. N  n=0

δn,m,k ≤ 1, ∀m ∈ M, ∀k ∈ K.

(7.16)

7.5 Sub-Optimal Resource Allocation

105

Algorithm 7.3 RB Allocation Scheme 1: Initiate δn,m,k = 1 and δn,m,k = 0, ∀n ∈ N, ∀m ∈ M, ∀k ∈ K, user set M ← M, and the RB set associated with eNB n Kn ← φ, ∀n ∈ N. 2: while M = φ do 3: for each user m ∈ M do  2   max 2   h  ← max h 4: δ m n,m,k n,m,k  n∈N,k∈K

5: 6: 7:

end for  2   m ∗ ← arg min h max m m∈M  2    {n ∗ , k ∗ } ← arg max δn,m ∗ ,k h n,m ∗ ,k  n∈N,k∈K

δn ∗ ,m ∗ ,k ∗ ← 1; δn,m ∗ ,k ∗ ← 0, ∀n ∈ N; δn ∗ ,m,k ∗ ← 0, ∀m ∈ M. M ← M − {m ∗ }, Kn ∗ ← Kn ∗ + {k ∗ } end while Reset user  set M ← M. while n∈N,m∈M ,k∈K δn,m,k > 0 do for each eNB max n ∈ N do Pn Pn ← max{1,|K n |} end for for each  user m ∈ M do if n∈N,k∈K δn,m,k = 0 then M ← M − {m} else   K N 2   δn,m,k pn |h n,m,k | ←B 20: Rm log2 1+ N0 B

8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19:

k=1

21: 22: 23: 24:

/R th κm ← R m m end if end for m ∗ ← arg min {κm }

25:

{n ∗ , k ∗ }

n=0

m∈M

 2    ← arg max δn,m ∗ ,k h n,m ∗ ,k  n∈N,k∈K

26: δn ∗ ,m ∗ ,k ∗ ← 1; δn,m ∗ ,k ∗ ← 0, ∀n ∈ N; δn ∗ ,m,k ∗ ← 0, ∀m ∈ M. 27: Kn ∗ ← Kn ∗ + {k ∗ } 28: end while

Proof The proof is presented in Appendix 3.



Theorem 7.2 indicates that it is preferred to choose only one optimal eNB to transmit for each user on its assigned RBs from the EE perspective, which is contradictory to our intuition of increasing the transmit diversity to improve the throughput from SE perspective [25]. According to Theorem 7.2, we develop the RB allocation algorithm, as shown in Algorithm 7.3. The RB allocation procedure can be divided into two parts. First, as optimizing the overall EE in (7.7a) can be realized by maximizing the minimum EE of individual user [26] and EE is strictly increasing with channel gain, we iteratively allocate one user with its most favorite RB if the user’s maximum channel gain is minimum, which is depicted from Lines 2 to 10. Then, in the subsequent RB

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7 Energy-Efficient Resource Allocation in Heterogeneous OFDMA Networks

Algorithm 7.4 FP-based TPC Algorithm 1: 2: 3: 4: 5: 6: 7: 8:

Determine ψ based on Algorithm 7.3. Initiate i = 0, λi = 0, and set tolerance . Solve the concave NLP P4 with λi to obtain the optimal ϕ, ϕi , based on PHR-AL method.  while F λi > do i ←i +1  λi ← ς Pt +PRs +ξ R with ϕi−1 Solve P4 with λi to obtain ϕi based on PHR-AL method. end while

allocation iteration, the user with the minimum ratio of potential throughput and required throughput, κm , occupies its most favorite RB among all unassigned ones to improve its throughput. The RB allocation iteration process will proceed until there are no available RBs, which is described from Lines 12 to 28. If the RB allocation result of Algorithm 7.3 satisfies all constraints of P3 simultaneously, i.e., κm ≥ 1, ∀m, the procedure of Algorithm 7.3 implies that there exists at least one feasible solution for the following TPC, which means that a feasible output from the sub-optimal algorithm is guaranteed. If the constraints of P3 cannot be satisfied simultaneously, i.e., ∃m, s.t. κm < 1, there exists the chance that there may be no feasible solution for the TPC with the output of Algorithm 7.3 as the output of Algorithm 7.3 may lie outside the feasible region of P3. It is noticed that the fact of not guaranteeing the feasible output from sub-optimal algorithm is due to the characteristic of problem decomposition. The same situation exists in [27, 28], where a feasible output from sub-optimal algorithm cannot be guaranteed. According to (7.7b) and (7.16), it is easy to conclude that Algorithm 7.3 needs to perform the RB allocation min{N + 1, M} · K times, i.e., n∈N,m∈M,k∈K δn,m,k = min{N + 1, M} · K . Hence, the computational complexity of Algorithm 7.3 is O (min{N + 1, M} · K ).

7.5.2 Transmit Power Control In previous subsection, we have presented a heuristic algorithm to perform the RB allocation by taking the per-user throughput requirement into account under the assumption that the total transmit power of each eNB is uniformly distributed on the RBs assigned by itself, as shown in Line 14 of Algorithm 7.3. Given the RB allocation results, we develop the TPC scheme to derive sub-optimal TPC through transforming the original MINLFP problem into a concave NLP problem. With given RB allocation results, we can transform P1 based on FP as P4 : max ϕ



R − λ (ς Pt + Ps + ξ R)

s.t. (7.7c) and (7.7d)



(7.17)

7.5 Sub-Optimal Resource Allocation

107

where λ is the non-negative parameter, which can be derived based on Dinkelbach method, as shown in Algorithm 7.4. Obviously, P4 is a concave NLP problem. With given λ , the concave NLP P4 can be solved based on PHR-AL method, as shown in Appendix 2. Given the values of δn,m,k ’s, we are only concerned with the value of pn,m,k with δn,m,k = 1 in P4. As a result, the size of interested pn,m,k ’s is equal to min{N + 1, M} · K . Hence, the computational complexity of Algorithm  7.4, which is dominated by that of PHR-AL method, is O (min{N + 1, M} · K )3 .

7.6 Performance Evaluation The system performance of the developed energy-efficient resource allocation schemes is thoroughly evaluated by means of extensive system-level simulations

Table 7.1 System-level simulation parameters Parameter Cell radius Carrier frequency Bandwidth Total number of RBs Shadowing User/eNB antenna number Antenna pattern Minimum distance between user and MeNB Minimum distance between user and SeNB Minimum distance between users MeNB power class SeNB power class User’s noise figure

Setting 250 m 2.0 GHz 10 MHz 50 RBs 8 dB log-normal 1 Omnidirectional ≥35 m ≥10 m ≥3 m 46 dBm (40 W) 30 dBm (1 W) 7 dB

Table 7.2 The CPU time (Second) of optimal and sub-optimal EE schemes with different network configurations RB K =2 K =3 K =4 K =5 K =6 K =7 K =8 number Optimal EE scheme Suboptimal EE scheme

6.05

11.23

17.86

49.88

87.82

309.87

969.41

0.0540

0.0546

0.0590

0.0606

0.0621

0.0631

0.0650

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7 Energy-Efficient Resource Allocation in Heterogeneous OFDMA Networks

Fig. 7.2 Energy efficiency of different resource allocation schemes. (N = 1 and M = 2)

Fig. 7.3 Network throughput of different resource allocation schemes. (N = 1 and M = 2)

Fig. 7.4 Cumulative distribution function of the number of inspected nodes in Algorithm 7.2. (N = 1 and M = 2)

7.6 Performance Evaluation

109

Table 7.3 The value of λ versus iteration times in Algorithm 7.1 RB Number Once Twice 3 times K K K K K K K

=2 =3 =4 =5 =6 =7 =8

0.026e6 0.034e6 0.043e6 0.068e6 0.083e6 0.073e6 0.085e6

0.061e6 0.076e6 0.088e6 0.14e6 0.16e6 0.17e6 0.19e6

0.071e6 0.089e6 0.099e6 0.16e6 0.18e6 0.20e6 0.21e6

Table 7.4 The value of λ versus iteration times in Algorithm 7.4 RB number Once Twice 3 times K K K K K K K

=2 =3 =4 =5 =6 =7 =8

0.026e6 0.034e6 0.035e6 0.042e6 0.065e6 0.073e6 0.081e6

0.061e6 0.074e6 0.081e6 0.087e6 0.13e6 0.17e6 0.18e6

0.071e6 0.087e6 0.092e6 0.096e6 0.15e6 0.19e6 0.20e6

4 times 0.071e6 0.089e6 0.099e6 0.16e6 0.18e6 0.20e6 0.21e6

4 times 0.071e6 0.087e6 0.092e6 0.096e6 0.15e6 0.19e6 0.20e6

based on 3rd Generation Partnership Project methodology [29]. The system parameters are given in Table 7.1. For sake of simplicity, √ we assume that the SeNBs lie on the circle centered at MeNB with a radius (3 − 3)r/2 [10], where r = 250 m is the radius of the cell, and the central angle between two SeNBs is 2π/N . The reciprocal of drain efficiency of power amplifier ς , dynamic circuit power consumption parameter ξ , and static circuit power consumption Ps are assumed as 1/0.38, 2W/Mbps, and 15W, respectively [4]. Figures 7.2 and 7.3 give the EE and network throughput of different resource allocation schemes for the downlink of heterogeneous OFDMA networks. The “EE: Optimal” and “EE: Sub-Optimal” results are obtained based on the optimal and suboptimal schemes presented in Sects. 7.4 and 7.5, respectively. The “SE: Optimal” and “SE: Sub-Optimal” results are derived by setting λ = 0 in Algorithm 7.1 and λ = 0 in Algorithm 7.4 respectively, as mentioned at the end of Sect. 7.3. The “EE: Relaxation” and “SE: Relaxation” results are obtained based on the binary constraint relaxation method [13, 30, 31]. It is noticed that the performance of sub-optimal scheme is very close to that of binary constraint relaxation method. Due to the exponential complexity of the optimal scheme, as shown in Fig. 7.4 and Table 7.2, the number of SeNBs, N , and the number of users, M, are constrained to be 1 and 2, respectively, in Figs. 7.2 and 7.3. In Fig. 7.4, the cumulative distribution function of the number of inspected nodes needed to reach the global optimal solution is plotted to depict the complexity of the BB method in Algorithm 7.2. In Table 7.2, the CPU time, which is proportional to the complexity of the algorithm, is adopted as a simple metric to evaluate the relative complexity to some extent, although it

110

7 Energy-Efficient Resource Allocation in Heterogeneous OFDMA Networks

Fig. 7.5 Spectral efficiency of sub-optimal schemes with varying SeNB number. (M = 10)

is very sensitive to hardware configuration and the possibility of other programs or tasks running at the same time. The convergence behaviors of Algorithms 7.1 and 7.4 are shown in Tables 7.3 and 7.4, respectively, by considering a snapshot of a scenario with N = 1 and M = 2. We can observe that both the Algorithms 7.1 and 7.4 converge very fast, namely in no more than 3 iterations, which is in line with the fact that Newton’s method has order-two convergence. Figure 7.2 shows that both the sub-optimal and optimal EE schemes improve the EE of the networks by more than 100% compared with the sub-optimal and optimal SE schemes. It is noticed that the EE improvements of sub-optimal and optimal EE schemes are at the cost of degraded network throughput, as shown in Fig. 7.3. In Fig. 7.2, the EE of sub-optimal EE scheme is close to that of optimal EE scheme, however, the computational complexity of the former is only a tiny fraction of that of the latter, as shown in Table 7.2. It is shown that the computational complexity of sub-optimal EE scheme increases slowly with the problem size and that of optimal EE scheme increases exponentially with the problem size. This implies that the sub-optimal EE scheme makes a good tradeoff between the EE performance and computational complexity. Figure 7.2 also shows that the EEs of all the schemes increase with the RB number due to the frequency selection diversity gain. Figure 7.5 shows the SE of the sub-optimal schemes with varying number of SeNBs. It further demonstrates that the EE enhancement of the sub-optimal EE scheme is at the price of degraded SE performance. In Fig. 7.5, we observe that the SE increases with the number of SeNBs. This can be explained directly by the fact that deploying more SeNBs would decrease the pathloss between one user and its serving eNB, and increase the channel capacity consequently. Figures 7.6 and 7.7 depict the EE of the sub-optimal schemes with varying number of SeNBs and varying number of users, respectively. In Fig. 7.6, we observe a significant EE performance improvement with the increase of the number of SeNBs. In Fig. 7.7, we observe an EE improvement with the increase of the number of users, which is resulted from the multiuser diversity gain. In both Figs. 7.6 and 7.7, it is

7.6 Performance Evaluation

111

Fig. 7.6 Energy efficiency of sub-optimal schemes with varying SeNB number. (M = 10)

Fig. 7.7 Energy efficiency of sub-optimal schemes with varying user number. (N = 10)

shown that the sub-optimal schemes with K = 25 outperform that with K = 15 in terms of EE performance. The reason is that the larger the number of RBs, the higher the frequency selection diversity gain. Furthermore, Figs. 7.5 and 7.6 imply that deploying more SeNBs in the heterogeneous OFDMA networks could improve the EE and SE simultaneously, which is in agreement with [6, 32]. According to (7.6), the upper bound of the EE here is 1/ξ = 0.5 Mbits/Joule. Figures 7.6 and 7.7 show that the EE of the developed sub-optimal EE scheme is close to the upper bound with the increase of the number of SeNBs and the number of users respectively, which further demonstrates the effectiveness of our developed scheme.

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7 Energy-Efficient Resource Allocation in Heterogeneous OFDMA Networks

7.7 Conclusions In this chapter, we have investigated the energy-efficient resource allocation problem for downlink heterogeneous OFDMA networks. Through FP, the original MINLFP problem is transformed into an equivalent parametric subtractive form, which is optimally solved using the Dinkelbach and BB methods. In order to decrease the computational complexity, we have further developed the sub-optimal scheme by performing RB allocation and TPC successively. Simulation results have been presented to demonstrate the effectiveness of the developed schemes and provide useful insights into the practical design of energy-efficient resource allocation in downlink heterogeneous OFDMA networks.

Appendix 1 Proof of Theorem 7.1 From the proof of convergence of the Dinkelbach method in [16], it is shown that λ = max {λ}. Consequently, 0 ≤ λ ≤ max {λ} = λ . From [16, 33], it is shown that λ = max η E E . ψ,ϕ

(7.18)

As R > 0 and ς Pt + Ps + ξ R > 0, η E E > 0, which means λ > 0. Now we prove λ < ξ1 by contradiction. Suppose λ ≥ ξ1 , then 1 − λ ξ ≤ 0 and     F λ = max R − λ (ς Pt + Ps + ξ R) ψ,ϕ    = max 1 − λ ξ R − λ (ς Pt + Ps ) < 0. ψ,ϕ

(7.19)

Obviously, (7.19) is contradictory to the fact that λ is the root of F (λ), and we can conclude that λ < ξ1 . From (7.18) and (7.19), (7.11) is proved.

Appendix 2 Solution to the Continuous Relaxations of P3 in Chap. 7 Based on PHR-AL Method An augmented Lagrangian function is a penalty function that depends on a penalty parameter, as well as on the Lagrange multiplier vectors associated with the constraints of the problem [34]. As augmented Lagrangian methods are very popular for solving NLP problems and one of the most famous augmented Lagrangian methods with inequality constraints is Powell-Hestenes-Rockafellar (PHR) method, here we adopt the augmented Lagrangian function of PHR directly to the NLP continuous

7.7 Conclusions

113

relaxations [34, 35] and use Newton’s method to solve the resulted unconstrained problem [35, 36]. It is noticed that the objective value of P3 is determined by the cross product of δn,m,k and pn,m,k , i.e., xn,m,k . Hence, we can solve P3 by only concerning the objective function (7.20) with (7.12b) and (7.12c) to derive the optimal solution X∗ firstly, and then construct the optimal ψ ∗ and ϕ ∗ based on X∗ while guaranteeing (7.7b), (7.12d), and (7.12e). As such, we reformulate P3 as ˜ : max P3 X





 R˜ − λ ς P˜t + Ps + ξ R˜

(7.20)

s.t. (7.12b) and (7.12c). With continuous relaxed variable δn,m,k , namely 0 ≤ δn,m,k ≤ 1, it is easy to verify that xn,m,k ≤ pn,m,k by replacing xn,m,k with δn,m,k pn,m,k . From (7.12d), we have δn,m,k ≥

xn,m,k . pnmax

(7.21)

Substituting pn,m,k = xn,m,k /δn,m,k , if xn,m,k > 0, into (7.12e) yields xn,m,k ≥

xn,m,k + pnmax δn,m,k − pnmax . δn,m,k

(7.22)

xn,m,k ≤ δn,m,k ≤ 1. pnmax

(7.23)

From (7.22), we have

∗ ∗ and δn,m,k based on X∗ is to set It is noticed that a simple way of deriving pn,m,k ∗ ∗ ∗ ∗ max max = pn and δn,m,k = xn,m,k / pn . We can observe that δn,m,k = xn,m,k / pnmax satisfies constraint (7.7b). Otherwise, constraint (7.12b) would be violated. Theorem 2 7.2 indicates that EE is strictly monotonically increasing in channel gain h n,m,k  , which means that it is reasonable to assign higher transmit power on the channel ∗ ∈ with higher channel gain from the EE perspective. As such, we can adjust pn,m,k    2   2 ∗ ∗ max     0, pn according to the strategy of pn,m,k ≤ pn ,m ,k , if h n,m,k ≤ h n ,m ,k , ∗ ∗ ∗ ∗ and δn,m,k = xn,m,k / pn,m,k satisfy (7.7b), while ensuring that the updated pn,m,k (7.12d), and (7.12e). Now we outline how to derive X∗ through PHR-AL method. The corresponding augmented Lagrangian of PHR is given by [34, 35] ∗ pn,m,k

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7 Energy-Efficient Resource Allocation in Heterogeneous OFDMA Networks

 + 2 N

 ρ θ n f n (X) + L ρ (X, θ , ν) =λ ς P˜t + Ps − (1 − λξ ) R˜ + 2 n=0 ρ     2 M ρ νm + gm (X) + + (7.24) 2 m=1 ρ M K max th ˜ where f n (X) = m=1 k=1 x n,m,k − pn , gm (X) = Rm − Rm , ρ is a positive penalty parameter, and θ = {θn } and ν = {νm } are the Lagrange multiplier vectors associated with f n (X) and gm (X), respectively. ˜ is now converted into From (7.24), the constrained optimization problem P3 an unconstrained optimization problem, which could be solved through Newton’s method [36]. Given the values of ρ, θ , ν, and initial X0 , the updated X is given by  ∇ L ρ (X, θ , ν)X=X0  X = X0 − 2 ∇ L ρ (X, θ , ν)

(7.25)

X=X0

where ∇ L ρ (X, θ , ν) =λς 1(N +1)M K ×1 − (1 − λξ ) ∇ R˜ + +





(ρ f n (X) + θn ) ∇ f n (X)

n∈Sn

(ρgm (X) + νm ) ∇gm (X)

(7.26)

m∈Sm

∇ 2 L ρ (X, θ , ν) = − (1 − λξ ) ∇ 2 R˜ + ρ +ρ

 m∈Sm



∇ f n (X) ∇ f n (X)T

n∈Sn

∇gm (X) ∇gm (X)T +



(ρgm (X) + νm ) ∇ 2 gm (X)

m∈Sm

(7.27) with 1(N +1)M K ×1 being the column vector of all ones with size of (N + 1)M K × 1, Sn = {n ∈ N |ρ f n (X) +θn > 0}, Sm = {m ∈ M |ρgm (X) + νm > 0 }, and superscript T denoting the matrix transpose operation. The detailed procedure of deriving X through PHR-AL method is shown in Algorithm 7.5, where θ k = {θn,k }, ν k = {νm,k }, ωk = {ωn,k },  k = {m,k }, parameter κ is used to check whether the penalty value needs to be increased, and ·∞ denotes the infinity norm. Similar to solving the continuous relaxation s, it is easy to solve sn ∈ S, the continuous relaxation of P3 with depth dsn , by the use of PHR-AL method.

7.7 Conclusions

115

Algorithm 7.5 Deriving X∗ through PHR-AL method 1: Initialize k ← 0, , ρk , θ k , ν k, and Xk .   θ ν . 2: ωn,k ← max f n (Xk ) , − ρn,kk and m,k ← max gm (Xk ) , − ρm,k k 3: while max {ωk ∞ ,  k ∞ } > do 4: Solve (7.24) with given ρk , θ k , ν k , and Xk through Newton’s method, as shown in (7.25), to obtain Xk+1 .     θ ν . 5: ωn,k+1 ← max f n (Xk+1 ) , − ρn,kk and m,k+1 ← max gm (Xk+1 ) , − ρm,k k   6: if k = 0 or max ωk+1 ∞ ,  k+1 ∞ < κ max {ωk ∞ ,  k ∞ } then 7: ρk+1 ← ρk 8: else 9: ρk+1 ← 2ρk 10: end if + +   11: θn,k+1 ← θn,k + ρk+1 f n (Xk+1 ) and νm,k+1 ← νm,k + ρk+1 gm (Xk+1 ) . 12: k ← k + 1 13: end while

Appendix 3 Proof of Theorem 7.2  2 The channel gain h n,m,k  is not involved into the downlink transmission with δn,m,k = 0, and consequently it has no impact on the total power consumption and network throughput. Here, we are only concerned with the impact of channel gain   h n,m,k 2 on EE with δn,m,k = 1.  2 Taking the first derivative of η E E with respect to channel gain h n,m,k  yields ∂η E E B (ς Pt + Ps ) pn,m,k >0  2 = P 2 α ln 2 ∂ h n,m,k 

(7.28)

 2 N δn,m,k pn,m,k h n,m,k  . It is shown from (7.28) that the EE, where α = N0 B + n=0  2 η E E , is a strictly monotonically increasing function of channel gain h n,m,k  . From  2  2 (7.28), letting h m,k   max h n,m,k  , we have n∈N

  N |h |2  log2 1 + Nm,k δ p n,m,k n,m,k 0B n=0 k=1 <   K N  |h m,k |2  P +ξB log2 1 + N0 B δn,m,k pn,m,k R + B

ηE E

K 

k=1

(7.29)

n=0

 where R = mM =1,m =m Rm and P = ς Pt + Ps + ξ R . Eq. (7.29) demonstrates that we can improve the EE by re-allocating the transmit power from multiple eNBs  N to one eNB associated with the maximum channel gain, which means that n=0 δn,m,k ≤ 1, ∀m ∈ M, ∀k ∈ K, from EE perspective.

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7 Energy-Efficient Resource Allocation in Heterogeneous OFDMA Networks

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Chapter 8

Energy-Efficient Power Control for D2D Communications

This chapter investigates the energy-efficient power control for device-to-device (D2D) communications. Section 8.1 introduces the motivation of developing energyefficient power control schemes for D2D communications. Section 8.2 presents the system model, and Sect. 8.3 formulates the energy-efficient power control problem. Sections 8.4 and 8.5 derive the optimal and sub-optimal solutions, respectively. Section 8.6 presents the simulation results, and Sect. 8.7 concludes this chapter.

8.1 Introduction In Chap. 7, we investigate the energy-efficient resource allocation in the networks, where a macro evolved Node B (eNB) and multiple small cell eNBs exist. It is noticed that besides the coexistence of macro cells and small cells, D2D communications have also been considered as a key component for future heterogeneous networks to meet the exponentially increased demand for high data rates [1–3]. In D2D communications, one user equipment (UE) transmits signals to its neighboring UE over a direct link between them instead of the cellular mode via an eNB. Therefore, D2D communications can simultaneously improve the network spectral efficiency (SE) and reduce the power consumption of UE, i.e., increase the energy efficiency (EE) [3, 4]. It has been shown that whether D2D pair could share the resource with cellular UE is affected by the system parameters, and the underlay mode is preferred especially when the cellular UE is closer to eNB than the D2D pair in the uplink [5]. Within underlay mode, transmit power has a two-fold effect on the system performance. The higher the transmit power, the stronger the intended received signal, and also the more severe the interference to other signals. Furthermore, the interference makes the Shannon capacity formula non-concave in transmit power [6, 7]. We notice that it is © Beijing Institute of Technology Press 2023 J. An et al., Sustainable Wireless Communications, https://doi.org/10.1007/978-981-19-0448-6_8

119

120

8 Energy-Efficient Power Control for D2D Communications

possible to further improve the resource efficiency or frequency reuse factor by concurrently enabling multiple D2D pairs to share the same resource [8], especially when the short distance characteristics of D2D communications are considered. Motivated by this, we investigate the energy-efficient power control through considering both the total EE and individual EE, where uplink resource blocks (RBs) allocated to one cellular UE are reused by multiple D2D pairs.

8.2 System Model We consider a single-cell cellular network,1 where M D2D pairs reuse the RBs allocated to one cellular UE2 and all the nodes are equipped with a single antenna, as shown in Fig. 8.1. Since uplink resource is preferred for D2D communications [9, 10], we assume that there are K contiguous RBs assigned to the cellular UE. For the sake of convenience, we denote the cellular link between cellular UE and eNB and the D2D link of D2D pair m as the zeroth and mth links, respectively. Denoting pm,k , 1 ≤ m ≤ M, and p0,k as the transmit powers of D2D pair m and cellular UE, respectively, on RB k, the received signal-to-interference-plus-noise ratio (SINR) of link m on RB k, 0 ≤ m ≤ M and 1 ≤ k ≤ K , is given by γm,k =

  h mm,k 2 pm,k M  m  =0,m  =m

  h mm  ,k 2 pm  ,k + N0

(8.1)

where h mm,k and h mm  ,k are the channel coefficients from the receiver of link m to the transmitter of link m and to the transmitter of link m  , respectively, on RB k, and N0 is the variance of zero mean additive white Gaussian noise. Here, perfect channel state information (CSI) of all the involved links is assumed at the eNB, which performs the energy-efficient power control based on the available CSI. To acquire perfect CSI, in addition to normal cellular measurement and reporting procedures, the D2D transmit UE can transmit probe signals, which are then measured at the D2D receive UE and the interference victims and reported to the eNB [11–13]. For example, D2D receive UE and the interference victims could measure the CSI based on the received sounding reference signal (SRS) from D2D transmit UE, and then report the CSI measurement to eNB through the physical uplink control channel (PUCCH) in long-term evolution (LTE) networks. In practical application, average based design 1

In single-cell networks, the intra-cell co-channel interference makes the resource allocation problem to be a non-convex “difference of two concave functions” (D.C.) programming problem. For multiple cell scenario, the co-channel interference consists of intra-cell interference and inter-cell interference, and the related resource allocation problem is still a D.C. programming problem, which means that the developed schemes in this chapter are also applicable for multiple cell scenario. 2 For the scenario that multiple D2D pairs reuse the RBs of multiple cellular UEs, the related energy-efficient power control problem is in the same form as the problem in this chapter.

8.2 System Model

121

Fig. 8.1 Illustration of system model. There exists interference between different links due to resource sharing

M

outperforms the instantaneous power control scheme in terms of lower signaling overhead. In average based design, the energy-efficient power control problem is a chance constrained program [14] that emphasizes on long-term performance of the networks, and its performance is upper bounded by that of our developed scheme with perfect CSI [15]. From (8.1), the achievable throughput of D2D pair m is Rm = B

K 

  log2 1 + γm,k

(8.2)

k=1

where B is the bandwidth of one RB. The power consumption of UE contains the power consumption of radio frequency power amplifier and that of other circuits incurred by signal processing and active circuit blocks [16], and the circuit power consumption can be modelled as the sum of a static term and a dynamic term [17], i.e., Pc = V Ileak + AC f V 2 , where V is the transistor’s supply voltage, Ileak is the leakage current, A is the fraction of gates actively switching, C is the circuit capacitance, and f is the clock frequency. Assuming that the frequency is dynamically scaled with the sum rate [18], the circuit power consumption of transmit UE of D2D pair m can be modelled as Pc,m = Ps,m + ξ Rm

(8.3)

where Ps,m is the static term, ξ is a constant denoting dynamic power consumption per unit throughput. As such, the power consumption of the transmit UE of D2D pair m is given by Pm = ς

K 

pm,k + Ps,m + ξ Rm

k=1

where ς is the reciprocal of drain efficiency of power amplifier.

(8.4)

122

8 Energy-Efficient Power Control for D2D Communications

Based on (8.2) and (8.4), the individual EE of D2D pair m is defined as ηm =

Rm Pm

(8.5)

η=

R P

(8.6)

and the total EE is defined as

M M where R = m=0 Rm and P = m=0 Pm . From (8.5) and (8.6), it is easy to conclude that both the total EE and individual EE are lower than 1/ξ . Obviously, the higher the dynamic circuit power consumption per unit throughput, the lower the EE.

8.3 Problem Formulation In this section, we will formulate the energy-efficient power control problem for D2D communications, where both the total EE and individual EE are considered.

8.3.1 Total EE In order to maximize the total EE while guaranteeing the quality of service (QoS) of each D2D pair with limited bandwidth and constrained transmit power, the power control problem for D2D communications underlaying uplink cellular networks is formulated as P1 : max η

(8.7a)

p

s.t.

K 

pm,k ≤ pmmax , 0 ≤ m ≤ M

(8.7b)

k=1

Rm ≥ Rmth , 0 ≤ m ≤ M

(8.7c)

   T where pm,k ∈ 0, pmmax , p = p0 , p1 ,. . . , pm , . . . , p M is the transmit power vec tor of size (M K + K ) × 1 with pm = pm,1 , . . . , pm,k , . . . , pm,K being the transmit power vector of D2D pair m on the K RBs, and pmmax and Rmth represent the maximum allowed total transmit power and the minimum required throughput of D2D pair m, respectively. Constraint (8.7c) guarantees the minimum required throughput of each user.

8.3 Problem Formulation

123

Similar to [11, 19], we enforce the minimum throughput constraint for each user instead of constraining the interference power directly in P1. For single subchannel scenario, enforcing a constraint on the throughput of one user is equivalent to constraining the interference power [19]. However, for the scenario of multiple subchannels, enforcing the minimum throughput constraint for each user, which may result in that some subchannels suffer from severe interference and the throughput on these subchannels are close to zero, is different from a constraint on the co-channel interference. Actually, constraining the interference power on each subchannel is more rigorous than enforcing the minimum throughput constraint for each user. Problem P1 is a non-concave fractional programming (FP) problem. By exploiting the property of FP [20, 21], it can be transformed into a parametric subtractive form as P2 : max {R − λP}

(8.8)

p

s.t. (8.7b) and (8.7c) where λ is a non-negative parameter, and R − λP =B (1 − λξ )

M  K 

M  M K     log2 1 + γm,k − λς pm,k − λ Ps,m .

m=0 k=1

m=0 k=1

m=0

Define F (λ) = max {R − λP} .

(8.9)

p

Function F (λ) is continuous and strictly monotonic decreasing in λ and has an unique root [21]. Notably, the optimal solution set p of P1 is the same as that of P2 with λ = λ , where λ is the root of F (λ) and can be derived using the Dinkelbach method. In addition, λ is equal to the optimal EE, i.e., λ = max η [20, 21]. Within p

Dinkelbach method, as shown in Algorithm 8.1, the value of λ increases in each iteration until convergence to λ . As η < ξ1 , we can conclude that 1 − λξ > 0. Based on the quotient property of logarithms, the objective function of P2 is reformulated as R − λP = f 1 ( p) − f 2 ( p) where

(8.10)

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8 Energy-Efficient Power Control for D2D Communications

f 1 ( p) =B (1 − λξ )

M  K 

log2 1 +

M  K 

pm,k − λ

m=0 k=1

f 2 ( p) =B (1 − λξ )

gmm  ,k pm  ,k

m  =0

m=0 k=1

− λς

M 

M 

Ps,m

m=0

⎛

M  K 

⎜ log2 ⎝1 +

⎞ M 

⎟ gmm  ,k pm  ,k ⎠

m  =0 m   =m

m=0 k=1

 2 with gmm  ,k = h mm  ,k  /N0 being the channel gain to noise ratio. Since 1 − λξ > 0, both f 1 ( p) and f 2 ( p) are concave. Hence, the objective function of P2 is the difference of two concave functions. Similarly, constraint (8.7c) is also the difference of two concave functions, which is h 1,m ( p) − h 2,m ( p) ≥ Rmth , 0 ≤ m ≤ M

(8.11)

where h 1,m ( p) =B

K  k=1

h 2,m ( p) =B

K  k=1

log2 1 + ⎛ log2 ⎝1 +

M 

gmm  ,k pm  ,k

m  =0 M 

⎞ gmm  ,k pm  ,k ⎠.

m  =0,m  =m

Consequently, P2 is a D.C. programming problem [22], from which we can derive the optimal solution to P1 due to the equivalence between P1 and P2. From P2, it is easy to conclude that the energy-efficient power control problem considered here encompasses the spectral-efficient power control problem as a special case with λ = 0.

8.3.2 Individual EE From the purely EE point of view, total EE is preferred. However, total EE views all the D2D pairs as a whole and tends to maximize the EE of a few links with good channels while the EE of the other links may be too low, i.e., they are not fairly treated. From the perspective of fairness, it is more reasonable to take the individual EE as an optimization objective to improve the minimum individual EE as much as possible [7, 16, 23]. To attain the max-min fairness among D2D pairs, the related power control problem is formulated as

8.3 Problem Formulation

125

P3 : max min ηm p

(8.12)

m

s.t. (8.7b) and (8.11). Problem P3 belongs to maximin generalized fractional programming (GFP) problem [24–26]. By resorting to GFP, we can equivalently transform P3 into a parametric subtractive-form problem as 

P4 : max min p

m

Rm − λ Pm



(8.13)

s.t. (8.7b) and (8.11). where λ is a non-negative parameter. The relationship between P3 and P4 is shown in the following lemma [26]. Lemma 8.1 Denoting the objective values of P3 and P4 as λ = max min p

m

Rm Pm

(8.14)

and     F  λ = max min Rm − λ Pm p

(8.15)

m

respectively, we have   (1) F  λ is decreasing in λ ; (2) F  (λ ) > 0 if and only if λ < λ ; hence, F  (λ ) ≤ 0; (3) If P3 has an optimal solution, then F  (λ ) = 0; (4) If F  (λ ) = 0, then problems P3 and P4 have the same set of optimal solutions. 

Proof The proof is presented in Appendix 1.

 Lemma 8.1 shows  that solving P3 is now equivalent to solving P4 with λ being   the root of F λ . In the following section, we will elaborate that the introduced non-negative parameter λ in P4 can be derived based on generalized Dinkelbach method [26], and the value of λ increases in each iteration until convergence. Since λ < ξ1 , we have 1 − λ ξ > 0. Using the quotient property of the logarithms, Rm − λ Pm is reformulated as K      Rm − λ Pm = 1 − λ ξ h 1,m ( p) − 1 − λ ξ h 2,m ( p) − λ ς pm,k − λ Ps,m . k=1

(8.16) As h 1,m ( p) and h 2,m ( p) are concave in p, (8.16) is a D.C. programming function. Since the minimum of D.C. functions is also a D.C. function [27], the objective

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8 Energy-Efficient Power Control for D2D Communications

  function of P4, min Rm − λ Pm , is a D.C. programming function. Consequently, m

P4 is a D.C. programming problem.

8.4 Optimal Power Control In Sect. 8.3, it has been shown that we can obtain the optimal solutions to the power control problems P1 and P3 through respectively solving the equivalent D.C. programming problems P2 and P4, which maximizes D.C. objective function under D.C. constraints. Usually, it is difficult to optimally solve a D.C. programming problem due to its non-convexity. One popular way to optimally solve the D.C. programming problem is to adopt the branch-and-bound (BB) method. In BB method, the optimal solution is obtained by solving a sequence of linear programming problems, which are approximations of the D.C. programming problem on a sequence of branch tree nodes. In this section, we will elaborate how to optimally solve problems P2 and P4 based on Dinkelbach and BB methods.

8.4.1 Total EE To solve P2, the value of λ shall be determined preliminarily. Here, we design a two-loop scheme to obtain the global optimal solution to P2, where the optimal λ and optimal p are derived in the outer and inner loops based on Dinkelbach and BB methods, respectively.

8.4.1.1

Outer Loop

Based on Newton’s method, Dinkelbach method is used to determine the optimal value of λ with given p iteratively, as shown in Algorithm 8.1, which produces an increasing sequence of λ [21] and converges to the global optimal value at a superlinear convergence rate [28]. In each iteration, the optimal pi is obtained with given λi , which is derived based on pi−1 . If the resultant |F (λi ) | is sufficiently small, i.e., |F (λi ) | ≤ , the iteration is terminated and the optimal p is obtained. Otherwise, we calculate new λi and start the next iteration.

8.4.1.2

Inner Loop

It is noticed that the key step in Algorithm 8.1 lies in solving the non-concave D.C. programming problem P2 with given λi , as shown on Line 6 of Algorithm 8.1, which will be outlined here based on BB method.

8.4 Optimal Power Control

127

Algorithm 8.1 Outer Loop of Deriving λ in P2 based on Dinkelbach Method 1: Set tolerance , and initialize i ← 0 and λi ← 0. 2: Solve the non-concave D.C. programming problem P2 with λi to obtain the optimal solution p, i.e., pi . 3: while |F (λi ) | >  do 4: i ← i + 1 5: λi ← PR with pi−1 6: Solve P2 with λi to obtain pi . 7: end while

Algorithm 8.2 Inner Loop of Solving P2 with Given λi based on BB Method 1: Set tolerance , and initialize global lower bound L = −∞, global upper bound U = ∞, upper bound set U ← φ, and region set S ← φ. 2: U = u s0 = F ˜ (s0 ) and ps0 = arg max F ˜ (s0 ). P2

3: 4: 5: 6: 7: 8:

P2

p

S ← S ∪ {s0 } and U ← U ∪ {u s0 }. if ps0 is feasible for P2 then   L = FP2 ps0 end if while U − L > do s ∗ ← arg max {u s } s∈S

9: Partition s ∗ along its longest edge into two subregions s1 and s2 . 10: S ← S\{s ∗ } and U ← U\{u s ∗ }. ˜ is solvable in subregion si , i ∈ {1, 2}, then 11: if P2 12: u si = F ˜ (si ) and psi = arg max F ˜ (si ). P2

13: 14: 15: 16: 17: 18:

p

P2

S ← S ∪ {si } and U ← U ∪ {u si }. if psi is feasible for P2 then    L = max L , FP2 psi end if end if U ← max {u s } s∈S

19: S ← S\{s} and U ← U\{u s }, if u s < L, ∀s ∈ S. 20: end while

The key idea of the BB method is to generate a sequence of asymptotically tight upper and lower bounds for the optimal value [29]. As its name indicates, the BB method basically branches the feasible region recursively into subregions, which is called branching process. Then, for each branch, it bounds the optimal value from above and below, which is called bounding process. Due to the non-concavity of P2 over the feasible region, BB method searches in different subregions to seek for the global optimal value rather than a local one [22], as shown in Algorithm 8.2.   In P2, the initial feasible region of power allocation is s0 = p 0 ≤ pm,k ≤ pmmax . In the branching process, the initial feasible region s0 is recursively branched into several subregions. The union of the subregions, S, contains the global optimal solution to P2. In S, the subregion associated with the highest upper bound is selected for further branching along its longest edge to minimize the total amount of computation [22, 29], as shown on Lines 8 and 9 of Algorithm 8.2. In each subregion, tight upper

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8 Energy-Efficient Power Control for D2D Communications

and lower bounds for P2 are needed. For the sake of convenience, we use FP (s) and FP ( p) to denote the optimal value of problem P in region s and the value of problem P with solution p, respectively, in Algorithm 8.2. In order to obtain an upper bound for P2, we derive an upper bound and a lower bound for f 1 ( p) and f 2 ( p) in (8.10) according to the geometric feature of concave function as  f 1u ( p) =

p−

pmin + pmax 2

T

    p + pmax pmin + pmax + f1 ∇ f 1 min 2 2 (8.17)

and T      f 2l ( p) = p − pmin ∇ f 2 pmax + f 2 pmin

(8.18)

respectively, where pmin ≤ p ≤ pmax . From (8.10), (8.17), and (8.18), we have R − λP ≤ f 1u ( p) − f 2l ( p) .

(8.19)

Similarly, an upper bound and a lower bound for h 1,m ( p) and h 2,m ( p) in (8.11) are given by  h u1,m

( p) =

p + pmax p − min 2

T

 ∇h 1,m

pmin + pmax 2



 + h 1,m

 pmin + pmax 2 (8.20)

and T      h l2,m ( p) = p − pmin ∇h 2,m pmax + h 2,m pmin

(8.21)

respectively. From (8.11), (8.20), and (8.21), we have Rm ≤ h u1,m ( p) − h l2,m ( p) .

(8.22)

Based on (8.19) and (8.22), we formulate a new problem as ˜ : max P2 p



f 1u ( p) − f 2l ( p)



s.t. h u1,m ( p) − h l2,m ( p) ≥ Rmth , 0 ≤ m ≤ M,

(8.23a) (8.23b)

(8.7b). ˜ is a linear programDue to the linearity of f 1u ( p), f 2l ( p), h u1,m ( p), and h l2,m ( p), P2 ming (LP) problem, which can be solved with sophisticated LP method [30]. It is ˜ As a result, shown from (8.22) that the feasible region of P2 is a subset of that of P2.

8.4 Optimal Power Control

129

˜ but not vice versa. From (8.19), a feasible point to P2 is also a feasible point to P2, ˜ as it is easy to conclude that an upper bound of P2 can be derived by solving P2, ˜ is feasible shown on Lines 2 and 12 of Algorithm 8.2. If the optimal solution to P2 for P2, a lower bound of P2 can be calculated easily, as shown on Lines 5 and 15 of Algorithm 8.2. Otherwise, we will not update the lower bound in BB method, which is realized through the if statements on Lines 4 and 14 of Algorithm 8.2. In Algorithm 8.2, the lower bound is updated by searching the maximum of local lower bounds, as shown on Line 15, and the upper bound is updated by splitting the subregion with the highest local upper bound into two new subregions to generate new tighter upper bound. As a result, the lower and upper bounds are nondecreasing and nonincreasing respectively, which is in agreement with [31]. If Algorithm 8.2 is terminated within limited iterations, we can immediately conclude that Algorithm 8.2 converges to the optimal solution. Otherwise, the longest edge of the subregion goes to zero. Consequently, the difference between the upper and lower bounds uniformly converges to zero, and Algorithm 8.2 would converge eventually [29, 32].

8.4.2 Individual EE According to [25], P4 is equivalent to the following problem P5 : max q

(8.24a)

p,q

s.t. Rm − λ Pm ≥ q, 0 ≤ m ≤ M, (8.7b) and (8.11).

(8.24b)

Problem P5 is also a D.C. programming problem. Due to the equivalence among P3, P4, and P5, solving P3 is now resorting to solving P5. In P5, a new variable, q, is introduced. From Lemma 8.1, we have the following corollary. Corollary 8.1 Denoting q  as the optimal value of q, we have q  = 0.

(8.25)

To solve P5, we also design a two-loop scheme, where the optimal λ and optimal { p, q} are derived in the outer and inner loops based on generalized Dinkelbach and BB methods, respectively.

8.4.2.1

Outer Loop

The procedures of generalized Dinkelbach method are similar to that of Dinkelbach method, as shown in Algorithm 8.3, whose convergence is proved in Appendix 2. The difference between Algorithms 8.1 and 8.3 lies in how to determine the optimal

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8 Energy-Efficient Power Control for D2D Communications

Algorithm 8.3 Outer Loop of Deriving λ in P5 based on Generalized Dinkelbach Method 1: Set tolerance , and initialize i ← 0 and λi ← 0. 2: Solve the non-concave D.C. programming problem P5 with λi to obtain solution p, i.e., pi .   3: while |F  λi | >  do 4: i ← i + 1 5: λi ← min RPmm with pi−1 m

6: Solve P5 with λi to obtain pi . 7: end while

values of λ and λ based on p, as shown on Line 5 of Algorithm 8.1 and Line 5 of Algorithm 8.3, respectively.

8.4.2.2

Inner Loop

In Algorithm 8.3, the key step also lies in solving the non-concave D.C. programming ˜ we can construct a new problem P5 with given λi in each iteration. Similar to P2, problem as follows. ˜ : max q u P5 u

(8.26a)

p,q

u l s.t. f 1,m ( p) − f 2,m ( p) ≥ q u ,

(8.26b)

(8.7b) and (8.23b). where      pmin + pmax pmin + pmax pmin + pmax T + f 1,m ∇ f 1,m 2 2 2 T      l f 2,m ( p) = p − pmin ∇ f 2,m pmax + f 2,m pmin 

u f 1,m ( p) =

p−

K      pm,k − λ Ps,m f 1,m ( p) = 1 − λ ξ h 1,m ( p) − λ ς

  f 2,m ( p) = 1 − λ ξ h 2,m ( p) .

k=1

˜ is also an LP problem. From P5, ˜ it is easy to get an upper bound and Problem P5 a lower bound for P5, based on which we can perform the BB method to solve P5. It is noticed that the procedure for solving P5 with given λi based on BB method is the same with that for solving P2 in Algorithm 8.2, and it is not reproduced here for the sake of brevity.

8.4 Optimal Power Control

131

8.4.3 Complexity Analysis Here, we take the total EE as an example to analyze the complexity of optimal solution in terms of the number of arithmetic operations. As the outer loop iteration converges at a superlinear convergence rate, the complexity of the optimal two-loop power control scheme is dominated by the inner loop. It is noticed that the complexity of inner loop is mainly determined by deriving the upper bound through solving an LP problem in each inspected subregion, as shown on Line 12 of Algorithm 8.2. Using an interior-point method,  the complexity of deriving  the upper bound in one subregion is O (M K + K )3 [30, 33]. However, since the number of inspected subregions of BB method in inner loop is hard to be estimated, the complexity of the inner loop cannot be exactly determined. Nevertheless, since the complexity of BB method increases exponentially with the problem size, the total complexity of optimal solution is exponential in the numbers of variables and constraints.

8.5 Sub-Optimal Power Control In the previous section, we developed two two-loop optimal power control schemes to maximize the total EE and individual EE, respectively. However, their complexity is unaffordable due to the branching process of BB method. Here, we will develop sub-optimal schemes with reasonable complexity.

8.5.1 Total EE The non-concavity of P2 is caused by the interference term involved into the capacity calculation. To decouple the interference from the Shannon capacity formula, we resort to interference temperature approach [34, 35], based on which we have P6 : max η˜ = p

R˜ P˜

(8.27a)

s.t. R˜ m ≥ Rmth , 0 ≤ m ≤ M M  m  =0,m  =m

(8.7b) where

  h mm  ,k 2 pm  ,k ≤ Im,k , 0 ≤ m ≤ M, 1 ≤ k ≤ K

(8.27b) (8.27c)

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8 Energy-Efficient Power Control for D2D Communications

R˜ =

M 

R˜ m

m=0

P˜ =ς

M  K  m=0 k=1

R˜ m =B

K 

pm,k +

log2 1 +

k=1

M 

Ps,m + ξ

m=0

M 

R˜ m

m=0

  h mm,k 2 pm,k Im,k + N0

and Im,k is the added constraint on the interference of D2D pair m received on RB k. Problem P6 is a non-concave FP problem. By exploiting the property of FP, P6 is equivalently reformulated as P7 : max R˜ − λ˜ P˜

(8.28)

p

s.t. (8.7b), (8.27b), and (8.27c) where λ˜ is a non-negative parameter. Obviously, P7 is a concave problem. In P7, the value of λ˜ can be derived based on Dinkelbach method, whose procedure is the same with that in Algorithm 8.1 and is not reproduced here for the sake of brevity. Given ˜ we can use sophisticated convex optimization method to solve P7 [30]. Due to λ, the equivalence between P6 and P7, we now have the optimal solution to P6. Assume that p6 is the optimal solution to P6. Due to the constraint (8.27c), we have Rm ≥ R˜ m . Consequently, p6 is also a solution to P1, and     max η˜ = η˜ p6 ≤ η p6 ≤ max η p

p

(8.29)

where η( p) and η( ˜ p) denote the values of η and η˜ with given p, respectively. Hence, P6 provides a lower bound for P1, which is in agreement with the result in [34]. In P6, the value of Im,k is crucial for providing a tight lower bound for P1. In the M    h mm  ,k 2 pm  ,k with pm  ,k ∈ p1 , where p1 is perfect scenario, i.e., Im,k = m  =0,m  =m

the optimal solution to P1, P6 is equivalent to P1. In the following, we will outline how to initialize and update the value of Im,k to tighten the lower bound provided from P6. In order to give an initial value for Im,k , we construct the following problem R p P s.t. Rm ≥ Rmth , 0 ≤ m ≤ M, (8.7b)

P8 : max

(8.30a) (8.30b)

8.5 Sub-Optimal Power Control

133

Algorithm 8.4 Updating of Im,k in P6 1: Set tolerance , and initialize i ← 0. i 2: Initialize Im,k based on the optimal solution of P8, as shown in (8.31).   M  K  i i−1  3: while m=0 k=1 Im,k − Im,k  >  or i = 0 do 4: i ← i + 1 i−1 into P6 and derive the optimal solution pi6 . 5: Substitute Im,k i 6: Similar to (8.31), construct Im,k based on pi6 . 7: end while

where 

R =

M 

Rm

m=0

P  =ς

M  K 

pm,k +

m=0 k=1

Rm =B

K 

log2

k=1



M 

Ps,m + ξ

m=0

M 

Rm

m=0

  h mm,k 2 pm,k 1+ . N0 

Since a solution to P1 is also a solution to P8 and PR  > PR , P8 provides an upper bound for P1. Solving P8 is similar to solving P6, where we can firstly convert P8 into an equivalent subtractive form and then solve the equivalent concave problem based on Dinkelbach and convex optimization methods. 0 , Assume that the optimal solution to P8 is p8 , the initial value of Im,k , Im,k 0 ≤ m ≤ M and 1 ≤ k ≤ K , can be obtained as follows. 0 = Im,k

M 

  h mm  ,k 2 pm  ,k

(8.31)

m  =0,m  =m 0 into P6 produces a lower bound for P1. In order where pm  ,k ∈ p8 . Substituting Im,k to tighten the lower bound, we can update Im,k based on the optimal solution of P6, as shown in Algorithm 8.4, where the updating process iterates until convergence. The proof of convergence of Algorithm 8.4 is presented in Appendix 3.

8.5.2 Individual EE Similar to P6, we can formulate a new problem to provide a lower bound for P3 based on interference temperature approach by adding extra constraint on the interference term as follows.

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8 Energy-Efficient Power Control for D2D Communications

P9 : max min η˜ m = p

m

R˜ m P˜m

(8.32)

s.t. (8.7b), (8.27b), and (8.27c) where P˜m =ς

K 

pm,k + Ps,m + ξ R˜ m .

k=1

Similar to P7, the equivalent subtractive form of P9 is a concave problem, and it can be solved through generalized Dinkelbach and convex optimization methods. The initialization and update process of Im,k is the same with that in Algorithm 8.4, which is not reproduced here for the sake of brevity.

8.5.3 Complexity Analysis The sub-optimal scheme comprises three loops, i.e., updating Im,k according to Algorithm 8.4 in outer loop, updating parameter λ˜ when solving the concave problem P7 in medium loop, and solving P7 with given λ˜ in inner loop. The complexity of updating Im,k is related to the times of “while” iterations in Algorithm 8.4. In the next section, we will show that Algorithm 8.4 would converge in very fewer iterations through simulations, namely, the complexity of outer loop is O (1). To update λ˜ , Dinkelbach method is adopted. As Dinkelbach method converges at a superlinear convergence rate, the number of iterations required to update λ˜ has complexity of O (1). To solve concaveP7, we can use the interior-point method, 1 whose complexity is O α 2 (α + β) β 2 , where α is the number of inequality constraints and β is the number of variables [7, 36]. Therefore, we can  conclude that the total complexity of the sub-optimal power control scheme is O (M K + K )3.5 .

8.6 Simulation Results In this section, the simulation results are presented to demonstrate the effectiveness of the developed energy-efficient power control schemes and examine the impacts of the network parameters on the EE performance of D2D communications. The system parameters are given in Table 8.1. We assume that the D2D pairs are uniformly distributed in the cellular networks with a cell radius of 100 m and the D2D link distance, i.e., the distance between the two UEs of one D2D pair, is d. The convergence behaviors of Algorithms 8.1, 8.3, and 8.4 are demonstrated by considering a snapshot of a scenario with K = 3, d = 10 m, and pmmax = 23 dBm,

8.6 Simulation Results

135

Table 8.1 Simulation parameters Parameter Cell radius Carrier frequency Shadowing UE/eNB antenna number Antenna pattern Minimum distance between UE and eNB Minimum distance between UEs UE power class UE’s noise figure Noise power spectral density

Setting 100 m 2.0 GHz 8 dB log-normal 1 Omnidirectional ≥35 m ≥3 m 23 dBm (200 mW) 7 dB −174 dBm/Hz

Table 8.2 The value of λ versus iteration times in Algorithm 8.1 M 2 4 6 8 Once Twice 3 times

3.44e6 3.62e6 3.62e6

2.87e6 3.18e6 3.18e6

2.53e6 2.61e6 2.61e6

2.18e6 2.39e6 2.39e6

Table 8.3 The value of λ versus iteration times in Algorithm 8.3 M 2 4 6 8 Once Twice 3 times

3.21e6 3.45e6 3.47e6

1.99e6 2.05e6 2.06e6

1.73e6 1.80e6 1.81e6

1.45e6 1.57e6 1.58e6

10 1.69e6 1.79e6 1.81e6

10 1.20e6 1.32e6 1.33e6

i−1 i Table 8.4 The difference between Im,k and Im,k versus iteration times in terms of     i i−1    i−1 − I I I  m k m,k m k m,k m,k

M

2

4

6

8

10

Once Twice

7.95e−2 2.04e−7

4.46e−2 6.79e−4

5.12e−2 2.13e−3

2.98e−2 4.97e−4

2.31e−1 1.71e−2

as shown in Tables 8.2, 8.3, and 8.4, respectively. Tables 8.2 and 8.3 show that both the Algorithms 8.1 and 8.3 converge very fast, namely in no more than two or three iterations, which is in line with the fact that Newton’s method has order-two convergence [37]. Table 8.4 also shows that Algorithm 8.4 converges very fast, namely in two iterations. From Tables 8.2, 8.3, and 8.4, we observe that the complexities of Algorithms 8.1, 8.3, and 8.4 are negligible compared with the arithmetic operations in the inner loops, as above mentioned in Sects. 8.4.3 and 8.5.3.

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8 Energy-Efficient Power Control for D2D Communications

Fig. 8.2 Cumulative distribution function of the number of inspected subregions for different scenarios

To depict the complexity of the inner loop in Algorithm 8.2, we investigate the cumulative distribution function (CDF) of the number of inspected subregions needed to reach the global optimal solution with a predefined tolerance value = U/100. To this end, four CDFs of the numbers of the inspected subregions for four scenarios with d = 10 m, pmmax = 23 dBm, and 5000 random snapshots are shown in Fig. 8.2. It is observed that the numbers of the inspected subregions vary significantly for different scenarios, such as the CDFs for four scenarios with 500 inspected subregions are 0.86, 0.31, 0.25, and 0.02, respectively. Obviously, the larger the problem size, the lower the CDF with given inspected subregion number, which is in agreement with the fact that the complexity of BB method is exponential in the problem size. Figure 8.3 shows the impact of RB number K on the EE. In Fig. 8.3, we can see that both the total EE and individual EE increase with the increase of K , which is

Fig. 8.3 Impact of RB number K on the EE. (M = 3, d = 10 m, and max = 23 dBm) pm

8.6 Simulation Results

137

Fig. 8.4 Impact of D2D link distance d on the EE. (M = 3, K = 4, and max = 23 dBm) pm

Fig. 8.5 Impact of maximum allowed transmit max on the EE. power pm (M = 3, K = 4, and d = 10 m)

due to the frequency diversity. The larger the value of K , the higher the frequency diversity, and hence the larger the EE. Figure 8.3 also shows that the total EE is higher than the individual EE, which is consistent with the definition of individual EE. Figure 8.4 plots the EE curves with variable D2D link distance d. It is obvious that d has a great influence on the EE performance. For both the total EE and individual EE schemes, their EE performance decreases dramatically with the increase of d, which is due to the fact that the larger the D2D link distance, the higher the transmit power is needed to compensate for the larger channel fading, and hence the lower the EE. As d increases from 5 m to 30 m, the optimal total EE and individual EE decrease about 69.3% and 79.4%, respectively.

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8 Energy-Efficient Power Control for D2D Communications

The effect of maximum allowed transmit power pmmax on the EEs is shown in Fig. 8.5. We observe that both the total EE and individual EE firstly increase with the increase of pmmax , and then approach a constant. This is due to the fact that the throughput increases only logarithmically with the transmit power while the transmit power consumption grows up linearly with the transmit power. As a result, the EE performance first strictly increases and then strictly decreases with transmit power [16]. This indicates that there exists a break point for the transmit power from the EE perspective, and the EE approaches peak value at the break point, i.e., between 20 and 23 dBm in Fig. 8.5, and the EE would be degraded if we further improve the transmit power beyond the break point, which is in agreement with the result in [16, 34]. When the maximum EE is achieved, the energy-efficient design is not willing to consume more power. From Figs. 8.3, 8.4 and 8.5, we observe that the EE performance of the sub-optimal scheme is close to that of optimal scheme, i.e., with a gap of less than 0.3 Mbits/Joule, which demonstrates the effectiveness of the sub-optimal scheme. However, as the complexity of the former is only a tiny fraction of that of the latter, we can conclude that the sub-optimal scheme makes a good tradeoff between the EE performance and complexity, and is promising for practical applications.

8.7 Conclusions In this chapter, we have investigated the energy-efficient power control for D2D communications underlaying cellular networks, where both the total EE and individual EE have been considered. It is shown that the individual EE and total EE are preferred from the fairness and purely EE points of view respectively. For practical design, it is preferred to optimize the EE of D2D communications while guaranteeing their QoS requirements to make a good tradeoff between SE and EE.

Appendix 1 Proof of Lemma 8.1 For the sake of convenience, we use Rm ( p) and Pm ( p) to indicate the values of Rm and Pm with given p, respectively. (1) Since Pm ( p) > 0, ∀ p and ∀m, we have Rm ( p) − λ1 Pm ( p) > Rm ( p) − λ2 Pm ( p), if λ1 < λ2 .

(8.33)

    min Rm ( p) − λ1 Pm ( p) > min Rm ( p) − λ2 Pm ( p) .

(8.34)

Hence, m

m

8.7 Conclusions

139

  Let p maximize F  λ2 , then     F  λ2 = min Rm ( p ) − λ2 Pm ( p ) m   < min Rm ( p ) − λ1 Pm ( p ) m     ≤ max min Rm ( p) − λ1 Pm ( p) = F  λ1 . p

m

(8.35)

(2) If F  (λ ) > 0, suppose p is the optimal solution to P4 such that     F  λ = min Rm ( p ) − λ Pm ( p ) > 0.

(8.36)

Rm ( p ) Rm ( p) ≤ max min = λ .  p m Pm ( p) Pm ( p )

(8.37)

m

Hence, λ < min m

Conversely, if λ < λ , suppose p is the optimal solution to P3 such that λ = min m

Rm ( p ) > λ . Pm ( p )

(8.38)

From (8.38), we have     F  λ = max min Rm ( p) − λ Pm ( p) p m   ≥ min Rm ( p ) − λ Pm ( p ) > 0. m

(8.39)

(3) Suppose that p is the optimal solution of P3 such that λ = min m

Rm ( p ) . Pm ( p )

(8.40)

Thus,   min Rm ( p ) − λ Pm ( p ) = 0. m

Since F  (λ ) ≤ 0, from (8.41) we have F  (λ ) = 0.

(8.41)

140

8 Energy-Efficient Power Control for D2D Communications

(4) From (8.40) and (8.41), it is shown that the optimal solution p of P3 is also the optimal solution to P4. Now, we assume that F  (λ ) = 0 and p is the optimal solution to P4. Thus,     F  λ = min Rm ( p ) − λ Pm ( p ) = 0.

(8.42)

m

Hence, min m

Rm ( p ) = λ . Pm ( p )

(8.43)

According to (8.14), it is easy to conclude that p is also the optimal solution to P3.

Appendix 2 Proof of Convergence of Algorithm 8.3 From the construction of λi , it is obvious that λi ≤ λ . R (p ) Given that λi = min Pmm ( pi−1) , we have m

i−1

    F  λi ≥ min Rm ( pi−1 ) − λi Pm ( pi−1 ) = 0.

(8.44)

m

  According to Lemma 8.1, if F  λi = 0, λi = λ . In such a case, the value of λ is given by λi , and pi−1 is the optimal solution to P5.     If F  λi > 0, we first prove that λi+1 > λi , where λi+1 = min RPmm (( ppi )) , as shown m

i

on Line 5 of Algorithm 8.3, with pi being the optimal solution of P5 with given λi , namely     F  λi = min Rm ( pi ) − λi Pm ( pi ) > 0. m

(8.45)

Thus, λi < min m

R m ( pi )  . = λi+1 Pm ( pi )

(8.46)

Now, similar to the proof of the convergence of Dinkelbach method [21], we can prove that lim λi = λ . i→∞

8.7 Conclusions

141

Appendix 3 Proof of Convergence of Algorithm 8.4 According to the updating protocol in Algorithm 8.4 and constraint (8.27c) in P6, we have i 0 ≤ Im,k =

M 

  h mm  ,k 2 pi

m  ,k

i−1 ≤ Im,k

(8.47)

m  =0,m  =m i−1 where pmi  ,k ∈ pi6 is the optimal solution to P6 with Im,k . From (8.47), we conclude  i  that sequence Im,k , ∀m and ∀k, is non-increasing and lower bounded. Hence, the i exists, and Algorithm 8.4 would converge eventually. limit of Im,k ∗ value of I and the optimal value of p with I ∗ Let I and p∗ be the converged  respectively, where I = Im,k . It is easy to demonstrate that (I ∗ , p∗ ) is a partial optimum of P6, which indicates that (I ∗ , p∗ ) is also a stationary point3 of P6 [38]. Hence, the developed interference temperature approach converges to a stationary point, but not necessarily to a local optimal point.

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13. L. Lei, Z. Zhong, C. Lin, X. Shen, Operator controlled device-to-device communications in LTE-advanced networks. IEEE Wireless Commun. 19(3), 96–104 (2012). 14. W.W.L. Li, Y.J. Zhang, A.M.C. So, M.Z. Win, Slow adaptive OFDMA systems through chance constrained programming. IEEE Trans. Signal Process. 58(7), 3858–3869 (2010). 15. S. Wang, M. Ge, W. Zhao, Energy-efficient resource allocation for OFDM-based cognitive radio networks. IEEE Trans. Commun. 61(8), 3181–3191 (2013). 16. C. Xiong, G.Y. Li, S. Zhang, Y. Chen, S. Xu, Energy-efficient resource allocation in OFDMA networks. IEEE Trans. Commun. 60(12), 3767–3778 (2012). 17. N.S. Kim, T. Austin, D. Blaauw, T. Mudge, K. Flautner, J.S. Hu, M.J. Irwin, M. Kandemir, V. Narayanan, Leakage current: Moore’s law meets static power. Computer 36(12), 68–75 (2003). 18. C. Isheden, G.P. Fettweis, Energy-efficient multi-carrier link adaptation with sum ratedependent circuit power, in IEEE Globecom IEEE(2010), pp. 1–6. 19. W. Zhao, S. Wang, Resource sharing scheme for device-to-device communication underlaying cellular networks. IEEE Trans. Commun. 63(12), 4838–4848 (2015). 20. C. Isheden, Z. Chong, E. Jorswieck, G. Fettweis, Framework for link-level energy efficiency optimization with informed transmitter. IEEE Trans. Wireless Commun. 11(8), 2946–2957 (2012). 21. W. Dinkelbach, On nonlinear fractional programming. Manage. Sci. 13(7), 492–498 (1967). 22. H. Al-Shatri, T. Weber, Achieving the maximum sum rate using D.C. programming in cellular networks. IEEE Trans. Signal Process. 60(3), 1331–1341 (2012). 23. Z. Li, S. Guo, D. Zeng, A. Barnawi, I. Stojmenovic, Joint resource allocation for max-min throughput in multicell networks. IEEE Trans. Veh. Technol. 63(9), 4546–4559 (2014). 24. S. Schaible, J. Shi, Recent developments in fractional programming: single-ratio and max-min case, in 3rd Inter. Conf. Nonlinear Analysis and Convex Analysis (2003), pp. 493–506. 25. C.R. Bector, S. Chandra, M.K. Bector, Generalized fractional programming duality: a parametric approach. J. Optim. Theory Appl. 60(2), 243–260 (1989). 26. J.P. Crouzeix, J.A. Ferland, S. Schaible, An algorithm for generalized fractional programs. J. Optim. Theory Appl. 47(1), 35–49 (1985). 27. H. Tuy, Convex Analysis and Global Optimization. Kluwer Academic, 1998. 28. T. Ibaraki, Parametric approaches to fractional programs. Math. Program. 26(3), 345–362 (1983). 29. P.C. Weeraddana, M. Codreanu, M. Latva-aho, A. Ephremides, Weighted sum-rate maximization for a set of interfering links via branch and bound. IEEE Trans. Signal Process. 59(8), 3977–3996 (2011). 30. S. Boyd, L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. 31. S. Boyd, J. Mattingley, Branch and bound methods. Available: https://see.stanford.edu/ materials/lsocoee364b/17-bb_notes.pdf. 32. V. Balakrishnan, S. Boyd, S. Balemi, Branch and bound algorithm for computing the minimum stability degree of parameter-dependent linear systems. Int. J. Robust Nonlin. Control. 1(4), 295–317 (1991). 33. F.A. Potra, S.J. Wright, Interior-point methods. J. Comput. Appl. Math. 124(1–2), 281–302 (2000). 34. D.W.K. Ng, E.S. Lo, R. Schober, Energy-efficient resource allocation in OFDMA systems with large numbers of base station antennas. IEEE Trans. Wireless Commun. 11(9), 3292– 3304 (2012). 35. S.M.H. Andargoli, K. Mohamed-pour, Weighted sum throughput maximisation for downlink multicell orthogonal frequency-division multiple access systems by intercell interference limitation. IET Commun. 6(6), 628–637 (2012). 36. A. Nemirovski, Interior point polynomial time methods in convex programming. Lecture Notes, Georgia Inst. of Technol., Atlanta, USA, 2004.

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Chapter 9

Energy-Efficient User Scheduling and Power Control for Multi-Cell OFDMA Networks Based on CDI

In this chapter, we investigate the energy-efficient user scheduling and power control for multi-cell orthogonal frequency division multiple access (OFDMA) networks based on channel distribution information (CDI). Section 9.1 introduces the motivation of developing energy-efficient user scheduling and power control with CDI. Section 9.2 presents the system model and formulates the problem. Section 9.3 develops the centralized joint user scheduling and power control algorithm, and Sect. 9.4 develops the decentralized power control scheme. Section 9.5 presents the simulation results, and Sect. 9.6 concludes this chapter.

9.1 Introduction OFDMA has been employed as an effective multiplexing scheme in the downlink of the 3rd Generation Partnership Project (3GPP) long-term evolution (LTE) systems due to its advantages of avoiding the intra-cell interference and mitigating the influence of multipath fading [1]. To meet the exponentially increasing demand for wireless data rate, the density of evolved Node B (eNB) is increased in LTE networks to achieve higher area spectral efficiency (SE) [2, 3], and the inter-cell interference (ICI) becomes more and more severe. To mitigate the ICI, eNB coordination is adopted [4]. It is noticed that joint user scheduling and power control in multi-cell OFDMA networks could significantly increase the system performance [5, 6]. Due to the exponential data growth, network power consumption becomes a nonnegligible problem from both the economic and environmental perspectives [7]. As such, energy efficiency (EE) has attracted more and more attention recently [8]. It is noted that channel state information (CSI) is usually required when optimizing EE [9–16]. However, it is hard to obtain the perfect CSI owing to channel variation over time, the channel estimation error, and the feedback delay. Considering the difficulty of obtaining perfect CSI, we can replace CSI by CDI, which remains unchanged for © Beijing Institute of Technology Press 2023 J. An et al., Sustainable Wireless Communications, https://doi.org/10.1007/978-981-19-0448-6_9

145

146

9 Energy-Efficient User Scheduling and Power Control for Multi-Cell …

a relatively long period of time and is easier to be obtained [17]. The updates of resource allocation based on CDI can be done at a log-normal shadowing time scale, and the CDI-based resource allocation scheme could reduce the signal processing energy consumption and signalling overhead significantly compared with the CSIbased scheme [18]. Motivated by this, we investigate the energy-efficient joint user scheduling and power control problem based on CDI in the chapter.

9.2 System Model and Problem Formulation We consider the downlink of a multi-cell OFDMA network, where N coordinated eNBs intend to transmit data to M users, based on LTE protocol. The network employs universal frequency reuse with K subcarriers, and both the eNBs and the users are equipped with a single antenna. Each user is associated with one eNB, and each subcarrier is assigned exclusively to at most one user in each cell. Let xn,m,k denote the transmit power from eNB n to user m on subcarrier k, and the received signal of user m on subcarrier k is z m,k =

N  √

xn,m,k L n,m h n,m,k sm,k +

n=1

M 

N  √

xn,m,k ˜ L n,m h n,m,k sm,k ˜ + n0,

m=1, ˜ m ˜ =m n=1

(9.1) where L n,m consists of the pathloss and log-normal shadow fading between eNB n and user m, h n,m,k represents the fast fading between eNB n and user m on subcarrier k, sm,k is the transmit symbol for user m on subcarrier k with unit energy, and n 0 is the additive white Gaussian noise with zero mean and variance σ 2 . From (9.1), if independent coding is carried out for different subcarriers, the instantaneous achievable SE (in bits/sec/Hz) of user m on subcarrier k is given by   Cm,k = log2 1 + SINRm,k , where SINRm,k =

N 2 n=1 x n,m,k G n,m |h n,m,k | N M 2 1+ m=1, x G ˜ n,m |h n,m,k | ˜ m ˜ =m n=1 n,m,k

(9.2)

with G n,m = L 2n,m /σ 2 . We assume

that only the CDI, namely, G n,m and the statistical property of h n,m,k , is available at each eNB. In such a scenario, the transmission outage occurs when rm,k > Cm,k , where rm,k is the actual SE  k. Therefore, the outage probability  of user m on subcarrier (OP) constraint, i.e., Pr SINRm,k < 2rm,k − 1 ≤ εm , where εm ∈ (0, 1) is a constant parameter close to zero to restrict the OP of user m, should be taken into consideration to guarantee the quality of service (QoS) of each user. For Rayleigh fading channels, h n,m,k ∼ CN (0, 1), ∀n, ∀m, and ∀k, is an independent complex Gaussian random variable, and the received signal-to-noise ratio  2 N xn,m,k G n,m h n,m,k  has an exponential distribution with parameter (SNR) n=1 N N n=1 x n,m,k G n,m if n=1 x n,m,k G n,m > 0 when each user is associated with exactly

9.2 System Model and Problem Formulation

147

N one eNB. Given user m and subcarrier k, we assume n=1 xn,m,k > 0. According to [18], the OP constraint has an analytical expression in Rayleigh fading channels with only CDI available, and is given by M 



(2rm,k − 1) ln 1 + N

m=1, ˜ m ˜ =m

N

n=1

n=1

xn,m,k ˜ G n,m



xn,m,k G n,m

+ N

2rm,k − 1

n=1

xn,m,k G n,m

+ ln (1 − εm ) ≤ 0. (9.3)

˜ = m. However, if xn,m,k = 0, It is noted that (9.3) holds even if xn,m,k ˜ = 0, ∀n, ∀m ∀n, we have rm,k = 0 and the OP rconstraint (9.3) is meaningless. After introducing m,k m with ε˜m = − ln (1 − εm ) > 0, we a new variable ym,k such that  N 2 x −1G = yε˜m,k n=1 n,m,k n,m have   N  xn,m,k G˜ n,m rm,k = log2 1 + , (9.4) ym,k n=1 where G˜ n,m = G n,m ε˜m . Comparing (9.4) with (9.2), ym,k can be viewed as the interference plus noise of user m on subcarrier k. Substituting ym,k into (9.3) yields M  m=1, ˜ m ˜ =m



N  ˜ xn,m,k ˜ G n,m ln 1 + ym,k n=1

 +

ε˜m − ε˜m ≤ 0. ym,k

(9.5)

Compared with (9.3), (9.5) is properly defined even if xn,m,k = 0, ∀n. Hence, we use (9.5) to represent the OP constraint. The total power consumption of the network could be given by [19, 20] Ptot = ς

N  M  K  n=1 m=1 k=1

xn,m,k + ξ B

M  K 

rm,k + Pc ,

(9.6)

m=1 k=1

where B is the bandwidth of one subcarrier, ς is the reciprocal of drain efficiency of power amplifier, ξ is the power consumption per unit throughput, and Pc is the static part of the circuit power consumption. After defining the EE as the ratio of the total network data rate to the total power consumption, the goal of the EE maximization is expressed as M K B m=1 k=1 r m,k . max  N  M  K M K ς n=1 m=1 k=1 xn,m,k + ξ B m=1 k=1 r m,k + Pc

(9.7)

148

9 Energy-Efficient User Scheduling and Power Control for Multi-Cell …

Since B and ξ are constant parameters, (9.7) could be equivalently transformed into min

ς

N M n=1

B

K

m=1 M m=1

k=1 x n,m,k K k=1 r m,k

+ Pc

M ⇔ max

ς

m=1 N M n=1 m=1

K

k=1 r m,k

K

k=1

. xn,m,k + Pc (9.8)

Combining (9.4), (9.5) and (9.8), the energy-efficient joint user scheduling and power control problem based on CDI is now formulated as M P1 :

max  N m=1 M x, y ς n=1 m=1 M  K 

K

k=1 r m,k

K

k=1

xn,m,k + Pc

xn,m,k ≤ Pnmax , ∀n,

s.t.

(9.9a) (9.9b)

m=1 k=1 K 

rm,k ≥ rmth , ∀m,

(9.9c)

x n,k ≤ 1, ∀n, ∀k, 0

(9.9d)

B

k=1

x m 0 ≤ 1, ∀m, (9.5), ∀m, ∀k.

(9.9e)





T In P1, x = xn,m,k , y = ym,k , x n,k = xn,1,k , · · · , xn,m,k , . . . , xn,M,k and T  K K K xm = represent the transmit k=1 x 1,m,k , · · · , k=1 x n,m,k , . . . , k=1 x N ,m,k power vector of eNB n on subcarrier k and the transmit power vector of eNBs for sending signals to user m, respectively, · 0 is zero norm, and Pnmax and rmth denote the maximum allowed total transmit power of eNB n and the minimum required throughput of user m, respectively. Constraints (9.9d) and (9.9e) mean that each subcarrier is exclusively assigned to at most one user in each cell and each user is served by at most one eNB, respectively. Although the equivalent channel response G˜ n,m is independent of the subcarriers, subcarrier assignment shall be taken into account in user scheduling. If the number of users is larger than that of eNBs, subcarrier assignment would make sure that there are more than one user served by a same eNB on different subcarriers, which is essential to guarantee the minimum throughput of each user. Moreover, constraint (9.9c) is inevitable to ensure that each user is associated with an eNB especially when some users have higher priority to access the eNBs.

9.2 System Model and Problem Formulation

149

It is obvious that P1 is a nonconvex problem due to the fractional objective function in (9.9a) and the OP constraint (9.5). Besides, P1 is mathematically intractable because of the cardinality constraints (9.9d) and (9.9e). Therefore, the global optimum is difficult to be obtained in practice. In the next section, we will show how to solve it suboptimally with affordable complexity.

9.3 Joint User Scheduling and Power Control In this section, we present an iterative algorithm to obtain a suboptimal solution of P1 based on the framework of successive convex approximation (SCA) through solving a sequence of subproblems, where the nonconvex objective function and constraints are replaced by suitable convex approximations [21]. In SCA, to find a local optima of a problem with continuously differentiable nonconvex function f (θ),1 we can resort to solving a sequence  of easier problems with the convex approximations of f (θ), denoted by f j (θ) . In the jth iteration, f j (θ) shall have the following three properties [22]: f j (θ) ≤ f (θ) , ∀θ,     f j θ ( j) = f θ ( j) ,     ∇ f j θ ( j) = ∇ f θ ( j) ,

(9.10a) (9.10b) (9.10c)

where θ ( j) is the solution of the ( j − 1)th iteration, and ∇ f (·) represents the gradient of f . If function f (θ) is not differentiable, the gradient of f could be replaced with the subgradient of f , denoted by ∂ f (·) [23]. Due to the fractional form of the objective function of P1, the OP constraint (9.5), and the cardinality constraints (9.9d) and (9.9e), it is difficult to approximate P1 by a convex problem directly. To circumvent the difficulty in fractional programming (FP), we first transform P1 into an equivalent non-fractional problem by resorting to the Charnes-Cooper transformation [14]. Specifically, we introduce a new posi−1    N M K , and let x˙n,m,k = xn,m,k w and tive variable w = ς n=1 m=1 k=1 x n,m,k + Pc y˙m,k = ym,k w. Obviously, w is upper bounded by Uw = P1c and lower bounded by −1   N L w = ς n=1 Pnmax + Pc . 



Substituting w, x˙ = x˙n,m,k , and y˙ = y˙m,k into P1, we can equivalently transform P1 into

If f (θ) is the objective function, the goal of the problem is to maximize f (θ). If f (θ) is the constraint function, the constraint has the form f (θ) ≥ 0. 1

150

9 Energy-Efficient User Scheduling and Power Control for Multi-Cell …

P2 : max w ˙ y˙ ,w x,

s.t.

M  K 

rm,k

(9.11a)

m=1 k=1 M  K 

x˙n,m,k ≤ Pnmax w, ∀n,

m=1 k=1 M  m=1, ˜ m ˜ =m



N  ˜ x˙n,m,k ˜ G n,m ln 1 + y˙m,k n=1

x˙ n,k ≤ 1, ∀n, ∀k, 0 x˙ m ≤ 1, ∀m, 0 ς

N  M  K 

(9.11b)  +

wε˜m − ε˜m ≤ 0, ∀m, ∀k, (9.11c) y˙m,k (9.11d) (9.11e)

x˙n,m,k + Pc w = 1,

(9.11f)

n=1 m=1 k=1

(9.9c), ∀m, 

T K where x˙ n,k = x˙n,1,k , · · · , x˙n,m,k , . . . , x˙n,M,k , x˙ m = k=1 x˙ 1,m,k , · · · ,  T K K , and rm,k is rewritten as k=1 x˙ n,m,k , . . . , k=1 x˙ N ,m,k  rm,k = log2

N  x˙n,m,k G˜ n,m 1+ y˙m,k n=1

 .

(9.12)

It is easy to conclude that constraints (9.11d) and (9.11e) are equivalent to constraints (9.9d) and (9.9e), respectively. The cardinality constraints (9.11d) and (9.11e) result in the intractability of P2 due to their nonconvexity and discontinuity. Conventional approaches to tackle such a problem are convex approximation and/or representing the zero norm as a sum of approximated indicator functions [24, 25], whose main drawback is that a new parameter is introduced to control the tightness of the approximation of the zero norm. It is noticed that the best value of the parameter is difficult to be determined since too large value may result in the numerical problems and too small value leads to a poor approximation. Different from approximation approaches, here we equivalently transform the cardinality constraints into the difference-of-convex constraints. Based on the norm

θ lgst,g = |θ|[1] + |θ|[2] + · · · + |θ|[g] ,

(9.13)

where |θ|[i] denotes the ith largest element of the absolute values of the entries in θ ∈ R G with G being any positive and g ≤ G, the cardinality constraint θ 0 ≤ g can be equivalently expressed as the difference of two norms [26]

θ lgst,g˜ − θ lgst,g = 0,

(9.14)

9.3 Joint User Scheduling and Power Control

151

where g ≤ g˜ ≤ G. We can explain (9.14) by the fact that θ lgst,g˜ − θ lgst,g is nonnegative, and is equal to zero if and only if the sum of the (g + 1)th to the gth ˜ largest absolute values is equal to zero [26]. Consequently, (9.11d) and (9.11e) can be equivalently represented by x˙ n,k − x˙ n,k ≤ 0, ∀n, ∀k, 1 ∞ x˙ m − x˙ m ≤ 0, ∀m. 1 ∞

(9.15a) (9.15b)

Due to the unboundedness of y˙m,k , the feasible set of P2 is not compact, which is desirable to strengthen the convergence properties of SCA [21]. To find the lower bound of y˙m,k , we could easily prove that the left-hand side (LHS) of (9.11c) is a non-increasing function in y˙m,k through taking its first derivative with respect to y˙m,k , and that its value is nonnegative with y˙m,k = w. Hence, to satisfy (9.11c), we have y˙m,k ≥ w ≥ L w .

(9.16)

Next, we derive the upper bound of y˙m,k . Since both the objective function of P2 and the LHS of (9.11c) are non-increasing with respect to y˙m,k , (9.11c) must hold with equality at the optimal solution of P2. Then constraint (9.11c) could be recast as 

  , w ≤ y˙m,k , (9.17) y˙m,k x˙n,m,k ˜ n,m ˜ =m where y˙m,k



x˙n,m,k ˜



 , w is the solution to the following equation n,m ˜ =m

M  m=1, ˜ m ˜ =m



N  ˜ x˙n,m,k ˜ G n,m ln 1 + y˙m,k n=1

 +

wε˜m − ε˜m = 0. y˙m,k

(9.18)



  Here, y˙m,k x˙n,m,k , w is a continuously differentiable function of ˜ n,m ˜ =m 

and w. Since (9.17) must hold with equality for any optimal soluxn,m,k ˜ n,m ˜ =m tion of P2, we could obtain the upper bound of y˙m,k by finding the upper bound of 

  y˙m,k x˙n,m,k ,w . ˜ n,m ˜ =m By utilizing the property of logarithm function, i.e., log (1 + θ) ≤ θ, the LHS of (9.18) is upper bounded by   

op Um,k y˙m,k , x˙n,m,k , w = ˜ n,m ˜ =m

M  m=1, ˜ m ˜ =m

N  ˜ x˙n,m,k wε˜m ˜ G n,m + − ε˜m . y˙m,k y˙m,k n=1

As both (9.19) and the LHS of (9.18) are non-increasing in y˙m,k , we have

(9.19)

152

9 Energy-Efficient User Scheduling and Power Control for Multi-Cell …

y˙m,k



x˙n,m,k ˜

M 

=



n,m ˜ =m , w

N 



u ≤ y˙m,k



x˙n,m,k ˜



n,m ˜ =m , w



x˙n,m,k ˜ G n,m + w ≤ U y˙m,k ,

(9.20)

m=1, ˜ m ˜ =m n=1



  u x˙n,m,k where y˙m,k ,w is the solution to the equation ˜ n,m ˜ =m   N  



 max max y˙m,k , x˙n,m,k G , w = 0, U = U P + 1 , ˜ n,m ˜ =m y˙m,k w n,m n n

op

Um,k and

n=1

the second inequality is obtained based on (9.11b). Consequently, we can add a bound constraint on y˙m,k to P2 based on (9.16) and (9.20) to make the feasible set of P2 compact, i.e., L w ≤ y˙m,k ≤ U y˙m,k , ∀m, ∀k,

(9.21)

which has no impact on the optimal solutions of P2. Based on (9.15a), (9.15b), (9.17), and (9.21), P2 is now rewritten as P3 : max

˙ y˙ ,χ,w x,

s.t.

K M  

wlog2

χ

m,k

 (9.22a)

w

m=1 k=1

N  χm,k x˙n,m,k G˜ n,m ≤1+ , ∀m, ∀k, w y˙m,k n=1

B

K  k=1

wlog2

χ

m,k

w



(9.22b)

≥ rmth w, ∀m,

(9.22c)

L χm,k ≤ χm,k ≤ Uχm,k , ∀m, ∀k,

(9.22d)

(9.11b), (9.11f), (9.15a), (9.15b), (9.17) and (9.21). 

In P3,  χ = χm,k is the new auxiliary variables, and L χm,k = L w and Uχm,k = max ˜ Uw 1 + max Pn G n,m are the lower bound and upper bound of χm,k , respecn

tively, to ensure that the feasible set of P3 is compact. From (9.12), rm,k is increasing with respect to the right-hand side (RHS) of (9.22b). In addition, the objective function of P2 and the LHS of (9.9c) are both nondecreasing with respect to rm,k . Therefore, it can be easily verified that P2 and P3 are equivalent since for any feasible solution of P3, we can increase the objective value of P3 by increasing χm,k until (9.22b) holds with equality. Constraint (9.22c) is derived through multiplying the both sides of (9.9c) by w. The lower bound and the upper bound of χm,k are derived as follows. By the fact that (9.22b) holds with equality at the optimum and the actual SE of user m on subcarrier k should be nonnegative, we have χm,k ≥ w ≥  L w = L χm,k . Since each user is associated with exactly one eNB, substituting (9.11b)  N x˙n,m,k G˜ n,m w ≤ w + w P max G˜ g(m),m ≤ and (9.16) into (9.22b) yields χm,k ≤ w + n=1 y˙m,k

g(m)

9.3 Joint User Scheduling and Power Control

153

    Uw 1 + max Pnmax G˜ n,m = Uχm,k , where g (m), 1 ≤ m ≤ M, denotes the serving n

eNB of user m, i.e., g (m) = n if user m is associated with eNB n. Up to now, solving P1 is replaced by solving P3. Dueto the  fact that the perspective function of f is concave if f is concave [27], wlog2 χwm,k is concave. However, P3 is still nonconvex due to constraints (9.15a), (9.15b), (9.17) and (9.22b). Approximating constraint (9.15a) in P3 would result in an overly conservative linearized constraint and lead to poor results, since there may be no user served by eNB n on subcarrier k. Fortunately, we can transform (9.15a) into a penalized form, and reformulate P3 as P4 : max

˙ y˙ ,χ,w x,

K M  

wlog2

χ

m,k

w

m=1 k=1



−ρ

N  K    x˙ n,k − x˙ n,k 1 ∞

(9.23)

n=1 k=1

s.t. (9.11b), (9.11f), (9.15b), (9.17), (9.21), (9.22b), (9.22c) and (9.22d), where ρ is a positive constant penalty coefficient [26]. Based on the theory of exact penalty functions, there exists a threshold ρ˜ > 0 of ρ such that, for any ρ ≥ ρ, ˜ P3 and P4 are equivalent [28]. It is noticed that large ρ would result in numerical problems, and it is usually difficult to obtain the exact value of ρ [29]. The exact penalty ρ for P4 is very hard to be obtained due to the minimum throughput constraint (9.22c). To find a reasonable surrogate of ρ, an exact penalty ρ for P4 without constraint (9.22c), which is denoted by P4 , is derived in Appendix 1. Based on ρ , we can choose ρ = cρ for P4, where c is a constant depending on the minimum required throughput of each user and the number of users M. Remark 1 In our numerical experiments, the penalty ρ for P4 is chosen based on following principles: the greater the number of users M is, the larger ρ is and the smaller the minimum required throughput of each user is, the larger ρ is. The reasons for these two principles could be explained as follows. Firstly, with the increasing of M, the probability that two users in one cell are assigned with the same subcarrier increases since the number of served users is upper bounded by N K in the network with N eNBs and K subcarriers. Secondly, if the minimum required throughput of each user is smaller, constraint (9.22c) could be satisfied with smaller value of x˙n,m,k , which results in the smaller penalty x˙ n,k 1 − x˙ n,k ∞ . In the following, we will elaborate on how to find a convex approximation of P4 at each iteration of SCA. Denoting ∂ · ∞ as the subgradient of · ∞ , namely  ∂ θ ∞ = arg max ω

 i

|θi | ωi :



 ωi = 1, 0 ≤ ωi ≤ 1, ∀i ,

(9.24)

i

where θ = {θi }, ω = {ωi }, and ωi is the subgradient of θi . Approximating θ ∞ by its affine minorant at θ 0 yields   

θ 1 − θ ∞ ≤ CAn θ θ 0 ,

(9.25)

154

9 Energy-Efficient User Scheduling and Power Control for Multi-Cell …

 T      x˙ ( j) , y˙ ( j) , CAn θ θ 0 = θ 1 − θ 0 ∞ −∂ θ 0 ∞ θ−θ 0 . Letting  χ( j) , w( j) represent the solution of the ( j − 1)th SCA iteration, the convex approximations of the penalty term of the objective of    function  of P4 andthe LHS  (9.15b) N K  ( j)  ( j) ˙ ˙ ˙ ˙ x x x and CA x , respecCA could be expressed as n=1   n n,k n m m n,k k=1 tively.  To approximate constraint (9.22b), we can recast (9.22b) as χm,k y˙m,k ≤    N w y˙m,k + n=1 x˙n,m,k G˜ n,m . Since the geometric mean is concave and the com-

where

position with an affine function preserves concavity [27], an approximation of (9.22b) is given by

CAs



   N       ( j) ( j)  χm,k , y˙m,k χm,k , y˙m,k ≤ w y˙m,k + x˙n,m,k G˜ n,m ,

(9.26)

n=1

      ( j)  ( j) y˙m,k χm,k   ( j) ( j) ( j) 1 1 χm,k − χm,k + 2 y˙m,k − where CAs χm,k , y˙m,k χm,k , y˙m,k = 2 ( j) ( j) χm,k y˙m,k    ( j) ( j) ( j) y˙m,k + χm,k y˙m,k is obtained through approximating χm,k y˙m,k by its first-order   ( j) ( j) Taylor expansion around point χm,k , y˙m,k .        ( j)  It is noted that the convex approximations CAn x˙ n,k  x˙ n,k , CAn x˙ m  x˙ (mj) and (9.26) satisfy the SCA condition (9.10), based on which we can convexly approximate all the constraints and the objective function of P4 except constraint (9.17). To convexly approximate constraint (9.17), we will design a convex upper bound 

 , w at each iteration of SCA in the of the implicit function y˙m,k x˙n,m,k ˜ n,m ˜ =m   (θ−θ¯ ) following. Since ln (1 + θ) ≤ ln 1 + θ¯ + , ∀θ ≥ 0, ∀θ¯ ≥ 0, the LHS of (9.18) 1+θ¯

is upper bounded by ( j) U y˙m,k

wε˜m = − ε˜m + y˙m,k



M 

ln 1 +

( j) m,m,k ˜

m=1, ˜ m ˜ =m



+

M 

N n=1

m=1, ˜ m ˜ =m

˜ x˙n,m,k ˜ G n,m − y˙m,k ( j) 1 + m,m,k ˜

( j)

m,m,k ˜

,

(9.27)

( j)

=

N

( j)

˜ x˙n,m,k ˜ G n,m

f,( j) y˙m,k



( j) x˙n,m,k ˜



 . Letting ,w ( j)

= y˙m,k n,m ˜ =m 

  equal to zero, an approximation of y˙m,k x˙n,m,k , w can be expressed ˜ n,m ˜ =m

where U y˙m,k as

( j) m,m,k ˜

n=1

( j)

yˆm,k =

f,( j)

y˙m,k

w ( j) υm,k

+

with

M 

N 

˜ x˙n,m,k ˜ G n,m  , ( j) 1 + m,m,k ˜

( j) m=1, ˜ m ˜ =m n=1 ε˜m υm,k

(9.28)

9.3 Joint User Scheduling and Power Control

where

( j) υm,k

=1+



M

m=1, ˜ m ˜ =m

155

( j)

 ˜  m,m,k  ( j) ε˜ m 1+m,m,k ˜

−

  ( j) ln 1+m,m,k ˜

( j)

. Since both U y˙m,k and

ε˜ m

( j)

the LHS of (9.18) are strictly decreasing with y˙m,k , yˆm,k is an upper bound of 

  

( j) y˙m,k x˙n,m,k , w . In addition, yˆm,k is a linear function of x˙n,m,k and ˜ ˜ n,m ˜ =m n,m ˜ =m 

  ( j) w. Hence, yˆm,k is a convex approximation of y˙m,k x˙n,m,k , w satisfying the ˜  n,m˜ =m  conditions (9.10a) and (9.10b). As the gradient of y˙m,k x˙n,m,k , w at point ˜ n,m ˜ =m    ( j) ( j) x˙n,m,k , w( j) is equal to that of yˆm,k , which is proved in Appendix 2, we ˜ n,m ˜ =m

can obtain a convex approximation of (9.17) by substituting (9.28) into (9.17) as w ( j)

υm,k

M 

+

N 

m=1, ˜ m ˜ =m n=1

˜ x˙n,m,k ˜ G n,m   ≤ y˙m,k . ( j) ( j) ε˜m υm,k 1 + m,m,k ˜

(9.29)

Based on (9.25), (9.26), and (9.29), we can obtain a convex approximation problem of P4. Unfortunately, there exist two disadvantages of using SCA to solve P4 directly. Firstly, due to constraints (9.15b) and (9.22c), it is difficult to find an initial feasible point, which is required for SCA and dominates the performance of SCA. Secondly, since constraint (9.15b) holds with equality at any feasible point of P4, the convex approximation is quite often poor and could lead to the infeasibility of convex subproblem [23]. Considering the two disadvantages of SCA, we present a generalized SCA algorithm through introducing a new slack variable to relax constraints (9.15b) and (9.22c), and to penalize the violation via ∞ norm, which will be elaborated as follows. After introducing a nonnegative slack variable t and exploiting (9.25), the convex subproblem at the jth iteration of the generalized SCA algorithm is expressed as P5 : max

˙ y˙ ,χ,w,t x,

K M   m=1 k=1

wlog2 

− SCT x˙ K 

( j)

χ

m,k

w



−ρ

N  K 

  ( j) CAnor m x˙ n,k , x˙ n,k

n=1 k=1

 , y˙ , χ , w( j) − ψ ( j) t

χ

( j)

m,k

( j)



+ t ≥ rmth w, ∀m, w k=1     CAn x˙ m  x˙ (mj) ≤ t, ∀m,

s.t. B

wlog2

(9.30a) (9.30b) (9.30c)

(9.11b), (9.11f), (9.21), (9.22d), (9.26) and (9.29), 2 2   where SCT x˙ ( j) , y˙ ( j) , χ( j) , w( j) = τx˙ x˙ − x˙ ( j) + τ y˙ y˙ − y˙ ( j) + τχ χ− 2 2 2  2 χ( j) + τw w − w( j) with positive parameters τx˙ , τ y˙ , τχ , and τw [21], and ψ ( j) > 0 2

156

9 Energy-Efficient User Scheduling and Power Control for Multi-Cell …

Algorithm 9.1 Generalized SCA Algorithm 1: Initialize Jmax ,  δ1 , δ2 , ψ (0) , and tolerances τ1 and τ2 .  2: Set initial point x˙ (0) , y˙ (0) , χ(0) , w(0) , index j ← 0, initial optimal objective value (0) ← ∞. 3: while j ≤ Jmax do   4: Solve P5 with x˙ ( j) , y˙ ( j) , χ( j) , w( j) to obtain the optimal solution   ( j+1) ( j+1) j+1) j+1) j+1) ( ( ( x˙ and the corresponding Lagrange multipliers , y˙ ,χ ,w ,t   ( j+1) j+1) ( α ,β .  ( j+1)   χm,k M K ( j+1) log 5: ( j+1) ← m=1 2 w( j+1) k=1 w   6: if ( j+1) − ( j)  ≤ τ1 and t ( j+1) ≤ τ2 then 7: Break. 8: end if   ( j+1) 9: λ¯ ← min λtotal + δ1 , τ3 10: if ψ ( j) ≥ λ¯ then 11: ψ ( j+1) ← ψ ( j) 12: else 13: ψ ( j+1) ← ψ ( j) + δ2 14: end if 15: j ← j + 1 16: end while

  is a penalty parameter. Adding the term SCT x˙ ( j) , y˙ ( j) , χ( j) , w( j) into the objective function, which has no impact on condition (9.10), is to guarantee the convergence of the generalized SCA algorithm. In P5, constraints (9.26), (9.29), and (9.30c) are the approximations of constraints (9.22b), (9.17), and (9.15b), respectively. Obviously, we can start from any infeasible point of P4 to solve P5 by using the developed algorithm, as shown in Algorithm 9.1, and the convergent solution of Algorithm 9.1 satisfies the generalized KarushKuhn-Tucker (KKT) conditions. For brevity, we omit the proof of the convergence of Algorithm 9.1, and similar proof can be found in [23]. the optimal solution of P5 with   In Algorithm 9.1, after obtaining ( j+1) ( j+1) ( j+1) ( j+1) ( j+1) at the jth SCA iteration, the iteration is ter, y˙ ,χ ,w ,t x˙ minated and the suboptimal solution is obtained if the change of the objective function of P3 and the slack variable t are smaller than the prescribed thresholds τ1 and τ2 , respectively, as shown from Lines 4 to 8. Since P5 is convex, ( j) = the first order generalized KKT condition  with respect to t is given by  ψ   j+1) j+1) j+1) j+1) ( j+1) ( ( ( ( M ( j+1) ( j+1) with α , , where λtotal = m=1 αm + βm = αm λtotal +    ( j+1) ( j+1) , and ( j+1) being the corresponding nonnegative Lagrange mulβ = βm tipliers related to (9.30b), (9.30c), and t ≥ 0, respectively. If t ( j+1) = 0 and δ1 ≤ ( j+1) coefficient ( j+1) , we have ( j+1) > 0 and ψ ( j) ≥ λtotal + δ1 , where the penalty ψ ( j+1) ( j+1) ( j) ( j) − x˙ + y˙ − y˙ + would not be updated. Furthermore, if τ3 = x˙ 2

2

9.3 Joint User Scheduling and Power Control

157

 ( j+1)  χ − χ( j) 2 + w( j+1) − w( j)  is relatively large, which means that the current solution is far away from the convergent solution, the penalty coefficient ψ will also remain unchanged, as shown from Lines 9 to 14. Otherwise, we will increase the penalty coefficient ψ by δ2 , as shown on Line 13. It is noticed that the SE maximization problem could also be solved by Algorithm 9.1 through setting w = 1 and removing constraint (9.11f). Moreover, if perfect CSI is available at the transmitter, Algorithm 9.1 can be also used to obtain a generalized KKT solution of the joint energy-efficient user scheduling and power control problem as follows. We first replace (9.5) in P1  2 N M   ≤ ym,k and w + and (9.17) in P4 with 1 + m=1, ˜ G n,m h n,m,k ˜ m ˜ =m n=1 x n,m,k  2 N M   ≤ y˙m,k , respectively, since ym,k in P1 could ˜ G n,m h n,m,k m=1, ˜ m ˜ =m n=1 x˙ n,m,k be viewed as the interference plus noise to user m on subcarrier   k. And then,  N x˙n,m,k G n,m |h n,m,k |2 and χwm,k ≤ 1 + we redefine rm,k and (9.22b) as log2 1 + n=1 y˙m,k N

x˙n,m,k G n,m |h n,m,k | y˙m,k

2

, respectively. In the ideal situation, amounts of elements of x˙ should be zero after convergence of Algorithm 9.1. Unfortunately, due to the numerical issue, a large number of elements of x˙ may be quite small positive values instead of zero, which has negative influence on the convergence criteria of Algorithm 9.1. In such a scenario, we can obtain the user scheduling results through setting the quite small elements of x˙ to zero and then perform the power control to further improve the EE. Specifically, if Algorithm 9.1 is terminated with j ≤ Jmax , a feasible solution of P1 is obtained by Algorithm 9.1. If the actual SE of we have

user m on subcarrier k is larger than the requirement,  g (m) = arg max xn,m,k , m = m ∪ k, and pg(m),k = max xn,m,k , where m ⊆ n=1

n

n

{1, 2, . . . , K }, 1 ≤ m ≤ M, is the set of subcarriers assigned by eNB g (m) to user m, and pn,k denotes the transmit power of eNB n on subcarrier k. Then power control could be performed based on the user scheduling solution, i.e., g (m) and m .

9.4 Decentralized Power Control Given user scheduling results, decentralized energy-efficient power control is more desirable in practice. In the following, we first formulate the power control subproblem of P5 and equivalently transform the power control problem at each iteration of SCA by a suitable one, which is amenable to applying the alternating direction method of multipliers (ADMM). Then a decentralized power control algorithm for EE maximization is developed through combining SCA and ADMM. With eNB association and subcarrier assignment

results  denoted by g (m) and m for user m, respectively, we have pg(m),k = max xn,m,k by the assumptions that n each user is associated with only one eNB. Suppose that an initial feasible solution of P3 is obtained in advance, the slack variable t in P5 could be removed. Through

158

9 Energy-Efficient User Scheduling and Power Control for Multi-Cell …

Algorithm 9.2 Centralized Power Control based on SCA

  1: Initialize q (0) , χ(0) , y˙ (0) , w(0) based on the solution of Algorithm 9.1 and set j ← 0 2: repeat   and obtain the optimal solution 3: Solve P6 with q ( j) , χ( j) , y˙ ( j) , w( j)   j+1) ( q ( j+1) , χ( j+1) , y˙ , w( j+1) . 4: j ← j + 1 5: until convergence

N defining the surrogate of x˙n,m,k as qg(m),k = pg(m),k w = n=1 x˙n,m,k , we can recast the power allocation subproblem of P5 without the slack variable t as N   

P6 : max

q, y˙ ,χ,w

wlog2

n=1 m∈n k∈m

 

s.t.

χ

m,k



(9.31a)

w

qg(m),k ≤ Pnmax w, ∀n,

(9.31b)

m∈n k∈m

       ( j) ( j) CAs χm,k , y˙m,k χm,k , y˙m,k ≤ w y˙m,k + qg(m),k G˜ g(m),m , ∀m, ∀k ∈ m ,

B



wlog2

χ

k∈m

w ( j) υm,k

ς

+

m,k



w

(9.31c) ≥ rmth w, ∀m,

N 

qn,k G˜ n,m   ( j) ( j) ˜ m υm,k 1 + n,m,k n=1,n=g(m) ε

N   

(9.31d) ≤ y˙m,k , ∀m, ∀k ∈ m ,

(9.31e)

qg(m),k + Pc w = 1,

(9.31f)

n=1 m∈n k∈m

(9.21), and (9.22d),



where q = qg(m),k , n = {m |g (m) = n} represents the setof users served by   ( j) q G˜ f,( j) ( j) ( j) ( j) eNB n, n,m,k = n,kf,( j)n,m with y˙m,k = y˙m,k qn,k , w( j) and qn,k being the y˙m,k

n=g(m)

( j)

optimal solution at the ( j − 1)th iteration of SCA, ∀n = g (m), and υm,k is redefined    ( j) ( j) ln 1+n,m,k N n,m,k   − . It should be noted that the LHS of as 1 + n=1,n=g(m) ( j) ε˜ m ε˜ m 1+n,m,k

(9.31e) is equivalent to that of (9.29) due to the fact that the nonzero summation terms in (9.31e) are quite the same as those in (9.29). By utilizing SCA, we can now solve energy-efficient power control problem, as shown in Algorithm 9.2, whose objective value is monotonically increasing and the solution converges to a KKT point of the non-convex power control subproblem of P3 [22].

9.4 Decentralized Power Control

159

Next we will elaborate how to solve P6 by ADMM at each iteration of SCA. Since the objective function and the constraints of P6 are not separable, ADMM can not be applied to P6 directly. Therefore, we first equivalently convert constraint (9.31e) into ⎧ N  ⎪ ln,m,k w ⎪ ⎨   ≤ y˙m,k , ∀m, ∀k ∈ m , (9.32a) + ( j) j) j) ( υm,k n=1,n=g(m) ε˜m υm,k 1 + (n,m,k ⎪ ⎪ ⎩ / n , ∀k ∈ m , (9.32b) ln,m,k = qn,k G˜ n,m , ∀m ∈ 

with ln,m,k being a new variable [16, 30]. Next, we introwhere l = ln,m,k m ∈ / n ,k∈m  

n  n duce auxiliary variables, i.e., wn , ln,m,k , and l , n,m,k ˜ m ∈ / ,k∈ n

m

n˜ =g(m),m∈n ,k∈m

n n , and ln,m,k are the local copies of w, ln,m,k , and ln,m,k for eNB n, where wn , ln,m,k ˜ ˜ respectively. Based on (9.32a), (9.32b), and the above introduced auxiliary variables, we define the local feasible set of eNB n at the jth iteration of SCA as

Fn( j)

  "     = q n , χn , ˙yn , η˜ n  qg(m),k ≤ Pnmax wn , m=n k∈m        ( j) ( j) CAs χm,k , y˙m,k χm,k , y˙m,k ≤ w y˙m,k + qn,k G˜ n,m , ∀m ∈ n , ∀k ∈ m , wn ( j)

υm,k

+

N 

l n˜ n,m,k  ≤ y˙m,k ,∀m ∈ n , ∀k ∈ m , ( j) ( j) n=1, ˜ n˜ =g(m) ε˜m υm,k 1 + n,m,k ˜

n = qn,k G˜ n,m , ∀m ∈ / n , ∀k ∈ m , ln,m,k    χm,k ≥ rmth wn , ∀m ∈ n , wn log2 B w n k∈ m

L w ≤ y˙m,k ≤ U y˙m,k , ∀m ∈ n , ∀k ∈ m ,

#

L χm,k ≤ χm,k ≤ Uχm,k , ∀m ∈ n , ∀k ∈ m ,

(9.33)



where q n = qn,k k∈n with n = {k |∃m, s.t.g (m) = n, k ∈ m } representing the  



set of active subcarriers in eNB n, χn = χm,k m∈n ,k∈m , ˙yn = y˙m,k m∈n ,k∈m , # " T  T 

n  n ˜ ˜ and η˜ n = l n , wn with l n = ln,m,k m ∈ . , ln,m,k ˜ / ,k∈ n

m

From (9.33), we can equivalently rewrite P6 as

n˜ =g(m),m∈n ,k∈m

160

9 Energy-Efficient User Scheduling and Power Control for Multi-Cell …

P7 :

max

q,χ, ˙y,η,η, ˜ ˜t

s.t.

N   

 wg(m) log2

n=1 m∈n k∈m

χm,k wg(m)



 q n , χn , ˙yn , η˜ n ∈ Fn( j) , ∀n,   Pc wn − 1 = t˜n , ∀n, qg(m),k + ς N m= k∈ 

n

N 

(9.34a) (9.34b) (9.34c)

m

t˜n = 0,

(9.34d)

n=1

η˜ n = η n , ∀n,

(9.34e)











 where q = q n , χ = χn , ˙y = ˙yn , η˜ = η˜ n , ˜t = t˜n , η = η n , and η n = T T l n , w with l n consisting of the corresponding global version of l˜n . Accordingly, the augmented Lagrangian function of P7 is   ˜ γ˜ ˜ η, ˜t , α, ˜ β, L A q, χ, ˙y, η,   N    χm,k = wg(m) log2 wg(m) n=1 m∈n k∈m ⎡ ⎛ ⎞ N    w − 1 P c n ⎣α˜ n ⎝ς − − t˜n ⎠ qg(m),k + N m∈n k∈m n=1 ⎞2 ⎤ ⎛  N 2 N

    c⎝ Pc wn − 1 c − − t˜n ⎠ ⎦ − β˜ qg(m),k + t˜n − t˜n ς 2 N 2 n=1 m∈ k∈ n=1 n

m

N ,  2   c T γ˜ n η˜ n − η n + , η˜ − η n − 2 n n=1

(9.35)

 ˜ and γ˜ = γ˜ n are where c > 0 is a positive penalty coefficient, and α ˜ = {α˜ n }, β, the Lagrange multipliers corresponding to (9.34c), (9.34d), and (9.34e), respectively. of Lagrange From the representations of η˜ and η n , it is obvious that γ˜ n is composed

n  n 

n   n n , and γ ˜ , and wn , , γ˜ n,m,k , l corresponding to l multipliers γ˜ n,m,k wn n,m,k ˜ n,m,k ˜ respectively. Now, we can resort to ADMM to solve P7. The general idea of ADMM is alternatively updating the local variables (i.e., q, χ, ˙y and η), ˜ the global variables (i.e., η and ˜t ), and the Lagrange multipliers [14]. Firstly, we update the local variables. From (9.34) and (9.35), the dth update of the local variables based on the framework of ADMM could be performed at each eNB by solving the following convex problem

9.4 Decentralized Power Control

 =

161

q (d+1) , χ(d+1) , ˙y(d+1) , η˜ (d+1) n n n n  

 

χm,k wn



wn log2 arg max (q n ,χn , ˙yn ,η˜ n )∈Fn( j) m∈n k∈m ⎛ ⎞   w − 1 P c n − t˜n(d) ⎠ qg(m),k + − α˜ n(d) ⎝ς N m∈n k∈m ⎞2 ⎛ c ⎝   Pc wn − 1 − − t˜n(d) ⎠ qg(m),k + ς 2 N m∈ k∈ n

m

T  2  c . η˜ n − η (d) − η˜ n − η (d) − γ˜ (d) n n n 2

(9.36)

Secondly, after exchanging the local copies among eNBs, the next step of ADMM is to update the global variables through N , 2 T   c  (d+1) (d+1) η ˜ γ˜ (d) η ˜ + − η − η n n n , n n 2 n=1

η (d+1) = arg min η

(9.37)

and ˜t

(d+1)

 N 2 N  (d) c  −α˜ n t˜n t˜n + t˜n + 2 n=1 ˜t n=1 n=1 ⎞2 ⎤ ⎛ −1 c ⎝   (d+1) Pc w(d+1) n − t˜n ⎠ ⎦ . + qg(m),k + ς 2 N m∈ k∈

= argmin β˜ (d)

N 

n

(9.38)

m

Obviously, (9.37) is an unconstrained quadratic optimization problem, and its closedform solution is   1 n,(d+1) 1  n,(d) g(m),(d+1) g(m),(d) (d+1) ln,m,k + ln,m,k , (9.39) ln,m,k = + γ˜ n,m,k + γ˜ n,m,k 2 c  N  1  1 w(d+1) = wn(d+1) + γ˜ w(d) . (9.40) N n=1 c n For (9.38), to derive its closed-form solution, we rewrite (9.38) by completing the square as ˜t

(d+1)

 N 2 N    2 (d+1) t˜n − s˜n = argmin + t˜n , ˜t

n=1

n=1

(9.41)

162

9 Energy-Efficient User Scheduling and Power Control for Multi-Cell …

where ⎛ s˜n(d+1) = ⎝ς

 

(d+1) qg(m),k

m∈n k∈m

⎞ Pc wn(d+1) − 1⎠ β˜ (d) − α˜ n(d) + . − N c

(9.42)

Setting the first derivative of (9.41) to zero yields t˜n(d+1) = s˜n(d+1) −

N 

t˜n˜(d+1) , ∀n.

(9.43)

n=1 ˜

From (9.43), the closed-form solution to (9.38) is derived as 1  (d+1) s˜n˜ . N +1 N

t˜n(d+1) = s˜n(d+1) −

(9.44)

n=1 ˜

Finally, the last step of ADMM is to apply dual ascent to update the Lagrangian multipliers as ⎛ α˜ n(d+1) =α˜ n(d) + c ⎝ς

 

(d+1) qg(m),k +

Pc wn(d+1)

m∈n k∈m

−1

− t˜n(d+1) ⎠ ,

c  (d+1) s˜ , N + 1 n=1 n   (d+1)

(d+1) η ˜ . =γ˜ (d) + c − η n n n

(9.45)

N

β˜ (d+1) =β˜ (d) + γ˜ (d+1) n

N

⎞

(9.46) (9.47)

Substituting (9.42), (9.44), and (9.46) into (9.45) produces β˜ (d) = α˜ n(d) , ∀n, and N g(m),(d) n,(d) + γ˜ n,m,k = 0 and n=1 γ˜ w(d) taking (9.39) and (9.40) into (9.47) yields γ˜ n,m,k ˜ n = 0. Hence, the global variables can be updated without the Lagrangian multipliers, i.e., (9.39), (9.40), and (9.42) are simplified to  1  n,(d+1) g(m),(d+1) ln,m,k + ln,m,k , 2 N 1  (d+1) = w , N n=1 n

(d+1) = ln,m,k

w(d+1)

s˜n(d+1) =ς

  m∈n k∈m

(d+1) qg(m),k +

Pc wn(d+1) − 1 . N

(9.48) (9.49) (9.50)

It is worth mentioning that the updates of t˜n(d+1) and w(d+1) are implemented by an average consensus algorithm, where the amount of required exchanged

9.4 Decentralized Power Control

163

Algorithm 9.3 Decentralized Power Control based on SCA and ADMM

  (0) 1: Initialize q (0) , χ(0) , y˙ (0) , η˜ (0) , η (0) , ˜t , α ˜ (0) , β˜ (0) , γ˜ (0) and set j ← 0. 2: repeat 3: Set d ← 0 4: repeat   n,(d+1) 5: eNB n, ∀n: Update local variables using (9.36) and send local copy ln,m,k 6:

7: 8: 9: 10: 11: 12:

m ∈ / n ,k∈m

to other eNBs. eNB n, ∀n: Update global variables using (9.44), (9.48) and (9.49) by average consensus algorithm. eNB n, ∀n: Update Lagrange multiplies using (9.45) and (9.47). d ←d +1 until ADMM converges Update the SCA parameters. j ← j +1 until SCA converges

information  is 2N (N − 1) [31]. After each eNB gathering the information of  n,(d+1) , where the overhead for exchanging is upper bounded by ln,m,k ˜ n˜ =g(m),m∈n ,k∈m

(d+1) K (N − 1), ln,m,k and the Lagrangian multipliers can be updated. Therefore, the total amount of the exchanged information at each ADMM iteration is upper bounded by N (N − 1) (K + 2), and eNBs can exchange these information through X2 interface [4]. Through information exchanging, we develop a decentralized energy-efficient power control algorithm based on the combination of SCA and ADMM, as shown in Algorithm 9.3. The convergence of Algorithm 9.3 is guaranteed by the following two reasons. Firstly, we solve the power control subproblem for EE maximization by solving a sequence of convex approximation problems based on SCA, whose convergence is studied in [22]. Secondly, for each SCA step, the approximation convex problem is solved in a decentralized fashion by resorting to ADMM. The convergence behavior of ADMM has been well investigated in [13]. It is noted that we could update the SCA parameters, as shown on Line 10 of Algorithm 9.3, without a centralized processing unit, which will be explained as follows. Firstly, at each eNB is based on the local

 of constraint (9.31c)( j+1)

 the update n = ln,m,k , n˜ = n, m ∈ n , G˜ n,m variables χn n and y˙ n n . Secondly, we have qn,k ˜ ˜ ˜ ( j+1) = wn , ∀n after the convergence of ADMM at the jth iteration k ∈ m and w ( j+1) can be derived at eNB n without any of SCA. Therefore, n,m,k ˜ n˜ =n,m∈n ,k∈m

other information exchanging, from which we conclude that the update of constraint (9.31e) could be performed at each eNB and Algorithm 9.3 could be implemented in fully decentralized manner.

164

9 Energy-Efficient User Scheduling and Power Control for Multi-Cell …

9.5 Simulation Results In this section, the simulation results are presented to evaluate the performance of the developed algorithms. A wireless cellular network consisting of N = 3 hexagon cells is considered. The cell radius is 500 m, the subcarrier spacing is 150 kHz, the carrier frequency is 2 GHz, the reference distance is 100 m, the path loss exponent is 3.8, and the log-normal shadowing is modeled by 10log10 L n,m ∼ N (0, 8). In the simulations, the drain efficiency of power amplifier 1ς , the dynamic circuit power consumption parameter ξ, and the static circuit power consumption Pc are set to 0.38, 2W/Mbps, and 10N W, respectively [19]. The tolerable OPs, i.e., εm , ∀m, are 0.1.

9.5.1 Convergence and Complexity of Algorithms The convergence results of Algorithms 9.1, 9.2, and 9.3 are presented in Figs. 9.1, 9.2 and 9.3, respectively. Figure 9.1 shows that Algorithm 9.1 converges within tens of iterations for different initial points, where the initial points of xn,m,k , are set to P max zero, equal power, i.e., Mn K , and random initialization, respectively. It is shown that the convergence rate and the performance of Algorithm 9.1 are sensitive to the initial points. Figure 9.2 shows the convergence rates of Algorithm 9.2 and the Dinkelbachbased method [9, 22] with three different channel realizations. Note that through the change of variables, i.e., q¯ g(m),k = ln pg(m),k and y¯g(m),k = ln ym.k , the OP constraint (9.5) can be transformed into a convex constraint since the logarithm of a sum of exponentials is convex [27]. Therefore, the method in [9, 22] could be also applied to our power control problem. Each marked point in the curve of Dinkelbach-based method represents the update of SCA parameters. As benchmark, we obtain the optimal results by using the non-commercial solvers for Mixed Integer Programming and Mixed Integer Nonlinear Programming [32]. We observe that Algorithm 9.2 may converge slower than the Dinkelbach-based method. However, Dinkelbach-based method is a two-stage approach, which is not efficient to implement a decentralized power control algorithm since it is not suitable for inter-operation sharing networks [14]. Besides, the method in [9, 22] is not suitable to solve the energy-efficient joint user scheduling and power control problem since the convex approximation of SE in [9, 22] may be meaningless if the signal-to-interference-plus-noise ratio (SINR) of user m on subcarrier k is zero, which results in log (0). The convergence of Algorithm 9.3 is shown in Fig. 9.3 with two different channel realizations. Figure 9.3 shows that the developed decentralized algorithm converges to the centralized solution within hundreds of iterations. The complexity of the developed algorithms depends on the number of iterations for convergence and the complexity of each iteration. A theoretical analysis for the number of iterations to reach convergence is very challenging [22], so we analyse the number of iterations for convergence numerically. From Figs. 9.1

9.5 Simulation Results

165

Fig. 9.1 Convergence of the Algorithm 9.1. (N = 3, max = 46 M = 6, K = 4, pm dBm, and rmth = 150 kbps)

0

Objective value of P5

−50

1.5

−100

1 −150 0.5

Zero Equal Random

−200

0 60

100

90

80

70

100

80

60

40

20

0

Iterations

Fig. 9.2 Convergence of the Algorithm 9.2. (N = 3, max = 46 M = 8, K = 4, pm dBm, and rmth = 150 kbps)

1.3 1.2

Objective value of P6

1.1 1 0.9 0.8 0.7

Optimal Alg.9.2 Dinkelbach

0.6 0.5 0.4

30

25

20

15

10

5

0

Iterations

0.12

Average Energy Efficiency (Mbits/Joule)

Fig. 9.3 Convergence of the Algorithm 9.3. (N = 3, max = 46 M = 10, K = 4, pm dBm, and rmth = 15 kbps)

0.11

0.1

0.09

0.08

Decentralized Centralized

0.07

0.06

100

200

300

400

500

Iterations

600

700

800

900

1000

166

9 Energy-Efficient User Scheduling and Power Control for Multi-Cell …

and 9.2, Algorithms 9.1 and 9.2 both converge within at most tens of iterations. Since the problem at each iteration is convex, it could be solved with polynomial complexity in the number of variables and constraints. Specifically, if the interior-point method is used  to solve the convex problem at each iter1

ation, then the complexity is O θ12 (θ1 + θ2 ) θ22 , where θ1 is the number of inequality constraints and θ2 is the number of variables [33]. Hence, the complexity 9.1   of solving the convex programming at each iteration of Algorithm 1 is O (2M (K + 1) + N + 2) 2 (M K (N + 4) + N + 2M + 4) (M K (N + 2) + 2)2 and that of 

1



Algorithm 9.2 is upper bounded by O (M (2K + 1) + N + 2) 2 (M (5K + 1) + N + 3) (3M K +1)2 . Figure 9.3 shows that Algorithm 9.3 converges within hundreds of iterations and the complexity of every eNB at each iteration mainly depends onthe convex programming (9.36), whose complexity is upper bounded by  1 2 O ((M + 1) (K +1)) 2 ((M + 1) (3K +1) + 1 − K ) (K (2M + 1) + 1) .

9.5.2 Performance Results Figure 9.4 plots the average EE of the developed schemes with the variable number of subcarriers K . As mentioned in Sect. 9.3, the final result, which is denoted by EE-EE, is obtained by first solving the energy-efficient joint user scheduling and power control problem based on Algorithm 9.1, then setting the quite small elements of x˙ to zero, and performing the power control to further improve the EE. Similarly, SE-EE and SE-SE represent the results that we first derive a joint spectral-efficient user scheduling and power control solution and then perform the power control for EE maximization and SE maximization, respectively. In Fig. 9.4, the term “SE-EED” indicates that the corresponding power control results are obtained based on the Dinkelbach method [9, 22]. The solid lines and the dashed lines represent the performance metrics of EE based on CDI and CSI, respectively, and the dotted lines and the dash-dot lines represent the performance metrics of SE based on CDI and CSI, respectively. Figure 9.4 shows that the average EEs of EE maximization schemes and SE maximization schemes are both monotonically increasing with respect to K . This observation is expected, since the greater the subcarrier K is, the higher the frequency selection diversity gains are. Figure 9.5 depicts the effects of the maximum allowed transmit power Pnmax on the average EE and the average sum rate. It is shown that the average EE of EE maximization schemes firstly increases, and then remains almost constant. This can be explained by the following facts. With low Pnmax , the average sum rates of the EE maximization schemes increase with Pnmax . However, when the increase speed of energy consumption is faster than that of the average sum rate, increasing the transmit power would result in lower EE. Hence, the EE maximization scheme would not utilize the extra available power to further improve the throughput [9], where the average sum rates of EE maximization schemes exhibit a floor as Pnmax increases.

9.5 Simulation Results

0.38 0.32

27

CSI:EE−EE:EE CSI:SE−EE:EE CSI:SE−SE:EE CSI:SE−EE−D:EE

CDI:EE−EE:EE CDI:SE−EE:EE CDI:SE−SE:EE CDI:SE−EE−D:EE

24 21

0.26

18

0.2

15

0.14

12

0.08

9

0.02 −0.04 −0.1

6

CDI:EE−EE:SE CDI:SE−EE:SE CDI:SE−SE:SE CDI:SE−EE−D:SE 3

4

5

6

7

Average Sum Rate (Mbits/s)

0.44

Average Energy Efficiency (Mbits/Joule)

Fig. 9.4 Impact of the number of subcarriers K on the average EE and on the average sum rate (N = 3, max = 46 dBm, M = 6, pm and rmth = 150 kbps)

167

CSI:EE−EE:SE CSI:SE−EE:SE 3 CSI:SE−SE:SE CSI:SE−EE−D:SE 8

9

0 10

0.2

12

0.16

10

0.12

8

0.08

6

4

0.04

CDI:EE−EE:EE CDI:SE−EE:EE CDI:SE−SE:EE CDI:SE−EE−D:EE CDI:EE−EE:SE CDI:SE−EE:SE CDI:SE−SE:SE CDI:SE−EE−D:SE

0

−0.04

−0.08 10

15

20

25

30

CSI:EE−EE:EE CSI:SE−EE:EE 2 CSI:SE−SE:EE CSI:SE−EE−D:EE CSI:EE−EE:SE 0 CSI:SE−EE:SE CSI:SE−SE:SE CSI:SE−EE−D:SE 35

40

Average Sum Rate (Mbits/s)

Fig. 9.5 Impact of maximum allowed transmit power Pnmax on the average EE and on the average sum rate. (N = 3, M = 8, K = 4, and rmth = 150 kbps)

Average Energy Efficiency (Mbits/Joule)

Number of Subcarriers K

−2 45

Maximum Allowed Transmit Power Pmax (dBm) n

In Fig. 9.6, we observe that the EE of the EE maximization schemes firstly increases and then decreases with M, which can be explained as follows. When M is relatively small, the average EE may increase with the increase of M due to the multiuser diversity. However, when M is large enough, the average EE would decrease with the increase of M since the ICI would deteriorate the EE performance eventually while guaranteeing the minimum required throughput of each user. From Figs. 9.4 and 9.5, it is obvious that there exists a performance gap between the CDI-based method and the CSI-based method. It should be pointed out that although the performance of the networks based on CSI is better than that based on CDI, the CDI-based resource allocation scheme could reduce the signal processing energy consumption and signalling overhead significantly compared with the CSI-based scheme [18]. From Figs. 9.4, 9.5, and 9.6, we conclude that the developed power control algorithm and the Dinkelbach-based method have the same performance if they start from the same initial point and the performance gap between EE-EE scenario and SE-EE scenario is not significant.

Fig. 9.6 Impact of the number of users M on the average EE and on the average sum rate. (N = 3, max = 46 dBm, and K = 4, pm rmth = 15 kbps)

0.13

8

0.11

7

0.09

6

0.07

5

0.05

4

0.03

3

CDI:EE−EE:EE CDI:SE−EE:EE CDI:SE−SE:EE CDI:SE−EE−D:EE

0.01

−0.01

3

4

5

CDI:EE−EE:SE CDI:SE−EE:SE CDI:SE−SE:SE CDI:SE−EE−D:SE 6

7

8

Average Sum Rate (Mbits/s)

9 Energy-Efficient User Scheduling and Power Control for Multi-Cell …

Average Energy Efficiency (Mbits/Joule)

168

2

9

1 10

Number of users M

9.6 Conclusions In this chapter, we have investigated the joint user scheduling and power control for EE maximization in downlink multi-cell multiuser OFDMA networks, where only the CDI is assumed available. We have formulated the EE maximization problem as a cardinality constrained combinatorial FP problem, and then solved a joint user scheduling and power control problem based on the framework of SCA, whose convergent solution satisfies the generalized KKT conditions. Given user scheduling solution, we have presented a decentralized power control algorithm to maximize EE through combining SCA and ADMM.

Appendix 1 Derivation of the Exact Penalty Coefficient

For we assume that there exists an optimal solution of P4 , denoted by  ∗ simplicity, ∗ ∗ ∗ x˙ , y˙ , χ , w , violating constraint (9.15a) for some n and k. From constraint N ∗ ∗ ∗ (9.15b), we have x˙n,m,k =0 if n = g (m) and n=1 x˙n,m,k = x˙ g(m),m,k G˜ G˜ g(m),m .   n,m      ∗ ∗ ∗ Let A∗ = (m, k) x˙ g(m),m,k = 0 and B ∗ = (m, k) x˙ g(m),m,k = 0, x˙ g(m),m,k = ∗ 2

x˙ g(m),k ∞ , the nonzero penalty term of the objective function of P4 could be expressed as

Here we assume that x˙ ∗g(m),k has a unique nonzero maximal element for any m and k. If there are several nonzero elements of x˙ ∗g(m),k equaling to x˙ ∗g(m),k , only one of the indices (m, k) such ∞ ∗ that x˙ g(m),m,k = x˙ ∗g(m),k belongs to B ∗ . For example, if ∃m˜ = m, such that g (m) ˜ = g (m) and ∞ ∗ ∗ ˙ ∗g(m),k , either (m, k) or (m, x˙ g(m),m,k = x˙ g(m), ˜ k) belongs to B ∗ . m,k ˜ = x

2

∞

9.6 Conclusions

169 N  K    ∗ x˙ − x˙ ∗ = n,k 1 n,k ∞ n=1 k=1



∗ x˙ g(m),m,k ,

(9.51)

(m,k)∈V ∗

where V ∗ = A∗ \B ∗ is the difference of sets A∗ and B ∗ . From constraint (9.11f), −1   ∗ + Pc . w∗ = ς (m,k) x g(m),m,k

Subsequently, we  could construct a new solution of P4 , denoted by  ∗∗ ∗∗ ∗∗ ∗∗ x˙ , y˙ , χ , w , as follows. We first derive the nonzero transmitted power x˙ ∗

y˙ ∗

∗ ∗ = g(m),m,k , (m, k) ∈ A∗ and ym,k = wm,k∗ , ∀m, ∀k. A new power vector x ∗∗ x g(m),m,k w∗ ∗ is obtained by setting x g(m),m,k = 0 for (m, k) ∈ V ∗ and remaining the rest of ele∗∗ = 0 if (m, k) ∈ B ∗ . Next, let ments of x ∗ unchanged. It is obvious that x g(m),m,k −1   ∗∗ ∗∗ ∗∗ ∗∗ ∗ w∗∗ = ς (m,k) x g(m),m,k + Pc , x˙n,m,k = xn,m,k w∗∗ , y˙m,k = ym,k w∗∗ , and χ∗∗ m,k =   N x˙ ∗∗ G˜ n,m . w∗∗ 1 + n=1 y˙n,m,k ∗∗ m,k

∗∗ ∗ From the rule of constructing the new solution, it is shown that xn,m,k ≤ xn,m,k and ∗∗ ∗ ∗ xn,m,k = 0 < xn,m,k if and only if n = g (m) and (m, k) ∈ V . Therefore, we have

w∗∗ =

ς

∗∗ x˙ g(m),m,k ∗∗ y˙m,k

 (m,k)

=

1 1 > w∗ =  ∗ , ∗∗ x g(m),m,k + Pc ς x g(m),m,k + Pc

(9.52a)

(m,k)

∗∗ x g(m),m,k ∗ ym,k

=

∗ x g(m),m,k ∗ ym,k

=

∗ x˙ g(m),m,k ∗ y˙m,k

, ∀(m, k) ∈ / V ∗,

∗∗ ∗∗ x˙ g(m),m,k = x g(m),m,k w∗∗ = 0, ∀(m, k) ∈ V ∗ ,

(9.52b) (9.52c)

∗∗ = x∗ with (m, k) ∈ / V ∗. where the second equality in (9.52b) holds since xn,m,k   ∗∗ ∗∗ ∗∗ n,m,k ∗∗ is also a feasible From (9.52), it could be easily verified that x˙  , y˙ , χ , w  solution to P4 . Compared with x˙ ∗ , y˙ ∗ , χ∗ , w∗ , the new constructed solution satisfies constraint (9.15a), ∀n and ∀k, which results in N  K    ∗∗ x˙ − x˙ ∗∗ = 0. n,k 1 n,k ∞

(9.53)

n=1 k=1

    ˙ y˙ , χ, w by Obj x, ˙ y˙ , χ, w . From Denote the objective value of P4 at point x, (9.51) and (9.53), we have

170

9 Energy-Efficient User Scheduling and Power Control for Multi-Cell …

    Obj x˙ ∗∗ , y˙ ∗∗ , χ∗∗ , w∗∗ − Obj x˙ ∗ , y˙ ∗ , χ∗ , w∗  ∗∗   ∗  M  K    χm,k χm,k ∗∗ ∗

∗ w log2 − w + ρ = log x˙ g(m),m,k 2 ∗∗ ∗ w w ∗ m=1 k=1 (m,k)∈V =

M  K    ∗∗ ∗∗  ∗ ∗ w rm,k − w∗rm,k + ρ x˙ g(m),m,k (m,k)∈V ∗

m=1 k=1

≥

M  K 

  ∗∗  ∗ ∗ + ρ w∗ rm,k − rm,k x˙ g(m),m,k (m,k)∈V ∗

m=1 k=1

⎛

⎞      ∗∗ ∗ ∗ ⎠ ∗ rm,k − =w∗ ⎝ − rm,k rm,k x˙ g(m),m,k + ρ / ∗ (m,k)∈V

=ρ



(m,k)∈V ∗

∗ x˙ g(m),m,k − w∗

(m,k)∈V ∗



(m,k)∈V ∗

∗ rm,k

(m,k)∈V ∗

∗  x˙ g(m),m,k G˜ g(m),m w∗ ∗ ln (2) (m,k)∈V ∗ y˙m,k (m,k)∈V ∗  ⎞ ⎛ max G˜ n,m n,m ⎟  ∗ ⎜ ≥ ⎝ρ − x˙ g(m),m,k , ⎠ ln(2) (m,k)∈V ∗

≥ρ



∗ x˙ g(m),m,k −

 ∗∗ where rm,k = log2 1 +

∗∗ x˙ g(m),m,k G˜ g(m),m ∗∗ y˙m,k

(9.54)

   x˙ ∗ G˜ g(m),m ∗ , and rm,k . The sec= log2 1+ g(m),m,k ∗ y˙ m,k

ond equality in (9.54) follows from the fact that (9.22b) must hold with equality at the optimum. The first inequality is obtained by (9.52a). The third and the forth inequalities are due to (9.52c) and (9.52b), respectively. According the property of logarithm function, i.e., ln (1 + θ) ≤ θ, the third inequality in (9.54) holds. The last inequality is derived from (9.21).   max G˜ n,m   n,m From (9.54), it is obvious that if ρ > ln(2) , Obj x˙ ∗∗ , y˙ ∗∗ , χ∗∗ , w∗∗ ≥     Obj x˙ ∗ , y˙ ∗ , χ∗ , w∗ , which is contradict to the assumption that x˙ ∗ , y˙ ∗ , χ∗ , w∗  is an optimal solution of P4 . Therefore, ρ =

max G˜ n,m n,m

ln(2)

is an exact penalty of P4 .

Appendix 2 Proof the Condition of SCA in Chap. 9   

Denote the LHS of (9.18) by the function f˜ y˙m,k , x˙n,m,k , w , which is abbre˜ n,m ˜ =m ˜ viated utilizing  to f . By  the implicit function theorem [34], the partial gradient of  , w with respect to x˙n,m,k ˜ = m, is given by y˙m,k x˙n,m,k ˜ ˜ ,m n,m ˜ =m

9.6 Conclusions

171 G˜ n,m

∂ y˙m,k ∂ f˜ ∂ y˙m,k G˜ n,m 1+m,m,k ˜  , = M =− =  w˜ ε m, m,k ˜ m ˜ ∂ x˙n,m,k ∂ x˙n,m,k ε˜m υm,k 1 + m,m,k ∂f ˜ ˜ + y˙m,k ˜ m=1, ˜ m ˜ =m 1+m,m,k ˜ (9.55) where υm,k = 1 +

M

m=1, ˜ m ˜ =m



   ˜ ln 1+m,m,k ˜  m,m,k  − ε˜ m ε˜ m 1+m,m,k ˜

and m,m,k ˜ =

N

n=1

˜ x˙n,m,k ˜ G n,m y˙m,k

.

The last equality in (9.55) is obtained by utilizing wε˜m = ε˜m − y˙m,k

M  m=1, ˜ m ˜ =m



 N  ˜ x˙n,m,k ˜ G n,m ln 1 + , y˙m,k n=1

(9.56)

which could be derived from (9.18). 

  Similarly, the partial gradient of y˙m,k x˙n,m,k , w with respect to w is ˜ n,m ˜ =m expressed as ε˜m ∂ f˜ ∂ y˙m,k ∂ y˙m,k = M =− m,m,k ˜ ∂w ∂w ∂ f˜ + m=1, ˜ m ˜ =m 1+m,m,k ˜

w˜εm y˙m,k

=

1 , υm,k

(9.57)

where the last equality holds due to (9.56).   

( j) From (9.55) and (9.57), we could conclude that if x˙n,m,k = x˙n,m,k ˜ ˜ n,m ˜ =m n,m ˜ =m 

  ( j) ( j) and w = w , the gradient of y˙m,k x˙n,m,k , w equals to that of yˆm,k . ˜ n,m ˜ =m

References 1. D. Lopez-Perez, A. Valcarce, G. de la Roche, J. Zhang, OFDMA femtocells: a roadmap on interference avoidance. IEEE Commun. Mag. 47(9), 41–48 (2009). 2. S.F. Yunas, M. Valkama, J. Niemelä, Spectral and energy efficiency of ultra-dense networks under differenct deployment strategies. IEEE Commun. Mag. 53(1), 90–100 (2015). 3. J. An, K. Yang, J. Wu, N. Ye, S. Guo, Z. Liao, Achieving sustainable ultra-dense heterogeneous networks for 5G. IEEE Commun. Mag. 55(12), 84–90 (2017). 4. R. Irmer, H. Droste, P. Marsch, M. Grieger, G. Fettweis, S. Brueck, H.-P. Mayer, L. Thiele, V. Jungnickel, Coordinated multipoint: concepts, performance, and field trial results. IEEE Commun. Mag. 49(2), 102–111 (2011). 5. T. Wang, L. Vandendorpe, Iterative resource allocation for maximizing weighted sum min-rate in downlink cellular OFDMA systems. IEEE Trans. Signal. Process. 59(1), 223–234 (2011). 6. S.-Y. Kim, J.-A. Kwon, J.-W. Lee, Sum-rate maximization for multicell OFDMA systems. IEEE Trans. Veh. Technol. 64(9), 4158–4169 (2015). 7. S. Buzzi, C.-L. I, T.E. Klein, H.V. Poor, C. Yang, A. Zappone, A survey of energy-efficient techniques for 5G networks and challenges ahead. IEEE J. Sel. Areas Commun. 34(4), 697–709 (2016).

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8. J.B. Rao, A.O. Fapojuwo, A survey of energy efficient resource management techniques for multicell cellular networks. IEEE Commun. Surveys Tuts. 16(1), 154–180 (2014). 9. L. Venturino, A. Zappone, C. Risi, S. Buzzi, Energy-efficient scheduling and power allocation in downlink OFDMA networks with base station coordination. IEEE Trans. Wireless. Commun. 14(1), 1–14 (2015). 10. S. He, Y. Huang, S. Jin, L. Yang, Coordinated beamforming for energy efficient transmission in multicell multiuser systems. IEEE Trans. Commun. 61(12), 4961–4971 (2013). 11. O. Tervo, L.-N. Tran, M. Juntti, Optimal energy-efficient transmit beamforming for multi-user MISO downlink. IEEE Trans. Signal. Process. 63(20), 5574–5588 (2015). 12. S. He, Y. Huang, L. Yang, B. Ottersten, Coordinated multicell multiuser precoding for maximizing weighted sum energy efficiency. IEEE Trans. Signal. Process. 62(3), 741–751 (2014). 13. S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trands Mcah. Learn. 3(1), 1–122 (2011). 14. Q.-D. Vu, L.-N. Tran, M. Juntti, E.-K. Hong, Energy-efficient bandwidth and power allocation for multi-homing networks. IEEE Trans. Signal. Process. 63(7), 1684–1699 (2015). 15. O. Tervo, L.-N. Tran, M. Juntti, Decentralized coordinated beamforming for weighted sum energy efficiency maximization in multi-cell MISO downlink, in IEEE Global Conf. Signal Inf. Process. IEEE(2015), pp. 1387–1391. 16. K.-G. Nguyen, Q.-D. Vu, M. Juntti, L.-N. Tran, Distributed solutions for energy efficiency fairness in multicell MISO downlink. IEEE Trans. Wireless. Commun. 16(9), 6232–6247 (2017). 17. W.-C. Li, T.-H. Chang, C.-Y. Chi, Multicell coordinated beamforming with rate outage constraint—part II: efficient approximation algorithms. IEEE Trans. Signal. Process. 63(11), 2763–2778 (2015). 18. S. Kandukuri, S. Boyd, Optimal power control in interference-limited fading wireless channels with outage-probability specifications. IEEE Trans. Wireless. Commun. 1(1), 46–55 (2002). 19. C. Xiong, G.Y. Li, S. Zhang, Y. Chen, S. Xu, Energy- and spectral-efficiency tradeoff in downlink OFDMA networks. IEEE Trans. Wireless. Commun. 10(11), 3874–3886 (2011). 20. K. Yang, S. Martin, C. Xing, J. Wu, R. Fan, Energy-efficient power control for device-to-device communications. IEEE J. Sel. Areas Commun. 34(12), 3208–3220 (2016). 21. G. Scutari, F. Facchinei, L. Lampariello, Parallel and distributed methods for constrained nonconvex optimization—part I: theory. IEEE Trans. Signal Process. 65(8), 1929–1944 (2017). 22. A. Zappone, L. Sanguinetti, G. Bacci, E. Jorswieck, M. Debbah, Energy-efficient power control: a look at 5G wireless technologies. IEEE Trans. Signal. Process. 64(7), 1668–1683 (2016). 23. H.A.L. Thi, V.N. Huynh, T.P. Dinh, DC programming and DCA for general DC programs. Advanced Computational Methods for Knowledge Engineering. Springer International Publishing, 2014, pp. 15–35. 24. H.L. Thi, T.P. Dinh, H. Le, X. Vo, DC approximation approaches for sparse optimization. Eur. J. Oper. Res. 244(1), 26–46 (2015). 25. G. Scutari, F. Facchinei, L. Lampariello, S. Sardellitti, P. Song, Parallel and distributed methods for constrained nonconvex optimization—part II: applications in communications and machine learning. IEEE Trans. Signal Process. 65(8), 1945–1960 (2017). 26. J.-Y. Gotoh, A. Takeda, K. Tono, DC formulations and algorithms for sparse optimization problems. Mathematical Engineering Technical Reports, Aug. 2015. 27. S. Boyd, L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. 28. G.D. Pillo, S. Lucidi, F. Rinaldi, An approach to constrained global optimization based on exact penalty functions. J. Global Optim. 54(2), 251–260 (2012). 29. T. Lipp, S. Boyd, Variations and extension of the convex-concave procedure. Available: https:// stanford.edu/~boyd/papers/pdf/cvx_ccv.pdf. 30. O. Tervo, H. Pennanen, D. Christopoulos, S. Chatzinotas, B. Ottersten, Distributed optimization for coordinated beamforming in multicell multigroup multicast systems: power minimization and SINR balancing. IEEE Trans. Signal Process. 66(1), 171–185 (2018). 31. L. Xiao, S. Boyd, Fast linear iterations for distributed averaging. Syst. Control Lett. 53, 65–78 (2014).

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Chapter 10

Energy-Efficient Base Station Association and Beamforming for Multi-Cell Multiuser Systems

This chapter investigates the energy-efficient base station (BS) association and beamforming for multi-cell multiuser systems. Section 10.1 introduces the motivation of investigating energy-efficient BS association and beamforming. Section 10.2 presents the system model and formulates the problem. Section 10.3 develops the centralized BS association and beamforming algorithm, and Sect. 10.4 develops the decentralized beamforming algorithm. Section 10.5 presents the numerical results, and Sect. 10.6 concludes this chapter.

10.1 Introduction In Chap. 9, we investigate the energy efficiency (EE) based on the channel distribution information (CDI) in the networks where the BS is equipped with one antenna. In this chapter, we further aim to maximize the EE based on CDI in the scenario where the BSs are equipped with multiple antennas. It is noted that to increase the system capacity, frequency reuse would severely deteriorate the performance experienced by the cell-edge users because of the increased interference [1–3], and coordinated beamforming, where the coordinated BSs share the knowledge of the spatial channels, is a practical approach for interference management [4]. Meanwhile, with the increasing number of BSs, the conventional greedy scheme, where each user is associated with the BS providing the highest signal, is no longer effective [5]. Therefore, the BS association becomes a significant design problem and joint BS association and coordinated beamforming is a systematic means to improve the performance of the systems [6]. Motivated by the above observations, we investigate the energy-efficient BS association and beamforming in coordinated multi-cell multiuser multiple-input single-output (MISO) downlink systems in this chapter.

© Beijing Institute of Technology Press 2023 J. An et al., Sustainable Wireless Communications, https://doi.org/10.1007/978-981-19-0448-6_10

175

176

10 Energy-Efficient Base Station Association and Beamforming …

10.2 System Model and Problem Formulation Consider a multi-cell multiuser downlink system consisting of M coordinated BSs and K users, where each BS is equipped with N antennas and each user has a single antenna. All the BSs share a common transmission bandwidth. Let wm,k ∈ C N ×1 and z k represent the transmit beamforming vector from BS m to user k and the data symbol for user k with unit energy, respectively. We assume that each user is allowed to access exactly one BS and each BS can serve multiple users. The received signal at the kth user is given by xk =

M 

H L m,k hm,k wm,k z k +

K M  

H L m,k hm,k wm,k˜ z k˜ + n 0 ,

(10.1)

˜ k=1, k˜ =k m=1

m=1

where the superscript “H ” denotes the matrix conjugate transpose, L m,k represents the large-scale fading composed of the path loss and the log-normal shadow fading between BS m and user k, hm,k ∈C N ×1denotes the instantaneous channel vector from BS m to user k, and n 0 ∼ CN 0, σ 2 is the independent complex additive white Gaussian noise with variance σ 2 . Assuming that each user k decodes its corresponding data symbol z k by treating the interference from other users as Gaussian noise, the instantaneous achievable spectral efficiency (SE) of the user k is given by ⎛

⎞  H 2   h G w m=1 m,k m,k m,k Ck = log2 ⎝1 +  H 2 ⎠ , K M   1+ k=1, ˜ ˜k=k m=1 G m,k hm,k wm,k˜ M

(10.2)

where G m,k = L 2m,k /σ 2 . In the following, we assume that each BS only has information of  the statistical H ∈ C N ×N , instead of hm,k , i.e., the channel covariance matrix Qm,k = E hm,k hm,k the instantaneous knowledge of hm,k . The channel covariance matrix could be measured through averaging the uplink and downlink channel measurements based on the channel reciprocity in time division duplexing (TDD) systems [7] and the channel covariance matrix needs to be fed back efficiently in frequency division duplexing (FDD) systems [8]. Note that the channel covariance matrix Qm,k may not have rank one since it is unrealistic to assume that all the transmit antennas are independent and identically distributed in Rayleigh fading channel and the eigenvalues of matrix Qm,k depend on the spatial correlation of multiple antennas [9]. It should be pointed out that if the angular spreads are not comparable at the transmitter and receiver, two main eigenvectors of the channel covariance matrix are sufficient to produce the strongest eigenmodes in the channel covariance matrix with high probability [8]. Therefore, the CDI-based scheme can reduce the amount of feedback significantly compared with channel state information (CSI) based scheme. However, due to the lack of instantaneous information of hm,k , each user may suffer from the transmission outage. Specifically, denote by rk the transmission SE

10.2 System Model and Problem Formulation

177

of user k, and the outage would occur if rk > Ck . To guarantee the probabilistic quality of service (QoS) of each user, the outage probability constraint Pr (rk > Ck ) ≤ εk , where εk is the maximum allowed outage probability of user k, should be considered in resource allocation. Suppose that hm,k has the complexGaussian distribution with covariance matrix being Qm,k , i.e., hm,k ∼ CN 0 N , Qm,k with 0 N being the N -by-1 zero vector. Due to the fact that each user associates with only one BS, the outage probability constraint of user k can be expressed as an analytical form [10], which is given by ⎛

(2rk − 1) ln ⎝1 + M

K 

M m=1

H ¯ Q w wm, k˜ m,k m,k˜

H ¯ m=1 wm,k Qm,k wm,k

˜ k=1, k˜ =k

+ M m=1

⎞ ⎠

2rk − 1 + ln (1 − εk ) ≤ 0, ¯ m,k wm,k wH Q

(10.3)

m,k

¯ m,k = G m,k Qm,k . where Q The power consumption of each BS consists of the power consumption of amplifier and that of other circuits incurred by signal processing and active circuit blocks [11]. The circuit power consumption could be divided into the static part and the throughput-dependent dynamic part. As a consequence, the total power consumption of the system can be expressed as Ptot = ς

K  K M  

wm,k 2 + ξ B rk + Pc , 2 k=1 m=1

(10.4)

k=1

where ς denotes the reciprocal of the power amplifier efficiency, B represents the system bandwidth, ξ is a constant accounting for the power consumption due to unit throughput, and Pc represents the static part of the circuit power consumption at all BS. To be more specific, Pc could be expressed as Pc = M (N Pa + Ps ), where Pa is the power radiation in each frequency chain, and Ps is the static power consumption of each BS which is independent of the number of transmit antennas. Here we assume that the dynamic circuit power consumption is linear with the throughput since the dynamic power consumption is caused by the charge and discharge of the capacitor [12] and it is reasonable to assume that the frequency varies linearly with the throughput [13]. Based on the definition of EE and (10.4), the aim of the energy-efficient resource allocation problem is represented as max

ς

k=1

K

k=1 r k ,

wm,k 2 + ξ B  K rk + Pc m=1 k=1 2

K M

B

which can be equivalently transformed into

(10.5)

178

min

10 Energy-Efficient Base Station Association and Beamforming …

ς

2

m=1 wm,k 2 K B k=1 rk

K M k=1

+ Pc

K ⇔ max

ς

K M k=1

k=1 r k

m=1

. (10.6)

wm,k 2 + Pc 2

From (10.6), it is obvious that the optimal solution of EE maximization problem is independent of the throughput-dependent power consumption. The problem of maximizing EE by joint BS association and beamforming is formulated as follows: K rk P1 : max  K  M k=1 (10.7a)

2 w,r ς k=1 m=1 wm,k 2 + Pc s.t.

K 

wm,k 2 ≤ P max , ∀m, m 2

(10.7b)

k=1

Brk ≥ rkth , ∀k,  k 0 ≤ 1, ∀k,

(10.7c) (10.7d)

(10.3), ∀k, 

2  H H H H H H where w = w1,1 , w1,2 , . . . , w1,K , w2,1 , . . . , w M,K , r = {rk },  k = w1,k 2 , . . . ,

T

wm,k 2 , . . . , w M,k 2 denotes the received power vector of user k with the super2 2 script “T ” being the matrix transpose, · 0 represents the zero norm counting the number of non-zero elements of the vector, Pmmax is the power budget of BS m, and rkth is the minimum throughput requirement of user k. Constraint (10.7d) guarantees that each user is associated with at most one BS. Due to the fractional form of the objective function of P1, the cardinality constraint (10.7d), and the outage probability constraint (10.3), P1 belongs to the class of the non-convex mixed integer nonlinear programming (MINLP) problems. As a result, it is extremely difficult to obtain the global optimum of P1 and low-complexity algorithm for P1 is more attractive in practice. In view of the intractability of P1, we develop a centralized joint BS association and beamforming algorithm to find a suboptimal solution of P1 with reasonable complexity in the next section.

10.3 Joint BS Association and Beamforming To tackle the non-convexity of the problem, we derive an iterative algorithm to solve P1 by utilizing the successive convex approximation (SCA) in the following. Note that there exist infinitely many convex approximations for any non-convex function and the impact of the various approximation functions could be evaluated based on several criteria, such as convergence speed, robustness, efficiency, and globality of computed solutions. Unfortunately, the choice of optimal approximation is still open [14]. In this chapter, we would find a suitable convex approximation of P1 based on the computational complexity. The total complexity of the SCA procedure depends

10.3 Joint BS Association and Beamforming

179

on the number of iterations for convergence and the complexity of each iteration. However, it is challenging to analyse the number of iterations in the SCA procedure theoretically [15]. Therefore, we only focus on the numerical complexity of the convex approximation at each iteration. In general, there are three factors having impact on the complexity of the convex programming. The first one is the form of the convex problem. For example, the second order cone programming (SOCP) could be solved more efficiently than the generic convex programming if a state-ofthe-art solver for SOCP is used [16]. The remaining two factors are the number of inequality constraints and the number of variables [17]. Note that a computationally efficient algorithm is more appealing in practical communication design, especially to the BS with limited computational ability. Based on the above reasons, we will find a computationally efficient convex approximation of P1, which is elaborated as follows. We first consider the following change of variables: ε˜ k 2rk − 1 = , ∀k, M H ¯ H ¯ Qm,k wm,k yk − ε˜ k m=1 wm,k m=1 wm,k Qm,k wm,k

M

(10.8)

M H ¯ Qm,k wm,k wm,k where yk is a new positive variable for user k such that yk > ε˜ k m=1 and ε˜ k = − ln (1 − εk ) > 0. Note that yk could be considered as the product of the total received power of user k and ε˜ k . From (10.8), we have 

M

r¯k = −rk = log2 1 −

m=1

H ˜ Qm,k wm,k wm,k

yk

 , ∀k,

(10.9)

˜ m,k = ε˜ k Q ¯ m,k . As such, rk is a function of yk and wm,k , ∀m, and P1 can where Q be solved through optimizing with respect to variables y = {yk } and w. Note that r¯k ≤ 0 is a nonpositive concave function of yk and wm,k , ∀m, which is explained ˜ m,k is a Hermitian positive semidefinite as follows. Since the correlation matrix Q  M

˜ wH Q

w

matrix, the quadratic-over-linear function m=1 m,kyk m,k m,k is convex. Considering that r¯k is a composition of logarithmic function log2 (1 − θ ) and the quadratic-overlinear function, and log2 (1 − θ ) is a nonincreasing concave function with respect to θ [18], it immediately follows that r¯k is a concave function. By substituting (10.9) into P1, we can reformulate P1 as the following problem P2 : min s w, y,s

s.t.

K 

r¯k

k=1 K 

(10.10a) ⎛

M

ln ⎝1 +

˜ k=1, k˜ =k

+

yk −

M

m=1

yk −

H ˜ Q w wm, k˜ m,k m,k˜

M

m=1

ε˜ k

H ˜ m=1wm,k Qm,k wm,k

H ˜ Qm,k wm,k wm,k

− ε˜ k ≤ 0, ∀k,

⎞ ⎠

(10.10b)

180

10 Energy-Efficient Base Station Association and Beamforming …

B r¯k + rkth ≤ 0, ∀k, 1 s ≤ K M ,

2 ς k=1 m=1 wm,k 2 + Pc

(10.10c) (10.10d)

(10.7b) and (10.7d), where s is an auxiliary variable. Note that r¯k , ∀k, is nonpositive and the objective function of P2 is nonincreasing with respect to s. As a result, constraint (10.10d) must hold with equality at the optimum of P2, from which we conclude that P2 is equivalent to P1. It is obvious that s is bounded by Ls =

ς

M

1

max m=1 Pm

+ Pc

≤ s ≤ Us =

1 , Pc

(10.11)

where the lower bound L s is derived through combining (10.7b) and the fact that (10.10d) is active at the optimal solutions. Consequently, adding constraint (10.11) to P2 does not affect the optimal solutions of P2. Motivated by the Charnes-Cooper transformation [19], we introduce two new variables w˙ m.k = swm,k , ∀m, ∀k and y˙k = s yk , ∀k. Through substituting w˙ = sw and y˙ = s y into P2, and combining (10.11), P2 could be equivalently transformed into K 

P3 : min s ˙ y˙ ,s w,

r˙k

k=1 K 

s.t.

 ln 1 +

˜ k=1, k˜ =k

+

(10.12a) M

H ˜ ˙ m, Q w˙ m=1 w k˜ m,k m,k˜ M H ˜ ˙ m,k Qm,k w˙ m,k s y˙k − m=1 w

ε˜ k s 2 − ε˜ k ≤ 0, ∀k, M H ˜ Qm,k w˙ m,k s y˙k − m=1 w˙ m,k

K 

w˙ m,k 2 ≤ P max s 2 , ∀m, m 2



(10.12b)

(10.12c)

k=1

Bs r˙k + rkth s ≤ 0, ∀k, ς

K  M 



w˙ m,k 2 + Pc s 2 − s ≤ 0, 2

(10.12d) (10.12e)

k=1 m=1



 ˙ k 0 ≤ 1, ∀k,

(10.12f)

(10.11),   M ˜ m,k w ˙ m,k w˙ H Q ˙k= is an equivalent form of r¯k and  where r˙k = log2 1 − m=1 m,k s y˙k 

2

2

2 T

w˙ 1,k , . . . , w˙ m,k , . . . , w˙ M,k . From (10.11), we conclude that s is a pos2

2

2

itive number and constraint (10.12f) is equivalent to constraint (10.7d). It is noted

10.3 Joint BS Association and Beamforming

181

that constraints (10.12c) and (10.12e) are convex and admit the second order cone (SOC) representations, which are given by

 H 

H

≤ P max s, ∀m,

w˙ , . . . , w˙ H m,K m

m,1

(10.13)

2

and

 

√  1 − s H 1+s

H ,

≤

ς w˙ , Pc s,

2 2

(10.14)

2

respectively. However, P3 is still nonconvex due to the objective function of P3 and constraints (10.12b), (10.12d), and (10.12f). Hence, to deal with the nonconvexity of P3, we would convexly approximate the nonconvex functions in P3 based on the SCA conditions in the next three steps. Firstly, let us recall that r¯k is a concave function with respect to yk and wm,k , ∀m. We have that s r˙k is the perspective function of r¯k and is a concave function of s, y˙k , and w˙ m,k , ∀m, since the perspective function of f (θ ) is convex (or concave) if f (θ ) is convex (or concave) [18]. Therefore, through utilizing  of  the first-order property any concave function f (θ), i.e., f (θ 0 ) ≤ f (θ) + Re (∂ f θ ∗ (θ)) H (θ 0 − θ ) , ∀θ , ∀θ 0 , where Re (·) is the real part of a complex number, s r˙k at the jth iteration of the SCA can be approximated by ( j) ok

=

M 

  H ( j) ( j) ( j) ˙ m,k + y˜k y˙k + s˜k s, Re w˜ m,k w

(10.15)

m=1 ( j)

˜ m,k w ˙ m,k −2s ( j) Q ( j) w˜ m,k = , ( j) χ

( j) s y˜k =

( j)

M

 H ( j) ˜ m,k w˙ ( j) w˙ m,k Q m,k

( j)

( j)

, s˜k =˙rk + k  M  ( j)  H j) (   ˜ m,k w ( j) ˙ m,k Q ˙ m,k m=1 w χk ( j) ( j) χk = with r˙k = log2 ( j) ( j) , χk ln 2s ( j) y˙k      M  ( j)  H ( j) ˜ m,k w˙ ( j) , and w˙ ( j) , y˙ ( j) , s ( j) being the solution ln 2 s ( j) y˙k − m=1 w˙ m,k Q m,k where

m=1

( j)

( j)

y˙k χk

of the ( j − 1)th SCA iteration. Then by approximating s r˙k with the linear func( j) tion ok , we can convexly approximate the objective function of P3 and constraint (10.12d). Secondly, to deal with the nonconvexity of the cardinality constraint (10.12f), the largest-g norm is adopted to transform the cardinality constraint into the differenceof-convex form [20], and constraint (10.12f) is equivalently expressed as [21]





 ˙ k ∞ ≤ 0, ∀k. ˙ k 1 − 

(10.16)

˙ k 1 is the composition of the nondecreasing function · 1 and It is noted that 



2 ˙ k is convex with respect convex quadratic functions w˙ m,k , ∀m. Therefore,  2

1

182

10 Energy-Efficient Base Station Association and Beamforming …

˙ k ∞ is also convex with respect to w. ˙ As to w˙ [18]. Similarly, we conclude that  a result, constraint (10.16) is a difference-of-convex representation of cardinality

˙ k ∞ by its affine minorization at point constraint (10.12f). Through replacing  H  ( j) H H H w˙ k , where w˙ k = w˙ 1,k , . . . , w˙ m,k , . . . , w˙ M,k , constraint (10.16) is approximated by 







( j)

( j) H ( j)

 ∗ ˙k ˙ k w˙ k − w˙ k ˙ k 1 − Re ∂w˙ k  −  ∞ ∞ 

H 



( j)

( j) ˙ k ≤ 0. ˙k w ˙ k 1 − Re ∂w˙ ∗k  ˙ k +  =  ∞



( j) ˙k In (10.17), ∂w˙ ∗k 

( j)

˙ k and is given by is the subgradient of · ∞ at 

∞



( j) ˙k ∂w˙ ∗k 

∞

(10.17)

∞

        H ( j) H ( j) H ( j) H

˙ 1,k , . . . , ωm ˙ m,k , . . . , ω M w ˙ M,k w = 2 ω1 w :   M M

 

( j) 2  ˙ m,k : ωm ∈ arg max ωm w ωm = 1, 0 ≤ ωm ≤ 1, ∀m . {ωm }∈R M

2

m=1

m=1

(10.18) Finally, since r˙k and the left-hand side (LHS) of constraint (10.12b) are nondecreasing and nonincreasing with respect to y˙k , respectively, constraint (10.12b) must be active at any optimal solution of P3. Hence, we can equivalently transform constraint (10.12b) into   ˙ s ≤ y˙k , y˙k w,

(10.19)

  ˙ s is the implicit function of variables w˙ and s is defined by the following where y˙k w, implicit equation K 

⎛ ln ⎝1 +

˜ k=1, k˜ =k

M

H ˜ ˙ m, ˙ Q w m=1 w k˜ m,k m,k˜ M H ˜ ˙ m,k Qm,k w˙ m,k s y˙k − m=1 w

⎞ ⎠+

s y˙k −

M

ε˜ k s 2

H ˜ ˙ m,k ˙ m,k Qm,k w m=1 w

− ε˜ k = 0.

(10.20) By utilizing the fact that ln (1 + θ ) ≤ ln (1 + θ0 ) + of the LHS of (10.20) is given by ( j)

U y˙k =

s y˙k − +

M

K 

ε˜ k s 2

m=1

( j)

1 + k,k˜

∀θ , ∀θ0 , an upper bound

K 

− ε˜ k +

H ˜ ˙ m,k Qm,k w˙ m,k m=1 w M H ˜ ˙ m, ˙ Q w m=1 w k˜ m,k m,k˜ M H ˜ ˙ m,k Qm,k w s y˙k − w˙ m,k

˜ k=1, k˜ =k

θ−θ0 , 1+θ0

˜ k=1, k˜ =k

  ( j) ln 1+ k,k˜

( j)

− k,k˜

,

(10.21)

10.3 Joint BS Association and Beamforming ( j) where k,k˜

 H ( j) ˜ m,k w˙ ( j) w˙ m,k˜ Q m,k˜  H f,( j)  M ( j) ( j) ˜ m,k w ˙ ˙ Q s ( j) y˙ − w

183

M

=

m=1

m=1

k

m,k

f,( j)

with y˙k

  ˙ ( j) , s ( j) . Note that both = y˙k w

m,k

( j) U y˙k

and the LHS of (10.20) are strictly decreasing with respect to y˙k . Therefore,   ( j) ˙ s through solving the equation U y˙k = 0, we can obtain an upper bound of y˙k w, expressed as ( j)

yˆk



H ˜ M K M H ˜   w˙ m, Q w˙    ˙ m,k ˙ m,k Qm,k w w s  ( j) ( j) k˜ m,k m,k˜   , ˙ s w ˙ ,s + w, = ( j) + ( j) ( j) s υk 1+ ˜ s m=1 ˜k=1,k˜ =k m=1 ε˜ k υk k,k

(10.22) 

  ( j) =1+ where . Since the quadratic-over( j) − ln 1 + ˜ k,k 1+ k,k˜    H ˜ ˙ Q w ˙ m,k w  ( j) ˙ s w˙ ( j) , s ( j) is a convex upper , ∀k, are convex, yˆk w, linear functions m,k m,k s           ( j) ˙ s w˙ ( j) , s ( j) = y˙k w, ˙ s and yˆk w, ˙ s at w˙ ( j) , s ( j) . Besides, we bound of y˙k w,     ( j) ˙ s w˙ ( j) , s ( j) meets the SCA condition in Appendix 1. can also prove that yˆk w, Based on the above reasons, constraint (10.19) can be convexly approximated by K

( j) υk

1 ˜ k=1, k˜ =k ε˜ k

( j)

yˆk



( j)

k,k˜

   ˙ s w˙ ( j) , s ( j) ≤ y˙k . w,

(10.23)

As expected, we can equivalently transform constraint (10.23) into an SOC constraint, which is given by

⎡ ⎤H

 

  H

⎣! 2 ( j) + 1 s, wˆ k , y˙k ⎦ ≤ s + y˙k .

j) (

υk

(10.24)

2

In

(10.24), ( j)

   H  H H  H H H ( j) ( j) ( j) ( j) ( j) ˆ k,2,1 , . . . , wˆ k,M,K ˆ k ( j) = w ˆ k,1,1 , wˆ k,1,2 , . . . , wˆ k,1,K , w w ( j)

1

( j)

˜ 2 w˙ ˜ , where c is defined as with wˆ k,m,k˜ = ck,k˜ Q m,k m,k k,k˜ ( j)

ck,k˜ = 1

⎧√ ⎨ '2, ⎩

2 , ( j) ( j) ε˜ k υk 1+ k,k˜

k˜ = k, k˜ = k,

(10.25)

˜ 2 is the square root of the positive semidefinite matrix Q ˜ . Specifically, let and Q m,k ) m,k ( H ˜ m,k be Um,k diag λ ˜ Um,k the eigenvalue decomposition of Q , where λn,Q˜ m,k n,Qm,k ) ( ˜ m,k and diag λ ˜ is a diagonal matrix is the nth largest eigenvalue of Q n,Qm,k

184

10 Energy-Efficient Base Station Association and Beamforming …

) ( 1 ˜2 = with the elements of vector λn,Q˜ m,k on the main diagonal, then we have Q m,k , *+ H Um,k Um,k diag λn,Q˜ m,k . Up to now, we have restricted our attention to the convex approximation of nonconvex functions in P3. However, it is worth mentioning that the compactness of the feasible region of P3 is desired to strengthen the convergence properties of the SCA [22]. From (10.11) and (10.12c), we conclude that all the optimization variables are compact except y˙k , ∀k. Hence, by using the fact that the first term of the LHS of (10.20) is nonnegative, we first derive the lower bound of y˙k as follows:

s y˙k −

M

ε˜ k s 2

m=1

H ˜ ˙ m,k Qm,k w w˙ m,k

≤ ε˜ k ⇔ y˙k ≥ s +

M H ˜  Qm,k w˙ m,k w˙ m,k ≥ L s = L y˙k . s m=1

(10.26) Next, to obtain the upper bound of y˙k , let us express an upper bound of the LHS of (10.20) as op Uk





˙ s = y˙k , w,

ε˜ k s 2 +

K

M

˜ k=1, k˜ =k M s y˙k − m=1

H ˜ ˙ m, Q w˙ m=1 w k˜ m,k m,k˜ H ˜ Qm,k w˙ m,k w˙ m,k

− ε˜ k ,

(10.27)

which is derived through utilizing the taylor approximation of logarithm function  op  ˙ s are at zero, i.e., log (1 + θ) ≤ θ . Since both the LHS of (10.20) and Uk y˙k , w, strictly decreasing with respect to y˙k , the upper bound of y˙k can be obtained by letting  op  ˙ s equal to zero, from which we have Uk y˙k , w, ⎞ H ¯ M K M H ˜    ˙ m,k˜ w˙ m, w˙ m,k Qm,k w˙ m,k ˜ Qm,k w k ⎠ y˙k ≤ s ⎝1 + + 2 2 s s m=1 ˜ k=1, k˜ =k m=1 ⎛

⎞

2 M K M

w˙ ˜ 2

w˙ m,k    m, k 2⎠ 2 ≤ s ⎝1 + λ1,Q˜ m,k + λ1,Q¯ m,k 2 2 s s m=1 ˜ k=1, k˜ =k m=1   M ( ( ))  max ≤ Us 1 + max max λ1,Q˜ m,k , λ1,Q¯ m,k Pm = U y˙k , ⎛

m

(10.28)

m=1

 op  ˙ s = 0, where the right-hand side of the first inequality is the solution to Uk y˙k , w, the second inequality follows from the definition of Rayleigh quotient, and the third inequality holds due to (10.11) and (10.12c). Now we can solve P3 through combining the convex approximations (10.15), (10.17), and (10.24). Before proceeding, it should be pointed out that there are two implementation issues involved in applying the SCA to P3 directly. The first one is that the SCA procedure needs a feasible initial point of P3, which is usually difficult to be determined because of constraints (10.12d) and (10.16). Besides, the choice

10.3 Joint BS Association and Beamforming

185

of the initial point has an impact on the performance of the SCA. The other issue is that the LHS of (10.16) is nonnegative and its corresponding convex approximation (10.17) may result in the infeasibility of the convex approximation problem at each iteration of the SCA. Therefore, we relax the convex approximation of (10.12d) and (10.16) by introducing a nonnegative auxiliary variable t, which are expressed as ( j)

Bok + rkth s ≤ t, ∀k,

(10.29)

and  



( j)

( j) H

 ˙k ˙ k w˙ k +  ˙ k 1 − Re ∂w˙ ∗k  ∞

∞

≤ t, ∀k,

(10.30)

respectively. Note that (10.30) can be rewritten as an SOC constraint by completing the square, which is expressed as 

⎡ ⎤

H 



( j) ( j)

˙ k − Re ∂w˙ ∗k  ˙ k w˙ k − t 1 + 

⎢ H ⎥ ∞ ∞

⎢w˙ , ⎥

⎣ k ⎦ 2



2 

H 

( j) ( j) ˙k ˙ k w˙ k −  1 + t + Re ∂w˙ ∗k  ∞ ∞ , ∀k. ≤ 2

(10.31)

To guarantee that the solution obtained by the SCA is a feasible point of P3, we should penalize the violation t in each iteration of the SCA. As a result, through combining (10.11), (10.26), and (10.28), the convex approximation problem of P3 at the jth iteration of the SCA is expressed as P4 : min

˙ y˙ ,s,t w,

K 

( j)

ok + ψ ( j) t

(10.32a)

k=1

s.t. L y˙k ≤ y˙k ≤ U y˙k , ∀k,

(10.32b)

(10.11), (10.13), (10.14), (10.24), (10.29), and (10.31), where ψ ( j) is the penalty coefficient. Obviously, P4 is an SOCP problem, which can be solved more efficiently by some SOCP solvers than a generic nonlinear program [16]. Then a solution of P3 satisfying the generalized Karush-KuhnTucker (KKT) conditions can be obtained by solving a sequence of P4, as summarized in Algorithm 10.1. The convergence of Algorithm 10.1 can be proved by the similar procedure in [14]. In [14], the condition for the convergence of the SCA is the strong convexity of the objective function of P4. However, the objective function of P4 is linear. To meet the condition for the convergence, we can

2  

2

add a quadratic term SCT w˙ ( j) , y˙ ( j) , s ( j) = τw˙ w˙ − w˙ ( j) + τ y˙ y˙ − y˙ ( j) + 2

2

186

10 Energy-Efficient Base Station Association and Beamforming …

Algorithm 10.1 Generalized SCA Procedure to Solve P3 1: Initialize Jmax , δ1 , δ2 , ψ (0) , andtolerances τ1 and τ2 . ˙ (0) , y˙ (0) , s (0) , index j ← 0, initial approximation of EE (0) ← ∞. 2: Set initial point w 3: while j ≤Jmax do  ( ) ( ) ( j+1) ( j+1) ˙ ( j+1) , y˙ ( j+1) , s ( j+1) , t ( j+1) and the Lagrange multipliers αk 4: Update w , βk   ˙ ( j) , y˙ ( j) , s ( j) . by solving P4 with w K ( j+1) 5: (j+1) ← k=1 o k j+1) j) ( (   6: if  − ≤ τ1 and t ( j+1) ≤ τ2 then 7: Break. 8: end if (   ) ( j+1) ( j+1) K 9: λ¯ ← min + δ1 , τ3 −1 + βk k=1 αk 10: if ψ ( j) ≥ λ¯ then 11: ψ ( j+1) ← ψ ( j) 12: else 13: ψ ( j+1) ← ψ ( j) + δ2 14: end if 15: j ← j + 1 16: end while

 2 τs s − s ( j) , where τw˙ , τ y˙ , and τs are positive parameters, to the objective func ( j) ( j) ( j) has no impact tion of P4. It is worth mentioning that adding SCT w˙ , y˙ , s on the SCA conditions. In our simulations, even if we set all the parameters τw˙ , 10.1 is still convergent. Note that adding a non-zero τ y˙ , and   τs as zeros, Algorithm

SCT w˙ ( j) , y˙ ( j) , s ( j) , P4 can be still solved by the SOCP through introducing a new variable [18]. It is noted that Algorithm 10.1 is terminated if both the linear approximation  K ( j+1) ok , and the relaxation variable t are less than the predefined of EE, i.e., k=1 thresholds τ1 and τ2 , respectively, as shown from Lines 4 to 8. The update of the penalty coefficient ψ ( j) shown from Lines 9 to 14 could be explained by the following reasons. Firstly, after P4 is solved, the first order generalized  KKT condition with  K  ( j+1) ( j+1) ( j+1) ( j) + ζ ( j+1) , where αk + βk , respect to t is expressed as ψ = k=1 αk ( j+1)

, and ζ ( j+1) are the nonnegative Lagrange multipliers related to constraints βk (10.29), (10.31), and t ≥ 0. Due to the complementary slackness condition, we have  K  ( j+1) ( j+1) ( j+1) ( j+1) ( j) + δ1 for any given = 0 if t > 0 and ψ ≤ k=1 αk + βk that ζ δ1 > 0, which means that ψ ( j) should be increased to let the value of t smaller. Sec

ondly, if the current point is far from the convergent point, i.e., τ3 = w˙ − w˙ ( j) + 2

 

( j)

y˙ − y˙ + s − s ( j)  is large enough, ψ ( j) remains unchanged even t ( j+1) > 0. 2 Note that Algorithm 10.1 can be also used to solve the joint BS association and beamforming for SE maximization by setting s = 1 and removing constraint (10.10d). Besides, if CDI is replaced with CSI, Algorithm 10.1 still works through redefining

10.4 Decentralized Beamforming

187

  M 2 G |h H x | and r¯k and the outage constraint (10.10b) as log2 1 −  K m=1 Mm,k m,k m,k 2 H 1+ k=1 ˜ m=1 G m,k | hm,k x m,k˜ |  H 2 K M   ≤ yk , respectively. 1 + k=1 ˜ m=1 G m,k hm,k x m,k˜ The complexity of the developed algorithms depends on the number of iterations required for convergence and the complexity at each iteration. Due to the difficulty of analysing the number of iterations for convergence analytically [15], we only analyse the complexity of solving the convex approximation problem at each iteration. Since the convex approximation at Algorithms 10.1 could  by the SOCP, the  be solved complexity of Algorithm 10.1 at each iteration is O M 3 K 3 N 3 [23].

10.4 Decentralized Beamforming In this section, we derive a decentralized beamforming algorithm for EE maximization, which is more appealing than the centralized one in practical networks [24]. Suppose that the BS association has been given and an initial feasible point of P3 is available in advance, the decentralized beamforming algorithm is designed by solving the beamforming subproblem of P4 in a decentralized manner at each iteration of the SCA. {1, 2, . . . , M}, 1 ≤ k ≤ K , denotes the BS assigned to user Assume that g (k) ∈ 

= m means that user k is associated with BS m. k, i.e., g (k) = arg max wm,k ˜ 2 m˜

Let μk = wg(k),k denote the beamforming vector for user k. Denote by μ˙ k = sμk the surrogate of w˙ g(k),k . The beamforming subproblem of P4 without the penalty term is expressed as   K  H   ( j) ( j) ( j) Re u˜ k u˙ k + y˜k y˙k + s˜k s

P5 : min

˙ y˙ ,s μ,

s.t.

(10.33a)

k=1

 2

u˙ k ≤ P max s 2 , ∀m, m 2 k∈m



Re ς

( j)

u˜ k

H

(10.33b)

   ( j) ( j) u˙ k + y˜k y˙k + s˜k + rkth s ≤ 0, ∀k,

M  

2

u˙ k + Pc s 2 − s ≤ 0, 2

(10.33c) (10.33d)

m=1 k∈m

s ( j)

υk

+

+

˜ g(k),k u˙ k u˙ kH Q + s

 k˜ ∈ / g(k)

 ˜ g(k) ,k˜ =k k∈

˜ g(k),k u˙ ˜ u˙ kH˜ Q k   ( j) ( j) ε˜ k υk 1 + k,k˜ s

˜ ˜ u˙ ˜ u˙ kH˜ Q g (k ),k k   ≤ y˙k , ∀k, ( j) ( j) ε˜ k υk 1 + k,k˜ s

(10.11) and (10.32b),

(10.33e)

188

10 Energy-Efficient Base Station Association and Beamforming …

Algorithm 10.2 Centralized Beamforming

  1: Initialize a feasible solution μ˙ (0) , y˙ (0) , s (0) and set j ← 0. 2: repeat     3: Update μ˙ ( j+1) , y˙ ( j+1) , s ( j+1) by solving P5 with μ˙ ( j) , y˙ ( j) , s ( j) . 4: j ← j + 1 5: until convergence

H  ˜ g(k),k u˙ ( j) −2s ( j) Q ( j) ( j) k where u˙ = u˙ 1H , . . . , u˙ kH , . . . , u˙ KH , u˜ k = with χk = ( j) χk  H   ( j) H  ˜ g(k),k u˙ ( j) Q s ( j) u˙ k k ( j) ( j) ( j) ( j) ( j) ( j) ( j) ˜ y˜k = ln 2 s y˙k − u˙ k , s˜k = r˙k + Qg(k),k u˙ k , ( j) ( j) 

( j) u˙ k

y˙k χk

H

˜ g(k),k u˙ ( j) Q k ( j) χk

, and u˙ ( j) being the optimal solution at the ( j − 1)th iteration of = {k |g (k) = m } denotes the set of users connected to BS m, and the SCA,  m  ( j)

k,k˜

( j)

=

H

( j)

˜ ˜ u˙ Q g (k ),k k˜  H . f,( j) ( j) j) ( ˜ g(k),k u˙ ( j) Q s y˙k − u˙ k k u˙ k˜

It is noted that the third term and the forth term in

constraint (10.33e) could be regarded as intra-cell interference and inter-cell interference, respectively. Then an energy-efficient beamforming solution is obtained by resorting to the SCA, as described in Algorithm 10.2. The convergence of Algorithm 10.2 is proved in Appendix B of [15]. In order to apply the alternating direction method of multipliers (ADMM) to solve P5 in a decentralized manner, we should first transform P5 into an equivalent problem, where the objective function and the constraints are separable with respect to each BS. More specifically, constraints (10.33d) and (10.33e) are converted into ⎧ 0   ⎪ 

2 2Pc ⎪ ⎪

⎪ ˙ s + p ˙ u + 1 s 2 + p˙ m2 , ∀m, (10.34a) ≥ 2ς + m k 2 ⎪ ⎨ k∈m M M ⎪  ⎪ ⎪ ⎪ p˙ m = 1, ⎪ ⎩

(10.34b)

m=1

and ⎧ ˜ g(k),k u˙ ˜  ˜ g(k),k u˙ k u˙ kH˜ Q ⎪ u˙ kH Q s k ⎪ ⎪   + + ⎪ ( j) ⎪ ( j) ( j) s ⎪ υ ⎪ 1 +

s ε ˜ υ k ˜ g(k) ,k˜ =k k k ⎪ k∈ ˜ k,k ⎪ ⎪ ⎨ 2 lk,  ˜  k  ≤ y˙k , ∀k, + ⎪ ⎪ ( j) ( j) ⎪ ⎪ ε˜ k υk 1 + k,k˜ s ˜ ∈ ⎪ k / g(k) ⎪ ⎪

1 ⎪ ⎪

⎪ ⎩ ˜2 ˙ u / g(k) ,

≤ lk,k˜ , ∀k˜ ∈

Q ˜ g (k˜ ),k k 2

(10.35a) (10.35b)

10.4 Decentralized Beamforming

189

 respectively, where { p˙ m } and lk,k˜ k˜ ∈ are the auxiliary variables. It can be shown / g(k) that (10.33d) and (10.34) are equivalent, whose proof is derived in Appendix 2. Due to the fact that the objective function of P5 is increasing with respect to y˙k , constraints (10.35a) and (10.35b) must hold with equality at the optimum of P5. Hence, constraint (10.33e) and (10.35) are equivalent. Through introducing the local m copies of s, p˙ m , lk,k˜ , and lk,k ˙ mm , lk, , and ˜ at BS m = g (k), which are denoted by sm , p k˜ m lk,k , respectively, we can define the local constraints of BS m at the jth iteration of ˜ the SCA as   *    2 ( j) 

u˙ k ≤ P max s 2 , Fm = μ˙ m , y˙ m , η˜ m  m m 2 k∈m      ( j) H ( j) ( j) u˙ k + y˜k y˙k + s˜k + rkth sm ≤ 0, ∀k ∈ m , Re u˜ k    2  2Pc   m 2 + p˙ m 2 ,

u˙ k + + 1 sm sm + p˙ m ≥ !2ς m 2 M k∈m

˜ g(k),k u˙ k u˙ kH Q + + j) ( sm

sm υk

˜ u˙ H u˙ Q k˜  g(k),k k˜ ( j) ( j) 1 + ˜ sm ˜ m ,k˜ =k ε˜ k υk k∈ k,k 

 2 lm ˜  k,k  + ≤ y˙k , ∀k ∈ m , ( j) ( j) 1 + ˜ sm ε˜ υ k˜ ∈ / m k k k,k

1

2

m

Q ˜ u˙ ≤ l ˜ , ∀k ∈ m , ∀k˜ ∈ / m ,

˜ k 

m,k

k,k

2

L s ≤ sm ≤ U s ,

,

L y˙k ≤ y˙k ≤ U y˙k , ∀k ∈ m ,

(10.36)

 ( )T m , where μ˙ m = μ˙ k k∈m , y˙ m = { y˙k }k∈m , and η˜ m = sm , p˙ mm , lk, k˜ k∈m ,k˜ ∈ / m  T  ( )T   T  m . Let η˜ = η˜ m and η = ηm with ηm = s, p˙ m , lk,k˜ k∈ ,k˜ ∈ , lk,k ˜ m / m k∈m ,k˜ ∈ / m T  T being the global version of η˜ m . lk,k ˜ k∈ ,k˜ ∈ / 

m



m

( j)

Based on the local feasible set Fm , P5 is equivalently transformed into P6 : min

˙ y˙ ,η,η ˜ μ,

s.t.

M



m=1

k∈m

   H  ( j) ( j) ( j) Re u˜ k u˙ k + y˜k y˙k + s˜k sm

  μ˙ m , y˙ m , η˜ m ∈ Fm( j) , ∀m, η˜ m = ηm , ∀m, (10.34b),

(10.37a) (10.37b) (10.37c)

190

10 Energy-Efficient Base Station Association and Beamforming …

  where μ˙ = μ˙ m and y˙ = y˙ m . It is obvious that both the objective function and the constraints of P6 are separable with respect to each BS now. Therefore, we can solve P6 by utilizing the ADMM. To be specific, we first express the augmented Lagrangian function of P6 as    M       ( j) H ( j) ( j) ˙ y˙ , η, ˜ η, α, ˜ β˜ = Re u˜ k L A μ, u˙ k + y˜k y˙k + s˜k sm m=1 k∈m

+

M   m=1

⎛ ⎛ ⎞ ⎞2  M M



 

  c c 2 T η˜ − η

η˜ m − ηm + β˜ ⎝ p˙ m − 1⎠ + ⎝ p˙ m − 1⎠ , α˜ m m m + 2 2 2 m=1

(10.38)

m=1

 where c is a penalty parameter, α˜ = α˜ m and β˜ are the Lagrange multipliers associated with constraints (10.37c) and (10.34b), respectively. According(to the ) ) definition ( m m m m , and α˜ k,k , of η˜ m , α˜ m can be decomposed as α˜ s , α˜ p˙m , α˜ k,k˜ ˜ k∈m ,k˜ ∈ / m ( k∈m ,k˜ ∈ / m ) m which are the Lagrange multipliers corresponding to sm , p˙ mm , lk, , and k˜ k∈m ,k˜ ∈ / m ( ) m lk,k , respectively. ˜ k∈m ,k˜ ∈ / m

The key scheduling of the ADMM is iteratively updating the local variables (i.e., ˜ ˙ y˙ , and η), ˜ the μ, the Lagrange multipliers (i.e., α˜ and β).  global variables (i.e., η), and  (d) (d) (d) (d) ˜ (d) (d) the solution of the (d − 1)th iteration Denoting by μ˙ , y˙ , η˜ , η , α˜ , β of the ADMM, the local variables at BS m can be updated through solving the following convex problem 

 μ˙ m (d+1) , y˙ m (d+1) , η˜ m (d+1)     ( j)  H  ( j) ( j) Re u˜ k u˙ k + y˜k y˙k + s˜k sm = arg min (μ˙ m , y˙ m ,η˜ m )∈Fm( j) k∈m T 

 c (d)

η˜ m − η(d) 2 . ˜ η −η + α˜ (d) m m m + m 2 2

(10.39)

Then after sharing the derived local variables η˜ (d+1) , we can update the global variables as M 

2  T   c 

(d+1)

(d+1) ˜ ˜ η α˜ (d) η + − η − η

m m m m m 2 2 η m=1  2  M  M  c  + β˜ (d) p˙ m − 1 + p˙ m − 1 . (10.40) 2 m=1 m=1

η(d+1) = arg min

It is noted that (10.40) is a problem of minimizing a quadratic function and has the analytical results, which are expressed as

10.4 Decentralized Beamforming

191

s (d+1) =

 M  1  (d+1) α˜ sm,(d) + sm , M m=1 c ⎛

(d+1) = lk, k˜

g (k˜ ),(d+1) ⎜ g(k),(d+1) + lk,k˜ ⎝lk,k˜

1 2

(10.41) ⎞ g (k˜ ),(d)   + α˜ k,k˜ ⎟ ⎠ , g (k) = g k˜ ,

c

g(k),(d)

+

α˜ k,k˜

(10.42) p˙ m(d+1) = pˆ m(d+1) −

M 1  (d+1) pˆ , ∀m, M + 1 m=1 m

(10.43)

where pˆ m(d+1)

=

p˙ mm,(d+1)

+

− β˜ (d) α˜ m,(d) p˙ m c

+ 1.

(10.44)

The final step of the ADMM is to replace the Lagrangian multipliers as   (d) (d+1)

(d+1) ˜ ˜ η , ∀m, α˜ (d+1) = α + c − η m m m m   1 M pˆ m(d+1) − 1 . β˜ (d+1)= β˜ (d) + c m=1 M +1

(10.45) (10.46)

Through substituting (10.41), (10.42), (10.43), and (10.46) into (10.45), we have M 

M   m,(d)   α˜ s = + c sm(d+1) − s (d+1)

α˜ sm,(d+1)

m=1

m=1

=

M 

α˜ sm,(d) −

m=1 g(k),(d+1)

α˜ k,k˜

M 

α˜ sm,(d) = 0,

(10.47)

m=1

g (k˜ ),(d+1) g (k˜ ),(d) g(k),(d) + α˜ k,k˜ =α˜ k,k˜ + α˜ k,k˜   g (k˜ ),(d+1) g(k),(d+1) (d+1) (d+1)

= 0, + c lk,k˜ − lk,k˜ + lk,k˜ − lk,k˜

α˜ m,(d+1) p˙ m

=

α˜ m,(d) p˙ m

+c



p˙ mm,(d+1)

−

p˙ m(d+1)



 α˜ m,(d) − β˜ (d) pˆ m(d+1) p˙ m − = +c −1 M +1 c   M 1  (d+1)

(d) = β˜ + c pˆ − 1 = β˜ (d+1) , ∀m. M + 1 m=1 m α˜ m,(d) p˙ m



(10.48)

M m=1

(10.49)

192

10 Energy-Efficient Base Station Association and Beamforming …

Algorithm 10.3 Decentralized Beamforming

  1: Initialize μ˙ (0) , y˙ (0) , η˜ (0) , η(0) , α˜ (0) , β˜ (0) and set j ← 0. 2: repeat 3: Set d ← 0. 4: repeat 5: Update local variables by (10.39) at each BS. 6: Update global variables by (10.43), (10.50), and (10.51). 7: Update Lagrange multiplies by (10.45). 8: d ←d +1 9: until ADMM converges 10: Update the SCA parameters. 11: j ← j + 1 12: until SCA converges

From (10.47), (10.48), and (10.49), the global variables can be updated without Lagrangian multipliers, and (10.41), (10.42), and (10.44) are simplified to 1  (d+1) s , M m=1 m     1 g(k),(d+1) g (k˜ ),(d+1) , g (k) = g k˜ , lk,k˜ = + lk,k˜ 2 M

s (d+1) =

(10.50)

(d+1) lk, k˜

(10.51)

pˆ m(d+1) = p˙ mm,(d+1) + 1, ∀m.

(10.52)

Note that the update of global variables could be implemented by the average consensus algorithm [25]. Then a decentralized beamforming scheme  is given by Algorithm 10.3. Let Nl represent the number of global variables lk,k˜ g(k)=g k˜ . Accordingly, () the total amount of exchanged information of the systems is 2 (M(M − 1)) + Nl , where the first term is used to update s (d+1) and p˙ m(d+1) . The convergence of Algorithm 10.3 is guaranteed due to the fact that the outer loop of Algorithm 10.3 is equivalent to Algorithm 10.2 and the inner loop of Algorithm 10.3 is the procedure of the ADMM, whose convergence is well studied in [26]. As a final remark, the update of SCA parameters, as shown on Line 10 of Algorithm 10.3, could be done at each BS since constraints (10.35a) and (10.35b) must hold with equality at the optimum of P6 and the solution obtained after the convergence of the inner loop in Algorithm 10.3 is the optimum of P6. The complexity analyses of the developed centralized and decentralized beamforming algorithms are similar with that of Algorithm 10.1. Therefore, both the complexity of Algorithm 10.2  iteration and that of Algorithm 10.3 at each  at each BS are upper bounded by O K 3 N 3 .

10.5 Numerical Results

193

10.5 Numerical Results In this section, we study the performance of the developed algorithms through numerical simulations. We assume that the system is composed of M = 3 hexagon cells and the radius of each cell is 500 m, as shown in Fig. 10.1. BSs 1, 2, and 3 cooperate with each other and the system bandwidth is 150 kHz. Since the users at the edge of the cells suffer from severe inter-cell interference, the users are uniformly distributed in the gray area of Fig. 10.1, where the width of the gray area is 50 m. The other simulation parameters are presented in Table 10.1 [11, 27], where N (0, 8)dB represents the log-normal distribution whose mean is 0 and deviation is 8dB. To model the channel covariance matrix Qm,k , the uniform linear antenna array with half-wavelength spacing is assumed at each BS, then the channel covariance matrix could be approximate by [28] ˙ n)ι ˜ k cos κm,k ) (π(n−    ˙ n) ˜ sin κm,k 2 exp− , Qm,k κm,k , ιk n˙ n˜ = expiπ(n− 2

(10.53)

√ where i = −1, κm,k is the central angle of the incoming rays from user k to BS m, ιk is the angular spread of local scatterers around user k, n˙ and n˜ are the row index and column index of matrix Qm,k , respectively. The convergence of Algorithm 10.1 is studied in Fig. 10.2, where the initial beamforming vectors are set to zero, equal power, and + random initialization, respectively. P max

In Fig. 10.2, the equal power means that wm,k = KmN 1 N , ∀m, ∀k, where 1 N is the N -by-1 vector whose elements are all 1, and the random initial points are sampled from the standard complex Gaussian distribution. Note that when the initial point is 0 N , the LHS of (10.13) is zero and the corresponding gradient is infinity, which incurs numerical problem in simulations. Therefore, constraint (10.13) should be equivalently transformed into

Fig. 10.1 System model and user distribution

1000 800 600 1 400 500m

(m)

200 0 −200

2

3

−400 50m

−600 −800 −1000

−500

0

(m)

500

1000

194

10 Energy-Efficient Base Station Association and Beamforming …

Table 10.1 Simulation parameters Parameters Carrier frequency Reference distance Path loss exponent Noise figure Shadow fading Power amplifier efficiency

Values 2 GHz 100 m 3.8 7 dB N (0, 8) dB 0.38

1 ς

Power consumption per unit throughput ξ Power radiation in each frequency chain Pa Static power consumption of each BS Ps Maximum allowed outage probability εk

Fig. 10.2 Convergence of Algorithm 10.1. (K = 6, N = 2, Pmmax = 35 dBm, and rkth = 22.5 kbps)

2W/Mbps 1W 10 W 0.1

0 0.021 0.02

Objective value of P4

−50 0.019 0.018

−100

0.017 80

85

95

90

100

0 −150

−50 −100

−200

−150

Zero Equal Random

−200 −250

−250 0

2 20

6

4 40

60

80

100

Iterations

 H 

H

w˙ , . . . , w˙ H , √cs s ≤ P max + cs s, ∀m, m,K m

m,1

(10.54)

2

where cs is a positive number. From (10.11), we have that both s and the LHS of (10.54) are positive even if the initial point is zero. Figure 10.2 shows that Algorithm 10.1 can be performed with any initial point and converges within tens of iterations. However, the initial point has an impact on the convergence rate and performance of Algorithm 10.1. Figure 10.3 compares the convergence results of Algorithm 10.2 with two algorithms in [29], denoted by Alg.CE and Alg.CA, respectively. In Alg.CE, the convex approximation at each iteration of the SCA is a generic convex programming, while the convex approximation in Alg.CA could be solved by the SOCP. However, Alg.CE and Alg.CA are designed based on CSI. To apply Alg.CE and Alg.CA to the CDI-based beamforming problem, the outage probability constraints in Alg.CE and

10.5 Numerical Results

195

Fig. 10.3 Convergence of Algorithm 10.2. (K = 8, N = 2, Pmmax = 35 dBm, and rkth = 12 kbps)

0.045

Objective value of P5

0.04

0.035

0.03

0.025

Alg.10.2 Alg.CE Alg.CA

0.02

0.015

10

0

30

20

40

50

70

60

80

Iterations −3

3.9

Average Energy Efficiency (Mbits/Joule)

Fig. 10.4 Convergence of Algorithm 10.3. (K = 6, N = 2, Pmmax = 35 dBm, and rkth = 15 kbps)

x 10

3.8 3.7 3.6 3.5 3.4 3.3

Decentralized Centralized

3.2 3.1

0

200

400

600

800

1000

1200

Iterations

Alg.CA are approximated by the method in Sect. 10.3. Figure 10.3 shows that Algorithm 10.2 converges within tens of iterations and the convergence rates among three algorithms are different. It should be pointed out that Algorithm 10.2 and Alg.CA could solve the problem more efficiently by the SOCP than Alg.CE [16, 29]. Figure 10.4 depicts the convergence rates of Algorithm 10.3 over two randomly generated channel realizations. It can be seen that Algorithm 10.3 requires hundreds of iterations to converge. Figure 10.5 investigates the effects of the maximum allowed transmit power Pmmax on the average EE. Due to the numerical issues, t in P4 is a small positive value rather than zero. From constraint (10.30), w˙ m,k , ∀m = g (k), may not be(exact zero

2 ) vector, so the final results should be obtained by letting g (k) = max w˙ m,k 2 m and then performing the beamforming. Accordingly, for the EE-EE scheme, the energy-efficient BS association solution is derived by Algorithms 10.1, and 10.2 is

Fig. 10.5 Impact of Pmmax on the average EE. (K = 9, N = 2, and rkth = 7.5 kbps)

10 Energy-Efficient Base Station Association and Beamforming … −3

8

Average Energy Efficiency (Mbits/Joule)

196

x 10

7.5 7 6.5 6 5.5 5 4.5 4 10

EE−EE SE−SE SE−EE SE−EE−CE SE−EE−CA 15

20

25

30

35

40

45

max

Maximum Allowed Transmit Power Pm (dBm)

performed for further improving EE. SE-SE and SE-EE mean that the BS association results are obtained based on the SE maximization and we derive the spectral-efficient and energy-efficient beamforming solutions, respectively. SE-EE-CE and SE-EE-CA represent that the corresponding beamforming results are obtained by Alg.CE and Alg.CA, respectively. Figure 10.5 shows that the average EE is not always increasing with respect to Pmmax . The reason is that using the extra power may increase the energy consumption significantly while the improvement of capacity is relatively small. Note that the average EE may decrease when Pmmax increases, which could be explained as follows. Although a larger Pmmax can increase the feasible region, Algorithms 10.1 and 10.2 only guarantee that the obtained solution satisfies the generalized KKT conditions and there may be more local optimal solutions when Pmmax is larger. From this figure, we can find that the developed algorithms and the compared algorithms almost achieve the same performance.

10.6 Conclusions In this chapter, we have investigated the energy-efficient joint BS association and beamforming problem in multi-cell multiuser MISO downlink systems. In view of the difficulties of solving the original problem optimally, we have developed a centralized joint BS association and beamforming algorithm to obtain a solution satisfying the generalized KKT conditions by resorting to the framework of the SCA, the differenceof-convex representation of the cardinality constraint, and the change of variables. The approximated convex problem at each iteration of the SCA has been solved more efficiently by the SOCP than the generic convex programming. Given the result of the BS association, a decentralized beamforming algorithm has been designed through combining the SCA and the ADMM.

10.6 Conclusions

197

Appendix 1 Proof of the SCA Condition in Chap. 10   For any equation F θ0 , . . . , θg , . . . θG  = 0, where θg ∈ R  = 0, ∀g, we can define a function h : RG → R such that θ0 = h θ1 , . . . , θg , . . . θG . According to the implicit function theorem [30], the partial derivative of h with respect  ( to)θg , ∀g,  is given by M H ˜ ∂θ0 ∂θ0 ∂ F k k ˜ = − ∂ F ∂θg . Define Ik˜ = m=1 w˙ m,k˜ Qm,k w˙ m,k˜ . Let f y˙k , Ik˜ , s represent the ∂θg k˜   ˙ s with respect to w˙ m,k˜ , k˜ = k, LHS of (10.20). Firstly, the partial derivative of y˙k w, is expressed as k ∂ y˙k ∂ y˙k ∂Ik˜ ∂ y˙k ∂ f˜ ˜ Qm,k w˙ m,k˜ = k = −2 ˙ m,k˜ ∂I ˜ ∂ w ˙ m,k˜ ∂w ∂ f˜ ∂Ikk˜ k 2 ˜ w˙ Q 1+ k,k˜ m,k m,k˜ s k,k˜ s3 + s yε˙˜ k−I k ˜ k=1, k˜ =k 1+ k,k˜ k k

= K

=

˜ m,k w˙ ˜ 2Q m,k   , ε˜ k υk 1 + k,k˜ s

(10.55)

   K Ikk˜

k,k˜ 1 where υk = 1 + k=1, − ln 1 + k,k˜ , k,k˜ = s y˙ −I k , and the last ˜ k˜ =k ε˜ k 1+ k,k˜ k k equality holds due to (10.20).   ˙ s of w˙ m,k is given by Secondly, the partial derivative of y˙k w, ∂ y˙k ∂Ikk ∂ y˙k ∂ f˜ ˜ ∂ y˙k Qm,k w˙ m,k = = −2 k ˙ m,k ∂w ∂Ik ∂ w˙ m,k ∂ f˜ ∂Ikk K

k,k˜ s2 + s yε˙˜ k−I ˜ m,k w˙ m,k k ˜ 2Q k=1, k˜ =k 1+ k,k˜ k k ˜ . Qm,k w˙ m,k = = 2 K s k,k˜ ε˜ k s 3 s + s y˙ −Ik ˜ k=1, k˜ =k 1+ ˜ k,k

k

(10.56)

k

  ˙ s with respect to s, we have Thirdly, for the partial derivative of y˙k w, ∂ y˙k ∂ f˜ ∂ y˙k =− = ∂s ∂ f˜ ∂s =

ε˜ k s −

ε˜ k sIkk s y˙k−Ikk

K

k,k˜ ε˜ k s 2 y˙k −2˜εk sIkk − y˙k k=1, ˜ k˜ =k 1+ k,k˜ s y˙k −Ikk K s k,k˜ s3 + s yε˙˜ k−I k ˜ k=1, k˜ =k 1+ k,k˜ k k

 − y˙k ε˜ k υk −

ε˜ k s 2 s y˙k −Ikk



=

2 y˙k − υk s

ε˜ υ s  k k k s y˙k − Ik υk − s 2 Ikk 1 = − 2− υk s s 2 υk K  Ik˜ Ik 1  k  , = − k2 − υk s ε˜ k υk 1 + k,k˜ s 2 ˜ ˜

(10.57)

k=1,k=k

where the third equation is derived through combining the definition of υk and (10.20). The last equality in (10.57) holds since

198

10 Energy-Efficient Base Station Association and Beamforming … K K    

k,k˜ ε˜ k s 2   = υ − ln 1 +

ε ˜ − , = ε ˜ k k k k,k˜ k 1 + k,k˜ s y˙k − Ik ˜ ˜ ˜ ˜ k=1,k=k

(10.58)

k=1,k=k

which can be obtained from (10.20).     ( j) ˙ s w˙ ( j) , s ( j) defined in (10.22) Comparing the partial gradients of yˆk w, with respect to w˙ m,k˜ , ∀m, ∀k˜ = k, w˙ m,k , ∀m, and s with (10.55), (10.56), and   ˙ w, s , i.e., (10.57), respectively, we conclude that the convex approximation of y ˙ k     ( j) ( j) ( j) ˙ s w˙ , s , meets the condition at the jth iteration of the SCA. yˆk w,

Appendix 2 Equivalence Between the Constraints It is not easy to show the equivalence between constraints (10.33d) and (10.34) directly. However, since (10.33d) is derived from the equivalent form of (10.10d), i.e., s≤

ς

K k=1

1 uk 22 + Pc

,

(10.59)

we would prove that (10.59) and (10.34) are equivalent in the following. By the fact that (10.59) must hold with equality at the optimum of P2, (10.59) is equivalently transformed into ⎧  Pc ⎪ ⎪ uk 22 + , ∀m, (10.60a) ⎪ pm ≥ ς ⎨ M k∈ m

⎪ 1 ⎪ ⎪ ⎩ s = M m=1

pm

,

(10.60b)

where pm , ∀m, is the auxiliary variable. Through introducing p˙ m = pm s, we can express (10.60) as ⎧  uk 2 Pc ⎪ 2 ⎨ p˙ m ≥ ς + s, ∀m, s M k∈m ⎪ ⎩ (10.34b).

(10.61a)

It is obvious that (10.34a) is the SOC representation of (10.61a), which completes the proof.

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Chapter 11

Summary and Outlook

This chapter summarizes the book and discusses the future directions to achieve sustainable wireless communications.

11.1 Summary In this book, we discuss various technologies to improve the performance of networks and to achieve sustainable wireless communications. Specifically, in Chaps. 2 and 3, we investigate the architecture of future wireless communication networks, where the sustainable ultra-dense heterogeneous networks (UDHN) and the sustainablity issues of non-orthogonal multiple access (NOMA) are discussed. In Chaps. 4–6, we investigate the reliable communications between different nodes, where the spacetime network coding (STNC) with maximal-ratio combining (MRC) and antenna selection (AS), and a compressive sensing (CS) based dynamic estimation algorithm for a unified laser telemetry, tracking, and command (TTC) system are discussed. In Chaps. 7–10, we investigate the energy-efficient resource allocations based on channel state information (CSI) and channel distribution information (CDI), respectively. The specific contributions of this book are as follows. 1. In Chap. 2, we have presented a potential network architecture for the UDHN, and then developed a generalized orthogonal/non-orthogonal random access scheme to improve the network efficiency while reducing the signaling overhead. Simulation results have demonstrated the effectiveness of the developed scheme. Furthermore, we have discussed some of the key challenges of the UDHN. 2. In Chap. 3, we have thoroughly examined the theoretical power regions of NOMA to show the minimum transmission power with fixed data rate requirement and demonstrated the energy efficiency (EE) performance advantage of NOMA over orthogonal multiple access. Then, we have explored the role of energy-aware © Beijing Institute of Technology Press 2023 J. An et al., Sustainable Wireless Communications, https://doi.org/10.1007/978-981-19-0448-6_11

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resource allocation and grant-free transmission in further enhancing the EE performance of NOMA. Based on this exploration, a hybrid NOMA strategy which reaps the joint benefits of resource allocation and grant-free transmission has been developed to simultaneously accomplish high throughput, large connectivity, and low energy cost. Furthermore, we have discussed some important and interesting future directions for NOMA designers to follow in the next decade. 3. In Chap. 4, we have investigated the STNC in cooperative multiple-input multipleoutput (MIMO) networks, where users communicate with a common destination with the aid of decode-and-forward (DF) relays. The transmit antenna selection and maximal-ratio combining (TAS/MRC) is adopted in user-destination and relay-destination links where a single transmit antenna that maximizes the instantaneous received signal-to-noise ratio (SNR) is selected and fed back to transmitter by receiver and all the receive antennas are combined with MRC. In the presence of perfect feedback, we have derived new exact and asymptotic closed-form expressions for the outage probability (OP) and the symbol error rate (SER) of STNC with TAS/MRC in independent but not necessarily identically distributed Rayleigh fading channels. To quantify the impact of delayed feedback, we have further derived new exact and asymptotic OP and SER expressions in closed form. Numerical and Monte Carlo simulation results have been provided to demonstrate the accuracy of our theoretical analysis and evaluate the impact of network parameters on the performance of STNC with TAS/MRC. 4. In Chap. 5, we have investigated the STNC with AS in the cooperative MIMO network, where users communicate with a common destination with the aid of DF relays. In this network, the best transmit/receive antenna pair with the highest SNR ratio is selected to perform the signal transmission and reception over the user-destination and relay-destination links. In order to quantify the performance degradation of STNC with AS due to time delay between instants of CSI estimation and data transmission, we have derived the SER and capacity expressions over flat Rayleigh fading channels. We have also examined the effect of spatial correlation on the performance of STNC with AS through obtaining new closedform expression for the asymptotic SER, which indicates that STNC with AS achieves the full diversity order over flat correlated Rayleigh channels regardless of the value of spatial correlation coefficient. Numerical and simulation results have been provided to demonstrate the accuracy of our theoretical analysis and evaluate the performance of STNC with AS. 5. In Chap. 6, we have developed a CS-based dynamic estimation algorithm to address the dynamic estimation problem in spaceborne intensity modulation direct detection (IMDD) based laser TTC systems. Specifically, we have designed the sparsity in transformation domain. Based on this design, a sparsity-aware algorithm for dynamic estimation through CS via sparse transformation has been developed. Extensive simulations have been conducted to demonstrate the effectiveness of the developed algorithm in terms of good accuracy. Moreover, analytical and simulation results have shown that the developed algorithm exhibits a better performance than conventional CS algorithms in terms of computational

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complexity, suggesting its attractiveness for spaceborne IMDD-based laser TTC systems. 6. In Chap. 7, we have investigated energy-efficient downlink resource allocation in heterogeneous orthogonal frequency division multiple access (OFDMA) networks, and formulated the EE maximization problem as a mixed-integer nonlinear fractional programming (MINLFP) problem with a non-concave nonlinear objective function and nonlinear constraints. By means of fractional programming and changing of variables, we have transformed the original MINLFP problem into an equivalent optimization problem in a parametric subtractive form, which is proved to be a concave mixed-integer nonlinear programming (MINLP) problem and is optimally solved by using Dinkelbach and branch-and-bound (BB) methods. In BB method, the concave MINLP problem is relaxed to a series of concave nonlinear programming problems and solved by the use of Powell-Hestenes-Rockafellar augmented Lagrangian method. The optimal solution can be used to benchmark the performance of sub-optimal solutions. As the computational complexity of BB method increases exponentially with problem size, we have further developed a sub-optimal two-step scheme, which first allocates the resource blocks and then performs the transmit power control, to give sub-optimal solution with much lower complexity. Simulation results have demonstrated the effectiveness of the developed schemes and shown that the developed sub-optimal two-step scheme is promising for practical applications as it makes a good tradeoff between EE performance and computational complexity. 7. In Chap. 8, we have investigated the energy-efficient power control for deviceto-device (D2D) communications underlaying cellular networks, where uplink resource blocks allocated to one cellular user equipment are reused by multiple D2D pairs and co-channel interference caused by resource sharing becomes a significant challenge. We have considered both the total EE and individual EE optimization problems, which are fractional programming and generalized fractional programming problems, respectively, and are hard to tackle due to their nonconcave nature. We have transformed them into equivalent optimization problems in parametric subtractive forms, which fit in a class of non-concave optimization methods known as difference of two concave functions programming, and then solved them using Dinkelbach and BB methods to give global optimal solutions. Due to the unaffordable complexity of the global optimal solution, we have further developed sub-optimal schemes through adding constraints on the interferences to convert the non-concave problems into concave ones and to give sub-optimal solutions with reasonable complexity. The sub-optimal solution gives a tight lower bound on the optimal EE. Simulation results have been presented to demonstrate the effectiveness of the developed schemes. 8. In Chap. 9, we have investigated the energy-efficient user scheduling and power control problem in downlink multi-cell multiuser orthogonal frequency division multiple access networks, where only the CDI is available. The resulting optimization problem is a combinatorial problem, which belongs to the class of cardinality constrained fractional programming problems subject to a maximum OP constraint. To solve the original problem, we have derived a centralized joint user

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scheduling and power control solution satisfying the generalized Karush-KuhnTucker (KKT) conditions by resorting to the successive convex approximation (SCA). In particular, the developed scheme can be used to solve the resource allocation problem subject to each user’s minimum throughput constraint even if all the subcarriers have the same channel gain. A decentralized power control scheme has been also developed to maximize the EE through combining SCA and alternating direction method of multipliers (ADMM). Simulation results have demonstrated the effectiveness of the developed schemes. 9. In Chap. 10, we have investigated the joint base station (BS) association and beamforming for EE maximization in coordinated multi-cell multiuser downlink systems. In particular, we assume that only the CDI is known to the BSs. The considered problem is difficult to be solved optimally due to the non-smooth and non-convex functions in the formulation. Therefore, we have developed an iterative suboptimal algorithm to solve the problem efficiently based on the SCA. More specifically, the convex approximation of the original problem at each iteration can be solved efficiently by the second-order cone programming and the solution obtained by the developed algorithm satisfies the generalized KKT conditions. To facilitate the implementation of decentralized beamforming, we have transformed the convex approximation problem at each iteration of the SCA into an equivalent form, which is amenable to applying the ADMM. By combining the SCA and the ADMM, a decentralized energy-efficient beamforming algorithm has been developed. Numerical results have been presented to show the performance of the developed algorithms.

11.2 Future Directions In this section, we discuss the future directions to achieve sustainable wireless communications.

11.2.1 Passive Communications Passive communications mainly refer to the backscatter communications and intelligent reflecting surface (IRS) enabled communications. In backscatter communications, the device can transmit data by modulating the received signals instead of generating carriers itself. In IRS-enabled communications, IRS can be used to reflect the signals to enhance the SNR of data transmission. Note that the typical backscatter devices and IRS are passive components, which results in a very low power consumption. Therefore, by integrating passive communications, the spectral efficiency (SE) and the EE can be improved, and passive communications have been expected as key components to achieve sustainable wireless communications. However, since the communication networks become more complicated when integrating

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passive communications into traditional communications, there are many issues to be investigated and how to effectively integrate passive communications technology into traditional communication networks is worth to be investigated.

11.2.2 Millimeter-Wave and Terahertz Communications The surging data rate, which is expected to reach Terabit per second in the sixth generation (6G) era, is very challenging by using current wireless technologies designed for microwave-band systems. Actually, the scarcity of available spectrum resources below 6 GHz extremely restricts the system capacity. Against this background, the millimeter wave (mmWave) band, ranging from 30 to 300 GHz, and terahertz (THz) band, ranging from 100 GHz to 10 THz, have triggered tremendous research interests, due to their ultra-large usable bandwidth. Signal processing at mmWave and THz frequencies is different from that at microwave frequencies. First, new hardware constraints, for example, low-resolution analog-to-digital converters (ADCs), hybrid precoding architectures, and imperfect phase shifters, arise due to the high operating frequency and large bandwidth. Second, the channels in the mmWave and THz band are different, which is mainly resulting from the small wavelength signal. For example, mmWave and THz channels contain a small number of propagation paths and show strong angular sparsity. Third, large antenna arrays are expected to be equipped at both the transmitters and receivers in mmWave and THz wireless systems, to combat the severe path loss. Thus, novel massive MIMO techniques which consider capabilities of hardware circuits and channel characteristics in the mmWave and THz band are imperative. Hence, many challenges remain in designing transmission strategies and deriving system performance for mmWave and THz wireless systems and they are worth to be investigated.

11.2.3 Space-Air-Ground Integrated Information Network The developments of future communication technologies and satellite services have prompted countries to accelerate the construction of space-air-ground integrated information network platforms. The space-air-ground integrated information network is made up with three sub-networks. The space-based network is composed of multiple satellites under different orbits, the air-based network is composed of various aircrafts, and the ground-based network is composed of cellular wireless networks, satellite ground stations, and mobile satellite terminals. Users, aircrafts, and various communication platforms in different layers are closely combined through inter satellite links and satellite to ground links. The space-air-ground integrated information network are designed to provide seamless network services to spatial, aerial, maritime, and ground users, satisfying the future network requirements in all-time, all-domain, and all-space communications and interconnected networking.

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However, in this network, frequent handovers will result in a decline of user experience as well as heavy signaling overheads. Therefore, developing fast and effective handover strategies among multi-layer heterogeneous networks to reduce latency and OP as well as improve the user experience is worth to be investigated.

11.2.4 Joint Radar and Communication Joint radar and communication (JRC) is supposed to support the wireless communication demands in the next ten years and is the key to realize the Internet of Things, self-driving vehicles, perceptive mobile networks, and other technologies. The JRC network shares a large number of hardware and signal processing modules, by transmitting a single joint-sensing-and-communication signal, which can double the SE of the system, reduce hardware costs, and save energy consumption. The signal transmitted by this network considers both requirements of radar detection and those of communication, and optimizes a joint performance metric. However, there are still many technical problems to be solved, including channel estimation and joint waveform design. A good joint waveform guarantees both stability of communication and robustness of sensing results, while it relies on fast channel estimation and processing results. The traditional channel estimation methods have small scales on calculation and high complexity, which, thus, cannot be applied to the large scale of antenna arrays in the 6G cellular network. The joint waveform is easy to generate interference, which leads to a decrease of SE instead of an enhancement. Therefore, new and efficient channel estimation and joint waveform design methods for JRC are worth to be investigated.

11.2.5 Artificial Intelligence-Enabled Wireless Communications Due to the rapid development of terminal devices and extremely diverse user demands, the future wireless communication networks are required to be cognitive and intelligent. However, the unprecedented complexity of future wireless networks makes the traditional methods challenging to satisfy the above-mentioned requirements. With the rapid development of artificial intelligence (AI) technologies, some AI technologies, like deep reinforcement learning (DRL), can be applied to allocate wireless network resources, such as transmit power, computing resources, beamforming vectors, etc. Moreover, deep learning approaches can be used for network operation and maintenance, such as channel estimation, anomaly detection, and traffic prediction, etc. These wireless network problems might be challenging or even impossible to solve by traditional approaches due to the deficit of model or algorithm. In contrast, the AI technologies can make use of the network data to solve these prob-

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lems. Nonetheless, the AI-based methods also face many challenges to deal with. For example, the training of deep learning or DRL might cost too much time and computation, and how the expert knowledge in wireless networks is utilized to assist the training needs to be investigated. In conclusion, the AI technology will play an important role in the future wireless networks and it is worth to be investigated.