Cooperative Communications and Networking: Technologies and System Design

Cooperative Communications and Networking Y.-W. Peter Hong • Wan-Jen Huang C.-C. Jay Kuo Cooperative Communications ...

Author: Y.-W. Peter Hong | Wan-Jen Huang | C.-C. Jay Kuo

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Cooperative Communications and Networking

Y.-W. Peter Hong • Wan-Jen Huang C.-C. Jay Kuo

Cooperative Communications and Networking Technologies and System Design Foreword by Georgios B. Giannakis

1C

Y.-W. Peter Hong National Tsing Hua University Department of Electrical Engineering Kuang-Fu Rd. Section 2 101 30013 Hsinchu Taiwan R.O.C. [email protected]

C.-C. Jay Kuo University of Southern California Viterbi School of Engineering McClintock Ave. 3740 90089-2564 Los Angeles California USA [email protected]

Wan-Jen Huang National Sun Yat-Sen University Lien-Hai Road 70 80424 Kaohsiung Institute of Comm. Engin. Taiwan R.O.C. [email protected]

ISBN 978-1-4419-7193-7 e-ISBN 978-1-4419-7194-4 DOI 10.1007/978-1-4419-7194-4 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010932009 © Springer Science+Business Media, LLC 2010 Portions of text appearing on pages 85–87, 104–106, 109–112, 158–159, 163–177, 196–225, 247–258, 274–288, 295–303, 307–313, 317–329, 336–357, 366–369, 376–391 used with permission © IEEE. See references for further details. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

“To my beloved parents, and my lovely wife and daughter.” – Peter “To my beloved husband, Wei-Feng, and my parents, Ching-Song and Pi-Hsia.” – Wan-Jen “I would like to dedicate this book to my beloved parents, wife and daughter.” – Jay

Foreword

Over the past years, there has been a considerable research activity in the ﬁeld of wireless cooperative communications and networking. By now, this ﬁeld has reached a mature level and there is a deﬁnite need to organize research results and present them in a uniﬁed manner. This is precisely the objective and strength of this book: to deliver in a coherent and consistent story the stateof-the-art in cooperative communications and networking. This grueling eﬀort will beneﬁt graduate students in advanced topics in wireless communications, active researchers to identify new directions, as well as designers to stay current for the latest ﬁndings. The authors of Cooperative Communications and Networking – Technologies and System Design have delivered extensive and fundamental results in this subject. From this vantage point, they were able to provide a timely and comprehensive view of this exciting topic. To allow for consistency and self-containment, the authors ﬁrst review wireless communications basics, with special emphasis on diversity techniques, capacity, and design tradeoﬀs. Building on these necessary foundations, simple illustrative cooperation models are ﬁrst analyzed followed by extensions comprising multiple relays and sources. The last part delineates how cooperation can be integrated with advanced wireless transmission technologies as well as how it permeates the beneﬁts of cooperation at the physical layer to higher-layers of the protocol stack. The end result is a book that not only paves the road for newcomers in the classroom and industry, but also serves as reference material for active researchers and network designers in this ﬁeld. Minnesota, USA, April 2, 2010

Georgios B. Giannakis

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Preface

Cooperative and relay communications have been one of the most widely explored topics in communications over the past few years. Although many underlying ideas originate from the early works on relay channels in the 1960’s, the technology has only recently been popularized by the works of Sendonaris, Erkip, and Aazhang as well as the works of Laneman, Tse, and Wornell in the late 1990’s and early 2000’s. The key idea is to have users cooperate in transmitting their messages to the destination, instead of operating independently and competing among each other for channel resources, as done in conventional networks. However, as the ﬁeld has progressed over the years, cooperative communications have now become a design concept rather than a speciﬁc transmission technology. This concept has revolutionized the design of wireless networks, allowing us to increase coverage, throughput, and transmission reliability even as conventional transmission techniques gradually reach their limits. In recent years, cooperative and relay technologies have also made its way toward next generation wireless standards, such as IEEE 802.16 (WiMAX) or LTE, and have been incorporated into many modern wireless applications, such as cognitive radio and secret communications. With its expanding scope of applications, it is necessary for engineers and researchers to have a more fundamental understanding of the design concept, thus motivating our work on this book project. With this book, we hope to provide a more systematic way into learning this subject. Our interest in this topic initiated from our works on asynchronous and opportunistic cooperative transmission schemes presented in the early 2000’s. At that time, we proposed and analyzed the so-called opportunistic large arrays (OLA) system, where all users in the network can participate in the cooperation and relay the source’s message in an opportunistic and uncoordinated fashion, whenever they are able to reliably decode the message. Since then, we have also been interested in the use of cooperative communications in sensor networks, focusing on exploiting its energy eﬃciency and its ability to utilize sensor dependency to improve communication eﬃciency. In recent years, we have studied extensively the design of multiuser cooperative sys-

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tems, examining both physical and MAC layer issues such as relay-assisted multiuser detection schemes and cooperative MAC protocols. We have found this subject to be extremely interesting and have been attracted by its rich set of applications. Yet, we realize that with more and more researchers working on this problem, the technology is becoming rather advanced and diverse. However, with this book, we hope to connect the dots between these advanced topics and provide readers with a more comprehensive view of the subject. The aim of this book is to provide the basic concepts of cooperative communications and relay technology to enable engineers or graduate students to have a clear grasp of this growing ﬁeld and have the basic knowledge to conduct advanced research and development in this area. The book is in no sense complete in terms of gathering all works in the ﬁeld, but contains basic concepts that allow readers to gain a fundamental understanding of the system and of the challenges that it often encounters. The contents of the book can be summarized as follows: • In Chapter 1, we give a brief introduction of the various cooperation and relay techniques that will be introduced in later chapters. We also provide a brief history of this technology and a survey of its role in next generation wireless standards. • In Chapter 2, we review basic wireless communication and MIMO techniques for readers that are not familiar with this ﬁeld. This knowledge is helpful in understanding the techniques to be introduced in other chapters of this book. • In Chapter 3, basic cooperative communication and relay techniques are introduced for a basic cooperative entity that consists of only two users and a common destination. • In Chapter 4, the cooperation and relay techniques are extended to systems with multiple relays. Here, the relays together form a distributed antenna array that can be used to exploit spatial diversity and multiplexing gains. • In Chapter 5, fundamental limits of cooperative and relay channels are described from the information-theoretic standpoint. • In Chapter 6, these techniques are applied to multiuser systems, where multiple cooperative entities may be transmitting simultaneously or where multiple sources may be accessing a common cooperative channel. • In Chapter 7, we describe how cooperative and relay technology can be integrated with other advanced wireless technology, such as OFDM and MIMO, and show how additional beneﬁts can be obtained by doing so. • In Chapter 8, we describe how cooperative advantages exploited in the medium access control (MAC) layer and how MAC policies should be designed to enhance cooperative communications in the physical layer. • In Chapter 9, we introduce several cross-layer and networking issues that may arise in cooperative networks, including routing, QoS, and security considerations.

Preface

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The authors would like to send special thanks to research assistants Ms. Shu-Hsien Wang, Mr. Jui-Yang Chang, and also to colleague Dr. ShihChin Lin for their extensive eﬀorts on the editing and proof-reading of this book. Without their help, we would never have been able to complete the manuscript. We would also like to thank Mr. Yung-Shun Wang, Ms. MiaoFeng Jian, and Ms. Chao-Wei Huang for their help on generating ﬁgures and illustrations, and also Mr. Meng-Hsi Chen, Mr. Ta-Yuan Liu, and Ms. Yi-Hua Lin for their help in proof-reading the manuscript. Hsinchu, Taiwan Kaohsiung, Taiwan Los Angeles, California, USA April, 2010

Y.-W. Peter Hong Wan-Jen Huang C.-C. Jay Kuo

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 An Overview on Cooperative Communications . . . . . . . . . . . . . 1 1.2 Brief History of Cooperative and Relay Channels . . . . . . . . . . . 5 1.3 Standardization of Cooperative Communication and Relay Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Book Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2

Review of Wireless Communications and MIMO Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Characteristics of Wireless Channels . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Path Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Shadowing Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Multipath Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Techniques to Exploit Spatial Diversity . . . . . . . . . . . . . . . . . . . . 2.2.1 Single-Input Multiple-Output (SIMO) System . . . . . . . 2.2.2 Multiple-Input Single-Output (MISO) System . . . . . . . 2.2.3 Multiple-Input Multiple-Output (MIMO) System . . . . 2.3 Capacity of Wireless Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Capacity of AWGN Channels . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Capacity of Flat Fading Channels . . . . . . . . . . . . . . . . . . 2.3.3 Capacity with Multiple Antennas . . . . . . . . . . . . . . . . . . . 2.4 Diversity-and-Multiplexing Tradeoﬀ . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 15 18 19 25 26 35 44 48 48 49 52 58 63

Two-User Cooperative Transmission Schemes . . . . . . . . . . . . . 3.1 Decode-and-Forward Relaying Schemes . . . . . . . . . . . . . . . . . . . . 3.1.1 Basic DF Relaying Scheme . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Selection DF Relaying Scheme . . . . . . . . . . . . . . . . . . . . . 3.1.3 Demodulate-and-Forward Relaying Scheme . . . . . . . . . . 3.2 Amplify-and-Forward Relaying Schemes . . . . . . . . . . . . . . . . . . .

67 67 68 78 81 87

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3.2.1 Basic AF Relaying Scheme . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Incremental AF Relaying Scheme . . . . . . . . . . . . . . . . . . . 3.3 Coded Cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Basic Coded Cooperation Scheme . . . . . . . . . . . . . . . . . . 3.3.2 User Multiplexing for Coded Cooperation . . . . . . . . . . . 3.4 Compress-and-Forward Relaying Schemes . . . . . . . . . . . . . . . . . . 3.5 Channel Estimation in Single Relay Systems . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88 100 102 103 106 114 115 120

4

Cooperative Transmission Schemes with Multiple Relays . 4.1 Orthogonal Cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Orthogonal Cooperation with AF Relays . . . . . . . . . . . . 4.1.2 Orthogonal Cooperation with DF Relays . . . . . . . . . . . . 4.2 Transmit Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Transmit Beamforming with AF Relays . . . . . . . . . . . . . 4.2.2 Transmit Beamforming with DF Relays . . . . . . . . . . . . . 4.3 Selective Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Selective Relaying with AF Relays . . . . . . . . . . . . . . . . . . 4.3.2 Selective Relaying with DF Relays . . . . . . . . . . . . . . . . . . 4.4 Distributed Space-Time Coding (DSTC) . . . . . . . . . . . . . . . . . . 4.4.1 Distributed Space-Time Coding with DF Relays . . . . . . 4.4.2 Distributed Space-Time Coding with AF Relays . . . . . . 4.5 Channel Estimation in Multi-Relay Systems . . . . . . . . . . . . . . . 4.5.1 Training Design for AF Multi-Relay Systems . . . . . . . . . 4.5.2 Training Design for DF Multi-Relay Systems . . . . . . . . . 4.6 Other Topics on Multi-Relay Cooperative Communications . . 4.6.1 Multi-Hop Cooperative Transmissions . . . . . . . . . . . . . . . 4.6.2 Asynchronous Cooperative Transmissions . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 126 127 133 135 135 141 146 147 150 153 153 161 168 168 173 178 178 187 190

5

Fundamental Limits of Cooperative and Relay Networks . . 5.1 Gaussian Relay Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Cut-Set Bound of Gaussian Relay Channels . . . . . . . . . . 5.1.2 Decode-and-Forward and Degraded Relay Channels . . . 5.1.3 Compress-and-Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Single-Relay Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Ergodic Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Diversity and Multiplexing Tradeoﬀs . . . . . . . . . . . . . . . . 5.3 Multi-Relay Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Upper Bound of Gaussian Multi-Relay Networks . . . . . 5.3.2 Lower bound of Gaussian Multi-Relay Networks and Asymptotic Capacity Results . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Multi-Relay Fading Channels . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 193 194 197 201 203 203 207 213 214 216 219 225

Contents

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Cooperative Communications with Multiple Sources . . . . . . 6.1 Time/Frequency-Division Multiple Access (TDMA/FDMA) . . 6.1.1 Round-Robin Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Opportunistic Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Code-Division Multiple Access (CDMA) . . . . . . . . . . . . . . . . . . . 6.2.1 Uplink CDMA with Designated Relays . . . . . . . . . . . . . . 6.2.2 Uplink CDMA with Shared Relays . . . . . . . . . . . . . . . . . 6.3 Space-Division Multiple Access (SDMA) . . . . . . . . . . . . . . . . . . . 6.4 Partner Selection Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Centralized Partner Selection Strategy . . . . . . . . . . . . . . 6.4.2 Decentralized Partner Selection Strategy . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227 227 229 234 237 238 247 254 259 260 267 268

7

Cooperation Relaying in OFDM and MIMO Systems . . . . . 271 7.1 Brief Review of OFDM Systems . . . . . . . . . . . . . . . . . . . . . . . . . 271 7.2 Resource Allocation in Pair-Wise Cooperative OFDM Systems274 7.2.1 Power Allocation in Pair-Wise Cooperative OFDM Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 7.2.2 Subcarrier Matching for Pair-Wise Cooperative OFDM Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 7.3 Cooperative OFDM Systems with Multiple Relays . . . . . . . . . . 282 7.3.1 Cooperative Beamforming for OFDM Multi-Relay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 7.3.2 Selective Relaying for OFDM Multi-Relay Systems . . . 288 7.4 Distributed Space-Frequency Codes . . . . . . . . . . . . . . . . . . . . . . . 292 7.4.1 Decode-and-Forward Space-Frequency Codes . . . . . . . . . 293 7.4.2 Amplify-and-Forward Space-Frequency Codes . . . . . . . . 301 7.5 Cooperation with MIMO Relays . . . . . . . . . . . . . . . . . . . . . . . . . . 305 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

8

Medium Access Control in Cooperative Networks . . . . . . . . 8.1 Cooperation with Slotted ALOHA . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Deﬁnition of Stability Region . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Stability Region of a Cooperative Pair . . . . . . . . . . . . . . 8.2 Collision Resolution Mechanisms in Cooperative Networks . . . 8.2.1 Network-Assisted Diversity Multiple Access (NDMA) . 8.2.2 Enhancements to NDMA with Relaying Users . . . . . . . . 8.3 Cooperation with CSMA/CA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Overview of the IEEE 802.11 MAC Protocol . . . . . . . . . 8.3.2 CoopMAC based on the IEEE 802.11 Protocol . . . . . . . 8.3.3 Analysis of CoopMAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Automatic Retransmission reQuest (ARQ) with Cooperative Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Throughput Optimal Scheduling Protocols for Cooperative Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

317 317 320 322 329 329 331 332 333 336 341 344 347

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8.5.1 Review of Throughput Optimal Control Policy for Non-Cooperative Networks . . . . . . . . . . . . . . . . . . . . . . . . 347 8.5.2 Throughput Optimal Control Policy for Cooperative Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 9

Networking and Cross-Layer Issues in Cooperative Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 QoS in Cooperative Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 QoS of a Simple Relay Network . . . . . . . . . . . . . . . . . . . . 9.1.2 QoS of a Cooperative Pair . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Routing in Cooperative Networks . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 General Formulation of Cooperative Routing . . . . . . . . . 9.2.2 Heuristic Algorithms for Cooperative Routing . . . . . . . . 9.3 Security Issues in Cooperative Networks . . . . . . . . . . . . . . . . . . . 9.3.1 Misbehavior in Relay Networks . . . . . . . . . . . . . . . . . . . . . 9.3.2 Security in Single-Relay Cooperative Networks . . . . . . . 9.3.3 Security in Multi-Relay Cooperative Networks . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

361 361 365 371 373 373 379 383 383 385 388 394

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

Acronyms

3GPP ACK AF ARQ AWGN BC BER BS BPSK CDF CDMA CF CoMP CP CRC CS/CB CSI CSMA CSMA/CA CTS DCF DF DFE DFT DIFS DMC DMT DS-CDMA DSTC EC eNodeB

3rd Generation Partnership Project acknowledgement amplify-and-forward automatic retransmission request additive white Gaussian noise broadcast channel bit error rate base station binary phase-shift keying cumulative distribution function code-division multiple access compress-and-forward coordinated multiple point cyclic preﬁx cyclic redundancy check coordinated scheduling/beamforming channel state information carrier sensing multiple access CSMA with collision avoidance clear-to-send distributed coordination function decode-and-forward decision feedback equalizer discrete Fourier transform DCF interframe space discrete memoryless channel diversity-and-multiplexing tradeoﬀ direct sequence code-division multiple access distributed space-time code equal-gain combining evolved node B

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FDMA i.i.d. JP KKT LD LMMSE LOS LTE MAC MAI MF MGF MIMO MISO ML MMSE MRC MS MUD NACK NAV NLOS NP OFDM OSTBC PCF PDF QoS QPSK RS RSC RTS SC SDP SDMA SER SIFS SIMO SINR SISO SNR SS STBC STC STTC

Acronyms

frequency-division multiple access independent and identically distributed joint processing Karush-Kuhn-Tucker linear dispersion linear minimum mean square error line-of-sight Long-Term Evolution multiple access channel or medium access control multiple access interference matched ﬁlter moment generating function multiple-input multiple-output multiple-input single-output maximum likelihood minimum mean square error maximal ratio combining or maximal ratio combiner mobile station multi-user detection negative acknowledgement network allocation vector non-line-of-sight non-polynomial orthogonal frequency-division multiplexing orthogonal space-time block code point coordination function probability density function quality of service quadrature phase-shift keying relay station recursive systematic convolutional ready-to-send selection combining semi-deﬁnite programming space-division multiple access symbol error rate short interframe space single-input multiple-output signal-to-interference-plus-noise ratio single-input single-output signal-to-noise ratio subscriber station space-time block code space-time code space-time trellis code

Acronyms

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SVD TDMA UE VQ WiMAX ZF

singular value decomposition time-division multiple access user equipment vector quantizer or vector quantization Worldwide Interoperability for Mircowave Access zero-forcing

Chapter 1

Introduction

Wireless communications have gained much popularity in recent years due to its ability to provide untethered connectivity and mobile access. However, before the turn of the century, many attempts to achieve reliable and high data-rate communication over the wireless channel have been unsuccessful due to multipath fading, shadowing, and path loss eﬀects. These eﬀects result in random variations of channel quality in time, frequency, and space, making it diﬃcult to employ conventional wireline communication techniques in the wireless environment. Not until the past two decades have people developed eﬀective transmit and receive diversity techniques to exploit diversity in diﬀerent channel dimensions, such as time, frequency, and space, and achieve the so-called diversity gains. With suﬃcient channel knowledge, one can devise power and bit allocation policies over time, frequency, and space that allocates more resources to channels that are more reliable and avoids transmitting in bad channels to preserve energy. Even when channel knowledge is not available, space-time or space-frequency codes can also be utilized to enhance the transmission reliability. In particular, advances in theory on multiple-input multiple-output (MIMO) [17, 47, 53] systems have made it desirable to embed multiple antennas on modern wireless transceivers, in order to achieve spatial diversity gains. However, as the size and cost of wireless devices are limited for many applications, e.g., in sensor networks or for cellular phones, placing multiple antennas on a single terminal may not be practical. In this case, cooperating with other nodes in the network to form a distributed antenna system becomes a desirable and promising alternative. This is achieved by the so-called cooperative communications.

1.1 An Overview on Cooperative Communications Cooperative communications, as popularized by the works [36, 37, 41, 42], allow users in the system to cooperate by relaying each other’s messages to the

Y.-W. Peter Hong et al., Cooperative Communications and Networking, DOI 10.1007/978-1-4419-7194-4_1, © Springer Science+Business Media, LLC 2010

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t

Fig. 1.1 Illustration of a pair-wise cooperative communication system.

destination. By doing so, users can eﬀectively form a distributed antenna array that emulates the spatial diversity gains achievable by centralized MIMO systems. An example of a pair-wise cooperative communication system is illustrated in Fig. 1.1, where the users are assumed to experience independent fading channels to the destination. Due to multipath fading, the signal-tonoise ratios (SNRs) at the destination may vary rapidly over time, causing communication outage whenever one of the users’ SNRs falls below the required level (as illustrated by the shaded regions in Fig. 1.1). However, if the two users can cooperate by relaying each others’ messages to the destination, communication outage will occur only when both users simultaneously experience poor channels, thereby improving the transmission reliability. Many cooperation strategies have been proposed in the literature based on diﬀerent relaying techniques, such as amplify-and-forward (AF) [36], decodeand-forward (DF) [36,42], selective relaying (SR) [36], coded cooperation [27], compress-and-forward (CF) [34] etc. When these schemes are employed in a pair-wise cooperating system as shown in Fig. 1.1, we can assume that, at each instant in time, only one user acts as the source while the other user serves as the relay that forwards the source’s message to the destination. The role between the source and the relay can be interchanged at any instant in time. An illustration of the DF, AF, and SR schemes are given in Fig. 1.2. As common in these schemes, cooperative transmissions are initiated by ﬁrst having the source (e.g., user 1) broadcast its message to both the relay and the destination. If the DF scheme is employed, the relay will decode and regenerate a new message to the destination in the subsequent time slot. At the destination, signals from both the source and the relay are combined to provide better detection performance. As an extension to the DF scheme, the message generated by the relay can be re-encoded to provide additional error protection, and such a scheme can also be referred to as coded cooperation. If the AF scheme is employed, the relay simply ampliﬁes the received signal and forwards it directly to the destination without explicitly decoding the message. The SR scheme, on the other hand, is a dynamic scheme where relays are selected to retransmit the source message only if the relay path

1.1 An Overview on Cooperative Communications

Decode-and-Forward (DF) Xs Source

3

Amplify-and-Forward (AF) Xs

1

1 1 1 0

Source

1

1

1 1 1 0

0 0

1 1 1

1

Relay

1

1 0 0

Destination

1

0 0

0

1 1

Yr

Xr=Xs

Relay

(a) Decode-and-Forward

1

1

0

Destination

0

0

Xr=aYr

(b) Amplify-and-Forward

Selective Relaying (SR) Method Xs Source

1

1 1 1 0

1 0 0

s-d channel > s-r-d channel

Destination

Xs 1

Source

1 1 1 0

1

1

Relay

0 0

0 1 1

Destination

1

Relay

Source

s-d channel < s-r-d channel

1 1 1

1

Relay

1 0 0

Destination

0

Xr=Xs

(c) Selective Relaying Fig. 1.2 Examples of basic cooperative communication schemes, namely, decode-andforward, amplify-and-forward, and selective relaying.

is suﬃciently reliable. This scheme can be applied on top of both DF and AF schemes to improve cooperation eﬃciency. Among the many cooperation schemes proposed in the literature, DF, AF, and SR schemes are the most basic and widely adopted. More sophisticated schemes, such as the CF scheme, can also be devised by exploiting the statistical dependencies between the messages received at the relay and the destination but require higher implementation complexity. These schemes will be described in Chapter 3. Most cooperation strategies involve two phases of transmission: the coordination phase and the cooperative transmission phase. Coordination is especially required in cooperative systems since the antennas are distributed among diﬀerent terminals, as opposed to that in centralized MIMO systems. Although extra coordination may reduce bandwidth ineﬃciency, the cost is often compensated for by the large diversity gains experienced at high SNR. Speciﬁcally, coordination can be achieved either by direct inter-user communication or by the use of feedback from the destination. Based on the information obtained through coordination, cooperating partners will compute and transmit messages so as to reduce the transmission cost or enhance the detection performance at the receiver. The cooperative communication schemes described above can be readily extended to a large network by having one user serve as the source and the remaining users serve as relays at each time instant, as shown in Fig. 1.3. The relays together form a distributed antenna array that is able to achieve

4

1 Introduction

u1=f1(x1)

y2

yL

.....

Source

Relay 2

.....

x

Relay 1

Relay L

u2=f2(x2) yd .....

y1

Destination

uL=fL(xL) (feedback)

Fig. 1.3 Illustration of multi-relay cooperative communication system.

spatial diversity and multiplexing gains [5,57,59] similar to that of centralized MIMO systems. These techniques include distributed space-time coding [3,29, 30,43,56], distributed beamforming [2,31,33], and antenna selection strategies [7,16,58]. However, the cost of coordination may increase with the number of cooperating users and, thus, eﬃcient inter-user or feedback communication strategies must be devised to make cooperation worthwhile. In the early literature on cooperative communications, most works consider the simpliﬁed scenario where only one user in the system acts as the source while all the other users serve as relays to the source. When applying these schemes to systems with multiple sources, as illustrated in Fig. 1.4, one must assume that the sources can access the relays through orthogonal channels and that the relays have suﬃcient energy and bandwidth resources for all users. If this is not the case, many multiuser problems may arise in both the physical and higher network layers. From the physical layer perspective, this results in multiple access interference (MAI) that may eventually dominate the BER performance and, thus, cause the diversity gains to diminish. Moreover, with limited energy and bandwidth resources at the relays, eﬃcient resource allocation policies must be devised to ensure high performance gains for all users. From the medium access control (MAC) layer perspective, scheduling or random access protocols must be developed to help resolve contention among those competing for the cooperative channel. In fact, even when cooperative advantages do not exist in the physical layer, it has been shown in [23] that MAC layer cooperation can still be beneﬁcial. For relay and cooperative communication technology to be fully deployed in modern communication systems, many high-layer issues remain to be addressed. The most evident among all is the issue of resource allocation between the source-relay, relay-destination, and source-destination links. This

1.2 Brief History of Cooperative and Relay Channels

5

r1

r2

s1

..

s2

..

..

..

..

rL

sK-1 sK Fig. 1.4 Illustration of cooperative systems with multiple sources.

can be particularly diﬃcult in a cellular system that involves multiple relay stations, each serving a diﬀerent subset of users. Moreover, user mobility is also a major issue as cooperative links may not be stable if cooperating users/relays are mobile and handoﬀ between relay stations may be frequent. Other issues such as buﬀer management and cooperative queuing must be addressed to determine how the relay packets should be treated at each relay or cooperative user; security and authentication mechanisms must be developed to combat the eﬀect of malicious or unauthorized relays. In multihop networks, ﬁnding the optimal cooperative route is known to be NP-hard and, thus, eﬃcient cooperative routing protocols must be designed. Some of these issues will also be addressed in this book (c.f. Chapters 8 and 9), but research in these areas are still in the preliminary stage.

1.2 Brief History of Cooperative and Relay Channels The main idea behind cooperative communications stems from the study of relay channels, initially proposed by van der Meulen in [49, 50]. In these works, general three terminal communication problems have been formulated and bounds have been provided for the capacity of relay channels (i.e., the channel that consists of a source transmitting to a destination with the help of a relay). Later in 1979, Cover and El Gammal published further results on relay channels in [10], where signiﬁcantly improved inner and outer bounds were derived. This is considered the most prominent work on relay channels up till this date since many of the results still could not be superseded. The studies on relay channels has continued into the 1980s where eﬀorts to extend

6

1 Introduction

the results of [10] to multiple relays have been made in [4,14] and the capacity of semideterministic relay channels have been derived in [15]. However, since the early 1980s, the interest in this area has decreased considerably, possibly due to theoretical diﬃculties as well as practical challenges in implementation. In fact, after several decades, the fundamental performance limits (e.g., the capacity) of the relay channel still remain to be unknown. Interestingly, during this time, important progress has been made on related problems, such as broadcast channels (BC) [6, 11, 20, 21, 39], multiple access channels (MAC) [12,13,18], and multiterminal source coding [44,54]. In relay channels, one can view the transmission from the source to the relay and the destination as a BC while the transmission from the source and the relay to the destination can be viewed as a MAC. Moreover, since the data transmitted by the relay is correlated with that at the source, the relay channel problem is also intimately related with multiterminal source coding problems. Hence, the results obtained in these areas have contributed substantially to the understanding of relay channels today. In particular, the works on MAC with generalized feedback, e.g., [9, 32], are said to include relaying and cooperative communication as special cases since information at the relay can be considered as information provided through feedback. These works have set a solid foundation for more recent works in these ﬁelds, such as [8, 28, 48, 51], and has led to further advances in relay channels, e.g., [24, 35, 52, 55]. In earlier years, research on relay channels have appeared mostly in the information theory community, focusing on the fundamental limits of the channel. However, with the advances in MIMO [17, 45–47] and coding techniques [19, 38] and due to the increased feasibility of practical implementation, interest in the communication and signal processing community has grown exponentially in recent years. Speciﬁcally, the exploding interest in cooperative communications was initiated by the works of Sendonaris et al in [42] and of Laneman et al in [36, 37], and also, from the signal processing community, of the works by Scaglione and Hong in [22, 41]. Since then, relay and cooperative communication has quickly been identiﬁed as an eﬀective scheme to achieve spatial diversity and to extend the coverage of high data rates. Numerous relay and cooperative transmission schemes have been rapidly developed in the communication and signal processing community and studies in the high-layer issues, such as medium access control, routing, or quality-of-service, have been identiﬁed by the networking community.

1.3 Standardization of Cooperative Communication and Relay Technology Cooperative communication and relay technologies have gradually made its way towards wireless standards, such as IEEE 802.16j and Long Term Evolution (LTE)-Advanced. Speciﬁcally, IEEE 802.16j [26], an amendment to IEEE

1.3 Standardization of Cooperative Communication and Relay Technology

7

802.16e mobile Worldwide Interoperability for Microwave Access (WiMAX) standard [25], supports relay functionalities that can be used to increase the throughput of cell-edge users and extend system coverage to the interior of buildings, to temporary locations, and also to within mobile transportation vehicles [40]. Relay stations (RSs) can also forward messages to other RSs further away from the base station (BS) to form a multihop relay network. Depending on whether or not mobile stations (MSs) are aware of the relays’ existence, relays in the IEEE 802.16j standard can be categorized as transparent and nontransparent relays. Transparent relays are used when subscriber stations (SSs) are able to receive control information directly from the BS. Thus, the RSs are used purely to enhance throughput of the SSs without transmitting additional control information. On the other hand, nontransparent relays can be used to serve MSs that are located outside or near the edge of the BS coverage by serving as a virtual BS to the MSs. In this case, the RS must be able to transmit control information and synchronization messages to the MS. With transparent relays, bandwidth allocation must be achieved through centralized scheduling while, with nontransparent relays, this can be done distributively at the RSs. Moreover, transparent relays must communicate with the BS and the MS using the same frequency band while nontransparent relays may utilize diﬀerent frequency bands on the two links. Although more ﬂexibility is allowed with nontransparent relays, increased implementation complexity is also required. Furthermore, cooperative relaying functionalities that enable the BS and the RSs to transmit cooperatively to the MS have also been deﬁned in IEEE 802.16j. In particular, three types of cooperative transmission schemes have been speciﬁed, namely, the cooperative source diversity, the cooperative transmit diversity, and the cooperative hybrid diversity techniques. In the cooperative source diversity scheme, the cooperating transmitters will simultaneously transmit the same signal to the MS using the same time-frequency channel to enhance the SNR at the MS. In the cooperative transmit diversity scheme, transmitters will together form a distributed antenna array where space-time codes (previously deﬁned in 16e [25]) can be used across the distributed antennas. The cooperative hybrid diversity scheme, which is a combination of the above two schemes, also employs space-time codes across the distributed RSs (or the BS), but two or more antennas may be transmitting identical signals in this case. In addition to IEEE 802.16j, relaying and cooperative transmission technologies are also being incorporated into the recent LTE-Advanced standard [1] speciﬁed by the 3rd Generation Partnership Project (3GPP). Here, the goal is also to extend coverage of high-data rates and improve group mobility, facilitate temporary network deployment, and increase cell-edge throughput. The relays in LTE-Advanced can be classiﬁed into inband or outband relays depending on whether or not the communication to the base station (or called eNodeB) and to the end user (or called User Equipment, UE) are over the same frequency band. The relays can also be classiﬁed as

8

1 Introduction

transparent or nontransparent depending on whether or not the UEs are aware of their existence, similar to IEEE 802.16j. Moreover, depending on the relay strategy, relays can also be part of a donor cell or control cells of its own. Although the LTE-Advanced standard has not yet been ﬁnalized, there is consensus that at least the Type 1 relay will be incorporated into the system. A Type 1 relay is an inband relay node that controls its own cell and, thus, sends its own synchronization signals, reference symbols, scheduling information etc. This type of relay consists of its own Physical Cell ID and appears to the UE as a separate base station. In any case, by connecting relays wirelessly to the eNodeB, backhaul wireline connections to the radio access network are no longer necessary and the cost of coverage extension is signiﬁcantly reduced. Besides the relay functionality, LTE-Advanced also may incorporate Coordinated Multiple Point (CoMP) transmission and reception [1], where the eNodeBs cooperate in communicating to and from the UE, to improve coverage of high data rates and cell-edge throughput. Speciﬁcally, on the downlink, coordinated transmission strategies can be classiﬁed into Joint Processing (JP) and Coordinated Scheduling/Beamforming (CS/CB). With JP, the data destined for the UE is available at each point in the CoMP cooperating set (i.e., eNodeBs that participate in the coordinated transmission). Two types of JP transmission can be employed, namely, the Joint Transmission scheme, where data is transmitted simultaneously from multiple points of the cooperating set, e.g., to improve the received signal quality and/or cancel active interference from other data transmission, and the Dynamic Cell Selection scheme, where data is transmitted from one point at a time. The former scheme can be based on, e.g., distributed beamforming or space-time coding, while the latter scheme is based on antenna selection (c.f. Chapter 4). With CS/CB, data is only available at the serving cell, but scheduling or beamforming decisions can be made in coordination with other cells to avoid interference. Although CoMP does not involve relaying, cooperation is achieved by employing similar strategies as those in Chapters 4 and 6, except that the transmitting terminals are linked through wireline connections. Overall, regardless of the speciﬁc relay and/or cooperation strategy that will be employed in IEEE 802.16j and LTE-Advanced, these examples demonstrate the importance of relay and cooperative communication technology in next generation wireless networks.

1.4 Book Outline The aim of this book is to serve as the main reference for a graduate course on cooperative communications and also for engineers working in this area. While cooperative communication technology has evolved tremendously over

1.4 Book Outline

9

the years, we focus on describing the basic techniques that have been proposed and hopefully prepare readers for further studies in the ﬁeld. In Chapter 2, an introduction to wireless communications is provided, including brief reviews of fading channels, diversity techniques, channel capacity, and the concept of diversity-multiplexing tradeoﬀ (DMT). In Chapter 3, basic cooperative communication techniques for two-user systems are described. In these schemes, one user serves as the source while the other user serves as the relay that helps forward the source’s messages to the destination. Basic techniques such as decode-and-forward (DF), amplifyand-forward (AF), and coded cooperation (CC) schemes will be described in detail. The concept of compress-and-forward (CF) will be brieﬂy discussed while readers will be referred to Chapter 5 for more details. Channel estimation schemes for the single relay scenario will also be described. In Chapter 4, cooperative transmission schemes that involve multiple relays is described. Similar to the case with a single relay, we assume that, at each instant in time, only one user serves as the source while the other users can be viewed as a distributed antenna array that relays the sources message cooperatively to the destination. The DF and AF schemes can be extended to the multi-relay scenario by assuming the availability of orthogonal channels at each relay. When orthogonal channels do not exist, the distributed antenna array formed by the relays can be used to emulate many MIMO techniques subject to individual constraints at the relays or limitations due to lack of coordination. Depending on the level of channel state information (CSI) available at the source and relays, diﬀerent schemes such as transmit beamforming, selective relaying, or distributed space-time coding schemes can be employed. Channel estimation schemes for the case of multiple relays are also discussed. In addition, asynchronous cooperation schemes are also introduced to show how cooperation can still be helpful without full coordination among the relays. Furthermore, in the case where some relays are located beyond the transmission range of the source, multihop relaying schemes will also be discussed to demonstrate how cooperative transmissions can eﬀectively extend the coverage of wireless transmission over a wide area. In Chapter 5, information-theoretic aspects of the relay channel are introduced. The basic results on the relay channel capacity given by Cover and El Gamal will be reviewed and, then, some results on large relay networks will be given. Since one of the advantages of cooperative communications is to emulate the spatial diversity and multiplexing gains provided by conventional MIMO systems, we also discuss some results on the diversity-multiplexing tradeoﬀ (DMT) of cooperative relay networks. In Chapter 6, a cooperative relay system with multiple sources competing for the use of the cooperative channel is discussed. We discuss the cases where cooperative users operate under time-division multiple access (TDMA), frequency-division multiple access (FDMA), code-division multiple access (CDMA), and space-division multiple access (SDMA) schemes. In these cases, resource allocation among the diﬀerent users and between sources

10

1 Introduction

and relays are important to ensure high performance for all users. Moreover, in the case of CDMA and SDMA, eﬀective mitigation of multiple access interference is critical to preserve the spatial diversity gains obtained through cooperation. Finally, partner selection algorithms will also be introduced to show how users can eﬃciently select cooperating partners. In Chapter 7, the use of cooperative relaying on top of advanced communication systems such as OFDM and MIMO systems is discussed. In the case of multicarrier systems, frequency diversity can be exploited in addition to the spatial diversity gains by allowing users to cooperatively allocate their power across subcarriers. When perfect CSI is not available, space-frequency codes are introduced to exploit the diversity gains. Moreover, when multiple antennas are available at each terminal, further spatial diversity gains can be exploited in addition to the cooperative gains. The diﬀerence compared to single-antenna cooperative systems is that each terminal now has full control of a subset of antennas and, thus, lesser restrictions are imposed on the transmission schemes that can be developed. In Chapters 8 and 9, MAC, networking, and cross-layer issues regarding cooperative communications are discussed. Speciﬁcally, in Chapter 8, we show how cooperation in the physical layer may aﬀect MAC layer designs and what cooperative advantages can be exploited in the MAC layer. These issues are addressed with studies on the fundamental slotted ALOHA protocol, the CSMA/CA protocol, and centralized scheduling schemes. In Chapter 9, we address networking or cross-layer issues such as quality-of-service (QoS) considerations, cooperative routing for multihop networks, and security issues. It is shown that existing MAC or network layer protocols must be modiﬁed or redesigned to fully exploit the advantages of cooperation in the physical layer and to explore further advantages in the MAC layer and above.

References 1. 3rd General Partnership Project: Technical speciﬁcation group radio access network: Further advancements for E-UTRA physical layer aspects (Release 9). Tech. Rep. 36.814 (V9.0.0) (2010) 2. Abdallah, M.M., Papadopoulos, H.C.: Beamforming algorithms for information relaying in wireless sensor networks. IEEE Transactions on Signal Processing 56(10), 4772–4784 (2008) 3. Anghel, P., Leus, G., Kaveh, M.: Distributed space-time cooperative systems with regenerative relays. IEEE Transactions on Wireless Communications 5(11), 3130–3141 (2006) 4. Aref, M.R.: Information ﬂow in relay networks. Ph.D. thesis, Stanford University (1980) 5. Azarian, K., El Gamal, H., Schniter, P.: On the achievable diversity-multiplexing tradeoﬀ in half-duplex cooperative channels. IEEE Transactions on Information Theory 51(12), 4152–4172 (2005) 6. Bergmans, P.P., Cover, T.M.: Cooperative broadcasting. IEEE Transactions on Information Theory 20(3), 317–324 (1974)

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7. Bletsas, A., Khisti, A., Reed, D.P., Lippman, A.: A simple cooperative diversity method based on network path selection. IEEE Journal on Selected Areas in Communications 24(3), 659–672 (2006) 8. Caire, G., Shamai, S.: On the achievable throughput of a multiantenna Gaussian broadcast channel. IEEE Transactions on Information Theory 49(7), 1691–1706 (2003) 9. Carleial, A.B.: Multiple-access channels with diﬀerent generalized feedback signals. IEEE Transactions on Information Theory 28(6), 841–850 (1982) 10. Cover, T., El Gamal, A.: Capacity theorems for the relay channel. IEEE Transactions on Information Theory 25(5), 572–584 (1979) 11. Cover, T.M.: Broadcast channels. IEEE Transactions on Information Theory 18(1), 2–14 (1972) 12. Cover, T.M., El Gamal, A., Salehi, M.: Multiple access channels with arbitrarily correlated sources. IEEE Transactions on Information Theory 26(6), 648–657 (1980) 13. Cover, T.M., Leung, C.S.K.: An achievable rate region for the multiple-access channel with feedback. IEEE Transactions on Information Theory 27(3), 292–298 (1981) 14. El Gamal, A.: On information ﬂow in relay networks. In: Proceedings of IEEE National Telecommunications Conference, vol. 2, pp. D4.1.1–D4.1.4. Miami, FL (1981) 15. El Gamal, A., Aref, M.: The capacity of the semideterministic relay channel. IEEE Transactions on Information Theory IT-28(3), 536 (1982) 16. Fareed, M.M., Uysal, M.: On relay selection for decode-and-forward relaying. IEEE Transactions on Wireless Communications 8(7), 3341–3346 (2009) 17. Foschini, G.J.: Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas. Bell System Technical Journal 1, 41–59 (1996) 18. Gaarder, N., Wolf, J.: The capacity region of a multiple-access discrete memoryless channel can increase with feedback. IEEE Transactions on Information Theory 21(1), 100–102 (1975) 19. Gallager, R.G.: Low density parity check codes. Ph.D. thesis, Massachusetts Institute of Technology (1963) 20. Gel’fand, S.I., Pinsker, M.S.: Capacity of a broadcast channel with one deterministic component. Problemy Peredachi Informatsii 16(1), 24–34 (1980) 21. Han, T.S., Costa, M.H.M.: Broadcast channels with arbitrarily correlated sources. IEEE Transactions on Information Theory 33(5), 641–650 (1987) 22. Hong, Y.-W., Scaglione, A.: Energy-eﬃcient broadcasting with cooperative transmissions in wireless sensor networks. IEEE Transactions on Wireless Communications 5(10), 2844–2855 (2006) 23. Hong, Y.-W. P., Lin, C.-K., Wang, S.-H.: Exploiting cooperative advantages in slotted ALOHA random access networks. to appear in IEEE Transactions on Information Theory (2010) 24. Host-Madsen, A., Zhang, J.: Capacity bounds and power allocation for wireless relay channels. IEEE Transactions on Information Theory 51(6), 2020–2040 (2005) 25. IEEE Standard 802.16e-2005: IEEE Standard for Local and Metropolitan Area Networks - Part 16: Amendment for Physical and Medium Access Control Layers for Combined Fixed and Mobile Operation in Licensed Bands (2005) 26. IEEE Standard 802.16j-2009: IEEE Standard for Local and Metropolitan Area Networks - Part 16: Air interface for broadband wireless access systems – Multihop relay speciﬁcation (2009) 27. Janani, M., Hedayat, A., Hunter, T.E., Nosratinia, A.: Coded cooperation in wireless communications: Space-time transmission and iterative decoding. IEEE Transactions on Signal Processing 52(2), 362–371 (2004) 28. Jindal, N., Vishwanath, S., Goldsmith, A.: On the duality of Gaussian multiple-access and broadcast channels. IEEE Transactions on Information Theory 50(5), 768–783 (2004) 29. Jing, Y., Hassibi, B.: Distributed space-time coding in wireless relay networks. IEEE Transactions on Wireless Communications 5(12), 3524–3536 (2006)

12

1 Introduction

30. Jing, Y., Jafarkhani, H.: Distributed diﬀerential space-time coding for wireless relay networks. IEEE Transactions on Communications 56(7), 1092–1100 (2008) 31. Jing, Y., Jafarkhani, H.: Network beamforming using relays with perfect channel information. IEEE Transactions on Information Theory 55(6), 2499–2517 (2009) 32. King, R.C.: Multiple access channels with generalized feedback. Ph.D. thesis, Stanford University (1978) 33. Koyuncu, E., Jing, Y., Jafarkhani, H.: Distributed beamforming in wireless relay networks with quantized feedback. IEEE Journal on Selected Areas in Communications 26(8), 1429–1439 (2008) 34. Kramer, G., Gastpar, M., Gupta, P.: Capacity theorems for wireless relay channels. In: Proc. 41st Annu. Allerton Conf. Communications, Control, and Computing, pp. 1074–1083. Monticello, IL (2003) 35. Kramer, G., Gastpar, M., Gupta, P.: Cooperative strategies and capacity theorems for relay networks. IEEE Transactions on Information Theory 51(9), 3037–3063 (2005) 36. Laneman, J.N., Tse, D.N.C., Wornell, G.W.: Cooperative diversity in wireless networks: Eﬃcient protocols and outage behavior. IEEE Transactions on Information Theory 50(12), 3062–3080 (2004) 37. Laneman, J.N., Wornell, G.W.: Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks. IEEE Transactions on Information Theory 49(10), 2415–2425 (2003) 38. MacKay, D.J.C.: Good error-correcting codes based on very sparse matrices. IEEE Transactions on Information Theory 45(2), 399–432 (1999) 39. Marton, K.: A coding theorem for the discrete memoryless broadcast channel. IEEE Transactions on Information Theory 25(3), 306–311 (1979) 40. Peters, S.W., Heath, Jr., R.W.: The future of WiMAX: Multihop relaying with IEEE 802.16j. IEEE Communications Magazine pp. 104–111 (2009) 41. Scaglione, A., Hong, Y.-W.: Opportunistic large arrays: Cooperative transmission in wireless multihop ad hoc networks to reach far distances. IEEE Transactions on Signal Processing 51(8), 2082–2092 (2003) 42. Sendonaris, A., Erkip, E., Aazhang, B.: User cooperation diversity–Part I: System description” and “User cooperation diversity–Part II: implementation aspects and performance analysis. IEEE Transactions on Communications 51(11) 1927–1938 and 1939–1948 (2003) 43. Sirkeci-Mergen, B., Scaglione, A.: Randomized space-time coding for distributed cooperative communication. IEEE Transactions on Signal Processing 55(10), 5003–5017 (2007) 44. Slepian, D., Wolf, J.: Noiseless coding of correlated information sources. IEEE Transactions on Information Theory 19(4), 471–480 (1973) 45. Tarokh, V., Jafarkhani, H., Calderbank, A.: Space-time block codes from orthogonal designs. IEEE Transactions on Information Theory 45(5), 1456–1467 (1999) 46. Tarokh, V., Seshadri, N., Calderbank, A.: Space-time codes for high data rate wireless communication: performance criterion and code construction. IEEE Transactions on Information Theory 44(2), 744–765 (1998) ˙ 47. Telatar, I.E.: Capacity of multi-antenna Gaussian channels. European Transactions on Telecommunications 10(6), 585–595 (1999) 48. Tse, D.N.C., Viswanath, P., Zheng, L.: Diversity-multiplexing tradeoﬀ in multipleaccess channels. IEEE Transactions on Information Theory 50(9), 1859–1874 (2004) 49. van der Meulen, E.C.: Transmission of information in a T-terminal discrete memoryless channel. Ph.D. thesis, Department of Statistics, University of California, Berkeley, CA (1968) 50. van der Meulen, E.C.: Three-terminal communication channels. Advances in Applied Probability 3(1), 120–154 (1971) 51. Vishwanath, S., Jindal, N., Goldsmith, A.: Duality, achievable rates, and sum-rate capacity of Gaussian MIMO broadcast channels. IEEE Transactions on Information Theory 49(10), 2658–2668 (2003)

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52. Wang, B., Zhang, J., Host-Madsen, A.: On the capacity of MIMO relay channels. IEEE Transactions on Information Theory 51(1), 29–43 (2005) 53. Winters, J.H.: On the capacity of radio-communication systems with diversity in a Rayleigh fading environment. IEEE Journal on Selected Areas in Communications 5(5), 871–878 (1987) 54. Wyner, A., Ziv, J.: The rate-distortion function for source coding with side information at the decoder. IEEE Transactions on Information Theory 22(1), 1–10 (1976) 55. Xie, L.-L., Kumar, P.R.: An achievable rate for the multiple-level relay channel. IEEE Transactions on Information Theory 51(4), 1348–1358 (2005) 56. Yiu, S., Schober, R., Lampe, L.: Distributed space-time block coding. IEEE Transactions on Communications 54(7), 1195–1206 (2006) 57. Yuksel, M., Erkip, E.: Multiple-antenna cooperative wireless systems: a diversity- multiplexing tradeoﬀ perspective. IEEE Transactions on Information Theory 53(10), 3371–3393 (2007) 58. Zhao, Y., Adve, R., Lim, T.: Improving amplify-and-forward relay networks: optimal power allocation versus selection. In: Proceedings on the IEEE International Symposium on Information Theory (ISIT), pp. 1234–1238 (2006) 59. Zheng, L., Tse, D.N.C.: Diversity and multiplexing: A fundamental tradeoﬀ in multiple-antenna channels. IEEE Transactions on Information Theory 49(5), 1073– 1096 (2003)

Chapter 2

Review of Wireless Communications and MIMO Techniques

To facilitate the understanding of the cooperative strategies introduced throughout this book, we provide in this chapter a brief review of wireless communications and MIMO techniques. First, we introduce some basic characteristics of the wireless environment, including path loss, shadowing, and multipath fading, and describe popular diversity techniques that are often used to combat these channel eﬀects. Then, we review the fundamental limits of Gaussian AWGN channels, fading channels, and multiple-input multiple-output (MIMO) channels. Finally, we conclude with a discussion on the diversity-and-multiplexing tradeoﬀ (DMT) for MIMO systems. More detailed discussions on these topics can be found in [11] and [37].

2.1 Characteristics of Wireless Channels The development of wireless communications have progressed tremendously over the past few decades due to advances in wireless hardware technology and also because of the large demand for mobile access. Compared with conventional wireline communications, the signal transmitted over wireless channels may suﬀer dramatic attenuation, large delays, and severe signal distortion. These non-ideal eﬀects are often characterized by three factors, i.e., path loss, shadowing, and multi-path fading. In this section, we will brieﬂy describe what these factors are and their eﬀect on the performance of wireless communication systems.

2.1.1 Path Loss Consider a point-to-point wireless communication system where a transmitter communicates with a receiver by sending an electromagnetic signal through

Y.-W. Peter Hong et al., Cooperative Communications and Networking, DOI 10.1007/978-1-4419-7194-4_2, © Springer Science+Business Media, LLC 2010

15

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2 Review of Wireless Communications and MIMO Techniques

the wireless medium. The strength of the signal attenuates as it traverses the medium and, thus, becomes weaker as the propagation distance increases. The amount of degradation in the signal strength with respect to the distance can be characterized by the ratio between the transmit power, Pt , and the receive power, Pr , which is denoted by PL =

Pt . Pr

This ratio is used to quantify the eﬀect of path loss and the value may depend on the geographic environment as well as certain radio properties, such as the spaciousness of the environment, the transmission distance, the radio wavelength, heights of the transmitter and receiver, ... etc. The path loss is usually represented in the decibel scale, i.e., PL(dB) = 10 log10

Pt . Pr

(2.1)

Several path loss models are given in the following.

Free Space Propagation When the radio propagates in free space, a line-of-sight (LOS) path may exist between the transmitter and the receiver, while the signal reﬂections may be negligible. In this case, the strength of the received signal can be expressed as [21] 2 λc Pr = Pt Gt Gr , (2.2) 4πd where d is the geographic distance between the transmitter and the receiver, Gt and Gr are the antenna gains at the transmitter and receiver, respectively, and λc is the wavelength of the electromagnetic signal. Here, the power of the received signal decays with the square of the transmission distance d.

Two-Ray Model In addition to free space propagation, we can also consider a slightly more realistic model that consists of one LOS path and one path obtained through ground reﬂection, as shown in Fig. 2.1. Since two propagation paths are considered, the model is often referred to as the two-ray model. Let us denote by d the distance between the transmitter and the receiver, and denote by ht and hr the eﬀective heights of the transmit and receive antennas, respectively. Moreover, as shown in Fig. 2.1, we can deﬁne dLOS as the length of the LOS path, dG as the distance between the transmit antenna and the reﬂecting point, and dG as the distance between the reﬂecting point and the receive

2.1 Characteristics of Wireless Channels

17

Fig. 2.1 Two-ray propagation model.

antenna. Assume that the distance d is much greater than the heights of the transmit and receive antennas (i.e., d ht , hr ) so that d ≈ dLOS ≈ dG + dG and so that the received signal power can be approximated as [23, 27] Pr ≈

ht hr d2

2 Gt Gr Pt .

(2.3)

Due to the interference between the LOS and the reﬂection paths, the received signal power decays with the fourth power of the distance d when the distance between the transmitter and the receiver is suﬃciently large.

A General Path-Loss Model To facilitate analytical studies on wireless communication systems, it is often convenient to use simpler path loss models, with the help of experimental data to characterize key parameters. Speciﬁcally, a simple path loss model often adopted in practice is the following: Pr = Pt K

d d0

−α ,

(2.4)

where d0 is the reference distance, K is a constant related to the antenna gain and the average channel attenuation, and α is the path-loss exponent. The constant K can be obtained from the empirical average of the receive power at the reference distance d0 . Due to the scattering phenomena in the antenna near ﬁeld, the simpliﬁed model in (2.4) is valid only when the distance d is greater than the reference distance d0 . The reference distance is usually 1-10 meters indoors and 10-100 meters outdoors [11]. The value of the path loss exponent α depends on the propagation environment and usually ranges between 2 and 6. The actual value of the path-loss exponent is obtained by

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2 Review of Wireless Communications and MIMO Techniques

ﬁtting the prescribed model to the empirical data. The parameters for several typical environments are listed as follows [9, 11, 20, 23, 30, 35, 36]. • • • • • • • •

Free space: α = 2 Urban macrocells: 3.7 ≤ α ≤ 6.5 Urban microcells: 2.7 ≤ α ≤ 3.5 Oﬃce building (same ﬂoor): 1.6 ≤ α ≤ 3.5 Oﬃce building (multiple ﬂoors): 2 ≤ α ≤ 6 Store: 1.8 ≤ α ≤ 2.2 Factory: 1.6 ≤ α ≤ 3.3 Home: α ≈ 3

The simpliﬁed path loss model can be expressed in decibel scale by d Pr(dBm) = Pt(dBm) + K(dB) − 10α log10 . d0

(2.5)

More details of the modeling and of the eﬀect of the path loss phenomena in wireless communications can be found in [11, 23].

2.1.2 Shadowing Eﬀect In addition to the power loss caused by free-space attenuation, the radio waves may also be distorted by the obstacles that appear along the transmission paths. These obstacles may absorb part of the signal energy, resulting in signal strength degradation or cause random scattering. The eﬀects may vary slowly over time due to the relative motion between the transmitter, the receiver, and near-by obstacles along the propagation path, such as buildings, trees, vehicles, or airplanes. This slow-varying power variation is called the shadowing eﬀect and is considered as a type of large-scale (or macroscale) fading. Based on the experimental measurements, the variation on the receive signal strength can be modeled as a log-normal random variable with probability density function (PDF) given by [2, 7] (10 log10 ψ − μψdB )2 ξ fψ (ψ) = √ , ψ > 0, (2.6) exp − 2σψ2 dB 2πσψdB ψ where ξ = 10/ ln(10) is a constant. The received signal strength in decibel, i.e., ψdB 10 log10 ψ, is a Gaussian random variable with mean μψdB and variance σψ2 dB . To jointly consider both the path loss and shadowing eﬀects, we may simply combine the log-normal distributed shadowing eﬀect with the average path loss as Pr d − ψdB . = 10 log10 K − 10α log10 (2.7) Pt (dB) d0

2.1 Characteristics of Wireless Channels

19

Power loss: Pr/Pt (dB)

Path Loss Shadowing Effect Multipath Fading

Distance between Tx and Rx: log(d/d0) Fig. 2.2 Illustration of path loss, shadowing eﬀect and multipath fading.

2.1.3 Multipath Fading In wireless communication systems, signals are transmitted through environments full of reﬂectors or scatters and, thus, the signals eventually arriving at the receiver are often superpositions of signals from multiple propagation paths. The signals arriving from diﬀerent paths may add up either constructively or destructively at the receiver and, thus, the signal strength may ﬂuctuate rapidly over time, space, and frequency. This is the so-called multipath fading eﬀect, which results in fast and small-scale (or microscale) amplitude and phase distortion. The various eﬀects of path loss, shadowing, and multipath fading on the received signal strength is illustrated in Fig. 2.2. We can see that the long-term average receive power decreases with the propagation distance while large-scale and small-scale perturbation on the receive power also exist due to shadowing and multipath fading, respectively.

Continuous-Time Passband Model In general, fading channels in wireless systems can be modeled as linear timevarying systems. Let us consider a rich scattering environment where the received signal can be modeled as a superposition of signals received from

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2 Review of Wireless Communications and MIMO Techniques

diﬀerent reﬂecting or scattering path. We assume that the transmitter, the receiver and/or the reﬂectors and scatters in the environment are in motion and, thus, the number of propagation paths, the signal propagation distances, and the channel gains may all vary with time. Let x(t) be the signal transmitted by the transmitter. The time-varying signal observed by the receiver can be expressed as

N (t)

y(t) =

ai (t)x(t − τi (t)),

(2.8)

i=1

where N (t) is the number of propagation paths for the transmitted signal at time t, and ai (t) and τi (t) are the attenuation and the propagation delay of the i-th path at time t. Here, the received signal can be expressed as the output of a linear time-varying system where ∞ y(t) = h(τ, t)x(t − τ )dτ. (2.9) −∞

The channel impulse response is given by

N (t)

h(τ, t) =

ai (t)δ(τ − τi (t)),

(2.10)

i=1

and the channel frequency response is given by

N (t)

H(f, t) =

ai (t)e−j2πf τi (t) .

(2.11)

i=1

The attenuation and delay depend on the propagation distance of each path.

Continuous-Time Baseband Equivalent Model In practical wireless systems, the radio spectrum is typically divided into parallel sub-bands and each transmitter’s signal must be limited to a certain frequency band, say [fc − W/2, fc + W/2], where fc is the center frequency and W is called the signal bandwidth. For example, the Global System for Mobile Communications (GSM) standard uses 890 − 915 MHz band for the cellular uplink (i.e., mobile to base station) but allocates bandwidth of only W = 200 KHz to each mobile’s transmissions. The signals are “up-converted” to the center frequency fc before transmission and is “down-converted” back to baseband (fc = 0) at the receiver in order to perform further processing. Since the up-conversion and down-conversion of signals do not aﬀect the baseband signal processing conducted at the transceivers, the cooperative signal processing techniques discussed in later chapters will mostly be introduced

2.1 Characteristics of Wireless Channels

21

Fig. 2.3 Illustrations of the power spectral density of a bandpass signal and its corresponding complex baseband equivalent signal.

using equivalent baseband signal models. The conversion from a continuoustime passband signal model to a continuous-time baseband equivalent model can be described as follows. Let x(t) be a narrowband bandpass signal with bandwidth W and center frequency fc , which can be expressed as √ x(t) = 2Re xb (t)ej2πfc t , (2.12) where xb (t) is the complex baseband equivalent signal of x(t). The baseband equivalent signal xb (t) is obtained by down-converting the positive portion of the spectrum of x(t) to the baseband. Therefore, in the frequency domain, the power spectral density (PSD) of xb (t) can be expressed as

√ 2S(f + fc ), f + fc ≥ 0, (2.13) Sb (f ) = 0, f + fc < 0, where S(f ) is the power spectral density (PSD) of x(t). The relation between the two PSDs S(f ) and Sb (f ) are illustrated in Fig. 2.3. By substituting (2.12) into (2.8), we can obtain ⎫ ⎡⎧ ⎤ (t) ⎨N ⎬ √ y(t) = 2Re ⎣ ai (t)xb (t − τi (t))e−j2πfc τi (t) ej2πfc t ⎦ , (2.14) ⎩ ⎭ i=1

and express the baseband equivalent version of received signal as

N (t)

yb (t) =

ai,b (t)xb (t − τi (t)),

(2.15)

i=1

where ai,b (t) = ai (t)e−j2πfc τi (t) is the complex attenuation of the i-th path.

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2 Review of Wireless Communications and MIMO Techniques

Discrete-Time Baseband Equivalent Model In designing signal processing techniques at the transceivers, it is often more convenient to consider the discrete-time signal model, which is typically obtained by sampling the continuous-time baseband signal. When xb (t) is bandlimited to W/2, the baseband continuous-time signal can be uniquely represented by the samples obtained at a sampling rate of W or higher. That is, given the discrete-time signal x[n] = xb (n/W ), for all n, the continuous-time signal can be reconstructed as xb (t) = x[n]sinc(W t − n), (2.16) n

where sinc(t) = sin(πt)/(πt). Consequently, the equivalent baseband model for the output signal can also be written in terms of the samples {x[n]} as yb (t) =

N (t)

x[m]

m

ai,b (t)sinc(W t − W τi (t) − m).

(2.17)

i=1

The sample of yb (t) at time t = n/W can be expressed by x[m] ai,b (n/W )sinc(n − m − W τi (n/W )) y[n] yb (n/W ) = m

=

i

x[n − ]

ai,b (n/W )sinc( − W τi (n/W ))

i

x[n − ]h [n],

(2.18)

where {h [n], ∀ } is the impulse response of the discrete-time baseband equivalent channel observed at time t = n/W with the -th tap given by

N (t)

h [n] =

ai,b (n/W )sinc( − W τi (n/W )).

(2.19)

i=1

Note that equation (2.18) can be also treated as a convolution between the time-varying channel sequence {h [n]} and the signal sequence {x[n]}.

Channel Coherence Time and Coherence Bandwidth Two important channel parameters have often been used to describe the temporal and spectral characteristics of the fading channel, namely, coherence time and coherence bandwidth. These parameters can be used to describe how fast the channel statistics vary over time and frequency, respectively.

2.1 Characteristics of Wireless Channels

23

Speciﬁcally, suppose that the time-domain and the frequency-domain representations of the baseband equivalent channel impulse response can be expressed as N (t) hb (τ, t) = ai,b (t)δ(τ − τi (t)), (2.20) i=1

and

N (t)

Hb (f, t) =

ai,b (t)e−j2πf τi (t)

(2.21)

i=1 N (t)

=

ai (t)e−j2πfc τi (t) e−j2πf τi (t) ,

(2.22)

i=1

respectively. The phase of each path (e.g., φi,b (t) = 2πfc τi (t) for the i-th path) can vary dramatically even for a small variation in the propagation delay τi (t) since the center frequency fc is typically large. The time-variation of τi (t) causes the phase φi,b (t) to change, resulting in a frequency-shift which we refer to as the Doppler eﬀect. The Doppler shift of the i-th path is then deﬁned as the amount of frequency-shift that occurs due to this time variation, i.e., 1 d Di φi,b (t). 2π dt By considering the case where dτi (t)/dt = vi /c, where c is the velocity of light (i.e., the case where the propagation length of the i-th path varies with velocity vi over time), the Doppler shift is given by Di = fc vi /c. In multipath environments, the channel may contain multiple paths that experience diﬀerent frequency shifts. Hence, we can deﬁne the so-called Doppler spread as the maximum diﬀerence between the Doppler shifts of diﬀerent paths, i.e., Ds = max |Di − Dj |. i,j

(2.23)

Since the phase of the i-th path, i.e., φi,b (t), rotates by π/2 every 1/(4Di ) seconds, we can say that the impulse response hb (τ, t), which consists of multiple paths, may vary considerably over each interval of duration 1/(4Ds ) seconds. In other words, we can also say that the equivalent channel impulse response remains roughly the same over an interval that is much shorter than 1/(4Ds ) seconds. The coherence time of the channel is then deﬁned as Tc = 1/(4Ds ). In practical systems, messages are often encoded into blocks of symbols and the encoding often depends on whether or not the channel varies over the symbol block. When the coherence time Tc is much larger than the transmission time of a symbol block, we say that the system is experiencing slow

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2 Review of Wireless Communications and MIMO Techniques

fading (or sometimes called block fading). In this case, the channel coeﬃcient h [n] in (2.18) is often treated as a constant h throughout the symbol block. When Tc is much smaller than the transmission time of the symbol block, we say that the system is experiencing fast fading, in which case the channel coeﬃcients may vary over each symbol period. By observing the channel frequency response in (2.22), one can see that, at any given time t, the phase diﬀerence between the i-th path and j-th path at frequencies, say f1 and f2 , can vary by π when |f1 − f2 | = 1/(2|τi (t) − τj (t)|). This shows that, as the multiple paths are more spread out in time, the faster the frequency response will vary over diﬀerent frequencies. Let us deﬁne the delay spread as the maximum diﬀerence between the propagation delays of diﬀerent paths, i.e., Td = max |τi (t) − τj (t)|. (2.24) i,j

In this case, we can say that the frequency response of the channel will have notable diﬀerences over frequencies separated by 1/(2Td). Therefore, the coherent bandwidth can be deﬁned as Wc = 1/(2Td). Based on the channel variation characteristics in the frequency domain, we say that the channel is ﬂat fading if the coherence bandwidth is much greater than the signal bandwidth, i.e., Wc W , and say that it is frequency selective, otherwise. In the case of ﬂat fading, the discrete-time channel (2.18) can be simpliﬁed by a single tap so that the received signal can be expressed as y[n] = h[n]x[n]. (2.25) In the case of frequency selective fading, the channel is represented by multiple taps as given in (2.18).

Statistical Models for Fading Coeﬃcients Due to the large amount of uncertainties in the environment, it is convenient to model the channel statistically with a distribution given to each tap of the channel, i.e., the coeﬃcients {h [n]} in (2.18). For simplicity, we assume in this section that the channel taps are independent and identically distributed (i.i.d.). More general models that consider correlation among channel taps can be found in [16]. Speciﬁcally, multipath fading in a rich scattering environment can be viewed as a collection of a large number of independent non-line-of-sight (NLOS) components. If the LOS path does not exist between the transmitter and receiver, each tap of the discrete channel impulse response in (2.19) will be a superposition of a large number of i.i.d. complex random variable. Thus, by central limit theorem, the real and imaginary parts of the channel coeﬃcients can be approximated as i.i.d. zero-mean Gaussian

2.2 Techniques to Exploit Spatial Diversity

25

random variables. In this case, the envelope of the received signal will be Rayleigh distributed with PDF given by 2 2x x f (x) = exp − , x ≥ 0, (2.26) Ω Ω where Ω is the average received power determined by the path loss and shadowing phenomenon, and the power of the received signal will be exponentially distributed with mean Ω. If there exists a strong LOS path, the real and imaginary parts of the channel coeﬃcients will again be i.i.d. Gaussian distributed but with mean that is no longer zero. Thus, the amplitude of the received signal is Rician distributed with PDF [24] 2x(K + 1) K(K + 1) (K + 1)x2 f (x) = exp −K − I0 2x , x ≥ 0, Ω Ω Ω (2.27) where Ω is the averaged received power in Rician fading, K is the Rician factor, and I0 (x) is the modiﬁed Bessel function of the ﬁrst kind of zeroth order deﬁned by 2π 1 I0 (x) = ex cos θ dθ. 2π 0 A more general channel model that better ﬁts the experimental measurements of diﬀerent environments is the Nakagami fading model [19]. In the so-called Nakagami-m fading channel, the envelop of the received signal is given by 2mm x2m−1 mx2 f (x) = , x ≥ 0, (2.28) exp − Γ(m)Ωm Ω where m ≥ 0.5 is the fading parameter, Γ is the Gamma function. When m = 1, Nakagami distribution is reduced to Rayleigh distribution and, when m = ∞, it is equivalent to the case without fading, i.e., a deterministic channel. In this book, we use Rayleigh fading as the representative scenario in many analysis on cooperative systems. Many of these analysis have also been conducted with Nakagami or other fading models in the literature. Readers will be referred to the appropriate literature when these topics arise (c.f. Chapter 4).

2.2 Techniques to Exploit Spatial Diversity In wireless systems, transmission failures occur mostly when the channel is in deep fade, resulting in the so called communication outage. To overcome this eﬀect, one can exploit diﬀerent kinds of diversity techniques in space,

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2 Review of Wireless Communications and MIMO Techniques

Fig. 2.4 Illustration of a SIMO system.

time, and frequency. In this section, we focus our discussions on the concept of spatial diversity since it is fundamental to the so-called cooperative diversity, which is to be discussed throughout this book. Spatial diversity can be exploited when the transmitter or the receiver have multiple antennas. Let us consider the ﬂat-fading scenario, where no inter-symbol interference occurs over time, and assume that the antennas are located suﬃciently apart from each other so that the channel coeﬃcients between diﬀerent transmitter and receiver antennas are statistically independent. Spatial diversity gains can be achieved with either precoding at the transmitter or signal combining at the destination. In the following, we will introduce these techniques for the three diﬀerent scenarios, namely, single-input multiple-output (SIMO), multipleinput single-output (MISO), and multiple-input multiple-output (MIMO) systems. These techniques can also be extended to systems with diversity in other dimensions, such as the case of multipath or frequency selectivity.

2.2.1 Single-Input Multiple-Output (SIMO) System When the receiver is equipped with multiple antennas, we can take advantage of spatial diversity at the receiver to enhance system performance. Consider the system where a single-antenna transmitter sends data to a receiver with Nr antennas, as shown in Fig. 2.4. Let x[n] be the symbol transmitted in the n-th symbol period and assume that E |x[n]|2 = 1. The signal received at the k-th antenna of the receiver in the n-th symbol period can be expressed as √ yk [n] = P hk x[n] + wk [n], (2.29) where P is the transmit power, hk is the channel coeﬃcient observed by the k-th receive antenna, and wk [n] ∼ CN (0, σk2 ) is the additive white Gaussian

2.2 Techniques to Exploit Spatial Diversity

27

Fig. 2.5 Illustration of a linear combiner in a SIMO system.

noise (AWGN) at the k-th antenna. The channel coeﬃcient hk can be written in terms of its amplitude |hk | and phase φk so that, for k = 1, 2, · · · , Nr , hk = |hk |ejφk . The SNR at the k-th antenna is then deﬁned as γk

P |hk |2 . σk2

(2.30)

Suppose that the instantaneous channel state information (CSI), i.e., the set of channel coeﬃcients {h1 , h2 , · · · , hNr }, is known at the receiver. Before performing signal detection, the receiver will linearly combine the received symbols y1 [n], y2 [n], · · · , yNr [n] with the respective weighting factors α1 , α2 , · · · , αNr , as shown in Fig. 2.5, to obtain the signal z[n] =

Nr

αk yk [n].

k=1

The values of weighting factors can be determined according to diﬀerent signal combining techniques [3–5, 22]. Some common techniques are introduced in the following.

Equal-Gain Combining (EGC) One technique that is often considered is the equal-gain combiner (EGC), where the signals received on the Nr antennas are each multiplied by a complex weighting factor that compensates for the phase rotation of the channel. The complex weighting factors are given by

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2 Review of Wireless Communications and MIMO Techniques

αk = e−jφk ,

for k = 1, 2, · · · , Nr .

(2.31)

This technique achieves phase coherence at the receiver and, thus, increases considerably the received signal strength. Note that the magnitudes of the weighting factors |α1 |,|α2 |,· · · , |αNr | are the same and do not depend on the SNR values of all links. This reduces the complexity of the scheme compared to the maximal-ratio combining (MRC) scheme to be introduced later on. The output of the equal-gain combiner (EGC) is given by zEGC [n] =

Nr

αk yk [n]

k=1

=

Nr k=1

√ = P

√

e−jφk N r

P |hk |ejφk x[n] + wk [n]

|hk | x[n] +

k=1

Nr

e−jφk wk [n].

(2.32)

k=1

Hence, the resulting SNR at the output of the equal-gain combiner can be given by 2 √ Nr |h | x[n] E P k k=1 γEGC = 2 Nr −jφk E k=1 e wk [n] P =

Nr k=1

Nr

2

|hk |

E[|x[n]|2 ]

E[|e−jφk wk [n]|]2 2 Nr P k=1 |hk | = . Nr 2 k=1 σk k=1

(2.33)

The outage probability of a SIMO system with EGC is demonstrated in Fig. 2.6 for the case of Rayleigh fading. The PDF of γEGC does not yield a closed-form in this case and, thus, the outage probability can only be evaluated numerically. In the computer simulation, the channel coeﬃcients are assumed to be i.i.d. circularly symmetric Gaussian random variables with zero mean and unit variance. The noise variances at diﬀerent receive anten2 nas are assumed to be identical so that σk2 = σw , ∀k. The SNR on the x-axis 2 is deﬁned as SNR P/σw . The threshold to determine outage events is set as γ0 = 1. We can see, in Fig. 2.6, that, due to co-phasing of the received signals, the decay rate of the outage probability will increase with the number of receive antennas, exploiting the spatial diversity at the receiver.

2.2 Techniques to Exploit Spatial Diversity

29

0

10

Nr=1 N =2 r

N =4

−1

r

10

Nr=8 N =10

Outage Probability

r

−2

10

−3

10

−4

10

0

5

10

15 SNR (dB)

20

25

30

Fig. 2.6 Outage probabilities of the equal-gain combining scheme.

Selection Combining (SC) Although the EGC scheme increases the SNR considerably at the receiver by co-phasing the signals, it is often diﬃcult to achieve in practice since the phase of the channel varies rapidly over time and is diﬃcult to track. When the signals cannot be perfectly co-phased, their addition may result in destructive interference and loss in spatial diversity. An alternative approach is to adopt the selection combining (SC) scheme where only the signal with the highest SNR among those received on the multiple antennas is utilized for detection. In this case, the weighting factors can be expressed as

1, if γk > γk , ∀k = k, αk = (2.34) 0, otherwise, where γk P |hk |2 /σk2 . Only the weight associated with the antenna with the highest SNR is equal to 1 while that of other antennas are equal to 0. The resulting SNR at the selection combiner (SC) output is equal to γSC =

max

k=1,··· ,Nr

γk .

(2.35)

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2 Review of Wireless Communications and MIMO Techniques

For example, let us consider the case with Rayleigh fading where the channel coeﬃcients are assumed to be i.i.d. complex Gaussian with mean 0 and variance σh2 , i.e., hk ∼ CN (0, σh2 ), for all k, and the channel gains |hk |2 , for all k, are assumed to be exponentially distributed with mean 2 2 E[|hk |2 ] = σh2 . By letting σ12 = · · · = σN = σw , the received SNR at the r 2 2 k-th antenna, i.e., γk = P |hk | /σw , will also be exponentially distributed 2 with mean γ = P σh2 /σw . The PDF is given by fγk (u) =

1 − uγ e , γ

u ≥ 0, for k = 1, 2, · · · , Nr .

The outage probability, i.e., the probability that γSC falls below the required SNR at the receiver, say γ0 , can be computed as Pr (γSC

Nr ≤ γ0 ) = Pr max γk ≤ γ0 = Pr (γk ≤ γ0 ) k

k=1

Nr = 1 − e−γ0 /γ k=1

Nr . = 1 − e−γ0 /γ

(2.36)

2 1, the outage When the transmit SNR is suﬃciently high, i.e., γ = P σh2 /σw probability can be approximated as

Pr (γSC ≤ γ0 ) ≈

γ0 γ

Nr

= γ0Nr e−Nr log γ .

By deﬁning the diversity order as d − lim

γ→∞

log Pr (γ ≤ γ0 ) , log γ

we can see that the SC scheme achieves diversity order of Nr , which is the number of receive antennas. In Fig. 2.7, we show the outage probability of the SIMO system with SC, where the simulation parameters are given as in Fig. 2.6. We can indeed observe that the diversity order of Nr is achieved at high SNR.

Maximal-Ratio Combining (MRC) Although EGC and SC utilize channel state information (CSI) to determine their weighting factors, the weighting factors applied in these schemes are not optimized in any sense. To fully exploit the spatial diversity provided by the multiple receive antennas, it is desirable to choose weighting factors that

2.2 Techniques to Exploit Spatial Diversity

31

0

10

Nr=1 N =2 r

N =4

−1

r

10

Nr=8 N =10

Outage Probability

r

−2

10

−3

10

−4

10

0

5

10

15 SNR (dB)

20

25

30

Fig. 2.7 Outage probabilities of the selection combining scheme.

maximize the receive SNR, which in order minimizes the outage probability. The scheme that achieves this task is referred to as the maximal-ratio combining (MRC) scheme. Speciﬁcally, given the instantaneous CSI, the weighting factors of the maximal-ratio combiner (MRC) can be given by αk = h∗k /σk2 = |hk |e−jφk /σk2 , for k = 1, . . . , Nr .

(2.37)

Here, the signals are weighted according to their local channel quality and are co-phased to achieve phase-coherent addition of signals at the receiver. The output of the MRC is given by zMRC [n] =

Nr k=1

√ = P

αk

√ P hk x[n] + wk [n]

N r

αk hk

x[n] +

k=1

√ = P

N r |hk |2 k=1

The resulting SNR is computed as

σk2

Nr

αk wk [n]

k=1

x[n] +

Nr h∗

k

k=1

σk2

wk [n].

(2.38)

32

2 Review of Wireless Communications and MIMO Techniques

2

N r

γMRC =

N r

2

P |hk | |hk | P k=1 k=1 = 2 Nr Nr |hk |2 /σk2 E (h∗k /σk2 )wk [n] k=1 2

/σk2

2

/σk2 =

Nr

γk , (2.39)

k=1

k=1

which is essentially the sum of the received SNRs at all antennas. In fact, one can show that the MRC scheme achieves the maximum SNR among all linear combining techniques. Theorem 2.1. The output SNR of the linear combiner z[n] =

Nr

αk yk [n]

k=1

is maximized with the set of coeﬃcients αk = c · h∗k /σk2 , for k = 1, . . . , Nr ,

(2.40)

where c is an arbitrary constant. Proof. Given the weighting factors α1 , α2 , · · · , αNr , the output of the combiner can be expressed as N Nr Nr r √ √ z= αk P hk x + wk = P αk hk x + αk wk . k=1

k=1

k=1

and the SNR is given by N r P | k=1 αk hk |2 P |αT h|2 , = γ = Nr 2 2 αH Σα k=1 |αk | σk

(2.41)

where α = [α1 , α2 , · · · , αNr ]T , h = [h1 , h2 , · · · , hNr ]T , and Σ = diag(σ12 , σ22 , 2 · · · , σN ). By Cauchy-Schwarz inequality, we can show that r γ= ≤

P |(Σ1/2 α∗ )H (Σ−1/2 h)|2 P |αT h|2 = αH Σα |Σ1/2 α|2 P |Σ1/2 α|2 |Σ−1/2 h|2 |Σ1/2 α|2

= P |Σ−1/2 h|2 =

Nr

γk .

(2.42)

k=1

The equality holds when α = cΣ−1 h∗ , where c is an arbitrary constant. That is, the SNR is maximized when αk = c · h∗k /σk2 , for k = 1, . . . , Nr . Suppose that the channels are i.i.d. Rayleigh distributed with mean σh2 2 2 and that the noise variances are identical, i.e., σ12 = · · · = σN = σw . In r

2.2 Techniques to Exploit Spatial Diversity

33

this case, the SNR of each link will be i.i.d. exponentially distributed with 2 mean γ = P σh2 /σw and, thus, the SNR at the output of the MRC, i.e., γMRC = γ1 + γ2 + · · · + γNr , can be modeled as a chi-squared random variable with 2Nr degrees of freedom. The mean and variance of γMRC is given by Nr γ and 2Nr γ, respectively. The PDF of γMRC can be written as fγMRC (u) =

uNr −1 e−u/γ , γ Nr (Nr − 1)!

u ≥ 0.

Consequently, the outage probability can be computed as

γ0

Pr (γMRC ≤ γ0 ) =

fγMRC (u)du = 1 − e

−γ0 /γ

0

Nr (γ0 /γ)k−1 k=1

(k − 1)!

.

(2.43)

By taking the Taylor expansion of the exponential term such that e−γ0 /γ =

∞ (−1)k (γ0 /γ)k k=0

k!

,

the term corresponding to the n-th power of γ0 /γ in (2.43), for 0 ≤ n ≤ Nr −1, can be computed as −

n (−1)k (γ0 /γ)k k=0

k!

(−1)k · 1(n−k) (γ0 /γ)n−k · =− (n − k)! k!(n − k)! k=0 n γ0 n = −(−1 + 1) γ n

γ0 γ

n

= 0. On the other hand, the term corresponding to the Nr -th power of γ0 /γ is computed as Nr Nr Nr Nr 1 (−1)k · 1(Nr −k) γ0 (−1)k · 1(Nr −k) γ0 + − = − k!(Nr − k)! γ k!(Nr − k)! Nr ! γ k=1 k=0 Nr 1 γ0 = . Nr ! γ Hence, the outage probability can be expressed as Nr Nr +1 γ0 1 γ0 Pr (γMRC ≤ γ0 ) = . +O Nr ! γ γ

(2.44)

At high SNR, i.e., when γ 1, the ﬁrst term in (2.44) dominates and, thus, the outage probability can be further approximated as

34

2 Review of Wireless Communications and MIMO Techniques

0

10

Nr=1 N =2 r

N =4

−1

r

10

Nr=8 N =10

Outage Probability

r

−2

10

−3

10

−4

10

0

5

10

15 SNR (dB)

20

25

30

Fig. 2.8 Outage probabilities of the maximal ratio combining scheme.

Pr (γMRC

γ Nr ≤ γ0 ) ≈ 0 e−Nr log γ = Nr !

γ0 2 (Nr !)1/Nr P σh2 /σw

Nr .

This shows that the MRC scheme achieves diversity order of Nr , similar to that of the SC scheme. However, a coding gain of (Nr !)1/Nr can be obtained, allowing the MRC scheme to achieve the same outage probability with (Nr !)1/Nr times less power than that required for the SC scheme. In Fig. 2.8, we show the outage probability of the MRC scheme with various number of receive antennas. The simulation parameters are the same as those considered in Figs. 2.6 and 2.7. By comparing the outage probabilities achieved with the MRC and the SC schemes, we can see that the MRC requires (Nr !)1/Nr times less power than the SC scheme in order to achieve the same outage probability. In other words, although both schemes achieve the same diversity order, the MRC scheme achieves an additional coding gain 10 equal to N log10 (Nr !) dB. The coding gain is obtained by the coherent comr bining as well as the optimal weighting of the received signals. The advantage increases with the number of receiver antennas, which can be veriﬁed by comparing the curves in Figs. 2.7 and 2.8. Furthermore, by comparing between Figs. 2.6 and 2.8, we can see that diﬀerence between the outage performance of the EGC and the MRC scheme is only within 1-2 dB since both schemes coherently combine the signals at the receiver.

2.2 Techniques to Exploit Spatial Diversity

35

Fig. 2.9 Illustration of a MISO system

2.2.2 Multiple-Input Single-Output (MISO) System When the transmitter is equipped with multiple antennas, the data symbols can be distributed among multiple transmit antennas to exploit spatial diversity at the transmitter. Here, we consider a system with Nt antennas at the transmitter and only a single antenna at the receiver, as shown in Fig. 2.9. Let {x[n]} be the sequence of data to be transmitted and assume that the data symbols are i.i.d. over time with zero mean and unit variance. Depending on the speciﬁc transmit diversity scheme, the data is ﬁrst preprocessed to form a sequence of transmit symbol vectors {s[n]}, where s[n] = [s1 [n], s2 [n], . . . , sNt [n]]T is the vector of symbols to be transmitted over the Nt antennas in the n-th symbol period. The transmitted symbols are assumed to satisfy the sum power constraint E[|s|2 ] =

Nt

E[|sk [n]|]2 ≤ 1.

(2.45)

k=1

The signal obtained at the receiver during the n-th symbol period is given by Nt √ y[n] = P hk sk [n] + w[n], k=1

where P is the total transmit power, hk ∼ CN (0, σh2 ) is the channel coeﬃcient between the k-th transmit antenna and the receiver, and w[n] is the AWGN 2 with zero mean and variance σw . Based on the diﬀerent level of CSI at the transmitter, diﬀerent signal processing techniques can be employed to exploit spatial diversity. These techniques are introduced in the following.

36

2 Review of Wireless Communications and MIMO Techniques

Fig. 2.10 Linear precoding (e.g., transmit beamforming) in a MISO system.

Transmit Beamforming (Full CSI) In the case of transmit beamforming, the data in each symbol period is multiplied by a set of weighting coeﬃcients that precompensate for the channel eﬀects before transmission. An illustration of transmit beamforming (or, the more general linear precoding scheme) is given in Fig. 2.10. Let β1 , β2 , · · · , βNt be the weighting factors imposed on the Nt antennas, respectively. The signal transmitted on the k-th antenna is given by sk [n] = βk x[n], where the transmission power is 2 √ Pk = E P sk [n] = P · |βk |2 . With the linearly precoded transmit symbols, the signal observed at the receiver can be written as y[n] =

Nt √

P hk βk x[n] + w[n].

(2.46)

k=1

The corresponding SNR at the receiver is then given by

γ=

2 Nt h k βk P k=1 2 σw

.

(2.47)

t When full instantaneous CSI (i.e., the channel coeﬃcients {hk }N k=1 ) is Nt available at the transmitter, the set of weighting coeﬃcients {βk }k=1 can be chosen to maximize the receive SNR given in (2.47) subject to the power constraint given in (2.45) [14]. The optimization problem can be formulated as follows:

2.2 Techniques to Exploit Spatial Diversity

37

2 Nt P k=1 h k βk

max

(2.48)

2 σw

β1 ,...,βNt

subject to

Nt

|βk |2 ≤ 1.

(2.49)

k=1

Similar to the proof of Theorem 2.1, we can show, by utilizing the CauchySchwarz inequality, that N 2 N N Nt t t t 2 2 = h k βk ≤ |hk | |βk | |hk |2 , k=1

k=1

k=1

k=1

where the equality holds when βk = c · h∗k , for k = 1, . . . , Nt . By choosing c to satisfy the power constraint in (2.49), we have h∗ βk = k Nt

k =1

|hk |2

, for k = 1, . . . , Nt .

(2.50)

The corresponding SNR at the receiver is given by

γBFM =

2 Nt hk h∗k P N t 2 k=1 |h | k k =1 2 σw

=

Nt P |hk |2 k=1

2 σw

.

(2.51)

Note that the SNR achieved with transmit beamforming is equal to the sum of the SNRs between each transmit antenna and the receiver, as if each transmit antenna takes turns in transmitting to the receiver at diﬀerent instants in time and utilizing full power P in each transmission. This shows that transmit beamforming is able to achieve an Nt -fold performance gain compared to the SISO system. It is also worthwhile to note that the SNR achieved with transmit beamforming is the same as that obtained with MRC in SIMO systems and, thus, the outage performances will be identical as well. In fact, with full CSI at the transmitter, the MISO system can be viewed as a dual of the SIMO system and, thus, transmit beamforming is also referred to as the transmit MRC scheme. Although transmit beamforming is able to achieve a signiﬁcant gain in SNR, the need of instantaneous CSI at the transmitter have hindered its use in practical systems. Several works in the literature have been done to address these issues. For example, one way to achieve this task is to have the receiver, which is able to estimate the CSI, compute the optimal beamforming coeﬃcients and send it to the transmitter through a dedicated feedback channel. Due to rate limitations on the feedback channel, the set of optimal beamforming coeﬃcients should be quantized before it is sent out through the

38

2 Review of Wireless Communications and MIMO Techniques

feedback channel [17, 18, 26, 38]. The codebook of the vector quantizer (VQ) should be designed to accurately represent the set of beamforming coeﬃcients. These coeﬃcients should be computed periodically with a frequency that depends on the time-variation of the channel. When the instantaneous CSI is not attainable, e.g., when the channel varies rapidly, the beamforming coeﬃcients can instead be derived based on the statistics (rather than the instantaneous realizations) of the channel, as discussed in [25, 41].

Antenna Selection (Partial CSI) Suppose that the transmitter is only able to obtain knowledge of the channel amplitude but not the channel phase information. This case may arise in practice since the phase varies much faster than the channel amplitude and, thus, is more diﬃcult to estimate. Without accurate knowledge of the phase information, the signals transmitted by diﬀerent antennas cannot be co-phased at the receiver. In this case, it may be more desirable to transmit only on the antenna with the best channel in order to avoid destructive interference. This is referred to as the antenna selection scheme [28, 29]. The concept of the antenna selection is similar to that of the selection combining scheme in SIMO systems. Assume that the k ∗ -th antenna is the one experiencing the highest instantaneous SNR, i.e., P |hk |2 k ∗ = arg max , (2.52) 2 k σw and is selected to transmit so that βk = 1, for k = k ∗ , and βk = 0, for k = k ∗ . In this case, the signal obtained at the receiver is √ y[n] = P hk∗ x[n] + w[n], and the resulting SNR is given by γAS =

P |hk∗ |2 P |hk |2 = max . 2 2 k σw σw

(2.53)

The outage performance of the antenna selection scheme in MISO systems is the same as that of selection combining in SIMO systems since they achieve the same receive SNR. In practice, where channel can only be estimated at the receiver, antenna selection can be performed at the receiver and the index of the antenna with the best channel can be fed back to the transmitter.

2.2 Techniques to Exploit Spatial Diversity

39

Space-Time Coding (No CSI) When channel information is not available at the transmitter, spatial diversity cannot be fully exploited by encoding data only over the spatial dimension, as done in the previous linear precoding schemes, e.g., transmit beamforming and antenna selection. However, they can be exploited by encoding the data over both space and time, which results in the so called space-time coding (STC) scheme. The key idea can be demonstrated through the well-known Alamouti code example [1], as described below. Let us consider a MISO system with Nt = 2 antennas at the transmitter and a single antenna at the receiver. Let x[1] and x[2] be two consecutive information symbols and let E[|x[i]|]2 = 1, for i = 1, 2. By employing the Alamouti space-time coding scheme, the codewords transmitted by the two antennas over the two consecutive symbol periods can be written as s [1] s2 [1] x[1] x[2] S 1 = , (2.54) −x∗ [2] x∗ [1] s1 [2] s2 [2] where sk [n], for k = 1, 2 and n = 1, 2, is the n-th symbol transmitted on the k-th antenna. Let P be the sum transmit power in each symbol period. We assume that P is equally allocated among transmit antennas since no CSI is available at the transmitter. The signals observed at the receiver in these two consecutive symbol periods are given by P y[1] = (h1 s1 [1] + h2 s2 [1]) + w[1] 2 P (h1 x[1] + h2 x[2]) + w[1], = 2 P (h1 s1 [2] + h2 s2 [2]) + w[2] y[2] = 2 P (−h1 x∗ [2] + h2 x∗ [1]) + w[2], (2.55) = 2 where hk is the channel coeﬃcient between the k-th transmit antenna and the 2 receiver, and w[n] ∼ CN (0, σw ) is the AWGN. Notice that the data symbols in y[2] are taken with respect to the complex conjugate of the original symbols. Therefore, by taking the complex conjugate of y[2], we can deﬁne the received signal vector as P h 1 h2 y[1] w[1] x[1] y= ∗ + = y [2] w∗ [2] x[2] 2 h∗2 −h∗1 P Hx + w, (2.56) = 2 where

40

2 Review of Wireless Communications and MIMO Techniques

h 1 h2 H= h∗2 −h∗1

(2.57)

is the eﬀective channel matrix, x = [x[1], x[2]]T is the vector of data symbols, and w = [w[1], w∗ [2]]T is the AWGN vector with zero-mean and covariance 2 matrix Cw = σw I. By assuming that each symbol vector x in the signal 2 constellation X are transmitted with equal probability, the optimal decision on x can be given by the maximum likelihood (ML) detector, where the detected symbol vector is given by ˆ = arg max f (y|x) = arg min y − x x∈X 2

x∈X 2

P Hx 2

2

.

Here, f (y|x) is the conditional density function of y given x. Notice that the ML detector reduces to the minimum distance detector in the AWGN channel. One disadvantage of the ML detector is that the decoding complexity increases exponentially with the length of the data vector. However, this issue is intelligently addressed with the Alamouti STC scheme, where the eﬀective channel matrix in (2.57) can be shown to be orthogonal, i.e., HH H = (|h1 |2 + |h2 |2 )I2×2 .

(2.58)

In this case, we can multiply the received vector by the channel matrix to obtain z = HH y P H H Hx + HH w = 2 P (|h1 |2 + |h2 |2 )x + w, = 2

(2.59)

where w = HH w is a zero-mean Gaussian noise vector with covariance matrix Cw = E[wwH ] = E[HH wwH H] 2 I2×2 )H = HH (σw 2 (|h1 |2 + |h2 |2 )I2×2 . = σw

(2.60)

Notice that the ﬁrst and second entries of z (denoted by z[1] and z[2]) depend only on the symbols x[1] and x[2], respectively, and the entries of the noise HH w are uncorrelated. Therefore, the joint ML detection on x can be decoupled into two separate ML detectors performed on the scalar variables x[1] and x[2]. The optimal ML detector is given equivalently as 2 P 2 2 x ˆ[1] = arg min z[1] − (|h1 | + |h2 | )x[1] 2 x[1]∈X

2.2 Techniques to Exploit Spatial Diversity

and

41

2 P 2 2 xˆ[1] = arg min z[2] − (|h1 | + |h2 | )x[2] . 2 x[1]∈X

The key to achieving reduced decoding complexity is to construct spacetime codes that achieve orthogonality in the eﬀective channel matrix H. The resultant SNR of each data symbol in (2.59) can be computed as P (|h1 |2 + |h2 |2 )2 2 (|h |2 + |h |2 ) 2σw 1 2 P (|h1 |2 + |h2 |2 ) = 2 2σw γ 1 + γ2 . = 2

γAla =

(2.61)

Compared with the transmit beamforming scheme, the SNR achieved with the Alamouti scheme is reduced by one halve. The decrease in SNR is expected since the transmitter does not have knowledge of the instantaneous CSI in the STC schemes. When considering Rayleigh fading channels, the SNR γAla can be modeled as a chi-square random variable with 4 degrees of freedom. Hence, it can be shown that the outage probability decays approximately as (1/¯ γ )2 , when E[γ1 ] = E[γ2 ] = γ¯ . This shows that diversity order of two is achieved with the Alamouti STC scheme, where instantaneous knowledge of the CSI is not required. Moreover, there is no loss in spectral eﬃciency compared to the SISO case since, on the average, one data symbol is transmitted in each symbol period. In the above example, we can see that spatial diversity is achieved with the STC scheme by encoding data symbols in both space and time so that each symbol can be transmitted over diﬀerent antennas in diﬀerent symbol periods. Two classes of space-time codes have been studied the most in the literature, namely, the class of space-time block codes (STBC) [1,13,15,32] and the class of space-time trellis codes (STTC) [33,39]. In STTC, the transmitted symbols are generated from a trellis diagram and thus the symbols contain memory. In STBC, the data symbols are encoded block-by-block into a sequence of T × Nt code matrices, where each code matrix is transmitted by Nt transmit antennas over T consecutive symbol periods. The decoding of both STTC and STBC is achieved with the ML detector (using the Viterbi algorithm in the case of STTC), which requires high decoding complexity. Let us consider in more detail the STBC scheme, where each block of M data symbols is encoded into a T × Nt codeword ⎤ ⎡ s1 [1] s2 [1] · · · sNt [1] ⎢ s1 [2] s2 [2] · · · sNt [2] ⎥ ⎥ ⎢ (2.62) S⎢ . .. ⎥ . .. . . ⎣ .. . . . ⎦ s1 [T ] s2 [T ] · · · sNt [T ]

42

2 Review of Wireless Communications and MIMO Techniques

Here, Nt is the number of transmit antennas and T is the length of codeword over time. Each codeword must satisfy the transmit power constraint E[ S 2F ] ≤ T , where S 2F = i,j |Si,j |2 represents the Frobenius norm of matrix S. In this case, the coding rate of the STC is equal to M/T symbols per channel use, which is equal to 1 in the case of Alamouti codes given in (2.54). Let y = [y[1], y[2], · · · , y[T ]]T be the vector of signals received over T symbol periods given by P y= Sh + w, (2.63) Nt where h = [h1 , h2 , · · · , hNt ]T is the vector of channel coeﬃcients on the Nt antennas and w is a T × 1 Gaussian white noise vector with correlation ma2 trix σw IT ×T . When each codeword is transmitted equally likely, the optimal decoding scheme is given by the ML decoding, i.e., ˆ = arg min y − S S∈S

P Sh Nt

2

,

(2.64)

where S is the set of all possible codewords. The ML decoding method can be easily extended to the case with multiple receive antennas. Although ML decoding is optimal, the decoding complexity in general increases exponentially with M . As demonstrated in the example on Alamouti codes, the decoding complexity can be reduced by considering codewords that allow the eﬀective channel matrix to be orthogonal. This is achieved with code matrices whose columns are mutually orthogonal. This class of STBC is referred to as the orthogonal STBC (OSTBC) [1, 32]. One of the most well-known example among this class of codes is the Alamouti code mentioned previously. For a rate-one OSTBC design, where Nt = T = M (e.g. Alamouti code), the SNR at the receiver can be written as γOSTBC =

Nt P |h |2 . 2 Nt σw =1

We can see that, with Nt transmit antennas in a MISO system, OSTBC achieves the same diversity order as transmit beamforming but with an SNR loss of 10 log10 Nt dB. Moreover, due to its orthogonal design, the optimal ML decoding can be decoupled into M scalar ML decoding operations through a simple linear transformation, which demands complexity that increases only linearly with M . Although OSTBC ideally is able to achieve full diversity as described above, it is not always possible to ﬁnd a full-rate and full diversity code for arbitrary number of transmit antennas, especially when complex constellations are used. In fact, it has been shown that full-rate full diversity OSTBC exists only for Nt = 2, i.e., the Alamouti code [32]. To exploit spatial diver-

2.2 Techniques to Exploit Spatial Diversity

43

sity in more general cases, one may consider other STCs that can achieve full diversity with higher rate. In the following, we will introduce two general criteria for STC design. Suppose we consider the general STBC matrix given in (2.62) with the optimal ML detector given in (2.64). In this case, the pairwise error probability (PEP) between Si and Sj , where j = i, that is, the probability that the receiver detects Sj when Si was transmitted, is given by ⎛ ⎞ 2 2 P P Sj h < y − Si h ⎠ Pr(Si → Sj |h) = Pr ⎝ y − Nt Nt ⎛ ⎞ 2 P (Sj − Si )h < w 2 ⎠ = Pr ⎝ w − Nt ' P hH Di,j h =Q 2 2Nt σw P hH Di,j h , (2.65) ≤ exp − 2 4Nt σw where Di,j = (Sj − Si )H (Sj − Si ) is the non-negative deﬁnite distance matrix between codeword pairs Si and Sj . The inequality in (2.65) fol2 lows from the fact that Q(x) ≤ e−1/2x . Denote the eigenvalue decompoH sition of Di,j as Di,j = UΛU , where U is a Nt × Nt unitary matrix and Λ = diag(λ1 , λ2 , · · · , λNt ) is a diagonal matrix with non-negative entries. The PEP conditioned on the channel realization is then given by Nt P hH UΛUH h P |b |2 λ = , (2.66) Pr(Si → Sj |h) ≤ exp − exp − 2 2 4Nt σw 4Nt σw =1

where b = [b1 , b2 , · · · , bNt ]T = UH h. Assume that the channel coeﬃcients are i.i.d. circularly symmetric complex Gaussian distributed, i.e., h ∼ CN (0, σh2 INt ×Nt ). Since b comes from a unitary transform of h, b will have the same distribution as h and, thus, {|b |2 , ∀ } will be i.i.d. exponentially distributed with mean σh2 . Therefore, the PEP averaged over channel statistics is given by P |b |2 λ Pr(Si → Sj ) ≤ E exp − 2 4Nt σw =1 −1 Nt P λ σh2 1+ . = 2 4Nt σw Nt

=1

At high SNR, the PEP can be approximately upper bounded by

(2.67)

44

2 Review of Wireless Communications and MIMO Techniques

Pr(Si → Sj )

−1 r P λ σ 2 =1

h 2 4Nt σw

=

P σh2 2 4Nt σw

−r r

λ−1 ,

(2.68)

=1

where r is the rank of the distance matrix Di,j . As SNR increases, it is observed from (2.68) that the overall error probability of the STC scheme decays exponentially with rate no less than rmin , which is the minimum rank of the distance matrices among all distinct codeword pairs. To optimize the error performance of the STC, two design criteria based on the PEP are given as follows [8]: 1. Rank criterion: The STC should be designed such that the minimum rank of the distance matrix between any pair of distinct codewords is as large as possible. An STC is able to achieve maximum diversity if the distance matrices among all pairs of distinct codewords are full rank. 2. Determinant criterion: The STC should be designed such that the minimum of the product of non-zero eigenvalues of all distance matrices is made as large as possible. When the distance matrices are full rank, this is equivalent to maximizing the minimum determinant of the distance matrices of all distinct codeword pairs. Since the minimum rank of distance matrices determines the diversity gain and the product of non-zero eigenvalues aﬀects only the coding gain, one should place more emphasis on the rank criterion when designing space-time codes.

2.2.3 Multiple-Input Multiple-Output (MIMO) System When both the transmitter and the receiver are equipped with multiple antennas, both precoding at the transmitter and signal combining at the receiver can be employed to exploit the addition degrees of freedom. Suppose that there are Nt and Nr antennas at the transmitter and receiver, respectively, as shown in Fig. 2.11. Let s1 [n], s2 [n], · · · , sNt [n] be the symbols to be transmitted simultaneously over the Nt antennas in the n-th symbol period and assume that Nt (2.69) E |sj [n]|2 ≤ 1. j=1

The signals observed at the Nr receive antennas are given, respectively, by √ y1 [n] = P (h1,1 s1 [n] + h2,1 s2 [n] + · · · + hNt ,1 sNt [n]) + w1 [n], √ y2 [n] = P (h1,2 s1 [n] + h2,2 s2 [n] + · · · + hNt ,2 sNt [n]) + w2 [n], .. .. . . √ yNr [n] = P (h1Nr s1 [n] + h2Nr s2 [n] + · · · + hNt Nr sNt [n]) + wNr [n],

2.2 Techniques to Exploit Spatial Diversity

45

Fig. 2.11 Multi-input-multi-output (MIMO) channel model.

where P is the total transmit power, hi,j is the channel coeﬃcient between the 2 i-th transmit antenna and the j-th receive antenna and wj [n] ∼ CN (0, σw ) is the AWGN occurred at the j-th receive antenna. Let s[n] = [s1 [n], s2 [n], · · · , sNt [n]]T and y[n] = [y1 [n], y2 [n], · · · , yNr [n]]T be the input and output vectors of the MIMO channel. By writing the received signal in vector form, we have √ y[n] = P Hs[n] + w[n], (2.70) where H is an Nr × Nt channel matrix with each element being [H]ij = hj,i and w[n] = [w1 [n], w2 [n], · · · , wNr [n]]T is an additive Gaussian noise vector 2 with zero mean and covariance matrix σw INr ×Nr . With full CSI at both the transmitter and the receiver, we may exploit the knowledge of H to design the precoding scheme and the signal combining scheme at the transmitter and the receiver, respectively. Suppose that the rank of H is M , where M ≤ min(Nt , Nr ), it is possible to transmit M symbols at a time such that the receiver is able to successfully separate all transmitted symbols. Let us take the singular value decomposition (SVD) of the channel matrix to obtain H = UΣVH , (2.71) where U and V are Nr × Nr and Nt × Nt unitary matrices, and Σ is a Nr × Nt diagonal matrix composed of singular values of H in a decreasing order. Since the rank of H is M , there are M nonzero singular values, say μ1 ≥ μ2 ≥ · · · ≥ μM , on the diagonal of Σ. Let {x1 , x2 , · · · , xM } be a set of transmitted symbols. By assuming that H is known at the transmitter, the symbols can be precoded as s = Vx,

(2.72)

where x = [x1 , x2 , · · · , xM , 0, · · · , 0]T satisﬁes the transmit power constraint

46

2 Review of Wireless Communications and MIMO Techniques

x1

s1

x2

s2

.. .

V

xM

sNt

y1

. . .

.. .

. . .

. . .

.. . H

z1

y2

z2

UH

.. .

zM

yNr

Fig. 2.12 Illustration of the precoding at the transmitter and linear combining at the receiver in MIMO channels.

¹1

x1 x2

.. . xM

z1

¹2

.. . ¹M

z2

.. . zM

Fig. 2.13 Illustration of equivalent parallel channels.

E s 2 = E Vx 2 = E x 2 ≤ 1. Please note that we have dropped the symbol index n here simply for the ease of notation. At the receiver end, we exploit the SVD of H to separate the transmitted symbols. This is achieved by taking √ z = UH y = UH P Hs + w √ = P UH (UΣVH )Vx + w √ = P Σx + w, (2.73) where w = UH w is an equivalent noise vector with the covariance matrix 2 E[wwH ] = σw I. Since the matrix Σ and the covariance matrix of w are both diagonal, we have produced equivalently M parallel channels with uncorrelated noise as shown in Fig. 2.13. The parallel channels are also called the eigen-channels or eigen-modes of the MIMO channel. The output of the k-th eigen-channel is given by

2.2 Techniques to Exploit Spatial Diversity

zk =

√ P μk xk + w k ,

47

for k = 1, 2, · · · , M.

(2.74)

Thus, each non-zero diagonal element, say the k-th diagonal term μk , of Σ is the eﬀective channel coeﬃcient of the k-th parallel channel. Under the MIMO transmission scheme, if the number of nonzero elements in x is greater than rank(H) = M , some symbols are not received at the receiver since the symbols xM+1 , xM+2 , · · · , xMt are mapped to the null space of the channel matrix H by the linear transformations at the transmitter and the receiver. The transmitted symbols over the eigen-channels can be allocated with diﬀerent power and encoded with diﬀerent rates. The power allocated to the diﬀerent eigen-channels can be further optimized according to the eﬀective channel coeﬃcients {μk } and based on diﬀerent optimization criteria, such as the maximum capacity criterion [c.f. Section 2.3] or the minimum pairwise error probability criterion etc. Assume that the distance between any two transmit antennas or between any two receive antennas is suﬃciently large such that all channel coeﬃcients are independently distributed. More speciﬁcally, assume that {hi,j } are i.i.d. complex circularly symmetric Gaussian random variables with zero mean and variance σh2 , for i = 1, 2, · · · , Nt , and j = 1, 2, · · · , Nr . In the following, we will show an Nt × Nr MIMO channel is able to achieve diversity order of Nt Nr when CSI is available at both the transmitter and the receiver. Given the CSI, the transmitter is able to apply the aforementioned SVD technique to decompose the MIMO channel as M = rank(H) orthogonal eigen-channels. Consider an extreme case √ that a common symbol is transmitted over all eigenchannels, i.e., xk = x/ M , where E[|x|2 ] = 1. The signals received over all eigen-channels are combined with the MRC at the receiver, which results in the SNR at the output as γMIMO =

Nt Nr M P |μk |2 P |hi,j |2 = , 2 2 M σw M σw m=1 i=1 j=1 P σ2

which is chi-squared distributed with mean γ = σ2h and 2Nt Nr degrees of w freedom. The equality comes from the fact that summation of {|μi |2 }, which are eigenvalues of HHH , equals to the trace of HHH . Similar to (2.44), the corresponding outage probability can be simpliﬁed with Taylor expansion as in (2.44) and is expressed as Nt Nr Nt Nr +1 1 M γ0 M γ0 Pr (γMIMO ≤ γ0 ) = (2.75) . +O (Nt Nr )! γ γ In high SNR regime, γ 1, the second term in (2.75) can be eliminated and the outage probability decays with SNR to the Nt Nr order. Hence, it shows that, when the channels between each transmit antenna and receive antenna are i.i.d., the diversity order of the MIMO channel is Nt Nr .

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2 Review of Wireless Communications and MIMO Techniques

However, it is sometimes prohibited for the transmitter to know the instantaneous CSI when the channel varies faster. If only the statistics of the channel coeﬃcients are available at the transmitter, it is still possible to exploit the correlation matrix of the channel coeﬃcients to design the pre-coder and decoder at the transmitter and receiver and the number of transmitted symbols allowed depends on the rank of the correlation matrix. When no CSI is available at the transmitter, we can still exploit spatial diversity by employing space-time codes at the transmitter, as shown in the MISO system, with the penalty of reduced coding gain at the receiver.

2.3 Capacity of Wireless Channels In this section, we study the fundamental limits of wireless channels in terms of its Shannon capacity [31], i.e., the maximum rate achievable between the transmitter and the receiver with arbitrary small error probabilities. The channel capacity is often used to characterize the limitations of a communication channel. In general, for a channel with input X and output Y , the capacity can be expressed as [6, (7.1)] C = max I(X; Y ),

(2.76)

p(x)

) ( p(x,y) is the mutual information between X where I(X; Y ) E log2 p(x)p(y) and Y . By this formula, one can view capacity as the maximum amount of information that the channel output can provide about the input message, when optimized over all possible input distributions p(x). In the following, we will brieﬂy introduce the capacity of AWGN and wireless fading channels, and then extend to the case with multiple antenna transmitters and/or receivers.

2.3.1 Capacity of AWGN Channels Consider a single-input single-output AWGN channel with the input-output relations given by √ y = P x + w, where x is the complex channel input with unit power constraint E[|x|]2 ≤ 1, P is the transmit power employed by the transmitter, and w is the addi2 tive Gaussian noise with zero-mean and variance σw seen at the receiver. In AWGN channels, given the input x, the channel output y is Gaussian √ 2 distributed with mean P x and variance σw . By maximizing the mutual

2.3 Capacity of Wireless Channels

49

information over the input distribution, the capacity of AWGN channels is given by C = max I(X; Y ) p(x) P = log2 1 + 2 σw = log2 (1 + SNR) bits per channel use,

(2.77) (2.78)

2 where SNR = P/σw is the transmit SNR at the transmitter, which equals to the SNR at the receiver in AWGN cases. Since the channel capacity is a function of SNR, the achievable transmission rate can be enhanced by increasing the value of the transmit SNR.

2.3.2 Capacity of Flat Fading Channels Consider the wireless fading scenario where the channel quality varies with time and space due to the eﬀects of path-loss, shadowing and multipath fading eﬀect, as introduced in previous sections. Here, we consider ﬂat fading channels for simplicity. Since the channel coeﬃcient is a random parameter, the capacity of a fading channel is random as well. For a given channel realization with channel coeﬃcient h, the channel output is given by √ y = P hx + w, where x is the channel input with unit power constraint E[|x|]2 ≤ 1, P is the 2 transmit power, and w is the AWGN with zero-mean and variance σw . Under a given channel state h, the channel capacity can be expressed as * + P |h|2 C = log2 1 + = log2 1 + SNR|h|2 . (2.79) 2 σw To further take into consideration the random variations of the channel, we can consider two capacity measures, one is the ergodic capacity, which is the rate achieved by encoding each message over multiple channel states, and the other is the outage capacity, which is the rate achieved under a constraint on the outage probability. These measures are introduced in the following.

Ergodic Capacity Ergodic capacity is deﬁned as the capacity achieved by encoding across multiple channel realizations. In the case of Gaussian inputs, the ergodic capacity can be characterized as

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2 Review of Wireless Communications and MIMO Techniques

C = E|h|2 log2 (1 + SNR|h|2 ) ∞ = f|h|2 (u) log2 (1 + SNRu)du.

(2.80)

0

When CSI is only available at the receiver, ergodic capacity can be achieved by coding across suﬃciently large number of channel realizations. When the CSI is also available at the transmitter and the transmission power is ﬁxed, the ergodic capacity will be the same as that in the case with only receiver-side CSI. In general, with transmitter-side CSI, power can be optimally allocated over time, using the water-ﬁlling algorithm, to further increase the ergodic capacity. However, no signiﬁcant gains can be obtained by doing so [12].

Outage Capacity Outage capacity is another popular performance measure for communication techniques in fading channels. Outage is the event that the message cannot be reliably decoded at the receiver. From an information theoretic standpoint, given the transmission rate R and the channel realization h, we say that outage * occurs if +the channel capacity is less than the transmission rate, i.e., log2 1 + SNR|h|2 < R. Hence, outage probability under the channel realization h can be expressed as a function of the transmission rate as given below: * + pout (R) Pr log2 (1 + SNR|h|2 ) < R * + = Pr SNR|h|2 < 2R − 1 . (2.81) The -outage capacity is then deﬁned as the maximum transmission rate that can be achieved over the fading channel such that the outage probability is smaller than , i.e., C =

arg max {R:R≥0,pout (R)≤ }

pout (R)

(2.82)

where ∈ [0, 1] is a constant threshold. The outage probability can be calculated by using the PDF of the channel coeﬃcient or the received SNR. For example, in the Rayleigh channel where the envelope of channel coeﬃcient is Rayleigh distributed and the receive SNR SNR|h|2 is exponentially distributed with mean SNRσh2 , the outage probability of the Rayleigh channel given the transmission rate R is given by * + pout (R) = Pr SNR|h|2 < 2R − 1 R 2 −1 . = 1 − exp − SNRσh2

2.3 Capacity of Wireless Channels

51

3.5

3

AWGN capacity Rayleigh, Ergodic Capacity Rayleigh, 10% Outage Capacity

C (bits/sec/Hz)

2.5

2

1.5

1

0.5

0 0

1

2

3

4

5 SNR (dB)

6

7

8

9

10

Fig. 2.14 Ergodic capacity and outage capacity in a fading channel and capacity of an AWGN channel.

In general, outage capacity can be obtained from the inverse of the outage probability function, i.e., p−1 out (·), or can be numerically found from the outage probability curve. The capacity of an AWGN channel and the ergodic and outage capacities of a Rayleigh fading channel are compared in Fig. 2.14. The variance of the fading channel coeﬃcient is set as one. We can see that, in the case of Rayleigh fading, the 10%-outage capacity is much smaller than the ergodic capacity. Moreover, we can also see that, even though the average SNR is the same for both the fading and the AWGN channels, the ergodic capacity of the fading channel is still inferior to that of the AWGN channel capacity. This is because, even though the SNR can be occasionally high in the fading case, the gain in capacity in these cases are not proportional due to the concavity of the logarithmic function. This follows mathematically by the Jensen’s inequality when comparing the two capacity expressions. From the deﬁnition of outage probability when the channel input is with the rate C , reliable transmission can be done with outage probability . The outage capacity is more practical to measure the channel capacity, since it provides a certain degree of guarantee on the reliable communication. In the following, we consider both outage capacity and ergodic capacity of channels with multiple transmit and/or receive antennas.

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2 Review of Wireless Communications and MIMO Techniques

12

10

C (bits/sec/Hz)

8

6

4 Nt=1, Nr=1 N =1, N =2

2

t

r

Nt=1, Nr=3 Nt=1, Nr=4 0 0

5

10

15 SNR (dB)

20

25

30

Fig. 2.15 Ergodic capacity of a single-input multiple-output (SIMO) channel.

2.3.3 Capacity with Multiple Antennas Consider a SIMO system with a single antenna at the transmitter and Nr antennas at the receiver. Here, we assume that the AWGN at each receiver 2 is i.i.d. Gaussian with zero-mean and variance σw . The received signals at all antennas can be combined using the MRC to achieve the maximum receive SNR. The resulting SNR is given by γ=

Nr P |hk |2 k=1

2 σw

= SNR

Nr

|hk |2 ,

j=1

2 is the SNR at the transmitter, hk is the channel coeﬃwhere SNR = P/σw cient between the transmit antenna and the k-th receive antenna. In a fading r channel, the channel coeﬃcients {hk }N k=1 are random. Given the set of channel coeﬃcients {h1 , h2 , · · · , hNr }, the channel capacity of the SIMO channel is given by Nr 2 C = log2 1 + SNR (2.83) |hk | . k=1

The receive SNRs obtained in Rayleigh and Rician fading channels take on central and non-central chi-squared distributions, respectively. The ergodic capacity or outage capacity can then be found from the distribution of the received SNR. The ergodic capacity and the 10%-outage capacity of a

2.3 Capacity of Wireless Channels

53

12 N =1, N =1 t

r

N =1, N =2 t

10

r

Nt=1, Nr=3 Nt=1, Nr=4

C (bits/sec/Hz)

8

6

4

2

0 0

5

10

15 SNR (dB)

20

25

30

Fig. 2.16 Outage capacity of a single-input multiple-output (SIMO) channel.

SIMO channel under Rayleigh fading are illustrated in Figs. 2.15 and 2.16, respectively. The variances of all channel coeﬃcients are set to 1. The ﬁgures show that channel capacity increases linearly with the transmit SNR (in dB scale) and the increasing rate of channel capacity is the same regardless of the number of receive antennas. In a MISO system with Nt transmit antennas and one receive antenna, the transmitted signals are linearly combined at the receiver and the maximal received SNR depends on the availability of CSI at the transmitter. When full CSI is available at the transmitter, the data symbol can be multiplied by a set of beamforming factors to maximize the received SNR. The resulting SNR at the receiver is the same with the received SNR of a SIMO system. Thus, the ergodic capacity and outage capacity of a MISO system with full CSI at the transmitter have the same performance as the SIMO system in Figs. 2.15 and 2.16. On the other hand, when the channel information is not available at the transmitter, we may apply space-time codes to exploit spatial diversity. The resulting SNR is only (1/Nt ) times that in the case with full CSI at the transmitter. Therefore, given the channel coeﬃcients {hk }, the channel capacity of a MISO system is given by Nt SNR (2.84) C = log2 1 + |hk |2 , Nt k=1

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2 Review of Wireless Communications and MIMO Techniques

10 9 8

C (bits/sec/Hz)

7 6 5 4 3 Nt=1, Nr=1 2

N =2, N =1

1

Nt=3, Nr=1

t

r

Nt=4, Nr=1 0 0

5

10

15 SNR (dB)

20

25

30

Fig. 2.17 Ergodic capacity of a multiple-input single-output (MISO) channel.

where hk is the channel coeﬃcient between the k-th transmit antenna and the receive antenna. Under the Rayleigh or Rician fading assumption, the receive SNR of a MISO system is also chi-squared distributed and we can use a similar method to ﬁnd the ergodic capacity and outage capacity. For the Rayleigh fading environment, the ergodic capacity and the outage capacity of a MISO system without CSI at the transmitter are shown in Figs. 2.17 and 2.18, respectively. In the simulations, the variances of all channel coeﬃcients are set to 1. Similar to the results in SIMO systems, the capacity of diﬀerent numbers of transmit antennas increase with the transmit SNR at the same rate in the MISO system. Compared with the results of the SIMO system shown in Figs. 2.15 and 2.16, we can see that the capacity of MISO systems without CSI at the transmitter is degraded due to the decreased SNR at the receiver. Moreover, it is worthwhile to note that the ergodic capacity of MISO systems without CSI at the transmitter is upper bounded as N Nt t SNR SNR 2 2 E log 1 + ≤ log 1 + |hk | E |hk | Nt Nt k=1

k=1

= log (1 + SNR) = CAWGN . In a MIMO system where the transmitter and the receiver have Nt and Nr antennas, respectively, the channel capacity depends on the availability of CSI at the transmitter, too. When the transmitter has channel information, the input symbols can be further designed to achieve the optimal transmission

2.3 Capacity of Wireless Channels

55

9 N =1, N =1 t

8 7

r

N =2, N =1 t

r

Nt=3, Nr=1 Nt=4, Nr=1

C (bits/sec/Hz)

6 5 4 3 2 1 0 0

5

10

15 SNR (dB)

20

25

30

Fig. 2.18 Outage capacity of a multiple-input single-output (MISO) channel.

rate. Let us consider a static channel state H, where the (j, i)-th element {H}j,i = hi,j is the channel coeﬃcient between the i-th transmit antenna and the j-th receive antenna. With full CSI at transmitter, the channel capacity is given by [10, 34] P C= max , (2.85) log2 det INr ×Nr + 2 HRs HH Rs :Tr(Rs )=1 σw where Rs = E[ssH ] is the covariance of the input signals, Tr(Rs ) = 1 indicates the power constraint, and the optimization is performed over the semi-positive deﬁnite covariance matrix of the input symbols. Recall that, in Section 2.2, we take the SVD of the channel matrix to decompose the MIMO system into a set of parallel channels. Let the SVD of the channel matrix be H = UΣVH , where U and V are Nr × Nr and Nt × Nt unitary matrices, and Σ is a diagonal matrix composed of singular values in a decreasing order. As shown in Figs. 2.12 and 2.13, by transmitting the pre-coded signals s = Vx at the transmitter and decoding with z = UH x at the receiver, the MIMO channel can be decomposed into M = rank(H) parallel channels. The k-th decomposed channel can be represented by √ zk = P μk xk + w k , for k = 1, 2, · · · , M, where μk is the eﬀective channel coeﬃcient of the k-th decomposed channel and is also k-th element of the diagonal matrix Σ. With the pre-coding and

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2 Review of Wireless Communications and MIMO Techniques

decoding schemes, the mutual information between the pre-coder input x and decoder output z equals to the sum of capacity of M parallel channels, i.e., I(x; z) =

M k=1

Pk |μk |2 , log2 1 + 2 σw

(2.86)

√ where Pk = E[| P xk |2 ] is the equivalent transmission power of the k-th parallel channel. The scheme is equivalent to the case where the input covariance is set as P · Rs = VPVH , where P = diag(P1 , P2 , . . . , PM , 0, . . . , 0). To satisfy the power constraint, we have Tr(P · Rs ) = Tr(P) = P1 + P2 + · · · + PM = P.

(2.87)

The powers allocated on the M inputs of the parallel channels can be further optimized to achieve the maximum channel capacity. That is, the optimization problem can be formulated as max

M

P1 ,··· ,PM

k=1

Pk |μk |2 , log2 1 + 2 σw

subject to P1 + P2 + · · · + PM ≤ P ; Pk ≥ 0, ∀k. To ﬁnd the optimal solution, we use the Lagrangian function given as M M Pk |μk |2 −λ log2 1 + Pk − P , (2.88) L(P1 , · · · , PM , λ) = 2 σw k=1

k=1

where λ is a Lagrange multiplier set to meet the sum power constraint. The optimum set of transmission powers can be found by taking the ﬁrst derivative of the Lagrangian function with respect to P1 , P2 , · · · , PM . Moreover, since the transmission powers {Pk , ∀k} are non-negative, the optimal power allocation must satisfy the Karush-Kuhn-Tucker (KKT) conditions where, for k = 1, · · · , K, it follows that ∂L(P1 , · · · , PM , λ) = 0, ∂Pk if Pk > 0, and

∂L(P1 , · · · , PM , λ) ≤ 0, ∂Pk

if Pk = 0. To be more speciﬁc, if the ﬁrst derivative of the Lagrange function with respect to Pk is negative at the point where Pk = 0, the maximum value of the Lagrange function occurs at a negative value of Pk and the Lagrangian function will monotonically decreasing with respect to Pk for Pk ≥ 0. The optimal value of Pk in this case is 0. Otherwise, the optimal Pk is positive and can be found by setting the ﬁrst derivative to zero. The ﬁrst derivative

2.3 Capacity of Wireless Channels

57

Fig. 2.19 Illustration of waterﬁlling solution in (2.90).

of L(P1 , · · · , PM , λ) with respective to Pk is given by −1 Pk |μk |2 |μk |2 ∂L(P1 , · · · , PM , λ) = (log2 e) · 1 + · 2 − λ. 2 ∂Pk σw σw Following the KKT conditions, the optimal transmission power is , 2 ≤ λ, 0, if (log2 e)|μk |2 /σw 2 Pk = log2 e , σw − , otherwise. λ |μk |2 + σ2 , k = 1, 2, · · · , M, = η − w2 |μk |

(2.89)

(2.90)

where η = log2 e/λ is a constant set to meet the sum power constraint in (2.87) and (x)+ = max(x, 0). The value of η can be found by the water-ﬁlling algorithm, as illustrated in Fig. 2.19, and the optimal solution in (2.90) is referred to as the water-ﬁlling solution. With the optimal pre-coder and the optimal decoder, the channel capacity is given by η|μk |2 C= . (2.91) log2 2 σw k:Pk >0

Consider the fading channel for example again, if the transmitter has knowledge of the instantaneous channel state H, the covariance Rs of channel input symbols can always be adapted accordingly to achieve the channel capacity. Therefore, the ergodic capacity of a MIMO system with full CSI at the transmitter is given by

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2 Review of Wireless Communications and MIMO Techniques

C = EH

max

Rs :Tr(Rs )=1

*

log2 det INr ×Nr + SNRHRs H

H

+

.

(2.92)

If the transmitter knows only the channel distribution, the covariance matrix of the channel input symbols can be set to maximize the average channel capacity by exploiting the channel statistics. Therefore the ergodic capacity of the MIMO system with statistical CSI at the transmitter is given by * + C= max EH log2 det INr ×Nr + SNRHRs HH . (2.93) Rs :Tr(Rs )=1

On the contrary, if the transmitter does not have any channel information, it can simply set the input symbols’ covariance by Rs = N1t INt ×Nt and the resulting ergodic capacity is given by SNR H C = EH log2 det INr ×Nr + . (2.94) HH Nt It can be veriﬁed that, in high SNR regime, the ergodic capacity can be approximated by [10] C ∼ min(Nt , Nr ) log2 SNR + O(1),

(2.95)

which implies that in a MIMO system where no CSI is available at the transmitter, the ergodic capacity increases linearly with the transmit SNR (in dB) with rate min(Nt , Nr ). It is worthy to note that the increasing rate of the ergodic capacity in MISO systems does not depend on the number of transmit antennas since min(Nt , Nr ) = 1. For MIMO systems where no CSI is available at the transmitter, the ergodic capacity of various numbers of transmit antennas and receive antennas are compared in Fig. 2.20. Each channel coefﬁcient is i.i.d. Gaussian with CN (0, 1). It can be observed that the increase in the ergodic capacity is proportional to the minimum number between the numbers of transmit antennas and receive antennas. Besides, without CSI at the transmitter, increasing the number of receive antennas is more eﬃcient than increasing that of transmit antennas.

2.4 Diversity-and-Multiplexing Tradeoﬀ A fundamental concept in MIMO wireless communications is the inherent tradeoﬀ between diversity and multiplexing gains. It has been shown that, to increase diversity gains, one must tradeoﬀ certain degrees of freedom available in the system and sacriﬁce the ability to achieve higher transmission rates. Similarly, in order to achieve a faster increase in transmission rate, with respect to the transmit power, one must tradeoﬀ the exponential decay rate of the error probability. These concepts are introduced in this section.

2.4 Diversity-and-Multiplexing Tradeoﬀ

59

35 N =1, N =1 t

30

25

r

N =1, N =2 t

r

Nt=2, Nr=1 Nt=2, Nr=2 N =4, N =4

C (bits/sec/Hz)

t

r

20

15

10

5

0 0

5

10

15 SNR (dB)

20

25

30

Fig. 2.20 Ergodic capacity of a multiple-input multiple-output (MIMO) channel where the CSI is only available at the receiver.

Let us start with a simple SISO system with a Rayleigh-faded channel. The signal received at the receiver is given by √ y[m] = P hx[m] + w[m], (2.96) where P is the transmission power, h ∼ CN (0, 1) is the channel coeﬃcient which is a complex Gaussian random variable, w[m] is the AWGN with zero 2 mean and variance σw , and x[m] is the transmitted symbol with unit power constraint, i.e., E[|x[m]|2 ] ≤ 1. Suppose that the transmitted symbols are modulated with pulse amplitude modulation (PAM). For instance, the PAM constellation points with data rate R = 1 bits/sec/Hz are {±1} (i.e., BPSK modulation), and the average symbol error probability is given by ( ) 1 SNR 2 pe = E|h|2 Q 1− 2|h| SNR = 2 1 + SNR ' 1 (d/2)2 1− , (2.97) = 2 1 + (d/2)2 √ 2 where SNR = P/σw is the transmit SNR and d = 2 SNR is the Euclidean distance between the two signal points. In general, for√a PAM scheme with √ rate R, there are 2R points ranging from − SNR to + SNR on the constellation, and the minimum distance between any two constellation points can

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2 Review of Wireless Communications and MIMO Techniques

be approximated by

√ 2 SNR , 2R when R is large. The error probability for the SISO system with rate-R PAM can be approximated in terms of the minimum distance by ' ' 2 (Dmin /2)2 Dmin 1 1 1− = 1− . (2.98) pe ≈ 2 2 (Dmin /2)2 + 1 2 Dmin +4 Dmin ≈

At high SNR, the minimum distance Dmin is suﬃciently large and, by Taylor approximation, we can further approximate the error probability in (2.98) as ' 2 1 1 22R−2 Dmin pe ≈ 1− ≈ 2 = . (2.99) 2 2 Dmin + 4 Dmin SNR From (2.99), we can ﬁnd that the error probability is a function of the data rate R and the SNR. If the transmit SNR is increased by 4 times (i.e., P = 4P ) and the data rate R is ﬁxed, the minimum distance is enlarged and the error probability are quartered, i.e., pe = pe /4. On the other hand, we may ﬁxed the error probability, i.e., the minimum distance, and we can double the number of points in the constellation to increase the transmission rate by one, i.e., R = R + 1. Therefore, the increase in the transmit power can either decreases the error probability by one quarter or increases the transmission rate by one. Hence, we can have the transmission rate increased by ΔR < 1 and the error probability decreased to pe > pe /4 by using time-sharing among the two extreme cases. As the increase in data rate approaches one, we trade the error performance for extra data rates. From this example, it is observed that there is a tradeoﬀ between the decrease in error probability and the increase in data rate as the transmit SNR increases. The tradeoﬀ is termed the diversity-and-multiplexing tradeoﬀ (DMT). Before taking a deeper look at the DMT, let us deﬁne the diversity gain and multiplexing gain mathematically. Deﬁnition 2.1. The diversity gain of a system with the average error probability in terms of the transmit SNR, i.e., pe (SNR), is deﬁned by d = − lim

SNR→∞

log pe (SNR) . log SNR

(2.100)

The diversity gain of a system characterizes how the average error probability decreases with the SNR. To be more speciﬁc, a communication system achieves diversity gain d if, in the high SNR regime, the average error probability scales as pe (SNR) ∼ c · SNR−d , for some constant c that is independent of the SNR. That is, in the high SNR regime, increasing the transmit SNR by 3 dB leads to the decrease of

2.4 Diversity-and-Multiplexing Tradeoﬀ

61

error probability by c · 2−d . In the literature, the error probability in (2.100) is sometimes replaced with the outage probability. Both probabilities have been used to deﬁne diversity gains. For example, consider a SISO channel with PAM modulation, the maximum diversity is dmax = 1 since the error probability decays with the inverse of SNR as given in (2.99). For a MIMO system with Nt transmit antennas and Nr receive antennas, the maximum diversity gain in general is shown to be d∗max = Nt × Nr , as given in (2.75) The notion of multiplexing gain is deﬁned as follows. Deﬁnition 2.2. The multiplexing gain of a system with the achievable rate in terms of transmit SNR, i.e., R(SNR), is deﬁned by r = lim

SNR→∞

R(SNR) . log SNR

(2.101)

To be speciﬁc, in the high SNR regime, the transmission rate of a communication system with multiplexing gain r scales as R(SNR) ∼ r · log(SNR). In other words, for a system with multiplexing gain r, each 3 dB increase in the transmit SNR leads to the increase of transmission rate by r bits/sec/Hz. The multiplexing gain characterizes the proportional increase of transmission rate with respect to log SNR when the asymptotic decay rate of the error probability (or the outage probability) does not change. The maximum multiplexing gain is achieved when the error probability is ﬁxed at any SNR. For example, the maximum multiplexing gain for a SISO system with PAM modulation equals to rmax = 1/2. For a MIMO channel with Nt transmit antennas, Nr receive antennas, and no CSI at the transmitter, the ergodic capacity at high SNR under Rayleigh fading has been given in (2.95). The ergodic capacity can be viewed as the achievable rate for an arbitrary small probability of errors. From the approximation in (2.95), the maximum multiplexing gain of a MIMO channel equals to ∗ rmax = min(Nt , Nr ).

When Nt = Nr = 1, the maximum multiplexing gain is one, which means that the uncoded transmission scheme modulated with PAM does not achieve the highest multiplexing gain of the SISO channel. By the PAM example, we know that we can not achieve maximum diversity gain and maximum multiplexing gain at the same time. For a channel with ∗ a given multiplexing gain r ≤ rmax , the transmission rate at high SNR is R(SNR) = r · log SNR and the diversity gain can be obtained from the error performance or outage performance of the system. The relationship between the diversity gain and the multiplexing gain is mathematically deﬁned as follows.

62

2 Review of Wireless Communications and MIMO Techniques

Deﬁnition 2.3. Given a multiplexing rate r, a diversity gain d(r) is achieved if, in high SNR regime, R(SNR) = r · log(SNR) and

log pe (SNR) . log SNR

d(r) = − lim

SNR→∞

(2.102)

(2.103)

Consider a SISO system with PAM modulation, for example, the maximum diversity gain is dmax = 1 and the maximum multiplexing gain is rmax = 1/2. In this case, a multiplexing gain of r ≤ 1/2 can be achieved by time-sharing among the two cases. The resulting diversity gain is given by d(r) = 1 − 2r,

0 ≤ r ≤ 1/2.

For a general SISO system with Rayleigh fading coeﬃcient h ∼ CN (0, 1), the error probability for a given rate R = r log SNR is . / pe = Pr log(1 + |h|2 SNR) < r log SNR 0

SNRr − 1 2 = Pr |h| < SNR SNRr − 1 = 1 − exp − SNR 1 ≈ . SNR1−r Hence, the optimal diversity-and-multiplexing tradeoﬀ of a general SISO channel, under the Rayleigh fading environment, is given by d∗ (r) = 1 − r,

r ∈ [0, 1].

(2.104)

For a MISO system with Nt transmit antennas, the error probability decays Nt times faster than that in a general SISO system. The diversity-andmultiplexing tradeoﬀ for the MISO system is given by d∗ (r) = Nt (1 − r),

r ∈ [0, 1].

One can draw the points (r, d∗ (r)), for r ∈ (0, 1), on a graph to gain knowledge of the relationship between diversity and multiplexing gains. For SISO and MISO system, the optimum DMT is just a straight line. The endpoints of the line are the point with maximum diversity gain and the point with maximum multiplexing gain. On the other hand, for a MIMO system with Nt transmit antennas and Nr receive antennas, the achievable diversity-and-multiplexing tradeoﬀ is proven to be [40]

References

63

d∗ (r) = (Nt − r)(Nr − r),

r = 0, 1, · · · , min(Nt , Nr )

(2.105)

By time-sharing of the cases with integer multiplexing gains, the curve of the maximum DMT is piecewise-linear connecting the points (r, (Nt − r)(Nr − r)),

r = 0, 1, · · · , min(Nt , Nr ).

With r → 0, the maximum diversity gain Nt Nr is achieved with very low multiplexing gain. As r → min(Nt , Nr ), we have the maximum multiplexing gain but the diversity gain is 0. Moreover, for a given diversity order, the multiplexing gain is increased by one only when both numbers of the transmit and receive antennas are increase by one simultaneously.

References 1. Alamouti, S.M.: A simple transmit diversity technique for wireless communications. IEEE Journal on Selected Areas in Communications 16(8), 1451–1458 (1998) 2. Bernhardt, R.: Macroscopic diversity in frequency reuse radio systems. IEEE Journal on Selected Areas in Communications 5(5), 862–870 (1999) 3. Beverage, H.H., Peterson, H.O.: Diversity receiving system of RCA communications, inc., for radiotelegraphy. In: Proceedings of IRE, pp. 531–561 (1931) 4. Brennan, D.: Linear diversity combining techniques. Proceddings of the IEEE 91(2), 331–356 (2003) 5. Brennan, D.G.: On the maximal signal-to-noise ratio realizable from several noisy signals. In: Proceedings of IRE, p. 1530 (1955) 6. Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2 edn. Wiley-Interscience (2006) 7. Cox, D., Murray, R., Norris, A.: 800 MHz attenuation measured in and around suburban houses. AT&T Bell Laboratory Technical Journal 63(6), 921–954 (1984) 8. Duman, T.M., Ghrayeb, A.: Coding for MIMO Communication Systems, John-Wiley & Sons Ltd. (2007) 9. Erceg, V., Greenstein, L., Tjandra, S., Parkoﬀ, S., Gupta, A., Kulic, B., Julius, A.A., Bianchi, R.: An empirically based path loss model for wireless channels in suburban environments. IEEE Journal on Selected Areas in Communications 17(7), 1205–1211 (1999) 10. Foschini, G.J.: Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas. Bell System Technical Journal 1, 44–59 (1996) 11. Goldsmith, A.: Wireless Communications. Cambridge University Press (2005) 12. Goldsmith, A.J., Varaiya, P.P.: Capacity of fading channels with channel side information. IEEE Transactions on Information Theory 43(6), 1986–1992 (1997) 13. Hassibi, B., Hochwald, B.: High-rate codes that are linear in space and time. IEEE Transactions on Information Theory 48(7), 1804–1824 (2002) 14. Jafar, S.A., Goldsmith, A.: On optimality of beamforming for multiple antenna systems. In: Proc. on IEEE International Symposium on Information Theory (ISIT), pp. 321 (2001) 15. Jafarkhani, H.: A quasi-orthogonal space-time block code. IEEE Transactions on Communications 49(1), 1–4 (2001) 16. Jakes, W.C.: Microwave Mobile Communications, 2 edn. IEEE Press (1994)

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17. Love, D.J., Heath, Jr., R.W., Strohmer, T.: Grassmannian beamforming for multipleinput multiple-output wireless systems. IEEE Transactions on Information Theory 49(10), 2735–2747 (2003) 18. Mukkavilli, K., Sabharwal, A., Erkip, E., Aazhang, B.: On beamforming with ﬁnite rate feedback in multiple-antenna systems. IEEE Transactions on Information Theory 49(10), 2562–2579 (2003) 19. Nakagami, N.: The m-distribution, a general formula for intensity distribution of rapid fading. Statistical Methods in Radio Wave Propagation, Edited by W. G. Hoﬀman, Oxford, England (1960) 20. Owen, F., Pudney, C.: Radio propagation for digital cordless telephones at 1700 MHz and 900 MHz. IEEE Electronics Letters 25(1), 52–53 (1989) 21. Parsons, D.: The Mobile Radio Propagation Channel. Willey (1994) 22. Peterson, H.O., Beverage, H.H., Moore, J.: Diversity telephone receiving system of RCA communications, inc. In: Proceedings of IRE, pp. 562–584 (1931) 23. Rappaport, T.: Wireless Communications, 2nd edn. Prentice-Hall (2001) 24. Rice, S.: Mathematical analysis of random noise. AT&T Bell Laboratory Technical Journal 23(2), 282–333 (1944) 25. Roh, J.C., Rao, B.D.: Multiple antenna channels with partial channel state information at the transmitter. IEEE Transactions on Wireless Communications 3(2), 677–688 (2004) 26. Roh, J.C., Rao, B.D.: Transmit beamforming in multiple-antenna systems with ﬁnite rate feedback: a VQ-based approach. IEEE Transactions on Information Theory 52(3), 1101–1112 (2006) 27. Rustako, Jr., A.J., Amiray, N., Owens, G., Roman, R.: Radio propagation at microwave frequencies for line-of-sight microcellular mobile and personal communications. IEEE Transactions on Vehicular Technology 40(1), 203– 210 (1991) 28. Sanayei, S., Nosratinia, A.: Antenna selection in MIMO systems. IEEE Communications Magazine 42(10), 68–73 (2004) 29. Sanayei, S., Nosratinia, A.: Asymptotic capacity analysis of transmit antenna selection. In: Proc. on IEEE International Symposium on Information Theory (ISIT), pp. 241 (2004) 30. Seidel, S., Rappaport, T., Jain, S., Lord, M., Singh, R.: Path loss, scattering, and multipath delay statistics in four European cities for digital cellular and microcellular radiotelephone. IEEE Transactions on Vehicular Technology 40(4), 721–730 (1991) 31. Shannon, C.E.: A mathematical theory of communication. Bell System Technical Journal 27(3), 379–423, 623–656 (1948) 32. Tarokh, V., Jafarkhani, H., Calderbank, A.: Space-time block codes from orthogonal designs. IEEE Transactions on Information Theory 45(5), 1456–1467 (1999) 33. Tarokh, V., Seshadri, N., Calderbank, A.: Space-time codes for high data rate wireless communication: performance criterion and code construction. IEEE Transactions on Information Theory 44(2), 744–765 (1999) ˙ 34. Telatar, I.E.: Capacity of multi-antenna Gaussian channels. European Transactions on Telecommunications 10(6), 585–595 (1999) 35. de Toledo, A.F., Turkmani, A.M.D.: Propagation into and within buildings at 900, 1800, and 2300 MHz. In: Proc. on IEEE Vehicular Technology Conference (VTC), pp. 633– 636 (1992) 36. de Toledo, A.F., Turkmani, A.M.D., Parsons, J.D.: Estimating coverage of radio transmission into and within buildings at 900, 1800, and 2300 MHz. IEEE Personal Communications Magazine 5(2), pp. 40–47 (1998) 37. Tse, D., Viswanath, P.: Fundamentals of Wireless Communication. Cambridge University Press (2005) 38. Xia, P., Giannakis, G.B.: Design and analysis of transmit-beamforming based on limited-rate feedback. IEEE Transactions on Signal Processing 54(5), 1853–1863 (2006)

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39. Yan, Q., Blum, R.: Improved space-time convolutional codes for quasi-static slow fading channels. IEEE Transactions on Wireless Communications 1(4), 563–571 (2002) 40. Zheng, L., Tse, D.N.C.: Diversity and multiplexing: A fundamental tradeoﬀ in multiple-antenna channels. IEEE Transactions on Information Theory 49(5), 1073– 1096 (2003) 41. Zhou, S., Giannakis, G.: Optimal transmitter eigen-beamforming and space-time block coding based on channel mean feedback. IEEE Transactions on Signal Processing 50(10), 2599–2613 (2002)

Chapter 3

Two-User Cooperative Transmission Schemes

Cooperative communications refer to systems or techniques that allow users to help transmit each other’s messages to the destination. Most cooperative transmission schemes involve two phases of transmission: a coordination phase, where users exchange their own source data and control messages with each other and/or the destination, and a cooperation phase, where the users cooperatively retransmit their messages to the destination. A basic cooperation system consists of two users transmitting to a common destination, as illustrated in Fig. 3.1. At any instant in time, one user acts as the source while the other user serves as the relay. In the coordination phase (i.e., Phase I), the source user broadcasts its data to both the relay and the destination and, in the cooperation phase (i.e., Phase II), the relay forwards the source’s data (either by itself or by cooperating with the source) to enhance reception at the destination. The two users may interchange their roles as source and relay at diﬀerent instants in time. To enable such cooperation among users, diﬀerent relay technology can be employed depending on the relative user location, channel conditions, and transceiver complexity. In this chapter, we introduce some of the basic cooperative relaying techniques such as decode-and-forward (DF) [18], amplify-and-forward (AF) [18], coded cooperation (CC) [10, 12, 14], and compress-and-forward (CF) [9, 17, 20, 21] etc. Moreover, for practical purposes, we also brieﬂy discuss how channel estimation can be performed in cooperative systems to enable coherent detection at the destination.

3.1 Decode-and-Forward Relaying Schemes Decode-and-forward (DF) relaying schemes refer to cases where the relay explicitly decodes the message transmitted by the source and forwards a newly generated signal to the destination, as illustrated in Fig. 3.2. These schemes are also known as regenerative relaying schemes, which have been widely

Y.-W. Peter Hong et al., Cooperative Communications and Networking, DOI 10.1007/978-1-4419-7194-4_3, © Springer Science+Business Media, LLC 2010

67

68

3 Two-User Cooperative Transmission Schemes

Fig. 3.1 A simple protocol of a two-user cooperative network.

adopted in the literature, including those employed in conventional multihop networks. In this section, we introduce three variants of DF relaying schemes: the basic DF relaying scheme (Basic DF), the selection DF relaying scheme (Selection DF), and the demodulate-and-forward (DeF) relaying scheme. In Basic DF, the relay is assigned to forward the source message in Phase II, given that it has successfully decoded it in Phase I. However, in Selection DF, the source is allowed to retransmit the message by itself in Phase II if the relay is not able to successfully decode the message in Phase I. Moreover, in cases where the relay is not able to perform channel decoding or encoding, either due to limited transceiver capability or lack of knowledge on the channel codebook, one can employ the DeF relaying scheme where the relay demodulates and forwards the message on a symbol-by-symbol basis. These schemes are described in the following.

3.1.1 Basic DF Relaying Scheme In the basic DF relaying scheme, the source ﬁrst transmits a message to both the relay and the destination in Phase I. Then, if the relay is able to successfully decode the message, it will regenerate the same message and

3.1 Decode-and-Forward Relaying Schemes

69

Fig. 3.2 Illustration of decode-and-forward (DF) relaying schemes.

forward it to the destination in Phase II. Assume that, in each cooperative transmission, an equal amount of time is allocated to Phase I and Phase II. Speciﬁcally, let xs = [xs [0], xs [1], . . . , xs [M − 1]] be the length-M code word transmitted by the source in Phase I and let E |xs [m]|2 = 1 for all m. Due to the broadcast nature of the wireless medium, both the relay and the destination will receive a noisy version of the signal, which are given by yr [m] = hs,r Ps xs [m] + wr [m], (3.1) (1) (1) yd [m] = hs,d Ps xs [m] + wd [m], (3.2) respectively, for m = 0, 1, · · · , M − 1. Here, Ps is the source transmission power, hs,r and hs,d are the channel coeﬃcients of the source-to-relay (s-r) and the source-to-destination (s-d) links, respectively, and wr [m] ∼ CN (0, σr2 ) (1) and wd [m] ∼ CN (0, σd2 ) are the additive white Gaussian noise (AWGN) at the relay and the destination, respectively. The channel is assumed to remain constant over the transmission of a codeword but varies independently and identically from block-to-block. With suﬃciently long codewords, one can invoke the channel coding theorem and assume that decoding at the relay is successful if and only if the transmission rate is no greater than the capacity of the s-r link, which is given by Cs,r (γs,r ) = log2 (1 + γs,r )

bits per channel use,

(3.3)

where γs,r = Ps |hs,r |2 /σr2 . If the desired average end-to-end rate is R, the codeword xs must be encoded with rate 2R since it will be transmitted twice (i.e, once by the source and once by the relay) throughout the transmission process. In this case, an outage occurs on the s-r link if 2R > Cs,r (γs,r ). When the relay is able to correctly decode the message, i.e., when 2R ≤ Cs,r (γs,r ), it will re-encode the message into a codeword xr using the same

70

3 Two-User Cooperative Transmission Schemes

codebook1 , such that xr = xs , and retransmit it to the destination in Phase II. Thus, the signal received at the destination in Phase II is given by (2) (2) yd [m] = hr,d Pr xs [m] + wd [m], for m = 0, 1, · · · , M − 1, (3.4) where Pr is the relay transmission power, hr,d is the channel coeﬃcient of the (2) relay-to-destination (r-d) link, and wd [m] ∼ CN (0, σd2 ) is the AWGN at the destination in the m-th symbol period of Phase II. In the two-phase cooperation scheme, two copies of the message will be received at the destination — one in Phase I (i.e., (3.2)) and another in Phase II (i.e., (3.4)). When the distance between source and destination is large, the signal received on the s-d link will be negligible and, thus, can be discarded at the destination. However, if both signals have comparable strength, they can be properly combined to enhance detection at the destination. In the following, we shall refer to the former as the case without diversity combining (i.e., Case I) and the latter as the case with diversity combining (i.e., Case II). The outage probability for both cases are analyzed in the following and the optimal power allocation between source and relay is derived under diﬀerent assumptions on the channel state information (CSI).

Case I: Without Diversity Combining In the case without diversity combining, the destination performs detection based only on the signal received from the relay in Phase II. The scheme is identical to conventional multi-hop (or, in this case, dual-hop) transmissions. In order to successfully transmit a codeword over both s-r and r-d links in this case, the rate of the codeword must be bounded by the capacity of both links, i.e., 2R ≤ min {log2 (1 + γs,r ) , log2 (1 + γr,d )} , (3.5) where γs,r = Ps |hs,r |2 /σr2 and γr,d = Pr |hr,d |2 /σd2 . Hence, the average endto-end achievable rate in Case I (without diversity combining) is given by CBasicDF,I (γs,r , γr,d ) =

1 min {log2 (1 + γs,r ) , log2 (1 + γr,d )} . 2

(3.6)

Hence, an outage occurs when R > CBasicDF,I . Let us consider the Rayleigh fading scenario, where hs,r , hr,d and hs,d are independent circularly symmetric complex Gaussian random variables 2 2 2 with zero mean and variances ηs,r , ηr,d , and ηs,d , respectively. The outage probability can be computed as follows:

1 In this section, we assume that the relay employs the same channel codebook as the source. Information-theoretic studies that may involve diﬀerent codebooks at the source and the relay will be discussed in Chapter 5.

3.1 Decode-and-Forward Relaying Schemes

71

pout = Pr (min {log2 (1 + γs,r ) , log2 (1 + γr,d )} < 2R) = 1 − Pr (min {log2 (1 + γs,r ) , log2 (1 + γr,d )} ≥ 2R) = 1 − Pr (log2 (1 + γs,r ) ≥ 2R, log2 (1 + γr,d ) ≥ 2R) . Since hs,r and hr,d are assumed to be independent with distributions hs,r ∼ 2 2 CN (0, ηs,r ) and hr,d ∼ CN (0, ηr,d ), respectively, it follows that γs,r and γr,d are exponentially distributed with mean γ s,r

2 Ps ηs,r σr2

and γ r,d

2 Pr ηr,d , σd2

respectively. Thus, the outage probability can be further evaluated as * + * + pout = 1 − Pr γs,r ≥ 22R − 1 Pr γr,d ≥ 22R − 1 2R 2R 2 −1 2 −1 exp − . (3.7) = 1 − exp − γ s,r γ r,d Given the total power constraint Ps + Pr ≤ 2P , we can set Ps = 2βP and 2 Pr = 2(1 − β)P , for some β ∈ (0, 1), and let σr2 = σd2 = σw . For example, if β = 1/2, then we have Ps = Pr = P . Then, at high SNR (i.e., when 2 SNR P/σw 0), the outage probability can be approximated as 1 22R − 1 22R − 1 22R − 1 1 . (3.8) + = + pout ≈ 2 2 γ s,r γ r,d SNR 2βηs,r 2(1 − β)ηr,d This shows that the diversity order in Basic DF without diversity combining is equal to 1. It is worthwhile to remark that, in the case of direct transmission, the outage probability is given by * + pdirect = Pr (log2 (1 + γs,d ) < R) = Pr γs,d < 2R − 1 out R 2R − 1 1 2 −1 · 2 , ≈ (3.9) = 1 − exp − γ s,d SNR ηs,d 2 2 2 where γs,d = P |hs,d |2 /σw and γ s,d = E[γs,d ] = P ηs,d /σw . By comparing (3.8) with (3.9), one can see that no diversity gain is obtained over direct transmission by employing multi-hop transmission (i.e., Basic DF without diversity combining). Even though this may be the case, multi-hop transmissions may still be useful in practice where the minimum adaptable transmission rate and the maximum transmission power are limited. Although diversity gains cannot be obtained in this case, one can still perform power allocation to improve the coding gains. Speciﬁcally, given the total power constraint Ps + Pr ≤ 2P , the optimal power allocation between Ps and Pr can be found by minimizing the outage probability in (3.7). Due to monotonicity of the exponential function, the optimization problem (c.f. [8])

72

3 Two-User Cooperative Transmission Schemes

can be formulated as follows: min

Ps ,Pr

1 1 + 2 /σ 2 2 /σ 2 Ps ηs,r P η r r,d r d

subject to Ps + Pr ≤ 2P

and Ps , Pr ≥ 0.

(3.10) (3.11)

By introducing Lagrange multipliers, the optimal power allocation can be derived as Ps = 2P ·

2 2 /σd2 ηr,d ηs,r /σr2 and P = 2P · . r 2 /σ 2 + η 2 /σ 2 2 /σ 2 + η 2 /σ 2 ηs,r ηs,r r r r,d d r,d d

(3.12)

2 Thus, by substituting (3.12) into (3.7) and by assuming that σr2 = σd2 = σw , the minimum outage probability can be expressed as 1 22R − 1 1 pout = 1 − exp − (3.13) + 2 2 /σ 2 P ηs,r ηr,d /σd2 r 1 22R − 1 1 . (3.14) = 1 − exp − + 2 2 SNR ηs,r ηr,d

Notice that, when the optimal power allocation is found by minimizing the outage probability, only the statistics of the channel must be known. However, when instantaneous CSI is available at the source and the relay, one can further derive the optimal power allocation to maximize the dualhop capacity CBasicDF,I (γs,r , γr,d ). In this case, the optimal power allocation is determined by solving the following optimization problem [31]:

0 1 Ps |hs,r |2 Pr |hr,d |2 min log2 1 + , log 1 + (3.15) max 2 Ps ,Pr 2 σr2 σd2 subject to Ps + Pr ≤ 2P and Ps , Pr ≥ 0.

(3.16)

One can observe that, to maximize the dual-hop capacity given in (3.15), the power constraint Ps + Pr ≤ 2P must be satisﬁed with equality. Thus, as Ps increases, the ﬁrst term of (3.15) will increase while the second term decreases, and vice versa as Ps decreases. Since either term inside the minimum of (3.15) cannot be increased without decreasing the other, the objective is maximized when both terms are equal, i.e., when Ps |hs,r |2 Pr |hr,d |2 = . σr2 σd2 The optimal power allocation under instantaneous CSI is given by Ps = 2P ·

|hr,d |2 /σd2 |hs,r |2 /σr2 + |hr,d |2 /σd2

(3.17a)

3.1 Decode-and-Forward Relaying Schemes

73

0

10

−1

Outage Probability

10

−2

10

−3

10

No Cooperation Basic DF I, OPA, Statistical CSI Basic DF I, OPA, Instantaneous CSI 0

5

10

15 20 Transmit SNR (dB)

25

30

Fig. 3.3 Performance comparison of the basic DF relaying schemes in Case I (without diversity combining).

and Pr = 2P ·

|hs,r |2 /σr2 . |hs,r |2 /σr2 + |hr,d |2 /σd2

(3.17b)

2 Furthermore, by assuming that σr2 = σd2 = σw , the resulting dual-hop capacity is instead given by 1 2SNR log2 1 + . 2 1/|hs,r |2 + 1/|hr,d|2

Notice that, in the case of direct transmission, the capacity is given by * + log2 1 + SNR · |hs,d |2 . Let us consider, for example, the case where |hs,d |2 ∝ 1/dα and |hs,r |2 , |hr,d |2 ∝ 1/( d2 )α (i.e., the case where the relay is located in the middle of the source and the destination). By utilizing the approximation log(1 + x) ≈ x, for x 1, one can show that a gain of 2α−1 can be obtained at low SNR when taking the ratio between the capacity of Basic DF (without diversity combining) and that of direct transmission. In Fig. 3.3, we show the outage probabilities versus transmit SNR (i.e., 2 SNR = P/σw ) for the basic DF relaying scheme without diversity combining.

74

3 Two-User Cooperative Transmission Schemes

Here, the rate is set as R = 1 bits/sec/Hertz. Let ds,r , dr,d , and ds,d be the distances between s-r, r-d, and s-d, respectively. The channel coeﬃcients are as−α sumed independent with distribution hs,r = CN (0, d−α s,r ), hr,d = CN (0, dr,d ), −α and hs,d = CN (0, ds,d ), where path loss exponent α is set as 3. In this experiment, we assume that the relay is located in the middle between source and destination, so that ds,d = 1 and ds,r = dr,d = 1/2. Setting ds,d = 1 means that the results are normalized by the performance of the s-d link. The Statistical CSI curve is obtained by employing the optimal power allocation (OPA) in (3.12) and the Instantaneous CSI curve is obtained by the optimal power allocation in (3.17). One can observe that, since no diversity gain is obtained, all three curves decrease at the same rate with respect to SNR. Moreover, the Basic DF scheme without diversity combining may not even be able to outperform direct transmission due to its lack of bandwidth eﬃciency (when transmitting the same codeword twice in a cooperative transmission period). However, signiﬁcant power gains can be observed when optimal power allocation is derived under instantaneous CSI.

Case II: With Diversity Combining In the previous case, only the signal transmitted by the relay is utilized for detection at the destination. However, due to the broadcast nature of the wireless medium, the signal transmitted by the source in Phase I will also be received at the destination and, thus, can be combined with the signal received from the relay to increase diversity. This is especially useful when the quality of the s-d link is comparable to that of the r-d link. First, recall that, in the basic DF scheme, the relay is allowed to forward the source’s message in Phase II only if it is able to successfully decode the message in Phase I. If this is the case, the destination will receive two copies of the signal, as given in (3.2) and (3.4), which can be collected into the vector √ (1) (1) P h [m] yd [m] w d yd [m] = = √ s s,d xs [m] + , (3.18) (2) (2) Pr hr,d yd [m] wd [m] for m = 0, . . . , M −1. The signal model is similar to conventional single-input multiple-output (SIMO) systems and the maximal ratio combiner (MRC) can be employed to maximize the receive SNR at the destination. Speciﬁcally, the signals √ received in Phase I and Phase II are multiplied by the weight √ coeﬃcients Ps h∗s,d and Pr h∗r,d , respectively, to obtain √ √ Ps h∗s,d Pr h∗r,d yd [m] * + = Ps |hs,d |2 + Pr |hr,d |2 xs [m] + w ˜d [m],

y˜d [m] =

(3.19)

3.1 Decode-and-Forward Relaying Schemes

75

√ √ (1) (2) where w ˜d [m] = Ps h∗s,d wd [m] + Pr h∗r,d wd [m] ∼ CN (0, σd2 (Ps |hs,d |2 + 2 Pr |hr,d | )) is the eﬀective noise at the output of the MRC. The SNR at the output of the MRC is given by γBasicDF,II =

Ps |hs,d |2 Pr |hr,d |2 + = γs,d + γr,d , 2 σd σd2

(3.20)

where γs,d = Ps |hs,d |2 /σd2 and γr,d = Pr |hr,d |2 /σd2 . Given that the relay successfully decodes the message, the rate achievable in Phase II of the DF scheme (with diversity combining) is given by log2 (1 + γs,d + γr,d ) .

(3.21)

However, in order to have the relay successfully decode the message in Phase I, the rate transmitted by the source must be less than the capacity of the s-r link, i.e., log2 (1 + γs,r ), where γs,r = Ps |hs,r |2 /σr2 . Consequently, the maximum achievable end-to-end rate is given by CBasicDF,II (γ) =

1 min {log2 (1 + γs,r ) , log2 (1 + γs,d + γr,d )} , 2

(3.22)

where γ = (γs,r , γs,d , γr,d ). Therefore, the outage probability of the basic DF relaying scheme in Case II (i.e., with diversity combining) can be computed as 1 pout = Pr min {log2 (1 + γs,r ) , log2 (1 + γs,d + γr,d )} < R 2 1 log2 (1 + γs,r ) < R = Pr 2 1 1 log2 (1 + γs,r ) ≥ R Pr log2 (1 + γs,d + γr,d ) < R + Pr 2 2 + * + * + * 2R 2R = Pr γs,r < 2 − 1 + Pr γs,r ≥ 2 − 1 Pr γs,d +γr,d < 22R − 1 In the Rayleigh fading scenario, the SNR γs,r is exponentially distributed 2 with mean γ s,r = Ps ηs,r /σr2 . Hence, we have pout

2R 2R * + 2 −1 2 −1 + exp − Pr γs,d + γr,d < 22R − 1 . = 1 − exp − γ s,r γ s,r (3.23)

To further evaluate pout , it is necessary to obtain the distribution of γs,d + γr,d . In general, for any two independent exponential random variables U and V , the CDF of W = U + V can be computed as [18]

76

3 Two-User Cooperative Transmission Schemes

Pr (W ≤ w) =

⎧ ( ⎨1 − ⎩

) − μwv v , for μu = μv e + μvμ−μ wu −μ 1− 1+ w , for μu = μv = μ μ e

− μwu μu μu −μv e

where μu and μv are expectations of U and V , respectively. Since γs,d and 2 γr,d are independent exponential random variables with mean Ps ηs,d /σd2 and 2 2 Pr ηr,d /σd , respectively, we can utilize the above result to show that * + Pr γs,d + γr,d < 22R − 1 ⎧ ⎨ 1 − γ s,d exp − 22R −1 − γ r,d exp − 22R −1 , if γ s,d = γ r,d γ s,d −γ r,d γ s,d γ r,d γ r,d s,d −γ2R = 2R ⎩ if γ s,d = γ r,d . 1 − 1 + 2 γ −1 exp − 2 γ −1 , s,d

s,d

(3.24) By substituting (3.24) into (3.23), the outage probability can be expressed as ⎧ 2R ( 2R γ s,d 2 −1 ⎪ 1 − exp − exp − 2 −1 ⎪ ⎪ γ s,r γ s,d −γ ⎨ r,d 2R ) γ s,d γ r,d pout = if γ s,d = γ r,d exp − 2 γ −1 , + γ −γ r,d s,d r,d ⎪ ⎪ 2R 2R 2R ⎪ ⎩ 1 − 1 + 2 −1 exp − 2 −1 − 2 −1 , if γ s,d = γ r,d . γ γ γ s,d

s,r

s,d

(3.25) At high SNR (i.e., when γ s,d , γ s,r , γ r,d 0), we can adopt the ﬁrst order Taylor approximation (where e−x ≈ 1 − x for x suﬃciently small) to obtain an approximation of the outage probability as ⎧ 22R −1 ⎨ if γ s,d = γ r,d γ s,r , 2 pout ≈ 2R 2R 2 −1 ) ( 1 ⎩ 2 −1 + + γ 1 , if γ s,r = γ r,d . γ γ γ s,r

s,d

s,r

s,d

By letting Ps = 2βP and Pr = 2(1 − β)P (0 < β ≤ 1) and by assuming that 2 σr2 = σd2 = σw , we can further show that ⎧ 22R −1 1 2 2 ⎨ if βηs,d

= (1 − β)ηr,d 2βη 2 SNR , (3.26) pout ≈ 22R −1 1 s,r * 1 + 2 2 ⎩ 2βη2 SNR + O SNR2 , if βηs,d = (1 − β)ηr,d . s,r

2 where SNR = P/σw . This shows that the diversity order of the basic DF relaying scheme is still equal to 1 even when diversity combining is employed at the destination. This is due to the fact that the relay must ﬁrst be able to successfully decode the message before the cooperative transmission can take place. Thus, the achievable rate is limited by the capacity of the s-r link. However, this can be improved upon if we allow the destination to decode

3.1 Decode-and-Forward Relaying Schemes

77

the message based only on the signal received from the source, which leads to the so called selection DF relaying scheme discussed in the following section. To further minimize outage probability, we can also derive the optimal power allocation that minimizes the expression in (3.24). However, a closedform of the solution is not easily attainable in this case and, thus, numerical methods must be utilized instead. On the other hand, when instantaneous CSI is available at the transmitters, one can also employ the optimal power allocation to maximize the achievable end-to-end rate given in (3.22). Specifically, given the total power constraint Ps + Pr ≤ 2P , the optimal power allocation can be found by solving the following optimization problem [31]:

0 1 Ps |hs,r |2 Ps |hs,d |2 Pr |hr,d |2 min log2 1 + , log2 1 + max + Ps ,Pr 2 σr2 σd2 σd2 subject to

Ps + Pr ≤ 2P and Ps , Pr ≥ 0.

Since the achievable rate is maximized when Ps + Pr = 2P , we can substitute Pr with 2P − Ps and omit the factor constant 1/2 to simplify the problem as 0

Ps |hs,r |2 Ps (|hs,d |2 −|hr,d|2 ) 2P |hr,d|2 , log2 1+ . + max min log2 1+ 0≤Ps ≤2P σr2 σd2 σd2 (3.27) When |hs,d |2 ≥ |hr,d |2 , one can observe that both terms inside the minimum operation in (3.27) are monotonically increasing with respect to the power Ps and, thus, the achievable rate is maximized by choosing Ps = 2P and Pr = 0. This implies that, when the quality of the s-d link is better than that of the r-d link, one need not rely on help from the relay. On the other hand, when |hs,d |2 < |hr,d |2 , the ﬁrst term inside the minimum operation increases with Ps while the second term decreases with Ps . In this case, the objective is maximized when both terms are equal and, thus, the optimal power allocation is given by Ps = 2P · and Pr = 2P ·

|hs,r |2 /σr2

|hr,d |2 /σd2 + (|hr,d |2 − |hs,d |2 )/σd2

|hs,r |2 /σr2 − |hs,d |2 /σd2 . 2 |hs,r | /σr2 + (|hr,d |2 − |hs,d |2 )/σd2

(3.28a)

(3.28b)

By comparing the results of Case I and Case II, i.e., (3.17) and (3.28), one can observe that more power is allocated to the source when diversity combining is employed at the destination. The reason is that, when diversity combining is employed, the signal transmitted by the source will be utilized for detection at both the relay and the destination and, thereby, is more essential to achieving a higher performance in the cooperative transmission. In Fig. 3.4, we compare the outage probabilities of the basic DF relaying strategies with and without diversity combining at the destination. The

78

3 Two-User Cooperative Transmission Schemes

0

10

−1

Outage Probability

10

−2

10

No Cooperation Basic DF I, OPA, Statistical CSI Basic DF I, OPA, Instantaneous CSI Basic DF II, OPA, Statistical CSI Basic DF II, OPA, Instantaneous CSI

−3

10

0

6

12 Transmit SNR (dB)

18

24

Fig. 3.4 Performance comparison of basic DF relaying schemes.

experiments are performed with optimal power allocations (OPA) based on instantaneous CSI and statistical CSI (obtained by numerically minimizing outage probability). The simulation scenario is the same as that in Fig. 3.3. We can see that the diversity order is the same for all schemes but a power gain is observed when optimal power allocation based on instantaneous CSI is employed and/or when diversity combining is performed at the destination.

3.1.2 Selection DF Relaying Scheme In basic DF relaying schemes, the outage performance is limited by the quality of the s-r link since the relay is required to successfully decode the source’s message. However, the diversity order can be improved if the strategy does not insist on having the relay participate in the cooperative transmission of Phase II. Speciﬁcally, in the selection DF relaying scheme (as proposed in [18]), the source can choose to retransmit the message itself in Phase II if the relay was not able to successfully decode the message in Phase I. We assume that the source can obtain knowledge of the instantaneous CSI on the s-r link and, thus, is able to infer whether or not the relay has successfully decoded the message.

3.1 Decode-and-Forward Relaying Schemes

(a) Phase I

79

(b) Phase II

Fig. 3.5 Illustration of the selection DF relaying scheme.

The selection DF relaying scheme takes on two phases of transmission, as illustrated in Fig. 3.5. In Phase I, the source transmits a length-M codeword xs = [xs [0], . . . , xs [M − 1]] to both the relay and the destination, similar to the basic DF scheme. The received signal at the relay and the destination are given, respectively, by (3.1) and (3.2). Similarly, to achieve the end-to-end rate R, the source must encode the message with rate 2R and, thus, the relay is able to successfully decode the message only if 2R < log2 (1 + γs,r ). If the relay is able to successfully decode the message, it will regenerate the same codeword xr = xs and forward it to the destination. Otherwise, the source will retransmit the codeword xs in Phase II, without help from the relay. The signal received at the destination in Phase II is given by , √ (2) hr,d Pr xs [m] + wd [m], if γs,r ≥ 22R − 1 (2) yd [m] = (3.29) √ (2) hs,d Pr xs [m] + wd [m], if γs,r < 22R − 1 (2)

where γs,r Ps |hs,r |2 /σr2 is the SNR on the s-r link and wd [m] ∼ CN (0, σd2 ) is the AWGN at the destination in Phase II. Notice that the source also transmits with power Pr in Phase II so that the same total power Ps +Pr = 2P is utilized in both cases. It is worthwhile to remark that, here, we are utilizing only the simple repetition code where the codeword transmitted by either the source or the relay in Phase II is a repetition of the codeword transmitted in Phase I. In general, more powerful codes can be employed (see Chapter 5) but the repetition code is suﬃcient to achieve full diversity order of 2 under selection DF relaying. Suppose that the source or relay’s address is appended to the message in Phase II so that the destination can determine the identity of the transmitter and impose appropriate weighting coeﬃcients in the MRC. Then, the eﬀective SNR at the output of the MRC can be obtained as , (1) γs,d + γr,d , if γs,r ≥ 22R − 1, γSDF (3.30) (1) (2) γs,d + γs,d , if γs,r < 22R − 1,

80

3 Two-User Cooperative Transmission Schemes (1)

(2)

where γs,d Ps |hs,d |2 /σd2 , γs,d Pr |hs,d |2 /σd2 are the SNRs on the s-d link in Phases I and II, respectively, and γr,d Pr |hr,d |2 /σd2 is the SNR on the r-d link. Hence, the achievable end-to-end rate of the selection DF scheme is given by ⎧ ⎨ 1 log2 1 + γ (1) + γr,d , if γs,r ≥ 22R − 1, s,d 2 CSDF (γ) = (3.31) ⎩ 1 log2 1 + γ (1) + γ (2) , if γs,r < 22R − 1, s,d s,d 2 (1)

(2)

where γ = (γs,r , γr,d , γs,d , γs,d ). Under Rayleigh fading assumptions, the outage probability of the selection DF relaying scheme can be computed as * + 1 (1) 2R pout = Pr γs,r ≥ 2 − 1 Pr log2 1 + γs,d + γr,d < R 2 + * 1 (1) (2) 2R log2 1 + γs,d + γs,d < R , + Pr γs,r < 2 − 1 Pr 2 * + (1) = Pr γs,r ≥ 22R − 1 Pr γs,d + γr,d < 22R − 1 * + (1) (2) + Pr γs,r < 22R − 1 Pr γs,d + γs,d < 22R − 1 . (3.32) 2 and Since |hs,r |2 and |hs,d |2 are exponential random variables with mean ηs,r 2 ηs,d , respectively, we have

2R−1 * + 2 Pr γs,r < 22R − 1 = 1 − exp − , γ s,r

(3.33)

and 22R − 1 (1) (2) Pr γs,d + γs,d < 22R − 1 = Pr |hs,d |2 < (Ps + Pr )/σd2 22R − 1 , = 1 − exp − (1) (2) γ s,d + γ s,d (1)

(2)

(3.34) (1)

2 2 where γ s,d = Ps ηs,d /σd2 and γ s,d = Pr ηs,d /σd2 . Moreover, since γs,d and 2 γr,d are independent exponential random variables with mean Ps ηs,d /σd2 and (1)

2 /σd2 , the probability Pr(γs,d + γr,d < 22R − 1) can also be derived usPr ηr,d ing the results in (3.24). By substituting these results into (3.32), the outage probability can be further evaluated as

3.1 Decode-and-Forward Relaying Schemes

pout

81

⎧ γ (1) exp−c/γ (1) γ exp −c/γ ⎪ ( s,d s,d r,d r,d ) c ⎪ ⎪ 1 − exp − + (1) (1) ⎪ γ s,r γ s,d −γ r,d γ r,d −γ s,d ⎪ ⎪ ( ⎪ ) ⎪ ⎪ (1) ⎪ ⎨ 1 − exp − γ c − exp − (1) c (2) , if γ s,d =

γ r,d s,r γ s,d +γ s,d = ⎪ c c ⎪ ⎪ exp − γ c − (1) 1 − 1 + (1) ⎪ s,r γ s,d γ s,d ⎪ ⎪ ( ⎪ ) ⎪ ⎪ (1) ⎪ ⎩ 1 − exp − γ c − exp − (1) c (2) , if γ s,d = γ r,d γ s,d +γ s,d

s,r

where c = 22R − 1. Assume that Ps = 2βP , Pr = 2(1 − β)P , and σr2 = σd2 = 2 σw . By ﬁrst order Taylor approximation, the outage probability at high SNR can be approximated as ⎧ (1) (22R −1)2 1 ⎨ if γ s,d = γ r,d , 2 η2 SNR2 4βηs,r s,d ( ) pout ≈ (22R −1)2 1+β , (3.35) (1) 1 1 ⎩ 2 2 η2 + η2 SNR2 if γ s,d = γ r,d 4β η s,d

s,r

s,d

2 where SNR = P/σw . Hence, the diversity order of the selection DF scheme is equal to 2. This is achievable since the transmission is no longer contingent upon the successful decoding at the relay and the source message can be transmitted opportunistically through one of two independent fading paths, i.e., the direct path or the relay path. In Fig. 3.6, we compare the outage probability of Selection DF scheme with that of Basic DF with diversity combining. Here, we assume that the relay is located in the middle of the source and the destination, similar to that in Figs. 3.3 and 3.4. We can see that Selection DF indeed achieves diversity order 2 and, thus, the outage probability decreases twice as fast as that of Basic DF schemes. In Fig. 3.7, we show the outage probabilities versus the distance ds,r = 1 − dr,d for the various schemes described above 2 with SNR = P/σw = 20 dB. The variances of channel coeﬃcients are set as −α 2 −α 2 2 ηs,r = ds,r , ηr,d = d−α r,d , and ηs,d = ds,d , and the path loss exponent equals to 3. We can see that the outage probability of the basic DF scheme increases as the distance between the source and the relay increases. This is expected since the performance of the basic DF scheme is limited by the quality of the s-r link. However, in the selection DF scheme, the diversity gain allows the outage probability to remain at a low level even when the quality of the s-r link deteriorates. Moreover, we can observe that the outage probability is minimized when ds,r ≈ 0.5.

3.1.3 Demodulate-and-Forward Relaying Scheme In the DF schemes described previously, the relay forwards the source’s message only if it is able to successfully decode the message in Phase I. However,

82

3 Two-User Cooperative Transmission Schemes

0

10

−1

Outage Probability

10

−2

10

−3

10

−4

10

No Cooperation Basic DF II, OPA, Statistical CSI Basic DF II, OPA, Instantaneous CSI Selection DF 0

6

12 Transmit SNR (dB)

18

24

Fig. 3.6 Performance comparison of selection DF relaying and basic DF relaying schemes with diversity combining when the relay node is located in the middle of the source and the destination nodes.

in many applications, channel decoding may not be desirable at the relays either due to limited transceiver capabilities or due to lack of knowledge of the channel codebook. In this case, the signals transmitted by the source can only be detected or demodulated on a symbol-by-symbol basis. With no error protection, the forwarded symbols may be incorrect and, thus, the error probability must be taken into consideration in the receiver design at the destination. The studies of DF in uncoded systems will be referred to as demodulate-and-forward (DeF) relaying schemes [2, 27, 28]. In the DeF relaying scheme, cooperation also takes on two phases of transmission. In Phase I, the source transmits a block of symbols xs = [xs [0], . . . , xs [M − 1]]T to both the relay and the destination, where the received signals are given by (3.1) and (3.2), respectively. Suppose that binary phase shift keying (BPSK) modulation is employed at the transmitters such that xs [m] ∈ {−1, +1}, for all m. The maximum likelihood (ML) detector at the relay can be expressed as * . /+ x ˆs [m] = sgn h∗s,r yr [m] , (3.36) where

3.1 Decode-and-Forward Relaying Schemes

83

No Cooperation Basic DF II, OPA, Statistical CSI Basic DF II, OPA, Instantaneous CSI Selection DF −2

Outage Probability

10

−3

10

−4

10

0.1

0.2

0.3

0.4

0.5 ds,r

0.6

0.7

0.8

0.9

Fig. 3.7 Performance comparison of selection DF relaying and basic DF relaying schemes with diversity combining in terms of the distance between the source and relay node.

⎧ ⎪ ⎨1, sgn(u) = 0, ⎪ ⎩ −1,

if u > 0, if u = 0, if u > 0,

is the sign function. Thus, the bit error rate (BER) is given by + *εr (γs,r ) Pr (ˆ xs [m] = xs [m]) = Q 2γs,r ,

(3.37)

where γs,r = Ps |hs,r |2 /σr2 is the SNR on the s-r link and the Q-function deﬁned by ∞ 2 1 Q(u) = √ e−t /2 dt, (3.38) 2π u i.e., the complementary CDF of a standard normal random variable. In Phase II, the relay forwards the detected symbol to the destination, regardless of whether or not it was detected correctly. Hence, the signal received at the destination can be given by (2) (2) yd [m] = hr,d Pr xr [m] + wd [m], (3.39) where xr [m] = x ˆs [m], for m = 0, . . . , M − 1. Given knowledge of the error probability at the relay, the ML detector can also be employed at the destination, combining the signals received in both Phase I and Phase II. Since

84

3 Two-User Cooperative Transmission Schemes

the detection is performed on a symbol-by-symbol basis, we shall omit the time index m in the following discussions. Let ⎛ 2 ⎞ √ (1) − h P x ⎟ y s,d s s d 1 ⎜ (1) fY (1) yd hs,d , xs = exp ⎝− ⎠ 2 2 d πσd σd (1)

be the conditional PDF of yd xs and let

fY (2) d

given the channel hs,d and the source symbol

⎛ 2 ⎞ √ (2) − h P x ⎟ y r,d r s + d 1 ⎜ (2) yd hs,r ,hr,d , xs = [1 − εr (γs,r )] 2 exp ⎝− ⎠ 2 πσd σd

*

⎛ 2 ⎞ √ (2) + h P x r,d r s ⎟ 1 ⎜ yd + εr (γs,r ) 2 exp ⎝− ⎠ πσd σd2

(3.40)

(2)

be the conditional PDF of yd given the channels hs,r , hr,d and the source symbol xs . The ﬁrst and second terms correspond to the case that a correct decision is made at the relay and the opposite case where an incorrect decision (1) (2) is made. Given the received signals in both phases, i.e., yd and yd , the ML detector can be derived as (1) (2) x ˆs = arg max fY (1) yd hs,d , x fY (2) yd hs,r , hr,d , x x∈{−1,+1}

d

= arg max ln fY (1) x∈{−1,+1}

d

d

(2) (1) yd hs,d , x + ln fY (2) yd hs,r , hr,d , x d

4 5 ( 4 5 (1) (2) = arg max 2Re ωs,d yd x + ln (1 − εr (γs,r )) exp 2Re ωr,d yd x x∈{−1,+1}

4 5) (2) +εr (γs,r ) exp −2Re ωr,d yd x ,

(3.41)

√ √ where ωs,d = h∗s,d Ps /σd2 and ωr,d = h∗r,d Pr /σd2 are the coeﬃcients of the combiner at the destination. When the relay is noiseless such that εr (γs,r ) = 0, for all γs,r , the ML detection reduces to the MRC combiner adopted in (3.19). However, for a larger constellation size, the conditional PDF in (3.40) and the ML detector in (3.41) becomes prohibitively complex. Moreover, due to possible error at the relay, the average symbol error probability and, thus, the diversity order of the uncoded DeF scheme can not be easily evaluated. In the following, we introduce the suboptimal cooperative MRC detector proposed in [28] and show how it can achieve full diversity even with the suboptimal design. Since full diversity is achievable by a sub-

3.1 Decode-and-Forward Relaying Schemes

85

optimal receiver, it must also be achievable with the ML detector as well. Suboptimal Cooperative MRC (C-MRC) Detector [28] The main complexity of the ML detector is the need to consider all possible combinations of errors occurring at the relay, e.g., in (3.40) and (3.41). To simplify the computations, one can consider a suboptimal scheme where the ML detector at the destination is aimed at detecting the relayed symbol xr (instead of the actual source symbol xs ), regardless of whether or not the detection at the relay is correct. In this case, the error probability of the relay link (i.e., s-r-d link) will be given by pˆe,relay (γs,r , γr,d ) =εr (γs,r ) [1 − εd (γr,d )] + [1 − εr (γs,r )] εd (γr,d ) =Q( 2γs,r ) 1 − Q( 2γr,d ) + 1 − Q( 2γs,r ) Q( 2γr,d ). If the relay link is viewed as an eﬀective single-hop AWGN channel, an eﬀective SNR for the relay link can be deﬁned as γeﬀ

2 1 −1 Q (ˆ pe,relay ) , 2

(3.42)

which is obtained by inverting the relation in (3.37) with the BER of the relay path, i.e., pˆe,relay (γs,r , γr,d ). Intuitively, the signal received from the relay in Phase II can viewed as an eﬀective AWGN channel with signal model given by (2) (2) (2) yd [m] = hr,d Pr xr [m] + wd [m] = heﬀ Ps xs [m] + w ˜d [m], √ √ (2) ˜d [m] where heﬀ = Pr hr,d / Ps is the eﬀective s-d channel in Phase II and w is the eﬀective noise that incorporates the detection error at the relay. The noise variance is given by σ ˜d2 = Ps |heﬀ |2 /γeﬀ = Pr |hr,d |2 /γeﬀ . It has been shown in [28] that 3.24 γmin − < γeﬀ ≤ γmin , (3.43) ν where γmin min{γs,r , γr,d } and ν = 2 (in the case of BPSK). Hence, γeﬀ can be closely approximated by γmin at high SNR. Taking into consideration the relay errors, the C-MRC receiver [28] combines the signals in Phase I and Phase II with the combiner coeﬃcients √ √ √ ∗ Ps h∗s,d Ps h∗eﬀ γeﬀ Pr hr,d ωs,d = and ω = = . eﬀ σd2 σ ˜d2 γr,d σd2 Thus, the signal at the C-MRC output can be expressed as

86

3 Two-User Cooperative Transmission Schemes (1)

(2)

yd = ωs,d yd + ωeﬀ yd ,* √ + √ (1) (2) ωs,d hs,d Ps + ωeﬀ hr,d Pr xs + ωs,d wd + ωeﬀ wd , = * √ √ + (1) (2) ωs,d hs,d Ps − ωeﬀ hr,d Pr xs + ωs,d wd + ωeﬀ wd ,

if x ˆr = xs , if x ˆr = −xs .

The ML detector based on the combiner output is given by - 2 x ˆs = arg max yd − ωs,d hs,d Ps + ωeﬀ heﬀ Ps x . x∈{−1,+1}

Since BPSK symbols are real-valued, it suﬃces to take into consideration only the real part of the combiner output, i.e., Re{yd }, whose noise variance is given by σ 2 = (|ws,d |2 + |weﬀ |2 )σd2 /2. The BER of the C-MRC receiver can be computed as pe (γs,r , γr,d , γs,d ) = [1 − εr (γs,r )]

√ +2 yd + (ωs,d hs,d + ωeﬀ heﬀ ) Ps √ dy exp − 2σ 2 2πσ 2 0 * √ +2 ∞ yd + (ωs,d hs,d − ωeﬀ heﬀ ) Ps 1 √ dy + εr (γs,r ) exp − 2σ 2 2πσ 2 0 ' ' 2(γs,d + γeﬀ )2 2(γs,d − γeﬀ )2 = 1 − Q( 2γs,r ) Q +Q( 2γs,r )Q 2 /γ 2 /γ γs,d +γeﬀ γs,d +γeﬀ r,d r,d ∞

1

*

(3.44) To derive the achievable diversity order, let us ﬁrst bound the ﬁrst term in (3.44) as ' 2(γs,d + γeﬀ )2 ≤Q 1 − Q( 2γs,r ) Q 2(γs,d + γeﬀ ) , 2 /γ γs,d +γeﬀ r,d which follows from the fact that γeﬀ ≤ γr,d . By utilizing the lower bound 2 in (3.43) and by applying the Chernoﬀ bound (where Q(x) ≤ 12 e−x /2 ), the average of the ﬁrst term over γs,r , γr,d , and γs,d can be further bounded as 2(γs,d + γeﬀ ) Eγs,r ,γr,d ,γs,d Q 1 Eγ [exp (−γs,d )] Eγs,r ,γr,d [exp (− min{γs,r , γr,d })] e1.62 2 s,d * + e1.62 γ s,r + γ r,d +* +. = * 2 1 + γ s,d γ s,r + γ r,d + γ s,r γ r,d

≤

Let us consider the case where Ps = 2βP , Pr = 2(1 − β)P , and σr2 = σd2 = 2 With SNR = P/σw , the right-hand-side can be approximated by k1 /SNR2 ,

2 σw .

3.2 Amplify-and-Forward Relaying Schemes

87

as SNR → ∞, for some constant k1 . Similarly, by applying the Chernoﬀ bound on the second term, the average of the second term in (3.44) can also be bounded as ' 2(γs,d − γmin )2 Eγs,r ,γr,d ,γs,d Q( 2γs,r )Q 2 /γ γs,d +γeﬀ r,d (γs,d − γeﬀ )2 1 1 exp(−γs,r ) exp − 1γs,d ≥γmin ≤ Eγs,r ,γr,d,γs,d 2 /γ 2 2 γs,d +γeﬀ r,d 1 exp(−γs,r )1γs,d <γmin , + Eγs,r ,γr,d ,γs,d 2 By evaluating the expectations and utilizing γmin min{γs,r , γr,d }, one can show that the average of the second term is also approximated by k2 /SNR2 , as SNR → ∞, for some constant k2 (see [28, Appendix B] for details). Hence, the C-MRC detector achieves diversity order of 2 despite its suboptimal design. In Fig. 3.8 (i.e. [28, Figure 7]), the BER of the C-MRC is shown along with that of the optimal ML detector and the λ-MRC scheme proposed in [19,25]. √ (2) In λ-MRC, the combiner coeﬃcient for yd is given by weﬀ = λh∗r,d Pr , where λ is optimally chosen to minimize the BER. In the experiment, three diﬀerent sets of average SNRs are considered, i.e., (γ s,r , γ r,d , γ s,d ) = (SNR + 30dB, SNR, SNR), (SNR, SNR+30dB, SNR), and (SNR, SNR, SNR). One can observe, from Fig. 3.8, that the performance of C-MRC bounds that of ML and λMRC. Hence, both ML and λ-MRC schemes are also able to achieve diversity order of 2 as well.

3.2 Amplify-and-Forward Relaying Schemes In amplify-and-forward (AF) relaying schemes, the relay ampliﬁes the analog signal received from the source and forwards it to the destination (without explicitly decoding or demodulating the messages or symbols) as shown in Fig. 3.9. These schemes are also referred to as non-regenerative relaying schemes. Here, relays need not have knowledge of the encoding or modulation schemes employed at the source. Moreover, in addition to its low complexity, AF schemes are also desirable when the quality of the s-r link is not suﬃcient to guarantee reliable decoding at the relay. In this case, amplifying the analog signal preserves soft information that can be further exploited at the destination. In this section, we describe two variants of AF relaying schemes, i.e., the basic AF relaying scheme (with and without diversity combining at the destination) and the incremental AF relaying scheme. The outage probabilities of these schemes will also be analyzed.

3 Two-User Cooperative Transmission Schemes

BER

88

Fig. 3.8 BER comparison of C-MRC and λ-MRC detectors in demodulate-and-forward c relaying scheme (From Wang, Cano, and Giannakis. [2007] IEEE ).

Fig. 3.9 Illustration of amplify-and-forward (AF) relaying schemes.

3.2.1 Basic AF Relaying Scheme In the basic AF relaying scheme, the relay forwards a scaled version of the received signal to the destination, regardless of the s-r link quality. Speciﬁcally, in Phase I, the source transmits the symbol block xs = [xs [0], . . . , xs [M − 1]] to both the relay and the destination, where the received signals are given by

3.2 Amplify-and-Forward Relaying Schemes

89

yr [m] = hs,r Ps xs [m] + wr [m], (1) (1) yd [m] = hs,d Ps xs [m] + wd [m],

(3.45) (3.46)

for m = 0, . . . , M − 1. Similar to that in (3.1) and (3.2). In Phase II, the relay ﬁrst scales the received signal in (3.45) to yield a normalized transmit vector xr with E[|xr [m]|2 ] = 1, for all m. If the instantaneous channel gain |hs,r |2 is known at the relay, the relay can multiply the received signal yr [m] by the gain 1 1 Gv = = -P |h |2 + σ 2 s s,r r E |yr [m]|2 |hs,r |2

(3.47)

to obtain xr [m] = Gv yr [m] ' Ps 1 = hs,r xs [m] + wr [m]. (3.48) Ps |hs,r |2 + σr2 Ps |hs,r |2 + σr2 Notice that the gain Gv depends on the s-r channel hs,r and, thus, varies in diﬀerent transmission intervals. Thus, this scheme is often referred to as the variable-gain AF relaying scheme. Then, with power Pr , the relay forwards the signal xr to the destination, where the received signal can be expressed as (2) (2) yd [m] = hr,d Pr xr [m] + wd [m] ' ' Ps Pr Pr (2) hs,r hr,d xs [m]+ hr,d wr [m]+wd [m], = 2 2 Ps |hs,r | +σr Ps |hs,r |2 +σr2 (3.49) for m = 0, . . . , M − 1. The signals arriving at the destination can be utilized for detection with or without diversity combining. The performance of these two cases will be discussed in the following.

Case I: Without Diversity Combining (2)

In the case without diversity combining, only the signal yd will be utilized for detection at the destination. By (3.49), the received SNR in this case can be computed as γBasicAF,I =

Ps Pr 2 2 Ps |hs,r |2 +σr2 |hs,r | |hr,d | Pr σr2 2 2 Ps |hs,r |2 +σr2 |hr,d | + σd

=

γs,r γr,d , γs,r + γr,d + 1

(3.50)

90

3 Two-User Cooperative Transmission Schemes

where γs,r = Ps |hs,r |2 /σr2 and γr,d = Pr |hr,d |2 /σd2 . Thus, the maximum achievable end-to-end rate is given by γs,r γr,d 1 CBasicAF,I (γs,r , γr,d ) = log 1 + . (3.51) 2 γs,r + γr,d + 1 Given that the end-to-end transmission rate is R, the outage probability is given by * + pout = Pr γBasicAF,I < 22R − 1 . (3.52) This probability depends on the distribution of the eﬀective SNR, i.e., γBasicAF,I , which is diﬃcult to evaluate in closed-form. However, at high SNR, the constant 1 in the denominator in (3.50) can be omitted and the eﬀective SNR can be approximated as γBasicAF,I ≈

γs,r γr,d = γs,r + γr,d

1 γs,r

+

1 γr,d

−1 .

(3.53)

That is, at high SNR, the eﬀective SNR (i.e., γBasicAF,I ) can be approximated by the harmonic mean of the SNRs on each hop. In Rayleigh fading scenar2 2 2 ios, where hs,r ∼ CN (0, ηs,r ), hr,d ∼ CN (0, ηr,d ) and hs,d ∼ CN (0, ηs,d ), the 2 SNRs γs,r and γr,d are exponentially distributed with mean γ s,r = Ps ηs,r /σr2 2 and γ r,d = Pr ηr,d /σd2 , respectively. Therefore, the distribution of the approximated expression of γBasicAF,I in (3.53) is given by [6] −1 1 1 +
3.2 Amplify-and-Forward Relaying Schemes

91

0

10

−1

Outage Probability

10

−2

10

−3

10

No Cooperation Basic AF I, OPA, Statistical CSI Basic AF I, OPA, Instantaneous CSI −4

10

0

5

10

15 20 Transmit SNR (dB)

25

30

Fig. 3.10 Performance comparison of basic AF relaying schemes without diversity combining.

Notice that the outage probability in (3.56) is identical to that of the basic DF scheme without diversity combining (see (3.7)). By utilizing the ﬁrst order Taylor approximation, we can further approximate the outage probability as 1 22R − 1 22R − 1 pout ≈ , · + 2 2 2βηs,r 2(1 − β)ηr,d SNR where we set Ps = 2βP and Pr = 2(1 − β)P . Hence, the diversity order of the basic AF relaying scheme without diversity combining is equal to 1. Given the total power constraint Ps + Pr ≤ 2P , the optimal power allocation among the source and the relay can be found by minimizing the outage probability, when only the channel statistics are known, or by maximizing the achievable end-to-end rate, when instantaneous CSI is available. When only the channel statistics are known, the optimal power allocation is diﬃcult to obtain since a closed-form expression of the outage probability is not available. However, at high SNR, we have shown that the approximated outage probability expression, as given in (3.56), is identical to that in the basic DF scheme and, thus, the optimal power allocation can also be approximated by (3.12). On the other hand, when the instantaneous CSI is available, the power allocation problem can be formulated as follows:

92

3 Two-User Cooperative Transmission Schemes

max

Ps ,Pr

Ps |hs,r |2 /σr2 · Pr |hr,d |2 /σd2 1 log 1 + 2 Ps |hs,r |2 /σr2 + Pr |hr,d |2 /σd2 + 1

subject to Ps + Pr ≤ 2P and Ps , Pr ≥ 0.

(3.57) (3.58)

Due to the monotonicity of the logarithmic function and the fact that the objective is maximized when the total power constraint is met with equality, the optimization problem can be reformulated as follows [30]: max Ps

Ps |hs,r |2 /σr2 · (2P − Ps )|hr,d |2 /σd2 Ps |hs,r |2 /σr2 + (2P − Ps )|hr,d |2 /σd2 + 1

subject to 0 ≤ Ps ≤ 2P.

(3.59) (3.60)

By taking the ﬁrst derivative of the objective in (3.59) with respect to Ps and setting it to zero, the optimal power allocation can be found as 1 + 2P |hr,d |2 /σd2 · 2P , (3.61a) Ps = 1 + 2P |hr,d |2 /σd2 + 1 + 2P |hs,r |2 /σr2 1 + 2P |hs,r |2 /σr2 · 2P Pr = . 1 + 2P |hr,d|2 /σd2 + 1 + 2P |hs,r |2 /σr2

(3.61b)

Notice that the powers allocated to the source and the relay are proportional to the squared root of 1 + 2P |hr,d |2 /σd2 and 1 + 2P |hs,r |2 /σr2 , respectively. In Fig. 3.10, we show the outage probability versus the transmit SNR, i.e., 2 SNR = P/σw , for the basic AF schemes without diversity combining. The experiments are also conducted by using the simulation parameters given for that in Fig. 3.3 and by assuming that the relay is located in the middle of the source and the destination. The power allocation for the case with instantaneous CSI (see Instantaneous CSI curve) and the case with only channel statistics (see Statistical CSI curve) are given by (3.61) and (3.12), respectively. Since no diversity combining is utilized at the destination, only diversity order 1 can be obtained and, thus, little improvement is observed (from a physical-layer standpoint) compared to the case with no cooperation (i.e., direct transmission). In fact, since two phase transmission is employed, the bandwidth eﬃciency decreases by 1/2 in the case of relaying and, thus, the outage performance may be worse than direct transmission. However, as described in previous sections, relaying may still be desirable from a practical standpoint. In Fig. 3.11, we show the outage probability versus the distance between source and relay, i.e., ds,r = 1 − dr,d , for the case with SNR = 20dB. The simulation environment is the same as that in Fig. 3.7. One can observe that the optimal location of the relay is exactly in the middle of the source and destination. This shows that the eﬀective SNR in (3.50) approaches its maximum when the SNRs of the s-r link and the r-d link are suﬃciently balanced.

3.2 Amplify-and-Forward Relaying Schemes

93

−1

10

Outage Probability

No Cooperation Basic AF I, OPA, Statistical CSI Basic AF I, OPA, Instantaneous CSI

−2

10

−3

10

0.1

0.2

0.3

0.4

0.5 ds,r

0.6

0.7

0.8

0.9

Fig. 3.11 Performance comparison of basic AF relaying schemes without diversity combining in terms of the distance between the source and relay node.

Case II: With Diversity Combining In the case with diversity combining, the signals received in Phases I and II, i.e., (3.46) and (3.49), can be optimally combined at the destination using MRC to obtain the output signal √ Ps Pr ∗ ∗ Ps h∗s,d (1) Ps |hs,r |2 +σr2 hs,r hr,d (2) y˜d = · y + · yd . (3.62) d Pr 2 σ2 + σ2 σd2 |h | r d Ps |hs,r |2 +σ2 r,d r

The eﬀective SNR at the output of the MRC is given by Ps |hs,d |2 Ps |hs,r |2 /σr2 · Pr |hr,d |2 /σd2 + 2 σr Ps |hs,r |2 /σr2 + Pr |hr,d |2 /σd2 + 1 γs,r γr,d . = γs,d + γs,r + γr,d + 1

γBasicAF,II =

(3.63)

The maximum achievable end-to-end transmission rate for the basic AF relaying scheme with diversity combining is thus given by 1 γs,r γr,d CBasicAF,II (γs,r , γr,d , γs,d ) = log2 1 + γs,d + . (3.64) 2 γs,r + γr,d + 1

94

3 Two-User Cooperative Transmission Schemes

Similar to previous cases, an outage occurs when the transmission rate R is greater than CBasicAF,II . Under Rayleigh fading assumptions, the outage probability can be computed as γs,r γr,d 2R pout = Pr γs,d + ≤2 −1 γs,r + γr,d + 1 ∞ γs,r γr,d 2R ≤ 2 − 1γs,d = u fγs,d (u)du Pr γs,d + = γs,r + γr,d + 1 0 22R −1 1 γs,r γr,d u ≤ 22R − 1 − u du, Pr exp − = γ s,d γ s,d γs,r + γr,d + 1 0 (3.65) where we utilized the fact that γs,d is exponentially distributed with mean 2 γ s,d Ps ηs,d /σd2 . At high SNR, the probability inside the integral can be approximated by γs,r γr,d γs,r γr,d 2R 2R Pr ≤ 2 − 1 − u ≈ Pr ≤2 −1−u γs,r + γr,d + 1 γs,r + γr,d 2R 2(22R −1−u) 2 −1−u 22R −1−u 2(22R −1−u) exp − K1 − = 1− γ s,r γ r,d γ s,r γ r,d γ s,r γ r,d 2R 2 − 1 − u 22R − 1 − u , (3.66) ≈ 1 − exp − − γ s,r γ r,d where the last equality comes from the approximation K1 (x) ≈ 1/x [1, pp. 375, eq. (9.6.9)], for x suﬃciently small. By substituting (3.66) into (3.65), the outage probability can be approximated at high SNR as 2R 2 −1−u 22R −1−u u 1 − exp − du exp − − γ s,d γ s,d γ s,r γ r,d 0 2R 2 −1 1 = 1 − exp − − γ s,d 1 − γ s,d /γ s,r − γ s,d /γ r,d 2R 2R 2 −1 2 −1 22R −1 − exp − . (3.67) × exp − − γ s,r γ r,d γ s,d

pout ≈

22R −1

1

2

By applying the second order Taylor approximation, where e−x ≈ 1 − x + x2 for x suﬃciently small, the outage probability can be further approximated by

3.2 Amplify-and-Forward Relaying Schemes

95

0

10

−1

10

Outage Probability

−2

10

−3

10

−4

10

No Cooperation Basic AF I, OPA, Statistical CSI Basic AF I, OPA, Instantaneous CSI Basic AF II, OPA, Statistical CSI Basic AF II, OPA, Instantaneous CSI

−5

10

−6

10

0

6

12 Transmit SNR (dB)

18

24

Fig. 3.12 Performance comparison of basic AF relaying schemes with respect to transmit 2. SNR, i.e., SNR = P/σw

1 22R − 1 22R − 1 22R − 1 + 2 γ s,d γ s,r γ r,d 2R 2 1 1 1 2 −1 1 , = + 2 2 2 8 SNR βηs,d βηs,r (1 − β)ηr,d

pout ≈

(3.68)

2 where we set Ps = 2βP , Pr = 2(1 − β)P , and σr2 = σd2 = σw . Hence, the diversity order of the basic AF relaying scheme with diversity combining is equal to 2, i.e., the outage probability decreases twice as fast as that of direct transmission or the case without diversity combining. Even though noise is ampliﬁed along with the signal at the relay in AF relaying schemes, the achievable diversity gain in the basic AF scheme is larger than that of the basic DF scheme since the cooperative transmission is not contingent upon the successful decoding at the relay (in fact, no decoding is performed at the relay). This is analogous to that observed in DeF relaying schemes where full diversity is also achieved by having the relay forward the detected symbols regardless of whether or not they were correctly detected. When only channel statistics are known, the optimal power allocation can be found to minimize the outage probability in (3.67). Unfortunately, a closedform expression of the optimal power allocation is not easily attainable in this case and, thus, one must resort to numerical methods. On the other hand, when the instantaneous CSI is known, the optimal power allocation can be

96

3 Two-User Cooperative Transmission Schemes

found by maximizing the achievable end-to-end rate [30]. The optimization problem can be formulated as follows: 1 Ps |hs,d |2 Ps |hs,r |2 /σr2 · Pr |hr,d |2 /σd2 max log 1 + (3.69) + Ps ,Pr 2 σd2 Ps |hs,r |2 /σr2 + Pr |hr,d |2 /σd2 + 1 subject to Ps + Pr ≤ 2P and Ps , Pr ≥ 0.

(3.70)

For notational simplicity, let us assume, without loss of generality, that the noise variances are σr2 = σd2 = 1. Then, due to monotonicity of the logarithmic function and the fact that the total power constraint must be satisﬁed with equality, i.e., Ps + Pr = 2P , the optimization problem can be recast as max Ps

Ps |hs,d |2 +

Ps |hs,r |2 · (2P − Ps )|hr,d |2 Ps |hs,r |2 + (2P − Ps )|hr,d |2 + 1

subject to 0 ≤ Ps ≤ 2P.

(3.71) (3.72)

By taking the derivative of the objective function and setting it to zero, the optimal power allocation, as derived in [30], are given by Ps =

2P |hs,r |2 |hr,d |2 + 2P |hs,d |2 |hr,d |2 + |hs,d |2 , 1+2P |h |2 2 2 f (h) + 1+2P |hs,r 2 |hs,r | |hr,d | f (h) | r,d

(3.73a)

Pr =

2P |hs,r |2 |hr,d |2 − 2P |hs,d |2 |hs,r |2 − |hs,d |2 , 1+2P |h |2 2 |h 2 f (h) f (h) + 1+2P |hr,d |h | | s,r r,d 2 s,r |

(3.73b)

where f (h) |hs,r |2 |hr,d |2 + |hs,d |2 |hr,d |2 − |hs,r |2 |hs,d |2 , when the following conditions are satisﬁed: (C1) 2P |hs,r |2 |hr,d |2 − 2P |hs,d |2 |hs,r |2 − |hs,d |2 > 0; (C2) |hs,r |2 |hr,d |2 + |hs,d |2 |hr,d |2 − |hs,r |2 |hs,d |2 > 0. On the other hand, when the conditions are not satisﬁed, direct transmission from the source to the destination achieves a higher rate and, thus, the optimal power allocation is given Ps = 2P and Pr = 0. It is worthwhile to note that, when hs,d = 0, the solution in (3.73) reduces to that obtained in (3.61) for the case without diversity combining. Compared to the solution in (3.61), more power is allocated to the source since it contributes to the reception at both the relay and the destination and, thus, has a larger impact on the overall system performance. Similar observations have also been found in the basic DF schemes as well. 2 In Fig. 3.12, we plot the outage probability versus SNR = P/σw , where 2 2 2 σd = σr = σw , for the basic AF schemes with and without diversity combining. The simulations are performed by using the parameters given for Fig. 3.3 and by assuming that the relay is located in the middle of the source and the destination. When diversity combining is employed at the destina-

3.2 Amplify-and-Forward Relaying Schemes

−1

Outage Probability

10

97

No Cooperation Basic AF I, OPA, Statistical CSI Basic AF I, OPA, Instantaneous CSI Basic AF II, OPA, Statistical CSI Basic AF II, OPA, Instantaneous CSI

−2

10

−3

10

−4

10

0.1

0.2

0.3

0.4

0.5 ds,r

0.6

0.7

0.8

0.9

Fig. 3.13 Performance comparison of basic AF relaying schemes in terms of distance between the source and relay node.

tion, the optimal power allocation (OPA) is obtained by (3.73), assuming knowledge of the instantaneous CSI. One can observe from Fig. 3.12 that the outage probabilities obtained with diversity combining decreases twice as fast as that obtained without diversity combining as the transmit SNR, i.e., SNR, increases. Moreover, when instantaneous CSI is available, a gain of approximately 2dB can be obtained compared to the case where powers are optimally allocated based on statistical CSI. With simply statistical CSI, OPA can be obtained numerically by to minimize outage probability approximated by (3.67). Furthermore, in Fig. 3.13, we plot the outage probabilities versus the s-r distance, i.e., ds,r = 1−dr,d , for the case with SNR = 20dB. One can observe that the optimal relay location is approximately in the middle, between the source and destination, except for the case with diversity combining. This is because, with optimal power allocation in case II, more power is allocated to the source compared to that in (3.61), but the SNR on the s-r and the r-d links must be balanced in order to achieve a larger eﬀective SNR (see descriptions to Fig. 3.11). Hence, the relay must be located closer to the destination. Extensions to Basic AF with Fixed Gain Relays In the previous discussions on Basic AF, we assumed that the relay has knowledge of the s-r channel, i.e., hs,r , and, thus, is able to impose a variable

98

3 Two-User Cooperative Transmission Schemes

gain Gv on the received signal yr [m]. However, if only the mean of the channel 2 gain ηs,r is known, the relay can instead multiply the received signal by a ﬁxed gain 1 1 Gf = (3.74) = 2 2 + σ2 E [|yr [m]| ] Ps ηs,r r to obtain ' xr [m] = Gf yr [m] =

Ps 1 hs,r xs [m] + wr [m], 2 + σ2 Ps ηs,r 2 + σ2 r Ps ηs,r r

where the constraint E[|xr [m]|2 ] = 1 is also satisﬁed. This is referred to as the ﬁxed-gain AF relaying scheme. Then, with power Pr , the relay forwards the signal xr to the destination, where the received signal can be expressed as (2) (2) yd [m] = Pr hr,d xr [m] + wd [m] ' ' Ps Pr Pr (2) = h h x [m]+ h w [m]+wd [m], 2 +σ 2 s,r r,d s 2 +σ 2 r,d r Ps ηs,r P η s s,r r r for m = 0, . . . , M − 1. Suppose that MRC is employed at the destination to combine the signals received in Phases I and II. The eﬀective SNR at the output of the MRC is given by ﬁxed γBasicAF,II = γs,d +

γs,r γr,d . γr,d + (γ s,r + 1)

(3.75)

Notice that the expression (3.75) is similar to that in (3.63) but with the term γs,r in the denominator of the second term replaced with the expected value γ s,r . The outage probability can be computed as pout

= Pr γs,d + = 0

22R −1

γs,r γr,d 2R ≤2 −1 γr,d + (γ s,r + 1) 1 γs,r γr,d u ≤ c − u du, Pr exp − γ s,d γ s,d γr,d + b

(3.76)

where c 22R − 1 and b 1 + γ s,r . The probability inside the integral can be further evaluated as

3.2 Amplify-and-Forward Relaying Schemes

Pr

99

γs,r γr,d ≤c−u γr,d + b 0

∞ 1 c−u b y dy 1 − exp − 1+ exp − = γ r,d γ r,d γ s,r y 0 ∞ 1 c−u y (c − u)b dy =1− exp − exp − − γ r,d γ s,r γ r,d γ s,r y 0 ' ' b(c − u) b(c − u) c−u (a) K1 2 = 1−2 exp − γ s,r γ r,d γ s,r γ s,r γ r,d (b) c−u , ≈ 1 − exp − γ s,r

where (a) follows from [7] and [4, eq. (3.324.1)] and (b) follows from the approximation that K1 (x) ≈ 1/x [1, pp. 375, eq. (9.6.9)], for x suﬃciently small. By substituting this into (3.76), the outage probability can be approximated by c−u u 1 − exp − du, ≈ exp − γ s,d γ s,d γ s,r 0 2R 2 −1 1 = 1 − exp − − γ s,d γ s,d (1/γ s,d − 1/γ s,r ) 2R 2R 2 −1 2 −1 − exp . × exp − γ s,r γ s,d

pout

22R −1

1

(3.77)

Furthermore, by taking the second order Taylor approximation, where e−x ≈ 2 1 − x + x2 for x suﬃciently small, the outage probability can be further approximated by 1 22R − 1 22R − 1 2 γ s,d γ s,r 2R 2 1 1 2 −1 = 2 η2 , 8 SNR β 2 ηs,d s,r

pout ≈

(3.78)

2 where we set Ps = 2βP , Pr = 2(1 − β)P , and σr2 = σd2 = σw . This shows that diversity order 2 is still achievable even when the instantaneous CSI is not available at the relay. However, a loss in coding gain may be observed compared to the case with variable gain relays. Moreover, in the case of ﬁxed gain relays, the instantaneous output powers at the relay may be too large at times when the instantaneous s-r channel gain is large. In practical systems, where the gain of the power ampliﬁer is limited, this may result in clipping errors that may reduce the beneﬁts of cooperation.

100

3 Two-User Cooperative Transmission Schemes

Fig. 3.14 Illustration of incremental AF relaying scheme.

3.2.2 Incremental AF Relaying Scheme In the previous subsection, we have shown that the basic AF relaying scheme is able to achieve full diversity when signals received in both Phase I and Phase II are combined for detection at the destination. However, the relaying schemes introduced in previous sections (including the DF schemes) may not be bandwidth eﬃcient, compared to direct transmission, since the same codeword is transmitted twice over the entire cooperative transmission period. To improve upon this, one can employ the incremental relaying scheme [18], as illustrated in Fig. 3.14, where Phase II of the cooperative transmission is utilized only when the source transmission fails in Phase I. This leads to higher bandwidth eﬃciency, since the second transmission phase is not always required, and can be achieved with a simple feedback mechanism at the destination. The incremental relaying scheme is described below for the AF system, but can be easily applied to the DF system as well. Following previous relaying schemes, the source ﬁrst transmits a block of symbols xs = [xs [0], . . . , xs [M − 1]]T in Phase I and the signals received at the relay and the destination are given by (3.45) and (3.46), respectively. In Phase I, the destination makes an attempt to decode the message based on (1) (1) (1) the received signal yd = [yd [0], . . . , yd [M −1]]T . To achieve an end-to-end rate of R, the codeword transmitted in Phase I must encode the message with rate 2R in case a second transmission is required. If the destination is able to successfully decode the message based only on the signal received in Phase I, the destination will broadcast an ACK (i.e., acknowledgement) message to inform the source and the relay of this success. In this case, the relay will discard the signal received from the source and the source will proceed in transmitting a new message. Note that this event occurs when 2R ≤ log2 (1 + γs,d )

3.2 Amplify-and-Forward Relaying Schemes

101

and achieves end-to-end rate of 2R when it occurs. On the other hand, if direct transmission fails, i.e., when 2R > log2 (1 + γs,d ), the destination will broadcast a NACK (i.e., negative acknowledge) message to inform both the source and the relay of this failure. Only in this case will the relay participate in the cooperative transmission and forward a scaled version of its received signal to the destination in Phase II. The signals received at the destination in both phases are given similarly by (3.46) and (3.49). The eﬀective SNR at the output of the MRC is given by (3.63). In this case, the transmission rate is equal to R, since two phases of transmission are utilized, and decoding at the destination is successful only if 1 γs,r γr,d R ≤ log2 1 + γs,d + . 2 γs,r + γr,d + 1 Clearly, the transmission rate of the incremental relaying scheme varies depending on the quality of the s-d link. If γs,d ≥ 22R − 1, the transmission rate will be as high as 2R but, if γs,d < 22R − 1, the rate will be only equal to R. Hence, the average transmission rate is computed by * + * + R = 2R · Pr γs,d ≥ 22R − 1 + R · Pr γs,d < 22R − 1 2R 2 −1 , (3.79) = R + R exp − γ s,d 2 /σd2 . Notice that the average rate R is greater than R, where γ s,d Ps ηs,d but can approach 2R as the SNR γ s,d of the direct link increases. Hence, the bandwidth eﬃciency is improved compared to previous relaying schemes. In the incremental relaying scheme, an outage occurs only when both the direct transmission in Phase I and the cooperative transmission in Phase II fail simultaneously. Therefore, the outage probability is given by γs,r γr,d 2R 2R pout = Pr γs,d < 2 − 1, γs,d + <2 −1 γs,r + γr,d + 1 γs,r γr,d < 22R − 1 , (3.80) = Pr γs,d + γs,r + γr,d + 1

where the second equality follows from the fact that the ﬁrst event includes the second event

0 . / γs,r γr,d γs,d < 22R − 1 ⊇ γs,d + < 22R − 1 . γs,r + γr,d + 1 Notice that the outage probability of the incremental relaying scheme is the same as that obtained in the basic AF relaying scheme with diversity combining and, thus, achieves the same diversity order of 2. However, it is worthwhile to remark that, even though the spectral eﬃciency is increased without sacriﬁcing the diversity gain, a simple feedback from the destination is required.

102

3 Two-User Cooperative Transmission Schemes

0

10

−1

10

Outage Probability

−2

10

−3

10

−4

10

No Cooperation Basic AF II, OPA, Statistical CSI Basic AF II, OPA, Instantaneous CSI Incremental AF

−5

10

−6

10

0

6

12 Transmit SNR (dB)

18

24

Fig. 3.15 Performance comparison of incremental AF relaying and basic AF relaying schemes in terms of transmission power.

In Fig. 3.15, we plot the outage probabilities versus the transmit SNR 2 (i.e., SNR = P/σw ) for the incremental AF relaying scheme as well as basic AF relaying schemes with diversity combining. The parameters adopted in the simulations are the same as those corresponding to Fig. 3.3. The relay node is assumed to be located in the middle, between the source and the destination. For the incremental AF relaying scheme, we assume that the source and the relay transmit with equal power, i.e., Ps = Pr = P . We can see that, although all AF relaying schemes considered here achieve the same diversity order, the incremental AF relaying scheme has an additional coding gain due to its increased bandwidth eﬃciency.

3.3 Coded Cooperation Coded cooperation (CC) can be viewed as a generalization of DF relaying schemes where more powerful channel codes (other than simple repetition codes used in the DF schemes) are utilized in both phases of the cooperative transmission. When using repetition codes, the same codeword is transmitted twice (either by the source or the relay) and, thus, bandwidth eﬃciency is decreased by one half. Instead, in coded cooperation schemes [10–12, 14, 22], diﬀerent portions of the same message are transmitted in the two phases.

3.3 Coded Cooperation

103

Speciﬁcally, the source message is encoded in the ﬁrst portion of the codeword that is transmitted by the source in Phase I and incremental redundancy (e.g., in the form of extra parity symbols) can be transmitted in the second portion of the codeword by either the source or the relay in Phase II. In the following, coded cooperation schemes are described using binary error-correction codes, but the concept can be extended naturally to non-binary codes.

3.3.1 Basic Coded Cooperation Scheme Let us consider a basic two-user coded cooperation scheme where one user serves as the source while the other user serves as the relay that forwards the source’s data to the destination. An illustration of the basic coded cooperation scheme is given in Fig. 3.16. Suppose that each source message can be encoded into a rate-R codeword with length N , i.e., x = [x[0], . . . , x[N − 1]]T , and is transmitted over the two phases of the cooperative transmission. The ﬁrst N1 symbols of the codeword, i.e., x(1) = [x[0], . . . , x[N1 − 1]]T , encodes the source message with rate R1 > R while the remaining N2 = N − N1 symbols, i.e., x(2) = [x[N1 ], . . . , x[N1 + N2 − 1]]T , contains additional parity symbols for the encoded message. Although the message can be decoded alone by the ﬁrst portion of the codeword x(1) , the combination of x(1) and x(2) together form a more powerful rate-R codeword. In Phase I, the source transmits the ﬁrst N1 symbols of the codeword, i.e., x(1) , and the signal received at the relay and the destination are given by yr [m] = hs,r Ps x[m] + wr [m], (1) (1) yd [m] = hs,d Ps x[m] + wd [m], for m = 0, . . . , N1 − 1. Let us deﬁne κ = N1 /N such that the rate in Phase I is R1 = R/κ. If the relay is able to successfully decode the message, it will generate the remaining N2 symbols of the codeword, i.e., x(2) , and transmit it to the destination in Phase II. If not, the source will transmit the remaining N2 symbols by itself. Notice that outage occurs on the s-r link when log2 (1 + γs,r ) < R/κ. Similar to the selection DF relaying scheme, we assume that the source has knowledge of the s-r channel and, thus, is able to infer whether or not the relay successfully decodes the message. The signal received at the destination in Phase II can be expressed as, , √ (2) hr,d Pr x[N1 + m] + wd [m], if γs,r ≥ 2R/κ − 1, (2) yd [m] = √ (2) hs,d Pr x[N1 + m] + wd [m], otherwise, for m = 0, . . . , N2 − 1, where we assume that both source and relay use power Pr in Phase II. The destination decodes the message based on the concatenation of signals received in both phases, i.e.,

104

3 Two-User Cooperative Transmission Schemes

(a) Phase I

(b) Phase II

Fig. 3.16 Illustration of basic coded cooperation with a single relay.

)T ( (1) (2) yd = (yd )T (yd )T , (1)

(1)

(1)

(2)

(2)

(2)

where yd = [yd [0], . . . , yd [N1 − 1]]T and yd = [yd [0], . . . , yd [N2 − 1]]T . If the signal in Phase II is received from the relay, the achievable rate is given by (a)

CCC (γs,d , γr,d ) = κ log2 (1 + γs,d ) + (1 − κ) log2 (1 + γr,d ) . If the signals in both phases are received from the source, the achievable rate is given by (b) CCC (γs,d ) = log2 (1 + γs,d ) . Diﬀerent from schemes based on repetition codes, the rate achievable with coded cooperation is not decreased by 1/2. Since N2 out of N total symbols are transmitted by the relay, the ratio 1 − κ characterizes the relay’s level of participation in the cooperative transmission. It is worthwhile to remark that coded cooperation can be achieved with a wide variety of channel codes, including block codes, convolutional codes, or the combination of the two. The symbols in the two portions of the codeword can be partitioned through puncturing, product codes, or other forms of concatenation Examples of coded cooperation using rate-compatible punctured convolutional (RCPC) codes are given in [10,11] and those using turbo codes are given in [14]. The repetition-based DF schemes can be considered as a special case of coded cooperation with N1 = N2 = M and xs = x(1) = x(2) = xr . To evaluate the overall outage probability, we ﬁrst derive the conditional outage probabilities for two separate cases, i.e., Case (a), where the relay successfully decodes the message, and Case (b), where the relay does not successfully decode the message. In Case (a), the relay will forward the second portion of the codeword in Phase II and the destination will decode based on the concatenation of signals received in both phases. Let Os,r {log2 (1 + γs,r ) < R/κ}

3.3 Coded Cooperation

105

c be the event that outage occurs on the s-r link and let Os,r be the complementary event of Os,r . The conditional outage probability in Case (a) can be derived as follows: c + * (a) pout = Pr κ log2 (1 + γs,d ) + (1 − κ) log2 (1 + γr,d ) < R Os,r κ (1−κ) = Pr (1 + γs,d ) (1 + γr,d ) (3.81) < 2R .

The second equality follows from the fact that γs,d and γr,d are independent of γs,r . In Case (b), the source transmits the second portion of the codeword itself since the relay was not able to decode it successfully. The conditional outage probability in this case is given by * + (b) pout = Pr (log2 (1 + γs,d ) < R | Os,r ) = Pr γs,d < 2R − 1 .

(3.82)

By taking an average over the two cases, the average outage probability can be computed as (a) (b) pout = Pr γs,r ≥ 2R/κ − 1 pout + Pr γs,r < 2R/κ − 1 pout (3.83) κ (1−κ) = Pr γs,r ≥ 2R/κ − 1 Pr (1 + γs,d ) (1 + γr,d ) < 2R * + + Pr γs,r < 2R/κ − 1 Pr γs,d < 2R − 1 . (3.84) Let us deﬁne the event 4 5 κ (1−κ) A (1 + γs,d ) (1 + γr,d ) < 2R 0

2R/(1−κ) − 1 a(γs,d ) . = γr,d < (1 + γs,d )κ/(1−κ) For the event A to occur, it must be the case that (1+γs,d )κ/(1−κ) < 2R/(1−κ) (or, equivalently, γs,d < 2R/κ − 1) since γr,d is strictly positive. Hence, under Rayleigh fading assumptions, we have

106

3 Two-User Cooperative Transmission Schemes

κ (1−κ) Pr (1 + γs,d ) (1 + γr,d ) < 2R = Eγs,d ,γr,d [1A ] 2R/κ −1 a(x) 1 1 x y dydx exp − exp − = γ s,d γ s,d γ r,d γ r,d 0 0 2R/κ −1 1 a(x) x = 1 − exp − dx exp − γ s,d γ s,d γ r,d 0 R/κ 2R/κ −1 −1 1 2 x a(x) = 1 − exp − − dx, exp − − γ s,d γ s,d γ s,d γ r,d 0 78 9 6

Ψ(γ s,d ,γ r,d ,R,κ)

(3.85) where the integral above is deﬁned as Ψ(γ s,d , γ r,d , R, κ). By substituting (3.85) into (3.84), the average outage probability can be further evaluated as R/κ R/κ −1 −1 2 2 pout = exp − 1 − exp − − Ψ(γ s,d , γ r,d , R, κ) γ s,r γ s,d R R/κ −1 2 −1 2 1 − exp − . (3.86) + 1 − exp − γ s,r γ s,d 2 Suppose for notational simplicity that Ps = Pr = P and σr2 = σd2 = σw . 2 At high SNR, i.e., when SNR = P/σw 0, the outage probability can be approximated by [12] * +* + 2R − 1 2R/κ − 1 1 1 Λ (R, κ) +O pout ≈ + 2 2 2 η2 ηs,d ηs,d ηr,d SNR2 SNR3 s,r

where R/(1−κ) R/κ 1−κ 2 − 1−2κ 2 +1, κ = 12 , κ = 12 . R ln 2 · 22R+1 − 22R +1,

, Λ (R, κ) =

κ 1−2κ

(3.87)

This shows that coded cooperation also is able to obtain full diversity while achieving higher bandwidth eﬃciency.

3.3.2 User Multiplexing for Coded Cooperation In the cooperation schemes based on DF, AF, and CC, we assumed that, at any time instant (or in any single channel), only one user serves as the source while the other user servers as the relay that forwards the source’s message to

3.3 Coded Cooperation

107

frequency

Phase I Phase II

User 1 tx

User 2 relays

User 2 tx

User 1 relays

User 1 tx

User 2 relays

time

frequency

(a) TDMA Phase I Phase II

User 2 tx

User 1 relays

User 2 tx

User 1 tx

User 2 relays

User 1 tx

time (b) FDMA Fig. 3.17 Illustration of multiplexing schemes in pairwise cooperative systems.

the destination. However, in a pair-wise cooperative system, both users may have their own data to transmit and, thus, their transmissions must be multiplexed in either time, frequency, or code-space (e.g., using TDMA, FDMA, or CDMA schemes) to allow for interference-free access of the cooperative channel. For example, in TDMA, the two users can take turns in playing the role of the source, as shown in Fig. 3.17(a); in FDMA, the two users can interchange their roles on two diﬀerent frequency channels, as shown in Fig. 3.17(b). Since each channel can be treated independently, the discussions provided in previous sections, thus, focus only on the relay operations in a single channel. However, joint consideration of the operations in both channels may yield simpliﬁed cooperation protocols. In the following, we describe a user multiplexing scheme proposed in [12] that exploits this concept in CC systems. Consider a system that consists of two users, e.g., user 1 and user 2, that cooperate in transmitting both of their messages to the destination. In Phase I, the users’ messages are transmitted through orthogonal channels and are received simultaneously by each other as well as the destination. Suppose that cyclic redundancy check (CRC) codes are applied on the messages so that each user can determine whether or not the messages received from its partner is decoded correctly. If a user succeeds in decoding the message, the user will forward the second portion of its partner’s codeword to the destination; if

108

3 Two-User Cooperative Transmission Schemes

(a) Case (a)

(c) Case (c)

(b) Case (b)

(d) Case (d)

Fig. 3.18 Illustration of the four signal transmission cases in two-user transmission coded cooperation.

not, the user will instead transmit the second portion of its own codeword. The diﬀerence with the original CC scheme is that, in this case, whether or not the source user transmits part of its own codeword in Phase II, depends only on whether or not it successfully decodes its partner’s message in Phase I. Therefore, knowledge of the s-r channel or feedback from the relay is no longer required and, thus, the protocol is easier to implement in practice. Let hi,j be the channel coeﬃcient of the link from user i to user j. Suppose that reciprocity is not satisﬁed on the inter-user channel such that h1,2 and h2,1 are independent. Moreover, for simplicity, assume that both users 2 transmit with the same power P and that σr2 = σd2 = σw . In Rayleigh fading scenarios, the channel coeﬃcients h1,2 , h2,1 , h1,d , and h2,d are assumed to be 2 2 2 2 complex Gaussian with mean 0 and variances η1,2 , η2,1 , η1,d , and η2,d , respectively. Four possible cases may occur throughout the transmission process, as illustrated in Fig. 3.18. In Case (a), both users successfully decode each other’s message and transmit additional parity for the other user in Phase II.

3.3 Coded Cooperation

109

In Case (b), neither user succeeds in decoding the other user’s message and, thus, transmits the second portion of its own codewords. In Case (c), user 2 successfully decodes user 1 but user 1 does not successfully decode user 2. As a result, no user will transmit the second portion of user 2’s codeword. Case (d) is the same as Case (c) but with the role of user 1 and user 2 reversed. Let us deﬁne 4 5 O1,2 {log2 (1 + γ1,2 ) < R/κ} = γ1,2 < 2R/κ − 1 as the event that outage occurs on the channel from user 1 to user 2, where 2 γ1,2 P |h1,2 |2 /σw , and deﬁne 4 5 O2,1 {log2 (1 + γ2,1 ) < R/κ} = γ2,1 < 2R/κ − 1 as the event that outage occurs on the channel from user 2 to user 1, where 2 γ2,1 P |h2,1 |2 /σw . The outage probabilities for these cases are evaluated separately in the following. Case (a): In Case (a), both users are able to successfully decode their partner’s message in Phase I, which occurs with probability qa = Pr (O1c ∩ O2c ) = Pr γ1,2 ≥ 2R/κ − 1 · Pr γ1,2 ≥ 2R/κ − 1 R/κ R/κ −1 −1 2 2 exp − , = exp − γ 1,2 γ 2,1

(3.88)

2 2 where γ i,j = E[γi,j ] = P ηi,j /σw , for i, j ∈ {1, 2}. In this case, each user will cooperate by transmitting the second portion of its partner’s codeword in Phase II. Therefore, similar to (3.85), the conditional outage probabilities for users 1 and 2 are given respectively by (a)

pout,1 = Pr (κ log2 (1 + γ1,d ) + (1 − κ) log2 (1 + γ2,d ) < R) κ (1−κ) = Pr (1 + γ1,d ) (1 + γ2,d ) < 2R R/κ −1 2 − Ψ(γ 1,d , γ 2,d , R, κ) =1 − exp − γ 1,d and (a)

pout,2 = Pr (κ log2 (1 + γ2,d ) + (1 − κ) log2 (1 + γ1,d ) < R) R/κ −1 2 − Ψ(γ 2,d , γ 1,d , R, κ), =1 − exp − γ 2,d

110

3 Two-User Cooperative Transmission Schemes

2 2 2 where γi,d P |hi,d |2 /σw and γ i,d = E[γi,d ] = P ηi,d /σw , for i = 1, 2.

Case (b): In Case (b), both users fail to decode their partner’s message in Phase I, which occurs with probability qb = Pr(O1,2 ∩ O2,1 ) R/κ R/κ −1 −1 2 2 = 1 − exp − 1 − exp − . γ 1,2 γ 2,1 Since neither of the two users successfully decode their partners’ messages, they both transmit the second portion of their own codewords in Phase II. The conditional outage probabilities for user 1 and user 2 are given respectively by R 2 −1 (b) pout,1 = Pr (log2 (1 + γ1,d ) < R) = 1 − exp − γ 1,d and (b) pout,2

R 2 −1 . = Pr (log2 (1 + P γ2,d ) < R) = 1 − exp − γ 2,d

Case (c): In Case (c), the message transmitted by user 1 is successfully decoded by user 2 in Phase I, but user 2’s message is not successfully decoded at user 1. The probability that this event occurs is given by R/κ R/κ −1 −1 2 2 c qc = Pr(O1,2 ∩ O2,1 ) = exp − 1 − exp − . γ 1,2 γ 2,1 Then, in Phase II, both users transmit simultaneously the second portion of user 1’s codeword in their respective channels. The second portion of user 2’s codeword is not transmitted by any user in Phase II and, thus, decoding of it’s message at the destination can be performed based only on the signal received in Phase I. Following similar derivations as in (3.85), the conditional outage probability for user 1 can be computed as (c)

pout,1 = Pr (κ log2 (1 + γ1,d ) + (1 − κ) log2 (1 + γ1,d + γ2,d ) < R) = Pr (1 + γ1,d )κ (1 + γ1,d + γ2,d )(1−κ) < 2R 2R −1 R 1 x b(x) 2 −1 − dx, exp − − = 1 − exp − γ 1,d γ 1,d γ 1,d γ 2,d 0 78 9 6 Φ(γ 1,d ,γ 2,d ,R,κ)

3.3 Coded Cooperation

111

where the integral above is deﬁned as Φ(γ 1,d , γ 2,d , R, κ) and b(x)

2R/(1−κ) − 1 − x. (1 + x)κ/(1−κ)

The conditional outage probability of user 2 is obtained more easily as R/κ 2 −1 (c) . pout,2 = Pr (κ log2 (1 + γ2,d ) < R) = 1 − exp − γ 2,d Case (d): Case (d) is exactly the same as Case (c) but with the roles of user 1 and user 2 reversed. The probability that is event occurs is given by R/κ R/κ + * −1 −1 2 2 c exp − . qd = Pr O1,2 ∩ O2,1 = 1 − exp − γ 1,2 γ 2,1 Similar to Case (c), the conditional outage probabilities of user 1 and user 2 are given respectively by R/κ −1 2 (d) pout,1 = 1 − exp − γ 1,d

and (d) pout,2

2R − 1 = 1 − exp − γ 2,d

− Φ(γ 2,d , γ 1,d , R, κ).

By considering the above four cases, the average outage probability of user i, for i ∈ {1, 2}, can be obtained by (a)

(b)

(c)

(d)

pind out,i = qa · pout,i + qb · pout,i + qc · pout,i + qd · pout,i , where the superscript “ind” indicates that independent (or non-reciprocal) inter-user channels are considered. At high SNR, where SNR 0, it has been shown in [12] that the average outage probability can be approximated as R/κ 2 − 1) (2 1 1 Λ(R, κ) ind pout,i ≈ +O , + 2 2 2 η2 ηi,d ηi,d ηj,d SNR2 SNR3 i,j where i = j ∈ {1, 2} and Λ(R, κ) was deﬁned in (3.87). This shows that the user multiplexing scheme described above achieves diversity order of 2 for both users, even without the help of feedback from the relay. It is worthwhile to note that, when reciprocity holds on the inter-user channel (i.e., when h1,2 = h2,1 ), only Cases (a) and (b) will occur and, thus, the outage probability reduces to that derived in (3.86). At high SNR, the outage probability of user i can be approximated as

112

3 Two-User Cooperative Transmission Schemes

Fig. 3.19 10% outage capacity versus mean uplink SNR for various inter-user channel variances. Both the reciprocal and independent inter-user channels are considered (From c Hunter, Sanayei, and Nosratinia with modiﬁed labels. [2006] IEEE ).

prec out,i

1 ≈ SNR2

(2R − 1)(2R/κ − 1) Λ(R, κ) 1 +O , + 2 2 2 η2 ηi,d ηi,d ηj,d SNR3 i,j

(3.89)

where i = j ∈ {1, 2}. Hence, diversity order of 2 is also achieved in this case. Moreover, one can observe that the outage probability in this case is lower than that achieved with independent inter-user channels. In Fig. 3.19 (i.e. [12, Figure 5]), the 10% outage capacities achieved by coded cooperation under independent and reciprocal inter-user channels are compared. The case with no user cooperation is also shown as a basis for comparison. The mean uplink SNRs are assumed to be the same for both users, i.e., γ 1,d = γ 2,d , and the statistics of the inter-user channels in both 2 2 directions are also assumed to be identical, i.e., η1,2 = η2,1 . The outage capacities are given for 3 diﬀerent scenarios: the scenario with noiseless inter-user 2 2 channels (i.e., η1,2 = η2,1 = ∞), the scenario where the inter-user and up2 2 2 2 link channels have the same variance (i.e., η1,d = η2,d = η1,2 = η2,1 ), and the scenario where the inter-user SNR is 10dB less than the uplink SNR. We can observe that coded cooperation performs better over reciprocal interuser channels. The reason is that, with independent inter-user channels, the second portion of a certain user’s codeword will not be transmitted by any

3.3 Coded Cooperation

113

user in either Case (c) or Case (d), thereby increasing the outage probability. Such negligence of a certain user’s parity symbols in Phase II does not occur with reciprocal inter-user channels since the inter-user transmissions are either both successful or both unsuccessful. In Fig. 3.19, we can see that coded cooperation provides 4 − 7 dB performance gain compared to direct transmission for outage capacity R = 1. In the CC scheme with independent inter-user channels, the conditional outage probabilities increase considerably for Cases (c) and (d) due to negligence of a certain user’s parity symbols in Phase II of these cases. To avoid this loss in performance, one can employ a strategy that allows each user to use a portion of its transmission power to forward its own codeword, even when it successfully decodes its partner’s message. This scheme is referred to as the space-time coded cooperation (STCC) scheme [12, 14]. More speciﬁcally, upon successfully decoding its partner’s message, a user will only utilize power (1 − β)P to forward the second portion of its partner’s codeword while utilizing the remaining βP to transmit its own codeword. The codewords of diﬀerent users are transmitted on their respective channels. Similarly, four possible cases may occur throughout the transmission process: Case (a) where both users successfully decode their partner’s messages, Case (b) where both user fail to decode their partner’s messages, Case (c) where only user 2 decodes its partner’s message, and Case (d) where only user 1 successfully decodes. In contrast to the previous scheme, no codeword is completely neglected in Cases (c) and (d) since each user will expend a portion of its power to forward its own message, even when it successfully decodes its partner’s message. Space-time coded cooperation was ﬁrst proposed in [14] to enable the use of coded cooperation over fast fading channels. In fast fading environments, the channels in the both phases of the transmission process are assumed to be independent and, thus, utilizing the help from its partner may not always be advantageous. Reserving part of the resource to transmit its own parity symbols can thus reduce the probability of outage. In Fig. 3.20 (i.e., [12, Figure 4]), the outage probability of the space-time coded cooperation scheme is plotted for the case with reciprocal inter-user channels and the case with independent inter-user channels. The rate is set as R = 1/2 b/s/Hz. The mean uplink SNR for both user are assumed to be equal and the variances of the inter-user channels in both directions are also assumed to be equal. The channel parameters are given the same as those used in Fig. 3.19. One can observe that the original CC scheme achieves a lower average outage probability compared to STCC. However, when Cases (c) or (d) occur, STCC is able to avoid the signiﬁcant increase in outage probability caused by the negligence of a certain user’s codeword.

114

3 Two-User Cooperative Transmission Schemes

Fig. 3.20 Performance comparison of coded cooperation and space-time coded cooperation over various slow fading channels (From Hunter, Sanayei, and Nosratinia with modc iﬁed labels. [2006] IEEE ).

3.4 Compress-and-Forward Relaying Schemes Compress-and-forward (CF) relaying schemes refer to cases where the relay forwards quantized, estimated, or compressed versions of its observation to the destination. In contrast to DF or CC schemes, the relay in CF schemes need not decode perfectly the source message but need only to extract, from its observation, the information that is most relevant to the decoding at the destination. The amount of information extracted and forwarded to the destination depends on the capacity of the r-d link. In fact, it has been shown in [17] that CF can outperform DF when the relay is farther from the source (i.e., decoding is less reliable at the relay) and is closer to the destination (i.e., more information can be conveyed to the destination through the r-d channel). Moreover, CF also provides a more general form of compression compared to the simple scaling done in AF. The works on CF stem from the fundamental studies on relay channels [3, 17] and have mostly been investigated from the information-theoretic perspective. Detailed discussions of these schemes will be delayed till Section 5.1.3, but the basic concepts are introduced in the following.

3.5 Channel Estimation in Single Relay Systems

115

Fig. 3.21 System model of compress-and-forward relaying scheme.

The CF relaying scheme also takes on two phases of transmission, as illustrated in Fig. 3.21. In Phase I, the source transmits a message to both the relay and the destination, where the received signals are denoted by Yr and (1) Yd , respectively; in Phase II, the relay compresses Yr or extracts from Yr the information that is most useful for the decoding at the destination. The compression at the relay must be performed with only statistical knowledge (1) of the signal received at the destination, i.e., Yd , which relates to the works on distributed source coding [26] or source coding with side information [29]. One approach to achieve this task is the use of Wyner-Ziv coding (WZC) at the relay. The basic framework pioneered by Wyner and Ziv in [29] is illustrated in Fig. 3.22. Here, the lossy source coding problem is considered where the source W is compressed under a given distortion constraint. It has been shown in [29] that, with knowledge of the joint distribution of the input W and side information S (which is only available at the decoder), the input W can be compressed with a rate that is smaller than that attainable without side information. WZC has been applied to many diﬀerent applications, e.g., distributed video coding, and can implemented using practical coding schemes. When applying WZC to the relay problem shown in Fig. 3.21, the relay observation Yr can now be viewed as the input W and the signal re(1) ceived at the destination, i.e., Yd , can be viewed as the side information S that is available at the decoder. The capacity of the r-d link determines the maximum compression rate at the relay and, thus, aﬀects the distortion of the reconstructed signal Yˆr . More practical studies on CF relaying schemes can be found in [9, 15, 20, 21].

3.5 Channel Estimation in Single Relay Systems In previous sections, DF, AF, CC, and CF relaying schemes have been introduced by assuming diﬀerent levels of transmitter CSI and by assuming perfect CSI at the receiver for coherent detection. In practice, the CSI required

116

3 Two-User Cooperative Transmission Schemes

Fig. 3.22 Block diagram of Wyner-Ziv coding.

to achieve these tasks must be obtained through actual channel estimation. Without an accurate estimate of the channels, the beneﬁts of cooperation may decrease substantially. However, the problem is less complex in the case of DF, CC, and CF schemes, where decoding is performed at both the relay and the destination. In these cases, the channel coeﬃcients of each point-to-point link (i.e., s-r, r-d, and s-d links) can be estimated separately at the relay and the destination using conventional channel estimation schemes [16]. However, in AF relaying schemes, the eﬀective channel received at the destination is a combination of the s-r and the r-d channels and, thus, the estimation performances may diﬀer from that in point-to-point channels. In this section, we brieﬂy touch upon these issues by summarizing the results presented in [23]. Let us consider the basic AF relaying model presented in Section 3.2 that consists of a source, a relay, and a destination. In each time slot, the source ﬁrst transmits a symbol to the relay in the ﬁrst half of the time slot while the relay ampliﬁes and forwards the symbol to the destination in the second half of the time slot. Suppose that the symbol xs [m] is transmitted by the source in the m-th symbol period and that E |xs [m]|2 = Ps , where the source power has been incorporated into the symbol xs [m] to simplify notations. The signal received at the relay during the m-th symbol period can be expressed as yr [m] = hs,r [m]xs [m] + wr [m] 2 2 where hs,r [m] ∼ CN (0, ηs,r ) and wr [m] ∼ CN (0, σw ). Diﬀerent from previous sections, we assume here the more general case where the channel coeﬃcients, e.g., hs,r [m] may vary from symbol to symbol based on a certain correlation model. The relay then transmits the signal G[m]yr [m] to the destination, where the received signal is given by

yd [m] = hr,d [m]G[m]yr [m] + wd [m] = G[m]hr,d [m]hs,r [m]xs [m] + G[m]hr,d [m]wr [m] + wd [m], 2 2 where hr,d [m] ∼ CN (0, ηr,d ) and wd [m] ∼ CN (0, σw ). The value of the amplifying gain depends on the speciﬁc AF scheme that is employed. For variable gain relaying, we have

3.5 Channel Estimation in Single Relay Systems

117

Fig. 3.23 An example of pilot-selection for channel estimation with Np = 4.

' G[m] =

' Pr = E [|yr [m]|2 |hs,r [m]]

Pr , 2 Ps |hs,r [m]|2 + σw

(3.90)

and, for ﬁxed gain relaying, we have ' ' Pr Pr = G, G[m] = 2 2 2 E [|yr [m]| ] Ps ηs,r + σw

(3.91)

which is constant over all symbol periods. Notice that the relay power Pr is also incorporated into the channel gains here. In order to perform coherent detection on the symbol transmitted through the s-r-d path, the destination must have knowledge of the eﬀective channel coeﬃcient heﬀ [m] = G[m]hr,d [m]hs,r [m]. Even though the s-d channel may also be of interest when considering diversity combining, it can be easily obtained using conventional point-to-point channel estimation schemes and, thus, will not be considered in the following. Speciﬁcally, we shall focus on the estimation of heﬀ [m], which is no longer Gaussian due to the multiplication of channel coeﬃcients and which may be aﬀected by the additional noise propagation term. √ (p) Suppose that a binary pilot symbol, say xs [ ] ∈ {± Ps }, is inserted every Tp symbol periods, where is a position of the pilot and the superscript (p) represents the fact that it is a pilot symbol. To obtain an accurate estimate of the channel, the destination takes the Np nearest pilot signals (with respect to the symbol of interest) to perform channel estimation. For example, if Np = 4, the destination will select the two pilot signals directly before and after the symbol of interest, as shown in Fig. 3.23. The signal received during each pilot symbol period is ﬁrst normalized by its corresponding training symbol to ( )T ˆ (p) [ ] = y (p) [ ]/x(p) ˆ (p) = h ˆ (p) [ 1 ], h ˆ (p) [ 2 ], . . . , h ˆ (p) [ Np ] obtain h s [ ]. Let h eﬀ

d

eﬀ

eﬀ

eﬀ

eﬀ

be the vector of Np normalized pilot signals used for channel estimation. In

118

3 Two-User Cooperative Transmission Schemes

conventional point-to-point channels, one of the most widely adopted channel estimation schemes is the minimum mean square error (MMSE) estimator. However, this estimator is diﬃcult to obtain for the AF relay channel due to the non-Gaussian nature of the eﬀective channel coeﬃcient and the noise propagation eﬀect. Therefore, we instead employ the Linear MMSE (LMMSE) estimator which can also be eﬀective as shown in the following. By employing the LMMSE estimator, the estimate of the channel at time m can be expressed as ˆ eﬀ [m] = r h h

ˆ (p) eff heff

ˆ (p) [m] · R−1 ·h eﬀ ˆ (p) ˆ (p) heff heff

(3.92)

) ) ( ( ˆ (p) )H and R ˆ p ˆ (p) = E h ˆ (p) (h ˆ (p) )H . where rh hˆ (p) [m] = E heﬀ [m] · (h eﬀ eﬀ eﬀ heff heff eff eff The cross correlation between the eﬀective channel and a normalized pilot signal is ( ) * ˆ (p)∗ [ ] = E G[m]hr,d [m]hs,r [m] G[ ]h∗ [ ]h∗ [ ] E heﬀ [m]h r,d s,r eﬀ +) ∗ (p) +G[ ]h∗r,d [ ]wr∗ [ ]/x(p) [ ] + w [ ]/x [ ] s d s and the correlation between two normalized pilot signals is ( ) ˆ (p) [ ]h ˆ (p)∗ [ ] E h eﬀ eﬀ ( (p) = E (G[ ]hr,d [ ]hs,r [ ] + G[ ]hr,d [ ]wr [ ]/x(p) s [ ] + wd [ ]/xs [ ])

) ∗ (p) ×(G[ ]h∗r,d [ ]h∗s,r [ ] + G[ ]h∗r,d [ ]wr∗ [ ]/x(p) s [ ] + wd [ ]/xs [ ]) .

For ﬁxed gain relaying, the amplifying gain G[m] is constant regardless of the time index m. In this case, we have ( ) ˆ (p)∗ [ ] = G2 Rh h [m − ]Rh h [m − ] E heﬀ [m]h s,r s,r r,d r,d eﬀ =

Pr Rhs,r hs,r [m − ]Rhr,d hr,d [m − ], 2 (1 + γ σw s,r )

2 2 where γ s,r = Ps ηs,r /σw is the average SNR of the s-r link, and

3.5 Channel Estimation in Single Relay Systems

119

( ) ˆ (p) [ ]h ˆ (p)∗ [ ] E h eﬀ eﬀ = G2 Rhs,r hs,r [ − ]Rhr,d hr,d [ − ] +

2 2 σw G2 ηr,d

Ps Pr Rh h [ − ]Rhr,d hr,d [ − ] = 2 σw (1 + γ s,r ) s,r s,r +

δ( − ) +

2 σw δ( − ) Ps

2 γ r,d σw σ2 δ( − ) + w δ( − ), (1 + γ s,r )Ps Ps

2 2 where γ r,d = Pr ηr,d /σw is the average SNR of the r-d link. For variable gain relaying, the amplifying gain then depends on the source-relay channel during each symbol period and, thus, must be estimated at the relay as well. We may also take the LMMSE estimate of this channel as

ˆ s,r [m] = r ˆ (p) [m] · R−1 ˆ (p) h ·h s,r ˆ (p) ˆ (p) hs,r h hs,r hs,r

s,r

(3.93)

( )T (p) (p) (p) ˆ (p) ˆ (p) ˆ (p) ˆ (p) where h and hs,r [ ] = yr [ ]/xs [ ]. s,r = hs,r [ 1 ], hs,r [ 2 ], . . . , hs,r [ Np ] By substituting this into the amplifying gain G[m], the correlation between the eﬀective channel coeﬃcients can be approximated as [24] ) P ( e1/γ s,r E1 (1/γ s,r ) r (p)∗ ˆ 1− Rhs,r hs,r [m − ]Rhr,d hr,d [m − ] E heﬀ [m]h eﬀ [ ] ≈ Ps γ s,r and ( ) P e1/γ s,r E1 (1/γ s,r ) r (p) ˆ (p)∗ ˆ 1− Rhs,r hs,r [ − ]Rhr,d hr,d [ − ] E heﬀ [ ]heﬀ [ ] ≈ Ps γ s,r +

2 γ r,d σw σ2 e1/γ s,r E1 (1/γ s,r )δ( − ) + w δ( − ), Ps γ s,r Ps

where E1 (z) =

z

∞

e−t dt t

is the exponential integral function [5]. Example: Downlink Relay System Let us consider a downlink system where the base station (BS) transmits a message to a mobile station (MS) through a relay station. Depending on the mobility of the relay, two scenarios can be considered, namely, the case with stationary relays and the case with mobile relays. Following the model given in [13] and [23], the temporal correlations of the channels in the case of stationary relays are given by

120

3 Two-User Cooperative Transmission Schemes

2 Rhs,r hs,r [k] = E hs,r [ + k]h∗s,r [ ] = ηs,r J0 (2πfs,r k)

(3.94a)

2 Rhr,d hr,d [k] = E hr,d [ + k]h∗r,d [ ] = ηr,d J0 (2πfr,d k),

(3.94b)

and

where J0 (x) is the zeroth order Bessel function of the ﬁrst kind and fs,r , fr,d are the Doppler frequencies associated with the channels hs,r [ ] and hr,d [ ], respectively. In the case of mobile relays, the channel correlations are given by 2 Rhs,r hs,r [k] = E hs,r [ + k]h∗s,r [ ] = ηs,r J0 (2πfs,r k) (3.95a) and 2 J0 (2πfs,r k)J0 (2πfr,d k). Rhr,d hr,d [k] = E hr,d [ + k]h∗r,d [ ] = ηr,d

(3.95b)

To obtain reliable estimates of the channels, the pilots must be placed accord1 ing to the Nyquist sampling criterion so that Tp ≤ 2fmax Ts , where fmax is the maximum Doppler frequency of the estimated channel and Ts is the symbol duration. Notice that fmax of the eﬀective channel is fs,r + fr,d for stationary relays and is 2fs,r + fr,d for mobile relays. It is interesting to note that, in the DF scheme, the source-to-relay and the relay-to-destination channels are estimated separately and pilots need only be placed at twice the maximum Doppler frequency of the two links, which is max{fs,r , fr,d } for stationary relays and fs,r + fr,d for mobile relays. That is, the pilots required in the AF scheme is generally much larger than that of the DF scheme. However, the DF scheme require more complex operations at the relay. By assuming Np = 4, Tp = 5, and equal average SNR on each hop (i.e., γ s,r = γ r,d ), the BER performance versus the SNR per hop are shown in Fig. 3.24 and Fig. 3.25 (see [23]). We can see that the performance loss due to channel estimation errors in the AF case is comparable with that in the DF case. This shows that it is indeed suﬃcient to estimate the combined channel (instead of to estimate each channel separately) in the AF case. Moreover, the ﬁgures show no loss of diversity gain due to channel estimation errors.

References 1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1965) 2. Chen, D., Laneman, J.N.: Modulation and demodulation for cooperative diversity in wireless systems. IEEE Transactions on Wireless Communications 5(7), 1785–1794 (2006) 3. Cover, T., El Gamal, A.: Capacity theorems for the relay channel. IEEE Transactions on Information Theory 25(5), 572–584 (1979) 4. Gradshteyn, I.S., Ryzhik, I.M., Jeﬀrey, A. (eds.): Table of Integrals, Series, and Products, 5 edn. Academic Press (1994)

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Fig. 3.24 For Np = 4 and Tp = 5, the BER vs SNR per hop (i.e., γ s,r = γ r,d ) is shown for the case with stationary relays (i.e. fs,r Ts = 0.001 and fr,d Ts = 0.01). (From Patel c and St¨ uber. [2007] IEEE ).

Fig. 3.25 For Np = 4 and Tp = 5, the BER vs SNR per hop (i.e., γ s,r = γ r,d ) is shown uber. for the case with mobile relays (i.e. fs,r Ts = fr,d Ts = 0.01). (From Patel and St¨ c [2007] IEEE ).

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3 Two-User Cooperative Transmission Schemes

5. Gradshteyn, I.S., Ryzhik, I.M., Jeﬀrey, A., Zwillinger, D. (eds.): Table of Integrals, Series, and Products, 7 edn. Academic Press (2007) 6. Hasna, M.O., Alouini, M.-S.: End-to-end performance of transmission systems with relays over Rayleigh-fading channels. IEEE Transactions on Wireless Communications 2(6), 1126–1131 (2003) 7. Hasna, M.O., Alouini, M.-S.: A performance study of dual-hop transmissions with ﬁxed gain relays. IEEE Transactions on Wireless Communications 3(6), 1963 (2004) 8. Hong, Y.-W., Huang, W.-J., Chiu, F.-H., Kuo, C.-C.J.: Cooperative communications in resource-constrained wireless networks. IEEE Signal Processing Magazine 24(3), 47–57 (2007) 9. Hu, R., Li, J.: Practical compress-and-forward in user cooperation: Wyner-Ziv cooperation. In: Proceedings on the IEEE International Symposium on Information Theory (ISIT) pp. 489-493 (2006) 10. Hunter, T., Nosratinia, A.: Cooperative diversity through coding. In: Proceedings on the IEEE International Symposium on Information Theory (ISIT) pp. 220 (2002) 11. Hunter, T.E., Nosratinia, A.: Diversity through coded cooperation. IEEE Transactions on Wireless Communications 5(2), 283–289 (2006) 12. Hunter, T.E., Sanayei, S., Nosratinia, A.: Outage analysis of coded cooperation. IEEE Transactions on Information Theory 52(2), 375–391 (2006) 13. Jakes, W.C.: Microwave Mobile Communications, 2 edn. IEEE Press (1994) 14. Janani, M., Hedayat, A., Hunter, T.E., Nosratinia, A.: Coded cooperation in wireless communications: Space-time transmission and iterative decoding. IEEE Transactions on Signal Processing 52(2), 362–371 (2004) 15. Jiang, J., Thompson, J.S., Grant, P.M., Goertz, N.: Practical compress-and-forward cooperation for the classical relay network. In: Proceedings of the 17th European Signal Processing Conference (EUSIPCO), pp. 2421–2425. Glasgow, Scotland (2009) 16. Kay, S.M.: Fundamentals of Statistical Signal Processing: Estimation Theory, vol. I. Prentice Hall PTR (1993) 17. Kramer, G., Gastpar, M., Gupta, P.: Cooperative strategies and capacity theorems for relay networks. IEEE Transactions on Information Theory 51(9), 3037–3063 (2005) 18. Laneman, J.N., Tse, D.N.C., Wornell, G.W.: Cooperative diversity in wireless networks: Eﬃcient protocols and outage behavior. IEEE Transactions on Information Theory 50(12), 3062–3080 (2004) 19. Laneman, J.N., Wornell, G.W.: Energy-eﬃcient antenna sharing and relaying for wireless networks. In: Proceedings of IEEE Wireless Communications and Networking Conference (WCNC), pp. 7-12 (2000) 20. Liu, Z., Uppal, M., Stankovic, V., Xiong, Z.: Compress-forward coding with BPSK modulation for the half-duplex gaussian relay channel. In: Proceedings of the IEEE ISIT, pp.2395 -2399 (2008) 21. Liu, Z., Uppal, M., Stankovic, V., Xiong, Z.: Compress-forward coding with BPSK modulation for the half-duplex gaussian relay channel. IEEE Transactions on Signal Processing 57(11), 4467–4481 (2009) 22. Nosratinia, A., Hunter, T.E., Hedayat, A.: Cooperative communication in wireless networks. IEEE Communications Magazine 42(10), 74–80 (2004) 23. Patel, C.S., St¨ uber, G.L.: Channel estimation for amplify and forward relay based cooperation diversity systems. IEEE Transactions on Wireless Communications 6(6), 2348–2356 (2007) 24. Patel, C.S., St¨ uber, G.L., Pratt, T.G.: Statistical properties of amplify and forward relay fading channels. IEEE Transactions on Vehicular Technology 55(1), 1–9 (2006) 25. Sendonaris, A., Erkip, E., Aazhang, B.: User cooperation diversity–Part I: System description” and “User cooperation diversity–Part II: implementation aspects and performance analysis. IEEE Transactions on Communications 51(11), 1927-1938 and 1939-1948 (2003) 26. Slepian, D., Wolf, J.: Noiseless coding of correlated information sources. IEEE Transactions on Information Theory 19(4), 471–480 (1973)

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27. Su, W., Sadek, A.K., Liu, K.J.R.: Cooperative communications protocols in wireless networks: Performance analysis and optimum power allocation. Wireless Personal Communications 44(2), 181–217 (2008) 28. Wang, T., Cano, A., Giannakis, G.B., Laneman, J.N.: High-performance cooperative demodulation with decode-and-forward relays. IEEE Transactions on Communications 55(7), 1427–1438 (2007) 29. Wyner, A., Ziv, J.: The rate-distortion function for source coding with side information at the decoder. IEEE Transactions on Information Theory 22(1), 1–10 (1976) 30. Zhang, J., Zhang, Q., Shao, C., Wang, Y., Zhang, P., Zhang, Z.: Adaptive optimal transmit power allocation for two-hop non-regenerative wireless relay system. In: Proceedings of IEEE 59th Vehicular Technology Conference, vol. 2, pp. 1213- 1217(2004) 31. Zhang, Q., Zhang, J., Shao, C., Wang, Y., Zhang, P., Hu, R.: Power allocation for regenerative relay channel with Rayleigh fading. In: Proceedings of IEEE 59th Vehicular Technology Conference, vol. 2, pp. 1167– 1171 (2004)

Chapter 4

Cooperative Transmission Schemes with Multiple Relays

In this chapter, cooperative transmission schemes are introduced for networks that consist of more than two users. Following the assumptions made in the previous chapter, we assume that, at each instant in time, only one user acts as the source while the other users serve as relays that help forward the source’s message to the destination. In this case, the relays can together form a distributed antenna array and adopt conventional MIMO signal processing techniques, such as beamforming, antenna selection, or space-time coding etc., to enhance communication performance. As the number of relays increases, more radio resources and more degrees of freedom can be pooled together and utilized jointly to assist the source’s transmission. However, to exploit these advantages, one must overcome the challenges posed by the individual resource constraints and the lack of coordination among relays. In Section 4.1, we ﬁrst introduce cooperation schemes where relays are assumed to transmit over orthogonal channels. These orthogonal cooperation schemes can be implemented without strict synchronization requirements but result in low bandwidth eﬃciency. Then, in Sections 4.2–4.4, we introduce non-orthogonal cooperation schemes where all relays are assumed to share a common channel. Under diﬀerent CSI assumptions, we introduce three cooperative transmission schemes, namely, distributed beamforming (BF), selective relaying (SR), and distributed space-time coding (DSTC). Each of these schemes can be incorporated with both AF and DF relays. The CSI can be obtained through channel estimation at the destination, which will be discussed in Section 4.5. More advanced multi-relay cooperation strategies such as multi-hop and asynchronous transmission schemes will also be discussed in Section 4.6.

Y.-W. Peter Hong et al., Cooperative Communications and Networking, DOI 10.1007/978-1-4419-7194-4_4, © Springer Science+Business Media, LLC 2010

125

126

4 Cooperative Transmission Schemes with Multiple Relays

Fig. 4.1 System model of a two-hop synchronous cooperation system.

4.1 Orthogonal Cooperation Consider a cooperative network that consists of L + 1 users, one acting as the source and L serving as the relays, as shown in Fig. 4.1. Let us denote the source by s, the destination by d, and label the relays from 1 to L. In orthogonal cooperation schemes, we assume that the source and the relays each transmit over an orthogonal time or frequency channel. Similar to the two-user scenario described in the previous chapter, all nodes are assumed to be half-duplex so that cooperative transmission must be conducted over two phases. In Phase I, the source ﬁrst transmits the symbol vector xs = [xs [0], . . . , xs [M − 1]]T (with E[|xs [m]|2 ] = 1, ∀m) to both the relays and the destination. The signals received at relay and the destination in the m-th symbol period are given by y [m] = hs, Ps xs [m] + w [m], (4.1) (1) (1) yd [m] = hs,d Ps xs [m] + wd [m], (4.2) where Ps is the source transmission power, hs, and hs,d are the channel coeﬃcients between source and relay (i.e., the s- link) and between source and destination (i.e., the s-d link), respectively, and w [m] ∼ CN (0, σ2 ) and (1) wd [m] ∼ CN (0, σd2 ) are the additive white Gaussian noise (AWGN) at relay

and the destination, respectively. In Phase II, the relays belonging to a certain cooperative set D will each forward the source’s message over an orthogonal channel. That is, relay

∈ D will transmit, in its corresponding channel, the symbol vector x = [x [0], . . . , x [M − 1]]T = f (y ) to the destination, where E[|x [m]|2 ] = 1, ∀m, and f (·) is the relay function that depends on the speciﬁc cooperative strategy. The signal received by the destination in the channel corresponding to relay is given by

4.1 Orthogonal Cooperation (2,)

yd

127

[m] = h,d

(2,) P x [m] + wd [m],

(4.3)

where P is the transmission power of relay , h,d is the channel coeﬃ(2,) cient between relay and the destination (i.e., the -d link), and wd [m] ∼ 2 CN (0, σd ) is the AWGN at the destination in the channel corresponding to relay . All channel coeﬃcients are assumed to be independent of each other. The superscript (2, ) is used to highlight the fact that the signals from different relays are received over diﬀerent time or frequency channels. In the following, we discuss both AF- and DF-based orthogonal cooperation schemes and analyze their performance under repetition-based coding schemes. These schemes can be viewed as direct extensions of the basic AF and DF schemes described in Chapter 3.

4.1.1 Orthogonal Cooperation with AF Relays In the AF orthogonal cooperation scheme, we assume that all relays participate in the cooperative transmission regardless of the quality of the received signal. Therefore, the cooperative set is given by D = {1, 2, · · · , L}. The signal forwarded by relay during Phase II is given by y [m] y [m] x [m] = , =2 E[|y | [m]] Ps |hs, |2 + σ2

(4.4)

for m = 0, . . . , M − 1. Suppose that the source and the relays are each allocated an equal-length time slot for the transmission of their codewords. Thus, a total of L + 1 orthogonal time slots are required throughout the cooperative transmission. The signals received at the destination in Phase I and in the -th time slot of Phase II are given by (1) (1) yd [m] = Ps hs,d xs [m] + wd [m], and (2,)

yd

[m] = ' =

-

(2,)

P h,d x [m] + wd

[m]

P Ps h,d hs, xs [m]+ Ps |hs, |2 +σ2

' P (2,) h,d w [m]+wd [m], Ps |hs, |2 +σ2

respectively. By employing diversity combing at the destination, the signals received over the L + 1 time slots can be combined using the maximal-ratio combiner (MRC), which yields the output signal

128

4 Cooperative Transmission Schemes with Multiple Relays

√ L Ps h∗s,d (1) y:d [m] = y [m] + d σd2

P Ps h∗ h∗ Ps |hs, |2 +σ2 ,d s,

P |h |2 σ2 =1 Ps |hs, |2 +σ2 ,d

+ σd2

(2,)

yd

[m].

(4.5)

The eﬀective SNR at the output of the MRC is given by γeﬀ = γs,d +

L =1

γs, γ,d , γs, + γ,d + 1

(4.6)

where γs,d = Ps |hs,d |2 /σd2 , γs, = Ps |hs, |2 /σ2 , and γ,d = P |h,d |2 /σd2 . With a given instantaneous CSI, the rate achievable with the AF orthogonal cooperation scheme can be expressed as L γs, γ,d 1 CAF (γ) = log2 1 + γs,d + , (4.7) L+1 γs, + γ,d + 1 =1

where γ = [γs,d , γs,1 , . . . , γs,L , γ1,d , . . . , γL,d ]. The unit is given by bits per channel use (bpcu). Given the average transmission rate R, the outage probability can be computed as pout = Pr (CAF (γ) < R) L γs, γ,d 1 log2 1 + γs,d +
(4.8)

=1

Let us consider the Rayleigh fading scenario where hs,d , {hs, }L =1 , and {h,d }L =1 are independent zero mean circularly symmetric complex Gaussian 2 2 2 L random variables with variances ηs,d , {ηs, }L =1 , and {η,d }=1 , respectively. In this case, the average link SNRs γs,d , γs, , and γ,d , for = 1, . . . , L, will be independent and exponentially distributed random variables with mean 2 2 2 γ s,d = Ps ηs,d /σd2 , γ s, = Ps ηs, /σ2 , and γ ,d = P η,d /σd2 , respectively. Then, by replacing the summation in (4.8) with the maximum among the L terms, the outage probability can be upper bounded by

4.1 Orthogonal Cooperation

pout

129

γs, γ,d R(L+1) <2 ≤ Pr γs,d + max −1 γs, + γ,d + 1 c γs, γ,d 1 −u/γ s,d du, e Pr max =
(4.9) (4.10)

(∗)

where c 2R(L+1) −1. Following derivations similar to (3.54)-(3.56), the term (∗) inside integral can be approximated and upper bounded at high SNR by −1 1 1 γs, γ,d < c − u ≈ Pr Pr +
c L 1 −u/γ s,d 1 1 · ≤ + (c − u)L e du γ s, γ ,d γ 0 s,d =1 c L 1 1 1 · ≤ + (c − u)L du γ s, γ ,d γ 0 s,d =1 L 1 (2R(L+1) − 1)L+1 1 1 . ≤ + L+1 γ s,d γ s, γ ,d

(4.12)

=1

Let us consider the case where the source and the relays’ transmission powers as well as the noise variances at all receivers are the same, i.e., Ps = 2 2 P1 = P2 = · · · = PL = P and σd2 = σ12 = · · · = σL = σw . In this case, the bound of the outage probability can be expressed as [41] R(L+1) L+1 L −1 1 2 1 1 1 , (4.13) pout ≤ 2 2 + η2 L+1 SNR ηs,d ηs, ,d =1

2 where the transmit SNR is deﬁned as SNR = P/σw . This shows that AF orthogonal cooperation schemes are able to achieve diversity order of

d − lim

SNR→∞

log pout ≥ L + 1. log SNR

(4.14)

130

4 Cooperative Transmission Schemes with Multiple Relays

Since only L + 1 independent fading paths are available between the source and the destination, we say that the system achieves full diversity. When full instantaneous CSI is available at the source and the relays, the transmission power can be allocated to maximize the achievable rate given in (4.7), which is equivalent to maximizing the eﬀective SNR since log2 (1 + x) is a monotonically increasing function of x. To simplify our discussions, let us assume that the source power Ps is ﬁxed and focus on the power allocation among relays. Thus, given the total relay power constraint P ≤ Pr , we ﬁnd the optimal power allocation by solving the following optimization problem: max

{P ,∀}

subject to

L =1 L

Ps P |hs, |2 |h,d |2 /(σ2 σd2 ) Ps |hs, |2 /σ2 + P |h,d |2 /σd2 + 1

(4.15)

P ≤ Pr and P ≥ 0, ∀ .

(4.16)

=1

Notice that, since the objective function given in (4.15) can be expressed as Ps |hs, |2 Ps2 |hs, |4 /σ4 + Ps |hs, |2 /σ2 , − 2 σ Ps |hs, |2 /σ2 + P |h,d |2 /σd2 + 1 we can reformulate the problem as min

{P ,∀}

subject to

L =1 L

Ps2 |hs, |4 /σ4 + Ps |hs, |2 /σ2 . Ps |hs, |2 /σ2 + P |h,d |2 /σd2 + 1

(4.17)

P ≤ Pr and P ≥ 0, ∀ .

(4.18)

=1

The Lagrangian of the optimization problem can be written as L L Ps2 |hs, |4 /σ4 + Ps |hs, |2 /σ2 +λ P − Pr . L({P , ∀ }, λ) = Ps |hs, |2 /σ2 + P |h,d |2 /σd2 + 1 =1

=1

By the Karush-Kuhn-Tucker (KKT) conditions, the optimal power allocation can be derived as [41] '

Ps2 |hs, |4 /σ4 + Ps |hs, |2 /σ2 Ps |hs, |2 /σ2 + 1 P = μ − 2 |h,d |2 /σd |h,d |2 /σd2 ⎛' ⎞+ 2 +γ γs, γ s, + 1 s, ⎠ , μ− =⎝ |h,d |2 /σd2 |h,d |2 /σd2

+ (4.19)

(4.20)

4.1 Orthogonal Cooperation

131

ĐŚŝĞǀĂďůĞZĂƚĞ;ďŝƚƐͬƐĞĐͬ,ǌͿ Throughput (bps)

Average Throughput for Four Schemes

10

10

10

0

−1

ŝƌĞĐƚ direct ŝƌĞĐƚ W EPA KWǁŝƚŚŝŶƐƚ͘^/ OPA−CSI KWǁŝƚŚƐƚĂƚ͘^/ OPA−STA

−2

−20

−15

−10

−5

0 SNR (dB)

5

10

15

20

Fig. 4.2 Average achievable rate of orthogonal AF relaying schemes with L = 3 relays. c (From Zhao, Adve, and Lim with modiﬁed labels; [2007] IEEE.)

√ + for = 1, 2, · · · , L, where (a) = max(a, 0) and μ = 1/ λ is set to satisfy the power constraint P ≤ Pr . On the other hand, when only statistical CSI is available, the optimal relay power allocation can be found by minimizing the upper bound of the outage probability given in (4.12). The problem can be formulated as follows: L L σ2 σd2 σ2 σd2 min ≡ min log 2 + P η2 2 + P η2 P ,∀ P ,∀ Ps ηs, Ps ηs, ,d ,d =1

subject to

L

=1

P ≤ Pr , P ≥ 0, ∀ .

=1

By introducing the Lagrange multiplier and by applying the KKT conditions, the optimal power allocation can be obtained as

132

4 Cooperative Transmission Schemes with Multiple Relays Outage Probability of Four Schemes, M 3 10

Outage Probability

10

10

10

10

−1

OPA−CSI AP−AF ŝƌĞĐƚ W EPA AP−AF Direct KWǁŝƚŚŝŶƐƚ͘^/ KWǁŝƚŚƐƚĂƚ͘^/ OPA−STA AP−AF

−2

−3

−4

−5

10

12

14

16

18

20

SNR TdB)

Fig. 4.3 Outage probability of orthogonal AF relaying schemes with L = 3 relays. (From c Zhao, Adve, and Lim with modiﬁed labels; [2007] IEEE.)

⎛ ⎞+ 2 η 2 /(σ 2 σ 2 ) · λ + P 2 η 4 /σ 4 2 4Ps η,d s s, /σ2 Ps ηs, s, d ⎠ P = ⎝ − 2 /σ 2 2 /σ 2 2η,d 2η,d d d ⎛ =⎝

2 /σ 2 · λ + γ 2 4γ s, η,d s, d 2 /σ 2 2η,d d

⎞+ γ s, ⎠ , − 2 2η,d /σd2

(4.21)

(4.22)

where λ is a constant satisfying the power constraint. In Figs. 4.2 and 4.3 (adopted from [41]), the average achievable rates (in bits per channel use) and the outage probabilities are shown for AF orthogonal cooperation systems with L = 3 relays. The performance curves are 2 plotted against the transmit SNR deﬁned by SNR = P/σw , and compared among four cases: (i) the case with equal power allocation (EPA), (ii) the case with optimal power allocation based on instantaneous CSI (OPA with inst. CSI), (iii) the case with optimal power allocation based on statistical CSI (OPA with stat. CSI), and (iv) the case with no cooperation (Direct). The source and total relay powers are given by Ps = P and Pr = LP for cooperative transmissions and the source power is set as Ps = (L + 1)P for direct transmission. The optimal power allocation with instantaneous CSI is given by (4.20) and that with statistical CSI is given by (4.21). Here, dis-

4.1 Orthogonal Cooperation

133

tance between source and destination is set as ds,d = 1 and the relays are placed within a unit circle centered in the middle of source and destination. The channel coeﬃcients hs,d , hs, , and h,d have distribution CN (0, 1/dα s,d ), α CN (0, 1/dα s, ), and CN (0, 1/d,d ), where α = 2.5 and di,j is the distance between nodes i and j, for i,j ∈ {s, 1, . . . , L, d}. The noise variances at all 2 nodes are assumed to be equal, i.e., σ2 = σd2 = σw . In Fig. 4.2, we can see that OPA with instantaneous CSI provides a gain of approximately 2 dB at low SNR compared to EPA and OPA with statistical CSI. However, the achievable rate may not be as high as that of direct transmission because of the loss in bandwidth eﬃciency when assigning each relay an orthogonal channel. In Fig. 4.3, we can see that OPA with instantaneous CSI provides a gain of approximately 1.5 dB compared to EPA and OPA with statistical CSI. Moreover, we can observe that power allocation based on statistical CSI does not improve much over EPA. Although direct transmission achieves a higher average rate, its outage probability is easily outperformed by the AF orthogonal cooperation schemes since diversity order of L + 1 is attainable in the latter case.

4.1.2 Orthogonal Cooperation with DF Relays Orthogonal cooperation schemes can also be applied using DF relays. To study this case, we assume that each relay can determine whether or not it is able to successfully decode the source’s message, e.g., through a cyclic redundancy check (CRC), and will forward the message only if it is able to do so. If a total of L + 1 orthogonal time slots are utilized to perform the cooperative transmission, as in the AF scheme, the message transmitted in Phase I must be coded with rate (L + 1)R in order to achieve an average end-to-end rate of R. In this case, the cooperative set D, i.e., the set of relays that successfully decode the message, is given by D = { : log2 (1 + γs, ) ≥ (L + 1)R} , where γs, = Ps |hs, |2 /σ2 . In Phase II, the relays in D, e.g. relay , will forward the codeword x = xs to the destination over orthogonal time slots. The signal received at the destination, in the channel corresponding to relay

, is given by (2,) (2,) (4.23) yd [m] = P h,d xs [m] + wd [m]. for all ∈ D and for m = 0, . . . , M − 1. By combining the signals received over all channels using the MRC, we obtain at the output the eﬀective SNR given by |hs,d |2 |h,d |2 γeﬀ = Ps + P = γs,d + γ,d . (4.24) 2 2 σd σd ∈D

∈D

134

4 Cooperative Transmission Schemes with Multiple Relays

Since L + 1 orthogonal time slots are used to perform the cooperative transmission, the achievable rate would be given by 1 CDF (γ) = log2 1 + γs,d + (4.25) γ,d , L+1 ∈D

where γ = [γs,d , γs,1 , . . . , γs,L , γ1,d , . . . , γL,d ]. Although diversity gains can be achieved similar to that in the AF scheme, the bandwidth eﬃciency is reduced by the number of orthogonal time slots that are needed. However, this can be improved upon when CSI is available at the relays. Speciﬁcally, when instantaneous CSI is available at the relays, the optimal power allocation can be found by maximizing the achievable rate (or, equivalently, the eﬀective SNR) at the destination. Given the total relay power L constraint =1 P ≤ Pr , the problem can be formulated equivalently as a linear programming given as follows: max

P ,∀

subject to

∈D L

P

|h,d |2 σd2

P ≤ Pr and P ≥ 0, ∀ .

(4.26)

(4.27)

=1

The optimal solution is to allocate all available transmission power to the relay with the maximum “weight”, i.e.,

Pr , if |h,d |2 ≥ |h ,d |2 ∀ , P = (4.28) 0, otherwise. In other words, the optimal policy is to have only the relay with the best relay-to-destination channel forward the source’s message. If this policy is employed, a total of only 2 time slots will be required, which improves considerably the bandwidth eﬃciency. The resulting scheme is similar to the selective relaying scheme to be discussed in Section 4.3.2 and, thus, readers are referred to this section for further details on the performance analysis. In orthogonal cooperation schemes, the signals transmitted by diﬀerent relays are received in separate orthogonal channels and, thus, can be processed separately at the receiver. This loosens considerably the synchronization and coordination requirements across relays. However, the main disadvantage is in the loss of bandwidth eﬃciency as the number of relays increases. In the following sections, we describe various cooperative transmission schemes based on conventional MIMO signal processing techniques. These schemes achieve high bandwidth eﬃciency by having relays share a common channel in Phase II, but may be subject to strict synchronization constraints.

4.2 Transmit Beamforming

135

4.2 Transmit Beamforming In multi-relay cooperative systems, as shown in Fig. 4.1, the relays can be viewed as a distributed antenna array and be used to perform many conventional MIMO transmission schemes. Speciﬁcally, when full instantaneous CSI is available at the relays, the forwarded signals can be aligned in phase to maximize the receive SNR at the destination. This is often referred to as distributed transmit beamforming, which can be done with both AF and DF relays as described in the following subsections.

4.2.1 Transmit Beamforming with AF Relays In AF transmit beamforming schemes, we assume that all relays have perfect knowledge of both source-to-relay and relay-to-destination channels, and are able to utilize this knowledge to achieve phase coherent transmission. Specifically, in Phase I of the transmission scheme, the source ﬁrst broadcasts symbol vector xs = [xs [0], . . . , xs [M − 1]] to both the relays and the destination, where the received signals are given by (4.1) and (4.2), respectively. Then, in Phase II, each relay, say relay , ﬁrst - normalizes the received signal by E[|y [m]|2 ] to obtain x [m] = y [m]/ E[|y [m]|2 ] and multiplies it by a complex beamforming coeﬃcient β before forwarding it to the destination. The signal forwarded by relay is thus given by y [m] y [m] β x [m] = β , for = 1, 2, · · · L. = β E[|y [m]|2 ] Ps |hs, |2 + σ2 The beamforming coeﬃcient is chosen such that the relay power is normalized to P , i.e., E[|β x [m]|2 ] = |β |2 = P . By assuming that the relay transmissions are perfectly synchronized in Phase II, the signal received at the destination can then be written as (2)

yd [m] =

L

(2)

h,d β x [m] + wd [m]

=1

√ L L β hs, h,d Ps β h,d (2) = x [m]+ w [m]+wd [m], s 2 2 2 2 Ps |hs, | + σ Ps |hs, | + σ =1 =1 (4.29) (2)

where wd [m] ∼ CN (0, σd2 ) is the AWGN at the destination. We assume that noise at the relays and the destination are independent of each other. The signals received at the destination in Phases I and II are combined using the MRC, which yields at its output the eﬀective SNR

136

4 Cooperative Transmission Schemes with Multiple Relays

γeﬀ = γs,d + γsrd , where γs,d = Ps |hs,d |2 /σd2 and

γsrd

2 L Ps β h h =1 Ps |hs, 2 s, ,d |2 +σ = . 2 2 L 1 2 β h σ + σ 2 ,d 2 d =1 Ps |hs, | +σ

(4.30)

Notice that γsrd is a function of the beamforming vector β = [β1 , β2 , · · · , βL ]T and, thus, is sometimes expressed as γsrd (β) in the following. L 2 Given the total relay power constraint L =1 P = =1 |β | ≤ Pr , the L beamforming coeﬃcients {β }=1 can be chosen to maximize the eﬀective SNR given above. By assuming that the source power Ps is ﬁxed, this is equivalent to maximizing the SNR of the signal transmitted over the s-r-d path, i.e., γsrd (β). To ﬁnd the optimal β, we ﬁrst notice that γsrd (β) can be expressed alternatively as : : H h∗ hT β |hTeﬀ β|2 β eﬀ eﬀ γsrd (β) = H = 2 β Σβ + σd2 : : H Σ + σd 2 I β β

β

(4.31)

: = β/ β is a unit-norm vector representing the direction of the where β beamforming vector β, T √ √ √ Ps hs,1 h1,d Ps hs,2 h2,d Ps hs,L hL,d = ,,··· , 2 Ps |hs,1 |2 + σ12 Ps |hs,2 |2 + σ22 Ps |hs,L |2 + σL

heﬀ

is the eﬀective channel vector and 2 |h2,d |2 σ22 |hL,d |2 σL |h1,d |2 σ12 , ,··· , Σ = diag 2 Ps |hs,1 |2 + σ12 Ps |hs,2 |2 + σ22 Ps |hs,L |2 + σL is the eﬀective noise covariance matrix. The optimal beamforming vector is thus given by β opt = arg max γsrd (β) = arg max

β 2 ≤Pr

2 =1

β

: H h∗ hT β : β eﬀ eﬀ 2 H σ : : Σ + Pdr I β β

(4.32)

where β 2 in the denominator of (4.31) is replaced with Pr since the SNR increases monotonically with β 2 . σ2 Since Σ + Pdr I is a symmetric and positive deﬁnite matrix, it can be decomposed as σ2 Σ + d I = UUH . Pr

4.2 Transmit Beamforming

137

: the SNR of the s-r-d path can be expressed as Then, by letting v = UH β, γsrd (β) =

2 vH U−1 h∗eﬀ hTeﬀ U−H v T −H v = h U eﬀ vH v

v

≤ hTeﬀ U−H 2

v

v

2

= hTeﬀ U−H 2 ,

(4.33)

where the inequality in (4.33) follows from the Cauchy-Schwartz inequality. The equality holds when v = c1 U−1 h∗eﬀ , for any nonzero constant c1 . Hence, the optimal beamforming vector is given by β

opt

−H

=U

−1 σd2 v = c1 Σ + I h∗eﬀ . Pr

(4.34)

By choosing the constant c1 to satisfy the total relay power constraint β 2 = Pr , i.e., ; < Pr c1 = < , −2 = σ2 hTeﬀ Σ + Pdr I h∗eﬀ the maximum SNR on the s-r-d path becomes γsrd (β

opt

)= =

U−1 h∗eﬀ 2 L =1

=

hTeﬀ

−1 σd2 Σ+ I h∗eﬀ Pr

Ps Pr |hs, |2 |h,d |2 /σ2 σd2 γs, γ :,d = , Ps |hs, |2 /σ2 + Pr |h,d |2 /σd2 + 1 γs, + γ :,d + 1 L

=1

(4.35) where γs, = Ps |hs, |2 /σ2 the SNR on the s- path and γ :,d = Pr |h,d |2 /σd2 is the SNR on the -d path when the relay transmits with total relay power Pr (which diﬀers from γ,d in the transmit power). The -th term in (4.35) can be viewed as the SNR obtained when the total relay transmission power Pr is allocated to a single relay . Notice that the beamforming coeﬃcient imposed by relay is equal to a positive constant times the complex conjugate of the combined channel hs, h,d , i.e., √ Ps h∗s, h∗,d c1 β = |h |2 σ2 = c h∗s, h∗,d , 2 σd 2 2 ,d Ps |hs, | + σ 2 + 2 Ps |hs, | +σ

Pr

which compensates for the phase rotation on both the s- and the -d links. As a result, the signals transmitted by the relays will add up coherently at the destination, resulting in a signiﬁcant increase in SNR. The SNR expression in (4.35) is similar to that of transmit beamforming in conventional MISO systems, as described in Chapter 2. The achievable rate is then given by

138

4 Cooperative Transmission Schemes with Multiple Relays

L γs, : γ,d 1 . CAF (γ) = log2 1 + γs,d + 2 γs, + γ :,d + 1

(4.36)

=1

Please recall that the eﬀective SNR obtained in the AF orthogonal cooperation scheme is given by γsrd =

L =1

γs, γ,d . γs, + γ,d + 1

(4.37)

This expression is similar to (4.35) but with γ,d replaced with γ :,d = Pr γ,d /P . This shows that the transmit beamforming scheme achieves Pr /P times increase in the relay-to-destination SNR. When power is equally allocated to the relays in the orthogonal cooperation scheme, i.e., P1 = . . . = PL = Pr /L, this gain is Pr /P = L. However, it is important to remark that, in practice, the relays may instead be operating under individual power constraints P1 , P2 , . . ., PL . In this case, the beamforming coeﬃcients can only be chosen to co-phase the signals arriving at the destination while transmitting with maximum power. Hence, the beamforming coeﬃcient of relay is given by β =

- h∗s, h∗,d . P |hs, h,d |

(4.38)

By substituting (4.38) into (4.31) and by assuming that σ1 = . . . = σL = σd , the eﬀective SNR obtained under the individual power constraint can be computed as ( )2 L γ γ /(γ + 1) s, ,d s, =1 ind γsrd = L . (4.39) =1 γ,d /(γs, + 1) + 1 The SNR obtained in this case is in general less than that in (4.35) but can perform close to the case with total relay power constraints when the relay channels have similar statistics. To further improve system performance, the source power Ps and the total relay power Pr can be further allocated to maximize the achievable rate in (4.36) (or, equivalently, the SNR in (4.35)) subject to a total power constraint Ps + Pr ≤ Ptot . Let Pr = τ Ptot and Ps = (1 − τ )Ptot , where 0 ≤ τ ≤ 1. The optimal value of τ cannot be obtained in closed form but can be obtained through numerical search. An approximate value of τ found by maximizing a lower bound to (4.35) has been derived in [35] as ⎛

⎞+

1 τ =⎝ − L 2

=1 min

|hs,d |2 /σd2 Ptot |hs, |2 |h,d |2 /σ2 σd2 Ptot |hs, |2 |h,d |2 /σ2 σd2 , 1+Ptot |h,d |2 /σ2 1+Ptot |hs, |2 /σ2 d

⎠ , (4.40)

4.2 Transmit Beamforming

139

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Fig. 4.4 Error probability of AF transmit beamforming schemes is shown for cooperative systems with L = 3 relays and full instantaneous CSI. (From Yi and Kim [35] with modiﬁed c labels; [2007] IEEE.)

which is less than 1/2. This implies that more than half the power should be allocated to the source since it contributes to the reception at both the source and the relays. In Fig. 4.4 (adopted from [35]), the average bit error rate of AF transmit beamforming is shown for three cases: the suboptimal τ in (4.40), the optimal τ obtained through numerical search, and τ = 0.5. Also shown for comparison is the case with-equal power allocation (EPA) among the relays, that is, the case with β = Pr /L for all . The experiments are performed for a network with L = 3 relays. The source is assumed to employ BPSK modulation, i.e., xs [m] ∈ {−1, +1}, and the channel coeﬃcients hs, and h,d are assumed to be i.i.d. with distribution CN (0, 1). We can observe that AF transmit beamforming achieves a signiﬁcant gain in performance compared to the equal power scheme. Moreover, we can also see that power allocation among source and relays do not have a signiﬁcant impact on the BER performance. It is interesting to note that the advantage of beamforming exists only when channel phase information is available at the relays and that the relays can co-phase their signals at the destination. If only knowledge of the statistical CSI is available at the relays, the signals transmitted by the diﬀerent relays will cause random constructive or destructive interference at the

140

4 Cooperative Transmission Schemes with Multiple Relays

destination, in which case, it is better to just let a single relay forward the message with total power Pr , resulting in the selective relaying scheme. For example, let us consider the case where the relays have perfect knowledge of the source-to-relay channel, i.e., hs, , but has only statistical knowledge of the relay-to-destination channel, i.e., h,d . In this case, the optimal beamforming coeﬃcients can only be obtained by maximizing the average SNR 2 L Ps β h h Eh,d =1 Ps |hs, 2 s, ,d 2 | +σ γ srd = 2 L 2 1 2 Eh,d βh σ + σd =1 Ps |hs, |2 +σ2 ,d

L

= L

Ps =1 Ps |hs, |2 +σ2

1 =1 Ps |hs, |2 +σ2

2

2 |β hs, | η,d 2

2 + σ2 |β | σ2 η,d d

=

: β

H

: : H Λβ β Σ+

, : β

σd2

β 2 I

(4.41)

: = β/ β is the normalized beamforming vector and Λ and Σ where β are diagonal matrices with the -th diagonal entry being 2 σ2 η,d , Ps |hs, |2 +σ2

2 Ps |hs, |2 η,d Ps |hs, |2 +σ2

and

respectively. By letting 12 σd2 : v = c1 Σ + I β,

β 2

(4.42)

where c1 is chosen such that v 2 = 1, we have

β:

max 2

|β | ≤Pr

: : H Λβ β H σ2 : : Σ + Pdr I β β

−1 −1 σd2 2 σd2 2 Σ+ v I Λ Σ+ I v = max v: v 2 =1 Pr Pr = λmax (D), H

(4.43)

(4.44) (4.45)

where λmax (D) is the maximum eigenvalue of the matrix −1 −1 σd2 2 σd2 2 D= Σ+ I Λ Σ+ I Pr Pr γs,2 γ γs,L γ :1,d :2,d :L,d γs,1 γ , ,··· , = diag γs,1 + γ :,d + 1 γs,2 + γ :2,d + 1 γs,L + γ :L,d + 1

(4.46) (4.47)

2 and γ :,d = Pr η,d /σd2 . Since D is a diagonal matrix, the maximum eigenvalue corresponds to the maximum diagonal element. Suppose that

4.2 Transmit Beamforming

141

∗ = arg max

γs, γ :,d γs, + γ :,d + 1

is the position of the maximum diagonal element. Then, v will be a vector : with 1 in the ∗ -th entry and 0 elsewhere. By the relation between v and β in (4.42), we have : = Pr e∗ , β = Pr β where e is a canonical vector with 1 in the -th entry and 0 elsewhere. This result shows that, when co-phasing is not possible among relays, the optimal cooperative strategy is to let only the relay with the maximum eﬀective SNR, i.e., relay ∗ , forward the source’s message. This scheme is further described in the section on selective relaying.

4.2.2 Transmit Beamforming with DF Relays For DF transmit beamforming schemes, the relays ﬁrst decode the source message and then forward coherently the same codeword to the destination. If the relays are able to perform error-detection (e.g., by using CRC) and participate in the cooperative transmission only when they have successfully decoded the message, the beamforming coeﬃcients need only be chosen to compensate for the relay-to-destination channel. However, if error-detection cannot be performed (e.g., in the uncoded case), the probability of error at the relays must be taken into consideration in the beamforming design. Both cases will be discussed in the following.

DF Transmit Beamforming with Error-Detection at Relays Let us ﬁrst consider the case where error-detection is performed at the relays. In this case, the source transmits in Phase I the codeword xs to both the relays and the destination, where the received signals are also given by (4.1) and (4.2), respectively. Then, in Phase II, each relay will attempt to decode the message and will help forward the source’s message to the destination only if it is able to successfully decode the message. The set of successfully decoding relays in a given transmission period is given by D = { : log2 (1+γs, ) ≥ 2R}. Notice that the decodable set D depends on the channel realization, it may vary over diﬀerent transmission periods. In Phase II, the relays in the set D will simultaneously retransmit the same codeword xs to the destination using beamforming coeﬃcients {β , ∀ ∈ D}. The signal received at the destination can be written as (2) (2) yd [m] = h,d β xs [m] + wd [m], for m = 0, . . . , M − 1, (4.48) ∈D

142

4 Cooperative Transmission Schemes with Multiple Relays (2)

where wd ∼ CN (0, σd2 ) is the AWGN at the destination. Given the beam(2) forming coeﬃcients {β , ∀ ∈ D}, the SNR of the received signal yd [m] can be computed as 2 ∈D h,d β γsrd = . (4.49) σd2 The signal model in (4.48) and the SNR expression in (4.49) are identical to that of conventional MISO systems when the number of transmit antennas is equal to |D|. Thus, with each relay having instantaneous knowledge of the relay-to-destination channel coeﬃcients and under the total relay power constraint Pr , the optimal beamforming coeﬃcients can be found by maximizing the SNR in (4.49). This results in the solution [16] ' Pr β = h∗ for all ∈ D, (4.50) 2 ,d k∈D |hk,d | which is similar to that obtained in conventional MISO systems. By combining both the direct and relay paths, the eﬀective SNR at the output of the MRC can be given by 2 Ps |hs,d |2 ∈D Pr |h,d | γeﬀ = + = γs,d + γ :,d . (4.51) 2 2 σd σd ∈D

In Fig. 4.5, the outage probability of DF transmit beamforming schemes is shown for systems with diﬀerent numbers of relays, namely, L = 1, 2, 4, 6, and 8. We assume the error-detection is performed perfectly at the relays and, thus, the decodable set in each transmitting period is given exactly as D = { : log2 (1 + γs, ) ≥ 2R}. The optimal beamforming coeﬃcients for the relays in D are given by (4.50). The channel coeﬃcients hs, , h,d are assumed to be i.i.d. with distribution CN (0, 1) and hs,d is assumed to be CN (0, 1/23). The transmission rate R is 1 bit/sec/Hz. The transmit SNR is deﬁned as 2 2 SNR = P/σw , where P = Ps = ∈D P and σw = 1. One can observe from the ﬁgure that diversity gains are indeed improved as the number of relays increases.

DF Transmit Beamforming without Error-Detection at Relays On the other hand, if the relays are not able to perform error detection, e.g., in the uncoded case, they can only demodulate-and-forward the received signal on a symbol-by-symbol basis. As a result, the symbols that are forwarded by the relays may be incorrect and, therefore, the beamforming coeﬃcients must be designed taking into consideration the error probability at each relay. Let us consider the scenario where the source transmits using BPSK modulation (i.e., xs [m] ∈ {±1}) and that each relay has knowledge of its local

4.2 Transmit Beamforming

143

0

10

−1

Outage Probability

10

−2

10

−3

10

−4

10

No Cooperation L=1 L=2 L=4 L=6 L=8

−5

10

0

5

10 15 Transmit SNR (dB)

20

25

Fig. 4.5 Outage probability of DF transmit beamforming schemes with error detection at the relays.

error probability, e.g., pe, Pr (x = xs ) for relay . The symbol forwarded by relay in the m-th symbol period can be expressed as x [m] = θ [m]xs [m], where θ [m] ∈ {±1} is a Bernoulli random variable with Pr (θ [m] = 1) = 1 − Pr (θ [m] = −1) = 1 − pe, . We assume that error is independent among relays and over time. By applying beamforming coeﬃcients {β }L =1 , the signal received at the destination in Phase II can be expressed as (2)

yd [m] =

L =1

(2)

h,d β x [m] + wd [m] =

L

(2)

h,d β θ [m]xs [m] + wd [m].

=1

Note that θ [m] = −1 corresponds to an error event and θ [m] = 1 corresponds to a correct detection at relay . By averaging over all possible error patterns θ[m] = [θ1 [m], . . . , θL [m]], the SNR of the signal received in Phase II can be computed as

144

4 Cooperative Transmission Schemes with Multiple Relays

γsrd

( )2 (2) Eθ yd [m] xs [m] = ( )2 ) ( 2 (2) (2) Eθ yd [m] xs [m] − Eθ yd [m] xs [m] 2 L =1 h,d β (1 − 2pe, ) = 2 ( ) 2 (2) Eθ yd [m] xs [m] − L h β (1 − 2p ) ,d e, =1

(4.52)

(4.53)

(2)

since E[θ [m]] = 1 − 2pe, . The second moment of yd can be evaluated as ⎡ 2 ⎤ L ) ( 2 (2) h,d β θ [m] ⎦ + σd2 E yd [m] xs [m] = E ⎣ =1

=

L L

h,d h∗k,d β βk∗ E[θ [m]θk [m]] + σd2 . (4.54)

=1 k=1

where E[θ [m]θk [m]] =

(1 − 2pe, )(1 − 2pe,k ), for = k, 1, for = k.

(4.55)

Let β = [β1 , β2 , · · · , βL ]T be the vector of beamforming coeﬃcients. Given β, the average SNR in (4.53) can be expressed in a vector form as : : H h∗ hT β |hTeﬀ β|2 β eﬀ eﬀ γsrd (β) = H = , 2 β Rh β + σd2 − |hTeﬀ β|2 : : H Rh + σd 2 I − h∗ hT β β eﬀ eﬀ

β

: = β/ β is the normalized beamforming vector, where β

(4.56)

heﬀ = [h1,d (1 − 2pe,1 ), h2,d (1 − 2pe,2 ), · · · , hL,d (1 − 2pe,L )]T is the eﬀective s-r-d channel vector, and Rh is the eﬀective channel correlation matrix with the (k, )-th element equal to [Rh ]k, = h,d h∗k,d E[θ [m]θk [m]]. The SNR expression in (4.56) is similar to that obtained in AF transmit beamforming. By maximizing the SNR at the destination, the optimal beamforming vector can be found similarly as β opt = arg max γsrd (β) =

β 2 ≤Pr

Pr arg max 2 =1

β

: H h∗ hT β : β eﬀ eﬀ , (4.57) 2 H : : Rh + σd I − h∗ hT β β eﬀ eﬀ Pr

where β 2 in the denominator of (4.56) is replaced with Pr since γsrd (β) increases monotonically with β 2 . Following the derivations in Section 4.2.1

4.2 Transmit Beamforming

145

(see also [1]), we obtain the optimal beamforming vector as −1 σ2 β opt = c2 Rh + d I − h∗eﬀ hTeﬀ h∗eﬀ , Pr

(4.58)

where the constant c2 is chosen to satisfy the total relay power constraint

β 2 = Pr and, thus, is given by ; < Pr c2 = < . −2 = 2 σ hTeﬀ Rh + Pdr I − h∗eﬀ hTeﬀ h∗eﬀ The maximum SNR on the s-r-d path is given by γsrd (β

opt

)= =

hTeﬀ L =1

=

L =1

σ2

−1 σd2 ∗ T Rh + I − heﬀ heﬀ h∗eﬀ Pr

(4.59)

Pr |h,d |2 (1 − 2pe, )2 4Pr |h,d |2 pe, (1 − pe, ) + σd2

(4.60)

γ :,d (1 − 2pe, )2 4: γ,d pe, (1 − pe, ) + 1

(4.61) σ2

since Rh + Pdr I − h∗eﬀ hTeﬀ is a diagonal matrix with 4pe, (1 − pe, )|h,d |2 + Pdr in the -th diagonal entry. In Fig. 4.6, the bit-error-rate (BER) of DF transmit beamforming schemes are shown for systems with L = 1, 2, 4, 6, and 8. Here, we assume that the relays simply demodulate-and-forward the symbols while taking into consideration the error probabilities in the beamforming design. The simulation parameters are the same as that given in Fig. 4.5. We can see that the diversity gains are not as evident due to possible errors in the relayed symbols. It is important to remark that the optimal beamforming coeﬃcients derived in this section rely on instantaneous knowledge of the channel coeﬃcient on each link, which is typically provided through feedback from the destination. However, due to rate-limitations on the feedback channel, perfect CSI is not attainable in practice and, thus, the beamforming coeﬃcients can only be designed based on quantized or limited knowledge of the channel coeﬃcients. It is then important to determine the best information to provide through the limited feedback channel and ﬁnd robust designs for the beamforming coeﬃcients. For instance, in the case of quantized feedback, the destination encodes the channel information into a B-bit message that is transmitted to the relays at least once every coherence interval. The B-bit message is selected from a codebook of 2B codewords, e.g., [17, 40], where each codeword is used as a quantized representation of the instantaneous channel vector hr,d = [h1,d , . . . , hL,d ]T . The best quantized representation of the channel

146

4 Cooperative Transmission Schemes with Multiple Relays

0

10

−1

10

−2

Bit Error Rate

10

−3

10

−4

10

−5

10

No Cooperation L=1 L=2 L=4 L=6 L=8

−6

10

−7

10

0

5

10 15 Transmit SNR (dB)

20

25

Fig. 4.6 Bit error rate of DF transmit beamforming schemes without error-detection at the relays.

vector is chosen at the beginning of each coherence interval and the index of the codeword (which requires only B-bits to represent) is transmitted back to the relays. In addition to utilizing quantized channel feedback, one may also adopt perturbation-based random search algorithms to ﬁnd the optimal beamforming coeﬃcients [7, 25, 26]. In these schemes, the relays randomly perturb their beamforming coeﬃcients before each transmission and decide whether or not to retain these values based on a 1-bit feedback provided by the destination. By doing so, the beamforming coeﬃcients will gradually converge to their optimal values in a distributed manner. Readers are referred to [7, 17, 25, 26, 40] for further discussions on these issues.

4.3 Selective Relaying In the transmit beamforming schemes presented in the previous section, full instantaneous CSI, including the magnitude and phase of all channel coefﬁcients, must be available at the relays in order to compute the optimal beamforming coeﬃcients. Even though channel information can be obtained through feedback, it is still rather diﬃcult in practice to track the changes in phase and to co-phase the relays’ signals towards the destination, especially when random frequency or timing oﬀsets exist at distributed terminals. If

4.3 Selective Relaying

147

the phases of the relays’ signals are not aligned up to a certain accuracy, beamforming gains may no longer be attained and, in fact, destructive interference may occur at the destination. In this case, it may be favorable to have only one relay participate in each cooperative transmission, resulting in the so-called selective relaying or opportunistic relaying schemes [4–6]. Both AF and DF based selective relaying schemes are studied in the following.

4.3.1 Selective Relaying with AF Relays Let us ﬁrst consider the case of selective relaying with AF relays. Similar to previous schemes, selective relaying also takes on two phases of transmission. In Phase I, the source ﬁrst transmits the codeword xs to the relays and the destination and, in Phase II, exactly one relay is selected to forward the source’s message. The signals received by the relays and the destination in Phase I are also given by (4.1) and (4.2), respectively. Suppose that relay

is the one selected to forward the source’s message in the given transmission period. In this case, relay will transmit a normalized sequence of symbols given by y [m] x [m] = , E|y [m]|2 for m = 0, . . . , M − 1. By transmitting with full relay power Pr , the signals received at the destination in Phase II can be written as (2) (2) yd [m] = h,d Pr x [m] + wd [m], (4.62) for m = 0, . . . , M − 1. Notice that the selected relay transmits with power equal to the total relay power Pr since no other relay is allowed to transmit in the given transmission period. In this case, the source, the -th relay, and the destination together form a basic two-user cooperative system, similar to that studied in Chapter 3. The achievable rate can be expressed as Ps |hs,d |2 1 Ps Pr |hs, |2 |h,d |2 /σ2 σd2 CAF, (γ) = log2 1 + + 2 σd2 Ps |hs, |2 /σ2 + Pr |h,d |2 /σd2 + 1 :,d 1 γs, γ , (4.63) = log2 1 + γs,d + 2 γs, + γ :,d + 1 where γs, = Ps |hs, |2 /σ2 , γ :,d = Pr |h,d |2 /σd2 , and γ = [γs,d , γs,1 , . . . , γs,L , γ :1,d , . . . , γ :L,d ]. To maximize the achievable rate, the optimal relay can be selected as

∗ = arg max CAF, (γ) = arg max

γs, γ :,d . γs, + : γ,d + 1

(4.64)

148

4 Cooperative Transmission Schemes with Multiple Relays

This implies that the rate achievable by the AF selective relaying scheme is maximized by choosing the relay with the maximum eﬀective SNR. The resulting system capacity can be expressed as γs, γ :,d 1 CAF (γ) = log2 1 + γs,d + max . (4.65) 2 γs, + γ :,d + 1 2 By considering the Rayleigh fading scenario, where hs,d ∼ CN (0, ηs,d ), 2 2 hs, ∼ CN (0, ηs, ), h,d ∼ CN (0, η,d ), for = 1, 2, · · · , L, are assumed to be mutually independent, the SNRs γs,d , γs, , and γ :,d can be modeled as 2 2 exponential random variables with mean γ s,d = Ps ηs,d /σd2 , γ s, = Ps ηs, /σ2 , 2 and γ :,d = Pr η,d /σd2 . Therefore, the outage probability can be computed as

pout = Pr (CAF (γ) < R) γs, : γ,d < 22R − 1 = Pr γs,d + max γs, + γ :,d + 1 c γs, γ :,d 1 −u/γ s,d
(4.66)

=1

where c = 22R − 1. Following the derivations in (4.11), the outage probability can be further upper-bounded, at high SNR, by c L 1 1 1 −u/γ s,d (c − u) pout ≤ + e du (4.67) γ γ γ : 0 =1 s, s,d ,d c L 1 1 1 L ≤ (c − u) + du (4.68) γ s, γ γ : 0 s,d ,d =1 * 2R +L+1 L 2 −1 1 1 1 . (4.69) ≤ + L+1 γ s,d γ s, γ :,d =1 2 2 By letting Ps = Pr P and σ12 = · · · = σL = σd2 σw , we have 2R L+1 L 1 2 −1 1 1 1 , pout ≤ 2 2 + η2 L+1 SNR ηs,d ηs, ,d =1

2 where SNR = P/σw . This shows that full diversity can be achieved even with the simple selective relaying strategy described above. In Fig. 4.7, the outage probability of the AF selective relaying is shown for a system with L = 6 relays. To focus on the relay performances, we consider only the signal received over the relay path and assume that hs, and h,d are

4.3 Selective Relaying

149

0

10

−1

10

−2

Outage Probability

10

−3

10

−4

10

−5

10

−6

Single Relay Beamforming without Cophasing Selective Relaying

10

−7

10

0

5

10 15 Transmit SNR (dB)

20

25

Fig. 4.7 Outage probability of AF selective relaying with L = 6 relays compared with that of the single relay scenario and transmit beamforming scheme without phase compensation.

i.i.d. with distribution CN (0, 1). For comparison, the outage probability of the case with a single relay and the case of optimal beamforming without the ability to co-phase the signals. In the latter case, optimal beamforming coeﬃcients in (4.34) are used with the phase of the beamforming coeﬃcients set to 0. We can see that diversity gains are not attainable with beamforming when co-phasing is not achievable. On the other hand, full diversity is achieved by simply choosing the relay with the maximum eﬀective SNR to transmit. Due to these reasons, selective relaying have often been studied in practice in place of transmit beamforming. However, if co-phasing is possible, transmit beamforming achieves signiﬁcant coding gains over selective relaying. In the above, we assumed that a centralized controller is available to collect the CSI of channels and to perform the relay selection. However, this relay selection process can actually be implemented in a distributed fashion using the so-called opportunistic carrier sensing scheme [3,39]. In this scheme, each relay sets a timer at the beginning of a transmission period and emits a signal to all other relays whenever its timer reduces to 0. The relay that ﬁrst emits the signal will be allowed to transmit. By setting the timer to be inversely proportional to the local selection criterion (e.g., its local SNR), the relay that possesses the maximum selection criterion will be the one that obtains the right to transmit. The CSI required to compute the local selection criterion can be obtained by overhearing the control messages transmitted prior to the data transmission, e.g., the ready-to-send (RTS) and the clear-to-send

150

4 Cooperative Transmission Schemes with Multiple Relays

(CTS) packets exchanged between the source and the destination. Readers are referred to [3, 39] for further details.

4.3.2 Selective Relaying with DF Relays In DF selection relaying, all relays attempt to decode the source’s message in Phase I and act as candidate relays for selection in Phase II only if it has successfully decoded the message. The set of candidate relays in each transmission period can be represented by the decodable set D = { : log2 (1+ γs, ) ≥ 2R}. In selective relaying schemes, only one relay within the set D will be allowed to forward the source’s message. If relay is the one selected, it will transmit the symbol vector x = xs to the destination in Phase II. The signal received at the destination is thus given by (4.62). For simplicity, let us consider the case where no diversity combining is employed at the destination. Hence, the system reduces to a dual-hop transmission where the maximum achievable rate is limited by the minimum capacity among the source-to-relay and the relay-to-destination links. Given that relay was selected, the achievable rate can be computed as . / 1 CDF, (γ) = min log2 (1 + γs, ) , log2 (1 + γ :,d ) , (4.70) 2 where γs, = Ps |hs, |2 /σ2 , γ :,d = Pr |h,d |2 /σd2 , and γ = [γs,1 , . . . , γs,L , γ :1,d , . . . , γ :L,d ]. To maximize the achievable rate, the optimal relay can be selected as . / 1 min log2 (1 + γs, ) , log2 (1 + γ :,d ) 2 / . = arg max min γs, , γ :,d .

∗ = arg max

The achievable rate is then given by . / 1 CDF (γ) = log2 1 + max min {γs, , : γ,d } . 2

(4.71)

Under the Rayleigh fading scenario, the outage probability of the DF selection relaying scheme can be computed as

4.3 Selective Relaying

151

pout = Pr (CDF (γ) < R) / . = Pr max min {γs, , γ :,d } < 22R − 1

=

L

* + Pr min {γs, , γ,d } < 22R − 1

=1

=

L

* + 1 − Pr min {γs, , : γ,d } ≥ 22R − 1

=1

=

L * + * + 1 − Pr γs, ≥ 22R − 1 · Pr γ :,d ≥ 22R − 1 =1

=

L =1

22R − 1 22R − 1 1 − exp − − γ s, γ :,d

,

(4.72)

2 where γs, and γ :,d are exponentially distributed with means γ s, = Ps ηs, /σ2 2 and γ :,d = Pr η,d /σd2 , respectively, and (4.72) follows from the assumption that hs, and h,d are mutually independent. 2 2 Let us consider the case where Ps = Pr = P and σ12 = · · · = σL = σd2 = σw . 2 By assuming that SNR = P/σw 0, we can apply the ﬁrst order Taylor expansion to upper-bound the outage probability as L L L 22R − 1 1 22R − 1 22R − 1 1 . pout ≤ 2 /σ 2 + P η 2 /σ 2 = 2 + η2 Ps ηs, SNR ηs, r ,d d ,d =1 =1 (4.73) This shows that the DF selection relaying scheme can achieve diversity order of L (or even L + 1 if the s-d path is included). In Fig. 4.8, the outage probability of the DF selective relaying scheme is shown for a system with L = 6 relays and compared with that of the single relay scenario and the case of transmit beamforming without co-phasing. In the latter case, the beamforming coeﬃcients in (4.50) are used with the phase of each entry set to 0. The simulation parameters are similar to that in Fig. 4.7. Similarly, we can see that selective relaying can achieve higher diversity gain than that of transmit beamforming when the latter cannot be performed with perfect phase compensation. In the selective relaying scheme described above, the relay that maximizes the dual-hop capacity is selected to forward the source’s message. This approach is useful when the source can adapt its coding rate according to the instantaneous dual-hop capacity such that the average achievable rate is maximized. However, from the point of view of achieving full diversity, it is suﬃcient to have the source choose any relay path that is able to support the transmission rate. In this case, outage occurs when the decodable set D is

152

4 Cooperative Transmission Schemes with Multiple Relays

0

10

−1

10

Outage Probability

−2

10

−3

10

−4

10

−5

10

Selection Relaying Beamforming without Cophasing Single Relay

−6

10

0

2

4

6 8 Transmit SNR (dB)

10

12

Fig. 4.8 Outage probability of the DF selective relaying scheme with L = 6 relays compared with that of the single relay scenario and transmit beamforming without co-phasing.

empty or when none of the relays in D will be able to achieve the rate 2R on the -d link. The outage probability is therefore the same as that obtained through optimal power allocation in Section 4.1.2. Recall that, in Section 4.1.2, the relay with the best -d channel, i.e.,

∗ = arg max ∈D

1 log2 (1 + γ :,d ) = arg max γ :,d = arg max |h,d |2 , 2 ∈D ∈D

(4.74)

is selected to transmit. Given the decodable set D, the conditional outage probability under Rayleigh fading can be derived as 1 pout|D = Pr log2 1 + max γ :,d < R D ∈D 2 2R :,d < 2 − 1 D = Pr max γ ∈D + * Pr γ :,d < 22R − 1 D = ∈D

=

∈D

22R − 1 1 − exp − 2 /σ 2 Pr η,d d

.

(4.75)

4.4 Distributed Space-Time Coding (DSTC)

153

The outage probability averaged over all possible realizations of D is given by pout = Pr (D) pout|D . (4.76) D

where 1 1 Pr log2 (1 + γs, ) ≥ R · log2 (1 + γs, ) < R 2 2 ∈D ∈D / 22R − 1 22R − 1 1 − exp − . (4.77) exp − = 2 /σ 2 · 2 /σ 2 Ps ηs, Ps ηs,

Pr (D) =

∈D

Pr

∈D /

It has been shown in [5] that the outage probability in (4.76) equals to that in (4.72). That is, both selection relaying methods achieve the same outage performance and diversity order. In this section, we have shown how simple relay selection can achieve full diversity in a multi-relay network. However, selecting only one relay in each transmission period may not always be the best choice, e.g., in terms of energy eﬃciency, as discussed in [24,27]. One can incorporate a hybrid scheme where more than one relay is selected to transmit in Phase II of the cooperative scheme. The relays may together employ the transmit beamforming scheme introduced in the previous section or the distributed beamforming scheme to be discussed in the following section.

4.4 Distributed Space-Time Coding (DSTC) In the MISO system, one way to achieve spatial diversity without knowledge of CSI at transmitter is to employ the space-time coding (STC). Space-time coding can also help cooperative transmission systems with multiple relays enhance outage performance. However, since the antenna belonging to each relay is located apart, a class of space-time coding called distributed spacetime coding (DSTC) is more natural to be applied among the relays. Similar to previous sections, both DF and AF based DSTC are studied in the following.

4.4.1 Distributed Space-Time Coding with DF Relays Let us ﬁrst consider the case of DF relaying with error-detection at relays. In this case, all relays attempt to decode the source’s message, and the set of the relays which has correctly decoded the message, i.e., the set D in Section 4.3.2, perform distributed space-time coding to forward the message in Phase

154

4 Cooperative Transmission Schemes with Multiple Relays

II. The use of space-time coding among relays to gain spatial diversity was ﬁrst studied in [19]. In the literature, a sub-class of STC called orthogonal space-time block coding (OSTBC) was usually applied among relays because of low decoding complexity and tractable performance analysis. For example, in two-relay cooperative systems, the Alamouti’s scheme was often adopted to encode the source’s symbols at the relays. When OSTBC is directly applied at relays, each relay is treated as a transmit antenna of the source, and the space-time-coded symbol sent by each relay corresponds to each column of the codeword matrix. However, a major diﬀerence between such a DFbased cooperative systems and a MISO system is that the membership in the decodable set D depends on the CSI of all s- links and may vary with time. Thus, it is not obvious that the DF-based cooperative system is able to achieve full diversity if OSTBC is adopted at relays. Since the OSTBC codewords transmitted by relays are mutually orthogonal in time domain, the eﬀective SNR at the destination equals to the summation of -d link SNR over all relays ∈ D. Thus, for a given decodable set D, the SNR of signal received at the destination in Phase II is given by γsrd =

P |h,d |2 . σd2

(4.78)

∈D

Assume that, before applying DSTC, the relays adopt another codebook of error-correction codes to re-generate the source symbols, and the codebooks adopted at the relays and the source are diﬀerent, but with the same coding rate. In this case, the achievable rate in two phases for a given decodable set D equals to P |h,d |2 1 Ps |hs,d |2 1 C = log 1 + . (4.79) + log 1 + 2 σd2 2 σd2 ∈D

The corresponding outage probability obtained in [19] as pout = Pr (D) pout|D

(4.80)

D

where Pr(D) is the occurrence probability of the decodable set D and pout|D is the conditional outage probability given D. For simplicity, consider the case that transmission power are evenly allocated among all nodes in the absence of CSI, i.e., Ps = P1 = P2 = · · · = PL = P/(L+1), and noise variances are all 2 2 identical to σw . Denoting SNR = P/σw , in high SNR regime, the conditional outage probability give D can be approximated by [19] pout|D = Pr (C < R|D) |D|+1 1 1 22R − 1 2R · 2 − 1), ≈ 2 · A|D| (2 SNR/(L + 1) ηs,d η,d ∈D

4.4 Distributed Space-Time Coding (DSTC)

where An (t) =

1 (n − 1)!

0

1

155

wn−1 (1 − w) dw, 1 + tw

n > 0,

A0 (t) = 1, and |D| is the cardinality of the set D. As was given in (4.77), Pr (D) can approximated at high SNR as 22R − 1 22R − 1 · 1 − exp − SNR 2 Pr (D) = exp − SNR 2 L+1 ηs, L+1 ηs, ∈D ∈D / L−|D| 1 22R − 1 × ≈ 2 . SNR/(L + 1) ηs, ∈D /

Substituting the approximations to Pr (D) and pout|D into (4.80), the outage probability can be given by [19] L+1 1 1 1 22R − 1 2R . pout ≈ · − 1) · 2 A|D| (2 2 · 2 SNR/(L + 1) ηs,d η,d ηs, D

∈D

∈D /

(4.81) This approximation of the outage probability at high SNR shows that, the DSTC strategy is helpful to achieve full diversity order in the absence of CSI. If the statistical CSI is available, one may exploit this information to allocate transmission powers at relays, which can further improve the system performance. Readers are referred to [23] for further detail. In Fig. 4.9, the outage probabilities for DF DSTC systems with diﬀerent numbers of relays are compared. The solid lines are obtained from numerical integration based on (4.80), while the dashed lines are computed with (4.81). For simplicity, the channels are assumed Rayleigh faded, and all channel coeﬃcients and noises are of unit variance. The transmission rate R is 1 bit/Hz/sec. The results indicates that the approximation of outage probability in (4.81) is close to numerical results at high SNR, and the DF-based cooperative system with OSTBC is able to achieve achieves full diversity of order L + 1 for suﬃciently large SNR. Although orthogonal space-time block codes (OSTBC) can be directly applied to the relays, the existing OSTBC is usually designed for a rather small number of transmit antennas. In cooperative relay networks, the number of relays could be large, and the membership of decodable set D may vary with time due to channel fading, which results in further challenges to apply OSTBC at relays. When the number of relays is greater than the column number of the OSTBC matrix, some relays have to forward symbols using the same column of the OSTBC matrix. As a consequence, the cooperative system may merely achieve diversity order of one if the set of decodable relays happens to use the same column of the OSTBC matrix. Thus, it is impractical to directly generalize OSTBC in DF systems when the number of relays is very large. To eﬀectively achieve diversity for a large cooperative network where the set

156

4 Cooperative Transmission Schemes with Multiple Relays

2XWDJH3UREDELOLW\

615G%

Fig. 4.9 Outage probabilities for systems using DF DSTC with various number of relays. c (From Laneman and Wornell with modiﬁed labels; [2003] IEEE.)

of decodable relays is random and relatively small, a practical DSTC scheme was proposed in [36] by assigning each user a unique signature vector. More speciﬁcally, in a cooperative network with L relays, the signal forwarded by relay ∈ D during the m-th block of time can be expressed by x [m] = Pr /Nd A[m]g , ∀ ∈ D, (4.82) where Pr is the total relay power, Nd = |D| is the number of decodable relays, A[m] is a N × Nc space-time codeword which conveys information symbols and is universal for all relays, N is block length of the DSTC, and g is a Nc ×1 designated unit-norm signature vector for r . The value of Nc is a design parameter of the DSTC, which depends on the tradeoﬀ between achievable diversity gain and computational complexity. The space-time codeword A[m] has been normalized to meet power constraint, i.e., E AH [m]A[m] = INc . Assume that all channels are quasistatic and remain invariant within N symbol periods. In Phase II, the vector of symbols received in consecutive N symbol periods is given by Pr (2) (2) yd [m] = h,d A[m]g + wd [m] Nd ∈D Pr (2) A[m]GD hD + wd [m], (4.83) = Nd

4.4 Distributed Space-Time Coding (DSTC)

157

where GD is a Nc × Nd matrix with columns being signature vectors of all decodable relays, and hD is a Nd × 1 vector composed by channel coeﬃcients of corresponding -d links for all ∈ D. When the decodable set D and the CSI is known at destination, one may adopt maximum likelihood (ML) detection to obtain the message conveyed in the codeword A[m]. In the following, to focus our discussion on the DSTC applied to relay links, the signal received from the s-d link is ignored. Nevertheless, the ML decoding algorithm can be easily extended to the case with diversity combining. For a given decodable set D, the ML estimate of the codeword matrix is given by (2) ˆ A[m] = arg min yd [m] − A[m]∈A

Pr A[m]GD hD Nd

2

,

(4.84)

F

where A is the set of all possible codewords. With ML detection, the upper bound of pairwise error probability (PEP) that Ak is mistaken as Am can be obtained by Chernoﬀ bound [c.f. Chapter 2]. For simplicity, let h,d be 2 i.i.d. with distribution CN (0, ηr,d ). At high SNR, the PEP given the set of decodable relays D is upper bounded by [36] Pr(Ak → Am |D) ≤

2 Pr ηr,d 2 4Nd σw

−r(B)

r(B)

·

i=1

1 , λi (B)

(4.85)

where r(B) and λi (B) are respectively the rank and nonzero eigenvalue of matrix B, which is given by H B = GH D (Ak − Am ) (Ak − Am )GD .

One can observe, from (4.85), that the maximum diversity can be achieved when the matrix B has full rank, i.e., both GD and Ak − Am are full rank for any k = m. In this case, the DSTC can achieve maximum diversity order of d = r(B) = min(Nd , Nc ). When the block length Nc is small such that Nc < Nd , diversity order is limited by Nc regardless of the number of decodable relays. On the other hand, if Nc is large enough, diversity order equals to the cardinality of decodable set Nd . Although a large Nc is preferable to attain maximum diversity, the complexity ML decoding increases exponentially with Nc . If decoding complexity is a design factor of concern, a smaller Nc could be preferred. When the CSI is not available at destination, one may encode the space-time codewords diﬀerentially, e.g. A[m + 1] = V[m]A[m], to allow non-coherent detection at the destination. In this case, it has been shown in [36] that the PEP given a decodable relay set D is upper bounded by

2 Pr ηr,d 2 8Ndσw

−r(B)

r(B)

·

i=1

1 , λi (B)

158

4 Cooperative Transmission Schemes with Multiple Relays

which performs 3 dB worse than coherent case. Since the role of matrix B in the PEP upper bounds of diﬀerential DSTC is similar that in (4.85), the design criteria of DSTC described in Section 2.2.2 can be incorporated directly. If both GD and Ak − Am are full rank and Nd ≥ Nc , we have r(B) = Nc and, from (4.85), the coding gain of the DSTC is given by 2 ηr,d

N c

1/Nc

*

λi (Ak −

Am )GD GH D (Ak

− Am )

H

+

i=1

* * + * ++1/Nc 2 H det GD GH = ηr,d . D · det (Ak − Am ) (Ak − Am ) The expression of coding gain can be easily extended to the case of Nd < Nc . It is worthwhile to note that the joint design of space-time codewords A and signature vector G {g } can be decoupled as two individual problems: 1. Optimization of signature vectors: For all decodable set D of cardinality |D| = Nd , GD is full rank and the minimum of det(GD GH D ) among all possible decodable sets is maximized. 2. Optimization of space-time codeword set: For any pair of distinct spacetime codewords (Ak , Am ), the diﬀerence matrix Ak − Am is full rank and the minimum value of det((Ak − Am )H (Ak − Am )) among all codeword pairs is maximized. Note that design criterion of codeword set A coincides with the design criteria of space-time codes (STC) described in Section 2.2.2. Therefore, it is reasonable to adopt a well-designed STC directly. Furthermore, when Nc is not large, existing orthogonal STC is preferred to design the codeword set A due to its low decoding complexity. In the following, we will introduce the construction of signature vectors. Following the second design criterion described above, we would like to ﬁnd an optimal set of signature vectors G such that the minimum of det(GD GH D) among all possible decodable sets with |D| = Nd is maximized. However, the number of decodable relays is random and it is impractical to design signature vectors respectively for each possible number of relays Nd = 1, 2, · · · , L, especially when the number of relays is large. Therefore, one may simply choose one speciﬁc number of decodable relays as optimizing the signature vectors. Let us denote Na as the assumed number of decodable relays, which can be the average number of decodable relays. The optimization problem now becomes Gopt = arg max

det(GD GH D)

(4.86)

subject to g 2 = 1, 1 ≤ ≤ L.

(4.87)

G

min

D:|D|=Na

4.4 Distributed Space-Time Coding (DSTC)

159

Note that the optimal set of signature vectors Gopt is not unique since the H

H for arbivalue of det(GD GH D ) equals to det(GD GD ), where GD = UGD V trary Nc ×Nc and Na ×Na unitary matrices U and V, respectively. Therefore, local optima may exist in the optimization problem. However, if we adopt gradient algorithm to ﬁnd the optimum set, it was observed in [36] that, regardless of the initial values, the algorithm turns out to converge to the same maximin value of det(GD GH D ), which implies that all local maxima are also global maxima. Since there is no local optimum, one may adopt gradient algorithm to obtain the optimal set Gopt . If GD GH D is non-singular, the gradient of det(GD GH ) equals to D

∂ det(GD GH H −1 D) = det(GD GH GD . D ) · (GD GD ) ∂G∗D

(4.88)

However, it is hard to ensure GD being full rank along the process to get converged. To avoid singularities, the gradient is modiﬁed as H −1 det(GD GH GD , D + INc ) · (GD GD + INc )

where is a small positive constant such that XD GD GH D + INc is nonsingular. The gradient algorithm to ﬁnd an optimal set is as follows. 1. Initialization: Set iteration number i = 0, and generate a set of signature vectors with unit norm randomly for all relays. 2. Find the worst set of signature vectors: D[i] = arg min det(GD GH D) D:|D|=Na

3. Update signature vectors in set D[i] GD[i] [i + 1] = GD[i] [i] + μ[i] det(XD[i] ) · X−1 D[i] GD[i] [i] 4. For ∈ D[i], normalize g [i + 1] as g [i + 1]/ g[i + 1] . 5. Termination: Given a convergence parameter ε > 0, if H | det(GD[i] [i + 1]GH D[i] [i + 1]) − det(GD[i−1] [i]GD[i−1] [i])|

| det(GD[i] [i + 1]GH D[i] [i + 1])|

< ε,

terminate the iteration. Otherwise, i = i + 1 and goto step (2). In step (3), the step size μ[i] can be varying with iteration number. In [36], μ[i] equals to 10−2 in the ﬁrst iteration and is decreased gradually to μ[i] = 10−5 in the ﬁnal iteration. Consider a cooperative network with L = 50 relays and data rate in Phase II is 3 bits per channel use. Figure 4.10 [36, Figure 6] compares the BER of

160

4 Cooperative Transmission Schemes with Multiple Relays

Nd = 1 Nd = 2 Nd = 4 Nd = 5 Nd = 7

Fig. 4.10 Bit error rates for systems using DF DSTC in (4.82) with various number of c decodable relays. (From Yiu, Schober and Lampe with modiﬁed labels; [2006] IEEE.)

aforementioned DSTC scheme with various numbers of decodable relays Nd . The solid lines indicate the DSTC scheme where A is constructed from rate3/4 OSTBC with 16-QAM modulation and N = Nc = 4. The set of signature vectors are optimized for L = 50 and Na = Nc = 4. The dash-dotted lines represents another DSTC scheme where Alamouti code is adopted to set A with 8-PSK modulation, and design parameters are N = Nc = Na = 2. For comparison, the dashed line shows the case with four collocated antennas and rate-3/4 OSTBC is adopted with 16-QAM modulation. It shows that for the case Nc = Na = 4, diversity order of DSTC linearly increases with the number of decodable relays when Nd is not greater than Nc = 4. However, when the number of decodable relays exceeds four, no extra diversity gain is provided, but higher coding gain is observed with the help of more relays. When the number of decodable relays is less than two, both DSTC schemes attain the same diversity gain. However, the scheme with Alamouti codewords and 8PSK modulation outperforms since the Euclidean distances among signal points are larger with smaller signal constellation. The design of distributed space-time codes in (4.82) can also be applied to the scenario where the receiver adopt diﬀerential or non-coherent decoding algorithm [36]. To achieve higher coding gain, the set A can be obtained by space-time trellis codes (STTC) [37] instead of OSTBC. When the relays simply demodulate the received symbols without employing error-detection or error-correction coding, the forwarded space-time codewords among the relays contain a certain degree of uncertainty since the source symbols demodulated at the relays could be erroneous. Similar to the demodulate-and-

4.4 Distributed Space-Time Coding (DSTC)

161

forward relaying scheme mentioned in Chapter 3, the decoding error probabilities among the relays shall be considered at the destination to recover the source symbols. The space-time processing of demodulate-and-forward case and the corresponding demodulation strategy can be found in [2].

4.4.2 Distributed Space-Time Coding with AF Relays In AF-based cooperative systems, one may directly adopt the OSTBC at the relays to achieve spatial diversity gain without CSI at relays. However, as we mentioned in the previous section, existing design of OSTBC is not suitable for a network with large number of relays. Moreover, it requires extra resource to coordinate the OSTBC applied at the relays as long as one relay node joins or leaves the cooperative networks. One method to deal with this situation is to adopt linear dispersion (LD) codes where the code matrix at each relay does not depend on the total number of relays L. Compared with the OSTBC, the LD codes are especially suitable for the AF relaying networks since the LD code matrices do not need special algebraic structures and are more robust to noise at the relays [13, 14]. Let xs [m] be a vector of source symbols with length N . During Phase I, a block of source symbols xs [m] is transmitted by the source, and the block of symbols received at the

-th relay is given by y [m] = Ps hs, xs [m] + w [m], = 1, 2, · · · , L (4.89) 2 where hs, ∼ CN (0, ηs, ) is channel coeﬃcient of s- link, and w is the complex additive white Gaussian noise vector with zero mean and covariance 2 matrix E[w [m]w [m]H ] = σw IN ×N . The received symbols is then encoded using LD codes. First, let us consider a simple LD coding scheme where the signal forwarded by relay can expressed by ' Pr /L t [m] = (4.90) 2 + σ 2 A y [m], Ps ηs, w

where A is the space-time encoding matrix, which is assumed to be unitary. Note that ﬁxed-gain AF relaying is adopted in this section, which only requires statistical CSI of s- link at each relay . The format of encoding matrices {A } is quite arbitrary and can be randomly generated by each relay. Since A is unitary, the average transmit power at each relay is given by Pr /L. During Phase II, each relay transmits the block of encoded symbols t and the block of the symbols received at the destination is given by

162

4 Cooperative Transmission Schemes with Multiple Relays (2)

yd [m] =

L =1

=

L

(2)

h,d t [m] + wd [m] '

Ps Pr /L 2 + σ 2 hs, h,d A xs [m] Ps ηs, w =1 ' L Pr /L (2) + 2 +σ 2 h,d A w [m] + wd [m], PS ηs, w =1

(2)

where wd is the additive Gaussian noise vector at the destination with zero2 mean and covariance matrix σw IN ×N . Let hs,r = [hs,1 , hs,2 · · · , hs,L ]T and T hr,d = [h1,d , h2,d · · · , hL,d ] be vectors of source-relay and relay-destination channel coeﬃcients respectively, and the block of received symbols can be expressed in vector form as ' Ps Pr /L (2) (2) yd [m] = (4.91) 2 + σ 2 Xs [m]Hr,d hs,r + Wd [m], Ps ηs, w where Xs [m] = [A1 xs [m], A2 xs [m], · · · , AL xs [m]], Hr,d = diag(hr,d ), and (2) Wd [m]

=

L =1

'

Pr /L (2) 2 + σ 2 h,d A w [m] + wd [m] Ps ηs, w

2 is the eﬀective noise with covariance matrix σw 1+ L =1

Pr |h,d |2 /L 2 +σ 2 Ps ηs, w

IN ×N .

To focus on LD codes applied over the relay links, the signal received from direct link is ignored. The ML detection of the source symbols can be obtained by ˆ s [m] = arg min x x∈MN

(2) yd [m]

−

=1

' (2)

= arg min yd [m] − X

L

'

2

Ps Pr /L 2 + σ 2 hs, h,d A x Ps ηs, w 2

Ps Pr /L 2 + σ 2 XHr,d hs,r Ps ηs, w

(4.92)

where M is the set of alphabets in the signal constellation. The complexity of the ML decoding algorithm is proportional to |M|N , which is exponentially increased with the block length. To reduce the decoding complexity, one may apply sphere decoding to detect the block of symbols [12]. The error performance of above LD codes will be analyzed in the following. Consider the case that channel coeﬃcients of all source-relay links and 2 2 relay-destination links are i.i.d. with distribution CN (0, ηs,r ) and CN (0, ηr,d ). Given the channel state hs,r and hr,d , the PEP of mistaking xk as xm can be obtained from Chernoﬀ upper bound [14] as

4.4 Distributed Space-Time Coding (DSTC)

163

Pr(xk → xm |hs,r , hr,d ) ≤ (4.93) Ps Pr /L H H exp − hs,r Hr,d Mk,m Hr,d hs,r L 2 2 (σ 2 + P η 2 + (P /L) σw r s s,r w =1 |h,d | ) where Mk,m = (Xk − Xm )H (Xk − Xm ) is distance matrix between codeword pair (Xk , Xm ). Averaging over the channel coeﬃcients of all s- links, the resulting conditional PEP is given by ∞ hH 1 s,r hs,r Pr(xk → xm |hr,d ) ≤ × exp − L L 2 ηs,r −∞ π ηs,r Ps Pr /L H H exp − hs,r Hr,d Mk,m Hr,d hs,r dhs,r L 2 (σ 2 +P η 2 +(P /L) 2 σw s s,r r w =1 |h,d | ) −1 2 Ps Pr ηs,r /L H = det IL + Hr,d Mk,m Hr,d L 2 (σ 2 + P η 2 + (P /L) 2 σw s s,r r w =1 |h,d | ) −1 2 Ps Pr ηs,r /L = det IL + , (4.94) Mk,m Gr,d L 2 (σ 2 + P η 2 + (P /L) 2 σw s s,r r w =1 |h,d | ) where Gr,d = diag(|h1,d |2 , |h2,d |2 , · · · , |hL,d|2 ). The derivation of upper bound in (4.94) is still intractable since the channel coeﬃcients of all -d links have impacts on both numerator and denominator. By the law of large number, when the number of relays L is very large, we have L

2 |h,d |2 ≈ Lηr,d .

=1

Assume that L is very large, the upper bound of conditional PEP can then be approximated by

2 Ps Pr ηs,r /L Pr(xk → xm |hr,d ) det IL + 2 2 2 ) Mk,m Gr,d 2 σw (σw + Ps ηs,r + Pr ηr,d

−1 .

(4.95) With total power constraint Ps + Pr ≤ 2P , one may further allocate transmission powers to minimize the pairwise error probability, which is given by averaging the conditional PEP in (4.95) over channel coeﬃcients of all -d links. It turns out that the optimum power allocation that minimize PEP is equivalent to maximize the following fraction ζ

2 Ps Pr ηs,r /L 2 (σ 2 +P η 2 +(P /L) σw s s,r r w

L

=1 |h,d |

2)

≈

2 /L Ps Pr ηs,r . (4.96) 2 (σ 2 +P η 2 +P η 2 ) σw s s,r r r,d w

164

4 Cooperative Transmission Schemes with Multiple Relays

Note that ζ can be treated as a deterministic function of Ps and Pr when the number of relays is suﬃciently large. By letting Pr = 2P − Ps and taking ﬁrst derivative with respective to Ps , the optimum power allocation is then given by 2 /σ 2 1 + 2P ηr,d w Ps = · 2P (4.97a) 2 /σ 2 2 /σ 2 + 1 + 2P ηs,r 1 + 2P ηr,d w w 2 /σ 2 1 + 2P ηs,r w Pr = · 2P. (4.97b) 2 /σ 2 2 /σ 2 + 1 + 2P ηs,r 1 + 2P ηr,d w w With optimum power allocation in (4.97), the minimum value of ζ can be approximated at high SNR as 2 Ps Pr ηs,r /L (4.98) 2 (P η 2 + P η 2 ) σw s s,r r r,d ' ' −1 2 /σ 2 2 2 /σ 2 1 + 2P η LP ηs,r 1 + 2P ηs,r w w r,d 2 2 ηs,r 1 + + ηr,d 1 + = . 2 /σ 2 2 2 /σ 2 2σw 1 + 2P ηs,r 1 + 2P ηr,d w w

ζ≈

Returning to the PEP analysis, we can see that, with a large number of relays at high SNR, the PEP can be approximated by ( ) −1 Pr(xk → xm ) Ehr,d det (IL + ζMk,m Gr,d ) . 2 be the minimum nonzero eigenvalue of Mk,m , and we have Let σM 2 Mk,m − σM UM diag(Ir(Mk,m ) , 0)UH M

be positive deﬁnite, where r(Mk,m ) is the rank of Mk,m and columns of UM are eigenvectors of Mk,m corresponding to eigenvalues with decreasing order. Thus, the PEP can be further upper bounded by ( * +−1 ) 2 Pr(xk → xm ) Ehr,d det IL + ζσM diag(Ir(Mk,m ) , 0)Gr,d ⎡ ⎤ r(Mk,m ) * + −1 2 ⎦ 1 + ζσM |h,d |2 = Ehr,d ⎣

=1 ∞

*

1+

= 0

+−1 2 ζσM u

1 2 ηr,d

r(Mk,m ) e

2 −u/ηr,d

du

r(Mk,m ) 1 1 1 exp Ei − 2 2 = − (4.99) , 2 η2 2 η2 ζσM ζσM ζσM ηr,d r,d r,d

4.4 Distributed Space-Time Coding (DSTC)

where Ei (x) =

165

x

−∞

e−u du u

is an exponential integral function. Taking Taylor expansion, Ei (x) can be expressed as ∞ (−1)k xk Ei (x) = c + log(−x) + , k · k! i=1 where c is the Euler constant. When total transmission power P is extremely large such that log P 1, we have 1 1 exp ≈ 1, =1−O 2 2 ζσM ηr,d P and

1 −Ei − 2 2 ζσM ηr,d

* 2 2 + = −c + log ζσM ηr,d − O

1 P

≈ log (P ) ,

since the 1/P decays much faster than a constant and log P grows rapidly. Therefore, with optimum power allocation in (4.97), the PEP at extremely high SNR can be upper bounded by r(Mk,m ) 1 log(P ) 2 η2 ζσM r,d

Pr(xk → xm )

r(Mk,m ) 2 η η 2σw log P r,d s,r 2 2 ηs,r 1+ +ηr,d 1+ ≈ 2 η2 σ2 P Lηs,r ηs,r ηr,d r,d M 2 r(Mk,m ) 2 P) 2σw 1 1 −r(Mk,m )(1− log(log ) log P =P + (4.100) . 2 LσM ηs,r ηr,d

The last equality follows the fact that log(log P ) =

log(log P ) log(log P ) . log P = log P log P log P

From the approximated upper bound of pairwise error probability, it can be easily shown that the diversity order of LD codes equals to rmin (1 − log(log P )/ log P ), where rmin = min∀k=m r(Mk,m ) is the minimum rank of Mk,m among all distinct codeword pairs. When transmission power approaches inﬁnity, i.e., log P 1, 1/ log P decays much faster than 1/ log(log P ), and log(log P )/ log P goes to zero. Therefore, the diversity order is closely approximated by rmin . Suppose that {Mk,m } are full rank for all codeword pairs, we have rmin = min(L, N ). That is, as long as the block length is suﬃciently

166

4 Cooperative Transmission Schemes with Multiple Relays BER of networks with different T and R

−2

10

1 / T=R=5 1 / T=10 R=5 1 / T=R=7 1 / T=R=10

−3

10

−4

BER

10

−5

10

−6

10

−7

10

−8

10

20

21

22

23

24

25 Power (dB) 615G%

26

27

28

29

30

Fig. 4.11 Error Probability of AF DSTC scheme using linear dispersion coding with c diﬀerent numbers of relays L and block length N . (From Jing and Hassibi; [2006] IEEE.)

large, the relaying system is able to achieve full diversity at extremely high SNR. However, the price of a large block length N is the huge complexity to perform ML decoding algorithm. In Fig. 4.11, the performance of AF DSTC scheme in (4.90) with diﬀerent numbers of relays and codeword length are compared. In the computer 2 2 2 simulation, ηs,r = ηr,d = 1 and SNR 2P/σw . According to these results, we can see that if the number of relays remains unchanged, increasing the transmission codeword length doesn’t necessary improve the diversity gain. However, coding gain is still introduced. From the ﬁgure, one can see that the system with N = 10, L = 5 performs about 1 dB better than that with L = N = 5. In fact, we may adopt a more general LD code which is expressed by ' Pr /L ∗ t [m] = (4.101) 2 + σ 2 (A y [m] + B y [m]) , Ps ηs, w where A and B are N × N complex space-time encoding matrices with re: as a vector concatenated spect to y [m] and y∗ [m]. For conciseness, denote u : [{uT }, {uT }]T . by real and imaginary parts of a complex vector u, i.e., u

4.4 Distributed Space-Time Coding (DSTC)

167

The vector :t [m] can then be expressed by ' Pr /L {A } + {B } −{A } + {B } :t [m] = : [m]. y 2 + σ2 Ps ηs, w {A } + {B } {A } − {B } (4.102) The format of encoding matrices {A } and {B } is quite arbitrary in the framework of LD codes and can be randomly generated by each relay as well. For simplicity and fairness among the received symbols (i.e., each received symbols are retransmitted with equal power), it is assumed that the matrix {A } + {B } −{A } + {B } F {A } + {B } {A } − {B } is orthogonal and each column has unit norm [14]. During Phase II, after transmitting the block of encoded symbols t [m] at each relay, the block of symbols received at the destination is given by (2)

yd [m] =

L =1

=

L =1

(2)

h,d t [m] + wd [m] '

Ps Pr /L (2) ∗ 2 + σ 2 h,d (A y [m] + B y [m]) + wd [m], Ps ηs, w

(2)

where wd [m] is the additive Gaussian noise vector at the destination with 2 zero-mean and covariance matrix σw IN ×N . Separating the real and imaginary (2) (2) :d [m] can be expressed by parts of yd [m], y (2) :d [m] y

=

L

'

=1

Ps Pr /L :s [m] + Wd [m], 2 + σ 2 H,d F Hs, x Ps ηs, w

{h,d }IN −{h,d}IN , = {h,d }IN {h,d }IN {hs, }IN −{hs,}IN , = {hs, }IN {hs, }IN

where H,d Hs,

and Wd [m] is eﬀective Gaussian noise given by ' L Pr /L (2) : : d [m] + Wd [m] = w 2 + σ 2 H,d F w [m]. Ps ηs, w =1

Similar to (4.92), the source symbols can be detected with ML decoding method given by

168

4 Cooperative Transmission Schemes with Multiple Relays

ˆ [m] = arg min x x∈MN

(2) :d [m] y

−

L =1

'

2

Ps Pr /L : 2 + σ 2 H,d F Hs, x Ps ηs, w

.

(4.103)

When CSI is not available at receiver, the aforementioned DSTC can be generalized to a diﬀerential DSTC can be applied by letting xs [i] = U[i]xs [i− 1], where U[i] is an information-embedded unitary matrix. It has been shown that error performance of the diﬀerential DSTC only performs 3 dB worse than the coherent DSTC scheme [15].

4.5 Channel Estimation in Multi-Relay Systems In most cooperation schemes mentioned earlier, perfect knowledge of the channel coeﬃcients at the destination is often assumed. However, in practice, this must be obtained through channel estimation at the relay and/or the destination. Ideally, conventional channel estimation schemes devised for pointto-point communications can also be employed to estimate separately the channels of each source-to-relay and relay-to-destination links. The sourceto-relay channels can be estimated at each relay by having the source embed a pilot signal in its transmission and the relay-to-destination channels can be estimated at the destination using techniques devised for conventional MISO channels. However, several constraints of the cooperative network may render these schemes inapplicable. For example, in AF cooperative systems, what the destination needs to know is the combination of the source-to-relay and the relay-to-destination channels, instead of the two individual channel coeﬃcients. Therefore, if the two channels are estimated separately, additional energy or bandwidth resources are needed for the relays to forward their local channel estimates to the destination, making conventional point-to-point estimation policies ineﬃcient. Moreover, in both AF and DF networks, the distributed relays may operate under individual power constraints that may lead to diﬀerent training signal design than conventional MISO systems. Due to these reasons, new training methods must be proposed to cope with these constraints. Following the works in [8] and [9], we discuss in this section the optimal training design for both AF and DF multi-relay networks. The discussions are based on DSTC systems but the channel estimation schemes can certainly be applied to other multi-relay systems as well.

4.5.1 Training Design for AF Multi-Relay Systems Let us consider the AF multi-relay system that consists of a source s, L relays (labeled by 1, 2, . . ., L), and a destination d, as shown in Fig. 4.1. If the LD-DSTC is employed as described in the previous section, the sig-

4.5 Channel Estimation in Multi-Relay Systems

169

nals transmitted by the relays can be viewed as linear combinations of their respective received signals and the decoding at the destination will require knowledge of the combined source-to-relay and relay-to-destination channels, (∗) e.g., hs, · h,d , where the superscript (∗) indicates that the complex conjugate may or may not exist depending on the actual DSTC scheme. Instead of estimating hs, and h,d separately at the relay and the destination, the (∗) destination can estimate directly the eﬀective channel heﬀ, = hs, · h,d , for

= 1, . . . , L. Suppose that the source transmits a length-N training sequence z in Phase I with zH z ≤ Ps N Es and N ≥ L. The signals received at relay and the destination are given by y = hs, z + w , (1) yd

= hs,d z +

(4.104)

(1) wd ,

(4.105)

2 2 where the channel coeﬃcients hs, ∼ CN (0, ηs, ), hs,d ∼ CN (0, ηs,d ) are assumed to be quasi-static, i.e., they remain constant over the transmission of (1) 2 2 a data block, and w ∼ CN (0, σw IN ×N ) and wd ∼ CN (0, σw IN ×N ) are the AWGN at relay and the destination, respectively. Since hs,d can be estimated using conventional point-to-point channel estimation schemes, we shall focus our discussions on the estimation of the eﬀective source-relaydestination channel. Following the operations of LD-DSTC, relay then transmits in Phase II the signal (∗)

t = β B y

(4.106)

where y may be either y or y∗ depending on the speciﬁc DSTC design [14] and β = Ps η2P+σ2 is the amplifying gain at relay , assuming that the ﬁxed (∗)

s,

w

gain relaying scheme is employed. Here, B is a unitary precoding matrix (i.e., BH B = I) to be designed along with the training sequence z. The signal received at the destination is (2)

yd =

L

(2)

h,d t + wd

(4.107)

=1

=

L

heﬀ, β B z(∗) +

=1

L

(∗)

(2)

h,d β B w + wd

(4.108)

=1

= ZΛheﬀ + wd ,

(4.109)

2 ), heﬀ = [heﬀ,1 , . . . , heﬀ,L ]T , Λ = diag(β1 , . . . , βL ), where h,d ∼ CN (0, η,d (∗)

(∗)

Z = [B1 z1 , . . . , BL zL ],

170

4 Cooperative Transmission Schemes with Multiple Relays

L (2) (∗) (2) (∗) 2 wd ∼ CN (0, σw IN ×N ) and wd = =1 h,d β B w + wd . Similarly, z ∗ may be either z or z depending on the STC. For the AF scheme, the noise experienced at the destination will not be Gaussian due to eﬀects of the channel and, thus, the general MMSE estimator will not be easy to derive. However, we can instead employ the Linear MMSE (LMMSE) estimator given by ˆ eﬀ = E[heﬀ (y(2) )H ](E[y(2) (y(2) )H ])−1 y(2) , h d d d d (2)

(2)

(4.110)

(2)

where E[heﬀ (yd )H ] = Rheff ΛZH and E[yd (yd )H ] = ZΛRheff ΛZH + L 2 2 ˆ eﬀ , the σw ( =1 |β |2 η,d + 1)IN ×N . By deﬁning the error as Δheﬀ = heﬀ − h error covariance matrix can be written as Cov(Δheﬀ ) = E[Δheﬀ ΔhH eﬀ ] =

R−1 heff

+

1

−1

ΛZH ZΛ L 2( 2 2 σw =1 |β | η,d + 1)

.

(4.111)

The optimal training sequence z and unitary precoding matrices {B , ∀ } can be found by minimizing the total error variance. The optimization problem can be formulated as: * + min tr Cov(Δheﬀ ) (4.112) z,B ,∀

subject to

[ZH Z] ≤ Es , for = 1, . . . , L.

(4.113)

Following the approach in [8], we can ﬁrst simplify the problem by deﬁning D = ZH Z and by performing the optimization with respect to D. By doing so, the optimization problem becomes −1 1 −1 min tr (4.114) Rheff + ΛDΛ D ρ subject to

[D] ≤ Es , for = 1, . . . , L D 0,

(4.115) (4.116)

L 2 2 ( =1 |β |2 η,d + 1) and M N means that M − N is a where ρ σw positive semi-deﬁnite matrix. The above optimization problem is convex and can be solved using standard optimization toolbox. In fact, it can also be reformulated as a semi-deﬁnite programming (SDP) as shown in [8]. However, one may get more insights on the structure of the optimal solutions by the followings. Note that for any positive semi-deﬁnite matrix D, one can always ﬁnd a feasible solution for z and {B , ∀ }. That is, one can always ﬁnd z and {B , ∀ } that satisﬁes the following set of equations:

4.5 Channel Estimation in Multi-Relay Systems (∗)

(∗)

(zk )H BH k B z BH B

171

= [D]k, ,

(4.117)

= I,

(4.118)

for all , k = 1, . . . , L. Speciﬁcally, suppose that it is indeed possible to arrive at a solution D that is positive semi-deﬁnite. Then, one can decompose D as D = CH C, where C is an N × L matrix. Let c be the -th column of C. Then, z and {B , ∀ } will satisfy (4.117) and (4.118) when the following holds: (∗)

B z

= c

(∗)

(∗)

or B z = c

(4.119)

√ (∗) for = 1, . . . , L. Since both z and c have the same norm Es , due to (∗) the power constraint, c can be viewed as a rotation of z with rotation (∗) (∗) (∗) matrix B . To ﬁnd B , one can ﬁrst rotate both z and c to the vector √ Es e1 , where e1 is the N -dimensional vector with 1 in the ﬁrst element and √ (∗) 0 √ elsewhere. That is, ﬁnd U and V such that Uz = Es e1 and V c = Es e1 . Hence, we have B = (VH U)(∗) . To derive a closed-form expression, one can replace back D with ZH Z and obtain the Lagrange function −1 L * * + + 1 −1 H H + L(Z, {μ }) =tr Rheff + ΛZ ZΛ μ tr eH Z Ze − Es , ρ =1

(4.120) where e is the -th column of an L × L identity matrix. Since the Lagrange function is quadratic, we can ﬁnd the optimal Z by setting ∂L(Z, {μ })/∂Z = 0. Notice that ∂tr(X−1 ) = −X−T X−T = −X−2T (4.121) ∂X and, for G = Q + MH XH XM (where Q, M are constant matrices), that ∂tr(G−1 ) ∂tr(G−1 ) ∂tr(G−1 ) ∂Gi,j = = (4.122) ∂X ∂Xk, ∂Gi,j ∂Xk, k, i,j =

∂tr(G−1 ) i,j

∂Gi,j

(XM)H

i,k

∂tr(G−1 ) T M = X∗ M∗ ∂G k, ∗ ∗ −2T = −X M G MT k, .

M,j

(4.123) (4.124) (4.125)

172

4 Cooperative Transmission Schemes with Multiple Relays

Moreover, with straightforward derivation, one can also show that ∂tr(eT ZH Ze ) = Z∗ e eT . ∂Z

(4.126)

1 H By taking G = R−1 heff + ρ ΛZ ZΛ, M = Λ, and X = Z in (4.125) and by using (4.126), the derivative of the Lagrange function can be expressed as

−2T L ∂L(Z, {μ }) −1 ∗ 1 −1 H = Z Λ Rheff + ΛZ ZΛ Λ+ Z∗ μ e eT ∂Z ρ ρ =1 L −2T 1 1 −1 ∗ T H =Z μ e e − Λ Rheff + ΛZ ZΛ Λ . ρ ρ =1

That is, the optimal Z must satisfy L

μ e eT

=1

⇒R−1 heff and, thus,

−2T 1 1 −1 H = Λ Rheff + ΛZ ZΛ Λ ρ ρ

1 1 1 + ΛZH ZΛ = √ Λ 2 ρ ρ

L

(4.127)

−1 2 μ e eT

1

1

Λ2 Ω2

(4.128)

=1

1 −1 ZH Z = ρΛ−1 Ω 2 − R−1 heff Λ

(4.129)

where Ω = diag(β12 /ρμ1 , . . . , βL2 /ρμL ). The coeﬃcients {μ , ∀ } must be chosen to satisfy the power constraint [ZH Z], = Es , for all . In the special case where the eﬀective channels are spatially uncorrelated such that E[heﬀ, h∗eﬀ,k ] = 0 for all = k, the covariance matrix Rheff will be a diagonal and, by the power constraint, ZH Z must be equal to Es IL×L . It is worthwhile to remark that the closed-form solution in (4.129) only holds when it yields a positive semi-deﬁnite solution, i.e., ZH Z 0. If this is not the case, the solution can still be found by using standard numerical optimization toolboxes. In Fig. 4.12. the average MSE achievable by the optimal training design is shown and compared with that of random training sequences. We consider systems with L = 2, 4, and 8 relays and with training length ﬁxed as N = 8, that is, the total training power remains constant for diﬀerent values of L. The channel coeﬃcients and noise variables are both assumed to be complex 2 Gaussian with mean 0 and variance 1, and SNR P/σw . By random training, we mean that the unitary matrices B are randomly generated. In each case, we set the source power as Ps = P and let half of the relays have power P = 0.8P while the other half have P = P . From Fig. 4.12, we can see that the MSE of the channel estimate increases with L for a ﬁxed value of N . This is because less degrees of freedom are available to estimate the channel of each relay as L increases. This is also the reason that a more observable

4.5 Channel Estimation in Multi-Relay Systems

173

1

10

L=2, N=8, optimal L=4, N=8, optimal L=8, N=8, optimal L=2, N=8, random L=4, N=8, random L=8, N=8, random

0

10

Average MSE

−1

10

−2

10

−3

10

−4

10

0

5

10

15 SNR (dB)

20

25

30

Fig. 4.12 Channel estimation MSE versus SNR for both optimal training and random training with diﬀerent L, same N .

loss can be seen for random training as L increases. In this case, orthogonality among the relays training sequences are less likely to be achieved.

4.5.2 Training Design for DF Multi-Relay Systems Training in DF multi-relay systems are less complex than that in AF systems since the destination needs only to know the relay-to-destination channels, instead of the combined eﬀective s-r-d channel. However, the individual power constraints at the relays still pose great challenges in the design of optimal training signals (see, e.g., [9]). Speciﬁcally, let us also consider the system illustrated in Fig. 4.1. To coherently detect the source symbol at the relays, each relay needs only to know their local channels between itself and the source. This can be done simultaneously with a single training signal emitted by the source, as in conventional SISO channels. The challenge of the channel estimation problem lies in Phase II where the destination must estimate all relay-to-destination channels through the concurrent transmission of training signals from the relays. Let z be the length-N training signal emitted by relay with N ≥ L.

174

4 Cooperative Transmission Schemes with Multiple Relays

The signal received at the destination in Phase II is given by (2) yd

=

L

(2)

(2)

h,d z + wd = Zhr,d + wd ,

(4.130)

=1

where Z = [z1 , . . . , zL ] and hr,d = [h1,d , . . . , hL,d ]T ∼ CN (0, Rhr,d ), and (2) 2 I) is the AWGN vector. wd ∼ CN (0, σw Assume that the training signals must satisfy both the individual power constraints [ZH Z], ≤ N P = E , for all , and the total power constraint tr(ZH Z) ≤ N Pr = Etot . Two degenerate cases may occur: (i) if Etot ≥ L =1 E , the total power constraint becomes redundant and the relays should simply transmit with its maximum power; (ii) if Etot ≤ min E , then the individual power constraints become redundant and the problem reduces to the training design for conventional MISO systems. Therefore, in the following, L we shall focus on the interesting case where min E < Etot < =1 E . Following the approach given in the previous subsection and also in [9], we consider the LMMSE estimator, which is now optimal since both the channel and noise are Gaussian. The LMMSE estimator is given by ) ( )−1 ( (2) ˆ r,d = E hr,d (y(2) )H E y(2) (y(2) )H h yd d d d * +−1 (2) 2 = Rhr,d ZH ZRhr,d ZH + σw I yd

(4.131) (4.132)

and the resulting error covariance matrix can be expressed as Cov(Δhr,d ) =

−1 1 H R−1 + Z Z . hr,d 2 σw

(4.133)

The optimal training sequences transmitted by the relays can then be found by minimizing the MMSE. The optimization problem is formulated as follows: −1 1 H min tr (4.134) R−1 hr,d + 2 Z Z Z σw (4.135) subject to ZH Z , ≤ E , ∀ , * H + (4.136) tr Z Z ≤ Etot . By replacing ZH Z with D, one can show that the above problem is convex with respect to D. Again, as shown in [9], it can be converted to a SDP problem, which can be solved eﬃciently by interior point methods [31]. Let us consider the special case where channels experienced by diﬀerent 2 2 relays are independent. That is, we have Rhr,d = diag(η1,d , . . . , ηL,d ). In this H case, the solution of Z Z must be diagonal. To show this, let us recall the identity

4.5 Channel Estimation in Multi-Relay Systems

175

L −1 + * [A], tr A−1 ≥ ,

(4.137)

=1

where A is any L × L positive deﬁnite matrix and the equality holds if and only if A is diagonal. By this fact, we can see that, for any D = ZH Z, we : = diag(D) such that can always ﬁnd D ⎛ ⎞ −1 −1 L 1 1 1 −1 −1 : ⎝ ⎠ = . tr Rhr,d + 2 D ≤ tr Rhr,d + 2 D [D] 1 σw σw + 2, 2 =1

η,d

σw

(4.138) H : where [D], = Z Z , ε is the energy of relay ’s training signal. This implies that the training signal emitted by the relays should be orthogonal to each other when the channels are uncorrelated. In this case, the optimization problem reduces to the following power allocation problem: L

min

1

ε ,∀

2 =1 η,d

1 +

(4.139)

ε 2 σw

0 ≤ ε ≤ E , ∀ , and

subject to

L

ε ≤ Etot .

(4.140)

=1

The Lagrange function can be written as L L L L 1 + + λ ε − E μ (ε − E ) − ν ε , L= tot 1 + σε2 =1 η 2 =1 =1 =1 w

,d

(4.141) where λ, {μ , ∀ }, and {ν , ∀ } are the Lagrange multipliers. The KKT conditions can then be listed as follows: (c1)

L

ε ≤ Etot ,

(c2) ε ≤ E ,

=1

(c4) λ

L

(c3) ε , μ , ν , λ ≥ 0

ε − Etot

= 0,

(c5) μ (ε − E ) = 0,

(c6) ν ε = 0

=1

(c7) − 2 σw

1 1 2 η,d

+

ε 2 σw

2 + λ + μ − ν = 0.

To solve the optimization problem, let us ﬁrst multiply both sides of (c7) by ε to obtain

176

4 Cooperative Transmission Schemes with Multiple Relays

⎛ ε ⎝−

⎞ *

1

1 2 σw 2 η,d

+

+ λ + μ ⎠ = 0.

+ ε 2 2 σw

(4.142)

2* 1 4 2 4 2 /σw , then λ + μ ≥ η,d /σw ≥ 1/ σw + Notice that, if λ ≥ η,d η2 ,d

+ ε 2 . 2 σw

In this case, ε must be equal to 0. Otherwise, if ε > 0, then λ + μ will +2 2* 1 and, thus, (c7) will not hold. be strictly greater than 1/ σw + σε2 η2 w

,d

Similarly, we can also multiply both sides of (c7) by E − ε to obtain ⎛ ⎞ 1 (E − ε ) ⎝− * (4.143) +2 + λ − ν ⎠ = 0. 1 2 σw η2 + σε2 w

,d

+2 2* 1 , then it must also be true that λ − ν ≤ + σE2 Similarly, if λ ≤ 1/ σw 2 η,d w + 2* 1 E 2 . In this case, ε must be equal to its maximum value E . 1/ σw η2 + σ2 w ,d +2 2* 1 + σε2 Otherwise, if ε < E , then λ−ν must be strictly less than 1/ σw η2 ,d

and, thus, again (c7) will not hold. 2* 1 Now, we can also show that, for 1/ σw + η2 ,d

+ E 2 2 σw

w

4 2 < λ < η,d /σw , both

μ and ν must be 0. From (4.142), we know that ⎞ ⎛ 1 ε ⎝− * + + λ⎠ ≤ 0. ε 2 1 2 σw + σ2 η2 w

,d

2* 1 Therefore, if λ > 1/ σw + η2 ,d

+ E 2 , then ε < E must hold and, thus, σ2 w

μ = 0 by (c5). Moreover, by (4.143), we also know that ⎛ ⎞ 1 (E − ε ) ⎝− * + + λ⎠ ≥ 0. ε 2 1 2 σw + 2 2 σ η ,d

w

4 2 Therefore, if λ < η,d /σw , then ε > 0 and, thus, ν = 0 by (c6). Hence, it follows that 2 1 σw σ2 − * − 2w . + λ = 0 ⇒ ε = (4.144) + 2 λ η,d σ 2 21 + ε2 w η ,d

σw

In summary, the optimal power allocation is given by [9]

4.5 Channel Estimation in Multi-Relay Systems

177

Fig. 4.13 Illustration to the cave-ﬁlling strategy for (4.145).

⎧ ⎪ 0, ⎪ ⎪ ⎪ ⎪ 2 ⎨ σw λ − ε = ⎪ ⎪ ⎪ ⎪ E ⎪ ⎩

for λ ≥ 2 σw 2 , η,d

for

2 σw

*

for λ ≤

4 η,d 2 , σw

1 1 η2 ,d 2 σw

E + σ2 w

*

+2 < λ <

1

1 η2 ,d

E

+ σ2

4 η,d 2 , σw

(4.145)

+2 ,

w

5 5 4 4 2 /λ − σ 2 /η 2 , E or, equivalently, ε = min max 0, σw . This solution w ,d has a cave-ﬁlling interpretation, as illustrated in Fig. 4.13, where the power is ﬁlled into the valleys either until the ceiling - set by the individual power 2 /λ is reached. constraints are met or until the water-level σw In Fig. 4.14, we show the average MSE achievable by the training scheme described in (4.145) and compare it to the scheme where training energy is allocated proportionally among relays with respect to their individual energy L constraints, i.e., the energy of relay is given by ε = (E / i=1 Ei )Etot . We consider systems with L = 2, 4, or 8 number of relays and training sequence ﬁxed as N = 8. Both the channel and noise variances are set equal to 1. In the simulations, half of the relays are assumed to have individual power constraint equal to 0.4Ps and the other half equal to Ps . The SNR on the 2 x-axis is deﬁned as SNR Ps /σw . The total relay power constraint is set as 4/7 that of the sum of the individual power constraints over all relays. In Fig. 4.14, we can see that the MSE performance does not depend on L when the length of the training sequence is ﬁxed. This behavior has also been observed in conventional MISO systems. Moreover, we can also see that the described training scheme can outperform the proportional power allocation scheme, especially as the individual power constraints diﬀer and the total relay power constraint is more strict.

178

4 Cooperative Transmission Schemes with Multiple Relays

1

10

L=2, N=8, Optimum PA L=4, N=8, Optimum PA L=8, N=8, Optimum PA L=2, N=8, Proportional PA L=4, N=8, Proportional PA L=8, N=8, Proportional PA

0

10

Average MSE

−1

10

−2

10

−3

10

−4

10

0

5

10

15

20

25

30

SNR(dB)

Fig. 4.14 MSE of channel estimation versus SNR for both the optimized training scheme in (4.145) and the case of proportional power allocation.

4.6 Other Topics on Multi-Relay Cooperative Communications In previous sections, we focused on the design of two-hop multi-relay systems, assuming that perfect synchronization is achieved among the relays. In the following, we extend the concept of cooperation to multi-hop networks and introduce advanced techniques that can be adopted for asynchronous relays.

4.6.1 Multi-Hop Cooperative Transmissions In networks where the distance between source and destination is large, multihop transmissions can be used to compensate for the eﬀect of path loss and to extend the system coverage. In this section, the concept of both DF and AF cooperation will be extended to the multi-hop scenario and the outage probability will be analyzed to illustrate the advantages of cooperation.

4.6 Other Topics on Multi-Relay Cooperative Communications

179

Fig. 4.15 System model of an (L + 1)-hop cooperative network.

Multi-Hop Transmissions with DF Relays Let us consider a multi-hop network that consists of a source, a destination, and L intermediate relays forming a multi-hop path from source to destination, as shown in Fig. 4.15. For notational simplicity, we label the source as node 0, the intermediate relays as nodes 1, 2, . . ., L, and the destination as node L + 1. In conventional multi-hop networks, a message is transmitted from source to destination in a hop-by-hop fashion, going from nodes 0 to 1, nodes 1 to 2, nodes 2 to 3, so on and so forth, until it reaches the destination. The system can be viewed as a concatenation of DF links without diversity combining (c.f. Section 3.1). Therefore, transmission of a message from source to destination is successful only when all intermediate relay nodes successfully decode the message and pass it forward. Let h,k be the channel coeﬃcient between nodes and k, where , k ∈ {0, 1, . . . , L + 1}. Given that node transmits with power P , the SNR of the 2 2 signal received at node k will be equal to γ,k = P |h,k |2 /σw , where σw is the noise variance at each receiver. Based on the Rayleigh fading assumption, γ,k can be modeled as an exponential random variable with mean γ ,k = 2 2 2 P η,k /σw , where η,k is the variance of channel coeﬃcient h,k . We assume 2 that η,k is inversely proportional to dα ,k , where α is the path loss exponent and d,k is the distance between nodes and k. Given the rate R of the transmitted codeword, transmission on the link from to k will be successful only if R is less than the instantaneous per-hop capacity log2 (1 + γ,k ). This translates into a threshold condition on the SNR γ,k , which is to say that transmission from to k is successful only if γ,k is greater than a certain threshold γth . Therefore, outage probability can be derived as pout = Pr (min {γ0,1 , γ1,2 , · · · , γL,L+1 } < γth ) = 1 − Pr (γ0,1 > γth , γ1,2 > γth , · · · , γL,L+1 > γth ) = 1−

L

Pr (γ,+1 > γth ) ,

=0

where the SNRs γ,+1 , for = 0, . . . , L, are assumed to be independent. Under the Rayleigh fading assumption, the probability inside the product term can be approximated at high SNR as

180

4 Cooperative Transmission Schemes with Multiple Relays

Fig. 4.16 Illustration of a multi-hop cooperative network with diversity combining only at the destination.

γth γth ≈1− Pr (γ,+1 > γth ) = exp − γ ,+1 γ ,+1 and, thus, pout = 1 −

L+1 =0

L γth γth exp − ≈ . γ ,+1 γ ,+1 =0

This shows that the outage probability increases as the number of hops increases. However, in practice, relays in multi-hop networks will have the ability to buﬀer the message and retransmit it in later time slots if previous transmission attempts have failed. Hence, as long as the end-to-end delay is tolerable, the message will eventually reach the destination with high probability. In the following, let us consider a slightly more advanced scenario where diversity combining is performed at the destination, but the intermediate relays only receive from its immediate upstream node, as illustrated in Fig. 4.16. By assuming that all intermediate relays are able to successfully decode the source’s message, the outage probability at the destination L + 1 can be computed as L pout,L+1 = Pr (4.146) γk,L+1 < γth . k=0

To obtain a closed-form expression of this probability, let us consider the worst case scenario where the distribution of the SNRs on each link is equal to that of the worst link, e.g., the distribution of γ0,L+1 . In this case, the L summation γsum k=0 γk,L+1 becomes a chi-squared random variable with 2(L + 1) degrees of freedom and mean equal to L k=0 γ k,L+1 = (L + 1)γ 0,L+1 . Recall that a chi-squared random variable Y with 2ν degrees of freedom and mean 2νσ 2 has CDF given by

4.6 Other Topics on Multi-Relay Cooperative Communications

181

x ν−1 1 x k FY (y) = Pr(Y ≤ y) = 1 − exp − 2 . 2σ k! 2σ 2

(4.147)

k=0

In the worst case, the outage probability in (4.146) equals to pout,L+1

k L 1 γth γth = 1 − exp − . γ 0,L+1 k! γ 0,L+1 k=0

By Taylor expansion, we have L ∞ 1 −γth k 1 γth k . (4.148) pout,L+1 = 1 − k! γ 0,L+1 k! γ 0,L+1 k=0

k=0

It is worthwhile to note that, for 0 ≤ ≤ L, the -th power of γth /γ 0,L+1 is canceled out because −

k=0

(−1)k k!( − k)!

γth

γ 0,L+1

= −(1 − 1)

γth γ 0,L+1

= 0.

On the other hand, the (L + 1)-th power of γth /γ 0,L+1 equals to −

L+1 k=1

(−1)k k!(L + 1 − k)!

γth

L+1

γ 0,L+1

L+1 L+1 (−1)k γth 1 − = (L + 1)! k!(L + 1 − k)! γ 0,L+1 k=0 L+1 1 γth = . (L + 1)! γ 0,L+1

Therefore, at high SNR, the outage probability can be approximated by L+1 L+2 γth 1 γth pout,L+1 = +O (L + 1)! γ 0,L+1 γ 0,L+1 L+1 γth 1 . (4.149) ≈ (L + 1)! γ 0,L+1 Similarly, with diﬀerent expected SNR on each link, the outage probability in (4.146) can be approximated at high SNR by [29] pout,L+1

L L+1 γth 1 ≈ . (L + 1)! γ k,L+1 k=0

(4.150)

182

4 Cooperative Transmission Schemes with Multiple Relays

Fig. 4.17 Illustration of a multi-hop cooperative network with diversity combining at all nodes.

Since this probability is conditioned on the fact that all intermediate relays successfully decode the message, the end-to-end outage probability must then take into consideration the outage probability at each relay , which is given by pout, = Pr (γ−1, < γth ) = 1 − exp(−

γth ), γ −1,

(4.151)

for = 1, . . . , L. The end-to-end outage probability is then given by pout = 1 −

L+1

(1 − pout, ).

(4.152)

=1

Notice that, if each intermediate relay can also combine signals transmitted from all of its upstream nodes, the end-to-end outage probability can be further improved. In this case, the outage probability at each intermediate relay can be computed similar to that in (4.146) and (4.150). That is, by assuming that all upstream nodes have successfully decoded the message, the outage probability at node can be approximated at high SNR by −1 −1 1 γth pout, = Pr γk, < γth ≈ , (4.153)

! γ k, k=0

k=0

for = 1, . . . , L + 1. By substituting this into (4.152), we can then obtain the expression for the end-to-end outage probability with diversity combining at all nodes.

Multihop Transmissions with AF Relays Although most conventional multi-hop systems adopt DF relaying techniques, the multi-hop transmission can be achieved similarly by using AF relaying

4.6 Other Topics on Multi-Relay Cooperative Communications

183

techniques. Speciﬁcally, consider an AF multi-hop network, similar to that shown in Fig. 4.15, where each intermediate node ampliﬁes and forwards its received signal to its downstream nodes. We ﬁrst examine the case without diversity combining, where each node receives only from its immediate upstream node. Following the signal model given in Section 3.2, the signal received at node can be expressed as y = h−1, P−1 x−1 + w , (4.154) where x−1 = β−1 y−1

(4.155)

is the signal transmitted by node − 1 and w is the noise at node . The amplifying gain imposed by relay − 1 is given by β−1 =

1 1 . = 2 E[|y−1 |2 ] P−2 |h−2,−1 |2 + σw

By substituting the expressions of {yk }−1 k=1 in (4.154) and (4.155), we can obtain an equivalent expression for y as y = : h0, P0 xs + w : , where : h0, = h0,1

√ −1 hk,k+1 Pk βk hk,k+1 Pk = h0,1 2 Pk−1 |hk−1,k |2 + σw k=1 k=1 −1

is the eﬀective multi-hop channel between 0 and , and −1 −1 hk,k+1 βk Pk wm w : = w + m=1

= w +

−1 m=1

k=m

−1

√ hk,k+1 Pk wm 2 Pk−1 |hk−1,k |2 + σw k=m

is the eﬀective noise at node . Therefore, the eﬀective signal power at the destination can be computed as |: h0,L+1 |2 P0 = |h0,1 |2 P0

L k=1

= |h0,1 |2 P0

L k=1

|hk,k+1 |2 Pk 2 Pk−1 |hk−1,k |2 + σw γk,k+1 , γk−1,k + 1

(4.156)

184

4 Cooperative Transmission Schemes with Multiple Relays

2 where γk,k+1 = |hk,k+1 |2 Pk /σw is the SNR of the link between nodes k and k + 1. Moreover, the eﬀective noise variance at the destination can be computed as L L 2 P |h | k k,k+1 2 2 σw L+1 = σw 1 + 2 P |h |2 + σw m=1 k=m k−1 k−1,k L L γk,k+1 2 . (4.157) 1+ = σw γ +1 k−1,k m=1 k=m

By (4.156) and (4.157), the eﬀective SNR of the multi-hop channel between source and destination is given by γ0,L+1

> γk,k+1 γ0,1 L h0,L+1 |2 P0 |: k=1 γk−1,k +1 = = >L 2 γk,k+1 σw L+1 1+ L m=1 k=m γk−1,k +1 >L k=0 γk,k+1 ( ) = >L L+1 >m−1 −1,k + 1) (γ γ k k,k+1 m=1 k =1 k=m L+1 −1 m−1 1 1 = 1+ γ γk−1,k m=1 m−1,m k=1 L+1 −1 1 1+ −1 . = γk−1,k

(4.158)

(4.159)

(4.160)

(4.161)

k=1

To evaluate the outage probability of the AF multi-hop transmission scheme, we must ﬁrst obtain the distribution of the eﬀective SNR γ0,L+1 . Unfortunately, this is not easy to derive in closed form. However, at high SNR, we can approximate the eﬀective SNR as γ0,L+1 ≈

L+1 k=1

1 γk−1,k

−1 γapprox .

(4.162)

Given the SNR threshold γth , the end-to-end outage probability is given by [10] + * −1 −1 pout ≈ Pr (γapprox < γth ) = Pr γapprox > γth M1/γapprox (−s) −1 , (4.163) = 1−L s 1/γth

where L(·) is the inverse Laplace transform and M1/γapprox (s) = E[e−s/γapprox ] is the moment generating function (MGF) of 1/γapprox . By (4.162), the MGF of 1/γapprox is a product of the MGFs of 1/γ0,1 ,· · · , 1/γL,L+1. That is, for

4.6 Other Topics on Multi-Relay Cooperative Communications

185

$)6\VWHP ')6\VWHP

Fig. 4.18 Outage probability versus number of hops for AF and DF multi-hop systems c without diversity combining. (From Hasna and Alouini with modiﬁed labels; [2003] IEEE.)

Rayleigh fading channels, we have M1/γapprox (s) =

L+1

M1/γk−1,k (s),

(4.164)

k=1

'

where M1/γk−1,k (s) =

4s γ k−1,k

' K1

4s γ k−1,k

(4.165)

the MGF of 1/γk−1,k and K1 (·) is the ﬁrst order modiﬁed Bessel function of the second kind. For Nakagami fading channels with parameter mn , the MGF of 1/γk−1,k is given by mn /2 mn s mn s 2 (4.166) K mn 2 M1/γk−1,k (s) = Γ(mn ) γ k−1,k γ k−1,k where Kmn (·) is the mn -th order modiﬁed Bessel function of the second kind. In Fig. 4.18, the outage probability is shown for both DF and AF multi-hop transmission schemes without diversity combining. In the simulations, each 2 hop is assumed to have equal SNR, i.e., γ k−1,k = Pk−1 ηk−1,k /σw = γ, ∀k,

186

4 Cooperative Transmission Schemes with Multiple Relays

and the ratio between the SNR on each hop and the SNR threshold is chosen such that γ/γth = 25 dB. We can observe that DF outperforms AF with an increasing gap between the two curves as the number of hops increases. In AF multi-hop relay systems, the signals transmitted by each node on the multi-hop route can also be combined at the destination to enhance the receive SNR. By considering a multi-hop network of L intermediate relays, the destination will receive a total of L + 1 copies of the message, including that transmitted by the source. The signal received at the destination from node can be written as () () () yL+1 = : h0,L+1 P0 xs + w :L+1 , where () : h0,L+1 = h0,1

−1

βk hk,k+1 Pk

β h,L+1

P

k=1

√ h,L+1 hk,k+1 Pk = h0,1 2 h,+1 Pk−1 |hk−1,k |2 + σw k=1 is the eﬀective channel between source and destination in the -th time slot and h,L+1 () () w :L+1 = wL+1 + hk,k+1 βk Pk wm h,+1 m=1 k=m √ hk,k+1 Pk h,L+1 () = wL+1 + wm . 2 2 Pk−1 |hk−1,k | + σw h,+1 m=1 k=m The signal received at the destination over L + 1 time slots can be collected into the vector ⎡ (0) ⎤ ⎡ (0) ⎤ ⎡ (0) ⎤ : h0,L+1 yL+1 w :L+1 ⎢ : (1) ⎥ ⎢ (1) ⎥ ⎢ (1) ⎥ ⎢ h0,L+1 ⎥ :L+1 ⎥ ⎢ yL+1 ⎥ ⎢w ⎢ ⎢ ⎥ ⎥ ⎥ (4.167) yL+1 = ⎢ ⎢ .. ⎥ = ⎢ .. ⎥ P0 xs + ⎢ .. ⎥ ⎣ . ⎦ ⎣ . ⎦ ⎣ . ⎦ (L)

yL+1

(L) : h0,L+1

= hxs + w.

(L)

w :L+1 (4.168)

By employing MRC at the destination, the eﬀective SNR at the destination can be computed as γeﬀ = P0 hH R−1 w h, where Rw is the covariance matrix of the eﬀective noise vector w. Since L + 1 time slots are required for a network with L relays, the average end-to-end capacity is given by

4.6 Other Topics on Multi-Relay Cooperative Communications

187

Fig. 4.19 Outage probability for the AF cooperative multi-hop systems deploying diversity combining at the destination with various numbers of hops. (Figure regenerated from c Pavan Kumar, Bhattacharjee, Herhold and Fettweis; [2004] IEEE.)

CAF (h, Rw ) =

+ * 1 log2 det I + (hhH )R−1 w . L+1

(4.169)

Given the transmission rate R, the outage probability can be evaluated by pout = Pr (CAF (h, Rw ) < R). In Fig. 4.19 (i.e. [18, Fig. 4]), the outage probability of AF multi-hop transmissions (using diversity combining at the destination) is shown for systems with diﬀerent numbers of hops. The end-to-end transmission rate is ﬁxed as R = 1 bit/sec/Hz and the channel gains are assumed to be Rayleigh distributed with mean proportional to the path loss. We assume that the sourcedestination distance ds,d is ﬁxed and, thus, the distance between neighboring hops decreases as ds,d /(L + 1). As the number of hops increases, the rate achievable on each hop increases due to the reduced path loss and the destination will be able to receive more copies of the signal. However, the average end-to-end rate may decrease due to the increased number of time slots required to traverse the entire path.

4.6.2 Asynchronous Cooperative Transmissions In the cooperative strategies described in previous sections, we assumed that the relays are perfectly synchronized so that their transmission arrive either

188

4 Cooperative Transmission Schemes with Multiple Relays

concurrently (e.g., in beamforming and DSTC schemes) or perfectly separated in time (e.g., in orthogonal or multi-hop transmissions) at the destination. While the performance achieved under these scenarios provide great insight on the gain achievable with cooperation, they are often diﬃcult to realize in practice. Due to these reasons, asynchronous cooperation schemes have also received much attention in recent years, e.g., in [11,28,34]. As an example, we introduce in this subsection the opportunistic large arrays (OLA) system, where relays receive and retransmit messages in an asynchronous fashion whenever they are conﬁdent that the forwarded symbols or messages are of suﬃcient quality (based either on an SNR threshold or a CRC test). The source’s message propagates through the network like the chanting of ola in a sports stadium. The participants do not follow any central control but respond based only on their local observations of the signal in the environment. This scheme avoids the need for strict synchronization or coordination among cooperating terminals. Consider a network that consists of a source s, L relays (labeled 1 to L), and a destination d. To characterize the asynchronous behavior, we model the source signal as a continuous-time function x(t), which can be either an uncoded symbol or a complete codeword. After the source transmits x(t), whomever receives a signal with suﬃcient quality (e.g., with suﬃciently high SNR) will immediately retransmit the signal to other nodes. Depending on whether DF or AF strategies are adopted, the signal forwarded by the intermediate nodes can be either a regenerated signal, say x (t) = x(t) for relay

, or an ampliﬁed version of its received signal, say x (t) = β y (t). The relays that are not able to receive with suﬃciently high SNR will continue to receive from other intermediate nodes until the accumulated SNR exceeds the desired threshold. Consequently, the nodes transmit at random diﬀerent instants and the signal received at relay can be written as the accumulation of signals transmitted by all other nodes. The received signal is given by y (t) = hk, x (t − τk, ) + w (t), (4.170) k∈D

where hk, is the channel coeﬃcient between node k and node , τk, is the arrival time of the signal transmitted by node k, and w (t) is the additive white Gaussian noise at node with correlation function Rw (τ ) = E[w (t + τ )w∗ (t)] = N0 δ(τ ). The set D is the set of nodes that eventually are able to participate in the cooperative transmission. We assume that the channel is quasi-static and ﬂat fading so that the channel coeﬃcients remain constant over the transmission time of the signal x(t). Interestingly, since all nodes are forwarding the same signal x(t), the signal received at relay (e.g., in (4.170)) can be treated as an equivalent multipath signal, i.e., y (t) = : hs, (t) ∗ x(t) + w : (t), (4.171)

4.6 Other Topics on Multi-Relay Cooperative Communications

189

firing instant received signal

rest phase

receive phase

delay spread time signal spatial propagation

xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx

space downstream

upstream

Fig. 4.20 Illustration of the integrate-and-ﬁre model that describes the local node operc ations in OLA networks. (From Scaglione and Hong; [2003] IEEE.)

where : hs, (t) is the eﬀective multipath channel and w : (t) is the eﬀective noise. For example, in the DF case, we have : hs, (t) = hk, δ(t − τk, ) (4.172) k∈D

and w : (t) = w (t);

(4.173)

whereas, in the AF case, we have : hs, (t) equal to the product of channel coeﬃcients and amplifying gains of all upstream nodes, and w : (t) is noise accumulated from all upstream nodes. When linear modulation is employed, the signal x(t) can be expressed as a sequence of symbol waveforms given by x(t) =

∞

In gtx (t − nTs ),

n=−∞

where In = an + jbn is the complex data symbol and gtx (t) is the transmit ﬁlter. In this case, the eﬀective multipath channel generated by the intermediate relays will result in frequency selectivity and inter-symbol interference that can be exploited either by multi-carrier modulation or standard equalization techniques. The operations at each relay can be described by an integrate-and-ﬁre model as illustrated in Fig. 4.20, where the solid curve represents the aggregate signal received by the relay. Each relay observes the signal received from its upstream nodes until the accumulated signal yields suﬃcient SNR to perform reliable detection. The relay then utilizes standard equalization or RAKE receivers to exploit the multipath signal and retransmit the signal x(t) to other nodes in the network. Once the relay transmits (or “ﬁres”), it will shut down its receiver and remain silent for the remainder of the time

190

4 Cooperative Transmission Schemes with Multiple Relays

Fig. 4.21 Fractional spaced equalizer for asynchronous cooperative systems. (Modiﬁed c from [34] [2006] IEEE.

needed for x(t) to be propagated to the destination. The rest phase is used to avoid having the same signal x(t) being received again at the same node. Speciﬁcally, a fractionally-spaced decision-feedback equalizer, as shown in Fig. 4.21, has been proposed in [34] to combine the signals received from multiple relay paths and resolve the intersymbol interference that may arise in (2) high data-rate transmissions. Here, the eﬀective multipath signal yd received from the asynchronous relays is ﬁrst passed through a T /2-fractionally spaced (1) equalizer and then combined with the signal yd received on the direct path. The diversity gains can be increased if the signals transmitted by diﬀerent relays can be made more resolvable. In fact, this can be achieved artiﬁcially by employing random delays (or locally coordinated delays) at the relays. Readers are referred to [34] for further details. An alternative approach to exploit frequency selectivity is the use of multicarrier modulations. The advantage of this technique is that it can be implemented on top of many cooperative strategies proposed in this section, such as transmit beamforming, selective relaying, or DSTC etc. These issues will be discussed in detail in Chapter 7. Issues regarding asynchronous relays in DSTC systems can also be found in [20–22, 32, 38]. Further studies on the diversity-multiplexing tradeoﬀ of asynchronous cooperative systems can be found in [33] and issues regarding channel and delay estimation for asynchronous cooperation are given in [30].

References 1. Abdallah, M.M., Papadopoulos, H.C.: Beamforming algorithms for information relaying in wireless sensor networks. IEEE Transactions on Signal Processing 56(10),

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4772–4784 (2008) 2. Anghel, P., Leus, G., Kaveh, M.: Distributed space-time cooperative systems with regenerative relays. IEEE Transactions on Wireless Communications 5(11), 3130–3141 (2006) 3. Bletsas, A., Khisti, A., Reed, D.P., Lippman, A.: A simple cooperative diversity method based on network path selection. IEEE Journal on Selected Areas in Communications 24(3), 659–672 (2006) 4. Bletsas, A., Khisti, A., Win, M.Z.: Opportunistic cooperative diversity with feedback and cheap radios. IEEE Transactions on Wireless Communications 7(5), 1823–1827 (2008) 5. Bletsas, A., Shin, H., Win, M.Z.: Outage-optimal cooperative communications with regenerative relays. In: Proceedings of the Conference on Information Sciences and Systems (CISS) (2006) 6. Bletsas, A., Shin, H., Win, M.Z., Lippman, A.: Cooperative diversity with opportunistic relaying. In: Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC) (2006) 7. Fertl, P., Hottinen, A., Matz, G.: Perturbation-based distributed beamforming for wireless relay networks. In: Proceedings of the IEEE GLOBECOM, pp. 1–5 (2008) 8. Gao, F., Cui, T., Nallanathan, A.: On channel estimation and optimal training design for amplify and forward relay networks. IEEE Transactions on Wireless Communications 7(5), 1907–1916 (2008) 9. Gao, F., Cui, T., Nallanathan, A.: Optimal training design for channel estimation in decode-and-forward relay networks with individual and total power constraints. IEEE Transactions on Signal Processing 56(12), 5937–5949 (2008) 10. Hasna, M.O., Alouini, M.-S.: Outage probability of multihop transmission over Nakagami fading channels. IEEE Communications Letters 7(5), 216–218 (2003) 11. Hong, Y.-W., Scaglione, A.: Energy-eﬃcient broadcasting with cooperative transmissions in wireless sensor networks. IEEE Transactions on Wireless Communications 5(10), 2844–2855 (2006) 12. Hassibi, B., Vikalo, H.: On the sphere decoding algorithm: Part I, the expected complexity. IEEE Transactions on Signal Processing 53(8), 2806–2818 (2005) 13. Jing, Y., Hassibi, B.: Wireless networks, diversity and space-time codes. In: Proceedings of IEEE Information Theory Workshop, pp. 463–468 (2004) 14. Jing, Y., Hassibi, B.: Distributed space-time coding in wireless relay networks. IEEE Transactions on Wireless Communications 5(12), 3524–3536 (2006) 15. Jing, Y., Jafarkhani, H.: Distributed diﬀerential space-time coding for wireless relay networks. IEEE Transactions on Communications 56(7), 1092–1100 (2008) 16. Kim, J.-B., Kim, D.: Cooperative system with distributed beamforming and its outage probability. In: Proceedings of the IEEE 65th Vehicular Technology Conference, pp. 1638–1641 (2007) 17. Koyuncu, E., Jing, Y., Jafarkhani, H.: Distributed beamforming in wireless relay networks with quantized feedback. IEEE Journal on Selected Areas in Communications 26(8), 1429–1439 (2008) 18. Pavan Kumar, M.S., Bhattacharjee, R., Herhold, P., Fettweis, G.: Cooperative multihop relaying over fading channels. In: International Conference on Signal Processing and Communications (SPCOM), pp. 250– 254 (2004) 19. Laneman, J.N., Wornell, G.W.: Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks. IEEE Transactions on Information Theory 49(10), 2415–2425 (2003) 20. Li, Y., Xia, X.-G.: A family of distributed space-time trellis codes with asynchronous cooperative diversity. IEEE Transactions on Communications 55(4), 790–800 (2007) 21. Li, Y., Zhang, W., Xia, X.-G.: Distributive high-rate space-frequency codes achieving full cooperative and multipath diversities for asynchronous cooperative communications. IEEE Transactions on Vehicular Technology 58(1), 207–217 (2009)

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22. Li, Z., Xia, X.-G.: A simple Alamouti space-time transmission scheme for asynchronous cooperative systems. IEEE Signal Processing Letters 14(11), 804–807 (2007) 23. Luo, L., Blum, R.S., Cimini, L.J., Greenstein, L.J., Haimovich, A.M.: Decode-andforward cooperative diversity with power allocation in wireless networks. (There’re two paper with the identical names, conference and journal versions: In: Proceedings of the IEEE GLOBECOM, 5, pp. 3048-3052 (2005) IEEE Transactions on Wireless Communications 6(3), 793-799 (2007) ) 24. Madan, R., Mehta, N., Molisch, A., Zhang, J.: Energy-eﬃcient cooperative relaying over fading channels with simple relay selection. IEEE Transactions on Wireless Communications 7(8), 3013–3025 (2008) 25. Mudumbai, R., Hespanha, J., Madhow, U., Barriac, G.: Distributed transmit beamforming using feedback control. IEEE Transactions on Information Theory 56(1), 411–426 (2010) 26. Mudumbai, R., Brown III, D.R., Madhow, U., Poor, H.V.: Distributed transmit beamforming: challenges and recent progress. IEEE Communications Magazine 47(2), 102– 110 (2009) 27. Pun, M.-O., Brown III, D.R., Poor, H.V.: Opportunistic collaborative beamforming with one-bit feedback. In: Proceedings of the IEEE 9th Workshop on Signal Processing Advances in Wireless Communications (SPAWC) (2008) 28. Scaglione, A., Hong, Y.-W.: Opportunistic large arrays: Cooperative transmission in wireless multihop ad hoc networks to reach far distances. IEEE Transactions on Signal Processing 51(8), 2082–2092 (2003) 29. Si, J., Li, Z., Liu, Z., Lu, X.: Joint route and power allocation in cooperative-multihop networks. In: Proceedings of the IEEE international conference on circuits and systems for communications, pp. 114–118 (2008) 30. Tourki, K., Deneire, L.: Channel and delay estimation algorithm for asynchronous cooperative diversity. Wireless Personal Comunications 37, 361–369 (2006) 31. Vandenberghe, L., Boyd, S.: Semideﬁnite programming. SIAM review 38(1), 49–95 (1996) 32. Wang, D., Fu, S.: Asynchronous cooperative communications with STBC coded single carrier block transmission. In: Proceedings of IEEE Global Telecommunications Conference (GLOBECOM), pp. 2987–2991 (2007) 33. Wei, S.: Diversity-multiplexing tradeoﬀ of asynchronous cooperative diversity in wireless networks. IEEE Transactions on Information Theory 53(11), 4150–4172 (2007) 34. Wei, S., Goeckel, D.L., Valenti, M.: Asynchronous cooperative diversity. IEEE Transactions on Wireless Communications 5(6), 1547–1557 (2006) 35. Yi, Z., Kim, I.-M.: Joint optimization of relay-precoders and decoders with partial channel side information in cooperative networks. IEEE Journal on Selected Areas in Communications 25(2), 447–458 (2007) 36. Yiu, S., Schober, R., Lampe, L.: Distributed space-time block coding. IEEE Transactions on Communications 54(7), 1195–1206 (2006) 37. Yiu, S., Schober, R., Lampe, L.: Decentralized distributed space-time trellis coding. IEEE Transactions on Wireless Communications 6(11), 3985–3993 (2007) 38. Yu, Q., Zheng, J., Fu, T., Wu, K., Zhang, B.: Asynchronous cooperative transmission using distributed unitary space-frequency coded OFDM in mobile ad hoc networks. In: IEEE Future Generation Communication and Networking, vol. 2, pp. 291–296 (2007) 39. Zhao, Q., Tong, L.: Opportunistic carrier sensing for energy-eﬃcient information retrieval in sensor networks. EURASIP Journal on Wireless Communications and Networking 2005(2), 231–241 (2005) 40. Zhao, Y., Adve, R., Lim, T.: Beamforming with limited feedback in amplify-andforward cooperative networks. In: Proceedings of IEEE Global Telecommunications Conference (GLOBECOM), pp. 3457–3461 (2007) 41. Zhao, Y., Adve, R., Lim, T.J.: Improving amplify-and-forward relay networks: optimal power allocation versus selection. IEEE Transactions on Wireless Communications 6(8), 3114–3123 (2007)

Chapter 5

Fundamental Limits of Cooperative and Relay Networks

In this chapter, the fundamental limits of relay networks are introduced under diﬀerent channel settings, including Gaussian channels and wireless fading channels. For each scenario, we will study the information-theoretic capacities, diversity-multiplexing tradeoﬀs, and scaling laws of large networks accordingly. In Section 5.1, we ﬁrst examine the case of the single-relay Gaussian channel. When the relay is full-duplex, it can be shown that decodeand-forward (DF) and compress-and-forward (CF) schemes achieve capacity under certain scenarios; however, such results are not enjoyed in the more practical half-duplex relay network. In Section 5.2, we take the channel fading into consideration and discuss the fundamental limits under fast and slow fading scenarios. It will be shown that the DF protocol achieves ergodic capacity in fast fading channel while the dynamic decode-and-forward (DDF) protocol achieves the optimal diversity-multiplexing tradeoﬀ (DMT) when the multiplexing gain is smaller than one-half. In the ﬁnal section, we extend the information-theoretical results to networks with more than one relay. It can be shown that amplify-and-Forward (AF) is close to optimal (i.e., near capacity-achieving) in Gaussian channels when the number of relays is large. The DF and DDF protocols are also optimal for fast and slow fading channels under certain circumstances. Many of the results for multi-relay systems follow from the properties of the relay channel as well as that of conventional MIMO channels.

5.1 Gaussian Relay Channels The basic element of a cooperative system is a single relay system with one source, one relay, and one destination. This model was ﬁrst introduced in the pioneering work by van der Meulen in [7]. In this section, we will focus on the Gaussian single-relay channel where the relay and destination suﬀer from Gaussian noise. Although the general characterization of the relay channel

Y.-W. Peter Hong et al., Cooperative Communications and Networking, DOI 10.1007/978-1-4419-7194-4_5, © Springer Science+Business Media, LLC 2010

193

194

5 Fundamental Limits of Cooperative and Relay Networks

Fig. 5.1 General single relay channel.

capacity is not known up to this day, meaningful achievable regions have been derived as we describe in the following. We will also show some special cases of which the capacities are known.

5.1.1 Cut-Set Bound of Gaussian Relay Channels We ﬁrst introduce the model for general single relay channels of which Gaussian relay channels are included. Consider the relay channel illustrated in Figure 5.1 where there is one source, one destination, and a single intermediate node that relays information from the source to the destination. The channel input Xns = [Xs [1], Xs [2], . . . , Xs [n]] of length n is selected according to the message u, and satisﬁes the power constraint 1 ( n E Xs n

2

)

2 ) 1 ( = E Xs [i] ≤ Ps . n i=1 n

Let Yrn = [Yr [1], . . . , Yr [n]] be the observations made by the relay and let Xnr = [Xr [1], . . . , Xr [n]] be the signals produced by the relay based on these observations. The relay signals are functions of the past local observations, that is, Xr [i] = fr,i (Yr [i − 1], Yr [i − 2], . . . , Yr [1]), and satisfy the power constraint as 1 ( n E Xr n i=1 n

2

) =

n 2 ) 1 ( E Xr [i] ≤ Pr . n i=1

Let xs , xr , yr , and yd be the realizations of Xs , Xr , Yr and Yd , respectively. The memoryless relay channel is speciﬁed by p(yr , yd |xs , xr ), the conditional probability that yr and ys are observed given that xs and xr were transmitted. The considered channel is memoryless in the sense that (yr [i], yd [i]) depends on the past channel inputs {(xs [j], xr [j])}ij=1 only through (xs [i], xr [i]). That is, for any message u ∈ U, where U is the set of possible messages, any choice of codewords xns = [xs [1], · · · , xs [n]], and any relay code functions {fr,i }ni=1 ,

5.1 Gaussian Relay Channels

195

we have the joint probability mass function * + * + p yr [i], yd [i]u, xis , xir , yri−1 , ydi−1 = p yr [i], yd [i]xs [i], xr [i] , where xis = [xs [1], xs [2], · · · , xs [i]], xir = [xr [1], · · · , xr [i]], yri−1 = [yr [1], · · · , yr [i − 1]], and ydi−1 = [yd [1], · · · , yd [i − 1]]. Besides, there are two kinds of relay devices, that is, the full-duplex and half-duplex relays. The full-duplex relays are those that can transmit and receive at the same time and in the same frequency band, while the half-duplex relays cannot transmit and receive at the same time. The relays might be fullduplex if each relay has two antennas: one receiving and one transmitting. However, due to practical considerations such as the dynamic range of incoming and outgoing signals and the bulk of ferroelectric components like circulators, most cooperative protocols proposed in the literature (and also in Chapters 3 and 4) impose the half-duplex constraint on the relays. For half-duplex relays, this additional constraint is yr [i] = 0 if xr [i] = 0 ∀i.

(5.1)

Based on the observations, the destination computes an estimate of the message u ˆ. Let U be the message set and |U| is the cardinality of U, the rate of a length n code log2 |U| R= n is said to be achievable if for any > 0, there exists suﬃciently large n such that Pr(ˆ u = u) < . The capacity C is the supremum of the set of achievable rates. In this section, we will focus on Gaussian relay channels. Speciﬁcally, as the source broadcast its symbol through the Gaussian relay channel, the signal received at the relay and the destination can be written as Yr [i] = Xs [i] + Wr [i]

(5.2)

Yd [i] = Xs [i] + Xr [i] + Wd [i],

(5.3)

and respectively, where Wrn = [Wr [1], · · · , Wr [n]] consists of i.i.d. zero-mean Gaussian random variables with variance σr2 , i.e., Wr [i] ∼ N (0, σr2 ), and Wdn = [Wd [1], · · · , Wd [n]] consists of i.i.d. random variables with distribution N (0, σd2 ) and is independent of Wrn . It is easy to see that this channel belongs to the general relay channel described in above. Suppose that the relay is full-duplex, the cut-set bound is introduced for the full-duplex Gaussian relay channel to characterize a capacity upper bound.

196

5 Fundamental Limits of Cooperative and Relay Networks

Fig. 5.2 Relay channel with broadcast cut and multiple-access cut.

Theorem 5.1. The capacity of the Gaussian relay channel Cf with fullduplex relay is upper-bounded by Cf ≤

sup min{I(Xs , Xr ; Yd ), I(Xs ; Yr , Yd |Xr )},

(5.4)

p(xs ,xr )

where the supremum is over all jointly Gaussian distributions p(xs , xr ) that satisfy the power constraints Ps and Pr , respectively. This theorem comes from the concept of max-ﬂow min-cut. The information rate (ﬂow) across any boundary (cut) is less than the mutual information between the inputs on one side of the boundary and the outputs on the other side, conditioned on the inputs on the other side. From Figure 5.2, the term I(Xs , Xr ; Yd ) comes from the multiple-access cut and the term I(Xs ; Yr |Xr ) comes from the broadcast cut. And the information rate must be smaller than the ﬂow in the minimum cut. With proper modiﬁcations, this theorem can be easily applied to other channels, including the discrete memoryless channel (DMC). The details of the proof can be found in [3]. Extensions to networks with more than one relay nodes can be found in Section 5.3.1. As for the half-duplex Gaussian relay channel, without loss of generality, we assume that the relay is in the receive mode for a κn symbols and in the transmit mode for the rest. Let the superscripts “1” represent the parameters in the relay-receive mode, and the superscripts “2” represent the parameters in the relay-transmit mode. The capacity can be upper-bounded as given in the following theorem. Theorem 5.2. The capacity of the Gaussian relay channel Ch with halfduplex relay is upper-bounded by Ch ≤

sup

(1)

(2)

min{κI(Xs(1) ; Yd |Xr = 0) + (1 − κ)I(Xs(2) , Xr ; Yd ),

(1) (2) p(xs ,xs ,xr )

(1)

(2)

κI(Xs(1) ; Yr(1) , Yd |Xr = 0) + (1 − κ)I(Xs(2) ; Yd |Xr )}, (1)

(2)

where p(xs , xs , xr ) is the joint Gaussian distribution function that satisﬁes the power constraints Ps and Pr imposed on the source and relay, respectively. The proof of Theorem 5.2 is a simple extension of that of Theorem 5.1 with the half-duplex constraint in (5.1) [4].

5.1 Gaussian Relay Channels

197

5.1.2 Decode-and-Forward and Degraded Relay Channels The ﬁrst important achievable rate of relay channels is developed for the decode-and-forward protocol [2], which is tight for physically degraded discrete memoryless relay channels and some Gaussian relay channels with fullduplex relays. After decoding the message, the relay re-encodes the message and transmits a new codeword. For full-duplex relay, the achievable rate is Theorem 5.3. Suppose that the relay complies with the decode-and-forward protocol, the capacity of the full-duplex Gaussian relay channel is lowerbounded as Cf ≥

sup min{I(Xs , Xr ; Yd ), I(Xs ; Yr |Xr )},

(5.5)

p(xs ,xr )

where the supremum is taken over the joint Gaussian distribution function p(xs , xr ) that satisﬁes the power constraints Ps and Pr . Here we provide the basic concept of the proof. Suppose that the source uses block Markov coding scheme, that is, the codeword for block j depends on the current message uj and the previous message uj−1 . At the end of block j, when decoding message uj from the received signal at the relay yrn (j), the relay has past estimations of u1 , ..., uj−1 and the correct xnr (j − 1). The rate R must be less than I(Xs ; Yr |Xr ) to make decoding at relay correct. The relay encodes uj−1 into xnr (j) by using another codebook and retransmits it to the destination. With the information of correctly decoded xnr (j), the message uj−1 can be known at the destination. The channel is similar to a 2 × 1 MISO channel and the rate R must be less than I(Xs , Xr ; Yd ) to ensure that the destination can correctly decode the data. The source and the relay act as a collocated multiple antenna system. The details can be found in [3]. In the following, we will show that decode-and-forward is optimal for fullduplex degraded relay channel. To begin with, the degraded relay channel is deﬁned below. Deﬁnition 5.1 (Degraded Relay Channel). The relay channel is degraded if p(yr , yd |xs , xr ) = p(yr |xs , xr )p(yd |yr , xr ). (5.6) The degraded relay channel deﬁned above can be interpreted as the case where the destination receives a combination of the degraded version of the signal observed at the relay Yr , and the output of the relay Xr . More specifically, the relay channel is degraded if p(yd |xs , xr , yr ) = p(yd |xr , yr ), i.e., Xs → (Xr , Yr ) → Yd is a Markov chain. Due to this Markov chain, the upper bound in Theorem 5.1 and decode-and-forward lower bound in Theorem 5.3 coincide [2], then the capacity of the degraded relay channel is given in the following theorem.

198

5 Fundamental Limits of Cooperative and Relay Networks

Fig. 5.3 Degraded Gaussian relay channel.

Theorem 5.4. The capacity of the degraded Gaussian relay channel with fullduplex relay is Cdeg =

sup min{I(Xs , Xr ; Yd ), I(Xs ; Yr |Xr )}

(5.7)

p(xs ,xr )

where the supremum is taken over the joint Gaussian distribution p(xs , xr ) that satisﬁes the power constraints Ps and Pr . Let us take a closer look at the degraded Gaussian relay channel as illustrated in Figure 5.3. Speciﬁcally, the observations made by the relay and the destination are Yr [i] = Xs [i] + Wr [i] (5.8) and Yd [i] = Yr [i] + Xr [i] + Wd [i] = Xs [i] + Xr [i] + Wr [i] + Wd [i]

(5.9)

where Wrn = [Wr [1], . . . , Wr [n]] consists of i.i.d. zero-mean Gaussian random variables with variance σr2 , i.e., Wr [i] ∼ N (0, σr2 ), and Wdn = [Wd [1], . . . , Wd [n]] consists of i.i.d. random variables with distribution N (0, σd2 ) and is independent of Wrn . The formula in Theorem 5.4 can be explicitly expressed as the following. Theorem 5.5. The capacity of the real degraded Gaussian relay channel is √ 0

Ps +Pr +2β Ps Pr 1 (1−β 2 )Ps 1 Cdeg = max min log2 1+ , log2 1+ . 0≤β≤1 2 σr2 +σd2 2 σr2 Similarly to Theorem 5.3, the ﬁrst term is obtained by having the source and the relay transmit coherently over the Gaussian channel and the second term is to guarantee correct decoding of the source’s message at the relay. The parameter β can be treated as the correlation between Xs and Xr to specify the joint Gaussian distribution. According to the SNR of the source-relay link and relay-destination link, the results can be discussed by considering two diﬀerent cases. Case I:

For Ps /σr2 ≥ Pr /σd2 , we can show that

5.1 Gaussian Relay Channels

199

Fig. 5.4 General relay channel with feedback.

Ps + Pr + 2 (1 − β)Ps Pr βPs Ps Ps + Pr − ≤ 2 − 2 σr2 σr2 + σd2 σr σr + σd2 σ2

Ps (1 + σd2 ) Ps r ≤ 2 − = 0, σr σr2 + σd2 where the ﬁrst inequality results from the fact that β = 1 maximizes the left-hand-side of it. Hence, the capacity is given by Ps 1 Cdeg = log2 1 + 2 . 2 σr This is an improvement compared to the case with no relay, which yields Ps 1 Cno relay = log2 1 + 2 . 2 σr + σd2 Case II: For Ps /σr2 < Pr /σd2 , the capacity is maximized by choosing β ∗ such that Ps + Pr + 2 (1 − β)Ps Pr 1 βPs 1 = log2 1 + 2 . log2 1 + 2 σr2 + σd2 2 σr In this case, the capacity becomes Cdeg =

β ∗ Ps 1 log2 1 + 2 . 2 σr

Note that the capacity result in Theorem 5.4 can be easily applied to other degraded relay channels, with continuous or ﬁnite alphabets. The followings are two examples. Example 1: General Relay Channel with Feedback A general relay channel with feedback, as shown in Figure 5.4, can be considered as a special case of the degraded relay channel. More speciﬁcally, in this case, the relay observes (Yr , Yd ) which contains the observation Yd

200

5 Fundamental Limits of Cooperative and Relay Networks

made at the destination. Hence, by replacing Yr with (Yr , Yd ) in (5.7), we obtain the capacity of the general relay channel with feedback as sup min{I(Xs , Xr ; Yd ), I(Xs ; Yr , Yd |Xr )},

C=

(5.10)

p(xs ,xr )

where p(xs , xr ) is the joint distribution of Xs , Xr which satisﬁes the power constraints. Example 2: Reversely Degraded Relay Channel The reversely degraded relay channel presents the channel where the relay observes a degraded version of the observation made by the destination. The formal deﬁnition is given as the following. Deﬁnition 5.2 (Discrete Reversely Degraded Relay Channel). The relay channel deﬁned over ﬁnite alphabets (Xs × Xr , p(yr , yd |xs , xr ), Yr × Yd ) is reversely degraded if p(yr , yd |xs , xr ) = p(yd |xs , xr )p(yr |yd , xr ).

(5.11)

In this case, the relay will not be able to cooperate in forwarding the source’s message, but may transmit a symbol xr to facilitate the source’s transmission. Given that Xr [i] = xr , for all i, the capacity of the reversely degraded relay channel with discrete alphabet follows directly from the results of the pointto-point channel and is given by Crev = max max I(Xs ; Yd |xr ), xr ∈Xr p(xs )

where Xr is the ﬁnite alphabets of Xr and p(xs ) is the distribution of xs drawing from ﬁnite alphabets Xs . As for the half-duplex relay, the relay spends a fraction of κn symbol time to listen (Phase 1), and the rest to transmit (Phase 2). Moreover, the source can split the message into two parts to improve the performance, one part is transmitted directly to the destination without the help of the relay, and the other part is transmitted through the relay. This scheme is sometimes called as “partial decode-and-forward” since only part of the source message is decoded by the relay. The achievable rate is lower-bounded as follows. Theorem 5.6. The capacity of the Gaussian relay channel Ch with halfduplex relay is lower-bounded by Ch ≥

max

(1) (2) p(xs ,xs ,xr )

(1)

(2)

min{κI(Xs(1) ; Yd |Xr = 0) + (1 − κ)I(Xs(2) , Xr ; Yd ), (2)

κI(Xs(1) ; Yr(1) |Xr = 0) + (1 − κ)I(Xs(2) ; Yd |Xr )}, (5.12)

5.1 Gaussian Relay Channels (1)

201

(2)

where p(xs , xs , xr ) is the jointly Gaussian distribution which satisﬁes the power constraints Ps and Pr , respectively. On transmitting a message u, the message is split into two parts, say ur and ud , where the message ur is the message to be decoded by the relay (with rate Rr ), and ud is the one being direct transmitted to the destination (with rate Rd ), respectively. The overall rate is R = Rr + Rd . The basic concept is described as follows, while the details can be found in [4]. In Phase 1, the relay-receive mode, the source transmits ur for the relay to decode. And in Phase 2, the relay transmits ur while the source transmits ur and ud using the superposition coding. The channel at destination during Phase 2 is a two-user MAC channel. One virtual user has message ud , which is encoded (2),n in codeword xs , while the other user has message ur , which is encoded in n codeword xr . For message ud , if (2)

Rd < (1 − κ)I(Xs(2) ; Yd |Xr )

(5.13)

in Phase 2, it can be decoded correctly at the destination. Since ur is transmitted in Phase 1 (without cooperation) and Phase 2 (with cooperation), if (1) (2) Rr < κI(Xs(1) ; Yd |Xr = 0) + (1 − κ)I(Xr ; Yd ) (5.14) it can be decoded correctly at the destination. However, ur must be decoded correctly at the relay in Phase 1, which can be done if Rr < κI(Xs(1) ; Yr(1) |Xr = 0).

(5.15)

By adding (5.13) and (5.14) we get the ﬁrst term in the brackets in (5.12), while by adding (5.13) and (5.15) we get the second term in the brackets in (5.12), respectively.

5.1.3 Compress-and-Forward For the degraded relay channel, where the relay observes a signal with higher quality compared to the destination, the results given in Theorem 5.4 implies that the source and the relay should cooperate fully to transmit the message u. On the other hand, when the relay observes instead a degraded version (more noisy) of the signal observed at the destination (i.e., the reversely degraded relay channel), the relay should choose the best Xr to transmit in order to facilitate the source’s transmission. Both cooperation and facilitation yield achievable rate regions for the general relay channel but are attained by considering two rather extreme scenarios. However, even though the relay may observe a less informative signal, the observation may not be a degraded version of the output observed at the destination. In this case, an interme-

202

5 Fundamental Limits of Cooperative and Relay Networks

diate strategy often referred to as the compress-and-forward strategy [5] (or estimate-and-forward in [2]) which has been described brieﬂy in Chapter 3 and may be used to derive a better achievable rate in some cases. Speciﬁcally, in the case where the relay may not be able to completely decode the message transmitted by the source, full cooperation will not be possible. Yet, the observation Yr still contains useful information regarding the message u and should be forwarded to the destination. However, with limited channel capacity between the relay and the destination, the observation Yr cannot be forwarded with inﬁnite precision, but an estimate Yˆr may be transmitted instead. The achievable rate is given in the following theorem. Theorem 5.7. Suppose that the full-duplex relay complies with compressand-forward, the capacity is lower-bound as Cf >

sup

I(Xs ; Yd , Yˆr |Xr ),

p(xs ,xr ,yd ,yr ,ˆ yr )

and the lower bound is achievable subject to the constraint that I(Xr ; Yd ) ≥ I(Yr ; Yˆr |Xr , Yd ).

(5.16)

The supremum is taken over the joint Gaussian distribution of the form p(xs , xr , yd , yr , yˆr ) = p(xs )p(xr )p(yd , yr |xs , xr )p(ˆ yr |yr , xr ), which satisﬁes the power constraints. Theorem 5.7 also illustrates the multi-antenna reception behavior: the compress-and-forward rate I(Xs ; Yd , Yˆr |Xr ) can be interpreted as the rate for a 1×2 SIMO channel (one transmit antenna, two receive antennas) as long as the rate constraint is satisﬁed. The results obtained above can be interpreted as follows. Suppose that the destination is able to decode reliably the message transmitted by the relay and, thus, the relay’s transmitted signal Xr is assumed known at the destination. The signal Xr contains information regarding the estimate Yˆr . Suppose that Yd and Yˆr are observed at the destination, the rate achievable in the destination is given by I(Xs ; Yd , Yˆr |Xr ). The transmission is enabled by: 1) compressing Yr into the rate R0 = I(Yr ; Yˆr |Xr , Yd ) when Xr and Yd is available at the destination; 2) conveying the compressed information to the destination under the constraint that R0 does not exceed the mutual information between Xr and Yd , that is, R0 ≤ I(Xr ; Yd ). The compression can be done based on the results of source coding with side information [11]. While the source-destination pair incorporating a half-duplex relay that complies with the compress-and-forward protocol, similar to the decode-andforward protocol in Theorem 5.6, we can also combine the rate-splitting

5.2 Single-Relay Fading Channels

203

technique with the compress-and-forward protocol. One part of message is transmitted directly to the destination, and the other part is helped by the relay via the source coding with side information. The achievable rate of this two-phase transmission is quite complex and the readers can refer to [4] for the detail formulae. Note that unlike decode-and-forward, choosing jointly Gaussian distribution may not be the optimal distribution for the compressand-forward even in the Gaussian relay channel.

5.2 Single-Relay Fading Channels In wireless fading channels, with the help of a single relay, the received signals at the relay and the destination are given by Yd [i] = hs,d [i]Xs [i] + hr,d [i]Xr [i] + Wd [i],

(5.17)

Yr [i] = hs,r [i]Xs [i] + Wr [i], where hs,d [i] is the channel from source to destination, hr,d [i] is the channel from relay to destination, and hs,r [i] is the channel from source to relay at time slot i, respectively. The noise vectors Wdn = [Wd [1], · · · , Wd [n]] and Wrn = [Wr [1], · · · , Wr [n]] consist of i.i.d. circularly symmetric Gaussian random variables. Typically, there are two kinds of fading channels considered in the literature, the slow and fast fading channels. In the fast fading channels each channel gain is assumed to be an i.i.d. random process with respect to time index; while in the slow fading channels, the channel gains are random but ﬁxed within one codeword length. In both cases, the channel gains can be obtained perfectly at the receiver by assuming perfect channel estimation. However, due to the limited feedback channel bandwidth from the receiver to the transmitter, these channel coeﬃcients are hard to be known perfectly at the transmitter.

5.2.1 Ergodic Capacity In this section, we will discuss the fast fading relay channels with partial channel state information at the transmitter. Before that, let us focus on a simple case where channel gains remain the same during each codeword length, and the transmitter knows the channel gains perfectly. The results of this channel is very useful for understanding the capacity in the fast fading channel. By taking the distance between nodes into consideration, we set 1 hs,d = , dα s,d

204

5 Fundamental Limits of Cooperative and Relay Networks

where ds,d is the distance between source and distinction and α is an attenuation exponent of path loss eﬀect. Similarly, hs,r = 1/( dα s,r ) and α hr,d = 1/( dr,d ). For this channel, the result of section 5.1.1 can be directly applied as follows. Let the correlation coeﬃcient between Xs and Xr in Theorem 5.1 be β, the cut-set bound of the full-duplex relay channel is , 1 1 2 Cf ≤ max min log2 1 + Ps ( α 2 + α 2 )(1 − |β| ) , (5.18) 0≤β≤1 ds,r σr ds,d σd ? √ Ps Pr 2β Ps Pr . log2 1 + α 2 + α 2 + α/2 α/2 ds,d σd dr,d σd ds,d dr,d σd2 Similarly, the best decode-and-forward rate from Theorem 5.3 is , Ps Rdf = max min log2 1 + α 2 (1 − |β|2 ) , (5.19) 0≤β≤1 ds,r σr ? √ Ps Pr 2β Ps Pr . log2 1 + α 2 + α 2 + α/2 α/2 ds,d σd dr,d σd d d σ2 s,d

r,d

d

As for the compress-and-forward scheme, we use the sub-optimal setting by selecting all variables as Gaussian, and the Gaussian test channel [3] Yˆr = ˆ r where the compression error W ˆ r is assumed to be a zero-mean Yr + W ˆr and independent of all complex Gaussian random variable with variance N other random variables. The rate in Theorem 5.7 is then given by Ps Ps Rcf = log2 1 + (5.20) + α 2 , 2 ˆ ds,d σd dα s,r σr (1 + Nr ) with ˆr = N

Ps 2 dα s,r σr

+

Ps 2 dα s,d σd

Pr 2‘ dα r,d σd

+1 .

Now suppose the source, relay, and destination are aligned as in Figure 5.5, where ds,r = |d|, dr,d = |1 − d| and ds,d = 1. As the relay moves toward the source d → 0, Rdf coincides with the cut-set bound. The limiting capacity is √ Ps Pr 2 Ps Pr lim Cf = log2 1 + 2 + 2 + . d→0 σd σd σd2 While the relay moves toward the destination d → 1, Rcf coincides with the cut-set bound. The limiting capacity is 1 1 lim Cf = log2 1 + Ps . + 2 d→1 σr2 σd

5.2 Single-Relay Fading Channels

205

Fig. 5.5 Single relay on a line. (Figure regenerated and modiﬁed from Kramer, Gastpar, c and Gupta. 2005 IEEE.)

Now we consider the fast-fading channel with phase fading, that is, each channel gain has an random phase unknown to the transmitter as ejθs,d [i] hs,d [i] = , dα s,d where θs,d [i] is i.i.d in the time domain and uniformly distributed in [0, 2π). The hs,r and hr,d are deﬁned similarly. All random phases of the three nodes are independent of each other. Under the above channel setting, we have the following capacity result. Theorem 5.8. When the relay is close to the source such that Ps α ds,d σd2

+

Pr α dr,d σd2

≤

Ps , α ds,r σr2

(5.21)

the ergodic capacity for full-duplex relay channel is Ps Pr Cf = log2 1 + α 2 + α 2 . ds,d σd dr,d σd And the capacity is achieved by utilizing the decode-and-forward protocol. Proof. In fast-fading channel, the rate E[Rdf ] is achievable [5], where the expectation is over the random phases. This rate becomes

0 Ps max min log2 1 + α 2 (1 − |β|2 ) , R(β) , (5.22) 0≤β≤1 ds,r σr where R(β) = Eφs,d ,φr,d log2

√ Ps Pr 2Re(βej(φs,d −φr,d ) ) Ps Pr 1+ α 2 + α 2 + . α/2 α/2 ds,d σd dr,d σd ds,d dr,d σd2

From Jensen’s inequality [3],

206

5 Fundamental Limits of Cooperative and Relay Networks

Fig. 5.6 Rate comparisons of cooperative schemes in three-node fast fading relay channels. c (From Kramer, Gastpar, and Gupta with modiﬁed labels. 2005 IEEE.)

R(β) ≤ log2 E

√ Ps Pr 2Re(βej(φs,d −φr,d ) ) Ps Pr 1+ α 2 + α 2 + . α/2 α/2 ds,d σd dr,d σd ds,d dr,d σd2

It is easy to see that the right-hand-side of above equation equals to R(0), and R(β) ≤ R(0) (5.23) With (5.23) and (5.21), it is easy to check that the decode-and-forward achievable rate (5.22) is maximized at β = 0. From the same arguments, the cut-set bound is also maximized at β = 0 [5]. The capacity upper and lower bounds are (5.18) and (5.22) with β = 0 and they coincide. This concludes the proof. The ergodic capacity results can be extended to other fading processes when the channel phase uniformly distributed over time and [0, 2π), such as Rayleigh fading channels. Figure 5.6 plots the rates for the geometry of Figure 5.5 with α = 2 and Ps = Pr = 10. It also shows the rates of compress-andforward where the relay uses the same test channel as that for the no-fading case. The achievable rate using compress-and-forward is E[Rcf ] = Rcf as (5.20). Note that the compress-and-forward also matches the upper cut-set bound in some cases, and has good performance for all d. The “relay oﬀ” curve is the case where relay does not transmit and Pr = 0. The total power consumption is only half of that being used in other curves. However, the ergodic capacity results in Theorem 5.8 can not be extended to the half-duplex relay channels. Only some partial results can be obtained. Interested readers are referred to [4] [5] for these results.

5.2 Single-Relay Fading Channels

207

5.2.2 Diversity and Multiplexing Tradeoﬀs For the slow fading channel, or the quasi-static channel, the channel gains are ﬁxed during one codeword interval. When the channel gains are not known at the transmitter, the outage probability is a good performance metric. However, since relay channels can be treated as virtual MIMO channels, the more advanced diversity and multiplexing tradeoﬀ (DMT) developed originally for MIMO channels is worth to study. This tradeoﬀ studies the fundamental tradeoﬀ between the achievable diversity and multiplexing gains for a given number of spatial dimensions. The more relays participating in the cooperation, the more spatial dimensions can be exploited to enhance system performance. We will use the DMT as the performance metric here, with the background provided in Section 2.3. The performance results based on outage probabilities can be derived easily from the non-fading case (or slow-fading channels with full channel state information at the transmitter) provided in Section 5.2.1. Interested readers are referred Chapter 3 and [4] [5] for the details. Achieving the optimal DMT for the full-duplex relay channel is rather straightforward, for example, one can use the simple amplify-and-forward (AF) strategy in [1]. We then focus on the channel with a half-duplex relay. The half-duplex constraint is often the cause of spectral ineﬃciency in the cooperative protocol. The protocols designed based on half-duplex relays, such as the DF and AF protocols are initially proposed in [6] and further improved by [1]. In the DF cooperative network where the relay ﬁrst decodes the messages and then transmits a new codeword to the destination, the achievable DMT has been shown to outperform the one in the AF cooperative network. Indeed, the dynamic decode-and-forward (DDF) protocol achieves the optimal tradeoﬀ for a certain range of multiplexing gains. In the following, we will ﬁrst introduce the AF protocol and then the other complex DF protocols. Moreover, we study the DMT for those relay protocols. Let us review our channel model and introduce a general DMT upper bound. We consider the simple three-node cooperative network with a source, a relay, and a destination, each equipped with a single antenna as (5.17). The channels gains are assumed to be Exponentially distributed and known perfectly only at the corresponding receivers. For simplicity, the power constraints of source and relay are both set to P . For this channel, we have the following upper bound. Theorem 5.9. The optimal diversity gain for the three-node cooperative network (5.17) with a single relay is upper-bounded by d∗ (r) ≤ 2(1 − r), where 0 ≤ r ≤ 1 is the multiplexing gain.

208

5 Fundamental Limits of Cooperative and Relay Networks

The deﬁnitions of the diversity gain and multiplexing gain can be found in Section 2.3. This genie-aided upper bound is obtained when the relay is assumed to know the information message a priori, and the bound follows directly from [12].

DMT for AF Cooperative Networks We will introduce the nonorthogonal amplify-and-forward (NAF) protocol, which achieves the best DMT by the class of AF protocols. Under the halfduplex constraint, the AF cooperation must be divided into two phases. In Phase 1, the source ﬁrst transmits a codeword of length n to both the relay and the destination; while, in Phase 2, the relay ampliﬁes and forwards the message concurrently with the source in the following n − n symbol periods. From the channel model (5.17), the signal received at the destination over the two phases can be written as 0 hs,d A1 0 ¯s + yd = wr + wd , x (5.24) hr,d hs,r BA1 hs,d A2 hr,d B

¯ s ∈ Cn is the length-n complex source symbols, wr ∈ Cn is the where x AWGN at the relay in Phase 1, and wd ∈ Cn is the AWGN at the destination over both phases. Moreover, B ∈ C(n−n )×n is the linear AF matrix at the relay and A1 ∈ Cn ×n , A2 ∈ C(n−n )×(n−n ) are diagonal power allocation matrices at the source in Phase 1 and 2, respectively. These matrices are chosen to satisfy the average power constraint at the relay

|hs,r | P 2

n

|bji | |ai | + 2

i=1

2

σr2

n

|bji |2 ≤ P,

j = 1, . . . , n − n ,

(5.25)

i=1

where the (j, i)-th entry of matrix B is bji and A1 = diag(a1 , . . . , an ). The NAF protocol chooses n = n/2, A1 = I n2 × n2 , A2 = I n2 × n2 , and with b ≤

B = bI n2 × n2 P P |hs,r |2 +σr2 .

The protocol is nonorthogonal compared with the

Laneman-Tse-Wornell AF (LTW-AF) protocol proposed in [6] since the latter chooses A1 = I n2 × n2 , A2 = 0. LTW-AF is equivalent to the basic AF introduced in Section 3.2. And we will see that this relaxation of orthogonality improves the performance a lot. The upper bound of DMT for all AF networks was derived in [1] and is given as follows.

5.2 Single-Relay Fading Channels

209

Theorem 5.10. The optimal diversity gain for a cooperative network with a single AF relay is upper-bounded by d∗ (r) ≤ (1 − r) + (1 − 2r)+ , where function (x)+ = max{x, 0}. This upper bound can be achieved with the NAF protocol as follows. Theorem 5.11. The NAF protocol achieves the optimal DMT for the AF single-relay scenario as d∗ (r) = (1 − r) + (1 − 2r)+ . Proof. To derive the diversity order of the NAF protocol, we ﬁrst derive an asymptotic upper bound for the error probability that is averaged over the ensemble of Gaussian random codebooks by the ML decoder. It then follows that there must exists at least one codebook with error performance bounded by this average upper bound. And the diversity order achievable by this codebook is at least as good as the exponential order of the average upper bound. Suppose that the source uses a Gaussian codebook with codewords of length-n (n is taken to be even), and rate R = r log2 ρ, where ρ = P/σd2 is the SNR. We use the notation ·

f (ρ) ≤ g(ρ) to indicate that limρ→∞

log2 f (ρ) log2 g(ρ)

(5.26)

≤ 1, and the notation ·

f (ρ) = ρ−v

(5.27)

log2 f (ρ) = −v. log2 ρ The average error probability of the ML decoder, i.e., Pe (ρ), can be upper bounded as to represent the fact that lim

ρ→∞

Pe (ρ) = PO (R)Pe|O + Pe,Oc ≤ PO (R) + Pe,Oc ,

(5.28)

where O denotes the outage event. We will choose O to make Pe,Oc being dominated by PO (R). To derive the probability Pe,Oc , we ﬁrst focus on the conditional ML pairwise error probability given the channels, which can be bounded by [1] ·

Pe|hs,d ,hr,d ,hs,r ≤ ρ− 2

n

max{2(1−vs,d ),1−(vr,d +vs,r ),0}

· ρrn ,

210

5 Fundamental Limits of Cooperative and Relay Networks ·

for (vs,d , vr,d , vs,r ) ∈ R3+ , where ≤ is deﬁned in (5.26) and we rewrite channel gains with respect to the SNR as |hs,d |2 = ρ−vs,d , |hr,d |2 = ρ−vr,d , and |hs,r |2 = ρ−vs,r . Then the error probability averaged over the set of channel realizations that do not cause an outage (i.e., Oc ) is given by 4 n 5 · ρ− 2 max{2(1−vs,d ),1−(vr,d +vs,r ),0} · ρrn · ρ−vs,d −vr,d −vs,r Pe,Oc ≤ O c ∩R3+

× dvs,d dvr,d dvs,r ,

(5.29)

where the asymptotic PDF of the random variable vs,d is [1]

−∞ ρ , for vs,d < 0, · pvs,d = ρ−vs,d , for vs,d ≥ 0, ·

where = is deﬁned in (5.27). The PDF of vr,d and vs,r are similar to that of vs,d . Note that the right-hand-side of (5.29) is dominated by the term corresponding to the minimum error exponent, i.e., ·

Pe,Oc ≤ ρ−de (r) , where n [max{2(1−vs,d), 1−(vr,d +vs,r ), 0}−2r]+vs,d +vr,d +vs,r . (vs,d ,vr,d ,vs,r ) 2

de (r) =

inf

∈O c ∩R3+

Similarly, the outage probability PO (R) in (5.28) can be written asymptotically as · PO (R) = ρ−do (r) , where do (r) =

inf

(vs,d ,vr,d ,vs,r )∈O∩R3+

(vs,d + vr,d + vs,r ).

By choosing O = {(vs,d , vr,d , vs,r ) ∈ R3+ | max{2(1 − vs,d ), 1 − (vr,d + vs,r ), 0} ≤ 2r}, the exponent de (r) can be made arbitrarily large by choosing n as large as desired. Thus, the error probability in (5.28) is dominated by the outage probability PO (R) as ·

·

Pe (ρ) ≤ PO (R) + Pe,Oc ≤ PO (R) = ρ−do (r)

(5.30)

where do (r) can be shown to be (1 − r) + (1 − 2r)+ . And it concludes the proof.

5.2 Single-Relay Fading Channels

211

2

Non−coop NAF LTW−AF

1.8

Diversity Gain d(r)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.2

0.4 0.6 Multiplexing Gain r

0.8

1

Fig. 5.7 DMT comparisons of AF protocols in three-node cooperative channels. (Figure c regenerated and modiﬁed from Azarian, El Gamal, and Schniter. 2005 IEEE.)

In Figure 5.7, the DMT of the NAF protocol is compared with the AF protocol proposed by Laneman, Tse, and Wornell in [6] (that is, the LTW-AF protocol described previously) and the non-cooperative case (that is, direct transmission). The NAF protocol shows uniform dominance over the other schemes since it achieves the optimal DMT. Notice that, for multiplexing gains greater than 0.5, the DMT of the NAF scheme coincides with the non-cooperative scheme. This shows that the AF cooperative scheme cannot support multiplexing gain greater than 0.5 due to the half-duplex constraint.

DMT for DF Cooperative Networks We will introduce the dynamic decode and forward (DDF) protocol [1] and show that it achieves the optimal tradeoﬀ for multiplexing gains 0 ≤ r < 0.5. The DDF improves the LTW-DF scheme in [6], which achieve the DMT characterized by d(r) = 2(1 − 2r) (LTW-DF is the same as the basic DF scheme introduced in Section 3.1). Similarly, in the DDF scheme, the source transmits a codeword of length n and rate R. The relay ﬁrst listens to the source until the mutual information between the source message and the received symbols exceed nR. In this case, it has been shown in [1] that the relay will be able to decode the message with arbitrarily small error probability provided that n is suﬃciently large. Suppose that this occurs after n symbols have been received. The relay then decodes and re-encodes the message using an independent Gaussian

212

5 Fundamental Limits of Cooperative and Relay Networks

2

Genie Aided NAF LTW−DF DDF

1.8

Diversity Gain d(r)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.2

0.4 0.6 Multiplexing Gain r

0.8

1

Fig. 5.8 DMT comparisons of DF protocols in three-node cooperative channels. (Figure c regenerated and modiﬁed from Azarian, El Gamal, and Schniter. 2005 IEEE.)

codebook and transmits in the remaining n−n symbol intervals. The method is dynamic in the sense that the duration n for which the relay listens to the source depends on the instantaneous interuser channel condition. Let {Xs [i]}ni=1 and {Xr [i]}ni=n +1 be the signals transmitted by the source and the relay, respectively. The signals received at the destination can be written as

for i = 1, . . . , n hs,d Xs [i] + Wd [i], Yd [i] = (5.31) hs,d Xs [i] + hr,d Xr [i] + Wd [i], for i = n + 1, . . . , n. Since the Gaussian codebook is used, the number of symbol periods that the relay should listen to is A0

@ nR n = min n, . (5.32) log2 (1 + |hs,r |2 P/σr2 ) The achievable DMT with this protocol is given in the following theorem with the proof given in Appendix 5.1. Theorem 5.12. The DMT achieved with a single relay complying with DDF is given by

2(1 − r), if r ∈ [0, 0.5] d(r) = (5.33) (1 − r)/r, if r ∈ (0.5, 1]. In Figure 5.8, we plot the DMT curves for the DDF, the NAF, and the LTW-DF schemes. The DMT of the LTW-DF is 2(1 − 2r). The genie-aided

5.3 Multi-Relay Networks

213

Broadcast Cut

W1 Y1

Relay 1 Encoder

X1

Relay 2 Encoder

X2

W2 Y2 U

Encoder

Xs

WL YL

Wd Yd

Decoder

U

. . . . Relay L XL Encoder

Multiple Access Cut

Fig. 5.9 Gaussian relay channel with multiple relays.

scheme, where the relay is assumed to know a priori the message sent by the source, serves as an upper bound to the DMT of all schemes and is also plotted for comparison. One can see that the DDF scheme achieves the optimal DMT for 0 ≤ r ≤ 12 since it coincides with the DMT of the genieaided scheme. However, for 12 < r ≤ 1, a gap is observed between the curves of the DDF scheme and the genie-aided scheme. The optimal DMT for threenode networks in this region is still unknown. And the DDF performs better than the NAF. It results from the fact that AF forwards a noisy observation from the source while the DDF forwards a clean re-coded message.

5.3 Multi-Relay Networks Consider a large relay network consisting of one source, one destination, and L relay nodes as shown in Figure 5.9. Let u be the message transmitted by the source that is encoded into a length n codeword Xns = [Xs [1], Xs [2], . . . , Xs [n]], whose components are complex and satisﬁes the power constraint n 2 ) 1 ( E Xs [i] ≤ Ps . n i=1 At time i, relay observes Y [i] = hs, Xs [i] + W [i]

214

5 Fundamental Limits of Cooperative and Relay Networks

where Wn = [W [1], . . . , W [n]] consists of i.i.d. circularly symmetric Gaussian random variables of mean zero and variance σr2 . And the noises over diﬀerent relays are mutually independent. Based on the sequence of observations {Y [i]}ni=1 , relay generates the codeword Xn = [X [1], . . . , X [n]] in a causal fashion such that X [i] = f,i (Y [i − 1], Y [i − 2], . . . , Y [1]). The codewords satisfy the total power constraint as L n 2 ) 1 ( E X [i] ≤ Pr . n i=1 =1

The output of the relay channel, i.e., the signal received at the destination, is given by L Yd [i] = hs,d Xs [i] + h,d X [i] + Wd [i] =1

{Wd [i]}ni=1

where are i.i.d. circularly symmetric Gaussian random variables of mean zero and variance σd2 . In the section, the relays are assumed full-duplex. All channels are known perfectly at each node. Although there are small diﬀerences compared to the Gaussian relay channel described in Section 5.1 due to the additional channel gains, we still call this channel (with full channel state information at the transmitters) as the Gaussian multi-relay channel. The channels without full channel state information at the transmitters are considered in Section 5.3.3. Notice that the Gaussian multi-relay channel is not a degraded relay channel, thus the capacity is unknown for any ﬁnite value of the relay number L. However, meaningful upper and lower bounds can be obtained as shown in [8]. Furthermore, it was also shown in [8] that the upper and lower bounds may coincide for certain cases as L tends to inﬁnity, which yields the asymptotic capacity of the large Gaussian relay channel.

5.3.1 Upper Bound of Gaussian Multi-Relay Networks In Sections 5.3.1 and 5.3.2, we discuss the upper bound and lower bound of Gaussian relay networks with L relays. To focus on the discussion of the relay links, it is assumed that direct link is too weak to be considered. Further results on these bounds that take into account the direct link are developed in [8]. To derive the upper bound, the cut-set bound in Section 5.1.1 is generalized as the following. Corollary 5.1 (Cut-Set Bound [8]). Let Ri,j , for i, j ∈ {s, 1, 2, . . . , L, d}, be the achievable rate between sender i and receiver j. For any partitioning of the network into two sets S, S c ⊂ {s, 1, 2, . . . , L, d} such that S ∩ S c = ∅

5.3 Multi-Relay Networks

215

and S ∪ S c = {s, 1, 2, . . . , L, d}, the following inequality must be satisﬁed Ri,j ≤ max I(XS ; YS c |XS c ), pXS ,XS c

i∈S,j∈S c

where pXS ,XS c is the joint distribution of XS , XS c which satisﬁes the power constraints. Proof. Here we provide a simple “genie-aided arguments” to invoke the wellknown MIMO capacity result. Suppose that all the nodes in S can cooperate arbitrarily, and also the nodes in S c . The rate upper bound of this new cooperative system is an upper bound of the original one since the original one is merely one way to implement arbitrary cooperations. And the new cooperative system is simply a point-to-point MIMO channel whose rate upper bound is given by max I(XS ; YS c |XS c ). pXS ,XS c

And this concludes the proof. Two natural cuts can be considered as shown in Figure 5.9: one is the “broadcast cut”, where we take S = {s} and S c = {1, . . . , L, d}, and the other is the “multiple access cut”, where we take S = {s, 1, . . . , L} and S c = {d}. In the broadcast cut, the mutual information to be maximized is given by I(Xs ; Yd , Y1 , . . . , YL |X1 , . . . , XL ) = I(Xs ; Y˜d , Y1 , . . . , YL ) where Y˜d = Yd −

L

h,d X .

=1

In this case, the observations Y˜d , Y1 , . . . , YL can be viewed as the output of a single-input multiple-output (SIMO) channel and the capacity is given by [9] [10] L PS 2 ˜ CBC = max I(Xs ; Yd , Y1 , . . . , YL ) = log2 1 + 2 (5.34) |hs, | . pXs σr =1

where the maximum is over all input distribution to satisfy power constraints. And CBC serves as the broadcast cut-set bound. On the other hand, in the multiple access cut, the upper bound CMAC is obtained by maximizing the mutual information I(Xs , X1 , . . . , XL ; Y ) over all pXs ,X1 ,...,XL subject to the power constraints

216

5 Fundamental Limits of Cooperative and Relay Networks

2 ) 1 ( E Xs [i] ≤ Ps n i=1 n

and

L n 2 ) 1 ( E X [i] ≤ Pr . n i=1 =1

To simplify our problem, we relax the problem by considering instead the joint power constraint n L n 2 ) 2 ) 1 ( 1 ( E Xs [i] + E X [i] ≤ Ps + Pr . n i=1 n i=1 =1

The problem then reduces to a multiple-input single-output (MISO) channel with capacity CMAC , where the multiple-access cut-set bound CMAC satisﬁes L Ps + Pr 2 (5.35) |h,d | . CMAC = log2 1 + σd2 =1

Proposition 5.1 (Upper Bound). The capacity of a full-duplex Gaussian multi-relay channel as illustrated in Figure 5.9 is upper-bounded by Cf ≤ min(CBC , CMAC ),

(5.36)

where the broadcast cut-set bound CBC is given in (5.34) and the multipleaccess cut-set bound CMAC is given in (5.35).

5.3.2 Lower bound of Gaussian Multi-Relay Networks and Asymptotic Capacity Results To obtain a lower bound on the capacity, let us consider a particular case where the relays simply amplify and forward the signals received from the source to the destination, while satisfying the power constraints Ps and Pr . In this case, the channel between the source and the destination can be viewed as a point-to-point channel. Clearly, Cp < Cf , where Cp is the capacity of this point-to-point channel. Moreover, it is well-known that in the point-to-point channel, the separation theorem [3] holds, that is, R(Dachieve ) < Cp , where Dachieve is the minimum achievable distortion and R(·) is the ratedistortion function of the source with respect to the selected distortion measure. In this case, R(·) is the rate-distortion function of Gaussian source with mean-square error (MSE) distortion measure [3] and log2

Ps < Cp < Cf . Dachieve

(5.37)

5.3 Multi-Relay Networks

217

Now the goal is to ﬁnd Dachieve . Suppose that the source transmits an uncoding i.i.d. Gaussian sequence {Xs [i]} with variance Ps . The relay delays the received signal by one time unit in order to satisfy the causality constraint and scale it to their desired power level. The signal received at relay at time i is Y [i] = hs, Xs [i] + W [i] and the output at time i + 1 is ' X [i + 1] = e

jθ

P 2 |hs, | Ps

+ σr2

Y [i]

where P is the power emitted by relay and θ is the appropriately chosen phase rotation. In this section, we assume that the destination is not able to received signal directly transmitted by the source for simplicity. The signal received at the destination is Yd [i + 1] =

L

h,d X [i + 1] + Wd [i + 1]

(5.38)

=1

=

L

a (hs, Xs [i] + W [i]) + Wd [i + 1],

=1

'

where a = h,d e

jθ

P 2 |hs, | Ps

+ σr2

.

Suppose that a linear scale of the received signal, i.e., γYd [i + 1], is used to estimate the message Xs [i]. The mean square error can be written as E |Xs [i] − γYd [i + 1]|2 2 L L a hs, − γ a W [i] − γWd [i + 1] = E Xs [i] 1 − γ =1

=1

2 L L 2 2 2 2 = Ps 1 − γ a hs, + |γ| σd + σr |a | . =1

=1

By taking the derivative of the above with respect to γ and setting it to zero, the optimal value of γ can be obtained as ∗ L Ps h a s, =1 γopt = . 2 L L 2 2 2 Ps =1 hs, a + σd + σr =1 |a | The minimum achievable mean square error distortion is then given by

218

5 Fundamental Limits of Cooperative and Relay Networks

E |X[i] − γopt Y [i + 1]|2 L Ps σd2 + σr2 =1 |a |2 = L L Ps | =1 hs, a |2 + σd2 + σr2 =1 |a |2 Ps σr2

=

|hT a|2

a 2 +σd2 /σr2

Ps + σr2

where vectors h = [hs,1 , . . . , hs, ]T and a = [a1 , a2 , . . . , aL ]T are deﬁned to simplify notations. Now the goal is to choose a to further minimize the distortion. Following [5], by taking ' Pr ∗ a = h , B(L) s, where B(L) =

L =1

|hs, |2 (|hs, |2 Ps +σr2 ) , |h,d |2

Dachieve = L

we can achieve the distortion

( Ps σr2 +

B(L)σd2 2 Pr L =1 |hs, |

2 2 =1 |hs, | Ps + σr +

)

B(L)σd2 2 Pr L =1 |hs, |

.

Now we have the minimum distortion Dachieve . Then from (5.36) and (5.37), we know that R(Dachieve ) ≤ Cf ≤ min{CBC , CMAC }. With considering direct link, two capacity bounds can be obtained with similar derivation, and further results can be found in [8] for readers’ reference. In Fig. 5.10, we show CBC and R(Dachieve in terms of the number of relay nodes numerically. The value of CBC can be regarded as an upper bound of Cf , and one can observe the asymptotic diﬀerence between two bounds. In the example, the source and the destination are located at (-0.25, 0) and (0.25, 0), respectively, in a Cartesian coordinate system. Two capacity bounds in Fig. 5.10 is contributed by the relay links as well as direct link [8]. The relay nodes are uniformly distributed in a disk of unit area centered at the origin, except a dead zone which is a circle of radius 0.01 centered at the source. The channel coeﬃcient between nodes i and j is given by 1/dα i,j where di,j is the distance between the two nodes and α is the path loss exponent. The source transmission power is Ps = 10 and the relays follow total power constraint Pr = 10L. The noise variances are set as one. The bounds obtained from a single realization (dashed lines) and the average over 100 realization (solid lines) are both illustrated. The curves indicates the log2 L behavior of capacity and the diﬀerence between the upper and lower bounds converges as L goes to inﬁnity [8]. The convergence of the asymptotic diﬀerence is also found in the following theorem. Theorem 5.13. Given regularity conditions on the asymptotic properties of the relay power and channel parameters, it holds that

5.3 Multi-Relay Networks

219

Fig. 5.10 Numerical evaluation of upper and lower bounds of capacity versus number of c relay nodes. (From Gastpar and Vetterli with modiﬁed labels. 2005 IEEE.)

lim (CBC − R(Dachieve )) = ΔBC ,

L→∞

and lim (CMAC − R(Dachieve )) = ΔMAC ,

L→∞

where ΔBC , ΔMAC are constants that depend on the transmit power and channel parameters. The detail regularity conditions can be found in [5]. By showing that the asymptotic diﬀerence between the upper and lower bound converges to a constant, we can then obtain the asymptotic scaling behavior of Cf . The theorem implies that the capacity of large Gaussian relay channels scales with the minimum between the capacity of the broadcast and multiple-access cuts. The latter has closed form expressions as given in (5.36). Moreover, it also shows that the amplify-and-forward (AF) relay scheme is asymptotically optimal in the sense that it achieves the same scaling performances as the network capacity as L goes to inﬁnity.

5.3.3 Multi-Relay Fading Channels Now we consider channels without full channel state information at the transmitters while the receivers have the full channel state information. As in Sec-

220

5 Fundamental Limits of Cooperative and Relay Networks

tion 5.2, the fast and slow fading channels are considered. For the fast fading channels, we consider the phase fading model as in Section 5.2.1, that is, for each relay , ejθ,d [i] h,d [i] = , dα ,d where ejθ,d [i] is i.i.d. in the time domain and uniformly distributed in [0, 2π), d,d is the distance between the relay to destination, and α is the attenuation exponent. The channels hs,d and hs, , 1 ≤ ≤ L are deﬁned similarly. The random phase of each receiver is independent of each other and unknown to the corresponding transmitter. The capacity result of Theorem 5.8 in Section 5.2.1 can be extended as the following [5]. Theorem 5.14. If the relays are close to the source, the ergodic capacity for full-duplex phase fading relay channel with L relays is L Pr Ps Cf = max log2 1 + α 2 + , 2 Pr1 ,...,PrL ds,d σd dα l,d σd =1

where the maximization is over all power L allocation strategies over the relays satisfying the total power constraint =1 Pr < Pr . The formal mathematical descriptions on the distance constraint between relays and the source to validate this theorem can be found in [5, Theorem 7]. Again, DF achieves the capacity described in this theorem. Now we turn to the slow fading channels as in Section 5.2.2 and study the DMT for half-duplex channels. The channel model is a simple extension of the one in 5.2.2 with L relays and omitted here. First, the genie-aided diversity upper bound in Theorem 5.9 can be extended to multi-relay systems with L relays as the following theorem. Theorem 5.15. The diversity gain of the L-relay cooperative network is upper-bounded by d(r) ≤ (L + 1)(1 − r), where the multiplexing gain 1 ≥ r > 0. The NAF protocol in Theorem 5.11 can be easily extended to multiple relay cases (i.e., L ≥ 1). In these cases, each transmission is divided into L cooperative frames where the L relays take turns in relaying the codewords transmitted by the source. More speciﬁcally, as shown in Figure 5.11, relay

overhears the codeword transmitted by the source in the ﬁrst half of the

-th cooperative frame while the relay ampliﬁes and forwards the codeword in the second half of the frame. The achievable DMT was given in [1] and summarized in the following theorem. Theorem 5.16. The achievable diversity gain of the NAF protocol with L relays is characterized by

5.3 Multi-Relay Networks

221

Fig. 5.11 Nonorthogonal amplify-and-forward protocol with L relays. (Figure regenerated c and modiﬁed from Azarian, El Gamal, and Schniter. 2005 IEEE.)

d(r) = (1 − r) + L(1 − 2r)+ . The DDF scheme in Theorem 5.12 can also be extended to the case of multiple relays. Again, suppose that there are L relays in the network. Similar to the single relay scenario, all relays ﬁrst listen to the source at the beginning of each transmission block until the mutual information between the source message and the locally received signal exceeds nR, that is, they can successfully decode the message, and forward a new codeword. It is worthwhile to notice that the decoding time at each relay may be diﬀerent. The DMT achieved in this case has also been derived in [1] and is summarized in the following. Theorem 5.17. The diversity gain achieved by the DDF scheme with L relays is characterized by ⎧ 1 ⎨ (L + 1)(1 − r), for 0 ≤ r ≤ L+1 , L(1−2r) 1 d(r) = 1 + 1−r , for L+1 < r ≤ 0.5, ⎩ 1−r for 0.5 < r ≤ 1. r , From the above theorem, DDF is optimal when multiplexing gain r is low as 1/(L + 1) ≥ r > 0, since DDF achieves the genie-aided upper bound in Theorem 5.15. In Figure 5.12, we demonstrate the DMT curves for NAF, DDF, and genie-aided protocols with the number of relays being 4. It is observed that the diversity order of DDF drops rapidly as the multiplexing gain increases. Nevertheless, DDF is optimal with small multiplexing gain and it outperforms NAF for any multiplexing gain r ∈ (0, 1).

222

5 Fundamental Limits of Cooperative and Relay Networks

5

Genie Aided NAF DDF

4.5 Diversity Gain d(r)

4 3.5 3 2.5 2 1.5 1 0.5 0 0

0.2

0.4 0.6 Multiplexing Gain r

0.8

1

Fig. 5.12 Diversity-multiplexing tradeoﬀ for NAF, DDF, and genie-aided protocols with L = 4 relays. (Figure regenerated and modiﬁed from Azarian, El Gamal, and Schniter. c 2005 IEEE.)

Appendix 5.1 Proof of Theorem 5.12 The basic proof steps follow those in the proof of Theorem 5.11. We ﬁrst derive the error probability formula corresponding to (5.28). For a given channel realization, the conditional error probability of the ML decoder can be written as Pe|h = Pe,Erc |h + Pe,Er |h ≤ Pe|Erc ,h + PEr |h where h = [hs,d , hr,d , hs,r ]T and Er is the event that a decoding error occurs at the relay. As described previously in (5.32), the probability of error at the relay can be made arbitrarily small when the mutual information between the source message and the received signal exceeds nR, provided that n is suﬃciently large. Thus, the error probability is asymptotically bounded as Pe|h ≤ Pe|Erc ,h . By taking the average over channel realizations, the average error probability is given by ·

Pe ≤ Pe|Erc ≤ PO|Erc + Pe,Oc |Erc .

(5.39)

We neglect the lower-script Erc in the following texts. According to [1], suppose that the rate is R = r log2 ρ and the codeword length is n, the second term in (5.39) can be bounded as

5.3 Multi-Relay Networks

223

2

Fig. 5.13 Outage region for the DDF protocol with a single relay (f ≤ 0.5). (Figure c regenerated and modiﬁed from Azarian, El Gamal, and Schniter. 2005 IEEE.)

·

Pe,Oc ≤ ρ−de (r) , where n n (1 − vs,d )+ + 1 − (1 − min{vs,d , vr,d })+ − r n inf de (r) = n n v∈O c ∩R3+ (5.40)

+ (vs,d + vr,d + vs,r ) , and v = [vs,d , vr,d , vs,r ]. Therefore, since ·

PO = ρ−do (r) where do (r) =

inf

v∈O∩R3+

(vs,d + vr,d + vs,r ),

(5.41)

we may choose the outage event as

0 n 3+ n + + (1 − min{vs,d , vr,d }) ≤ r (5.42) O = v ∈ R (1 − vs,d ) + 1 − n n by observing from (5.40) and (5.41), so that the error probability in (5.39) is ·

dominated by the outage probability PO , i.e., Pe ≤ PO . In this case, we need only to determine the value of do (r). Let us consider four diﬀerent cases: (i) vs,d , vr,d ≥ 1; (ii) 0 ≤ vs,d ≤ 1, vr,d ≥ 1; (iii) vs,d ≥ 1, 0 ≤ vr,d ≤ 1; (iv) 0 ≤ vs,d , vr,d ≤ 1. Please note that, by (5.32), it follows that

224

5 Fundamental Limits of Cooperative and Relay Networks

2

Fig. 5.14 Outage region for the DDF protocol with a single relay (f > 0.5). (Figure c regenerated and modiﬁed from Azarian, El Gamal, and Schniter. 2005 IEEE.)

f

0

0

r n · r log2 ρ = min 1, . = min 1, n log2 (|hs,r |2 ρ) 1 − vs,r

(5.43)

The ﬁrst three cases all exceed the diversity order achieved by the genie-aided scheme, i.e., 2(1 − r) [1], and thereby cannot be the dominating factors in the DMT of the DDF scheme. Case (iv) is then of particular interest. In case (iv) where 0 ≤ vs,d , vr,d ≤ 1, the inequality in (5.42) is * + f (1 − vs,d ) + 1 − f (1 − vs,d ) = 1 − vs,d ≤ r ⇒ vs,d ≥ 1 − r, for vs,d ≤ vr,d , and is * + * + f (1 − vs,d ) + 1 − f (1 − vr,d ) ≤ r ⇒ f vs,d + 1 − f vr,d ≥ 1 − r, for vs,d ≥ vr,d . The possible values of (vs,d , vr,d ) are plotted in Figure 5.13 for f ≤ 12 and in Figure 5.14 for f > 12 . Moreover, for vs,r , it also holds by (5.43) that f = 1−vr s,r ≥ r and that vs,r = 1 − fr . Let us consider the two cases regarding the value of f . For r ≤ f ≤ 12 , as can be observed from Figure 5.13, inf (vs,d ,vr,d ,0)∈O

vs,d + vr,d

takes on the value 2(1 − r). This is equivalent to the DMT of the genieaided scheme. Thus we can always choose f = r such that vs,r = 0 and do (r) = 2(1 − r). On the other hand, for f > max{r, 12 }, the minimum value r of vs,d + vr,d is given by 1−r f . With vs,r = 1 − f , the diversity order is

References

225

do (r) =

inf f >max{r,0.5}

1+

1 − 2r . f

Then it can be checked that the results in Theorem (5.12) is valid as in [1].

References 1. Azarian, K., El Gamal, H., Schniter, P.: On the achievable diversity-multiplexing tradeoﬀ in half-duplex cooperative channels. IEEE Transactions on Information Theory 51(12), 4152–4172 (2005) 2. Cover, T., El Gamal, A.: Capacity theorems for the relay channel. IEEE Transactions on Information Theory 25(5), 572–584 (1979) 3. Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. WileyInterscience (2006) 4. Host-Madsen, A., Zhang, J.: Capacity bounds and power allocation for wireless relay channels. IEEE Transactions on Information Theory 51(6), 2020–2040 (2005) 5. Kramer, G., Gastpar, M., Gupta, P.: Cooperative strategies and capacity theorems for relay networks. IEEE Transactions on Information Theory 51(9), 3037–3063 (2005) 6. Laneman, J., Tse, D., Wornell, G.: Cooperative diversity in wireless networks: Eﬃcient protocols and outage behavior. IEEE Transactions on Information Theory 50(12), 3062–3080 (2004) 7. van der Meulen, E.C.: Three-terminal communication channels. Advances in Applied Probability 3, 120–154 (1971) 8. Gastpar, M., Vetterli, M.: On the capacity of large Gaussian relay networks. IEEE Transactions on Information Theory 51(3), 765–779 (2005) ˙ 9. Telatar, I.E.: Capacity of multi-antenna Gaussian channels. European Transactions on Telecommunications 10(6), 585–595 (1999) 10. Tse, D., Viswanath, P.: Fundamentals of Wireless Communication. Cambridge University Press (2005) 11. Wyner, A., Ziv, J.: The rate-distortion function for source coding with side information at the decoder. IEEE Transactions on Information Theory 22(1), 1–10 (1976) 12. Zheng, L., Tse, D.N.C.: Diversity and multiplexing: A fundamental tradeoﬀ in multiple-antenna channels. IEEE Transactions on Information Theory 49(5), 1073– 1096 (2003)

Chapter 6

Cooperative Communications with Multiple Sources

Previously, we focused on cooperative systems where only one user is allowed to act as the source at any instant in time while the other users serve as relays of the source. However, in multiuser systems, multiple sources may be accessing the cooperative channel simultaneously and, thus, multiple access strategies must be devised to separate their signals in either time, frequency, code, or space. In this chapter, we will examine diﬀerent multiple access schemes for cooperative communication systems, including time-division multiple access (TDMA), frequency-division multiple access (FDMA), code-division multiple access (CDMA), and space-division multiple access (SDMA) schemes. In TDMA/FDMA systems, sources transmit over orthogonal time or frequency channels, where radio resources must be properly allocated to fully exploit the advantages of cooperation. In CDMA or SDMA systems, users transmit simultaneously over diﬀerent code or spatial dimensions, but may experience multiple access interference (MAI) due to practical diﬃculties in achieving orthogonality among the diﬀerent dimensions. Therefore, precoding techniques at the relays or multiuser detection schemes at the destination must be employed to mitigate the MAI. In addition to the multiple access issues, we will also introduce diﬀerent partner selection policies and show how they can be used to further enhance the cooperative advantages in multiuser systems. Note that this chapter is devoted to the studies on multiple access techniques, which are used to multiplex transmissions from diﬀerent users. Issues involving user competition and collision resolution will be discussed in Chapter 8, where medium access control (MAC) policies are introduced.

6.1 Time/Frequency-Division Multiple Access (TDMA/FDMA) In multiuser cooperative systems, one may encounter diﬀerent network topologies depending on the role of the relays and the number of destinaY.-W. Peter Hong et al., Cooperative Communications and Networking, DOI 10.1007/978-1-4419-7194-4_6, © Springer Science+Business Media, LLC 2010

227

228

6 Cooperative Communications with Multiple Sources

(a) Designated Relays

(b) Shared Relays Fig. 6.1 Example of multi-source cooperative networks with diﬀerent relay functionalities.

tions. Speciﬁcally, we can consider the case of designated relays, where each source is served exclusively by one or multiple relays, as shown in Fig. 6.1(a), and the case of shared relays, where multiple sources are served by a common set of relays, as shown in Fig. 6.1(b). In the case of designated relays, the resources of a given relay are allocated completely to a single source and, thus, the system is simpler to implement. However, the diversity gain achievable in this case is often limited since the number of relays that can be designated to one particular source is usually small. In the case of shared relays, more than one source may be transmitting through a common set of relays and, thus, the resources at the relays must be properly allocated to maximize cooperative advantages received by the sources. Higher diversity gains can potentially be

6.1 Time/Frequency-Division Multiple Access (TDMA/FDMA)

229

achieved compared to the case of designated relays. Furthermore, we can also consider diﬀerent settings based on the number of destinations. Two scenarios will be considered in this chapter, i.e., the ad hoc scenario, where each source transmits to a diﬀerent destination, and the uplink scenario, where the sources transmit to a common destination (or base station). In TDMA or FDMA systems, each transmission entity is assigned an orthogonal time or frequency channel to conduct its transmission. In systems with designated relays, each source and its relays can form a transmission entity that is treated as a super-user and allocated a single orthogonal channel for transmission. The channel can be further divided into subchannels if the cooperation requires multi-phase transmissions. The rate achievable as a group, through cooperation, can then be viewed as the rate achievable by the transmission entity and, thus, the problem becomes identical to that of non-cooperative systems. Related studies on the use of TDMA/FDMA for systems with designated relays can be found in [2]. The problem is considerably more interesting when shared relays are employed. In this case, sources may compete for resources at the relays and, thus, eﬃcient resource allocation policies are essential to achieve high cooperative gains. These issues will be discussed for two TDMA scheduling schemes: round-robin scheduling, where the sources take turns in accessing the relays, and opportunistic scheduling, where the relays dynamically serve the source with the best eﬀective channel in each time slot. The discussions are focused on TDMA systems while extensions to FDMA systems are left as exercises for the readers.

6.1.1 Round-Robin Scheduling Round-robin scheduling is one of the simplest TDMA policies, where time is divided into equal-length time slots and each source takes turns in accessing the common set of relays. To simplify our discussions, we consider in this section the case where only one relay exists in the network and where the basic cooperative schemes introduced in Chapter 3 are employed. The case of multiple relays can also be examined similarly by considering the cooperative strategies introduced in Chapter 4. Moreover, it is worthwhile to note that, in round-robin scheduling, the users are scheduled according to the order of their indices (not the quality of their channels). Hence, the sources that are scheduled to transmit often will not be able to support high data rates and, thus, will aﬀect the overall system throughput. This eﬀect can be mitigated with proper resource allocation across time. Let us consider the ad hoc scenario shown in Fig. 6.2 where the sources are transmitting to diﬀerent destinations through a common relay. Suppose that there are K distinct source-destination pairs, denoted by {(sk , dk )}K k=1 , and that each source-destination pair is allocated 1/K share of time to transmit.

230

6 Cooperative Communications with Multiple Sources

Fig. 6.2 A cooperative TDMA system with a single relay serving multiple sourcedestination pairs.

Let hk,r and fk,k be the channel coeﬃcients between source sk and relay r and between source sk and destination dk , respectively, and let gr,k be the channel coeﬃcient between relay r and destination dk . By considering the basic DF relaying scheme, the capacity of the k-th source-destination pair under the assistance of the common relay is given by , (k) Psk |fk,k |2 1 Pr |gr,k |2 (DF) Ck log2 1 + , = min + 2K σd2 σd2 0 Ps |hk,r |2 1 log2 1 + k 2 , (6.1) 2K σr (k)

where Psk is the transmission power of source sk , Pr is the transmission power used by the relay when forwarding sk ’s messages, and σd2 and σr2 are the noise variances at the destinations and the relay, respectively. The additional 1/K factor comes from the fact that time is divided equally among K users. By deﬁning ϕk,k |fk,k |2 /σd2 , γr,k |gr,k |2 /σd2 , and ηk,r |hk,r |2 /σr2 as the SNRs on the sk -dk , r-dk and sk -r links, the capacity expression in (6.1) can be written as 4 5 1 (DF) Ck min log2 1 + Psk ϕk,k + Pr(k) γr,k , log2 (1 + Psk ηk,r ) . = 2K (6.2) Due to channel variations among diﬀerent users, the relay can dynamically adjust the powers used to forward the sources’ data in order to maximize the aggregate throughput of the K sources [21]. That is, given the instantaneous (k) CSI, the powers {Pr }K k=1 are chosen to maximize the sum rate of all sourceK (k) destination pairs subject to a sum power constraint k=1 Pr ≤ Pr . The optimal power allocation problem can be formulated as follows:

6.1 Time/Frequency-Division Multiple Access (TDMA/FDMA) K

max (k) K }k=1

{Pr

231

(DF)

Ck

(6.3)

Pr(k) ≤ Pr , Pr(k) ≥ 0, k = 1, 2, ..., K.

(6.4)

k=1

subject to

K k=1

By neglecting the second term inside the minimum in (6.2), the problem reduces to K

max (k) K }k=1

{Pr

log 1 + Psk ϕk,k + Pr(k) γr,k

(6.5)

Pr(k) ≤ Pr , Pr(k) ≥ 0, k = 1, 2, ..., K.

(6.6)

k=1

subject to

K k=1

The Lagrange function can be written as L({Pr(k) }K k=1 )

=

K

log 1 + Psk ϕk,k +

Pr(k) γr,k

+ μDF Pr −

k=1

K

Pr(k)

.

k=1

Then, by Karush-Kuhn-Tucker (KKT) conditions, the optimal power allocation can be computed as Pr(k) =

1 μDF

−

1 + Psk ϕk,k γr,k

+ , ∀k,

(6.7)

where (x)+ = max(x, 0) and μDF is a constant chosen to satisfy the total K (k) power constraint k=1 Pr ≤ Pr . This is in the form of a water-ﬁlling solution, where power is distributed among the various bins corresponding to diﬀerent users in a way that is analogous to the distribution of water when poured into a vessel (see Chapter 2.3). However, when taking into consideration the second term inside the minimization of (6.2), the capacity of each source-destination pair will be upper 1 bounded by the capacity of the sk -r link (i.e., 2K log2 (1 + Psk ηk,r )) regardless of the power allocated to the k-th source-destination pair. Hence, the (k) value of Pr should not exceed Psk (ηk,r − ϕk,k )/γr,k ; otherwise, extra power will be expended with no increase in the system sum rate. The optimal power allocation leads to a cave-ﬁlling solution with ceiling layer given by (1 + Psk ηk,r )/γr,k and base layer given by (1 + Psk ϕk,k )/γr,k , as illustrated in Fig. 6.3 for the case of 5 users. The water poured into the cave corresponding to each source-destination pair will be limited by its ceiling layer and will ﬂow into the cave corresponding to other pairs. The distance between the water-level (or the ceiling layer) and the base layer is the allocated power (k) Pr . Notice that the base layer in Fig. 6.3 is determined by the quality of

232

6 Cooperative Communications with Multiple Sources

Hj- 5 ( Hj+

Hj, ) <>

Hj* Hj)

k*

k)

k+

k,

k-

Fig. 6.3 Illustration of the cave-ﬁlling solution for a TDMA cooperative network with 5 source-destination pairs.

the sk -dk link while the ceiling layer is determined by the quality of the sk -r link. The optimal solution is given by Pr(k) =

0 +

1 + Psk ϕk,k 1 1 + Psk ηk,r − min , , ∀k, μDF γr,k γr,k

(6.8)

K (k) where 1/μDF is chosen to satisfy the total power constraint k=1 Pr ≤ Pr . More speciﬁcally, for users whose ceiling layers are lower than the waterlevel 1/μDF , e.g., users 1 and 2 in Fig. 6.3, the allocated relay power is given by Ps (ηk,r − ϕk,k ) Prk = k , (6.9) γr,k and the resulting capacity is given by (DF)

Ck

=

1 log2 (1 + Psk ηk,r ) , 2K

(6.10)

which means that the DF relay capacity is bounded by that of the sk -r link. For users whose ceiling layers are higher than 1/μDF , e.g., user 3 and user 4 in Fig. 6.3, the power is given by 1 + Psk ϕk,k 1 Prk = (6.11) − μDF γr,k and the corresponding capacity of the k-th source-destination pair is

6.1 Time/Frequency-Division Multiple Access (TDMA/FDMA) (DF)

Ck

=

1 log2 2K

γr,k μDF

233

.

(6.12)

However, if the user’s base layer is higher than 1/μDF (e.g., user 5 in Fig. 6.3) or even the ceiling layer (i.e., ϕk,k > ηk,r ), no power will be allocated at the relay to forward its data (i.e., Prk = 0). The round-robin based TDMA system can also be performed using the basic AF relaying technique. Utilizing the capacity expressions given in Chapter 3, we can also ﬁnd the optimal power allocation by maximizing the system sum rate. Speciﬁcally, when the basic AF relaying scheme is employed, the capacity of the k-th source-destination pair, i.e., (sk , dk ), can be expressed as (k) 1 Psk Pr ηk,r γr,k (AF) log2 1 + Psk ϕk,k + . (6.13) = Ck (k) 2K 1 + Psk ηk,r + Pr γr,k The optimal power allocation can be found by maximizing the sum rate of all sources as given below [21]: K

max (k) K }k=1

{Pr

(AF )

Ck

(6.14)

Pr(k) ≤ Pr , Pr(k) ≥ 0, ∀k.

(6.15)

k=1

subject to

K k=1

The Lagrange function can be written as K (k) K log 1 + Psk ϕk,k + L({Pr }k=1 ) = k=1

(k)

Psk Pr ηk,r γr,k

(k)

1 + Psk ηk,r + Pr γr,k K (k) . Pr + μAF Pr − k=1

By KKT conditions, the optimal power allocation is then given by

⎛ ak ⎜ −( bk

Pr(k) = ⎜ ⎝

+ 2) +

ak bk

2 +

4ak μAF

(1 +

2(ak + bk )

where Psk ηk,r /(Psk ηk,r + 1) , (1 + Psk ϕk,k )/γr,k γr,k , bk = Psk ηk,r + 1

ak =

⎞+ ak bk ) ⎟

⎟ , ⎠

(6.16)

234

6 Cooperative Communications with Multiple Sources

and μAF is a constant chosen to satisfy the sum power constraint in (6.15). The power allocation problem described above can also be studied with the more advanced relaying strategies introduced in Chapter 3 and for the multirelay cooperative strategies introduced in Chapter 4. However, these problems are left as exercises for interested readers. It is worthwhile to remark that, in multi-relay systems, individual power constraints at the relays may render the problem more diﬃcult and is an interesting topic for future investigation. The round-robin based TDMA system introduced in this subsection can be easily extended to FDMA systems, where users’ transmissions are multiplexed in frequency rather than in time. More speciﬁcally, in FDMA systems, the source-destination pairs are allocated orthogonal frequency bands with equal bandwidth and each frequency band is further divided into two sub-bands, used for Phase I and Phase II transmissions performed by the source and the relay, respectively. However, to maintain causality, the data forwarded by the relay in the second sub-band must be received in the previous time slot (in the ﬁrst sub-band). In this case, an additional time slot is needed for the source to initiate the transmission at the very beginning, but the average rate will not be aﬀected as long as the transmission proceeds over a suﬃciently large number of time slots. In frequency-selective environments, the power allocated to each source-destination pair can also be obtained similar to (6.8) or (6.16). Although the round-robin scheduler is easy to implement in practice, it does not fully exploit the spatial diversity existing in multiuser communication systems. In fact, instead of allocating less power to a source when it does not have a favorable channel, it is more beneﬁcial to simply schedule the user with a better channel to transmit. This leads to the opportunistic scheduling scheme introduced in the next subsection.

6.1.2 Opportunistic Scheduling Instead of scheduling users in a predetermined order (as in round-robin scheduling), the opportunistic scheduling scheme dynamically selects the best user (e.g., the user that yields the highest eﬀective SNR at its destination) to transmit in each time slot, exploiting the so called multiuser diversity. Let us consider a cooperative TDMA system that consists of K sources transmitting to their respective destinations over L shared relays. Assume that the channel remains constant over each cooperative transmission but varies independently from block to block. If the k-th source is served in the mth time slot, the signals received at the L relays in Phase I can be represented ( )T (k) (k) (k) (k) by the vector yr [m] = y1 [m], y2 [m], · · · , yL [m] , where yr(k) [m] =

-

Psk hk,r xk [m] + wr [m].

(6.17)

6.1 Time/Frequency-Division Multiple Access (TDMA/FDMA)

235

Fig. 6.4 Illustration of opportunistic scheduling with two source-destination pairs.

and hk,r = [hk,1 , hk,2 , · · · , hk,L ]T is the vector of channel coeﬃcients between source sk and the L relays, xk [m] is source symbol with unit energy, i.e., E[|xk [m]|2 ] = 1, ∀k, and wr [m] = [wr1 [m], . . . , wrL [m]]T ∼ CN (0, σr2 I) consists of the noise at the relays. In Phase II of the same time slot, the relays will together forward the source’s data to its respective destination. Suppose that an AF multi-relay (k) policy is employed such that the amplifying factor α is imposed at relay r when forwarding the data of source k. Let g,k be the channel coeﬃcient between relay r and destination dk . Then, the signal received at destination dk can be expressed as ydk [m] =

L

(k) (k)

g,k α y [m] + wd [m]

=1

=

T T Psk gr,k A(k) hk,r xk [m] + gr,k A(k) wr [m] + wd [m]

where gr,k = [g1,k , g2,k , · · · , gL,k ]T is the channel vector between the relays (k) (k) (k) and destination dk , A(k) = diag(α1 , α2 , · · · , αk ) is a diagonal matrix consisting of the amplifying coeﬃcients of source sk at the L relays, and wd [m] ∼ CN (0, σd2 ) is the AWGN at the destination. The instantaneous SNR achieved by the k-th source-destination pair in the m-th time slot is given by γk [m] =

T A(k) hk,r |2 Psk |gr,k . * + T A(k) A(k) H g∗ σd2 + σr2 gr,d r,k

(6.18)

To maximize the system sum rate, the source-destination pair with the highest instantaneous SNR will be scheduled to transmit in each time slot. An example with two users is illustrated in Fig. 6.4. We can see that, by select-

236

6 Cooperative Communications with Multiple Sources

M source/destination pairs with L relays at P/σ

20dB

1%−outage aggregate throughput [bps/Hz] RXWDJHDJJUHJDWHWKURXJKSXWELWVVHF+]

0

0 0 0 0 0 0

L relays 1XPEHURIUHOD\V

Fig. 6.5 Aggregate 1% outage throughput in terms of the number of relays L in adaptive TDMA cooperative systems (From Hammerstrom, Kuhn and Wittneben with modiﬁed c labels; [2004] IEEE ).

ing the user with the best SNR, the eﬀective SNR observed by the system remains at a high level. In Fig. 6.5 (obtained from [11]), the aggregate 1% outage throughput is shown with respect to the number of relays L for a cooperative network with K = 1 ∼ 6 source-destination pairs. In this experiment, the amplifying gain at relay r is given by ' Pr (k) ejφ , for k = 1, . . . , K, (6.19) α = Psk |hk, |2 + σr2 where φ is the random phase oﬀset at relay r that is assumed to be uniformly 2 distributed between 0 and 2π. It is also assumed that σr2 = σd2 = σw and 2 that the transmit SNRs of each source and the relays are set as Psk /σw = 2 L × Pr /σw = 20 dB for all k. We can observe that the aggregate throughput increases with the number of source-destination pairs because of the increased multiuser diversity. Moreover, we can see that the increase in the aggregate throughput also increases with the number of relays but saturates rapidly since the multiuser diversity exploited in this case already achieves signiﬁcant gains to the throughput and, thus, the additional cooperative diversity gains obtained by increasing the number of relays become limited.

6.2 Code-Division Multiple Access (CDMA)

237

The opportunistic TDMA scheduling introduced in this section allows the system to exploit multiuser diversity and achieve a signiﬁcant increase in the aggregate system throughput. However, this method does not ensure fairness and, thus, users with low average SNR will have little opportunity to transmit. A simple approach to achieving fairness is to limit the maximum number of time slots that can be allocated to each source-destination pair over a given period of time. More sophisticated issues can also be developed based on diﬀerent fairness criteria, e.g., max-min or proportional fairness etc [16].

6.2 Code-Division Multiple Access (CDMA) Code-division multiple access (CDMA) [1, 8, 26] refers to channel access schemes where sources share the same time and frequency channel but separate their signals by modulating them on diﬀerent signature codes. If the codes are orthogonal to each other, the receiver will be able to perfectly separate the signals by passing the aggregate signal from multiple users through matched ﬁlters that are associated with the codes of diﬀerent users. Since the code is unique to each user and the resulting waveform occupies a much wider bandwidth than the original signal, the CDMA codes are often referred to as signature or spreading codes (or waveforms). These schemes are thus considered as a subclass of the more general spread spectrum technology [23]. In the past two decades, CDMA technology has drawn much attention from both academia and industry due to its ability to resolve multipath signals, to enable easier frequency planning, and to increase user capacity. It is also the key enabling technology in many modern wireless communication systems such as cdmaOne and cdma2000. In cooperative networks, CDMA systems can be incorporated in a way similar to that of TDMA and FDMA systems if the signature waveforms are perfectly orthogonal to each other. Unfortunately, this is diﬃcult to guarantee in practice due to the lack of perfect synchronization at the receiver, non-ideal waveform design, or overloaded systems etc. As a result, both the relays and the destinations will be subject to multiple access interference (MAI) and signiﬁcant performance degradations will be observed, especially when conventional single-user detectors are used at the receivers. This loss in performance cannot be reduced by simply increasing the transmission power since the interference increases proportionally with the transmission power of the users. Hence, an error ﬂoor will exist as SNR increases. These eﬀects can be even more detrimental if the interfering users are closer to the destination than the source, resulting in the so called near-far eﬀect. To mitigate the MAI, multiuser detection (MUD) schemes, where the signals of multiple users are detected jointly at the receiver, have been investigated extensively in the literature for conventional non-cooperative systems [9, 17, 18, 25]. While the use of MUD have been avoided in the past due

238

6 Cooperative Communications with Multiple Sources

Fig. 6.6 Illustration of a CDMA uplink system with designated relays.

to its high complexity, it is no longer avoidable in cooperative systems since, without eﬀective mitigation of the MAI, the diversity gains that lead to faster decrease in error probability will diminish due to the error ﬂoor laid by the MAI. In this section, we will discuss the use of MUD in uplink cooperative CDMA systems for networks with designated and shared relays, respectively.

6.2.1 Uplink CDMA with Designated Relays Consider an uplink cooperative network that consists of K sources (denoted by s1 , · · · , sK ), K relays (denoted by r1 , . . ., rK ), and a single destination d (i.e., the base station), as shown in Fig. 6.6. Each source, say sk , is served by only one designated relay (i.e., rk ), but may still cause interference at other relays due to the broadcast nature of the wireless medium. The sourcerelay pair can be viewed as two mutually cooperating users that take turns in acting as the source and the relay. The signature waveforms of the sources are not assumed to be orthogonal of each other but are assumed to be linearly independent1 . The lack of orthogonality among the signature waveforms will cause MAI at both the relays and the destination. Hence, MUD must be employed at both locations in order to preserve the cooperative gains. Speciﬁcally, let us consider a two-phase cooperative transmission scheme. In Phase I, each source transmits a sequence of M symbols, modulated by its unique signature waveform. Suppose that xk [m] is the symbol transmitted by 1 The linear independence will be satisﬁed in practice with high probability. However, when it is not satisﬁed, the MUD can still be treated similarly by using the pseudo-inverse whenever the lack of linear independence leads to non-invertible matrices.

6.2 Code-Division Multiple Access (CDMA)

239

source sk in the m-th symbol period. Then, the signal transmitted by source sk can be written as the continuous time function M Psk xk [m]uk (t − mTs ),

xk (t) =

(6.20)

m=1

where Psk is the transmission power of source sk and uk (t) is the signature waveform of sk that is assumed to last over a symbol duration Ts . The signature waveform of source sk can be expressed mathematically as N 1 ck [n]ψ(t − nTc ), uk (t) = √ N n=1

(6.21)

where ck [n] ∈ {±1} is the n-th element of the spreading sequence assigned to source sk , N is the spreading gain, and ψ(t) is the normalized chip waveform with unit energy and chip duration Tc = Ts /N . Here, we assume that BPSK modulation is employed so that xk [m] ∈ {±1}. The results can be easily extended to systems with higher order modulations, e.g., QPSK or 64-QAM. Let us consider for simplicity the synchronous CDMA scenario where the users’ transmissions are assumed to arrive simultaneously at each receiver and are synchronized on the symbol level. Therefore, the the signal received at relay r and the destination d are given respectively by y (t) =

K M Psk hk, xk [m]uk (t − mTs ) + w (t)

(6.22)

m=1 k=1

and (1)

yd (t) =

K M (1) Psk fk,d xk [m]uk (t − mTs ) + wd (t),

(6.23)

m=1 k=1

where hk, and fk,d are the channel coeﬃcients of the sk -r and the sk -d (1) links, respectively, and w (t) and wd (t) are the AWGN at relay r and destination d, respectively, both with mean 0 and autocorrelation Rwr (τ ) = 2 Rwd (τ ) = σw δ(τ ). The synchronous assumption is applicable when the users are relatively close to each other and the signal bandwidth is suﬃciently small. Although a more practical study can be conducted based on the asynchronous scenario, the synchronous case is suﬃcient to demonstrate the eﬀects of MAI and the beneﬁts of using MUD in cooperative networks. In Phase II, each relay, say r , will decode and forward the symbols transmitted by its designated source s using the same signature waveform, i.e., u (t). By considering the uncoded DF relaying scheme, the symbol estimate x ˆ [m] forwarded by relay r at time m may possibly be incorrect due to the lack of error detection. Therefore, the signal received at the destination in

240

6 Cooperative Communications with Multiple Sources

Phase II can be written as (2)

yd (t) =

K M (2) Pr g,d x ˆ [m]u (t − mTs ) + wd (t),

(6.24)

m=1 =1

where Pr is the transmission power of relay r , g,d is the channel coeﬃcient (2) on the r -d link, and wd (t) is the AWGN with mean 0 and autocorrelation 2 Rwd (τ ) = σw δ(τ ). To detect the symbols at the relays, diﬀerent receiver structures can be employed, including both single-user and multiuser detectors. The simplest of these schemes is the single-user matched ﬁlter (MF) detector, where the received signal, e.g., y (t), is passed through a ﬁlter matched to the signature waveform of the source of interest, e.g., u (t). The corresponding MF output during the m-th symbol period can be written as

Ts

zMF [m] =

y (t + mTs )u (t)dt Psk ρk, hk, xk [m] + w [m], = Ps h, x [m] + 0

(6.25)

k=

where ρk, is the correlation between the signature waveforms uk (t) and u (t), i.e., Ts ρk, = uk (t)u (t)dt, 0

B Ts

and w [m] = 0 w (t+ mTs )u (t)dt is Gaussian distributed with mean 0 and 2 variance σw . The ﬁrst term in (6.25) contains the signal of interest at relay r while the second term contains the MAI from other users. The signal-tointerference-plus-noise ratio (SINR) of the MF output zMF at relay r is given by SINR =

Ps |h, |2 . 2 2 2 k= Psk ρk, |hk, | + σw

(6.26)

The single-user MF detector is optimal in AWGN channels but may experience signiﬁcant performance degradations in the presence of MAI. The eﬀect is particularly pronounced when interfering users are located closer to the destination or transmitting with much higher power than the source, resulting in the so-called near-far eﬀect . To mitigate MAI, more sophisticated multiuser detectors (MUDs) can be employed at the receivers to jointly detect all sources’ symbols instead of treating the signals transmitted by other sources as interference. To perform joint detection over all sources’ symbols, the signal received at each relay, say relay r , can ﬁrst be passed through a matched ﬁlter bank (MFB) consisting of K receive ﬁlters in parallel that are matched to the signature waveforms

6.2 Code-Division Multiple Access (CDMA)

241

u1 (t), u2 (t), . . ., uK (t). Suppose that x[m] = [x1 [m], x2 [[m], · · · , xK [m]]T is the vector of symbols transmitted by the K sources during the m-th symbol period. Then, the MFB output during this symbol period can be written as y [m] = RHs, x[m] + w [m], (6.27) + *where Hs, = diag Ps1 h1, , Ps2 h2, , · · · , PsK hK, is a diagonal matrix consisting of the transmit power times the channel coeﬃcients between the sources and relay r , the matrix R is the correlation matrix given by ⎡ ⎤ ρ1,1 ρ1,2 · · · ρ1,K ⎢ ρ2,1 ρ2,2 · · · ρ2,K ⎥ ⎢ ⎥ (6.28) R=⎢ . . ⎥, .. .. ⎣ .. . . .. ⎦ ρK,1 ρK,2 · · · ρK,K and w [m] is the AWGN at the output of the MFB with zero-mean and 2 correlation matrix E[w [m]w [m]H ] = σw R. By assuming that each symbol is transmitted with equal probability, the optimal MUD that minimizes the average symbol error probability can be derived by the maximum likelihood (ML) criterion, where the detected symbol vector at relay r is given by ˆ = arg max p (y [m]|x[m]) x x[m]

= arg min (y [m] − RHs, x[m]) R−1 (y [m] − RHs, x[m]) . H

(6.29)

x[m]

However, the possible values of x[m] and, thus, the complexity of the ML detector grows exponentially with the number of users, which is unaﬀordable to most mobile devices in practice. Alternatively, several suboptimal linear MUDs have been proposed to reduce MAI at a more reasonable computational cost. Two linear MUDs have been adopted the most in the literature: the decorrelating MUD [17] and the minimum mean square error (MMSE) MUD [18]. Speciﬁcally, in decorrelating MUDs, the received signal y [m] is multiplied by C = (RHs, )−1 to obtain zDEC [m] = C y = (RHs, )−1 y : [m], = x[m] + w

(6.30)

: [m] = (RHs, )−1 w [m] is the eﬀective noise at the output of the where w MUD. The received signal has been decorrelated in the sense that each element in zDEC depends only on the realization of one source’s symbol. Conse quently, zDEC can be viewed as K parallel single-user AWGN channels and a decision on each source’s transmitted symbol can be made based only on the corresponding entry in zDEC . However, at the output of the decorrelating receiver, the noise covariance matrix becomes

242

6 Cooperative Communications with Multiple Sources

−1 −1 −H −1 2 : [m]w : [m]H ] = H−1 E[w E[w [m]w [m]H ]R−1 H−H Hs, , s, R s, = σw Hs, R

which may yield large noise values if the matrix R is ill-conditioned. This is the so called noise enhancement problem similar to that experienced in zero-forcing equalizers (used to combat intersymbol interference). In MMSE MUDs, the linear ﬁlter C is chosen to minimize the MSE E |x[m] − C y [m]|2 . By taking the derivative of the MSE with respect to C and setting it to zero, we obtain C = Kxy K−1 yy ,

(6.31)

Kxy = E[x[m]y [m]H ] = HH s, R

(6.32)

2 Kyy = E[y [m]y [m]H ] = RHs, HH s, R + σw R.

(6.33)

where and The output of the MMSE MUD is given by [m] = C y [m]. zMMSE

(6.34)

When relay r is designated to source s , it will only be interested in the symbol transmitted by source s and, thus, only the -th element of zMMSE [m] is utilized for detection. The -th element of zMMSE [m] is given by MMSE MMSE −1 z, [m] = eH [m] = eH z Kxy Kyy y [m] * +−1 H 2 = Ps h∗, eH y [m], R RHs, Hs, R + σw R

(6.35) (6.36)

where e is the -th column of a K × K identity matrix. The resulting MSE is ( 2 ) MMSE MSE = E x [m] − z, [m] * +−1 H 2 = 1 − Ps |h, |2 eH Re R RHs, Hs, R + σw R and the SINR at relay r can be computed as −1 2 −1 Hs, HH Ps |h, |2 eH e s, + σw R SINR = −1 . 2 −1 Hs, HH 1 − Ps |h, |2 eH e s, + σw R

(6.37)

The MMSE MUD minimizes the total interference plus noise power at the receiver and, thus, serves as a good compromise between noise reduction and interference mitigation. It is important to note that linear receivers described above can be followed by either a hard decision device if DF relaying is employed or by a analog amplifying devices if AF relaying is adopted. These operations are described

6.2 Code-Division Multiple Access (CDMA)

243

in detail in the following. For notational simplicity, we shall omit the index on m since MUD is performed on a symbol-by-symbol basis and the operations do not vary with m.

(i) MUD with Decode-and-Forward Relays When considering the DF scheme, we assume that each relay, say relay r , performs a hard decision to detect the symbol of its corresponding source, i.e., x ˆ = sgn(z, ), and forwards it to the destination in Phase II of the transmission process. By considering the uncoded case, we assume that the detected symbols are forwarded at the relays without performing any error detection. Therefore, the symbol forwarded by relay r can be expressed as x ˆ = θ x , where θ = −1 if the detection at relay r is incorrect and θ = 1 if it is correct. Here, θ ∈ {±1} can be modeled as a Bernoulli random variable with distribution Pr (θ = −1) = 1 − Pr (θ = 1) = ε , where ε is the detection error probability at relay r . By treating the interference as Gaussian, the error probability can be approximated by + *√ ε ≈ Q 2SINR , where the SINR is given by (6.26) if the single-user MF receiver is applied, or is given by (6.37) if the MMSE MUD is applied at the relays. ˆ = [ˆ Let x x1 , xˆ2 , · · · , x ˆK ]T be the vector consisting of the detected symbols of their corresponding sources at the K relays. At the destination, the signals received in Phase I and Phase II are passed through a MFB corresponding to the signature waveforms u1 (t), u2 (t), . . ., uK (t), which yield the outputs (1)

(1)

(2)

(2)

yd = RFs,d x + wd

(6.38)

and ˆ + wd , yd = RGr,d x (6.39) respectively, where Fs,d = diag( Ps1 f1,d , Ps2 f2,d , · · · , PsK fK,d), Gr,d = (1) (2) diag( Pr1 g1,d , Pr2 g2,d , · · · , PrK gK,d ), and wd and wd are the AWGNs 2 at the destination in the two phases with distribution CN (0K×1 , σw R). Recall that fk, and g,d are the channel coeﬃcients of the sk -r and the r -d links. The signals received in both phases can be collected into the vector

244

6 Cooperative Communications with Multiple Sources (1)T

(2)T

Yd = [yd , yd ]T (1) wd Fs,d 0 x R 0 + = (2) ˆ x 0 Gr,d 0 R w

(6.40)

d

ˇ Hˇ ˇ x + w. ˇ R

(6.41)

By adopting the MMSE MUD at the destination, we can obtain the symbol estimate −1 zMMSE = E xYdH E Yd YdH Yd , d * + ˇ HR ˇ R ˇ HK ˇ xˇ xˇ H ˇ HR ˇ + σ2 R ˇ −1 Yd (6.42) = Kxˇx H w where Kxˇ xˇ

H ˇx ˇ = E x

I Θ Θ I

(6.43)

and Θ = diag(1 − 2ε1 , 1 − 2ε2 , · · · , 1 − 2εK ). The symbols in x are then detected by performing a hard decision on each entry of zMMSE . d (ii) MUD with Amplify-and-Forward Relays When considering the AF scheme, the signal forwarded by relay r can be expressed as z, x ˆ = , E [|z, |2 ] where * + 2 −1 −1 Hs, HH e E |z, |2 = Ps |h, |2 eH s, + σw R when the MMSE MUD is employed at the relay. Let us deﬁne the -th row of C in (6.31) as c, = E[x yH ] =

* +−1 H 2 Ps h∗, eH R RHs, Hs, R + σw R

such that z, = c, y as in (6.36). Suppose that the signals received in both phases at the destination are also passed through the MFB, similar to the DF (1) case, which yields the output yd in Phase I given by (6.38) and the output in Phase II given by

6.2 Code-Division Multiple Access (CDMA) (2)

245

(2)

ˆ + wd , yd = RGr,d x ⎛⎡

⎡

⎤

⎤⎞

w1 RHs,1 ⎢ w2 ⎥⎟ ⎜⎢ RHs,2 ⎥ ⎢ ⎥ ⎥⎟ ⎜⎢ (2) = RGr,d ACr ⎜⎢ ⎥ x + ⎢ .. ⎥⎟ + wd , .. ⎣ . ⎦⎠ ⎦ ⎝⎣ . RHs,K wK

where

(6.44)

(6.45)

+− 1 * A = diag E[|z1,1 |2 ], E[|z2,2 |2 ], · · · , E[|zK,K |2 ] 2

and Cr = diag (c1,1 , c2,2 , · · · , cK,K ) . The MFB outputs of both phases can again be collected into the vector (1) wd RFs,d x+ Yd = (2) , RGr,d ACr Ξ ˇ r + wd RGr,d ACr w ⎡

where

⎢ ⎢ Ξ=⎢ ⎣

RHs,1 RHs,2 .. .

⎤ ⎥ ⎥ ⎥ ⎦

RHs,K ˇ r = [w1T , w2 T , · · · , wK ]T is a K 2 × 1 noise vector with zero mean and and w 2 correlation matrix Rw ˇ r = σw diag(R, R, · · · , R). By employing the MMSE MUD at the destination, we can also obtain the symbol estimate −1 zd = E xYdH E Yd YdH Yd , where

H H H E xYdH = FH s,d R Ξ Cr AGr,d R

and E Yd YdH 2 RFs,d FH s,d R + σw R = H RGr,d ACr ΞFs,d R

(6.46)

H RFs,d ΞH CH r AGr,d R . H H 2 RGr,d ACr (ΞΞH+Rw ˇ r )Cr AGr,d R+σw R (6.47)

In Fig. 6.7 (obtained from [24], the BER performance of the single-user MF receiver and the MMSE MUD is shown for both the DF and the AF scenarios. Here, the system consists of 15 sources, 4 of which are served by their respective relays and the remaining 11 sources transmit directly to the BS without cooperation. The spreading gain is set as N = 15. The noncooperative (NC) case, where no users are served by relays, and the ideal cooperative (IC) case, where the source symbols are assumed to be available

246

6 Cooperative Communications with Multiple Sources

Fig. 6.7 BER performance of cooperative users in cooperative CDMA uplink systems c with designated relays (From Venturino, Wang and Lops; [2006] IEEE ).

at the relays, are also shown for comparison. We can observe that, in the DF case, no spatial diversity gain is achieved by employing the single-user MF receiver at the relays since the relay errors caused by the MAI will eventually dominate the overall BER. On the other hand, in the AF case, no hard decision is made at the relays and, thus, diversity gains can be preserved by employing MMSE MUD at the destination, even when the MF receiver is used at the relays. However, since the MAI is not suppressed at the relays if the single-user MF receiver is adopted, a large part of the relay power may be used to forward interference instead of the source’s message, resulting in loss of coding gains. When the MMSE receiver is employed at the relays, spatial diversity gain can be achieved for both AF and DF systems. So far, we assumed that full CSI is available at the relays when deriving the multiuser detectors. However, it is often prohibitive in practice to acquire the channel of all source-relay links. When only partial CSI is available (e.g., only the statistics of the channel are known), blind techniques, such as the blind MMSE estimator proposed in [24], can be applied alternatively to alleviate the MAI. Readers are referred to [24] for further details.

6.2 Code-Division Multiple Access (CDMA)

247

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Fig. 6.8 Illustration of a K-source CDMA uplink system with L shared relays.

6.2.2 Uplink CDMA with Shared Relays In the previous subsection, we assumed that each relay only forwards messages for a certain designated source. Therefore, the MUD at each relay is used only to detect the symbol of its corresponding source and the signals transmitted by other sources are suppressed. However, with designated relays, the spatial diversity provided by the multiple relay paths and the relays ability to jointly process the sources’ messages are not fully utilized. In this subsection, we consider instead the case of shared relays, where all relays can together forward the messages of all sources. In this case, the messages received from diﬀerent sources can be jointly processed and forwarded to the destination, instead of treating other sources’ signals as interference [13]. Consider a cooperative network with K source users (i.e., s1 , s2 , · · · , sK ) and L relay stations (i.e., r1 , r2 , · · · , rL ) that together forward messages from the sources to the destination as shown in Fig. 6.8. Since no relay is designated to any particular source, there is no constraints on the values of L and K. Again, we assume that the cooperation scheme takes on two phases of transmission and that all sources transmissions are synchronized (i.e., the synchronous CDMA scenario [25]). In Phase I, each source, say sk , transmits a block of M symbols using its unique spreading waveform uk (t). Then, similar to the previous case, the signals received at the relays and the destination are passed through the MFB with ﬁlters matched to the spreading waveforms of the K sources, i.e., u1 (t), u2 (t), · · · , uK (t), resulting in the MFB output given by y [m] = RHs, x[m] + w [m] and (1)

(1)

yd [m] = RFs,d x[m] + wd [m],

248

6 Cooperative Communications with Multiple Sources

respectively, where Fs,d = diag( Ps1 f1,d , Ps2 f2,d , · · · , PsK fK,d ), Hs, = diag( Ps1 h1, , Ps2 h2, , · · · , PsK hK, ), R is the correlation matrix of the (1) spreading waveforms, and w [m] and wd [m] are the AWGNs at relay r and 2 destination d with mean 0K×1 and correlation matrix σw R. The MFB output at the relays can then be used to perform an MMSE estimate on the sources’ symbols as given in (6.34). Depending on whether or not a hard decision is made at the relays based on the MMSE estimate, DF and AF relaying schemes can be adopted. In the following, we shall focus our discussions on the DF case while the AF case can be devised similarly. Suppose that the MMSE MUD is employed at the relays to detected the ˆ [m] = [ˆ sources’ symbols. The vector x x,1 [m], . . . , x ˆ,K [m]]T is used to denote the vector of symbols detected at relay r in the m-th symbol period, with the k-th element correspond to that of source sk . The symbols detected over the M symbol periods of a data block can be collected into the matrix ˆ = [ˆ ˆ [M ]]. The advantage of having shared relays is that each X x [1], · · · , x source can now be served by multiple relays and the cooperative strategies introduced for multi-relay systems in Chapter 4 can be employed. In general, the symbols forwarded by relay r can be expressed as ˆ ) = [t [1], · · · , t [M ]], T = κ(X ˆ . The function κ(·) dewhich is a function of the detected symbol matrix X pends on the speciﬁc cooperation strategy that is employed and, thus, will be referred to as the cooperative function. For example, if transmit beamformˆ , ing is performed, the transmitted symbols can be expressed as T = B X where B = diag(β,1 , . . . , β,K ) and β,k is the beamforming coeﬃcient imposed by r on source sk ’s symbols. The set of coeﬃcients β1,k , β2,k , . . ., βL,k across the L relays form the set of beamforming coeﬃcients for the relaying of sk ’s symbols, as described in Chapter 4.2. Selective relaying schemes can be expressed similarly with β,k > 0 if r is chosen to forward sk ’s symbol and β,k = 0, otherwise. Given the cooperation scheme, the m-th column of T , i.e., t [m] = [t,1 [m], · · · , t,K [m]]T , can be viewed as the vector of symbols transmitted by relay r during the m-th symbol period. The k-th element t,k [m] will be transmitted using the spreading waveform of source sk , i.e., uk (t). The set of cooperative functions has to satisfy power constraint L L H =1 Pr = =1 E[t [m] Rt [m]] = Pr . The signal received at destination d during Phase II is given by

(2) yd (t)

=

K M L

(2)

g,d t,k [m]uk (t − mTs ) + wd (t).

(6.48)

m=1 k=1 =1

By passing the signal through the MFB, we will obtain, in the m-th symbol period, the MFB output

6.2 Code-Division Multiple Access (CDMA) (2)

yd [m] =

L

249 (2)

g,d Rt [m] + wd [m],

(6.49)

=1 (2)

2 R). The MFB outputs obtained over the M where wd [m] ∼ CN (0K×1 , σw symbol periods can be collected into the matrix

(2)

Yd =

L

(2)

g,d RT + Wd ,

(6.50)

=1 (2)

(2)

(2)

where Wd = [wd [1], · · · , wd [M ]]. By employing MMSE MUD at both the relays and the destination, MAI can be reduced up to a certain degree, but may still dominate the BER performance at high SNR. However, since the relays are able to obtain an estimate of all sources’ symbols, precoding can be performed at the relays to further enhance the MUD’s ability to mitigate MAI. With proper design of the relay precoder, the sources’ signals can be decorrelated at the destination without noise enhancement, which is achieved by the so-called relay-assisted decorrelating multiuser detector (RAD-MUD) [13] described in the following.

Relay-Assisted Decorrelating MUD Conventionally, signal decorrelation have been performed by imposing a decorrelating ﬁlter R−1 either at the destination or at the relays. The former scheme leads to the decorrelating MUD which has been shown to cause noise enhancement at the destination. The latter scheme is referred to as the zero-forcing precoder [27] where R−1 is used to precompensate for the MAI that is to occur over the relay-to-destination link. However, the ZF precoding scheme results in power ampliﬁcation at the relays that degrades performance when the relay power is constrained. The two schemes perform identically under the same relay power constraint. In contrast to these schemes, RADMUD performs half the decorrelating operation at the relays and half at the destination [13]. By doing so, both the noise enhancement and the power ampliﬁcation problems will be avoided, as we show in the following. The key idea behind RAD-MUD is to decompose the matrix R−1 into two identical matrices and distribute them among the relays and the destination, respectively. The block diagram of the RAD-MUD is illustrated in Fig. 6.9. Speciﬁcally, when R is full rank, we can take the Cholesky decomposition [12] such that R = LLH , where L is a K × K lower triangular matrix. Then, at each relay, the output ˆ ) is pre-multiplied by the matrix L−H before of the cooperative function κ(X being forwarded to the destination. Therefore, the transmitted symbol matrix

250

6 Cooperative Communications with Multiple Sources

Fig. 6.9 The block diagram of RAD-MUD for multiuser forwarding CDMA systems. c (From Huang, Hong and Kuo with modiﬁed notations; [2008] IEEE ).

can now be expressed as

ˆ ). T = L−H κ(X

(6.51)

By (6.50), the MFB output at the destination can be expressed as (2)

Yd =

L

ˆ ) + W . g,d RL−H κ(X d (2)

(6.52)

=1

Then, the received signal in (6.52) is multiplied by L−1 (i.e., the other half of R−1 ), which yields the output ˘ (2) = L−1 Y(2) = Y d d

L

(2)

ˆ ) + W ˘ , g,d κ(X d

(6.53)

l=1 (2) (2) (2) ˘ (2) = L−1 W(2) = [w ˘ d [1], · · · , w ˘ d [M ]]. The noise term w ˘ d [m] = where W d d (2) L−1 wd [m] now has covariance matrix ) ( (2) (2) 2 ˘ d [m]H = σw ˘ d [m]w IK×K E w

and, thus, L−1 can be viewed as a pre-whitening ﬁlter at the destination. Since the noise covariance matrix does not depend on R−1 , the noise enhancement problem is avoided. Moreover, it is also possible to show that the power of the signal transmitted by the relays does not depend on R−1 (see [13]) and, thus, the power ampliﬁcation problem is also avoided. By employing the RADMUD scheme, the relay-to-destination channel is eﬀectively decomposed into K orthogonal channels, one for each source’s signal, and, thus, the multirelay cooperation strategies proposed in Chapter 4 can be employed without interference from other sources. However, it is worthwhile to note that, with limited resources at the relays, the sources may still need to compete for resources at the relays, which may again aﬀect their performance.

6.2 Code-Division Multiple Access (CDMA)

251

Finally, the signals received at the destination in Phases I and II, i.e., (1) (1) (1) ˘ (2) , can be combined to detect the sources’ Yd = [yd [1], . . . , yd [M ]] and Y d symbols. For simplicity, let us consider the case where the cooperative functions are linear, such as the case of transmit beamforming or selective relayˆ ) = B X ˆ ing. In these case, the cooperative function can be written as κ(X with B = diag(β,1 , · · · , β,K ) and, thus, only the signals received in the m(1) (2) ˘ d [m], will contain information regarding th symbol period, i.e., yd [m] and y the source symbol x[m]. These signals can be combined into the single vector (1) (1) RHs,d wd [m] yd [m] x[m] + yd [m] = = L , (6.54) (2) (2) ˘ d [m] ˘ d [m] y w =1 g,d B Θ where Θ = diag(θ,1 , · · · , θ,K ) with θ,k ∈ {−1, 1} being a Bernoulli random variable indicating whether or not the symbol xk [m] is correctly decoded at the relay r . Similar to the case with designated relays (see Section 6.2.1), we have Pr(θ,k = −1) = 1 − Pr(θ,k = −1) = ε,k , where ε,k is the detection error probability. By treating the interference as noise, the error probability can be approximated as ⎛; ⎞ < −1 < H H 2 2 −1 ek ⎟ ⎜< Psk |hk, | ek Hs, Hs, + σw R <2 ⎟. ε,k ≈ Q ⎜ (6.55) = −1 ⎝ ⎠ H H 2 2 −1 1 − Psk |hk, | ek Hs, Hs, + σw R ek When using the MMSE combiner, the destination computes the MMSE estimate of the data vector, i.e., zd [m] = Cd yd , where

(6.56)

−1 Cd = E x[m]ydH [m] E yd [m]ydH [m]

is the result of minimizing the MSE E[ ˆ xd [m]−Cd yd [m] 2 ]. Detection is then performed by taking the hard decision of each element in zd [m]. However, this method requires the computation of the inverse of E yd [m]ydH [m] , which requires complexity O(2K)3 . To reduce the computational complexity, a component-wise MMSE combiner can be applied as proposed in [13]. In this scheme, we ﬁrst compute the MMSE estimate of the source symbols based only on the signal received (1) on the direct path, i.e., yd [m], and combine it component-by-component (2) ˘ d , which is already a decorrelated signal. The MMSE estimate that with y (1) is computed based only on yd [m] is given by

252

6 Cooperative Communications with Multiple Sources

)−1 ) ( ( (1) (1) (1) (1) (1) zd [m] = E x[m]yd [m]H E yd [m]yd [m]H yd [m] −1 H 2 yd [m]. = HH s,d R(RHs,d Hs,d R + σw R) (1)

(6.57)

Since these operations involve only the computation of E[yd [m]yd [m]H ]−1 , the computational complexity is reduced to O(K 3 ). By (6.57), the k-th ele(1) ment of zd [m] can be expressed as (1)

(1)

(1)

zd,k [m] = [ΓD ]k,k xk [m] + ξk [m],

(6.58)

H 2 −1 −1 ) Hs,d and ξk [m] is the combined MAIwhere ΓD = HH s,d (Hs,d Hs,d +σw R plus-Gaussian-noise term. By exploiting the fact that xk [m] is independent (1) (1) of ξk [m] and that E[zd [m]zd [m]H ] = ΓD , one can show that ξk [m] has zero mean and variance [ΓD ]k,k − [ΓD ]2k,k . (1)

(2)

˘ d together into the vector Let us collect the k-th element of zd [m] and y (1) ξk [m] [ΓD ]k,k zd,k [m] (6.59) xk [m] + (2) = L yd,k [m] = (2) w ˘d,k [m] y˘d,k [m] =1 g,d β,k θl,k (1)

(2)

˘ d are then combined under the MMSE The k-th element of zd [m] and y criterion, resulting in the MMSE estimate −1 zk [m] = E xk [m](yd,k [m])H E yd,k [m](yd,k [m])H yd,k ,

(6.60)

where L H ∗ ∗ E xk [m](yd,k [m]) = [ΓD ]k,k , g,d β,k (1 − 2ε,k ) =1

and E yd,k [m](yd,k [m])H ⎡ [Γd ]k,k ⎢ =⎢ L ⎣ [Γd ]k,k g,d β,k (1 − 2ε,k ) =1

L

∗ ∗ [Γd ]k,k g,d β,k (1 =1

L L

⎤ − 2ε,k )

2 g,d h∗ ,d β,k β∗ ,k E [θ,k θ ,k ]+σw

⎥ ⎥. ⎦

=1 =1

Note that E [θ,k θ ,k ] is equal to 1 if = and is equal to (1−2ε,k )(1−2ε,k ), otherwise. Finally, the symbol transmitted by sk is detected by taking a hard decision on zk [m], i.e., xˆd,k [m] = sgn(Re{zk [m]}). In Figs. 6.10 and 6.11, the performance of cooperative MMSE-MUD and of RAD-MUD is shown for diﬀerent cooperation strategies, namely, transmit beamforming (BF), selective relaying (SEL) and distributed space-time coding (DSTC). Cooperative MMSE MUD refers to the case where conven-

6.2 Code-Division Multiple Access (CDMA)

253

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tional MMSE MUD is employed directly at the relays and the destination, without precoding at the relays; whereas RAD-MUD refers to the case where relay-assisted precoding is used to help the decorrelation at the destination, as described in (6.51). In these experiments, the number of sources and the number of relays are both equal to 8, i.e., K = L = 8 and the spreading gain of the spreading codes is also given by N = 8. Each source transmits with equal power Ps and the sum transmit power of the sources is equal to the sum transmit power of the relays, i.e., KPs = Pr = P . Speciﬁcally, in Fig. 6.10, the BER performances of both MUDs are shown for 3 transmit beamforming schemes: (i) beamforming based only on knowledge of the relay-to-destination channels (BF R-D); (ii) BF based on the combined source-to-relay and relay-to-destination channels (BF S-R-D); and (iii) BF with selective relaying, where BF is performed only over a selected group of relays whose receive SNR is above a certain threshold. Details of the cooperation strategies can be found in [13]. We can see that the RADMUD outperforms the Cooperative MMSE-MUD in all cases since it is able to decorrelate the sources’ symbols without noise enhancement or power ampliﬁcation. In fact, the advantages of RAD-MUD are more evident for cooperative strategies that achieve higher diversity gains since the error ﬂoor caused by MAI is lowered considerably. The gains provided by RAD-MUD can also be observed in systems employing DSTC. In fact, a gain of up to 6 dB can be observed at BER= 10−3 .

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6 Cooperative Communications with Multiple Sources

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6.3 Space-Division Multiple Access (SDMA) In addition to conventional TDMA, FDMA, and CDMA techniques, spacedivision multiple access (SDMA) techniques have also received much attention in recent years due to advances in MIMO technology. This scheme utilizes the adjustment of the antenna gains and phases (i.e., beamforming coeﬃcients) at the transmitter and/or the receiver to construct spatial radiation patterns towards the desired source or destination and spatial nulls at other sources’ or destinations’ locations to avoid interference. By treating the relays as a virtual antenna array, SDMA can also be employed by the relays to separate the signals received from simultaneously transmitting sources and forward them to their respective destinations without interference on others. To discuss these issues, we shall consider in this section the ad hoc scenario, where each source has a diﬀerent destination. Consider a cooperative network with K sources transmitting simultaneously to their respective destinations with the help of L relays, as shown in Fig. 6.12. The cooperation takes on two phases of transmission. In Phase I, all K sources simultaneously transmit their data symbols to the relays with no a priori signal separation, either in time, frequency, or code. Suppose that xs = [xs,1 , xs,2 , . . . , xs,K ]T is the vector consisting of the symbols transmitted by the K sources, and E[|xs,k |2 ] = Psk . Then, the signal received at each

6.3 Space-Division Multiple Access (SDMA)

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relay, say relay r , will be a mixture of the signals transmitted by the sources, i.e., K yr, = hk, xs,k + w , k=1 2 where hk, is the channel coeﬃcient between sk and r and w ∼ CN (0, σw ) is the AWGN at relay r . The signals received over the L relays can be gathered into the L × 1 vector given by

yr = [yr,1 , . . . , yr,L ]T = Hs,r xs + wr ,

(6.61)

where Hs,r is an L × K channel matrix with the ( , k)-th element being 2 {Hs,r },k = hk, and wr is the AWGN with distribution CN (0L×1 , σw IL×L ). Since no additional degree of freedom is available at each relay to resolve the mixture of signals, the relays would not be able to reliably detect the source symbols and, thus, DF schemes do not appear to be viable. However, one can consider the AF relaying scheme, where each relay forwards in Phase II a scaled version of the received signal to the destination. The signal forwarded by relay r can be deﬁned as xr, = a yr, . Let a = [a1 , a2 , · · · , aL ]T be the vector of gain coeﬃcients at the relays and let Da = diag(a1 , a2 , . . . , aL ). Thus, the symbols transmitted by all relays form the vector xr = [xr,1 , . . . , xr,L ]T = Da yr . The signals received at the K destinations in Phase II can be written as yd = [yd,1 , . . . , yd,K ]T = Gr,d xr = Gr,d Da Hs,r xs + Gr,d Da wr + wd ,

(6.62)

256

6 Cooperative Communications with Multiple Sources

where Gr,d is a K × L channel matrix with {Gr,d }k, = g,k (i.e., the channel coeﬃcient between relay r and destination dk ) and wd is the AWGN at the 2 destinations with distribution CN (0K×1 , σw IK×K ). By (6.62), it appears that the signals received at each destination is still a mixture of signals transmitted by all sources, which is the cause of MAI. However, the MAI can be mitigated via appropriate choices of the amplifying gains a1 , . . ., aL . (i) MMSE Gain Coeﬃcients At each destination, the received symbols will be scaled with a constant according to the vector of gain coeﬃcients at relays a to minimize the mean square error (MSE) between xs and a scaled version of yd . To determine the value of a, one can utilize the MMSE criterion where the minimum MSE between xs and the scaled version of yd given a is minimized subject to a total power constraint at the relays [4]. The problem can be formulated as follows: ) ( 2 min E xs − Dq yd , (6.63) a,q subject to E Da yr 2 ≤ Pr , (6.64) where Dq = diag(q1 , q2 , . . . , qK ) is a diagonal matrix containing in its k-th diagonal element the factor qk that is the scale imposed at destination dk (for the purpose of minimizing the MSE the estimate on source sk ’s transmitted symbol). The value of the vector q = [q1 , q2 , . . . , qK ] must be determined along with the amplifying gains a. The Lagrange function can be written as L(a, q, λ) = E xs − Dq yd 2 + λ(E Da yr 2 − Pr ), (6.65) where λ is the Lagrange multiplier. The optimal values of a and q can be found by taking the derivatives of L(a, q, λ) with respect to these vectors and setting them to zero. In general, the optimal values of a and q do not yield closed forms and must be obtained numerically [4]. However, a special case exists when q1 , q2 , . . ., qK , are identical. More speciﬁcally, when q1 = q2 = · · · = qK q (i.e., Dq = qI), the optimal value of a that minimizes the MSE is given by [4] ˜MMSE aMMSE = λ− 2 a 1

(6.66)

˜MMSE and the Lagrange multiplier λ are given by where the vector a ˜ MMSE = (B A∗ )−1 c, a and λ=

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6.3 Space-Division Multiple Access (SDMA)

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(ii) Zero-Forcing (ZF) Gain Coeﬃcients When the number of relays is much greater than the number of sourcedestination pairs such that L ≥ K(K − 1) + 1, the gain coeﬃcients can be chosen to completely eliminate the MAI at each destination. This leads to so-called zero-forcing (ZF) design [10] as described in the following. Let us ﬁrst rewrite the received signal in (6.62) as yd = Gr,d Da Hs,r xs + Gr,d Da wr + wd : s,d xs + w, : =H

(6.69)

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To eliminate the MAI at each destination, the matrix Da = diag(a) must : s,d is diagonalized. That is, we have to force all oﬀbe chosen such that H diagonal terms in Gr,d Da Hs,r to zero with the choice of Da (or a). By letting gm be the m-th row of Gr,d and let hk be the k-th column of Hs,r , the ZF condition can be expressed as gm Da hk = 0,

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∀k = m.

258

6 Cooperative Communications with Multiple Sources

The row vectors (gm hTk ), for all k = m, can be stacked together to form the K(K − 1) × L matrix HI . The ZF condition can then be expressed as HI a = 0,

(6.70)

which means that a should fall within the null space of HI . Let NHI be an L × [L − K(K − 1)] matrix whose columns form a basis for the null space of HI . The ZF solution can then be expressed as a linear combination of the columns of NHI , i.e., aZF = NHI q,

(6.71)

where q = [q1 , . . . , qL−K(K−1) ]T is a vector of the weighting coeﬃcients. The value of q can be further chosen to optimize system performance. By choosing aZF according to (6.71), the signal received at destination dk , i.e., the k-th term of yd , can be expressed as : s,d ]k,k xs,k + (gk aT )wr + wd , yd,k = [H

(6.72)

and, thus, the instantaneous SNR of the k-th source-destination pair can be computed as Ps |[Hs,d ]k,k |2 +. SNRk = 2 * (6.73) σw 1 + gkT aZF 2 To further improve system performance, the weighting vector q can be chosen to maximize the system sum rate. Speciﬁcally, the optimal weighting coeﬃcients are found by max q

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(6.74)

where * + ∗ ∗ 2 E Da yd 2 = qT NTHI Ps Hs,r HH s,r IL×L + σw IL×L NHI q . The optimal value of q does not have a close form and, thereby, must be found numerically. In Fig. 6.13, the average sum rate is shown for cooperative SDMA systems with K = 2 and K = 4 source-destination pairs. The performance is shown for both MMSE and ZF designs, i.e., using aMMSE and aZF derived from (6.66) and (6.71), respectively. In the simulations, source s1 and destination d1 are assumed to be closer to the relays such that E |h1, |2 = E |g,1 |2 = 100 for all , and the other to be equal distance from the relays users are assumed such that E |hk, |2 = E |g,k |2 = 1 for all and for all k = 1. The label “MMSEequal ” indicates the sum rate achieved by using MSE gain factors with qk = q for all k, and the label “MMSEind ” indicates the sum rate achieved by using the MMSE gain factors with optimized values of qk , which is diﬀerent

6.4 Partner Selection Strategies

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for each individual. We can observe that the MMSE relaying with optimized scaling factors at the destination performs the best. Also, one should note that ZF gain factors cannot be found when the number of relays is smaller than the number of source-destination pairs. However, it can achieve performance that is comparable with MMSE relaying when the number of relays is suﬃciently large.

6.4 Partner Selection Strategies When considering cooperation with designated relays in previous sections, we assumed that the relays cooperating with each source are assigned according to some predetermined partner selection strategy. However, the problem remains as to what the selection criterion should be and how the partners should be selected in order to fully exploit the cooperative advantages. This is diﬀerent from the study on selective relaying (see Chapters 4), where we assumed that only one user serves as the source and that it can select any other user to serve as its relay. In the partner selection problem, we are considering systems that may contain multiple sources at any given time instant. Once a relay (or some user willing to serve as a relay) is selected by a certain source,

260

6 Cooperative Communications with Multiple Sources

the relay will no longer be available for other sources in the case of designated relays, or will not be able to provide its full services if it is shared by more than one source. Hence, a favorable partner selection policy must take into consideration the cooperative advantages from a system-wide perspective. In pair-wise cooperative schemes, where users in each pair may take turns in acting as the source and the relay, the partner selection strategy must also allow cooperation to be mutually beneﬁcial for both users in the pair. In the following, both centralized and decentralized partner selection strategies will be introduced. We shall let N = {1, 2, . . . , N } be the set of users in the network, S ⊂ N be the set of users acting as the source at the given time instant, and R ⊂ N be the set of users willing to serve as relays.

6.4.1 Centralized Partner Selection Strategy Recall that, in selective relaying (SR) schemes [5, 6], we assumed that only one user is acting as the source (which is denoted by s) and that it is allowed to ask any user or subset of users in R to help forward its message. Therefore, SR schemes can be viewed as a special case of partner selection where there is only one source. One example given in Chapter 4.3 is the DF selective relaying scheme, where the source s chooses the relay that has the best relayto-destination channel among those in the decoding set D(s) = { ∈ R : 2 P |hs, |2 /σw ≥ 22R − 1}. Here, P is the transmission power of the source, R is the transmission rate, and h,d is the channel coeﬃcient between user and the destination. The user selected by s to serve as its relay is given by 2 r(s) = arg max P |h,d |2 /σw .

(6.75)

∈D(s)

However, when there are multiple users acting as sources, i.e., |S| > 1, performing the selection individually for each source based on (6.75) is suboptimal. This is because, by doing so, a relay (especially the one with the best relay-to-destination channel) may be selected by multiple sources and it transmission power must be divided to forward the messages of diﬀerent sources, reducing the cooperative beneﬁts received by each source. In this case, it may be better for a user in S to choose instead the second best relay so that contention of relay power is avoided. Hence, the optimal partner selection strategy must jointly consider the selection among all source users. Note that the set of sources S and the set of relays R may not be mutually