Low Complexity MIMO Detection

Low Complexity MIMO Detection Lin Bai • Jinho Choi Low Complexity MIMO Detection 123 Lin Bai School of Electronic...

Author: Lin Bai | Jinho Choi

134 downloads 1205 Views 2MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Low Complexity MIMO Detection

Lin Bai • Jinho Choi

Low Complexity MIMO Detection

123

Lin Bai School of Electronic and Information Engineering Beihang University F-627, New Main Building No.37 XueYuan Road, HaiDian District Beijing 100191 People’s Republic of China [email protected]

Jinho Choi School of Engineering Swansea University, Singleton Park Room 123, Digital Technium, Singleton Park Swansea SA2 8PP United Kingdom [email protected]

ISBN 978-1-4419-8582-8 e-ISBN 978-1-4419-8583-5 DOI 10.1007/978-1-4419-8583-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011943879 © Springer Science+Business Media, LLC 2012 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 our families and friends

Foreword

What is the most important emerging technology leading to high data rate wireless services? With scarce wireless spectrum, the use of multiple antennas is becoming the key foundation to achieve the requirement. My colleagues, Prof. Bai and Prof. Choi have worked on this topic for many years. They have made good achievements and published a number of papers within this topic. In this book, they share their key findings. With signal detection methods now representing a key application of signal processing methods to communication systems, this book provides a range of important techniques for signal detection when multiple transmitted and received signals are available. In this book, various optimal and suboptimal signal detection methods are explained in the context of multiple-input multiple-output (MIMO) systems, including list decoding and lattice reduction (LR)-aided detection, while various user selection schemes are also discussed within multiuser systems. Those techniques are then analyzed using performance analysis tools. With a carefully balanced blend of theoretical elements and applications, this book is ideal for both graduate students and practicing engineers in wireless communications. All the techniques introduced in this book are quite new. Furthermore, this book makes an easy-to-follow presentation from the elementary to the profound level. Beijing

Quan Yu Academician Chinese Academy of Engineering

vii

Preface

In order to improve the spectral efficiency in wireless communications, multiple antennas are employed at both transmitter and receiver sides, where the resulting system is referred to as the multiple-input multiple-output (MIMO) system. In MIMO systems, it is usually required to detect signals jointly as multiple signals are transmitted through multiple signal paths between the transmitter and the receiver. This joint detection becomes the MIMO detection. The MIMO detection can be performed by an exhaustive search method for the maximum likelihood (ML) detection. Unfortunately, although this method provides the optimal performance, it is impractical for a number of real systems since its complexity grows exponentially with the number of transmit antennas. For the case of MIMO channels in cellular systems where the transmitter is a base station and the receiver is a mobile terminal, since the receiver usually has a limited computing power for symbol detection, the use of ML detection based on an exhaustive search or those with high computational complexity becomes impossible. To avoid this prohibitively high computational complexity, computationally efficient suboptimal MIMO detection methods are investigated, including linear detectors that take the signals from the other antennas as the interference; but, poor performance is expected due to a high date error rate. Therefore, it is desired to develop MIMO detection methods that have near optimal performance as well as low computational complexity. In this book, we attempt to explain such low complexity MIMO detectors. So far, there are many existing books related to MIMO systems. To be different from those books, our book focuses on low complexity MIMO symbol detection itself. Although our book is very specific, we have adopted an easy-to-follow presentation from the elementary to the profound level. Furthermore, we include a number of recent research outcomes that are also useful for those experts in this area. Our group has worked on the design of low complexity MIMO detection for many years and has produced various new results on low complexity MIMO detection with the ideas of list decoding and lattice basis reduction. In addition, as an extension, multiuser MIMO and the corresponding strategies are also investigated. This book includes not only our research outcomes but also other recent research ix

x

Preface

outcomes that could be very useful to practitioners and postgraduate students who want to learn new outcomes of low complexity MIMO detectors in the field of wireless communications. This book systematically introduces the signal detection in MIMO systems. It has been written for the reader who wants to become an expert from a beginner in the field of MIMO detection. In addition, it is suitable for postgraduate students who have some fundamental knowledge of wireless communications, and for R&D personnel who works in MIMO area. Beijing Swansea

Lin Bai Jinho Choi

Acknowledgments

We would like to thank many people for supporting this work, in particular: Q. Yu (Chinese Academy of Engineering), J. Zhang (Beihang University), C. Chen (Peking University), W. Xing (Swansea University), J. He (Swansea University), C. Ling (Imperial College), and W. Guan (Swansea University). They helped us by providing valuable comments and proofreading for the remaining errors, typos, unclear passages, and weaknesses is ours. Special thanks go to those people who inspire and encourage us all the time: Q. Yu (Chinese Academy of Engineering) for guidance and encouragement as the mentor, J. Zhang (Beihang University) for generous support, C. Chen (Peking University) for long-term friendship, and many others including our students, J. Xie, Q. Li, and H. Liu, for useful discussions. Then, we want to express our appreciation to our parents, families, and friends. Without their support, we can barely make the achievement. Finally, we deeply thank Editor B. Kurzman at Springer and Project Manager E. Ahmad at SPi Global, who were always there with us, for their wonderful help during the completion of the book. This work has been supported by the China Postdoctoral Science Foundation and the China National 973 project under the grant no. 2009CB320403.

xi

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 MIMO Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Point to Point MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Multiuser MIMO .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Outline .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Part I

1 1 4 8 10

Point to Point MIMO

2 Background of MIMO Detection .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 ML Detection.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 Exhaustive Search Approach .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.2 Performance Analysis . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Linear Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 ZF Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 MMSE Detection . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.3 Performance Analysis . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 SIC Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.1 QR Factorization .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.2 ZF-SIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.3 MMSE-SIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.4 Ordering .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.5 Performance Analysis . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 BER Versus SNR Simulation Results . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Conclusion and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

15 15 16 16 17 20 20 21 22 23 24 24 28 29 31 36 42

3 List and Lattice Reduction-Based Methods . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 List-Based Detection .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 Detection Algorithms .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.2 Ordering .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.3 Subdetectors.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

43 43 43 46 48 xiii

xiv

Contents

3.1.4 Performance Analysis . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Lattice Reduction-Based Detection . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 MIMO Systems with Lattice . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.2 Lattice Reduction-Based MIMO Detection . . . . . . . . . . . . . . . . . . . 3.2.3 Lattice Reduction Schemes for Two Basis Systems . . . . . . . . . . 3.2.4 Gaussian Lattice Reduction for Two Basis Systems . . . . . . . . . . 3.2.5 LLL and CLLL Algorithms . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.6 Performance Evaluation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Conclusion and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

52 54 56 56 58 64 69 74 79 86 90

4 Partial MAP-Based Detection . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 MAP Detection .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Partial MAP Detection .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.1 The Case of 2 2 MIMO . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.3 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Partial MAP-Based List Detection . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 System Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 The Case of List Length Q D 1 . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.3 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.4 Algorithm of the Partial MAP-Based List Detection . . . . . . . . . 4.3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Conclusion and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

91 91 92 92 93 97 99 100 101 103 107 110 112

5 Lattice Reduction-Based List Detection . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Lattice Reduction-Based List Detection . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.1 Algorithm Description .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.2 Lattice Reduction-Based Detection .. . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.3 List Generation in the LR Domain .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.4 Impact of List Length.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.5 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.6 Components of the LR-Based List Detection . . . . . . . . . . . . . . . . . 5.1.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Error Probability-Based Column Reordering Criteria .. . . . . . . . . . . . . . . . 5.2.1 System Model with CRIS . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.2 Detection Algorithm with CRIS . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.3 OD-CRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.4 EP-CRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Conclusion and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

113 114 114 116 117 118 122 122 130 131 133 134 135 136 137 139

6 Detection for Underdetermined MIMO Systems . . . . .. . . . . . . . . . . . . . . . . . . . 141 6.1 Joint Detection for Underdetermined MIMO Systems . . . . . . . . . . . . . . . . 143 6.1.1 System Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 143

Contents

6.2

6.3

6.4

6.5 Part II

xv

6.1.2 Existing Approaches .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.3 Prevoting Cancellation-Based MIMO Detection .. . . . . . . . . . . . . Selection for Prevoting Vectors Depending on SubDetectors .. . . . . . . . 6.2.1 Selection Criterion with Linear Detector . .. . . . . . . . . . . . . . . . . . . . 6.2.2 Selection Criteria with LR-Based Linear and SIC Detectors.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Performance Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.1 Diversity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.2 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Simulation Results and Discussions . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.2 Discussion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Conclusion and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

144 146 147 148 148 150 150 157 158 158 161 165

Multiuser MIMO

7 Selection Criteria of Single User . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 User Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.1 Maximum Mutual Information Criterion . .. . . . . . . . . . . . . . . . . . . . 7.2.2 User Selection Criteria for ML Detector . .. . . . . . . . . . . . . . . . . . . . 7.2.3 User Selection Criterion for Linear Detectors .. . . . . . . . . . . . . . . . 7.2.4 User Selection Criteria for LR-Based Detectors . . . . . . . . . . . . . . 7.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Conclusion and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

169 169 170 171 172 175 177 181 183

8 Selection Criteria of Multiple Users . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 User Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.1 ML and Linear Selection Criteria . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.2 LR-Based Linear and SIC Selection Criteria . . . . . . . . . . . . . . . . . . 8.3 LR-Based Greedy User Selection Using an Updating Method . . . . . . . 8.3.1 LR-Based Greedy User Selection . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.2 A Complexity Efficient Method for LR Updating . . . . . . . . . . . . 8.4 Diversity Analysis and Numerical Results. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4.1 Diversity Gain Analysis from Error Probability.. . . . . . . . . . . . . . 8.4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.5 Conclusion and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

185 187 189 190 191 193 193 197 203 204 210 215

9 Conclusion of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 217 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 219 About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 225 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 227

List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4 Fig. 1.5

A SISO system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A 2 2 MIMO system. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Signal constellations and binary codes . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . MIMO detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A multiuser MIMO system . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Fig. 2.1

BER performance of ZF-SIC for each layer in a 4 4 MIMO system, where BPSK is used for signaling . . . . . . . . . . . . . . . . . . BER performance of conventional detectors in a 4-QAM 2 2 MIMO system . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER performance of conventional detectors in a 16-QAM 2 2 MIMO system . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER performance of conventional detectors in a 64-QAM 2 2 MIMO system . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER performance of conventional detectors in a 4-QAM 4 4 MIMO system . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER performance of conventional detectors in a 16-QAM 4 4 MIMO system . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER performance of conventional detectors in a 64-QAM 4 4 MIMO system . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 2.7

Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5

BER performance of different detectors in a 16-QAM 2 2 MIMO system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER performance of different detectors in a 16-QAM 4 4 MIMO system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER performance of list-based detectors using different types of subdetectors in a 16-QAM 4 4 MIMO system. . . . . . . . . . . The decision boundaries of ZF detection with a lattice generated by the basis H in (3.53) . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The decision boundaries of ZF detection with a lattice generated by the basis G in (3.54) . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2 2 4 6 9

35 39 39 40 40 41 41

54 55 55 60 61 xvii

xviii

List of Figures

Fig. 3.6

BER performance of various detectors in a 4-QAM 2 2 MIMO system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.7 BER performance of various detectors in a 16-QAM 2 2 MIMO system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.8 BER performance of various detectors in a 64-QAM 2 2 MIMO system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.9 BER performance of various detectors in a 4-QAM 4 4 MIMO system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.10 BER performance of various detectors in a 16-QAM 4 4 MIMO system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.11 BER performance of various detectors in a 64-QAM 4 4 MIMO system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5

Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4

Fig. 5.5

Fig. 5.6

Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 6.4

The lower bound of Pcond . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Bounds of Pcond for different list length with N1 D 2 .. . . . . . . . . . . . . . Bounds of Pcond for different list length with N1 D 4 .. . . . . . . . . . . . . . BER performance of various detection methods in Table 4.1 for a 16-QAM 2 2 system . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER performance of various detection methods in Table 4.1 for a 16-QAM 4 4 system . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER versus Eb =No of different MIMO detectors in a 4 4 MIMO system (N1 D N2 D 2) with 4-QAM signaling .. . BER versus Eb =No of different MIMO detectors in a 4 4 MIMO system (N1 D N2 D 2) with 16-QAM signaling .. . . . BER versus Eb =No of different MIMO detectors in a 4 4 MIMO system (N1 D N2 D 2) with 64-QAM signaling . BER versus Eb =No of different MIMO detectors in a 4 4 MIMO system (N1 D N2 D 2) with 4-QAM signaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER versus Eb =No of different MIMO detectors in a 4 4 MIMO system (N1 D N2 D 2) with 16-QAM signaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER versus Eb =No of different MIMO detectors in a 4 4 MIMO system (N1 D N2 D 2) with 64-QAM signaling . BER versus Eb =N0 of different detectors listed in Table 6.1 for 4-QAM, M D 4 and N D 2 . . . . . . .. . . . . . . . . . . . . . . . . . . . BER versus Eb =N0 of different detectors listed in Table 6.1 for 4-QAM, M D 4 and N D 3 . . . . . . .. . . . . . . . . . . . . . . . . . . . BER versus Eb =N0 of different detectors listed in Table 6.1 for 16-QAM, M D 3 and N D 2 . . . . .. . . . . . . . . . . . . . . . . . . . BER versus Eb =N0 of different detectors listed in Table 6.1 for 16-QAM, M D 4 and N D 3 . . . . .. . . . . . . . . . . . . . . . . . . .

87 87 88 88 89 89 98 106 107 110 111

131 132 133

138

139 140

160 161 162 163

List of Figures

xix

Fig. 6.5

BER versus Eb =N0 of Detector III and Detector XI listed in Table 6.1 for ve D f0; 0:02; 0:05g with 4-QAM, M D 4 and N D 2. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 164

Fig. 7.1

Block diagram for multiuser MIMO uplink channels of K users equipped per user with M transmit antennas and the BS equipped with N receive antennas . .. . . . . . . . . . . . . . . . . . . . BER performance of various multiuser MIMO systems with 4-QAM, M D N D 4, and K D 10 .. . . . . . .. . . . . . . . . . . . . . . . . . . . BER performance of various multiuser MIMO systems with 16-QAM, M D N D 4, and K D 10 . . . . . .. . . . . . . . . . . . . . . . . . . . BER performance of various multiuser MIMO systems with 64-QAM, M D N D 4, and K D 10 . . . . . .. . . . . . . . . . . . . . . . . . . .

Fig. 7.2 Fig. 7.3 Fig. 7.4

Fig. 8.1

Fig. 8.2

Fig. 8.3

Fig. 8.4

Fig. 8.5

Fig. 8.6

Fig. 8.7

Block diagram for multiuser MIMO uplink channels of K D 10 users equipped per user with P transmit antennas and the BS equipped with N receive antennas, while 4 users are selected to transmit signals to the BS during a time slot interval.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Block diagram of virtual antennas in a single user MIMO system, while 4 AS are selected among 10 AS to transmit signals to the BS during a time slot interval.. . . . . . . . . . . . BER versus Eb =N0 of the multiuser MIMO systems listed in Table 8.9 for the case of .M; P / D .4; 1/ and .M; P / D .2; 2/ (16-QAM, K D 5, N D 4) .. . .. . . . . . . . . . . . . . . . . . . . BER versus Eb =N0 of the multiuser MIMO systems listed in Table 8.9 for the case of .M; P / D .4; 1/ (16-QAM, K D 5, N D 4) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER versus Eb =N0 of the multiuser MIMO systems listed in Table 8.9 for the case of .M; P / D .2; 2/ (16-QAM, K D 5, N D 4) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER versus K of the multiuser MIMO systems listed in Table 8.9 for the case of .M; P / D .4; 1/ (16-QAM, Eb =N0 D 12 dB, N D 4) .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . BER versus K of the multiuser MIMO systems listed in Table 8.9 for the case of .M; P / D .2; 2/ (16-QAM, Eb =N0 D 12 dB, N D 4) .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

170 181 182 183

188

189

212

213

214

215

216

List of Tables

Received signal vectors of a 2 2 MIMO system . . . . . . . . . . . . . . . . Received signal vectors of a 2 users multiuser MIMO system under a certain user selection strategy .. . . . . . . . . . . . . . . . . . . .

11

Table 2.1 Table 2.2 Table 2.3 Table 2.4 Table 2.5 Table 2.6

Candidate vectors of 2 transmit antennas with 4-QAM . . . . . . . . . . . dml corresponding to s in Table 2.1 .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Detection errors at symbol and bit levels . . . . . . .. . . . . . . . . . . . . . . . . . . . ML detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . MMSE detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . MMSE-SIC detection . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

16 20 36 37 38 38

Table 3.1 Table 3.2

List generation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The average value of column swapping per iteration when the CLLL is employed for different MIMO channels (N D 8 and M D 2; 3; : : : ; 8).. . . . . . .. . . . . . . . . . . . . . . . . . . .

44

Different MIMO detection methods .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The average complexity of various detection methods in Table 4.1 for a 16-QAM 2 2 system . . . . . .. . . . . . . . . . . . . . . . . . . . The average complexity of various detection methods in Table 4.1 for a 16-QAM 4 4 system . . . . . .. . . . . . . . . . . . . . . . . . . . The average list length of the partial MAP-based list detection with different SNR for 16-QAM 2 2 and 4 4 MIMO systems, where N1 D N2 . . . . . . . .. . . . . . . . . . . . . . . . . . . .

109

Table 1.1 Table 1.2

Table 4.1 Table 4.2 Table 4.3 Table 4.4

Table 5.1 Table 5.2 Table 5.3 Table 5.4 Table 5.5

Signal and parameters for the LR-based detection in (5.5) and (5.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Complexity analysis of different detectors .. . . .. . . . . . . . . . . . . . . . . . . . Components of the LR-based list detection . . . .. . . . . . . . . . . . . . . . . . . . Gram–Schmidt algorithm . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Householder reflection algorithm .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

5

86

109 109

110

117 122 123 124 125 xxi

xxii



List of Tables

Complexity comparison of Gram–Schmidt algorithm and Householder reflection algorithm . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 125 Gaussian LR algorithm .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 126 L–R decomposition.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 127 Different detection methods for underdetermined MIMO systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 159 Complexity comparison of CSel for Detectors IX, X, and XI, listed in Table 6.1 .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 163 Complexity comparison of different detectors listed in Table 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 164

Matrices with the LR at each user selection.. . .. . . . . . . . . . . . . . . . . . . . Size reduction in CLLL . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Column swapping in CLLL . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Basis updating in UBLR (Part I).. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Basis updating in UBLR (Part II) .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The UBLR (based on the CLLL) algorithm at the mth user selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 8.7 The average value of in a multiuser MIMO system when the CLLL-based MMSE-SIC detector is used with the LRG and UBLRG user selection, where K D 10, N D 8, and .M; P / D .8; 1/ .. . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 8.8 The average value of in a multiuser MIMO system when the CLLL-based MMSE-SIC detector is used with the LRG and UBLRG user selection, where K D 10, N D 8, and .M; P / D .4; 2/ . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 8.9 Nine multiuser MIMO systems employed in the simulations .. . . . Table 8.10 The average complexity of multiuser MIMO systems listed in Subsect. 8.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

196 198 199 200 201 202

202

203 211 211

Acronyms

APP APRP AWGN AS BER BPSK BS cdf CLLL CMs CRC CRIS CSCG CSI DFE DMT DRC EP-CRC flops GS GSD ISI LAPPR LBR LLL LLR LR LRG MAP MD MDist

A posteriori probability A priori probability Additive white Gaussian noise Antenna subset Bit error rate Binary phase shift keying Base station Cumulative density function Complex-valued LLL Complex multiplications Column reordering criteria Column reordering index set Circular symmetric complex Gaussian Channel state information Decision feedback equalizer Diversity multiplexing trade-off Dimension reduction condition Error probability based CRC Floating point operation Gram–Schmidt Generalized sphere decoding Intersymbol interference Logarithms of a posteriori probability ratios Lattice basis reduced Lenstra–Lenstra–LovKasz Log-likelihood ratio Lattice reduction LR-based greedy Maximum a posteriori probability Max–min diagonal term Max–min distance xxiii

xxiv

ME MIMO ML MMI MMMSE MMSE MSE OD-CRC ODR OFDM PAM pdf PDR PEP PVC PVS QAM SDMA SER SIC SINR SISO SNR SSE SVP TSD-CR UBLR UBLRG UILS V-BLAST VLSI ZF

Acronyms

Max–min eigenvalue Multiple-input multiple-output Maximum likelihood Maximum mutual information Min–max mean square error Minimum mean square error Mean square error Orthogonality deficiency-based CRC Optimal decision region Orthogonal frequency division multiplexing Pulse amplitude modulation Probability density function Probability of dimension reduction Pairwise error probability Prevoting cancellation Postvoting vector selection Quadrature amplitude modulation Space division multiple access Symbol error rate Successive interference cancellation Signal to interference plus noise ratio Single-input single-output Signal-to-noise ratio Sum of squared error Shortest vector problem Tree search decoder-column reordering Updated basis LR UBLR-based greedy Underdetermined integer least squares Vertical Bell laboratories layered space–time Very large scale integration Zero forcing

Notations

A=a Ar =ar AT ; AH ; A ŒAp;q A.a W b; c W d / A.W; n/ A.n; W/ Tr.A/ det.A/ adj.A/ D.A/ min .A/ L.A/ EŒ CN .m; C/ log./ 0 kk kkF dˇc bˇc jˇj n

(Boldface upper/lower letters) complex-valued matrix/vector (Boldface upper/lower letters) real-valued matrix/vector Transpose, Hermitian transpose, Pseudo inverse, respectively The .p; q/th element of A The submatrix of A with the elements obtained from rows a; : : : ; b and columns c; : : : ; d The nth column vector of A The nth row vector of A The trace operation of a square matrix A Determinant of matrix A Adjoint of matrix A Length of the shortest nonzero vector of the lattice generated by A Minimum eigenvalue of A Lattice generated by A Statistical expectation Real and imaginary parts Inner product of two vectors a and b Complex Gaussian vector distribution with mean m and covariance C Natural logarithm Matrix with all entries of 0 2-norm The Frobenius norm The nearest integer to ˇ The closest integer which is smaller than ˇ Absolute value of scalar ˇ Set minus

xxv

xxvi

Notations

I ˚n k.1/ ; k.2/ ; : : : erfc.x/

An n n identity matrix The collection set of k.1/ ; k.2/ ; : : : Complementary error function of x, i.e., erfc.x/ D

f9x W f .x/g Z

e z dz There is at least one x such that a function of x, f .x/, is true Set of integer numbers

2

p2

R C1 x

Chapter 1

Introduction

1.1 MIMO Systems There is ever growing demand of wireless services of higher data rates. Unfortunately, a conventional single-input single-output (SISO) system where the transmitter and receiver are equipped with single antenna as shown in Fig. 1.1 could have limitations to support higher data services. The received signal of such a system can be expressed as y D hs C n; (1.1) where h, s, and n denote the channel gain, transmitted symbol, and noise, respectively. The channel capacity (in bits per second per Hz) can be given by jhj2 Ps C D E log 1 C 2 ! E jhj2 Ps log 1 C ; 2

(1.2)

where Ps D E jsj2 and 2 D E jnj2 as the expectation is carried out with respect to jhj2 . The inequality results from Jensen’s inequality. As the capacity grows logarithmically with the signal-to-noise ratio (SNR), Ps = 2 , in order to have a high transmission rate, we need to have either high SNR or wide bandwidth. In wireless communications, since there are always limitations to increase SNR due to propagation loss, the bandwidth should be wide enough to support high data rate services. However, the scarce wireless spectrum has posed a big challenge on wireless communication systems with increasing data rate demands. To improve the spectral efficiency [1] in wireless communications, multiple antennas are employed at both transmitter and receiver, where the resulting system is called the multipleinput multiple-output (MIMO) system [2]. A 2 2 MIMO system is illustrated in

L. Bai and J. Choi, Low Complexity MIMO Detection, DOI 10.1007/978-1-4419-8583-5 1, © Springer Science+Business Media, LLC 2012

1

2

1 Introduction

Fig. 1.1 A SISO system

Fig. 1.2 A 2 2 MIMO system

Fig. 1.2. Since each receive antenna is able to receive signals from both transmit antennas, the received signals of two antennas at the receiver are, respectively, given by y1 D h11 s1 C h12 s2 C n1 ; y2 D h21 s1 C h22 s2 C n2 ;

(1.3)

where hij , sj , and ni represent channel gain from the j th transmit antenna to the i th receive antenna, transmitted symbol from the j th transmit antenna, and additive noise at the i th receive antenna, respectively. Denote by superscript T the transpose operation. Let y D Œy1 y2 T . Using the principle of matrix multiplication, the received signal vector of such a MIMO system is given by y D Hs C n:

(1.4)

h h Here, we have the MIMO channel matrix H D 11 12 , the transmit signal vector h21 h22 s n s D 1 , and the noise vector n D 1 . s2

n2

Equation (1.4) can also be used to represent the received signal vector of a MIMO system with M transmit and N receive antennas. In this case, the channel matrix is given by 3 2 h11 h12 h1M 6 h21 h22 h2M 7 7 6 (1.5) HD6 : : :: 7 ; : :: :: 4 :: : 5 hN1 hN 2 hNM while the transmit signal vector and noise vector are denoted by s D Œs1 ; s2 ; : : : ; sM T and n D Œn1 ; n2 ; : : : ; nN T , respectively. The received signal vector, denoted by y D Œy1 ; y2 ; : : : ; yN T , can also be expressed as in (1.4).

1.1 MIMO Systems

3

As a channel modeling method, additive white Gaussian noise (AWGN) channel is normally considered to build up MIMO systems for theoretical analysis and simulations. With AWGN channel, the noise vector, n, is assumed to be a zeromean circular symmetric complex Gaussian (CSCG) random vector with EŒnnH D N0 I, where I and superscript H represent an identity matrix and the Hermitian transpose, respectively. Note that n CN .0; R/, denotes a zero-mean CSCG random vector with covariance matrix R. It was known that the capacity of MIMO channels in (1.4) can linearly increase with the minimum of the numbers of transmit and receive antennas under certain conditions (where the channel gains, hij , are independent zero-mean CSCG random variables) [3, 4]. Therefore, for a given bandwidth, the more antennas, the higher transmission rates can be achieved without increasing the transmission power significantly. To exploit this increased capacity, we need to employ efficient signal modulation schemes for MIMO systems. Pulse amplitude modulation (PAM) and quadrature amplitude modulation (QAM) have been developed as the modulation methods of digital signals with MIMO systems. For example, we can have the signal alphabet of A-ary PAM as S D fA C 1; A C 3; : : : ; 1; C1; : : : ; A 3; A 1g ;

(1.6)

where the symbol energy becomes Es D Letting AN D

p

A2 1 : 3

(1.7)

A, the signal alphabet of A-ary square QAM is given by

˚ ˚ S D a C jb j a; b 2 AN C 1; AN C 3; : : : ; 1; C1; : : : ; AN 3; AN 1 ; (1.8) where the symbol energy becomes Es D

2.A 1/ 2.AN2 1/ D : 3 3

(1.9)

The constellation points of square QAM can be considered in lattices with equal vertical and horizontal spacing. In digital telecommunications, they are converted into binary signals to be encoded and decoded. Let the symbol energy be not considered for convenience. The constellation points and their corresponding binary codes are showed in Fig. 1.3, where binary phase shift keying (BPSK), 4-QAM, and 16-QAM are considered. Example 1.1. Consider a MIMO system with 2 transmit and 2 receive antennas, where 4-QAM is used for singling. The channel is assumed to be an AWGN channel and not varying over L-symbol interval (i.e., block fading is considered). The SNR is defined as the energy per bit to the noise power spectral density ratio, Eb =N0 .

4

1 Introduction 4 BPSK 0000

3

4−QAM

16−QAM

0100

1000

1100

0101

1001

1101

0

1

2 00 0001

Quadrature

1 0

−1

0010

01

10

0110

11

1010

1110

1011

1111

−2 −3 −4 −4

0011

−3

−2

0111

−1

0 In−Phase

1

2

3

4

Fig. 1.3 Signal constellations and binary codes

Using MATLAB as the simulation platform, with random transmit signal vectors, received signal vectors of the MIMO system with a 8 dB SNR is formulated in Table 1.1.

1.2 Point to Point MIMO MIMO detection is to detect the transmit signal s from the received signal y under the knowledge of estimated channel state information (CSI). Note that CSI contains the information of H as well as the statistical properties of n in (1.4). However, by CSI, we often mean the knowledge of H (under the assumption that the statistical properties of n are available). In this book, since we focus on detection itself, we assume that CSI is perfectly known at the receiver. In other words, MIMO detection is to estimate the unknown transmit signal at the receiver, which is illustrated in Fig. 1.4. In MIMO systems, it is usually required to detect signals jointly as multiple signals are transmitted simultaneously from multiple transmit antennas. For coded MIMO systems, the diversity multiplexing trade-off (DMT) [5] is widely used to measure the performance. A system can be asymptotically characterized by the

1.2 Point to Point MIMO

5

Table 1.1 Received signal vectors of a 2 2 MIMO system close all; clear all; (1) Nt = 2; % Number of transmit antennas. (2) Nr = 2; % Number of receive antennas. (3) M = 4; % 4-QAM modulation. (4) L = 1000; % Length of transmit signal. (5) SNRdB = 8; % SNR in dB. (6) SNR = 10.O(SNRdB/10); (7) var = 1./(SNR*log2(M)); (8) ALP = [-1+i -1-i 1+i 1-i]*sqrt(3/(2*(M-1))); % Signal alphabet in QAM. (9) ALPbin = dec2bin([0:M-1],log2(M)); % Signal alphabet in binary coding. (10) s = ALP(randint(Nt,L,[1 M])); % Random transmit signal in QAM. (11) i = 0; (12) for u = 1:L (13) for v = 1:Nt (14) i = i+1; (15) sbin(i,:) = ALPbin(find(ALP==s(v,u)),:); % Random transmit signal in binary codes. (16) end (17) end (18) H = (randn(Nr,Nt)+sqrt(-1)*randn(Nr,Nt))/sqrt(2); % Channel matrix. (19) n = (randn(Nr,L)+sqrt(-1)*randn(Nr,L))*sqrt(var)/sqrt(2); % AWGN noise. (20) y = H*s+n; % Received signal.

spatial multiplexing gain r and spatial diversity gain d as the data rate R.SNR/ achieves [5] R.SNR/ D r; (1.10) lim SNR!1 log SNR and the average error probability Pe .SNR/ achieves

lim

SNR!1

Pe .SNR/ D d: log SNR

(1.11)

6

1 Introduction

Fig. 1.4 MIMO detection

A system may have a higher r, but a lower d , vice versa. However, it is not possible to increase both r and d simultaneously for given numbers of transmit and receive antennas. For uncoded systems, the spatial diversity can also be used as a performance metric [1] for various MIMO detection methods. Note that a full receive diversity gain is considered with its diversity order equals the number of receive antennas in MIMO systems. Using exhaustive search, a maximum likelihood (ML) detector can be employed to detect joint signals, and the optimal performance can be achieved with a full receive diversity order. Since the complexity of the ML detection grows exponentially with the number of transmit antennas, it is impractical in many applications. Instead, for MIMO detection, various computationally efficient approaches have been proposed. Linear detectors such as the zero forcing (ZF) and minimum mean square error (MMSE) detectors can be considered, in which the signals from the other antennas are considered as interfering signals. In general, they have low complexity, but the performance is not good enough for some cases, in particular, at a high SNR. Since linear detectors cannot provide reasonably good performance and a full receive diversity gain from multiple receive antennas, other approaches are considered. For example, the vertical Bell laboratories layered space time (V-BLAST), known as the successive interference cancellation (SIC), is proposed in [6]. It is shown that a MMSE detector with SIC can improve the performance, but suffers from the error propagation. By optimizing the order of the signal detection and cancellation, the error propagation can be reduced. Clearly, linear detectors and the ML detector have two desirable features we want to achieve, low computational complexity and optimal performance, respectively. This book aims to design MIMO detectors of near ML performance with comparable complexity to that of linear detectors. To achieve such design goals, we exploit several existing approaches and methods, including list and lattice reduction (LR)based detection. Since list-based detectors [7–13] construct a list of candidate vectors and then choose the best one as the final decision by exhaustive search, the computational complexity can be considerably reduced if the list length is short. By regarding

1.2 Point to Point MIMO

7

some bits/symbols as unreliable, the list is constructed in [8]. In [9], the list of the parallel detector is generated by employing a separate low-complexity detector for each possible value of the first symbol to be detected. In [12], a list sphere detector is proposed by considering the candidate list in a sphere. Furthermore, a family of list-based Chase detectors is proposed in [14–19]. The principle of the Chase detection is to separate the detection procedure into two layers. In the first layer, one symbol is chosen to be detected separately and a list of candidates for this symbol will be constructed. On the second layer, the contribution from the detected symbol is treated as the interference and will be canceled from the received signal. The residual signal will be detected by the subdetectors to decompose the remaining symbols. The final hard decision symbol vector is determined by MMSE over the concerned vectors. Note that different algorithms can be employed as the subdetectors in the detection. Taking the channel matrix as a basis for a lattice, various approaches based on the properties of lattice are considered for MIMO detection. Since a lattice can be generated by different bases or channel matrices, in order to mitigate the interference between multiple signals, we can find a nearly orthogonal basis or a matrix whose column vectors are nearly orthogonal to generate the same lattice. Based on the Lenstra–Lenstra–LovKasz (LLL) algorithm [20], a lattice-reduced matrix with a nearly orthogonal basis is generated. By employing various low complexity detectors (e.g., MMSE and MMSE-SIC detectors) with the lattice reduced matrix, the LR-based detection [21–28] is carried out, which can provide a full receive diversity gain with good performance. Furthermore, its complexity is significantly lower than that of the ML detector using an exhaustive search. Although the LR can be performed with a complex-valued channel matrix as in [25–27] or a real-valued one converted from the complex-valued one as in [21, 24], they can provide the same performance as shown in [25,27]. Since the LR with a complex-valued matrix has lower complexity [25], in general, we consider the LR with a complex-valued matrix. Some other approaches are also proposed to reduce the complexity for MIMO detection. By regarding the ML detection problem with a partial information of a posteriori probability (APP), the partial maximum a posteriori probability (MAP) principle is applied in [29] to reduce a higher-dimensional ML detection problem to two lower-dimensional subdetection problems to mitigate the intersymbol interference (ISI) and reduce the complexity. In [30], a complexity efficient LR-based list detector is studied to reduce a large MIMO detection problem into multiple small subdetection problems. A channel matrix is called square or tall if the number of transmit antennas M is equal to, or smaller than the number of receive antennas N . For most detectors, it is usually assumed that channel matrix is square or tall. However, there could be the cases where the channel matrices are fat (M > N ), which results in underdetermined or rank-deficient MIMO systems. Note that the LR-based detection is only considered for the cases of tall or square channel matrices. Although the list detection can be employed to such underdetermined MIMO systems, it cannot provide good performance with a full receive diversity gain. Therefore, some generalized sphere

8

1 Introduction

decoding (GSD) approaches [31–35] are developed for such MIMO systems. In [36], two suboptimal group detectors are introduced. A geometrical approachbased detector for underdetermined MIMO systems is studied in [37]. To further reduce the complexity, a computationally efficient GSD-based detector with column reordering is proposed in [38]. Example 1.2. Let us consider a MIMO system with 2 transmit and 2 receive antennas, where its channel is assumed to be an AWGN channel and 4-QAM is used for signaling. Without taking into account the symbolenergy, we can 1 C i . With have the transmit signal vector, for example, denoted by s D 1i 3:58 C 0:73i 1:35 C 0:71i and the noise vector the channel matrix H D 2:77 0:06i 3:03 0:21i 0:07 C 0:20i , the received signal vector becomes nD 0:18 0:27i y D Hs C n 4:87 C 5:12i D : 0:05 0:66i

(1.12)

In order to detect s from y, for example, we can multiply H1 by y in (1.12) as follows: H1 y D H1 Hs C H1 n D s C H1 n 1:01 C 1:02i : D 0:96 1:11i

(1.13)

1:01 C 1:02i From (1.13), we can observe that the most likely signal vector for 0:96 1:11i 1 C i is considered to be , which is the same as the transmit signal vector. Thus, 1i the MIMO detection has been successfully processed. It is also shown that as long as H1 n becomes negligible in terms of the detection performance, we can have H1 y s.

1.3 Multiuser MIMO A cellular system is an example of multiuser communication systems. In a cell, multiple users communicate with a base station (BS). In uplink channels, multiple users can access a common channel. If a BS is equipped with multiple antennas to

1.3 Multiuser MIMO

9

Fig. 1.5 A multiuser MIMO system

receive signals from multiple users, the resulting channel becomes a MIMO channel. In addition to antenna diversity gain, due to users’ different locations and channel conditions, it is possible to exploit another diversity gain, where the performance can be maximized by choosing the user of the best channel at a time. The resulting system and its corresponding diversity gain are called multiuser MIMO system [39] and multiuser diversity gain [40], respectively. Figure 1.5 shows a cellular system as a multiuser MIMO system. Suppose that a multiuser MIMO system with a BS and K users, where each user is equipped with the same number of multiple antennas. Within a time slot, if the best user can be selected to access a channel to transmit or receive signals, the average error probability Pe .SNR/ can achieve

Pe .SNR/ O D d K; SNR!1 log SNR lim

(1.14)

where SNR denotes the signal-to-noise ratio of the channel of the selected user. Here, d represents the spatial diversity gain exploited by multiple antennas, while KO denotes the multiuser diversity gain and KO K. If KO D K, it is said that a full

10

1 Introduction

multiuser diversity gain is obtained. In general, the user selection plays a key role in exploiting this diversity. For example, if a user is chosen randomly or independently from the channel conditions, the multiuser user diversity cannot be achieved. In general, SNR-based user selection criteria for multiuser diversity are studied to maximize the throughput [41, 42]. Hence, the user who has the highest channel capacity can be chosen in multiuser MIMO systems. Although SNR-based or throughput-based optimal user selection schemes are adopted for user selection, the actual performance can be different from the expected one if nonideal or suboptimal MIMO detectors are employed for joint detection. In [43], the error probability is considered for various user selection criteria that choose the user who has the smallest error probability for given MIMO detectors. The user selection criteria with the ML detector as well as other low complexity suboptimal detectors are derived. It is shown that near optimal performance with a full diversity gain (i.e., multiuser diversity and multiple antenna diversity) can be achieved using those user selection criteria with LR-based detectors. Example 1.3. Consider a 2 users multiuser MIMO system in uplink channels, where each user is equipped with 2 transmit antennas and the BS is equipped with 2 receive antennas. Each user has a 2 2 channel matrix and a 2 L signal matrix to be transmitted. Note that the channels are assumed to be AWGN channels and not varying over L-symbol interval (i.e., block fading channels). 4-QAM is used for signaling. Within this interval, one user is selected to transmit signal to the BS under a certain user selection strategy. Using MATLAB as the simulation platform, received signal vectors of the multiuser MIMO system is presented in Table 1.2. Here, we use Eb =N0 to define the SNR and assume that Eb =N0 D 8 dB.

1.4 Outline The rest of the book is organized as follows. In Part I, we present point to point MIMO systems and various low complexity detection methods. In particular, the background of conventional MIMO detectors is presented in Chap. 2, list and LR-based detection methods are introduced in Chap. 3, a computationally efficient partial MAP-based list detector is presented in Chap. 4, a LR-based list MIMO detector with the corresponding column reordering strategy is explained in Chap. 5, and a prevoting cancellation-based detection method is developed for underdetermined MIMO systems in Chap. 6.

1.4 Outline

11

Table 1.2 Received signal vectors of a 2 users multiuser MIMO system under a certain user selection strategy close all; clear all; (1) Nt = 2; % Number of transmit antennas. (2) Nr = 2; % Number of receive antennas. (3) M = 4; % 4-QAM modulation. (4) L = 1000; % Length of transmit signal. (5) SNRdB = 8; % SNR in dB. (6) SNR = 10.^(SNRdB/10); (7) var = 1./(SNR*log2(M)); (8) ALP = [-1+i -1-i 1+i 1-i]*sqrt(3/(2*(M-1))); % Signal alphabet. (9) s1 = ALP(randint(Nt,L,[1 M])); % Random transmit signal of the 1st user. (10) s2 = ALP(randint(Nt,L,[1 M])); % Random transmit signal of the 2nd user. (11) H1 = (randn(Nr,Nt)+sqrt(-1)*randn(Nr,Nt))/sqrt(2); % Channel matrix of the 1st user. (12) H2 = (randn(Nr,Nt)+sqrt(-1)*randn(Nr,Nt))/sqrt(2); % Channel matrix of the 2nd user. (13) n1 = (randn(Nr,L)+sqrt(-1)*randn(Nr,L))*sqrt(var(in))/sqrt(2); % AWGN noise of the 1st user. (14) n2 = (randn(Nr,L)+sqrt(-1)*randn(Nr,L))*sqrt(var(in))/sqrt(2); % AWGN noise of the 2nd user. Input: fH1,H2g :: : A certain user selection strategy ; Suppose that the 2nd user is selected; :: : Output: fH2,s2,n2g (15) y2 = H2*s2+n2; % Received signal.

Different user selection strategies will be discussed in Chap. 7.

In Part II, we introduce multiuser MIMO systems with a consideration of actual employed MIMO detectors, where different schemes for single user selection are discussed in Chap. 7 and those for multiple users selection are studied in Chap. 8. Finally, we conclude this book in Chap. 9.

Part I

Point to Point MIMO

In MIMO systems, the joint detection can be performed by the ML detector to obtain optimal performance. However, since its complexity grows exponentially with the number of transmit antennas, this detection method may not be used in practical systems. To avoid prohibitively high complexity, computationally efficient suboptimal MIMO detection methods are investigated, including the MMSE and ZF detectors which take the signals from other antennas as the interference. Although it shows that they have almost the lowest complexity, they cannot provide satisfactory performance, especially at a high SNR. Therefore, to find a detector that has the same complexity as that of the linear detectors, while providing the ML performance poses a big challenge in resent MIMO research. In this part, we present different computationally efficient algorithms with reasonably good performance for point to point MIMO detection, where two key ingredients are considered to develop these methods: list decoding and lattice reduction.

Chapter 2

Background of MIMO Detection

In this chapter, we introduce a system model for MIMO detection. Then, we present several well-known MIMO detection approaches, including the ML, linear, and SICbased detectors.

2.1 System Model Consider a MIMO system with M transmit and N receive antennas. Let Hl denote an N M channel matrix at symbol time l. It is assumed that a packet of length L is transmitted. Thus, l D 0; 1; : : : ; L 1. Denote by sm;l and yn;l the data symbol transmitted by the mth transmit antenna and the received signal at the nth receive antenna during the lth symbol interval, respectively. Assume that sm;l 2 Z C j Z and a common signal alphabet, denoted by S, is used for all sm;l . That is, sm;l 2 S Z C j Z, m D 1; 2; : : : ; M . Then, the received signal vector over a flat-fading MIMO channel is given by yl D Œy1;l y2;l yN;l T D Hl sl C nl ; l D 0; 1; : : : ; L 1;

(2.1)

where the superscript T denotes the transpose, sl D Œs1;l ; s2;l ; : : : ; sM;l T and nl DŒn1;l ; n2;l ; : : : ; nN;l T denote the transmit signal vector and the noise vector, respectively. Note that nl is assumed to be a CSCG random vector with EŒnl nH l D N0 I, where the superscript H represents the Hermitian transpose. Furthermore, it is assumed that the elements of channel matrix Hl are independent zero-mean CSCG random variables with variance h2 (in this variance term, the signal power is absorbed for convenience) and CSI is perfectly known at the receiver.


15

16

2 Background of MIMO Detection

According to [1], the MIMO channel capacity grows linearly with min.M; N /. Note that there is a fundamental trade-off between receive diversity gain and multiplexing gain [5]. Thus, we may prefer that M D N which results in that Hl is square.

2.2 ML Detection For the sake of simplicity, we will omit the time index l in (2.1) and rewrite the received signal vector as y D Hs C n: (2.2) The channel matrix, H, can also be written as H D Œh1 ; h2 ; : : : ; hM ;

(2.3)

where hm denotes the mth column vector of H. MIMO detection is to estimate the unknown transmitted signal vector, s, for given received signal vector, y, and the channel gain, H. Although we cannot predict the noise vector, n, we have the knowledge of all the possible combinations of s, which is regarded as the candidate vectors. For M transmit antennas, with the signal alphabet S, the number of candidate vectors is given by jSjM , where jSj denotes the size of the alphabet. For example, when a transmitter is equipped with 2 antennas and 4-QAM is employed for signaling, we have 42 D 16 possible candidates for s which is shown in Table 2.1. Note that the symbol energy is not included. Hence, using a certain modulation method, we can show that the number of candidate vectors grows exponentially with M .

2.2.1 Exhaustive Search Approach The ML detection can be carried out by exhaustively searching for all the candidate vectors and selecting the maximum likely one with the smallest error probability. Table 2.1 Candidate vectors of 2 transmit antennas with 4-QAM

s 1 2 3 4 5 6 7 8

1 i , 1 i , 1 i , 1 i , 1 C i , 1 C i , 1 C i , 1 C i ,

1 i 1 C i 1i 1Ci 1 i 1 C i 1i 1Ci

9 10 11 12 13 14 15 16

1 i, 1 i, 1 i, 1 i, 1 C i, 1 C i, 1 C i, 1 C i,

1 i 1 C i 1i 1Ci 1 i 1 C i 1i 1Ci

2.2 ML Detection

17

Let f .yjs/ denote the likelihood function of s for a given received signal vector y. Then, the best symbol vector under the ML detection is given by sml D arg max f .yjs/ s2S M

D arg min ky Hsk2 ; s2S M

(2.4)

where arg minx f .x/ (arg maxx f .x/) represents the argument of x that leads the function f .x/ to reach the minimum (maximum, respectively) value, and S A denotes the A-dimensional Cartesian product of S. Since an exhaustive search is carried out to find the ML vector and the number of candidate vectors for s is jSjM , the complexity grows exponentially with the number of transmit antenna, M . If the a priori probability (APRP) of s is available, the MAP detection can be formulated, which will be further introduced in Sect. 4.1. Let b D Œb1 b2 bMN T represent the bit level symbol vector of s, where the elements of b are binary and MN D M log2 jSj. Denoting the APRP of b by Pr.b/, the MAP detection is given by bmap D arg max Pr.bjy/ b

D arg max f .yjb/ Pr.b/: b

(2.5)

Furthermore, the APP of each bit is given by X Pr.bjy/ Pr.bi D C1jy/ D C

b2Bi

Pr.bi D 1jy/ D

X

Pr.bjy/;

(2.6)

b2Bi

where Bi˙ D

n o T b1 b2 bMN j bi D ˙1; bm 2 fC1; 1g; 8m ¤ i .

2.2.2 Performance Analysis A fully exploited spatial diversity gain is achieved by the ML detection, which equals the number of receive antennas, N . This diversity gain is derived by considering the pairwise error probability (PEP). Supposing that s1 is transmitted, while s2 is erroneously detected, from (2.4), the PEP is represented by P .s1 ! s2 / D Pr ky Hs2 k2 ky Hs1 k2 D Pr kH .s1 s2 / C nk2 knk2 (2.7) D Pr kHdk2 2< nH Hd ;

18


where d D s1 s2 and D aH b the inner product of two vectors a and b. To find two orthogonal vectors that generate the same lattice as H does, we define r1 D h1 r2 D h2 !h1 ;

(2.32)

where !D

hh2 ; r1 i kr1 k2

D

hh2 ; h1 i kh1 k2

(2.33)

in order to lead hr1 ; r2 i D rH 1 r2 D 0. With the linear relationship provided in (2.32), we can show that .h1 ; h2 / and .r1 ; r2 / can span the same subspace. Under the condition that ri is a nonzero vector for i 2 f1; 2g, from (2.32), it is derived as 1! 0 1 kr1 k 0 1! D Œq1 q2 0 kr2 k 0 1 kr1 k !kr1 k ; D Œq1 q2 0 kr2 k

Œh1 h2 D Œr1 r2

(2.34)

where qi D ri =kri k. From (2.34), the QR factorization is given by letting the or kr1 k !kr1 k . thogonal matrix Q D Œq1 q2 and the upper triangular matrix R D 0 kr2 k Note that with r2 D h2 and r1 D h1 !h2 , the other QR factorization of H can be obtained.

2.4.2 ZF-SIC Based on the QR factorization of the channel matrix H, the SIC method is proposed and analyzed in [45, 46]. We assume that H is square or tall, where M N .

2.4 SIC Detection

25

(1): For the case of M D N , H is factorized as H D QR 2

DQ

r1;1 6 0 6 6 : 4 :: 0 „

r1;2 r2;2 :: :

r1;M r2;M : :: : :: 0 rM;M ƒ‚

3 9 > = 7 > 7 M 7 5 > > ; ;

(2.35)

…

M

where Q of size M M is unitary and R of size M M is upper triangular. Here, rp;q denotes the .p; q/th entry of R. By multiplying QH , (2.2) is rewritten as x D QH y D Rs C QH n;

(2.36)

where QH n is a zero-mean complex Gaussian random vector. Since QH n and n have the same statistical properties, QH n can be used to denote n. We have (2.36) as x D Rs C n 2 6 6 6 4

x1 x2 :: :

3

+

2

r1;1 7 6 0 7 6 7D6 : 5 4 ::

xM

0

32 s1 r1;M 6 s2 r2;M 7 76 :: 7 6 :: :: : : 54 : 0 rM;M sM

r1;2 r2;2 :: :

3

2

7 6 7 6 7C6 5 4

n1 n2 :: :

3 7 7 7; 5

(2.37)

nM

where xk and nk denote the kth element of x and n, respectively. Thus, we have xM D rM;M sM C nM : xM 1 D rM 1;M sM C rM 1;M 1 sM 1 C nM 1 : :: : (2): For the case of M < N , H is factorized as H D QR

(2.38)

26


2

r1;1 6 0 6 6 : 6 :: 6 6 6 0 6 D Q6 0 6 6 :: 4 : 0

3 r1;2 r1;N r2;2 r2;N 7 7 :: 7 :: : : : : 7 : 7 7 0 rM;N 7 7 0 ::: 0 7 7 :: :: :: 7 : : : 5 0 0 ƒ‚

„

9 > > = M > > ; 9 ; = N M ;

(2.39)

…

M

T T N 0 where the N N matrix Q is unitary and the N M matrix R D R N (the M M submatrix R is upper triangular). From (2.36), the N 1 received signal vector x is given by 2

3

2

r1;1 7 6 0 6 7 6 6 7 6 : 6 7 6 :: 6 7 6 6 7 6 6 6 xM 7 D 6 0 7 6 6 6 xM C1 7 6 0 7 6 6 6 :: 7 6 :: 4 : 5 4 : 0 xN x1 x2 :: :

r1;2 r2;2 :: : 0 0 :: :

r1;M r2;M : :: : :: rM;M ::: 0 :: :: : :

0

0

3 7 72 7 s1 7 7 6 s2 76 76 : 7 4 :: 7 7 s 7 M 5

2

n1 n2 :: :

3

6 7 6 7 6 7 6 7 7 6 7 7 6 7 7 C 6 nM 7 : 7 5 6 6 nM C1 7 6 7 6 :: 7 4 : 5 nN 3

(2.40)

Furthermore, we have xN D nN :: : xM C1 D nM C1 xM D rM;M sM C nM xM 1 D rM 1;M sM C rM 1;M 1 sM 1 C nM 1 :: :

(2.41)

Since the received signals fxM C1 ; xM C2 ; : : : ; xN g do not have any useful information, we can simply ignore them. Then, (2.38) and (2.41) become the same. This results in a sequential detection procedure.

2.4 SIC Detection

27

Firstly, sM can be detected from xM as follows: 1. Let sQM D

xM rM;M

D sM C

nM : rM;M

(2.42)

˚

2. Denoting by S D s .1/ ; s .2/ ; ; s .K/ the signal alphabet of K-ary QAM, the hard decision of sM is given by ˇ ˇ2 sOM D arg min ˇs .k/ sQM ˇ : s .k/ 2S

(2.43)

It shows that there is no interference involved of this decision. Then, the contribution of sOM is to be canceled in detecting sM 1 from xM 1 . This sequential detection procedure is terminated till all the data symbols of s are detected. The mth symbol of s, sm , can be detected after canceling M m data symbols as um D xm

M X

rm;q sOq ;

m 2 f1; 2; : : : ; M 1g;

(2.44)

qDmC1

where sOq denotes the hard-decision estimate of sq from uq . Supposing that there has been no error detection occurred so far, sm is estimated as ˇ ˇ2 sOm D arg min ˇs .k/ sQm ˇ ; s .k/ 2S

where sQm D

um rm;m

D sm C

(2.45)

nm . rm;m

Since QH is used to perform a nulling processing in (2.36), a ZF decision feedback equalizer (DFE) over ISI channels has been carried out to perform the SIC detection. Consequently, this detection is regarded as the ZF-SIC detection. Note that as H is fat and M > N , the N M sized matrix R after QR factorization of H is not upper triangular, thus, the sequential detection-based SIC method cannot be employed. Example 2.2. Consider a 2 2 MIMO system with 4-QAM modulation method, where the channel matrix and the noise vector are given by 0:03 C 0:09i 0:12 C 0:67i 1:41 0:71i , respectively. and n D H D 0:08 0:11i 1:49 1:21i 1:42 C 1:63i 1i Supposing that the eighth candidate vector in Table 2.1, s D , is 1Ci transmitted, the received signal is shown as

28


y D Hs C n 2:7000 C 1:5800i : D 0:0100 C 0:2400i

(2.46)

Performing the QR factorization on H, we have H D QR; "

where Q D

0:0589 C 0:3290i 0:9083 0:2517i 0:7316 0:5941i 0:3022 C 0:1427i

(2.47)

#

"

and R D

2:0365 0

0:2462 C 1:6142i 2:1211

#

From (2.36), x is obtained by x D QH y 0:2108 0:8117i : D 2:0859 C 2:1886i

(2.48)

Using the method in (2.43), sO2 D 1 C i is estimated from sQ2 , where sQ2 D x2 =r2;2 D 0:9834 C 1:0318i . After sO2 is detected, its contribution is canceled from x1 as sQ1 D .x1 r1;2 sO2 / =r1;1 D 1:0170 1:0703i;

(2.49)

where the scheme in (2.45) is used to estimate sO1 D 1 i . With sO D 1i , a correctly detection has been performed with the SIC. 1Ci

sO1 sO2

D

2.4.3 MMSE-SIC In order to improve the performance, the background noise can be taken into account for linear filtering, which results in the MMSE-DFE based SIC (MMSESIC) detection. Two schemes are presented for this detection as follows. q iT h Scheme I: Define an extended channel matrix as Hex D HT NE0s I , while y and q iT h T n are also extended as yex D yT 0T and nex D nT NE0s sT , respectively. With the QR factorization, we have Hex D Qex Rex ;

(2.50)

:

2.4 SIC Detection

29

where Qex and Rex represent a unitary matrix and an upper triangular matrix, respectively. In (2.36), let y, H, n, Q, and R be replaced by yex , Hex , nex , Qex , and Rex , respectively. Then, the resulting system becomes xex D QH ex yex D Rex s C QH ex nex :

(2.51)

With (2.51), the MMSE-SIC detection is carried out by the sequential detection procedure given from (2.36) to (2.45). Scheme II: From (2.2), the other scheme is carried out by adopting the MMSE estimator, where the MMSE estimator for symbol s1 is given by hˇ ˇ2 i wmmse;1 D arg min E ˇs1 wH yˇ w

N0 1 N h1 ; I D HHH C Es

(2.52)

where hN 1 denotes the first column vector of HH . Then, a hard-decision operation is carried out to detect s1 from sO1;mmse D wmmse;1 y:

(2.53)

Assuming that s1 is successfully detected and its contribution is canceled from y, we have M X hm sm C n: (2.54) y1 D mD2

With y1 , the MMSE method can be employed to detect s2 . Repeating cancellation and MMSE estimation, the detection of the sm ’s can be performed.

2.4.4 Ordering In the SIC detection, the ordering of symbol detection plays a key role in mitigating the error propagation, where the overall performance would be decided by the magnitude of the diagonal terms in R. For example, consider the system presented in (2.34). We have r1;1 D kr1 k and r2;2 D kr2 k, where kr1 k2 D kh1 k2 and kr2 k2 D kh2 !h1 k2 D hh2 !h1 ; h2 !h1 i D hh2 ; h2 !h1 i

30


D kh2 k2 !hh2 ; h1 i D kh2 k2

jhh2 ; h1 ij2 kh1 k2

0

(2.55)

denote the magnitude of r1;1 and r2;2 , respectively. It shows that the overall performance can be improved as h1 and h2 are more orthogonal or less correlated, where a lower correlation is obtained by increasing kr2 k. Thus, in order to improve the performance of SIC detection, one can be found that the ordering is determined to maximize min fkh1 k; kr2 kg. Example 2.3. Consider two 2 2 matrices, H D Œh1 h2 D

5 1

2 3

and G D Œg1 g2 D

2 5 ; 3 1

(2.56)

which consist of the same column vectors but with different column ordering. For the SIC detection, we aim to analyze whether H or G is better in terms of the performance. According to (2.32) and (2.55), we have 2

2

hQ 2 D kh2 k2 jhh2 ; h1 ij 2 kh1 k

kQg2 k2 D kg2 k2

jhg2 ; g1 ij2 ; kg1 k2

(2.57)

and ˚

kh1 k; hQ 2 D f5:0990; 3:3340g

fkg1 k; kQg2 kg D f3:6056; 4:7150g :

(2.58)

˚ Since min kh1 k; hQ 2 < min fkg1 k; kQg2 kg, we can show that an optimal ordering of detection is carried out with the matrix G rather than H. In addition, the performance of the MMSE-SIC highly depends on the reliability of detected symbols in the early stages. To improve the performance, a preordering method for SIC detection is proposed and discussed in [46, 47]. Furthermore, a simple strategy is considered by selecting the first symbol to be detected that has the smallest MSE (i.e., equivalently, highest signal to interference plus noise ratio (SINR)) as

˚ k.1/ ; wk.1/ D arg

min

hˇ ˇ2 i min E ˇsk wH yˇ ;

k2f1;2;:::;M g w

(2.59)

2.4 SIC Detection

31

where k.1/ and wk.1/ denote the index of the first detected symbol and its corresponding MMSE filtering vector, respectively. Then, the cancellation is carried out as yO D y hk.1/ sOk.1/ ; (2.60) O , the next symbol to where sO k.1/ denotes a hard decision of sk.1/ from wH k.1/ y. With y be detected is found as hˇ ˇ2 i ˚

k.2/ ; wk.2/ D arg min min E ˇsk wH yO ˇ ; (2.61) k2I

w

where I D f1; 2; : : : ; M g n k.1/ and n denotes the set minus. The cancellation and MMSE filtering are repeated until all symbols are detected. We summarize the algorithm as follows: .a/ Let y0 D y, I0 D f1; 2; : : : ; M g, and k D 1. .b/ Perform the ordering and detection as ˚

k.m/ ; wk.m/ D arg min min E jsq wH ym1 j2 : q2Im1

w

(2.62)

.c/ The detected signal is canceled as ym D ym1 hk.m/ sOk.m/ :

(2.63)

.d / Let Im D Im1 n k.m/. .e/ If m < M , go to Step .a/. The algorithm is terminated as m D M .

2.4.5 Performance Analysis In order to study the performance of SIC detection, we assume that H is square (i.e., the size of H is M M ), while BPSK is used for signaling. The ZF-SIC is carried out to detect signals. From [48] and [49], the statistical properties of R in (2.35) are summarized as follows: Lemma 2.1. Denoting by rm;m the mth diagonal element of R, m D 1; 2; : : : ; M , we have .1/ The nonzero elements of R are independent random variables. .2/ jrm;m j2 is a chi-square distributed random variable with 2.M m C 1/ degrees of freedom and its probability density function (pdf) becomes f jrm;m j2 D where jrm;m j2 0.

2 1 jrm;m j.M m/ e jrm;mj ; .M m/Š

(2.64)

32


.3/ The nonzero off-diagonal elements, rm;l for l < m, are CSCG random variables with mean zero and unit variance. According to Lemma 2.1, a main disadvantage of SIC detection is that there could be more detection errors at lower layers and be propagated to higher layers. In order to provide a more precise analysis, the average BER of SIC detection is studied in [50]. Consider ˚ p p the system model presented in (2.37) with M D 2. For BPSK, let sm 2 Rm ; Rm . Since there is no cancellation in the second layer, the bit error probability of detecting s2 conditioned on r2;2 is given by 0s

Rm , N0

1

2 jr2;2 j R2 A: N0

Ps2 .R2 ; r2;2 / D Q @

Letting m D given by

2

(2.65)

the average BER obtained by taking the expectation over r2;2 is Ps2 .R2 / D

r 2 1 1 : 2 1 C 2

(2.66)

In [51], the function G.d; / D

for D

q

, .1C /

1 .1 / 2

d X k d 1 1 d 1Ck .1 C / k 2

(2.67)

kD0

provides the average BER over a d -fold diversity Rayleigh fading

channel with mean branch SNR . Using the function G.d; /, (2.66) becomes Ps2 .R2 / D G .1; 2 / :

(2.68)

The BER of the detection at the first layer depends on the cancellation, which is summarized as follows: (a) If the cancellation is correct, from (2.44), we have u1 D x1 r1;2 sO2 D r1;1 s1 C n1 ;

(2.69)

where the conditional bit error probability on r2;2 is given by 0s Ps2 .s1 D sO1 / D Q @

1 2 jr2;2 j2 R2 A : N0

(2.70)

2.4 SIC Detection

33

(b) If the cancellation is not correct, from (2.44), we have u1 D x1 r1;2 sO2 D r1;1 s1 C r1;2 .s2 sO2 / C n1 ; (2.71) p p where s2 sOp 2 D 2 R2 or 2 R2 with a half probability each. Assuming that s2 sO2 D 2 R2 in (2.71), the conditional bit error probability of detecting s1 becomes p p u1 > 0js1 D R1 ; r1;1 Pe s2 sO2 D 2 R2 D Pr < r1;1 p p D Pr jr1;1 j2 R1 < < r1;1 n1 C 2 R2 r1;2 jr1;1 (2.72) p Let nQ 1 D n1 C 2 R2 r1;2 . According to Lemma 2.1, we have nQ 1 CN .0; N0 C 4R2 /, then (2.72) is rewritten as 1 0s 2 p 2 R j jr 1;1 1A : (2.73) Pe s2 sO2 D 2 R2 D Q @ 4R2 C N0 In summary, from (2.70) and (2.73), the bit error probability of detecting s1 conditioned on r1;1 can be obtained as 0s 0s 1 1 2 2 2 R 2 R j j jr jr 1;1 1 1;1 1 A .1Ps2 .R2 // CQ @ A Ps2 .R2 / ; Ps1 .R1 ; r1;1 / D Q @ N0 4R2 C N0 (2.74) where Ps2 .R2 / denotes the probability of incorrect cancellation and .1 Ps2 .R2 // becomes the probability of correct cancellation. By taking the expectation over r1;1 and using the function G.d; /, the average BER is found as Ps1 .R1 / D G.2; 1 / .1 Ps2 .R2 // C G 2;

R1 4R2 C N0

Ps2 .R2 / :

(2.75)

Note that the approach in above can be generalized to find the average BER for higher layers (i.e., M > 2), which will be introduced as follows. For convenience, consider error patterns at the mth layer with binary numbers. Then, the set of the error patterns at the mth layer is defined by Um D

.2M mC1 / .2/ p.1/ m ; pm ; : : : ; pm

;

(2.76)

34

2 Background of MIMO Detection .1/

.2m /

.2/

where pm D Œ0 0 0T , pm D Œ0 0 1T , : : :, pm D Œ1 1 1T are .M m C 1/ 1 vectors consists of 0 or 1. For example, when M D 4 and m D 3, we have n o .1/ .2/ .3/ .4/ U3 D p3 ; p3 ; p3 ; p3 ; (2.77) .1/

.2/

.3/

.4/

where p3 D Œ0 0T , p3 D Œ0 1T , p3 D Œ1 0T , and p3 D Œ1 1T . Furthermore, it .i / shows that pm 2 Um for i D 1; 2; : : : ; 2M mC1 . For the sake of simplicity, we omit the index i . Thus, if the j th element of pm is zero or one, the j th detection becomes correct or incorrect, respectively. Note that from Um , we can easily generate Um1 as Um1 D

n

pTm 0

o T T T ˇˇ ; pm 1 ˇ pm 2 Um :

(2.78)

Lemma 2.2. Let vm D ŒRM RM 1 Rm T . At the mth layer, the bit error probability conditioned on rm;m and the previous error pattern pmC1 becomes s ! 2 R j 2jrm;m m : (2.79) Psm .errjrm;m ; pmC1 I Rm / D Q N0 C 4vTmC1 pmC1 Taking the expectation over rm;m , we have Psm .errjpmC1 I Rm / D E ŒPm .errjrm;m ; pmC1 I Rm / ! Rm : D G m; N0 C 4vTmC1 pmC1

(2.80)

Proof. After the cancellation, symbol of the mth layer is given by um D xm

M X

rm;q sOq C nm

qDmC1

D rm;m sm C nN m ; where nN m D

M X

rm;q .sq sOq / C nm :

(2.81)

(2.82)

qDmC1

Using the result of Lemma 2.1, we can show that nN m is a Gaussian random i h variable conditioned on pmC1 . Furthermore, we have E ŒnN m D 0 and E jnN m j2 D N0 C 4vTmC1 pmC1 . Thus, the bit error probability of detecting sm in (2.81) conditioned on rm;m and pmC1 is given by (2.79). From (2.79), we have (2.80), which completes the proof. t u According to Lemma 2.2, the average BER is obtained from the recursion as follows.

2.4 SIC Detection

35

100 1st Layer 2nd Layer 3rd Layer 4th Layer

BER for each layer

10−1

10−2

10−3

10−4

0

2

4

6

8

10

12

14

16

18

20

Eb /N0

Fig. 2.1 BER performance of ZF-SIC for each layer in a 4 4 MIMO system, where BPSK is used for signaling

Theorem 2.1. The average BER at the mth layer is given by Psm .Rm / D

X

Psm .errjpmC1 I Rm / PsmC1 .pmC1 / ;

(2.83)

pmC1 2PmC1

where ( Psm .pm / D

if pm D ŒpmC1 1T I Psm .errjpmC1 I Rm / PsmC1 .pmC1 / ; 1 Psm .errjpmC1 I Rm / PsmC1 .pmC1 / ; if pm D ŒpmC1 0T : (2.84)

Proof. Note that this derivation follows the same principle used to generate (2.75). Since the result is straightforward to be obtained, we omit the proof. t u In Fig. 2.1, the average BER curves of detection at each layer are presented by simulation result, where M D 4 and BPSK is used for signaling. The result shows that the error probability of the M th layer is lower than that of other layers. It also shows that each layer has the same diversity gain, which is decided by the M th layer due to the error propagation.

36


Although a full receive diversity gain cannot be achieved with the SIC detection, it enjoys a performance and complexity trade-off which will be shown in Sect. 2.5 by simulations.

2.5 BER Versus SNR Simulation Results In order to numerically analyze the performance of different MIMO detectors, at a certain SNR, the BER or symbol error rate (SER) can be considered as a performance metric in simulations. For example, consider a vector s of 10 symbols using 4-QAM is transmitted, while the vector sO is received with errors. Symbols are mapped into binary codes to obtain the BER. In Table 2.3, QAM symbols (without taking into account the symbol energy) and their corresponding binary codes of s and sO are presented. It shows that there are 3 out of 10 symbols and 4 out of 20 bits are falsely detected which lead to that SER D 0:3 and BER D 0:2, respectively. Note that for simulations, the MATLAB program of QAM symbols and their corresponding binary codes can be found in Table 1.1. In this section, we present the BER simulations of different conventional MIMO detectors. Using the received signals of MIMO channels in Table 1.1, we provide MATLAB programs of BER simulations for the ML, MMSE, and MMSE-SIC detection in Tables 2.4–2.6, respectively. We consider uncoded 16-QAM 2 2 and 4 4 MIMO systems for the simulations. The elements of the MIMO channels are generated as independent CSCG random variables with mean zero and unit variance. The SNR is defined by the energy per bit to the noise power spectral density ratio, Eb =N0 . Note that in order to obtain the average BER at a certain SNR, the Monte Carlo method need to be adopted in simulations. In Figs. 2.2–2.7, we show the BER performance of the ML, MMSE, and MMSESIC detectors for 2 2 and 4 4 MIMO systems, where 4-, 16-, and 64-QAM are used for signaling. In Fig. 2.3, it is observed that with the ML detection when BER drops from 103 to 104 , SNR increases by 5 dB, where an estimate of Table 2.3 Detection errors at symbol and bit levels

Os

s 1Ci 1i 1 C i 1i 1 i 1Ci 1Ci 1 i 1i 1Ci

10 11 00 11 01 10 10 01 11 10

1Ci 1i 1 i 1i 1 i 1Ci 1Ci 1i 1 C i 1Ci

10 11 01 11 01 10 10 11 00 10

2.5 BER Versus SNR Simulation Results

37

Table 2.4 ML detection (1) Input: fy,H,sbin,Nt,L,M,ALP,ALPbing % Output from Table 1.1. (2) dml = zeros(M,M,L); (3) kset = zeros(Nt,L); (4) ibin = 0; % Initialization of parameters. (5) for u = 1:L (6) for i1 = 1:M (7) for i2 = 1:M (8) dml(i1,i2,u) = (norm(y(:,u)-H*[ALP(i1);ALP(i2)]))O2; % ML function with an exhaustive search. (9) end (10) end (11) dmin = 1e8; % Define a small value. (12) for j1 = 1:M (13) for j2 = 1:M (14) if dml(j1,j2,u) < dmin (15) dmin = dml(j1,j2,u); % Find the index of detected symbols. (16) kset(1,u) = j1; (17) kset(2,u) = j2; (18) end (19) end (20) end (21) hats(1,u) = ALP(kset(1,u)); (22) hats(2,u) = ALP(kset(2,u)); % Map the detected signals in QAM. (23) hatsbin(ibin+1,:) = ALPbin(find(hats(1,u)==ALP),:); (24) hatsbin(ibin+2,:) = ALPbin(find(hats(2,u)==ALP),:); % Map the detected signals in binary codes. (25) ibin = ibin+2; (26) end (27) BER = mean(reshape(abs(sbin-hatsbin),1,Nt*L*log2(M))); % Generate the BER at bits level.

the receive diversity gain is 2 for a 2 2 MIMO channel. When BER drops from 104 to 105 , Fig. 2.6 shows that the SNR of ML detection increases by 2.5 dB, where the estimated receive diversity gain is 4 for a 4 4 MIMO channel. Since an exhaustive search is considered with the ML detector, the optimal performance with a full receive diversity gain of N is obtained at the cost of high complexity. While the MMSE has the lowest complexity, it provides the worst performance. We also note that the MMSE-SIC detector has a trade-off between the performance and complexity.

38

2 Background of MIMO Detection Table 2.5 MMSE detection (1) Input: fy,H,sbin,var,Nt,L,M,ALP,ALPbing % Output from Table 1.1. (2) dmmse = zeros(M,Nt*L); (3) kset = zeros(Nt,L); % Initialization of parameters. (4) tildes = (pinv(H’*H+ var*eye(Nt))*H’)*y; % MMSE estimator. (5) for j = 1:length(ALP) (6) dmmse(j,:) = reshape(abs(tildes-ALP(j)).O2,1,size(y,1)*size(y,2)); (7) end (8) [dmin,kset] = min(dmmse); % Obtain the index of detected symbols. (9) for q = 1:size(dmmse,2) (10) hatsbin(q,:) = ALPbin(kset(q),:); % Map the detected signals in binary codes. (11) end (12) BER = mean(reshape(abs(sbin-hatsbin),1,Nt*L*log2(M))); % Generate the BER at bits level. Table 2.6 MMSE-SIC detection (1) (3) (4) (4) (5) (6) (7) (8) (6) (8) (6) (7) (7) (8) (8) (9) (10) (11) (12) (25) (26) (27)

Input: fy,H,sbin,var,Nt,L,M,ALP,ALPbing % Output from Table 1.1. kset = zeros(Nt,L); ibin = 0; % Initialization of parameters. Hex = [H; sqrt(var)*eye(Nt)]; yex = [y; zeros(Nt,L)]; [Q,R] = qr(Hex); % QR factorization on channel matrix. tildes = Q’*yex; for u = 1:L for j = 1:M dsic2(j) = (abs(ALP(j)-(tildes(2,u)/R(2,2))))O2; end [dmin2,kset(2,u)] = min(dsic2); % Obtain the index of sO2 . hats(2,u) = ALP(kset(2,u)); % Map sO2 in QAM. for q = 1:M dsic1(q) = (abs(ALP(q)-(tildes(1,u)-hats(2,u)*R(1,2))/R(1,1)))O2; % Cancel the contribution of sO2 . end [dmin1,kset(1,u)] = min(dsic1); % Obtain the index of sO1 . hatsbin(ibin+1,:) = ALPbin(kset(1,u),:); hatsbin(ibin+2,:) = ALPbin(kset(2,u),:); % Map the detected signals in binary codes. ibin = ibin+2; end BER = mean(reshape(abs(sbin-hatsbin),1,Nt*L*log2(M))); % Generate the BER at bits level.


39

100 ZF MMSE ZF−SIC MMSE−SIC ML

10−1

BER

10−2

10−3

10−4

10−5

10−6

0

5

10

15

20

Eb /N0(dB)

Fig. 2.2 BER performance of conventional detectors in a 4-QAM 2 2 MIMO system 100 ZF MMSE ZF−SIC MMSE−SIC ML

10−1

BER

10−2

10−3

10−4

10−5

0

5

10

15

20

25

Eb /N0

Fig. 2.3 BER performance of conventional detectors in a 16-QAM 2 2 MIMO system

40

2 Background of MIMO Detection 100 ZF MMSE ZF−SIC MMSE−SIC ML

10−1

BER

10−2

10−3

10−4

10−5

0

5

10

15

20

25

30

Eb /N0 (dB)

Fig. 2.4 BER performance of conventional detectors in a 64-QAM 2 2 MIMO system 100

10−1

BER

10−2

10−3

10−4 ZF MMSE ZF−SIC MMSE−SIC ML

10−5

10−6

0

2

4

6

8

10

Eb /N0


12


41

100

10−1

BER

10−2

10−3


10−5

10−6

0

2

4

6

8

10

12

14

16

Eb /N0

Fig. 2.6 BER performance of conventional detectors in a 16-QAM 4 4 MIMO system 100

10−1

BER

10−2


10−4

10−5

4

6

8

10

12

14

16

18

20

Eb /N0


42


2.6 Conclusion and Remarks In this section, we presented three well-known approaches for MIMO detection. While exhaustive search can be used for optimal performance of the ML detection, the prohibitively high complexity makes it unrealistic to be employed. There are some suboptimal approaches which can provide relatively low complexity (e.g., MMSE and MMSE-SIC detectors). However, their performance is not comparable with that of the ML detector, especially at a high SNR. Therefore, it is necessary to find additional techniques for conventional approaches to improve the performance of these conventional suboptimal approaches. In the following chapter, we explain the list and LR-based detection.

Chapter 3

List and Lattice Reduction-Based Methods

In MIMO detection, the signals transmitted by multiple antennas can be detected jointly based on the ML principle for optimal performance. However, its complexity is prohibitively high for a large number of transmit antennas and a higher order modulation method. There are suboptimal approaches of low complexity. Unfortunately, they cannot exploit a full receive diversity gain, which results in the performance gap between the ML and suboptimal approaches (e.g., linear MMSE detection). This gap usually increases with SNR. Therefore, it is desirable to find some new detection methods that can provide near ML performance with low complexity which is comparable to that of linear detectors. In this chapter, we introduce two key ingredients: list decoding and lattice (basis) reduction, which can be used to develop different computationally efficient algorithms with reasonably good performance.

3.1 List-Based Detection Let us recall the SIC detection where the sequential decision and cancellation are carried out. As SIC detectors suffer from error propagation, list-based approaches [7–13] can be employed to mitigate error propagation by selecting multiple candidate symbols to build up a list for the final hard decision. In this section, we will explain a class of computational efficient list-based detectors. Specifically, we focus on the MIMO detection using Chase-decoding algorithms [14–19] that provide good performance with reasonable low complexity.

3.1.1 Detection Algorithms In this subsection, we first review the list-based Chase algorithm using the linear filter to perform a two-layer detection procedure, which is referred to as the linear-list detection. L. Bai and J. Choi, Low Complexity MIMO Detection, DOI 10.1007/978-1-4419-8583-5 3, © Springer Science+Business Media, LLC 2012

43

44

3 List and Lattice Reduction-Based Methods

Table 3.1 List generation

Initialization: SQ D S for q D 1 to Q .q/ 2 sM D arg minsM 2SQ jsM yj O .q/ SQ D SQ n sM end for

(1) (2) (3) (4) (5)

Recall the N 1 received signal vector over a MIMO channel as y D Œy1 ; y2 ; : : : ; yN T D Hs C n;

(3.1)

where the N M channel matrix H D Œh1 ; h2 ; : : : ; hM , the M 1 transmit signal vector s D Œs1 ; s2 ; : : : ; sM T , and the N 1 noise vector n D Œn1 ; n2 ; : : : ; nN T . Throughout this chapter, we assume s 2 S M , where S denotes a common signal alphabet for all symbols. Thus, each signal symbol has the same energy. In addition, n is assumed to be an independent zero-mean CSCG random vector with E nnH D N0 I. T Let H D ŒH hM and s D sT sM , where H D Œh1 ; h2 ; : : : ; hM 1 and s D Œs1 ; s2 ; : : : ; sM 1 T . Then, (3.1) is rewritten as y D Hs C hM sM C n;

(3.2)

where the sizes of submatrix H and subvector s are N .M 1/ and .M 1/ 1, respectively. Based on the system models in (3.1) and (3.2), the linear-list detection can be summarized as follows [15]: n o .1/ .2/ .Q/ .a/ Generate a list of Q candidate values for sM , say sM ; sM ; : : : ; sM , where .q/

O q D 1; 2; : : : ; Q. Here, Q jSj and sM denotes the qth closest symbol to y, yO D wH y and W D Œw ; : : : ; w represents the linear filter (ZF in (2.18) or 1 M M MMSE in (2.21)) for given H. The algorithm to build the list is presented in Table 3.1. .b/ By canceling the contribution of the symbol vector sM to y, using each candidate of sM in the list, a set of Q residual vectors fy.1/ ; y.2/ ; : : : ; y.Q/ g is generated as .q/

y.q/ D y hM sM :

(3.3)

.q/ .c/ Apply an independent subdetector y and obtain decision of the ˚ .1/ .2/ to each .Q/ (MIMO detectors that work for remaining M –K symbols s ; s ; : : : ; s square or tall MIMO channels can be used when K M N ). Let s.q/ D # " ˚ .1/ .2/ s.q/ .Q/ .q/ : As a result, the Q candidate hard decision vectors s ; s ; : : : ; s sM can be obtained.

3.1 List-Based Detection

45

˚ .d/ From the candidates s.1/ ; s.1/ ; : : : ; s.Q/ , obtain the final hard decision vector s that best represents the observation vector y in the sense of the sum of squared error (SSE) as y Hs.q/ 2 : sO D arg min (3.4) s.q/ 2fs.1/ ;:::;s.Q/ g Note that the performance of linear-list detection is degraded due to the interference. Using the Chase algorithm with the SIC, a class of SIC-list detection can be carried out to improve the performance by mitigating the impact of interference. However, it cannot be used for underdetermined MIMO systems (i.e., M > N ). Assume that N M . Using the QR factorization H D QR (i.e., N N matrix Q and N M matrix R are unitary and upper triangular, respectively) from (3.1), we have x D QH y D Rs C n: T T For the case of M D N , letting x D xT xM and n D nT nM (i.e., x and n are .M 1/ 1 subvectors of x and n, respectively), x is rewritten as

x xM

A c D 0 0 rM;M

s

sM

n C nM

;

(3.5)

where A, c, and rM;M denote a .M 1/.M 1/ triangular submatrix, an .M 1/1 subvector, and the .M; M /th entry of R, respectively. Alternatively, for the case of N > M , it can be easily obtained that 2

3 2 x A c 6 xM 7 6 0 0 rM;M 6 7 6 6 xM C1 7 6 0 0 0 6 7D6 6 : 7 6 : :: : 4 : 5 4 :: : xN

00

2

3 n 7 7 6 6 nM 7 7 6 nM C1 7 7 s C6 7; 7 6 : 7 7 sM : 4 : 5 5 3

0

(3.6)

nN

T T where x D xT xM xM C1 xN and n D nT nM nM C1 nN . According to (3.5) and (3.6), we have x D As C csM C n xM D rM;M sM C nM :

(3.7)

Now the SIC-list detection can be summarized as follows [16]: o n .1/ .2/ .Q/ .a/ Generate a list of Q candidate values for sM , say sM ; sM ; : : : ; sM , where .q/

Q jSj and sM denotes the qth closest symbol to xO M (using the method in 1 xM . Table 3.1), q D 1; 2; : : : ; Q. Here, xO M D rM;M

46


.b/ By canceling the contribution of the symbol ˚vector sM to x using each candidate of sM in the list, a set of Q residual vectors x.1/ ; x.2/ ; : : : ; x.Q/ is generated as .q/

x.q/ D x csM :

(3.8)

of the .c/ Apply an independent subdetector for each x.q/ and obtain"the decision # .q/ ˚ s remaining M 1 symbols s.1/ ; s.2/ ; : : : ; s.Q/ . Let s.q/ D .q/ . As a result, sM ˚ .1/ .2/ the Q candidate hard decision vectors s ; s ; : : : ; s.Q/ can be obtained. ˚ .d/ From the candidates s.1/ ; s.2/ ; : : : ; s.Q/ , obtain the final hard decision vector sO that best represents the observation vector x in the sense of the SSE as sO D arg

min

s.q/ 2fs.1/ ;:::;s.Q/ g

x Rs.q/ 2 :

(3.9)

Clearly, the complexity of the detection method in above becomes lower as a smaller list length Q is considered, while the performance highly depends on the decision of sM .

3.1.2 Ordering In order to improve the performance of list detection, sM must be reliably detected to avoid error propagation in detecting the subsequent symbols. Thus, the order of the symbols to be detected should be properly decided, where column vectors of H are reordered accordingly. Various ordering strategies are introduced in this subsection. For the linear list detector, a simple strategy to choose the first symbol to be detected is based on the maximum SINR or minimum MSE, which is shown as hˇ ˇ2 i ˇ ; (3.10) kO D arg min E ˇsk wH ky k2f1;:::;M g

where kO denotes the index of the symbol that has the smallest MSE and wkO denotes its corresponding linear filtering vector. Note that although the correct detection for the first symbol (first layer detection) can be performed with a high probability by maximizing SINR, the performance of the subdetection with H in (3.2) (second layer detection) is not taken into account. A trade-off between the performance of the first and second layer detection is discussed in [15]. In [16], the S-Chase detector has been studied to select the index (of the first symbol to be detected) as kO D arg

max

kD1;2;:::;M

khk km ;

(3.11)


47

where

( mD

1; 1;

Q> Q

3jSj 4 3jSj 4

:

From this, we can show that the symbol is chosen as the one with its corresponding column vector of matrix H that has either the minimum of maximum norm, depending on the list length Q. Although it is not an optimal solution, the complexity reduction is significant as there is no SNR calculation required. For the SIC list detection, the detection ordering can be found using a permutation matrix P and the received signal becomes y D HP sN C n;

(3.12)

where sN D P T s. With the QR factorization, we have HP D QR, where Q is unitary and R is upper triangular. With the permutation matrix P, two strategies of ordering can be formulated, namely BLAST ordering [52, 53] and B-Chase ordering [15]. By column swapping of the matrix H using the permutation matrix P, we can have M Š possible cases of P in total. In order to find the optimal P, one chooses the symbol with the largest SNR which is equivalent to minimizing the MSE, and the index of the first symbol to be detected is decided as " # 1 H 1 IC min H H kD1;2;:::;M N0 k;k " 1 # 1 H D arg min R R IC kD1;2;:::;M N0 k;k N R k;k ; D arg min

kO D arg

kD1;2;:::;M

(3.13)

1 N D I C 1 RH R . where ŒAk;k denotes the .k; k/-entry of matrix A and R N0 According to (3.13), the BLAST ordering is carried out to find the optimal P and an illustration of the algorithm to implement the approach is as follows.

(5)

INPUT: An M M channel matrix H OUTPUT: An M M permutation matrix P H0 D H; P D 0M M m0 D Œ1; 2; : : : ; M for i D 1 to M ŒQ; R D qr .Hi 1 /

1 N D I C 1 RH R R

(6) (7) (8)

N k;k kO D arg mink2f1;2;:::;M i C1g R O j D the kth element of mi 1 Put a 1 in the j th row and i th column of P

(1) (2) (3) (4)

N0

48


O element of mi 1 (9) mi is updated by deleting the kth O (10) Hi is updated by deleting the kth element of Hi 1 (11) end Since the original BLAST ordering requires O.M 4 / multiplications, some computational efficient algorithms were investigated in [52, 53] with O.M 3 / multiplications only. With the B-Chase algorithm [15], the first symbol to be detected is found by maximizing the minimum SNR. Let Pk denote the permutation matrix that arranges the kth symbol to be detected first (equivalent to that the kth column vector of H becomes the last one after ordering) while the remaining columns are arranged according to the BLAST ordering. We have HPk D Qk Rk ;

(3.14)

where again Qk and Rk are unitary and upper triangular, respectively. Define the effective SNR gain, GQ (that will be illustrated in the following subsection), which is a function of the list length Q. The SNRs of the first detected symbol and the remaining M 1 symbols are given by ˇ ˇ ˇ .k/ ˇ2 GQ ˇrM;M ˇ (3.15) SNRM;k D N0 and SNRm;k D .k/

ˇ ˇ ˇ .k/ ˇ2 ˇrm;m ˇ N0

; m 2 f1; 2; : : : ; M 1g;

(3.16)

.k/

respectively. Here, rM;M and rm;m represent the M th and mth diagonal elements of Rk , respectively. The index of symbol to be detected first is found by kO D arg

max

k2f1;2;:::;M g

min fSNR1;k ; SNR2;k ; : : : ; SNRM;k g :

(3.17)

Implementing (3.17) with Q > 1 also requires O.M 4 / multiplications, where there are M QR factorizations with BLAST ordering. Furthermore, when Q D 1 is considered with the B-Chase ordering, it becomes the conventional BLAST ordering since there is no SNR gain enjoyed for the detection of the first symbol and G1 D 1.

3.1.3 Subdetectors For linear-list and SIC-list detection, the subdetectors are applied to each vector yq in (3.3) and xq in (3.8), respectively. Assuming that the qth candidate symbol in the list is correct and to be canceled, the resulting signal after cancellation can be written as


49

yq D Hs C n for linear-list detection; xq D As C n

for SIC-list detection:

(3.18)

The ML detection can be adopted to detect the subvector s for the optimal performance at the expense of high complexity. However, due to the limited computing power in practical systems, lower complexity subdetectors (such as linear and SIC detectors) are preferable. The linear subdetector provides the most efficient solution in terms of computing requirement as only the following simple operation is required (taking MMSE as an example):

1 H q sO D HH H C N0 I H y for linear-list detection;

1 H q A x for SIC-list detection: (3.19) sO D AH A C N0 I Since the linear subdetector provides less-impressive performance, for the trade-off between the ML and linear schemes, the SIC subdetector is preferred. According to (3.18), on one hand, the SIC subdetector can be adopted with the QR factorization H D QR for the linear-list approach; on the other hand, for the SIC-list approach, since the matrix A is upper triangular already, the SIC subdetector can be directly employed with A. Example 3.1. Consider a 3 3 MIMO system with 4-QAM modulation, where the channel matrix and the noise vector are given by H D Œh1 ; h2 ; h3 D 2 3 2 3 0:5 0 1 1Ci 4i 1 i 5 and n D 4 0:4i 5, respectively. Suppose that the 4-QAM is 1:4 i 0 i used for signaling and the transmit signal vector is given by s D Œs1 ; s2 ; s3 T D 2 3 1i 4 1 C i 5. Then, the received signal becomes 1 C i 2 3 0:5 C i y D Hs C n D 4 1 C 1:4i 5: (3.20) 1:4 The linear filter of H using the ZF is given by 2 0:5 0:5i 0 H 1 D 4 0:5 C 0:5i WDH H H 1 0:5 C 0:5i 1

3 0:5 C 0:5i 0:5 0:5i 5: 0:5 C 0:5i

(3.21)

Denoting by w3 the third column vector of W, we have yO D wH 3 y D 0:25 0:15i;

(3.22)

50


n o .1/ .2/ .Q/ where a list of Q candidate values for s3 , i.e., s3 ; s3 ; : : : ; s3 , can be generated. .q/

O we have Letting Q D 2 and q D 1; 2, while s3 denotes the qth closet symbol to y, ( .1/ s3 D 1 i : (3.23) .2/ s3 D 1 C i .1/

On one hand, the contribution of s3 D 1 i is then canceled from the received signal and the subdetection problem is formulated as .1/

y.1/ D y h3 s3 D Hs.1/ C n; (3.24) h iT .1/ .1/ . Using the ZF as the subdetector to detect where H D Œh1 h2 and s.1/ D s1 ; s2 the subvector s.1/ in (3.24), we have

s.1/ D

1i : 1 C i

(3.25) .2/

On the other hand, after canceling the contribution of s3 D 1 C i , the subvector can be detected with the ZF detector as 1i .2/ s D : (3.26) 1Ci As a result, the two candidate hard decision vectors are given by 8 2 3 1i ˆ ˆ ˆ ˆ ˆ s.1/ D 4 1 C i 5 ˆ ˆ ˆ ˆ 1 i < : 2 3 ˆ ˆ ˆ 1i ˆ ˆ ˆ .2/ 4 1Ci 5 ˆ s D ˆ ˆ : 1 C i

(3.27)

Using the method in (3.4), the final hard decision is obtained as sO D s.2/ :

(3.28)

This completes the detection. Furthermore, it is straightforward to show that as Q D 1, a falsely decision is carried out with sO D s.1/ . Example 3.2. Consider the same MIMO system introduced in Example 3.1. Using the QR factorization, the received signal vector is given by y D QRs C n;

(3.29)


51

2

3 1:41 0:82 0:82i 5. Letting x D Œx1 ; x2 ; x3 T D 0:82

1:41 0:71i 4 where R D 0 1:22 0 0 QH y, one can be obtained that

2

3 0:99 C 1:70i x D Rs C n D 4 0:57 1:39i 5 : 0:20 C 0:12i

(3.30)

Denoting by r3;3 the .3; 3/th entry of R, we have 1 x3 D 0:25 0:15i; (3.31) xO 3 D r3;3 n o .1/ .2/ .Q/ from where a list of Q candidate values for s3 , i.e., s3 ; s3 ; : : : ; s3 , can be .q/

generated. Denote by s3 the qth closet symbol to xO 3 . With Q D 2, we have (

.1/

s3 D 1 i .2/

:

(3.32)

s3 D 1 C i .1/

Once we cancel the contribution of s3 D 1 i from the received signal, on one hand, the subdetection problem is formulated as .1/

x.1/ D x cs3 D As.1/ C n; T

(3.33)

1:41 0:71i 1:41 , and s.1/ D ,A D 0 1:22 0:82 0:82i

where x D Œx1 ; x2 , c D h iT .1/ .1/ s1 ; s2 . Using the SIC as the subdetector to detect s.1/ in (3.33), we have 1i : D 1 C i

.1/

s

(3.34) .2/

On the other hand, after canceling the contribution of s3 D 1 C i , the subvector can be detected with the SIC detector as 1i s.2/ D : (3.35) 1Ci

52


As a result, the two candidate hard decision vectors are obtained as 8 2 3 1i ˆ ˆ ˆ ˆ s.1/ D 4 1 C i 5 ˆ ˆ ˆ ˆ < 1 i 2 3 ˆ ˆ 1i ˆ ˆ ˆ ˆ s.2/ D 4 1 C i 5 ˆ ˆ : 1 C i

;

(3.36)

where (3.9) is employed to obtain the final hard decision as sO D s.2/ :

(3.37)

This completes the detection. With Q D 1, the SIC-list detection reduces to the conventional SIC detection, where an incorrect decision is made with sO D s.1/ .

3.1.4 Performance Analysis For the list-based detection, the performance can be improved with a larger list length Q due to a better chance of successful cancellation for the first symbol. However, the larger Q, the higher computational complexity for detection is expected. Let dQ denote the distance between the transmitted symbol and the nearest decision boundary for given list length Q. In order to see the impact of the list length on the performance, the effective SNR gain of list length Q is considered in [15, 54] and given by 2 dQ GQ D : (3.38) d1 j! Denote by s D jaje symbol using 4-QAM as the modulation ˚ the3transmit method, where ! 2 ˙ 4 ; ˙ 4 . Suppose that s D e j 4 is transmitted. With the list length Q D 1, the decision region of the list-based detection becomes the conventional decision region and j! 4 j < 4 , which results in d1 D p12 . With Q D 2, the successful detection in above takes place if j! 4 j < 2 , where

2 D d2 D 1. Therefore, the resulting effective SNR gain becomes G2 D dd21 2. Similarly, since the same minimum distance between the transmit symbol and decision boundary can be considered with Q D 3, we can show that G3 D 2. Using the same strategy, [15] and [54] show that for 16-QAM, we have G2 D 2, G3 D 2, G8 D 8, and G10 D 10; for 64-QAM, we have G4 D 4, G8 D 8, G18 D 20,


53

G33 D 40, and G48 D 58. It is noteworthy that for the cases of Q D 1 (conventional detection) and Q D jSj (full length), G1 D 1 and GjSj D 1, respectively, since there would be no decision boundary for the case of full length. Although the decision region-based SNR gain is considered as an approximate performance metric, as shown in [54], it provides an accurate result to a certain extent (for an error probability of 0.01, it is accurate within 1 dB for a 16-QAM detector with Q 2 f1; : : : ; 9g and for a 64-QAM detector with Q 2 f1; : : : ; 41g). The correlation between list length and SNR gain is further analyzed by Liu, Ling, and StehlKe in [55]. By viewing the channel matrix as the basis of a lattice, a list of candidate lattice points is built up, where the closest lattice point is found to perform the list-based detection. Taking the real-valued channel matrix transformation (as will be shown in (3.45)), the channel matrix in (3.1) is generated as a 2N 2M real-valued matrix and [55] shows that QD

16eM GQ

GQ =4 ;

GQ < 16M;

(3.39)

which represents the relation between GQ and Q. Furthermore, in order to achieve the near-ML performance, it can be found in [55] that the list length follows Q D .e0 /4M=0

(3.40)

under the condition that PF GQ D

16M ; 0

0 > 1;

(3.41)

where PF denotes the proximity factor in [56]. The computational complexity of list-based detection depends on the ordering, the list length, and the subdetection methods. Without taking into account the ordering, the computational complexity of the detection is linearly proportional to the list length Q. In general, the overall complexity is mainly affected by the type of the subdetector. Denote by Csub and Csel the complexity of the subdetector (e.g., ML, linear, or SIC detector) employed for the detection and the complexity of the ordering, respectively. The overall complexity of list-based detection is given by Cchase D QCsub C Csel :

(3.42)

Depending on different applications, we can choose different subdetectors for detection (ML detector is used for good performance and linear detector is used for low complexity), where the complexity and performance trade-off needs to be considered. We should also note that the list-based detection cannot provide a full receive diversity order, which is illustrated through the simulation results.

54

3 List and Lattice Reduction-Based Methods 100

BER

10−1

10−2

ZF ZF−SIC List, ZF−SIC sub−detector, Q= 2 List, ZF−SIC sub−detector, Q= 4 List, ZF−SIC sub−detector, Q= 8 ML

10−3

10−4 0

2

4

6

8

10

12

14

16

18

20

Eb /N0

Fig. 3.1 BER performance of different detectors in a 16-QAM 2 2 MIMO system

3.1.5 Simulation Results Figures 3.1 and 3.2 show the BER performance of various detectors for 2 2 and 4 4 uncoded MIMO systems, respectively. 16-QAM is employed for signaling. Note that the ZF–SIC detector is used as the subdetector for the linear-list detection, where the S-Chase algorithm is carried out to perform symbol ordering. In general, it shows that the list-based detectors has a significant performance improvement compared to conventional suboptimal detectors (i.e., ZF and ZF–SIC detectors). In Fig. 3.1, we can show that the performance of list-based detector with Q D 8 approaches that of the ML detector. Figure 3.2 shows that the list-based detectors do not suit for a large MIMO system, since there is a big gap compared to the ML performance. Thus, the list-based detection can be less efficient when there are more transmit antennas (i.e., more layers of interference). It is also noteworthy that the list-based detection cannot exploit a full receive diversity order. Figure 3.3 shows the impact of the choice of subdetector on the overall performance when the same list length, Q D 4, is used. Obviously, with the ML subdetector, the performance of the list-based detector approaches that of


55

100

10−1

BER

10−2

10−3

10−4 ZF ZF−SIC List, ZF−SIC sub−detector, Q= 2 List, ZF−SIC sub−detector, Q= 4 List, ZF−SIC sub−detector, Q= 8 ML

10−5

10−6

0

2

4

6

8 Eb /N0

10

12

14

16

Fig. 3.2 BER performance of different detectors in a 16-QAM 4 4 MIMO system 100

10−1

BER

10−2

10−3

10−4

ZF ZF−SIC List, ZF sub−detector, Q= 4 List, ZF−SIC sub−detector, Q= 4 List, ML sub−detector, Q= 4 ML

10−5

10−6

0

2

4

6

8 Eb /N0

10

12

14

16

Fig. 3.3 BER performance of list-based detectors using different types of subdetectors in a 16QAM 4 4 MIMO system

56


the conventional ML exhaustive search method. As expected, using a lower complexity subdetector such as the ZF and ZF–SIC detectors leads to an inevitable performance degradation.

3.2 Lattice Reduction-Based Detection In this section, we will introduce the lattice (basis) reduction or LR and its applications to MIMO detection [21–28]. In general, the LR [20] was developed to transform a basis to a nearly orthogonal one, which has been used for lattice decoding and deriving computationally efficient search algorithms to find the closest point. Taking the channel matrix as a basis for a lattice, the MIMO detection problem can be considered as a lattice decoding problem, where the LR is adopted to develop computational efficient detection schemes. In this section, MIMO detection that uses LR is referred to as the LR-based detection.

3.2.1 MIMO Systems with Lattice Consider a basis B consisting of M real-valued linearly independent basis vectors which is given by B D fb1 ; b2 ; : : : ; bM g :

(3.43)

Since a lattice can be generated from an integer linear combination of a basis, with B, we can have a lattice defined by ( D uju D

M X

) bm zm ; zm 2 Z ;

(3.44)

mD1

where Z denotes the set of integer numbers. Note that a lattice can be generated by different bases or matrices. Let us review the system model of MIMO channel presented in (2.2) with N M (note that the lattice-based MIMO systems are commonly carried out with N M ). There are three key elements to adopt such a MIMO system with lattice, which are shown as follows: (1) H becomes the basis (i.e., B in (3.43)). Thus, the basis vectors (i.e., column vectors) of H should be real-valued. (2) s is used to produce an integer linear combination of the basis in (3.44). Therefore, the elements of s need to be integer. (3) y becomes a vector in the lattice generated by the basis H.

3.2 Lattice Reduction-Based Detection

57

Using the real-valued matrix transformation [21, 24], H can be converted to the one with real-valued basis vectors (i.e., column vectors), and then (2.2) is rewritten as R.n/ R.s/ R.H/ J.H/ R.y/ ; (3.45) C D J.n/ J.s/ J.H/ R.H/ J.y/ where R./ and J./ denote the real and imaginary parts, respectively. Furthermore, T we define the real-valued vectors and matrix as yr D R.y/T J.y/T of size 2N 1, T T sr D R.s/T J.s/T of size 2M 1, nr D R.n/T J.n/T of size 2N 1, and Hr D R.H/ J.H/ of size 2N 2M . Then, (3.45) becomes J.H/ R.H/ yr D Hr sr C nr ;

(3.46)

where the real-valued matrix Hr can be used as a basis of lattice. Next, scaling and translation operations are carried out to transfer the elements of sr to consecutive integers. For example, suppose that sr 2 S 4 and S D f3; 1; 1; 3g. Let sNr D 12 .sr C 31/, where 1 D Œ1 1 : : : 1T . Thus, the elements of sNr are in the set of consecutive integers, denoted by Z. If we replace yr , sr , and nr in (3.46) by yN r D 12 .yr C 3H1/, sN r D 12 .sr C 31/, and nN r D 12 nr , respectively, the received signal vector can be rewritten as yN r D Hr sNr C nN r :

(3.47)

From (3.47), we can see that Hr sNr (or yN r , approximately) becomes a vector in the lattice generated by the basis Hr . Thus, the MIMO detection problem reduces to a search problem which finds a vector in the lattice. Although the lattice is considered with Hr initially, later research developed the lattice with a complex-valued H [25-27], where the complex-valued LR is used to improve the performance of MIMO detection. For the complex-valued LR, we can transfer the real and imaginary parts of the QAM symbol sk to consecutive integers, where sk denotes the kth element p of s. Denote by C D Z C j Z the set of complex consecutive integers, where j D 1. Let the symbol index k be omitted for the sake of simplicity. Using proper scaling and shifting, we can have f˛s C ˇg C;

s 2 S;

(3.48)

where ˛ and ˇ denote the scaling and shifting coefficients, respectively. For KQAM, the alphabet is given by S D fs D a C jbja; b 2 f.2P 1/A; : : : ; 3A; A; A; 3A; : : : ; .2P 1/Agg ; (3.49)

58


q 3Es where P D log22 K and A D . Here, Es D EŒjsj2 denotes the symbol 2.K1/ energy. In this case, the scaling and shifting coefficients that lead (3.48) to be satisfied become 1 2A 2P 1 ˇD .1 C j /: 2 ˛D

(3.50)

It is noteworthy that the pair of ˛ and ˇ is not uniquely determined. From (3.48), it can be easily shown that sN D ˛s C ˇ1 2 CM :

(3.51)

With y in (2.2), the complex-valued lattice-based MIMO system model is found as yN D ˛y C ˇH1 N D HNs C n;

(3.52)

where nN D ˛n. In addition, it shows that there is no performance difference between the real-valued and complex-valued lattice-based approaches [25, 27]. For the sake of convenience, throughput this section, we assume that signals and channels are complex-valued stated otherwise. Furthermore, Es denotes the unless symbol energy of s and E nnH D N0 I.

3.2.2 Lattice Reduction-Based MIMO Detection Since a lattice can be generated by different bases or channel matrices, in order to mitigate the noise and interference between multiple signals, we can find a matrix whose column vectors are nearly orthogonal to generate the same lattice. This technique is regarded as the LR. LR can be applied to MIMO systems to improve the performance of suboptimal MIMO detection, where the resulting detection methods are regarded as the LR-based MIMO detection [21–28]. In this subsection, we study the LR-based detection for MIMO systems. Consider two bases H and G that span the same lattice, where each column vector of a basis is an integer linear combination of the column vectors of the other basis. For example, if 2 1 (3.53) HD 1 1 and

GD

1 0 ; 0 1

(3.54)


we can easily show that

and

59

0 1 2 C D2 1 0 1

(3.55)

0 1 1 : C D 1 0 1

(3.56)

Thus, bases H and G have the same lattice. It is also shown that H D GU;

(3.57)

where U is a unimodular matrix. Then, the received signal of (2.2) can be rewritten as y D GUs C n D Gc C n;

(3.58)

where c D Us. With the unimodular matrix U that consists of integers, if s 2 CM , then we can have c 2 CM . However, for s consists of QAM symbols, i.e., sk , the scaling and shifting coefficients in (3.50) can be used to transfer the real and imaginary parts of sk into a consecutive integer set. Based on (3.58), since the received signal can be treated as the lattice points spanned by the basis (i.e., H or G), MIMO systems with lattice are developed, where conventional low complexity detectors (e.g., linear and SIC detectors) are able to be carried out to detect c. Note that although the ML detection can be applied to the lattice-reduced matrices, there is no performance gain due to an exhaustive search has already been carried out.

3.2.2.1 LR based Linear Detection The LR-based linear detectors [24] are carried out to detect c as cQ D WH y;

(3.59)

1 H G is used for the LR-based ZF detector where the linear filter WH D GH G

1 N0 H 1 H H H and W D G G C Es U U G is used for the LR-based MMSE detector. Suppose that s consists of QAM symbols. Let cN D UNs. According to (3.51), we can show that cN D ˛Us C ˇU1 D ˛c C ˇU1 2 CM

(3.60)

60


Fig. 3.4 The decision boundaries of ZF detection with a lattice generated by the basis H in (3.53)

after proper scaling and shifting. Thus, a hard decision of c is found by cO D

1 .b˛Qc C ˇU1e ˇU1/ ; ˛

(3.61)

where b:e represents the rounding operation. Then, the estimation of s is obtained from cO as sO D U1 cO : (3.62) It is noteworthy that the scaling and shifting operations of cO would be unnecessary of ZF detection if s 2 CM . In Figs. 3.4 and 3.5, we present decision boundaries 1 0 2 1 , and G D with the same lattice generated by two bases, H D 0 1 1 1 respectively. It is observed that the decision boundaries of Fig. 3.4 are narrow and small amounts of noise would lead to errors in detection. On the contrary, in Fig. 3.5, with an orthogonal basis of G, the wide decision boundaries may result in good performance in detection. Therefore, for suboptimal MIMO detection, a nearly orthogonal basis is desired to be employed to improve the performance. As shown in Fig. 3.5, the decision regions of ZF detection become the same as that of ML detection, which leads to that the ZF detection is able to provide the optimal performance, as an orthogonal basis or channel matrix is considered. Therefore, it becomes important to find a unimodular matrix U to produce an orthogonal or nearly orthogonal matrix G (i.e., column vectors of G are orthogonal or nearly orthogonal). To this end, the LR is employed to generate G from H, which will be introduced in the following subsections. The matrix G generated by the LR is regarded as the lattice-reduced matrix. Example 3.3. Consider a 2 2 channel matrix HD

2 1 1 1

(3.63)


61

Fig. 3.5 The decision boundaries of ZF detection with a lattice generated by the basis G in (3.54)

and an orthogonal matrix

GD

It can be given that

1 0 : 0 1

(3.64)

2 1 ; HDG 1 1 „ ƒ‚ …

(3.65)

U

where U is integer unimodular. Thus, H and G span the same lattice, where two 1 column vectors of G are more orthogonal than those of H. Denote by s D 1 0:4 the transmit signal vector and the noise vector, respectively. Then, and n D 0:3 the received signal becomes 1:4 : (3.66) y D Hs C n D 0:3 On one hand, let the conventional ZF detection be carried out to detect s as j

1 H m 2 1:7 sO D HH H ; (3.67) D H y D 2 2 which is an incorrect decision. On the other hand, if the ZF detection is employed with the orthogonal matrix G, letting y D G „ƒ‚… Us Cn, we have cO D

j

1 H m GH G G y D

c

1:4 0:3

D

1 : 0

(3.68)

62


Then, the estimation of s is converted from cO as sO D U1 cO D

1 : 1

(3.69)

This provides a correct solution.

3.2.2.2 LR–based SIC Detection For the LR-based ZF–SIC detector [21], the matrix G is QR factorized as G D QR;

(3.70)

where Q is unitary and R is upper triangular. Premultiplying QH to y in (3.58), we have QH y D QH .Gc C n/ D QH QRc C QH n D Rc C n;

(3.71)

as the statistical properties of QH n and n are the same. For the LR-based MMSE–SIC detector [25], the system presented in (3.46) is rewritten as # " # " H n y q D q N0 sC : (3.72) 0 0 I s N Es Es q iT q iT h h T N0 T T 0 Let yex D yT 0 , Hex D HT N I , and n D n . After ex Es Es s performing the LR with Hex , the lattice reduced matrix Gex can be found as Hex D Gex Uex , where Uex is unimodular. Then, (3.72) is rewritten as yex D Gex cex C nex ;

(3.73)

where cex D Uex s. Using the QR factorization of Gex D Qex Rex , the LR-based MMSE-SIC detection is carried out, where Qex and Rex are unitary and upper triangular, respectively. Pre-multiplying QH ex to yex in (3.73), we can have QH ex yex D Rex cex C nex :

(3.74)

H Since QH ex nex has the same statistical properties as nex , we use nex to denote Qex nex in (3.74). Then, the ZF-SIC and MMSE-SIC are carried out to detect c in (3.71) and cex in (3.74), respectively. Taking ZF-SIC as an example. Denote by ci , xj , and rp;q the


63

i th symbol of c, the j th symbol of x, and the .p; q/-th entry of R, respectively. The element of the last row, the M -th symbol of c, is detected first as cOM D

1 .b˛ cQM C ˇuM 1e ˇuM 1/ ; ˛

(3.75)

M where cQM D rxM;M and uk denotes the kth row vector of U. Then, its contributions in the second last row are canceled and the .M 1/-th symbol of c is detected. This operation is repeated up to the first row. The mth symbol of c is detected after canceling M m symbols as

1 .b˛ cQm C ˇum 1e ˇum 1/ ; (3.76) ˛ P O D M qDmC1 rm;q cOq and m 2 f1; 2; : : : ; M 1g. Let c cOm D

1 xM where cQm D rk;k

ŒcO1 ; cO2 ; : : : ; cOn T , the estimation of s is given by sO D U1 cO :

(3.77)

More details of the SIC detection can be found in Sect. 2.4. Note that instead of using an exhaustive search method in (2.43) and (2.45) for the conventional SIC, the rounding operation is used to estimate each symbol when the LR-based SIC is carried out, which leads to the reduced computational complexity. Example 3.4. Considering the same MIMO system discussed in Example 3.3, we have 2 1 1 0 2 1 D : (3.78) 1 1 0 1 1 1 „ ƒ‚ … „ ƒ‚ … „ ƒ‚ … H

G

U

Using the QR factorization 2:2361 1:3416 0:8944 0:4472 HD 0 0:4472 0:4472 0:8944 „ ƒ‚ …„ ƒ‚ …

R

Q

with the received signal

(3.79)

1:4 D Hs C n; 0:3 „ ƒ‚ …

(3.80)

y

on one hand, the conventional ZF–SIC detection is performed by multiplexing QH to y as

64


2:2361 1:3416 1:1180 s1 D C QH n: 0 0:4472 0:8944 s2 ƒ‚ … „ ƒ‚ … „

QH y

(3.81)

R

From (3.81), symbols are detected sequentially as follows: 0:8944 .1:1180 .1:3416 sO2 // sO2 D D 2 ) sO1 D D 2: (3.82) 0:4472 2:2361 Clearly, the conventional ZF–SIC cannot provide a correct solution. On the other hand, with the orthogonal matrix G, the LR-based ZF–SIC detection is carried out with the QR factorization 1 0 1 0 GD ; (3.83) 0 1 0 1 „ ƒ‚ … „ ƒ‚ … R

Q

and then

1 0 c1 1:4 D C QH n; c2 0 1 0:3 „ ƒ‚ … „ ƒ‚ …

(3.84)

R

QH y

where the sequential detection is performed as cO2 D b0:3e D 0

)

cO1 D b1:4e D 1:

(3.85)

Letting cO D ŒcO1 cO2 T , s is estimated as sO D U1 cO D

1 : 1

(3.86)

This provides a correct solution.

3.2.3 Lattice Reduction Schemes for Two Basis Systems The problem to find the best basis vectors for a given lattice in terms of the orthogonality is regarded as a nonzero shortest vector problem in a lattice or the NPhard problem [57], where the LLL algorithm is considered as an approximation with polynomial complexity. In this subsection, we explain the LR for MIMO systems with two real-valued basis vectors, where the two shortest vectors are generated. Different LR algorithms with real and complex valued channel matrices will be presented in the following subsections for MIMO detection.


65

In Sect. 2.4, we have introduced the QR factorization for two basis systems where H D Œh1 h2 . It has been shown that for the SIC detection, the best ordering of basis is determined with the maximized min fkh1 k; kr2 kg in (2.34). Instead of considering the subspace spanned by H D Œh1 h2 with the QR factorization, the lattice generated by the basis H is carried out with the LR. Using the linear relationship introduced in (3.57), the received signal vector is given by y D Hs C n D GUs C n D Gc C n;

(3.87)

where G D Œg1 g2 and the integer matrix U leads to that c 2 Z2 . Then, as explained in Subsect. 3.2.2, low-complexity detectors are carried out to detect signals from c rather than those from s. Denote by gQ 1 the component of g1 that is orthogonal to g2 , while gQ 2 is defined similarly. Note that using the method presented in (2.32), gQ 1 and gQ 2 are generated as gQ 1 D g1 gQ 2 D g2 !g1 ;

(3.88)

where ! D hgkg21;gk12i . Due to that H and G generate the same lattice, it is desired to find a certain G that can provide better performance of different suboptimal detection schemes, including linear and SIC-based detection. For example, with the SIC detection, since the effective SNR of detecting the s1 and s2 are kg1 k and kQg2 k, respectively, the best basis is given by max min fkg1 k ; kQg2 kg subject to H D GU: U

(3.89)

According to (3.89), the lattice-reduced basis G for suboptimal MIMO detection is generated. Furthermore, it is defined that Definition 3.1. The basis G D Œg1 g2 of a lattice is regarded as the latticereduced basis when g1 is the nonzero shortest vector in the lattice and g2 is the shortest vector that is not proportional to g1 in . Two theorems are provided to illustrate that Definition 3.1 is able to deliver the solution in (3.89) as follows: Theorem 3.1. Denote by a lattice that is generated by a two basis matrix, H D Œh1 h2 . Suppose there exists a matrix G D Œg1 g2 which leads to H D GU and U is integer unimodular. Letting g1 be the nonzero shortest vector in and g2 be the shortest vector that is not proportional to g1 in , we have ˚ min fkg1 k; kQg2 kg min kh1 k; hQ 2 :

(3.90)

66


Proof. Theorem 3.1 shows that it is desirable to transform H into G, which is proved by two parts: ˚ ˚ .1/ kg1 k min kh1 k; khQ 2 k .2/ kQg2 k min kh1 k; khQ 2 k : (3.91) Proof of part (1): With the same lattice generated by fh1 h2 g and fg1 g2 g, letting a1 ; a2 2 Z, we have g1 D a1 h1 C a2 h2 . Defining that hO 2 D h2 hQ 2 ? h2 , where hQ 2 is orthogonal to h1 and hO 2 , it can be given that kg1 k D ka1 h1 C a2 h2 k

D a1 h1 C a2 hQ 2 C hO 2 ka2 hQ 2 k:

(3.92)

Since a1 ; a2 2 Z, considering the case of j a2 j 1, we have kg1 k a2 hQ 2 khQ 2 k ˚ min kh1 k; hQ 2 :

(3.93)

Then, let us consider the case of a2 D 0 and a1 ¤ 0. It is given by kg1 k D j a1 j kh1 k kh1 k:

(3.94)

Note that the shortest vector g1 leads to that kg1 k kh1 k:

(3.95)

From (3.94) and (3.95), we can show that kg1 k D kh1 k ˚ min kh1 k ; hQ 2 :

(3.96)

This completes the proof of part (1). Proof of part (2): Since fh1 h2 g and fg1 g2 g generate the same lattice , it is given by kg1 k kQg2 k D kh1 k hQ 2 : From (3.95) and (3.97), it is derived that kQg2 k hQ 2 ˚ min kh1 k; hQ 2 : This completes the proof of part (2).

(3.97)

(3.98) t u


67

Theorem 3.1 shows that the basis G is able to provide better detection performance according to (3.89), the next question would be the ordering of vectors: whether G D Œg1 g2 or G D Œg2 g1 , which is illustrated by the following theorem. Theorem 3.2. Denote by a lattice that is generated by a two basis matrix G D Œg1 g2 . Letting g1 be the nonzero shortest vector in and g2 be the shortest vector that is not proportional to g1 in , we have min fkg1 k; kQg2 kg min fkg2 k; kQg1 kg :

(3.99)

Proof. Since gQ 1 is the component of g1 that is orthogonal to g2 in (3.88), and g1 is the shortest vector in , we have kg2 k kg1 k kQg1 k and then min fkg2 k; kQg1 kg D kQg1 k:

(3.100)

With kg1 k kQg1 k and (3.100), we only need to show that kQg2 k kQg1 k to identify the inequality of (3.99). To this end, using the method shown in (2.55), it is derived as kQg1 k2 D kg1 k2

j hg1 ; g2 i j2 kg2 k2

kQg2 k2 D kg2 k2

j hg2 ; g1 i j2 ; kg1 k2

(3.101)

which results in

j hg2 ; g1 i j2 D kg2 k2 kg1 k2 kQg1 k2

D kg1 k2 kg2 k2 kQg2 k2 :

(3.102)

Up to this point, we can show that kQg2 k2 kQg1 k2 D kg1 k2 kg2 k2

(3.103)

or kQg1 k2 D

kg1 k2 kQg2 k2 kg2 k2

kQg2 k2 This completes the proof.

(3.104) t u

So far, we have provided that the basis G according to Definition 3.1 is lattice reduced and can improve the performance of suboptimal detection. Furthermore, in order to obtain the lattice-reduced basis, Theorem 3.3 provides a sufficient condition.

68


Theorem 3.3. Denote by a lattice that is generated by a two basis matrix G D Œg1 g2 . If kg1 k < kg2 k and jhg1 ; g2 ij 12 kg1 k2 , then g1 is the nonzero shortest vector in and g2 is the shortest vector that is not proportional to g1 in . Proof. (1) Proof of “g1 is the nonzero shortest vector in ”: Let b D a1 g1 C a2 g2 be a vector in , where a1 ; a2 2 Z. It is derived as kbk2 D j a1 j2 kg1 k2 C j a2 j2 kg2 k2 C 2ha1 g1 ; a2 g2 i j a1 j2 kg1 k2 C j a2 j2 kg2 k2 C 2 j a1 a2 j hg1 ; g2 i

j a1 j2 C j a2 j2 C j a1 a2 j kg1 k2 :

(3.105)

Given that a1 and a2 do not equal to zero, simultaneously, with a1 ; a2 2 Z, we have j a1 j2 C j a2 j2 C j a1 a2 j 1:

(3.106)

From (3.105) and (3.106), we have kg1 k2 kbk2 for any b 2 and b ¤ 0. This completes the proof of part (1). (2) Proof of “g2 is the shortest vector that is not proportional to g1 in ”: Let b D a1 g1 C a2 g2 be a vector in n fcg1 ; c 2 Zg, where a1 ; a2 2 Z, a2 ¤ 0, and “n” denotes the set minus. Then, it follows that kbk2 D j a1 j2 kg1 k2 C j a2 j2 kg2 k2 C 2ha1 g1 ; a2 g2 i

C j a2 j2 kg1 k2 j a2 j2 kg1 k2

D j a2 j2 kg2 k2 kg1 k2 C j a1 j2 kg1 k2 C j a2 j2 kg1 k2 C 2ha1 g1 ; a2 g2 i:

(3.107)

˚ Since j a1 j2 kg1 k2 C j a2 j2 kg1 k2 C 2ha1 g1 ; a2 g2 i kg1 k2 , (3.107) is rewritten as

kbk2 j a2 j2 1 kg2 k2 kg1 k2 C kg2 k2 : (3.108) Given that j a2 j2 1 0 and kg2 k2 kg1 k2 0, we have kbk2 kg2 k2 : This completes the proof of part (2).

(3.109) t u

In addition, Theorem 3.3 can be easily extended for the cases of complex basis vectors as follows. Theorem 3.4. Denote by a lattice that is generated by a two basis matrix G D Œg1 g2 . If kg1 k < kg2 k, j R .hg1 ; g2 i/ j 12 kg1 k2 , and j J .hg1 ; g2 i/ j 12 kg1 k2 , then g1 is the nonzero shortest vector in and g2 is the shortest vector that is not proportional to g1 in .


69

3.2.4 Gaussian Lattice Reduction for Two Basis Systems Using the results provided in Subsect. 3.2.3, we can build the Gaussian LR algorithms for two-dimensional lattices as follows. Consider a real-valued matrix with two column vectors, which is given by H D Œh1 h2 under the condition that kh1 k kh2 k (column swapping is considered to provide this condition). In order to find a lattice reduced matrix of H, let G D Œg1 g2 , where g1 D h1 g2 D h2 ch1 :

(3.110)

Note that c 2 Z in (3.110) according to the principle of lattice. With the linear relationship between H and G, (3.110) is rewritten as (3.111) G D HU1 ; 1 c . Here, c is determined to (1) minimize where the unimodular matrix U1 D 0 1 the length of g2 and (2) minimize the absolute correlation between g1 and g2 , which can be further developed as cO D arg min kg2 k2

.1/

c2Z

D arg min kh2 ch1 k2 D

c2Z

hh2 ; h1 i kh1 k2

(3.112)

and .2/

cO D arg min j hh2 ch1 ; h1 i j2 c2Z

D arg min j hh2 ; h1 i ckh1 k2 j2 D

c2Z

hh2 ; h1 i ; kh1 k2

(3.113)

respectively. It shows that the minimization of the length of g2 is equivalent to the minimization of the absolute correlation between g1 and g2 . We can also show that g1 and g2 become orthogonal when c D hhkh21;hk12i , however, since c 2 Z, g1 ; and g2 may not be orthogonal and their correlation is given by hg2 ; g1 i D h .h2 ch O 1 / ; h1 i D hh2 ; h1 i ckh O 1 k2 hh2 ; h1 i D cO kh1 k2 : kh1 k2

(3.114)

70


j With cO D

hh2 ;h1 i kh1 k2

m , one can be shown that ˇ ˇ ˇ hh2 ; h1 i ˇ 1 ˇ ˇ ˇ kh k2 cO ˇ 2 ; 1

(3.115)

1 kg1 k2 : 2

(3.116)

and then (3.114) is rewritten as jhg2 ; g1 ij This fulfills Theorem 3.3.

10 a real-valued matrix of two column Denote by H D Œh1 h2 and T D 01 vectors and an identity matrix, respectively. The algorithm of real-valued Gaussian LR is summarized as follows:

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

INPUT: fH; Tg OUTPUT: fG; Tg if kh1 k > kh2 k Swap two columns in H and T, respectively end if if 2jhh2j; h1 ij >mkh1 k2 c D hhkh2 ;hk12i 1 1 c h2 D h2 ch1 and T D T 0 1 end if if kh1 k < kh2 k The algorithm is terminated with G H else Go to step (1) end if

Here, the correlation between two vectors and the length of vectors are checked at Step (4) and Step (8), respectively. If both the conditions are fulfilled, we have the lattice-reduced matrix G and an integer unimodular matrix T, where G D HT

(3.117)

and T D U1 . Example 3.5. Consider a real-valued channel matrix H D Œh1 h2 41 36 ; D 59 48

(3.118)


where

h1 D

41 59

71

and h2 D

36 : 48

(3.119)

1 0 . Using H and T as the input, the Gaussian LR 0 1 algorithm is performed as follows. Since kh1 k > kh2 k, we swap the basis vectors as 36 41 and h2 D ; (3.120) h1 D 48 59 0 1 . By checking the correlation between two vectors at Step (4), while T D 1 0 we have jhh2 ; h1 ij D 1:20 > 1=2: (3.121) kh1 k2

Let the identity T D

Given that

cD

the basis reduction is carried out as

h2 ( h2 and

TD

0 1 1 0

hh2 ; h1 i kh1 k2

D 1;

(3.122)

hh2 ; h1 i 5 D h 1 11 kh1 k2

1 0

0 1 D 1 1

1 : 1

By checking the length of updated vectors at Step (8), we have 36 5 48 > 11 ; „ ƒ‚ … „ ƒ‚ … kh1 k

(3.123)

(3.124)

(3.125)

kh2 k

which is not fulfilled. Then, go to Step (1) of the algorithm with the up-to-date fh1 ; h2 ; Tg. The Gaussian LR algorithm is performed iteratively until both the conditions (the correlation between two vectors and length of vectors) are fulfilled. In summary, the iterative basis reduction procedure is shown as 41 36 36 5 5 11 11 5 Œh1 h2 D ) ) ) 59 48 48 11 11 7 7 11 5 1 1 5 0 1 1 0 : (3.126) ) ) ) TD 6 1 1 6 1 1 0 1

72

Denote by


11 5 GD 7 11

5 and T D 6

1 1

(3.127)

the lattice-reduced matrix and the corresponding integer unimodular matrix, respectively, and we have G D HT. Note that the reduced basis has smaller length and lower correlation compared to those of the original one, which is illustrated as ( fkh1 k; kh2 kg D f71:85; 60g Original basis H ; hH 1 h2 1 kh1 kkh2 k ( fkg1 k; kg2 kg D f13:04; 12:08g : (3.128) Reduced basis G gH 1 g2 D 0:14 kg1 kkg2 k The real-valued Gaussian LR can be extended to the one with complex-valued lattice, which is regarded as the complex-valued Gaussian LR. Denote by H D Œh1 h2 a complex-valued matrix with two column vectors, where kh1 k < kh2 k. As in (3.111), there exists a lattice-reduced matrix consists of two complex-valued column vectors, G D Œg1 g2 . The difference would be that c is a complex integer and c 2 Z C j Z. Hence, we have cO D arg min jhh2 ch1 ; h1 ij2 c2ZCj Z

ˇ ˇ2 D arg min ˇhh2 ; h1 i ckh1 k2 ˇ D

c2ZCj Z

hh2 ; h1 i ; kh1 k2

(3.129)

where the rounding operation is carried out with complex numbers and bxe D bR.x/e C j bJ.x/e. From (3.114) and (3.129), we can show that hh2 ; h1 i hg2 ; g1 i D cO kh1 k2 ; (3.130) kh1 k2 and then

ˇ ˇ ˇ ˇ ˇR hh2 ; h1 i cO ˇ 1=2 ˇ ˇ 2 kh1 k ˇ ˇ ˇ ˇ ˇJ hh2 ; h1 i cO ˇ 1=2: ˇ ˇ 2 kh1 k

(3.131)

According to (3.130) and (3.131), we have 1 kg1 k2 2 1 jJ .hg2 ; g1 i/j kg1 k2 ; 2 and thus Theorem 3.4 is fulfilled. jR .hg2 ; g1 i/j

(3.132)


73

10 a complex-valued matrix of two Denote by H D Œh1 h2 and T D 01 column vectors and an identity matrix, respectively. The algorithm of complexvalued Gaussian LR is summarized as follows:

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

INPUT: fH; Tg OUTPUT: fG; Tg if kh1 k > kh2 k Swap two columns in H and T, respectively end 2 2 if 2jR .hh j 2 ; h1 i/j m > kh1 k or 2jJ .hh2 ; h1 i/j > kh1 k c D hhkh21;hk12i 1 c h2 D h2 ch1 and T D T 0 1 end if if kh1 k < kh2 k The algorithm is terminated with G H else Go to Step (1) end if

The major differences of the complex-valued Gaussian LR algorithm are: (a) at Step (4), the correlation is checked with real and imaginary parts; (b) at Step (5), the rounding operation is carried out with complex numbers. Example 3.6. Consider a complex-valued channel matrix H D Œh1 h2 27 C 38i 46 C 55i ; D 36 C 45i 47 C 63i where

h1 D

27 C 38i 36 C 45i

and h2 D

46 C 55i : 47 C 63i

(3.133)

(3.134)

By employing the complex-valued Gaussian LR, the iterative basis reduction procedure is shown as follows: 27 C 38i 19 C 17i 27 C 38i 46 C 55i Œh1 h2 D ) 36 C 45i 11 C 18i 36 C 45i 47 C 63i 11 C 4i 19 C 17i 19 C 17i 11 C 4i ) ) 14 C 9i 11 C 18i 11 C 18i 14 C 9i 1 0 1 1 1 1 3 1 TD ) ) ) : (3.135) 0 1 0 1 3 2 2 1

74


Denote by GD

11 C 4i 19 C 17i 14 C 9i 11 C 18i

and T D

3 2

1 1

(3.136)

the lattice-reduced matrix and the corresponding integer unimodular matrix, respectively, and we have G D HT. Note that the reduced basis has smaller length and lower correlation compared to the original one, which is clarified as ( Original basis H ( Reduced basis G

fkh ˇ 1 k;H kh2ˇkg D f74:12; 106:39g ; ˇ h1 h2 ˇ 1 ˇ kh1 kkh2 k ˇ fkg ˇ 1 k;H kg2ˇkg D f20:35; 33:09g : ˇ g1 g2 ˇ 0:31 ˇ kg1 kkg2 k ˇ D

(3.137)

3.2.5 LLL and CLLL Algorithms In order to find a matrix whose column vectors are nearly orthogonal to generate the same lattice, the LLL algorithm is proposed in [20]. Using the LLL algorithm, the LR can be performed for the M -basis MIMO system with the channel matrix of size N M , N M . Note that the LLL algorithm is initially designed with a real-valued matrix which can be transformed from a complex-valued matrix using the method in (3.45). However, later research developed a class of LLL algorithms with complex-valued matrices, which is referred to as the CLLL. In this subsection, we first introduce the LLL algorithm (real-valued) for the LR of MIMO channels. Using the method in (3.45), the 2N 2M real-valued matrix Hr is obtained from the N M complex-valued matrix H. Then, the LLL algorithm is carried out to transform the given basis, Hr , into a new basis consisting of nearly orthogonal basis vectors, Gr . The real-valued matrix Gr of size 2N 2M is regarded as the LLL-reduced matrix [20] if Gr is QR factorized as Gr D Qr Rr ;

(3.138)

where Qr of size 2N 2N is unitary (QTr Qr D IN ) and Rr of size 2N 2M is upper triangular. The elements of Rr satisfy the following inequalities: j ŒRr `; j

1 j ŒRr `;` j; 2

1 ` < 2M

(3.139)

and ıŒRr 21;1 ŒRr 2; C ŒRr 21; ;

D 2; : : : ; 2M;

(3.140)


75

where ŒRr p;q denotes the .p; q/th entry of Rr . The parameter ı is closely related to a quality-complexity trade-off [20]. Note that for the LLL and CLLL algorithms, ı is chosen from . 14 ; 1/ and . 12 ; 1/, respectively [27], while ı D 34 can be considered to meet a good quality–complexity trade-off. The LLL algorithm [20, 24] that generates the LLL-reduced matrix Gr from the real-valued channel matrix Hr is summarized as follows. The input and output of the algorithm are given by fHr g and fQr ; Rr ; Tr g, respectively.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

INPUT: fHr g OUTPUT: fQr ; Rr ; Tr g ŒQr Rr qr.Hr / size.Hr ; 2/ Tr I while for ` D 1 W 1 dRr . `; /=Rr . `; `/c if ¤ 0 Rr .1 W `; / Rr .1 W `; / Rr .1 W `; `/ Tr .W; / Tr .W; / Tr .W; `/ end if end for if ı.Rr . 1; 1//2 > Rr .; /2 C Rr . 1; /2 Swap the . 1/th and th columns in Rr and Tr .1;1/ ˛ D kRRr r.1W;1/k ˛ ˇ with D r .;1/ ˇ ˛ ˇ D kRrR.1W;1/k

(15) Rr . 1 W ; 1 W / Rr . 1 W ; 1 W / (16) Qr .W; 1 W / Qr .W; 1 W / T (17) maxf 1; 2g (18) else (19) C1 (20) end if (21) end while From the output, the LLL-reduced matrix is obtained as Gr D Hr Tr . Then, using the reduced matrix Gr and the corresponding unimodular matrix Tr , the LR-based linear or SIC detectors can be carried out to perform the LR-based MIMO detection (Subsect. 3.2.2 gives the details). Example 3.7. Consider a 4 4 complex-valued channel matrix 2

0 61 C i HD6 4 i 1Ci

3 0 1 0 1 i 1Ci7 7: i 1 0 5 1 1Ci i

(3.141)

76


In order to perform the LLL algorithm, H is transformed to a 88 real-valued matrix Hr as Hr D

R.H/ J.H/ J.H/ R.H/

D Œh1 2 0 61 6 60 6 6 61 D6 60 6 61 6 41 1

h2 h3 h4 h5 h6 h7 h8 0 1 0 1 0 0 1 0

1 0 1 1 0 1 0 1

0 1 0 0 0 1 0 1

0 1 1 1 0 1 0 1

0 0 1 0 0 1 0 1

3 0 17 7 0 7 7 7 17 7: 0 7 7 1 7 7 0 5 0

0 1 0 1 1 0 1 1

(3.142)

After the LR (using the LLL algorithm), the unimodular matrix Tr is given by 2

1 6 1 6 6 0 6 6 6 1 Tr D 6 6 0 6 6 0 6 4 0 0

3 3 1 1 1 0 0 0

1 1 0 1 1 1 0 0

0 1 0 0 1 1 0 0

1 1 0 0 0 1 0 0

1 1 0 0 1 1 0 1

0 0 0 0 1 1 0 1

3 1 0 7 7 0 7 7 7 0 7 7 37 7 3 7 7 1 5

(3.143)

1

and the LLL-reduced matrix Gr becomes Gr D Hr Tr D Œg1 2 0 61 6 60 6 6 60 D6 60 6 60 6 40 0

g 2 g3 g4 g 5 g 6 g7 g8 1 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 0 1 0

0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 1

0 0 0 0 0 1 0 0

3 0 07 7 07 7 7 07 7: 17 7 07 7 05 0

(3.144)


77

Here, the orthogonality deficiency [27] is carried out to compare the orthogonality between the original matrix and the LLL-reduced one, and then we have 8 det.HH Hr / ˆ < Hr W 1 Q8 rkh k2 D 0:9956 i i D1 : Orthogonal deficiency of HG det G ˆ . : Gr W 1 Q8 r r / D 0 2 i D1

kgi k

From this, we can see that the highly correlated matrix Hr was transformed to an orthogonal one Gr after the LR (using the LLL algorithm). We can further show that the number of column swapping in Step (13) is 14 in this transformation. In [25, 27], the CLLL algorithm is developed to straightforwardly perform the LR with a complex-valued matrix, where the extra transformation in (3.45) is not required. Comparing to the LLL, the CLLL can provide the same performance with half complexity, approximately. Therefore, it is desired to use the CLLL for LRbased detection for the complexity reduction. Consider a matrix G, which is generated from a complex-valued matrix H of size N M , using the CLLL algorithm. With the QR factorization of G D QR, where Q is unitary and R is upper triangular, G is CLLL-reduced if the elements of R.m/ satisfy the following inequalities [27]: j R.ŒR`; / j

1 1 j ŒR`;` j and j J.ŒR`; / j j ŒR`;` j; 2 2

1` jR.; /j2 C jR. 1; /j2 Swap the . 1/th and th columns in R and T

78


R.1;1/ ˛ D kR.1W;1/k ˛ ˇ with R.;1/ ˇ ˛ ˇ D kR.1W;1/k R. 1 W ; 1 W / R. 1 W ; 1 W / Q.W; 1 W / Q.W; 1 W / T maxf 1; 2g else C1 end if end while

(14) (15) (16) (17) (18) (19) (20) (21)

D

The major differences of the CLLL algorithm are: (a) at Step (6), the rounding operation is carried out with complex numbers; (b) at Step (12), an absolute operation is performed; (c) the unitary matrix is computed with complex numbers. Using the complex-valued unimodular matrix T and CLLL-reduced matrix G D HT, LR-based linear and SIC detectors can be perform to estimate c (Subsect. 3.2.2 gives the details). Note that in order to convert c to s, the proper scaling and shifting should be performed on both real and imaginary parts. Example 3.8. Consider the same 4 4 complex-valued channel matrix used in Example 3.7 as H D Œh1 h2 h3 h4 2 3 0 0 1 0 61 C i 1 i 1Ci7 7: D6 4 i i 1 0 5 1Ci 1 1Ci i

(3.147)

After the LR (using the CLLL algorithm), the unimodular matrix T is given by 2 3 i 1 1i 3Ci 61 i 1 7 1Ci 3 7 TD6 (3.148) 4 0 5 0 0 1 0 1 i 1 and the CLLL-reduced matrix Gr becomes G D HT D Œg1 2 0 60 D6 4i 0

g2 g3 g 4 0 1 0 0

0 0 0 i

3 1 07 7: 05 0

(3.149)


79

Using the orthogonality deficiency, we compare the orthogonality between the original matrix and the CLLL-reduced one as follows: 8 ˆ M . In summary, according to (3.167) and (3.173), the average error probability is upper bounded as " ˇ # knk2 ˇˇ 2 EH Pe;LR jH En Pr khmin k 2 ˇn cı " # N 1 2N En cNM knk cı2 N 1 .2N 1/Š 1 N : (3.175) D cNM .N 1/Š N0 cı2 Thus, the upper bound on Pe;LR jH in (3.175) results from the N th moment of chisquare random variable jjnjj2 , and the receive diversity order of the LR-based linear detection is greater than or equal to N . Note that N is also the maximum receive diversity order for the N M MIMO system. Thus, a full receive diversity order of N can be achieved with the LR-based linear detection. This completes the proof. t u For LR-based SIC detection, it can be deduced from [43] that the bound on its error probability results from the same moment of jjnjj2 as that of the LR-based linear detection. Theorem 3.6. The use of the LR-based SIC detection with an N M MIMO system (i.e., N M ) can exploit a full receive diversity order, which is N . Proof. Consider the lattice-reduced matrix G from H and the QR factorization of G D QR, where Q is unitary and R is upper triangular. Letting x D QH y in (3.71), we have x D Rc C n;

(3.176)

where c 2 ZM C j ZM , n D Œn1 n2 : : : nN , and nk represents the kth element of n. There would be no error at the M th layer of the LR-based SIC detection if jnM j < 12 or 4 jnM j2 < jŒRM;M j2 . Thus, there would be no error across all the jŒRM;M j layers of the LR-based SIC detection if 4 jnk j2 < jŒRk;k j2 , for k D 1; 2; : : : ; M . According to this, the probability of no error is lower bounded as

Pr .no error/ Pr 4 jnk j2 < jŒRk;1 j2 ; 8k T

D

M Y

Pr 4 jnk j2 < jŒRk;k j :

(3.177)

kD1

Since jnk j2 is a chi-square random variable with two degrees of freedom, we have 2

jŒRk;k j 4N 2 0 : Pr 4 jnk j < jŒRk;k j D 1 e

(3.178)


85

From (3.178) and (3.177), the error probability of LR-based SIC detection is given by 2! M jŒRk;k j Y 4N 0 1e Pe;LR jH 1 kD1

'e

mink

jŒRk;k j

2

4N0

(3.179)

as N0 tends to 0. In addition, from (3.145) and (3.146), we have ı jŒRk;k j2 jŒRk;kC1 j2 C jŒRkC1;kC1 j2 ; for k D 1; 2; : : : ; M 1: Supposing that ı D 1, the following inequalities are obtained as 1 jŒRk;k j2 ; jŒRkC1;kC1 j2 ı 4 1 M 1 min jŒRk;k j ı jŒR1;1 j2 : k 4

and

2

(3.180)

(3.181)

(3.182)

Since G D QR, it is straightforward to show jŒR1;1 j2 D kg1 k2 and kg1 k2

min kHdk2 ;

(3.183)

d2D; d¤0

˚ where g1 denotes the first column vector of G and D D d D s s0 js; s0 2 S M

ZM C j ZM . Here, we assume that s is transmitted, while s0 is erroneously detected. Then, from (3.182) and (3.183), we can obtain 1 M 1 min kHdk2 : (3.184) min jŒRk;k j2 ı k 4 d2D; d¤0 Furthermore, applying the approach used in Sect. 2.2, we have "

jŒRk;k j2 E exp min k 4N0

!#

X d2D; d¤0

!N

M 1 ı 14 det I C ddH 4N0

(3.185)

under the assumption of h D 1. Using (3.179) and (3.185), the error probability is given by !N

M 1 X ı 14 H det I C dd ; (3.186) Pe 4N0 d2D; d¤0

where a full receive diversity order of N is obtained by the LR-based SIC detection. This completes the proof. t u

86

3 List and Lattice Reduction-Based Methods Table 3.2 The average value of column swapping per iteration when the CLLL is employed for different MIMO channels (N D 8 and M D 2; 3; : : : ; 8) Average value of column swapping per iteration M 2 3 4 5 6 7 8 CLLL 0.2909 0.9029 1.8022 3.0633 4.7711 7.2925 12.1228

Another approach of diversity analysis is studied by Gan, Ling, and Mow in [25], where the proximity factor [58] is used to derive the bound on error probability. Define the proximity factors of LR-based ZF detection as i;ZF D sup

2 ./ ; kgi k2 sin2 i

(3.187)

where sup stands for the supremum that is taken over the lattice- reduced basis G and

i denotes the angle between gi and the linear subspace spanned by the rest M 1 basis vectors. Letting ZF D max1i M i; ZF , from [58], the error probability of the LR-based ZF detection with a given SNR is upper bounded as Pe .SNR/

M X

Pe;LD

i D1

SNR i;ZF

MPe;LD

SNR ; ZF

(3.188)

where LD denotes the lattice decoding. Furthermore, [25, Lemma 1] shows that M i p 1n 2 ˛ ; (3.189) p sin i 2C 2

1 where ˛ D ı 12 2. If M D 2 and ı D 1, we can have ZF 2, which agrees with the result derived in [21] (i.e., the maximum loss of 3 dB). Using the similar approach, the performance of LR-based SIC detection is also analyzed in [25]. From this, we can confirm that the LR-based detection can exploit a full receive diversity order with a countable SNR loss. In [25, 59, 60], the computational complexity of LR is studied. It is shown that the average complexity of LR follows O M 3 N log M . Moreover, the complexity of LR highly depends on the number of column swapping in Step (13) of the LLL and CLLL algorithms. In Table 3.2, the average number of column swapping per iteration is shown when the CLLL is employed for different MIMO channels (N D 8 and M D 2; 3; : : : ; 8).

3.2.7 Simulation Results In Figs. 3.6–3.11, the performance of the LR-based detection and the conventional detection are compared for uncoded 2 2 and 4 4 MIMO systems, where 4-, 16-, and 64-QAM are used for signaling. The CLLL algorithm is used to perform the


87

100 MMSE MMSE−SIC LR−based MMSE LR−based MMSE−SIC ML

10−1

BER

10−2

10−3

10−4

10−5

10−6

0

5

10

15

20

Eb /N0 (dB)

Fig. 3.6 BER performance of various detectors in a 4-QAM 2 2 MIMO system

100 MMSE MMSE−SIC LR−based MMSE LR−based MMSE−SIC ML

10−1

BER

10−2

10−3

10−4

10−5

0

5

10

15

20

Eb /N0


25

88

3 List and Lattice Reduction-Based Methods 100 MMSE MMSE−SIC LR−based MMSE LR−based MMSE−SIC ML

10−1

BER

10−2

10−3

10−4

10−5

0

5

10

15

20

25

30

Eb /N0 (dB)


10−1

10−2

BER

10−3

10−4

MMSE MMSE−SIC LR−based MMSE LR−based MMSE−SIC ML

10−5

10−6

0

2

4

6

8

10

Eb /N0


12


89

100

10−1

BER

10−2

10−3

10−4

10


−5

10−6

0

2

4

6

8

10

12

14

16

Eb /N0


100

10−1

BER

10−2

10−3


10−4

10−5

4

6

8

10

12

14

16

18

Eb /N0


20

90


LR. It is shown that the performance of the MMSE detection can be significantly improved by introducing the LR method. It is also known that the LR-based MMSE– SIC detection outperforms the LR-based MMSE detection, since the interference can be mitigated by using the SIC approach, especially with large MIMO systems (e.g., 4 4 MIMO), which is illustrated in Figs. 3.9–3.11. Furthermore, simulation results show that the LR-based detection can exploit a full receive diversity order, which is the same as the one obtained by the ML detection.

3.3 Conclusion and Remarks In this chapter, we have firstly introduced the list-based detection schemes for MIMO systems. It has been shown that the performance of conventional suboptimal detectors (e.g., linear and SIC detectors) can be improved by using a list and ordering. However, the list-based detectors cannot exploit a full receive diversity order, especially when a large MIMO system is considered (i.e., more layers of interference). Then, the lattice basis reduction and its application to MIMO detection are developed. It has been shown that the LR method can improve the performance of suboptimal detectors (e.g., linear and SIC detectors) with reasonably low complexity. More importantly, from theoretical and numerical results, we have shown that a full receive diversity gain can be exploited with the LR-based detection. To further improve the performance of the list and LR-based detection, different schemes and components are designed and then adopted, which will be introduced in the following chapters.

Chapter 4

Partial MAP-Based Detection

4.1 MAP Detection In [3,4], it has been shown that the capacity of wireless channels can be significantly improved by using MIMO systems. In MIMO systems, it is often desirable to use the ML detection to achieve the best performance. With the likelihood function and an available APP of transmitted signals, the MAP principle can be used to perform the ML detection. Recall a MIMO system with N transmit antennas and N receive antennas. The received signal vector over a flat-fading MIMO channel is written as y D Œr1 ; r2 ; : : : ; rN T D Hs C n;

(4.1)

where the N N matrix H is the channel matrix, the N 1 vector s D Œs1 ; s2 ; : : : ; sN T is the transmit signal vector, and the N 1 vector n D Œn1 ; n2 ; : : : ; nN T is the noise vector. Throughput this chapter, we assume that the components of h, s, and n are all complex-valued and n CN .0; N0 I/. Furthermore, it is assumed that a common alphabet S is used for all transmitted symbols, where s 2 S N . As introduced in (2.4), the ML detection is employed to find the symbol vector that maximizes the likelihood function as sOml D arg max f .yjs/; s2S N

(4.2)

where f .yjs/ represents the likelihood function of s under the condition that y is given. With the CSCG random vector n, we can show that exp .y Hs/H .N0 I/1 .y Hs/ ; (4.3) f .yjs/ D det .N0 I/


91

92

4 Partial MAP-Based Detection

and the ML detection becomes sOml D arg max .y Hs/H .N0 I/1 .y Hs/: s2S N

(4.4)

Denote by Pr.s/ the APRP of s. If Pr.s/ is available, the MAP detection is given by maximizing the APP, which can be shown as the following: sOmap D arg max Pr.sjy/ s2S N

D arg max f .yjs/ Pr.s/: s2S N

(4.5)

Note that if the symbols are equally likely, MAP detection reduces to ML detection, as Pr.s/ becomes a constant for all s 2 S N .

4.2 Partial MAP Detection Since the complexity of the ML detection grows exponentially with the number of transmit antennas, the ML detection approach is not practical for higher dimensional detection problems of large MIMO systems. In order to reduce the computational complexity of the MIMO detection, the V-BLAST approach [47] has been proposed by employing the nulling and cancellation operations with the SIC in MIMO systems. Due to error propagation, the SIC-based detection may not be able to provide the optimal performance. However, under a certain condition, the SIC can be used to solve the ML detection problem. In [29], the partial MAP principle is considered with the SIC to provide the ML performance with reduced complexity, where a higher dimensional detection problem is reduced to multiple lower dimensional subdetection problems.

4.2.1 The Case of 2 2 MIMO With the QR factorization H D QR, the received signal vector in (4.1) is rewritten as QH y D Rs C QH n ; „ƒ‚… „ƒ‚…

(4.6)

n

x

where Q is unitary and R is upper triangular. Note that n is used to denote QH n since they have the same statistical properties. Letting N D 2, from (4.6), we have

x1 x2

D

r1;1 r1;2 0 r2;2

s1 n C 1 ; s2 n2

(4.7)

4.2 Partial MAP Detection

93

where xi , si , and ni denote the i th elements of x, s, and n, respectively, and rp;q denotes the .p; q/th element of R. Since (4.7) can be rewritten as x1 D r1;1 s1 C r1;2 s2 C n1 x2 D r2;2 s2 C n2 ;

(4.8)

one can be found that x2 only contains the information from s2 . Thus, using the likelihood function, the ML detection problem in (4.7) is given by sOml D arg max f .x1 ; x2 js1 ; s2 / fs1 ;s2 g

D arg max f .x1 js1 ; s2 / f .x2 js2 / fs1 ;s2 g

(4.9)

under the condition that n1 and n2 are statistical independent. Consider the two-step SIC detection performed with (4.8). Firstly, s2 is detected from x2 , where the APP of s2 can be found with x2 as f .x2 js2 / ; s2 / sQ2 2S f .x2 jQ

Pr .s2 jx2 / D P

(4.10)

under the assumption that the s2 is equally likely. Note that Pr .s2 jx2 / can also be formulated as 1 2 Pr .s2 jx2 / / exp r s : (4.11) jx2 2;2 2 j N0 When the problem of detecting s1 and s2 with the observation of x1 in (4.8) is considered, Pr .s2 jx2 / can be used as the APRP of s2 , which is rewritten as Pr .s2 /. Then, a partial MAP detection problem is formulated from (4.8) as sOml D arg max f .x1 js1 ; s2 / Pr .s2 / fs1 ;s2 g

D arg max f .x1 js1 ; s2 / Pr .s2 / Pr .s1 / fs1 ;s2 g

D arg max Pr .s1 ; s2 jx1 / ; fs1 ;s2 g

(4.12)

since there is no prior information of s1 in this detection and s1 is assumed to be equally likely. From this, we can show that using the SIC approach, the ML detection problem of detecting fs1 ; s2 g from fx1 ; x2 g becomes a partial MAP detection problem of detecting fs1 ; s2 g from fx1 g.

4.2.2 General Case The approach in above can be extended with any value of N . Using the system model presented in (4.6), the received signal vector is given by

94


A1 A2 n1 r s1 D C ; z 0 B s2 n2 „ƒ‚… „ ƒ‚ … „ƒ‚… „ƒ‚… x

R

s

(4.13)

n

where the N1 1 vector r and N2 1 vector z denote the first and second subvectors of x, respectively, and N D N1 C N2 . Denote by si and ni the i th subvectors of s and n, respectively, while A1 , A2 , and B are the submatrices of R. Let n1 and n2 be the zero-mean Gaussian noise vectors and n1 H H D N0 I: E (4.14) n1 n2 n2 In addition, we assume that s1 and s2 are independent, where s1 2 S N1 and s2 2 S N2 . Previously, we have introduced the SIC detection with N2 D 1; however, the SIC detection can be adopted with N2 > 1. Rewrite (4.13) as r D A1 s1 C A2 s2 C n1 z D Bs2 C n2 :

(4.15)

With the subvector z, the subvector s2 can be detected individually without the interference from s1 , where the APP, i.e., Pr.s2 jz/, is available and found with z in (4.15) as f .zjs2 / (4.16) Pr .s2 jz/ D P s2 / Qs2 2S N2 f .zjQ and

1 k z Bs2 k2 : Pr .s2 jz/ / exp N0

(4.17)

Using the APP in above as the APRP denoted by Pr.s2 / to detect the s1 and s2 with the observation of r in (4.15), the partial MAP detection is formulated as sOml D arg max f .rjs1 ; s2 / Pr .s2 /: fs1 ;s2 g

Since the likelihood function is given by 1 2 f .rjs1 ; s2 / / kr .A1 s1 C A2 s2 /k ; N0

(4.18)

(4.19)

the partial MAP detection problem is derived as sOpmap D arg max f .rjs1 ; s2 / Pr .s2 / fs1 ;s2 g

D arg min

fs1 ;s2 g

1 kr .A1 s1 C A2 s2 /k2 log Pr .s2 / : N0

(4.20)


95

Theorem 4.1. The ML detection problem in (4.13) is identical to the partial MAP detection problem in (4.20), where Pr .s2 / is found by utilizing Pr .s2 jz/ in (4.17). Theorem 4.1 shows that with an available the APRP of s2 , the ML detection problem can be generated as a partial MAP detection problem. Next, we will discuss the solution of partial MAP detection problem in (4.20), where SIC can be implemented to reduce the complexity of MIMO detection under certain condition and probability. In Theorem 4.2, we introduce this condition, namely the dimension reduction condition (DRC). Theorem 4.2. Consider the system presented in (4.15). Suppose that s1 is equally likely and Pr .s2 / is available. Let sN2 D arg maxs2 Pr .s2 /, and assume that min s1 ;s2

1 kr .A1 s1 C A2 s2 /k2 C; N0

(4.21)

where C is a constant. With rN D r A2 sN2 , if min s1

1 Pr.Ns2 / ; kNr A1 s1 k2 C C min log N0 s2 ¤Ns2 Pr.s2 /

(4.22)

then the solution of the partial MAP detection problem in (4.20) becomes (

sO 1;pmap D arg mins1 sO2;pmap D sN2

1 N0

kNr A1 s1 k2

:

(4.23)

Note that the condition in (4.22) is regarded as the DRC. Proof. With the inequality in (4.21), we can easily show that min

s1 ;s2 ¤Ns2

1 1 1 C C min log : kr .A1 s1 C A2 s2 /k2 C log N0 Pr.s2 / s2 ¤Ns2 Pr.s2 /

(4.24)

From the DRC in (4.22), we have C C min log s2 ¤Ns2

1 1 1 min : kr .A1 s1 C A2 sN2 /k2 C log s1 N0 Pr.s2 / Pr.Ns2 /

(4.25)

Then, (4.24) and (4.25) lead that mins1 ;s2 ¤Ns2

1 N0

kr .A1 s1 C A2 s2 /k2 C log Pr.s1 2 /

mins1

1 N0

kr .A1 s1 C A2 sN2 /k2 C log Pr.N1s2 / ;

and thus, sN2 becomes the optimal solution. This completes the proof.

(4.26)

t u

96


According to Theorem 4.2, if Pr.s2 / is available and DRC in (4.22) is satisfied, the ML detection problem becomes a partial MAP detection problem, where a twostep detection procedure is summarized as follows: .1/

sO2;pmap D arg max Pr.s2 /

.2/

sO 1;pmap D arg max f .rjs1 ; sO2;pmap /:

s2 s1

(4.27)

More importantly, with the structure of upper block triangle MIMO in (4.13), when Pr.s2 / is the APP of s2 for given z, the SIC is used to reduce the ML detection problem of a higher dimension to the ML detection problems of a lower dimension if the DRC holds. According to Theorems 4.1 and 4.2, the solution of (4.27) becomes 1 kz Bs2 k2 N0

1 r A1 s1 C A2 sO2;pmap 2 : D arg min s1 N0

.1/

sO2;pmap D arg min

.2/

sO1;pmap

s2

(4.28)

Example 4.1. Consider the received signal vector of a 2 2 MIMO channel as

x1 r1;1 r1;2 n s1 D C 1 : x2 0 r2;2 s2 n2 „ƒ‚… „ ƒ‚ … „ƒ‚… „ƒ‚… x

s

R

(4.29)

n

0:8 0:7 , si 2 S D f3; 1; 1; 3g, ni N .0; N0 / and N0 D 0 0:5 0:4 (i D1, 2). With the impact of the noise, the received vector is given by x D Œ1:4 0:7T . Using the exhaustive search, the ML solution can be found as

Let R D

sOml D arg min kx Rsk2 s2S 2

3 : D 1

(4.30)

Alternatively, let the partial MAP detection be carried out with C D 0. Considering that 1 2 (4.31) Pr.s2 / / exp jx2 r2;2 s2 j ; N0 we have min log

s2 ¤Ns2

Pr.Ns2 / D 1:5; Pr.s2 /

(4.32)


97

where sN2 D 1. In addition, min

s1 2S

1 jx1 .r1;1 s1 C r1;2 sN2 /j2 D 0:225 N0

(4.33)

and the minimum is achieved when sN1 D 3. According to Theorem 4.2, the DRC in (4.22) is satisfied. As a result, the partial MAP detection provides the ML solution with reduced complexity, where sOpmap D ŒNs1 sN2 T D Œ3 1T .

4.2.3 Theoretical Analysis For the sake of simplification, we can have C D 0, since the term on the left hand side in (4.21) is greater than 0. Then, the DRC in (4.22) can be simplified as min s1

1 Pr.Ns2 / : kNr A1 s1 k2 min log N0 s2 ¤Ns2 Pr.s2 /

(4.34)

Denote by K the number of elements in S, i.e., the size of the signal alphabet. If DRC is satisfied, compared to the ML detection which requires the complexity of O.K N / for exhaustive search, the partial MAP detection reduces the complexity to O.K N1 C K N2 /, as s1 2 S N1 and s2 2 S N2 . Let LT D min log s2 ¤Ns2

Pr.Ns2 / Pr.s2 /

D min .log Pr.Ns2 / log Pr.s2 // ; s2 ¤Ns2

(4.35)

and define the probability that DRC in (4.34) is satisfied by Pcond

1 2 D 1 Pr min kNr A1 s1 k LT ; s1 N0

(4.36)

namely the probability of dimension reduction (PDR). Supposing that sN 2 is correctly detected, we have 1 1 Pr (4.37) kn1 k2 LT Pr min kNr A1 s1 k2 LT s1 N0 N0 and

Pcond

1 1 Pr kn1 k2 LT N0

:

(4.38)

98

4 Partial MAP-Based Detection 1 0.9 0.8

Lower bound of Pcond

0.7 0.6 0.5 0.4

N1=1 N1=2

0.3

N1=3 0.2

N1=4

0.1

N1=5 N1=6

0

0

2

4

6

8 LT

10

12

14

16

Fig. 4.1 The lower bound of Pcond

Since N10 kn1 k2 is a chi-square random variable with N1 degrees of freedom, it can be derived as Z LT x=2 N1 1 e x 1 2 1 Pr dx kn1 k LT D N N0 .N 1 /2 1 0 N 1 1 X 1 LT k : (4.39) D 1 e LT =2 kŠ 2 kD0

As a result, the lower bound of Pcond is found by Pcond

POcond , 1 e LT =2

N 1 1 X kD0

1 kŠ

LT 2

k :

(4.40)

Since POcond is the cdf, it increases with LT and decreases with N1 , which illustrates

that the probability of reducing the computational complexity from O K N to O K N1 C K N2 increases with LT and decreases with N1 . In addition, Fig. 4.1 shows the lower bound of Pcond in (4.40) with different values of N1 .

4.3 Partial MAP-Based List Detection

99

In (4.21), considering the case of C > 0, Pcond in (4.36) is rewritten as 1 Pcond D 1 Pr min kNr A1 s1 k2 LQ T ; s1 N0

(4.41)

where LQ T D LT C C > LT . This leads to that Pcond increases with C . Since the maximized C is given by Cmax D min s1 ;s2

1 kr .A1 s1 C A2 s2 /k2 ; N0

(4.42)

Q the optimized LQ T can be found as LT D LT C Cmax . Obviously, the complexity N to find Cmax exactly is O K . Therefore, C D 0 is always assumed to avoid the increase of the complexity throughout the chapter.

4.3 Partial MAP-Based List Detection We have introduced that the partial MAP detection is able to provide an optimal solution with reduced complexity if the DRC in (4.22) is satisfied. Note that the partial MAP detection chooses the candidate that has the maximum APP among the candidate set in the first subdetection,1 after separating the detection problem into two subdetection problems, dimensionally (i.e., as shown in (4.15)). In other words, the partial MAP detection discusses a problem with a single candidate in the first subdetection. However, considering the case that the DRC does not hold with the candidate sN2 as min s1

Pr.Ns2 / 1 ; kNr A1 s1 k2 > C C min log N0 Pr.s2 / s2 ¤Ns2

(4.43)

the partial MAP detection cannot provide an optimal solution. Interestingly, if we can derive a DRC with multiple candidates in the first subdetection, the partial MAP detection could provide an optimal solution with reduced complexity. With the partial MAP principle, the SIC can be implemented to reduce the complexity of MIMO detection under a certain DRC and probability [29]. In this section, we use a list decoding approach [7–13] to exploit the DRC and PDR of partial MAP detection with any list of the candidates (multiple candidates) involved in the first subdetection, then the best candidate can be chosen among the list in the second subdetection as the final decision. In particular, we focus on a family of listbased Chase strategies [14–19] to work with the partial MAP detection, where the partial MAP principle is used to choose candidate symbol vectors in the detection.

1

Obtain sN2 with the observation of z as sN2 D arg maxs2 Pr.s2 jz/.

100


This scheme is regarded as the partial MAP-based list detection. Furthermore, a computationally efficient algorithm is developed to solve such a detection problem in MIMO systems. It can be shown that compared to the conventional SIC list detection in Subsect. 3.1.1, the proposed scheme improves the performance with reasonable complexity.

4.3.1 System Model Review the received signal of an N N MIMO system in (4.1) as y D Hs C n:

(4.44)

Using the QR factorization [61] H D QR, rewrite the received signal as x D Rs C n;

(4.45)

where x D QH y and n D QH n. After that, as shown in (4.13), we have r A1 A2 n1 s1 D C ; z 0 B s2 n2 „ƒ‚… „ ƒ‚ … „ƒ‚… „ƒ‚… x

R

s

(4.46)

n

and from (4.15), we have r D A1 s1 C A2 s2 C n1 ; z D Bs2 C n2 :

(4.47)

From (4.47), the detection of s can be decomposed into the two subdetection of s1 and s2 , which have N1 and N2 elements, respectively. Due to no interference from s1 , the subdetection of s2 can be done independently, while the subdetection of s1 is not straightforward as s2 becomes an interfering signal through A2 . In the next subsection, we study the subdetection of s1 using the list decoding [7–10] and the partial MAP principle [29] to deal with the interference effectively. Note that we have assumed that the number of receive antennas is identical to that of transmit antennas for convenience. The QR factorization in (4.45) would also be possible when the numbers are different. As shown in Subsect. 4.2.2, the partial MAP principle can be applied to the subdetection of s1 by using the APRP of s2 , which can be obtained from the result of the subdetection of s2 . Here, we focus on the subdetection of s1 under the assumption that the APRP of s2 is available. While the SIC is used to mitigate the interference of s2 , we use the list decoding approach [7–10] to effectively deal with the interference of s2 . Basically, instead of exhaustive searching for all the possible decision vectors


101

in the detection problem, the list decoding creates a list of candidate vectors and then chooses the best candidate within the list for the final decision. In this section, we drive the partial MAP solution with the SIC, by generating a list of candidate decision vectors of s2 for the final decision. In order to derive the partial MAP-based listn detection, let us define the finite o .1/ .2/ .M / set of all the possible candidate vectors for s2 as s2 ; s2 ; : : : ; s2 , where M D jSjN2 denotes the number of all the possible candidate vectors (for e.g., M D 162 if the size s2 is 2 1 and 16-QAM is used), and the APRP of the candidate vectors are denoted by

where

.1/ .2/ .M / Pr s2 D s2 ; Pr s2 D s2 ; : : : ; Pr s2 D s2 ;

(4.48)

.1/ .2/ .M / : Pr s2 D s2 Pr s2 D s2 Pr s2 D s2

(4.49)

Moreover, ˇ that the partial APP for each candidate is available and denoted suppose .n/ ˇ by Pr s2 D s2 ˇ r , where ˇ ˇ .n/ ˇ f z ˇs2 .n/ ˇ Pr s2 D s2 ˇ r D P s2 / Qs2 2S N2 f .zjQ

and

ˇ 2 1 .n/ ˇ .n/ Pr s2 D s2 ˇ r / exp z Bs2 ; N0

(4.50)

(4.51)

for n 2 f1; 2; : : : ; M g. Denoting by Q the list length of the first subdetection, we develop the partial MAP-based list detection as follows.

4.3.2 The Case of List Length Q D 1 In order to develop the partial MAP based list detection with any list length Q, let us first analyze the conventional partial MAP detection with Q D 1, where the logarithms of a posteriori probability ratios (LAPPR) [62] is considered. When Q D 1, using the APRP of s2 , the partial MAP detection of s1 and s2 with the observation of r in (4.46) or (4.47) is defined as ˚ 1 1 sO1;pmap ; sO2;pmap D arg min ; kr .A1 s1 C A2 s2 /k2 C log s1 ; s2 N0 Pr.s2 /

(4.52)

102


where Pr.s2 / stands for the APRP of s2 . Furthermore, the LAPPR [62] is defined as ˇ .1/ ˇ Pr s2 D s2 ˇ r ˇ : L.s2 j r/ D log .n/ ˇ maxs.n/¤s.1/ Pr s2 D s2 ˇ r 2

2

If L.s2 j r/ 0, then we can have ˇ ˇ .1/ ˇ .n/ ˇ Pr s2 D s2 ˇ r max Pr s2 D s2 ˇ r ; .n/

(4.53)

.1/

(4.54)

s2 ¤s2

ˇ .1/ ˇ where Pr s2 D s2 ˇ r becomes the maximum among all the partial APPs, and .1/

thus the partial MAP solution of s2 is s2 . According to [29], a sufficient condition to make sure that L.s2 j r/ 0 is as follows: .1/ 2 Pr s2 D s2 1 .1/ : min r A1 s1 C A2 s2 log .2/ s1 N0 Pr s2 D s2

(4.55)

If (4.55) is satisfied, the partial MAP detection problem in (4.52) can be simplified as 2 1 .1/ .1/ s1 D arg min (4.56) r2 A1 s1 ; s1 N0 .1/

.1/

where r2 D r A2 s2 . The condition in (4.55) is identical to the DRC as we have introduced. It is to say, if we can have the DRC satisfied, then the N dimensional detection problem can be decomposed into an N1 dimensional and N2 dimensional n o .1/ .1/ subdetection problems which can reduce the complexity, and we decide s1 ; s2 as the partial MAP solution of fs1 ; s2 g. Furthermore, the PDR of (4.55) becomes 1 .1/ D s Pr s 2 2 2 1 .1/ A D Pr @min r A1 s1 C A2 s2 log .2/ s1 N0 Pr s2 D s2 0

Pcond

(4.57)

and the lower bound of Pcond can be further derived as Pcond

POcond , 1 e LT =2

N 1 1 X kD0

1 kŠ

LT 2

k ;

(4.58)


where LT D log

.1/ Pr s2 Ds2 .2/ Pr s2 Ds2

103

(see the previous section for more details). Clearly, the

PDR is the probability that the computational complexity of an N dimensional detection problem is reduced to the computational complexity of N1 dimensional and N2 dimensional subdetection problems.

4.3.3 General Case With the list length Q D m, where m 2 f1; 2; : : : ; M 1g, a general case can be considered. Note that the case of m D M will be explained later. With Q D m, the LAPPR can be given by ˇ ˇ .1/ ˇ .m/ ˇ Pr s2 D s2 ˇ r C C Pr s2 D s2 ˇ r ˇ : L.s2 j r/ D log .n/ ˇ maxs.n/ ¤s.1;2;:::;m/ Pr s2 D s2 ˇ r 2

(4.59)

2

If L.s2 j r/ 0, then we have ˇ ˇ .1/ ˇ .m/ ˇ Pr s2 D s2 ˇ r C C Pr s2 D s2 ˇ r

.n/

max

.1;2;:::;m/

s2 ¤s2

ˇ .n/ ˇ Pr s2 D s2 ˇ r ; (4.60)

where ˇ ˇ ˇ .1;2;:::;m/ ˇ .1/ ˇ .m/ ˇ Pr s2 D s2 ˇ r D Pr s2 D s2 ˇ r C C Pr s2 D s2 ˇ r

(4.61)

becomes and the partial MAP solution of s2 can be obtained from n the maximum, o .1/ .m/ sN2 2 s2 ; : : : ; s2 . Using the max–log approximation [63] principle, we can have the modified LAPPR as 9 8 = < 1 2 1 .1;2;:::;m/ O 2 j r/ min L.s r A1 s1 C A2 s2 C log .1;2;:::;m/ ; s1 : N0 Pr s2 D s2 9 8 = < 1 2 1 .n/ : min r A1 s1 C A2 s2 C log .n/ ; .n/ .1;2;:::;m/ : N 0 s1 ; s ¤s Pr s D s 2

2

2

2

(4.62)

104


O Note nthat more details o about the derivation can be found in [64–66]. If L.s2 j r/ 0, .1/ .m/ becomes the list of candidates for the partial MAP solution of then s2 ; : : : ; s2 s2 . Here, we suppose that min

s1 ; s2

1 kr .A1 s1 C A2 s2 /k2 C; N0

(4.63)

where C 0 is a constant. Furthermore, we can show that 9 8 = < 1 1 log : min log .n/ ; .mC1/ .n/ .1;2;:::;m/ : s ¤s Pr s D s Pr s D s 2

2

2

2

2

(4.64)

2

If 8 < 1 2 .1;2;:::;m/ min r A1 s1 C A2 s2 s1 : N 0 9 =

C log

1 1 C C log ; (4.65) .1;2;:::;m/ ; .mC1/ Pr s2 D s2 Pr s2 D s2

O 2 j r/ 0 and the partial MAP solution can be found. For the sake we can have L.s of simplification, we can have C D 0 and the DRCs are derived as .1/ Pr s D s 2 2 2 1 .1/ min r A1 s1 C A2 s2 log .mC1/ s1 N0 Pr s2 D s 2

.2/ 2 Pr s2 D s2 1 .2/ min r A1 s1 C A2 s2 log .mC1/ s1 N0 Pr s D s

2

2

:: :

.m/ 2 Pr s2 D s2 1 .m/ : min r A1 s1 C A2 s2 log .mC1/ s1 N0 Pr s2 D s

(4.66)

2

Note that if any condition (among m inequalities) in (4.66) is satisfied with the .m/ O partial MAP solution of s2 , denoted by sN2 D s2 , where m O 2 f1; 2; : : : ; mg, then the partial MAP detection problem in (4.52) can be simplified as

4.3 Partial MAP-Based List Detection .m/ O

s1 .m/ O

105

D arg min s1

2 1 .m/ O r2 A1 s1 ; N0

(4.67)

.m/ O

where r2 D r A2 s2 . Thus, the N dimensional detection problem can be decomposed into N1 dimensional and N subdetection problems to n 2 dimensional o .m/ O .m/ O reduce the complexity, and we decide s1 ; s2 as the partial MAP solution. Furthermore, for more than one of these conditions or inequalities in (4.66) satisfied, the candidate of s2 whose mins1 N10 k r .A1 s1 C A2 s2 / k2 C log Pr.s1 2 / achieves the minimum value can be chosen as the partial MAP solution. Let 2 1 .m/ N DmN D min (4.68) r A1 s1 C A2 s2 ; s1 N0 and LT mN

.m/ N Pr s2 D s2 ; D log .mC1/ Pr s2 D s2

(4.69)

where m N 2 f1; 2; : : : ; mg. Therefore, the probability of each condition satisfied in (4.66) is Pr .DmN LT mN /, and the PDR becomes Pcond D Pr ..D1 LT 1 / [ [ .Dm LT m //;

(4.70)

where [ denotes the union. Since it is difficult to obtain the exact value of Pcond , we derive the upper and SI lower bounds of it. Let Pcond denote the PDR under the assumption that the DRCs FC are statistical independent. In addition, let Pcond denote the PDR for the case of fully correlated DRCs. Then, we have SI FC Pcond Pcond Pcond :

(4.71)

SI FC and Pcond can be obtained as Furthermore, the lower bound of Pcond

SI Pcond

SI POcond D1

1 Pr k n k2 LTj N 0 j D1

and

FC FC POcond D 1 Pr Pcond

respectively. Since freedom, we have

m Y

1 N0

1 k n k2 LTj N0

(4.72)

;

(4.73)

k n k2 is a chi-square random variable with 2N1 degrees of

106

4 Partial MAP-Based Detection 1 0.9 0.8 0.7

Pcond

0.6 0.5 0.4 0.3 S.I. (Q = 4) Pcond

0.2

PS.I. cond (Q = 2) PF.C. cond (Q = 1)

0.1 0

0

1

2

3

4 5 6 LT1 = LT2 = LT3 = LT4

7

8

9

10

Fig. 4.2 Bounds of Pcond for different list length with N1 D 2

SI POcond D1

m Y

G.LTj ; N1 /

(4.74)

j D1

and FC D 1 G.LTj ; N1 /; POcond

(4.75)

PN1 1 1 LTj k FC where G.LTj ; N1 / D e LTj =2 kD0 , and j 2 f1; 2; : : : ; mg. Since POcond kŠ 2 is independent from the list length m, it can be used as the lower bound for the cases with any list length. Note that if m D M , it can be shown that LTM D .M /

log

Pr s2 Ds2 .M / Pr s2 Ds2

SI FC D 0, which leads to that POcond D POcond D 1. That is, since all

the candidates have been on the list, then the condition must be satisfied, which is obvious. SI FC Since POcond and POcond are the cdf, they increase with LT 1 and LT 2 and decrease SI FC with the number of receive antennas, N1 . The curves of POcond and POcond are shown in Figs. 4.2 and 4.3 with different list length Q and values of N1 . In these curves, we simply assume that LT 1 D LT 2 D LT 3 D LT 4 . According to Figs. 4.2 and 4.3,


107

1

PS.I. cond (Q = 4)

0.9

PS.I. cond (Q = 2) PF.C. cond (Q = 1)

0.8 0.7

Pcond

0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4 5 6 LT1 = LT2 = LT3 = LT4

7

8

9

10

Fig. 4.3 Bounds of Pcond for different list length with N1 D 4

we can see that the PDR increases with the list length. It can also be shown that the probability decreases with the number of antennas.

4.3.4 Algorithm of the Partial MAP-Based List Detection Using the list-based Chase algorithms in Subsect. 3.1.1, we develop the algorithm of the partial MAP-based list detection which is summarized as follows. (1) Among the N data symbols in the vector s, we select N2 symbols for the subvector s2 with ordering. According to [16], let in2 denote the index of the n2 th data symbols of s2 (n2 D 1; 2; : : : ; N2 and in2 2 f1; 2; : : : ; N g), which corresponds to the index of hj (the column vector of H and j 2 f1; : : : ; N g) that has the n2 th minimum or n2 th maximum norm among all the columns, if 3M Q 3M 4 or Q > 4 , respectively. (2) After ordering the symbols, the QR factorization of the channel matrix H is performed as H… D QR, where the N N matrix … represents the permutation matrix according to the symbol ordering. Here, the n2 th column of … is the in2 th column of the identity matrix. The final N1 columns of … are

108


determined according to the sorted-QR decomposition [67], which orders the weaker symbols to be detected later. (3) After that, with the received signal vectors given in (4.46) or (4.47), we find the APP of s2 as follows: (a) for j D 1 to M 2 .j / .j / .j / (b) dj D N10 z Bs2 ; s2 2 ˇN2 . Note that s2 is chosen with dj being the j th minimum among the cases. all 2 ˇ .j / ˇ .j / 1 (c) Pr s2 ˇ z D c exp N0 z Bs2 , where c is the normalization constant. (d) end

ˇ /ˇ (4) The APRP of s2 is updated by using the resulting Pr s.j 2 ˇ z in Step (3). With .j / the APRP of s2 , which is denoted by Pr s2 D s2 , the DRC can be verified as follows: (a) Letting m D 1, the DRC in (4.66) is verified. (b) If the DRC is satisfied, the partial MAP solution of s2 , which is denoted .m/ O by sN 2 D s2 .m O 2 f1; 2; : : : ; mg/, can be decided by choosing s2 that 2 .m/ O among all the minimizes mins1 N10 r A1 s1 C A2 s2 Clog 1 .m/ O Pr s2 Ds2

conditions in (4.66). (c) If the DRC is not satisfied, m D m C 1 and go back to Step (b). Note that the iterative method (called the iterative partial MAP-based list detection) is terminated until either the DRC satisfied or all of the candidates have been on the list. o n .m/ O .m/ O (5) From (4.67), the partial MAP solution of fs1 ; s2 g is generated as s1 ; s2 . The detected signals are reordered according to the original ordering. To consider the computational complexity, let K denote the number of the elements in S, i.e., the size of the signal alphabet. The complexity of the ML

detection that uses an exhaustive search is O K N . The SIC which uses the nulling and cancellation has the detection complexity of O.NK/ excluding the complexity associated with computing nulling filters’ coefficients. The proposed partial MAP based list detection has the complexity of O QK N1 C K N2 , as s1 2 S N1 and s2 2 S N2 . Since the SIC-list detector (in Subsect. 3.1.1) chooses one symbol for the first subdetector among the N received symbols, i.e., N2 D 1, it has the detection

complexity of O QK N 1 C K , as s1 2 S N 1 and s2 2 S. Then, the proposed partial MAP-based list detector has comparable complexity to the SIC-list detector when N2 D 1. Let us consider different detection methods in Table 4.1. As shown in Tables 4.2 and 4.3, empirical complexity of various detectors (in Table 4.1) for each symbol vector detection is listed, in terms of the average number of the floating point operation (flops). These systems are simulated with MATLAB-V5.3 on a PC. The


109

Table 4.1 Different MIMO detection methods MIMO systems with different detection methods I MMSE detector II MMSE–SIC detector III SIC-list detector using the ML subdetector (exhaustive search) with Q D 1 IV Partial MAP-based list detector using the ML subdetector (exhaustive search) V Partial MAP-based list detector using the ML subdetector (sphere decoding) VI ML detector (exhaustive search) VII ML detector (sphere decoding) Table 4.2 The average complexity of various detection methods in Table 4.1 for a 16-QAM 2 2 system Average flops (103 ) for 16-QAM 2 2 system Eb =N0 (dB) 0.0 4.0 8.0 12.0 16.0 20.0 I 0.240 0.240 0.240 0.240 0.240 0.240 II 0.704 0.704 0.704 0.704 0.704 0.704 III 0.752 0.752 0.752 0.752 0.752 0.752 IV 4.8719 3.6202 2.1472 1.3636 1.0822 1.0501 V 1.221 0.908 0.588 0.406 0.361 0.355 VI 13.827 13.827 13.827 13.827 13.827 13.827 VII 0.614 0.598 0.568 0.556 0.551 0.549 Table 4.3 The average complexity of various detection methods in Table 4.1 for a 16-QAM 4 4 system Average flops (103 ) for 16-QAM 4 4 system Eb =N0 (dB) I II III IV (N1 D 3; N2 D 1) IV (N1 D N2 D 2) V (N1 D 3; N2 D 1) V (N1 D N2 D 2) VI VII

0.0 0.644 2.0873 565.9 6291.8 1061.6 28.7475 41.775 11140 3.8933

4.0 0.644 2.0873 565.9 4451.7 502.64 19.9401 19.779 11140 3.2906

8.0 0.644 2.0873 565.9 2429.1 187.47 10.6409 7.3768 11140 2.9496

12.0 0.644 2.0873 565.9 1154.7 75.687 5.1543 2.9783 11140 2.7760

16.0 0.644 2.0873 565.9 632.8 48.767 2.3321 1.9190 11140 2.7058

20.0 0.644 2.0873 565.9 566.3 45.545 2.2592 1.7922 11140 2.6756

MATLAB command “flops” is used to count the number of flops. For detectors IV and V, it is confirmed that the complexity for the case of N1 D N2 D 2 is lower than that of N1 D 3 and N2 D 1 when N D 4. Thus, the complexity can be lower if N1 D N2 . Note that to compare with the sphere decoding, we also perform the sphere decoder as the subdetector of the proposed approach in Table 4.2. As shown in above, the complexity of the list detector depends on the list length, Q. If Q approaches M D K N2 , the complexity approaches that of the ML detector. Thus, it is desirable to have a small Q. As will be shown in Subsect. 4.3.5, the list length of the proposed partial MAP-based list detector decreases as the SNR increases.

110

4 Partial MAP-Based Detection 100

BER

10−1

10−2

10−3

10−4

Detector I Detector II Detector III Detector V Detector VII 0

2

4

6

8

10

12

14

16

18

20

Eb /N0

Fig. 4.4 BER performance of various detection methods in Table 4.1 for a 16-QAM 2 2 system Table 4.4 The average list length of the partial MAP-based list detection with different SNR for 16-QAM 2 2 and 4 4 MIMO systems, where N1 D N2 Average value of the list length Eb =N0 (dB) 0.0 4.0 8.0 12.0 16.0 20.0 2 2 MIMO 5.64598 4.00584 2.29326 1.29278 1.04493 1.01218 4 4 MIMO 23.3585 11.0593 4.1247 1.6653 1.073 1.0021

4.3.5 Simulation Results In this subsection, we consider 16-QAM 2 2 and 4 4 MIMO systems for simulations. The elements of MIMO channels are generated as independent complex Gaussian random variables with mean zero and unit variance. The SNR is defined by the energy per bit to the noise power spectral density ratio, Eb =N0 . According to the simulation results, we perform the detection methods in Table 4.1 to compare with the proposed approach. In Fig. 4.4, we show the BER simulation results for a 2 2 MIMO system, where N1 D N2 D 1. It is shown that the proposed partial MAP-based list detector outperforms the conventional SIC-list detector. Note that the proposed partial MAP-based list detector uses the approximation in (4.62). Due to this approximation, the performance is worse than that of the ML detection. Table 4.4 shows the


111

100

10−1

BER

10−2

10−3 Detector I Detector II Detector III Detector V (N1 = 3, N2 = 1) Detector V (N1 = N2 = 2) Detector VII

10−4

10−5

0

2

4

6

8

10

12

14

16

18

20

Eb /N0

Fig. 4.5 BER performance of various detection methods in Table 4.1 for a 16-QAM 4 4 system

average list length. We can observe that the list length decreases as Eb =N0 increases. Since the DRC can be satisfied with a shorter list length as Eb =N0 increases, the list length can be shorter. Moreover, in Fig. 4.5, the BER simulation results for a 4 4 MIMO system are also shown. We can see that the proposed partial MAP-based list detector has insignificant performance degradation from the ML performance and the SNR loss is less than 0.5 dB at a broad range of BER (up to BER D 103 ). The performance of the proposed partial MAP-based list detector with N1 D N2 D 2 is better than that with N1 D 3 and N2 D 1. Since the first subdetector is performed with 162 D 256 candidates and twofold diversity gain when N2 D 2, the reliability of the transmission is better than that with 16 candidates and onefold diversity gain when N2 D 1. Therefore, there would be less error propagation and the performance can be improved with a large N2 . The average list length for the 4 4 system is shown in Table 4.4. As Eb =N0 increases, the average list length becomes shorter. This indicates that the proposed partial MAP-based list detector can be computationally efficient when the target BER is sufficiently low, say BER D 104 for the case of N1 D N2 D 2, where the corresponding Eb =N0 is 14 dB.

112


4.4 Conclusion and Remarks Since the complexity of the ML detection becomes prohibitively high for large MIMO systems, it is often impractical. We showed that the partial MAP principle can be an effective means to reduce the computational complexity in conjunction with the list-based approach for the MIMO detection. It was also shown that the partial MAP-based list detection can perform better than the conventional list-based detection. Furthermore, in terms of the performance and complexity, it was shown that the proposed approach with N1 D N2 D 2 outperforms that with N1 D 3 and N2 D 1. Utilizing the benefits of the LR adopted in MIMO systems, the performance of the list-based detection can be significantly improved. In the next chapter, we will consider the use of LR together with list-based approach for low complexity MIMO detection. This scheme is referred to as the LR-based list detection. In addition, a column reordering strategy will be developed to further reduce the complexity for slow fading MIMO channels.

Chapter 5

Lattice Reduction-Based List Detection

Since most low-complexity suboptimal MIMO detectors could not exploit a full receive diversity gain [68], the LR-based MIMO detectors are developed to provide a full receive diversity gain with low computational complexity [56, 69, 70]. The LR-based MIMO detector is first discussed in [21], then more LR-based MIMO detectors are proposed in [24]. It has been shown that the LRbased MIMO detector using the MMSE-SIC approach, namely the LR-based MMSE-SIC detector, approaches the ML performance. An overview of LR-based detection can be found in [26] and a soft-decision with the detection is discussed in [71]. Since the LR algorithm requires polynomial complexity [20], the complexity increases relatively rapidly with the number of basis vectors. Therefore, the LRbased detection would have high computational complexity for a large MIMO system. It is noteworthy that the complexity can be reduced when a large MIMO detection problem is decomposed into multiple small MIMO subdetection problems using the SIC as in [29], where an LR-based detector is used to deal with subdetection problems. Due to the SIC, however, this approach would suffer from the error propagation. Since the overall performance is limited by the error propagation, there should be an approach to mitigate it. In this chapter, using the list decoding approach, the error propagation due to the SIC is mitigated in the LR-based subdetection. The resulting scheme is regarded as the LR-based list detection, which has low complexity as the number of basis vectors in the subdetection problem is small. Due to the characteristics of the list detection, there is a trade-off between the complexity and performance. In general, the longer the list length, the less error propagation (or the better performance) the proposed LR-based list detector experiences. However, the complexity increases with the list length.


113

114

5 Lattice Reduction-Based List Detection

5.1 Lattice Reduction-Based List Detection Considering a MIMO system equipped with N transmit and N receive antennas (although the proposed approach is also valid when there are more receive antennas than transmit antennas, we assume they are the same for convenience), the N 1 received signal vector is given by y D Hs C n;

(5.1)

where H, s, and n are the N N channel matrix, the N 1 transmitted signal vector, and the N 1 noise vector which is a zero-mean CSCG random vector with EŒnnH D N0 I, respectively. Denote by S the signal alphabet for symbols, i.e., sk 2 S, where sk represents the kth element of s and its size is denoted by jSj. With the QR factorization H D QR, where N N matrices Q and R are unitary and upper triangular, respectively, the received signal vector is rewritten as x D QH y D Rs C QH n:

(5.2)

Since the statistical properties of QH n are identical to that of n, QH n will be denoted by n in this chapter.

5.1.1 Algorithm Description The LR-based detectors [21–28] which have been introduced in Sect. 3.2 show excellent performance and their performance approaches that of the ML detector using an exhaustive search. In addition, it has been shown that the LR-based detectors can achieve a full receive diversity gain as the ML detector does. However, the complexity of the LR-based detectors grows significantly with the number of basis vectors of channel matrices. To avoid this problem, an LR-based list detection method is developed in this section. The main idea of this method is to break a higher dimensional MIMO detection problem into multiple lower dimensional MIMO subdetection problems. As the number of basis vectors in the lower dimensional MIMO subdetection is small, the complexity becomes lower. In order to perform the LR-based list detection, (5.2) can be rewritten as s1 A C n x1 D C 1 ; (5.3) 0 B x2 s2 n2 where xi , si , and ni denote the Ni 1 i th subvectors of x, s, and n, respectively, for i D 1; 2. It is noteworthy that N D N1 C N2 . From (5.3), we can have .i/

x2 D Bs2 C n2

.ii/

x1 D As1 C Cs2 C n1 ;

(5.4)

5.1 Lattice Reduction-Based List Detection

115

where two lower dimensional MIMO subdetection problems are carried out to detect s2 and s1 , sequentially. In the proposed LR-based list detection, the subdetection of s2 is carried out first using the LR-based detector. Then, a list of candidate vectors of s2 is generated and the subdetection of s1 is performed with the LR-based detector. The candidate vector in the list is used for the SIC to mitigate the interference from s2 . The proposed LRbased list detection is summarized as follows. .1/ According to (i) in (5.4), there is no interference from s1 in detecting s2 . Then, with the received signal x2 , the LR-based detection is performed as cQ 2 D LRD.x2 /;

(5.5)

where LRD denotes the operation of the LR-based detection and cQ 2 becomes the estimated vector of s2 in the corresponding LR domain. Note that more details of the LR-based detection can be found in the Subsect. 5.1.2. .2/ Generate a list of candidate vectors in the lattice reduced domain as C2 D ListLR .Qc2 /;

(5.6)

where ListLR denotes the operation that chooses the Q closest vectors to cQ 2 in the LR domain, and 1 Q jSjN2 . Note that the details of the list generation in the LR domain will be shown in the Subsect. 5.1.3. ˚ .1/ .2/ .Q/ the list of candidates of s2 , which is .3/ Denote by S2 D sO2 ; sO2 ; : : : ; sO2 converted from C2 (the list of candidates in the lattice reduced domain). .4/ With (ii) in (5.4), the LR-based detection of s1 is carried out after the SIC of the candidates in S2 as .q/ .q/ cQ 1 D LRD x1 COs2 ; (5.7) .q/

where sO2 denotes the qth decision vector of s2 from S2 and q D 1; 2; : : : ; Q. .q/ .q/ .5/ Denoting by sO1 the signal vector corresponding to cQ 1 in the LR domain, we have 3 2 .q/ O s 1 5: sO.q/ D 4 (5.8) .q/ sO2 Then, the final hard decision is obtained by sO D arg

min

qD1;2;:::;Q

x ROs.q/ 2 ;

(5.9)

where the best observation is represented in the sense of the SSE. In the following subsections, we will explain details of the proposed detection.

116


5.1.2 Lattice Reduction-Based Detection In this subsection, we describe the LR-based detection that is used in Steps (1) and (4). Consider the MIMO detection in Step (1) with the received signal as x2 D Bs2 C n2 :

(5.10)

With the scaling and shifting coefficients in (3.50), the received signal is transformed to d D ˛x2 C ˇB1 D B .˛s2 C ˇ1/ C ˛n2 D Bb C ˛n2 ;

(5.11)

where 1 D Œ1 1 : : : 1T and b D ˛s2 C ˇ1 2 CN2 . Using the LR algorithms that have been introduced in Sect. 3.2, we have B D BT;

(5.12)

where the unimodular matrix T (i.e., j det.T/j D 1) is found to make the column vectors of B shorter, and T D U1 . Note that as the basis vectors are complex, we can use complex LR algorithms or convert complex matrix into real matrix (i.e., CLLL or LLL in Subsect. 3.2.5). Then, (5.11) follows that d D BTT1 b C ˛n2 D Bc C ˛n2 ;

(5.13)

where c D T1 b. The MMSE filter to estimate c is given by h 2 i Wmmse D min E WH .d d/ .c c/ W

1 D BCov.c/BH C j˛j2 N0 I BCov.c/ 1 BTH ˛ 2 Es ; D BBH ˛ 2 Es C j˛j2 N0 I

(5.14)

where d D EŒd D ˇB1 c D EŒc D T1 ˇ1 Cov.c/ D j˛j2 T1 TH Es :

(5.15)

5.1 Lattice Reduction-Based List Detection Table 5.1 Signal and parameters for the LR-based detection in (5.5) and (5.7)

Step (1) Step (4)

117

x2 .q/ x1 COs2

B A

s2 s1

n2 n1

cQ2 .q/ cQ1

N2 N1

The estimation of s2 in the corresponding LR domain is given by cQ 2 D c C WH mmse .d d/ D c C ˛WH mmse x2 :

(5.16)

This has completed the LR-based detection in Step 1). Using the same detection method (i.e., (5.10)–(5.16)), the LR-based MMSE detection in Step (4) is carried out by replacing the signals and parameters shown in Table 5.1. Note that other approaches including the LR-based MMSE-SIC detector in Subsect. 3.2.2 can also be used.

5.1.3 List Generation in the LR Domain In order to improve the performance or mitigate the error propagation, a list of candidate vectors of s2 is built to detect s1 . With the conventional list-based detection, the candidate vectors in the list is ordered by using the ML metric as .1/ .2/ .jSjN2 / f x2 jOs2 f x2 jOs2 f x2 jOs2 (5.17) or

2 2 2 .1/ .2/ .jSjN2 / ; x2 BOs2 x2 BOs2 x2 BOs2

(5.18)

.q/

where sO2 denotes the symbol vector that provides the qth largest likelihood. Therefore, an optimal length Q list of s2 in Step (3) becomes o n .1/ .2/ .Q/ : (5.19) S2 D sO2 ; sO 2 ; : : : ; sO2 Note that the ordering operation in (5.17) or (5.18) requires an exhaustive search, which results in high computational complexity due to computing of kx2 Bs2 k2 for all s2 2 S N2 . In order to avoid this high computational complexity, a suboptimal list in the LR domain is employed in (5.6). Again, consider the MIMO detection problem in Step (1) as x2 D Bs2 C n2 ;

(5.20)

and let d D ˛x2 C ˇB1; b D ˛s2 C ˇ1;

(5.21)

118


where ˛ and ˇ denote the proper scaling and shifting coefficients. Using (5.12), we can see that the ML metric to construct the list is given by kd Bbk D kd Bck ;

(5.22)

where the right hand side in (5.22) is defined in the LR domain. Denote by sQ2 the signal vector in S N2 that corresponds to cQ 2 in (5.16) and assume that sQ 2 is sufficiently .1/ close to sO2 in (5.18). Then, we can have the following approximation: d ' BQc2 ;

(5.23)

from where the ML metric to construct the list in the LR domain becomes kd Bck ' kBQc2 Bck D kQc2 ckBH B ;

(5.24)

p and kTkG D TH GT denotes the Mahalanobis distance. Hence, the list in the LR domain (i.e., C2 in (5.6)) is generated as n o C2 D c2 j kQc2 ckBH B < rB .Q/ ;

(5.25)

where rB .Q/ denotes the radius of an ellipsoid centered at cQ 2 that contains Q elements in the LR domain. Moreover, with the orthogonal basis vectors in the LR domain, the list of c2 is approximated as C2 ' fc2 j kQc2 ck < r.Q/g ;

(5.26)

where r.Q/ > 0 denotes the radius of a sphere centered at cQ 2 that contains Q elements. In this case, generating the list according to (5.26) requires lower complexity since there are no additional matrix-vector multiplications. Note that a necessary condition of (5.26) is that the basis vectors in the LR domain should be orthogonal or nearly orthogonal. Since the LR is able to provide the nearly orthogonal basis vectors in the LR-based detection (i.e., B in (5.12) nearly orthogonal), the resulting list in (5.26) would be a good approximation.

5.1.4 Impact of List Length The list length, Q, plays a key role in the trade-off between the complexity and performance of the proposed LR-based list detection. Denote by Pe .S2 / and Pe .C2 /, the error probability that S2 and C2 do not have the correct vector of s2 and c2 , respectively. If Q increases, then Pe .S2 / or Pe .C2 / decreases. Thus, it is desirable to have a long list length or a large Q for better performance. However, it requires


119

an expense of higher computational complexity. In this subsection, we focus on the impact of the list length on the performance, while the complexity issue will be analyzed in Subsect. 5.1.5. Denote by s02 and c02 the transmitted signal vector and the corresponding vector in the LR domain, respectively. Then, the error probability is given by Pe .C2 / D Pr c02 … C2 D Pr cQ 2 c02 BH B > rB .Q/ : (5.27) For the approximation of rB .Q/, consider a 2N2 -sphere in which there are Q lattice points in CN2 , where the radius of this sphere is denoted by rNB .Q/. Assume that the volume of this sphere is equal to the sum of Q volumes of the fundamental regions associated with the lattice points in C2 or Q V .B/, where V .B/ represents the volume of the Voronoi region for the generator matrix B. Note that this assumption would be valid as Q is sufficiently large. Since the volume of an n-sphere with a radius r is given by Vn .r/ D

n=2 r n ; n2 C 1

(5.28)

where Vn .r/ D QV .B/ and n D 2N2 , the squared radius rNB .Q/ can be found as follows: 1

rNB2 .Q/

.QV .B/ N2 Š/ N2 : D

(5.29)

As a result, (5.27) is approximated as Pe .C2 / ' Pe .Q/ 2 D Pr cQ 2 c02 BH B > rNB2 .Q/ :

(5.30)

Letting c D c02 in (5.13), with (5.22), we have d D Bc02 C ˛n2 ' BQc2 ;

(5.31)

where we assume that cQ 2 is sufficiently close to the ML solution that has the minimum distance between d and BQc2 (note that this assumption is considered in building C2 ). It is derived as cQ 2 c0 2 H ' j˛j2 kn2 k2 2 B B D

j˛j2 N0 2 2N2 ; 2

(5.32)

120


where 2n denotes a chi-square random variable with n degrees of freedom. Based on (5.32), the error probability in (5.30) is approximated as Pe .C2 / ' Pe .Q/ ' Pr

22N2

>

2rNB2 .Q/ j˛j2 N0

! ;

(5.33)

where two random variables are considered in this error probability: 22N2 due to the back-ground noise and rNB2 .Q/ due to fading. As we consider MIMO fading channels, B becomes a random matrix. Since V .B/ depends on the random matrix B, with B BT and det.T/ D ˙1, we have q V .B/ D

det BH B

p det .TH BH BT/ p D det .BH B/ D

D V .B/:

(5.34)

Theorem 5.1. Suppose that the elements of N N matrix H in (5.1) are independent zero-mean circular-complex Gaussian random variables with variance unit. For the upper triangular matrix B, there exists two matrices that satisfy B2 D Q2 B;

(5.35)

where B2 is an N2 N2 random matrix whose elements are independent zeromean circular complex Gaussian random variables with variance unit and Q2 is an N2 N2 matrix whose column vectors are orthonormal. According to Theorem 5.1, it is provided that det BH B D det BH 2 B2 ;

(5.36)

where BH 2 B2 is a Wishart matrix. From (5.34) and (5.36), we define ZDV

1 N2

DV

1 N2

.B/

.B/ H 2N1 D det B2 B2 2 :

(5.37)


121

Then, substituting (5.29) and (5.37) into (5.33), the conditional error probability is given by Pe;cond .Z/ D Pr 22N2 > ZjZ D 1

.N2 ; Z/ ; .N2 /

where .x/ denotes the Gamma function, .n; x/ D lower incomplete Gamma function, and is defined by

(5.38) Rx 0

zn1 e z dz denotes the

1

2 .QN2 Š/ N2 : D j˛j2 N0

(5.39)

Using A-QAM as the modulation method, we have 1 4A2 A1 D ; 6Es

j˛j2 D

(5.40)

and then (5.39) is rewritten as 1

12Es .QN2 Š/ N2 : D .A 1/N0

(5.41)

Since the conditional error probability in (5.38) is the tail of the chi-square distribution with the threshold Z, the error probability is given by Pe .Q/ D E ŒPe;cond .Z/ :

(5.42)

In addition, the mean value of Z is found as 2N1 2 EŒZ D E det BH B 2

D

NY 2 1 pD0

N2 p C .N2 /

1 N2

:

(5.43)

Consequently, the impact of Q through has been analyzed in (5.38). Since 1 is proportional to Q N2 , the improvement of the error probability by increasing Q would be slow, especially with a large N2 .

122

5 Lattice Reduction-Based List Detection Table 5.2 Complexity analysis of different detectors Detector MMSE ML LR-based List 2 2 2 N N .5N C 12/=8 C QN 2 CMs 2.N C 1/N N jS j PN 1 C2LR N C nD1 2 .N n C 1/2 2 4-QAM 160 4,096 n/a n/a 16-QAM 160 1:049 106 64-QAM 160 2:684 108 n/a QD8 n/a n/a 262 Q D 12 n/a n/a 326 Q D 16 n/a n/a 390

5.1.5 Complexity Analysis Letting N1 D N2 D N2 , we compare the computational complexity of the LR-based list detector to that of the MMSE and ML detectors, where the exhaustive search is used for the ML detection. Note that we ignore all the additional operations while taking into account the complex multiplications (CMs) only. Denote by LR N the complexity of the LR operation with basis- N2 . Let N D 4 as 2 usual. The complex-valued Gaussian LR algorithm in Subsect. 3.2.4 can be carried out with the two 2 2 submatrices, A and B, for low complexity. With the Gaussian LR, on each submatrix, the number of column swapping is generally less than 3 in most cases, and thus LR2 6 (CMs). For our proposed detector, the LR algorithm is carried out twice only for the two sub-matrices, while one QR factorization of H is required in the P detection process. The Householder transformation for this N 1 factorization requires nD1 2 .N n C 1/2 CMs. Let LR2 D 6. In Table 5.2, we show the LR-based list detection is computationally efficient as the complexity is slightly higher than that of the MMSE detector, when Q D 8, 12, and 16, respectively. In addition, we will show in Subsect. 5.1.7 that the LR-based list detection can provide near ML performance with the Q valued in above.

5.1.6 Components of the LR-Based List Detection This subsection presents the components implemented in the LR-based list detection. For convenience, in Table 5.3, we outline the steps required for the proposed LR-based list detection with respect to implementation. In addition, we provide the log-likelihood ratio (LLR) for each bit detected, namely the soft-bit generation in Subsect. 5.1.6.6.


123

Table 5.3 Components of the LR-based list detection Subsect. Components Implementation 5.1.6.1 QR decomposition H D QR 5.1.6.2 Gaussian LR B D BT A D AT0 1 5.1.6.3 Matrix inversion BBH ˛ 2 Es C j˛j2 N0 I 1 AAH ˛ 2 Es C j˛j2 N0 I 5.1.6.4

5.1.6.5

MMSE filtering

in Step .1/ in Step .4/

cQ2 D c2 C ˛WH mmse;2 x2

in Step .1/

.q/ cQ1

in Step .4/

D c1 C

.q/ ˛WH N1 mmse;1 x

2 ' fc2 j kQc2 ck < r.Q/g

List in LR domain

in (5.3) in Step .1/ in Step .4/

in Step .2/

5.1.6.1 QR Decomposition The QR decomposition of H is carried out in (5.3) as H D QR; where

A C RD : 0 B

(5.44)

(5.45)

The QR decomposition [72] is preferred to Cholesky decomposition due to the numerical stability. Although the QR operation is required once only for each channel update in our detection algorithm, it still provides significant computational complexity as the operation is carried out to the channel matrix of full size (N N ). Therefore, we review different algorithms of QR decomposition, namely the Gram– Schmidt (GS) and the Householder transformation.

Gram–Schmidt Using the GS procedure, the QR decomposition of H is carried out as H D QR, where Q is unitary and R is upper triangular. However, since the GS algorithm requires costly square-root and division operations, which leads to high computational complexity, a modified version of the GS is developed [73]. The details of the modified GS are discussed in [74, 75]. Denoting by ŒRp;q and qi the .p; q/-th entry of R and the i -th column vector of Q, respectively, the GS algorithm is summarized in Table 5.4. Since the GS is accurate to the floating-point precision, the problem of quantization and round-off errors is not ignorable for fixed-point arithmetic, which leads to the loss in accuracy (e.g., loss in the orthogonality of Q) [74]. In [76], it has been shown that the orthogonalization error eo in fixed-point version of the GS algorithm

124


Table 5.4 Gram–Schmidt algorithm

(1) (2) (3) (4)

Initialization: Q D H,R D 0 for n D 1 to N p ŒRn;n D qH n qn q qn D ŒRn

(5) (6) (7) (8) (9)

for j D n C 1 to N ŒRn;j D qH n qj qj D qj ŒRn;j qn end for end for

n;n

is bounded by the product of condition number .H/ of matrix H and machine precision , which is e o D I Q H Q .N / .H/ ;

(5.46)

where .N / denotes a low degree polynomial in N depending only on details of computer arithmetic. This implies that for a well-conditioned matrix, fixed-point architecture for the GS is still accurate to the integer multiples of the machine precision . However, for ill-conditioned matrices, the computed Q can be very far from orthogonal. Therefore, based on unitary transformations, numerically more favorable schemes are considered.

Householder Transformation Instead of the conventional methods, the use of unitary transformations is to alleviate the numerical problem such as requirement of high number precision (i.e., large silicon area in fixed-point very large scale integration (VLSI) implementation is required). The reason for this more favorable behavior is that unitary transformations do not alter the length of a vector and thus cannot lead to an excessive increase in dynamic range or to an enhancement of quantization noise. Based on unitary transformations, the Householder reflection algorithm recursively applies a sequence of unitary transformations QH i to matrix H as .n/ R.nC1/ D QH nR ;

(5.47)

where R.1/ D H. Each transformation will eliminate more subdiagonal entries until finally we have R D RN 1 H H D QH N 1 QN 2 : : : Q1 H;

(5.48)

5.1 Lattice Reduction-Based List Detection Table 5.5 Householder reflection algorithm

125

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Initialization: Q.0/ D I,R.1/ D H for n D 1 to N 1 qN n D rn C krn k 1 H N n D I 2 qn qn2 Q kqn k I 0 Pn D n1 N 0 Qn .n/ ŒRH nC1 D Pn R .n/ .n1/ Q D Pn Q end for QH D Q.N 1/

Table 5.6 Complexity comparison of Gram–Schmidt algorithm and Householder reflection algorithm Algorithm Gram–Schmidt Householder reflection Division N N 1 Square root N N 1 PN PN 1 2 n1 .N n C 1/2 CMs 2N 2 C 2 nD1 N.N n/

where the unitary matrix is given by H H QH D QH N 1 QN 2 : : : Q1 :

(5.49)

Denoting by ri the i -th column vector of R, the Householder reflection algorithm is summarized in Table 5.5. The complexity of the two methods of QR decomposition is compared in Table 5.6. The Householder reflection algorithm provides a slightly lower number of CMs, divisions, and square-root operations compared to the Gram–Schmidt algorithm. For instance, when N D 4, the CMs of Gram–Schmidt and Householder reflection algorithms are 80 and 78, respectively. Furthermore, for fixed-point implementation, the Householder reflection algorithm is considered to be more stable. For hardware implementations, .K 2 C K.K C 1/=2/ “words of memory” (i.e., the amount of memory required to store one complex number) are required to store matrices Q and R at the output of the QR decomposition operation.

5.1.6.2 Gaussian Lattice Reduction (Revised) The Gaussian LR is employed in Steps (1) and (4) of the LR-based list detection as B D BT and A D AT0 ; respectively, where T and T0 are unimodular matrices.

(5.50)

126 Table 5.7 Gaussian LR algorithm


(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

Input: .b1 ; b2 ; V / 0 1 1 0 Let J D and T D 1 0 0 1 i D0 do if kb1 k > kb2 k swap b1 and b2 T D TJ end if if jhb2 ; b1 ij > 12 2 ;b1 i tO D b hb e kb1 k2 O b2 D b t b1 2 1 t TDT 0 1 end if i Di C1 while .kb1 k < kb2 k/ && .i V / return .b1 ; b2 ; T/

Let N1 D N2 D 2. In the proposed LR-based list detection, the LR is applied to the submatrices A and B of size 2 2 only. This basis-2 LR can be carried out using the simple Gaussian LR method in Subsect. 3.2.4 for the computational efficiency. As shown in Subsect. 5.1.5, the number of column swapping in the Gaussian LR algorithm required for the majority of channel matrices is less than 3. Thus, by limiting the maximum number of column swapping to a small number (e.g., 2 is reasonable), the overall performance remains almost the same. For the implementation purpose, the maximum number of column swapping to V can be fixed, and then the revised Gaussian LR algorithm is developed to secure the low complexity. Letting B D Œb1 b2 , as an example, the revised Gaussian LR algorithm is presented in Table 5.7. Note that the revised Gaussian LR algorithm runs until the maximum iteration i D V , the number of CMs required for the Gaussian LR is 4V . In addition, six words of memory are required to store data of the unimodular matrix at the output.

5.1.6.3 Matrix Inversion The matrix inversion is carried out in (5.14) for the LR-based MMSE subdetection in Steps (1) and (4). Note that the dominant complexity component in obtaining the MMSE filtering weight matrices is the matrix inversion operation, for example, is given by .BBH ˛ 2 Es C j˛j2 N0 I/1 . For the case of N1 D N2 D 2, the size of matrices to be inverted is only 2 2 which leads to reasonably low computational

5.1 Lattice Reduction-Based List Detection Table 5.8 L–R decomposition

127

(5)

Initialization: L D D,R D I for i D 1 to M for j D 1 to M Pj 1 ŒRj;i D ŒLj;i kD1 ŒLj;k ŒRk;j ŒR ŒLj;i D ŒRj;i

(6) (7)

end for end for

(1) (2) (3) (4)

i;i

complexity. For instance, a 2 2 matrix D D

d1;1 d1;2 d2;1 d2;2

can be easily inverted

using adjoint method as D

1

1 D d1;2 d2;1 d1;1 d1;1

d2;2 d1;2

d2;1 ; d1;1

(5.51)

which requires 1 division and 6 CMs. For an M M sized matrix D, where M > 2, the complexity of inversion operation may vary depending on different implementation methods. We list some typical methods as follows.

Adjoint Method Using the adjoint method, the inversion of matrix D is given by D1 D

adj .D/ ; det .D/

(5.52)

where the approximated number of CMs is up to scale in 2B [77] as CMs a2B C B 2 C B:

(5.53)

L-R Decomposition Using the L–R decomposition, matrix D is decomposed into a lower-triangular matrix L and an upper-triangular matrix R as D1 D R1 L1 :

(5.54)

Denoting by ŒAp;q the .p; q/-th element of the matrix A, the L–R decomposition algorithm is summarized in Table 5.8, where the number of CMs becomes 4.M 3 M /=3.

128


QR Decomposition Using the QR decomposition, the matrix D is inverted as D1 D R1 QH ;

(5.55)

where R is upper-triangular and Q is unitary. If the Gram–Schmidt algorithm is used for QR decomposition, the total CMs required for matrix inversion becomes .9M 3 C 10M 2 M /=6. Generally, the matrix inversion algorithms require a high number precision which gives rise to a large silicon area in fixed-point VLSI implementations. The two main reasons for these numerical requirements are shown as follows: (a) The use of costly operations (i.e., square root and divisions) leads to a significant increase of the dynamic range for some intermediate variables. (b) The desire to replace repeated divisions by multiplications with the corresponding inverse in order to reduce the number of costly operations. Unfortunately, multiplications often result in an enhancement of the quantization noise, which requires a high fixed-point precision. Therefore, based on the QR decomposition with modified Gram–Schmidt algorithm, a VLSI architecture has been proposed in [75] to deal with numerical problems for fixed-point implementation. It shows that for a 4 4 channel matrix, the architecture was able to achieve a clock rate of 277 MHz with a latency of 18 time units and area of 72K gates using 0:18-m CMOS technology, which provides a better solution compared to previously known architectures. In other directions, the architecture can be designed for reducing number of matrix inversions, which is well-suited to the systems with multiple channels to be processed (i.e., MIMO orthogonal frequency division multiplexing (OFDM) systems [77, 78]).

5.1.6.4 MMSE Filtering This MMSE filtering operation is carried out in Steps (1) and (4) of the LR-based list detection. In Step (1), the MMSE filtering operation to estimate c2 is applied to the received signal vector x2 as cQ 2 D c2 C ˛WH mmse;2 x2 ;

(5.56)

where c2 D U1 2 ˇ1. In Step (4), Q times of same operation are applied to the received signal vector x1 as .q/

.q/

N1 ; cQ 1 D c1 C ˛WH mmse;1 x .q/

.q/

q D 1; 2; : : : ; Q;

(5.57)

N 1 D x1 COs2 . Note that Q operations in (5.57) where c1 D U1 1 ˇ1 and x can be carried out in parallel. The parallel structure often allows low latency and high throughput. The most complex steps can then be processed in a single cycle,


129

however, at the expense of large silicon area. In addition, with parallel structure, memories need to be implemented based on register files for sufficient access bandwidth. Thus, trade-off between latency/throughput and silicon area needs to be considered. The weight matrices Wmmse;1 and Wmmse;2 are precalculated and stored in the preprocessing stage. Note that only 8 words of memory are needed for this storage requirement. Memory-wise, there are 2Q words required to store the outputs ˚ cQ 1 ; cQ 2 ; : : : ; cQ Q .

5.1.6.5 List in LR Domain The list of candidate vectors in LR domain is formed in Step (2) as C2 ' fc2 j kQc2 ck < r.Q/g ;

(5.58)

c D T1 .˛s2 C ˇ1/ :

(5.59)

where

Thus, the alphabet of signal in LR domain, c, varies depending on channel, T. However, with Gaussian LR, T becomes 1 t ; TD 0 1

(5.60)

where t is an integer. Since the maximum number of column swapping in the Gaussian LR algorithm is limited to V D 2 or 3 only, it is easy to obtain a known set of t and then T. Thus, a look-up table can be formed for the alphabet of c2 after the Gaussian LR algorithm is carried out to submatrix B, where memory is required to store this precalculated data. For instance, it requires V jSj words of memory to store the alphabet of c2 . Furthermore, 2Q words are required to build C2 .

5.1.6.6 Soft-Bit Generation In order to perform the channel coding, the LLR of each bit is required, where the soft-bits are generated. The probability of the qth candidate sO.q/ in the list is given by

.q/ 1 .q/ 2 D CQ exp ; (5.61) x ROs Pr sO N0 where the normalization constant CQ is found as 8 9 =1

< X 1 2 x ROs.q/ exp CQ D : ; N 0 qD1;2;:::;Q

(5.62)

130


and

X

Pr sO.q/ D 1:

(5.63)

qD1;2;:::;Q

Denote by bO .q/ the bit level symbol vector that represents sO .q/ , where the elements of bO .q/ are binary andthe size of bO .q/ is SN 1 for SN D N log2 jSj. Furthermore, we have sO.q/ D M bO .q/ , where M.:/ is the mapping operator. Then, the probability of bO .q/ is given by

2 1 .q/ .q/ O O D CQ exp Pr b x RM b N0

(5.64)

and the soft LLR of the i th bit of bO .q/ (i.e., bi for i D 1; 2; : : : ; SN ) is found as P O .q/ Ob.q/ 2BiC Pr b ; (5.65) ƒ .bi / D log P O .q/ Ob.q/ 2B Pr b i

n o

T where Bi˙ D b1 b2 : : : bSN jbi D ˙1; bm 2 fC1; 1g ; 8m ¤ i .

5.1.7 Simulation Results The BER performance of the LR-based list MIMO detection (using the LR-based MMSE subdetection) with different list length Q is considered in this subsection. We assume 4 4 MIMO systems of N1 D N2 D 2 are employed, while 4, 16, and 64-QAM are used for signaling. The elements of MIMO channel matrices are generated as independent CSCG random variables with mean zero and unit variance, while the SNR is defined as energy per bit to the noise power spectral density ratio, Eb =No . In Fig. 5.1, we show the BER performance of the LR-based list detection with the list length Q equals 1, 2, and 4, when 4-QAM is used for signaling. It is given that the performance can be close to that of the ML detection as Q D 8. The BER performance of the LR-based list detection with 16-QAM signaling is presented in Fig. 5.2. In this case, near ML performance can be achieved when Q D 12, the SNR loss of the LR-based list detection compared to the ML performance is less than a half dB at a BER of 104 when Q D 12. According to (5.41), since decreases with A, it is reasonable to consider a larger list length when A increases to achieve good performance. The simulation results with 64-QAM are considered in Fig. 5.3, which again confirms that the LR-based list detection can provide the near ML performance with low complexity. At a BER of 104, the SNR loss is less than a half dB compared to the ML performance when Q D 16. It is observed that as the SNR increases, the SNR loss increases in general. However, with a sufficiently large list length

5.2 Error Probability-Based Column Reordering Criteria

131

10−1

10−2

BER

10−3

10−4 ML LR−based list (Q = 1) LR−based list (Q = 2) LR−based list (Q = 4) LR−based list (Q = 8) MMSE

10−5

10−6

0

2

4

6

8

10

12

Eb/No (dB)

Fig. 5.1 BER versus Eb =No of different MIMO detectors in a 44 MIMO system (N1 D N2 D 2) with 4-QAM signaling

Q (e.g., 8 for 4-QAM, 12 for 16-QAM, and 16 for 64-QAM), this loss can be reduced as the list length can exploit the trade-off between the performance and complexity. It is also noteworthy that the complexity of the proposed scheme with the sufficiently large list length is comparable to that of the MMSE detection, which has been illustrated in Table 5.2.

5.2 Error Probability-Based Column Reordering Criteria Using the LR and list detection techniques in the SIC architecture, the LR-based list detection algorithm is proposed in Subsect. 5.1.1 by decomposing a large MIMO detection problem into multiple small MIMO subdetection problems, where the complexity grows linearly with its list length. It has been shown that the LR-based list MIMO detection can provide the near ML performance with a sufficiently large list length. However, the good performance is not guaranteed with a small list length (low complexity). Therefore, it is desired to develop a strategy to improve the performance of the LR-based list detection when a small list length is considered.

132

5 Lattice Reduction-Based List Detection 100

10−1

BER

10−2

10−3

10−4

MMSE LR−based list (Q = 1) LR−based list (Q = 4) LR−based list (Q = 8) LR−based list (Q = 12) ML

10−5

10−6

0

2

4

6

8

10

12

14

16

Eb/No (dB)


Note that column swapping of channel matrices can result in different performance in the LR-based list detection. For example, consider the channel matrix H N After the QR decomposition, we have H D QR and its column reordered one H. N R, N DQ N where and H N N A C N D A C : (5.66) and R RD 0 BN 0 B N B ¤ B, N In this N which leads to A ¤ A, N and C ¤ C. Then, there would be R ¤ R, case, the performance of LR-based subdetection could vary depending on different lattice reduced submatrices, especially when a small list length is considered for low computational complexity. In this chapter, in order to obtain an optimal order of columns in terms of the error probability of subdetection, we propose column reordering criteria (CRC) for a given MIMO sub-detector employed. Through simulations, we show our proposed CRC can significantly improve the performance of LR-based list detection with a small list length (low complexity) for slow fading MIMO channels.


133

100

10−1

BER

10−2

10−3 MMSE LR−based list (Q = 1) LR−based list (Q = 4) LR−based list (Q = 8) LR−based list (Q = 16) ML

10−4

10−5

4

6

8

10

12

14

16

18

20

Eb/No (dB)


5.2.1 System Model with CRIS In this subsection, we briefly review the LR-based list detection [30] with the column reordering index set (CRIS), K. Consider again a MIMO system equipped with N transmit and N receive antennas. Let sn denote the data symbol to be transmitted by the nth transmit antenna, n D 1; 2; : : : ; N . Note that sn 2 S, where S denotes a common signal alphabet. Denote by yn the received signal at the nth receive antenna. The received signal vector over a flat-fading MIMO channel with signal reordering is given by y D Œy1 ; : : : ; yN T D HK sK C n;

(5.67)

where the N N channel matrix HK D hk.1/ ; hk.2/ ; : : : ; hk.N / , the transmit signal

T vector sK D sk.1/ ; sk.2/ ; : : : ; sk.N / , and the noise vector n D Œn1 ; n2 ; : : : ; nN T which is a zero-mean CSCG random vector with EŒnnH D N0 I. Note that hk.n/ denotes the k.n/ -th column vector of HK and the CRIS is denoted by

134


˚ K D k.1/ ; : : : ; k.N / , which is a permutation of f1; 2; : : : ; N g. Throughout the section, we assume that the CSI is perfectly known at the receiver.

5.2.2 Detection Algorithm with CRIS Let the QR factorization of HK be H K D Q K RK ;

(5.68)

where N N matrices QK and RK are unitary and upper-triangular, respectively. From (5.67), we have x D QH Ky D RK sK C n:

(5.69)

˚ ˚ Defining two sub-CRIS’s by K1 D k.1/ ; : : : k.N1 / and K2 D k.N N2 C1/ ; : : : k.N / for N1 C N2 D N , we have xD

n1 x1 s ; sK D K1 ; and n D : x2 sK2 n2

(5.70)

Note that x2 , sK2 , and n2 are N2 1 sub-vectors of x, sK , and n, respectively. Then, x is rewritten as x1 AK1 C sK 1 n D C 1 : (5.71) x2 sK2 n2 0 BK2 From (5.71), a two-layer LR-based list detection approach is carried out with the CRIS, which is briefly summarized as follows: .a/ Perform an LR-based detector on x2 to obtain cO 2 , which is an estimated vector of sK2 in LR domain. By choosing the Q closest vectors to cO 2 , a list of candidate .1/ .2/ .Q/ vectors C2 for cO 2 is generated. Note that C2 D fOc2 ; cO 2 ; : : : ; cO 2 g, where .1/ .2/ .Q/ k cO 2 cO 2 kk cO 2 cO 2 k k cO 2 cO 2 k. .b/ Denote n by S2 the list ofocandidates for sK2 , which is mapped from C2 . Here, .1/ .2/ .Q/ S2 D sOK2 ; sOK2 ; : : : ; sOK2 for Q jSjN2 . Let .q/

.q/

x1 D x1 COsK2 ;

(5.72) .q/

where q D 1; 2; : : : ; Q. Performing an LR-based detector on x1 to estimate .q/ sK1 in the LR domain, the resulting vector is denoted by cO 1 .

5.2 Error Probability-Based Column Reordering Criteria .q/

135

.q/

.c/ Mapping cO 1 to sOK1 . Let

2 sOK D 4 .q/

.q/ sOK1 .q/

sOK2

3 5:

(5.73)

Then, the final hard decision is given by sOK D arg

.q/

min .1/

.q/

.Q/

OsK 2fOsK ;:::;OsK g

k x RK sOK k2 :

(5.74)

As the list in a/ is generated in LR domain, the LR-based subdetection is carried N K1 and BN K2 the out to detect sK2 and sK1 in two layers, sequentially. Denote by A lattice reduced matrices of AK1 and BK2 , respectively. The performance of the LRbased list detection highly depends on: 1. The list length Q. N K1 and BN K2 . 2. The level of orthogonality of A 3. The performance (in terms of the error probability) of LR-based subdetection employed to detect sK1 and sK2 . Since the impact of Q (on performance and complexity) has been well-discussed in Sect. 5.1, in this section, we aim to improve the performance of the LR-based list detection with a fixed Q. Furthermore, it ˚ has been known that for a channel N K1 ; BN K2 , which results in different matrix HK , different K may lead to different A performance. In order to obtain an optimal CRIS K to improve the performance, based on the well-known orthogonality deficiency [27], we first introduce a CRC, namely the orthogonality ˚ deficiency based CRC or OD-CRC, to generate the most orthogonal N K1 ; BN K2 . Then, by taking an actually employed subdetector into account, A we develop an error probability based CRC or EP-CRC. We will compare the performance of the two CRCs in simulations. It is noteworthy that the performance of LR-based list detection highly depends on the reliability of the detected sK2 , under the fact that the error probability of the first layer detection plays a key role in overall performance. Therefore, both OD-CRC and EP-CRC perform into two layers.

5.2.3 OD-CRC Letting KN D f1; 2; : : : ; N g, the sub-CRIS K2OD is obtained as K2OD D arg min OD.BN K2 /; N K2 K

(5.75)

136


where the orthogonality deficiency function is given by det DH D OD .D/ D 1 QL 2 j D1 k dj k

(5.76)

for matrix D D Œd1 ; : : : ; dL [27]. Let KN ( KN n K2OD , where “ n ” denotes the set minus. Then, the sub-CRIS K1OD becomes N K1 /: K1OD D arg min OD.A N K1 K

(5.77)

Take BN K2 as an example. Although better performance of LR-based detection could be obtained with a more orthogonal BN K2 , since the lattice reduced sub-matrix is already nearly orthogonal within a certain limit for any K2 thanks to the LR, the use of OD-CRC may not improve the performance significantly. Therefore, we consider the EP-CRC.

5.2.4 EP-CRC By extending the vector selection criteria proposed in [43,69], a two-layer selection strategy based EP-CRC is developed to minimize the error probability for a given MIMO subdetector. Supposing the LR-based linear subdetector is used to detect both sK1 and sK2 , in the first layer, the EP-CRC is carried out to obtain the subCRIS K2EP as N K2EP D arg max min BN H (5.78) K BK2 : 2

N K2 K

With KN ( KN n K2EP , in the second layer, the K1EP is given by H NK ; N A KEP D arg max min A 1 1

K1

N K1 K

(5.79)

where min ./ denotes the minimum eigenvalue operator. As will be discussed in Sect. 7.2, this criterion is regarded as the max–min eigenvalue (ME), which is known to minimize the error probability for a given lattice reduced matrix. For the LR-based SIC subdetection, another criterion developed in Sect. 7.2, namely the max–min diagonal term (MD) criterion [43,69], is carried out. Consider N K1 D QA RA , where QB and QA the QR factorization of BN K2 D QB RB and A .K / .K / are unitary while RB and RA are upper-triangular. Denote by bp;p2 and aq;q1 the .p; p/-th element (for p D 1; 2; : : : ; N2 ) of RB and the .q; q/-th element (for q D 1; 2; : : : ; N1 ) of RA , respectively. In the first layer, the EP-CRC is carried out to obtain the sub-CRIS K2EP as ˇ ˇ ˇ .K2/ ˇ K2EP D arg max min ˇbp;p (5.80) ˇ : N K 2 K

p


Let KN ( KN n K2EP . Then, K1EP is given by K1EP D arg max

N K1 K

ˇ ˇ ˇ .K1 / ˇ min ˇaq;q ˇ : q

137

(5.81)

5.2.5 Simulation Results In this subsection, we present simulation results with 4 4 MIMO systems. The elements of channel matrices are generated as independent CSCG random variables with mean zero and variance unit. The SNR is defined as the energy per bit to the noise power spectral density ratio, Eb =No . In Figs. 5.4, 5.5, and 5.6, respectively, 4, 16, and 64-QAM are used for signaling. Different MIMO detection methods are considered, including: (1) MMSE detection. (2) ML detection. (3) LR-based list detection (using the LR-based MMSE detection as the subdetection and N1 D N2 D 2). (4) OD-CRC employed for the LR-based list detection in (3): LR-based list C ODCRC. (5) EP-CRC employed for the LR-based list detection in (3): LR-based list C EPCRC. In terms of BER versus SNR, in Fig. 5.4, we show simulation results of five different detection schemes in above with 4-QAM signaling, where the LR-based list detection is employed for two cases of Q D 1 and 2. It shows that by using the OD-CRC, the performance of LR-based list detection cannot be improved in general. Nevertheless, with the EP-CRC, a more than 2 dB improvement is observed at BER D 104 compared to that without CRC. Moreover, we can see that the LRbased list detection with the EP-CRC has a SNR loss of a half dB at a range of BER D 103 to 105 compared to the ML performance. The BER performance of different detection schemes with 16-QAM signaling is presented in Fig. 5.5. The LR-based list detection is employed for the cases of Q D 1 and 4. Using the EP-CRC, we can show that the performance of the LRbased list detection with Q D 4 can be close to that of the ML detection, where the SNR loss is less than one dB at a BER of 104. In Fig. 5.6, with 64-QAM signaling, the performance of the LR-based list detection is again confirmed. It shows that the EP-CRC can significantly improve the performance of the detection. The SNR loss of the LR-based list detection compared to the ML performance is less than one dB at a BER of 104 , when Q D 4 and EPCRC is used. Since the complexity of the LR-based list detection has been well-discussed in Subsect. 5.1.5, here, we analysis the computational complexity of the EP-CRC only. N K1 and BN K2 , Denote by CME;A and CME;B the complexity of the ME operation on A

138

5 Lattice Reduction-Based List Detection 10−1

10−2

BER

10−3

MMSE ML LR−based list (Q = 1) LR−based list + OD−CRC (Q = 1) LR−based list + EP−CRC (Q = 1) LR−based list (Q = 2) LR−based list + OD−CRC (Q = 2) LR−based list + EP−CRC (Q = 2)

10−4

10−5

10−6

0

2

4

6 Eb/No (dB)

8

10

12


respectively. The overall complexity of EP-CRC is given by CEPCRC D

NY 2 1

N1 Y

uD0

vD1

.N u/CME;B C

vCME;A :

(5.82)

Note that EP-CRC is only carried out once for a channel matrix. For slow fading channels, since the coherence time is long, the extra computational complexity required for EP-CRC per each symbol detection would be negligible. Under the assumption that the channel is not varying in the duration of 1,000 transmitted symbol vectors. Taking the different MIMO detection methods presented in Fig. 5.5 as an example, we compare the computational complexity for each symbol vector detection in terms of the average number of flops.1 The flops per symbol vector of different detection methods are listed as follows:

1

We simulate these systems using MATLAB-V5.3 on a PC. The MATLAB command “flops” is used to count the number of flops.

5.3 Conclusion and Remarks

139

100

10−1

BER

10−2

10−3 MMSE ML LR−based list (Q = 1) LR−based list + OD−CRC (Q = 1) LR−based list + EP−CRC (Q = 1) LR−based list (Q = 4) LR−based list + OD−ORC (Q = 4) LR−based list + EP−ORC (Q = 4)

10−4

10−5

10−6

0

2

4

6

8

10

12

14

16

Eb/No (dB)


1. 2. 3. 4.

MMSE detection D 458. ML detection D 11,140. LR-based list detection D 956. LR-based list C EP-CRC D 976.

5.3 Conclusion and Remarks In this chapter, we proposed EP-CRC for LR-based list detection. We compared OD-CRC with our proposed EP-CRC. It showed that with our proposed EP-CRC, the performance of LR-based list detection is significantly improved. Furthermore, the near optimal performance can be achieved with a small list length by employing EP-CRC, where low complexity is guaranteed. So far, we have introduced different low-complexity detection methods for MIMO systems with square or tall channel matrices. However, it is not straightforward to develop a detection method that can be adopted to underdetermined MIMO systems to achieve good performance and low complexity at the same time.

140

5 Lattice Reduction-Based List Detection 100

10−1

BER

10−2

MMSE ML LR−based list (Q = 1) LR−based list + EP−CRC (Q = 1) LR−based list + OD−CRC (Q = 1) LR−based list (Q = 4) LR−based list + OD−CRC (Q = 4) LR−based list + EP−CRC (Q = 4)

10−3

10−4

10−5

4

6

8

10

12

14

16

18

20

Eb/No (dB)


For this sake, we will study the prevoting cancellation based detection for such MIMO systems in the next chapter.

Chapter 6

Detection for Underdetermined MIMO Systems

In MIMO systems, the channel matrix is called fat, square, or tall matrix if the number of transmit antennas M is greater than, equal to, or smaller than the number of receive antennas N . According to [1], the MIMO channel capacity can be approximated as CMIMO ' min.M; N /CSISO ;

(6.1)

where CSISO denotes the channel capacity of SISO channels. Thus, with regard to capacity, we may prefer a square channel matrix (i.e., M D N ), where different detection methods for such MIMO systems have been well- discussed in previous chapters of the book. However, as we need to employ a lower order modulation method due to limited receiver’s complexity (e.g., mobile terminals in a cellular system), we can consider a fat channel matrix (i.e., M > N ), because the spectral efficiency per transmit antenna can be lower as CMIMO N D CSISO < CSISO : M M

(6.2)

For this reason, in this chapter, we focus on underdetermined MIMO systems. Note that throughout this chapter, we assume that different symbols transmitted by M transmit antennas are linear independent from others within a time slot. For symbol detection in underdetermined MIMO systems, different techniques are considered. Instead of exhaustive searching for all the possible decision vectors as in the ML detection, list-based detectors [7–13] create a list of candidate decision vectors and then choose the best candidate as their final decision. In [14–19], a family of list-based Chase detectors is proposed and has been discussed in Sect. 3.1. Since the Chase detection cannot achieve a full receive diversity order, especially when underdetermined MIMO systems are considered, GSD approaches [31–35] were developed. Moreover, two suboptimal group detectors are introduced in [36] and a geometrical approach-based detector for underdetermined MIMO systems is studied in [37]. In order to further reduce the complexity, a computationally efficient L. Bai and J. Choi, Low Complexity MIMO Detection, DOI 10.1007/978-1-4419-8583-5 6, © Springer Science+Business Media, LLC 2012

141

142

6 Detection for Underdetermined MIMO Systems

GSD-based detector with column reordering is proposed in [38], namely “tree search decoder-column reordering” (TSD-CR). However, their complexity is still considerably high in some applications. On the other hand, the LR-based detector is only applicable to the case of tall or square channel matrices [24, 27]. Hence, it is desired to develop a detector that can be employed to fat channel matrices and has the near optimal performance with reasonably low complexity, especially for a lower order modulation method. For underdetermined MIMO systems, in this chapter, a prevoting vector cancellation (PVC)-based MIMO or PVC-MIMO detection is discussed. The main idea of the proposed detector is to divide the transmitted symbols into two groups as follows: 1. First, one or more reference symbols are selected out of all the transmit symbols as the prevoting vector (the residual symbols constitute the postvoting vector) and all the possible candidate symbols for the prevoting vector are considered. For example, if two symbols are selected for the prevoting vector and 4-QAM method is used, there are 4 4 D 16 possible candidate symbol vectors to be considered. 2. Then, for each candidate prevoting vector, its contribution (as the interference) is canceled from the received signal and the remaining symbol estimates are obtained by a subdetector (which could be a linear detector or LR-based detector introduced in previous chapters) operating on size-reduced square subchannels. 3. The final hard-decision symbol vector is obtained by taking the one that minimizes the Euclidean distance metric among the candidate vectors. Note that the size of prevoting vectors is determined to generate square subchannels, where the LR-based detectors can be employed for the subdetection. For example, for a 2 4 channel matrix, two symbols are selected for the prevoting vector and the resulting size-reduced subchannel matrix becomes a 2 2 square matrix. Furthermore, in this chapter, theoretical and numerical results show that the proposed approach using an LR-based detector for the subdetection can achieve a full receive diversity order. In [43], user selection criteria are considered for multiuser MIMO systems, where a single user is selected to transmit signals to a BS at a time. Note that different user selection criteria are developed for different MIMO detectors in order to maximize the performance. By viewing multiuser MIMO as virtual antennas in a single user MIMO system, the user selection problems can be regarded as the transmit antenna selection problems. In this chapter, we extend the selection criteria in [43] to support multiple antennas (transmit symbols) at a time for the PVCMIMO detection, where there are more transmit antennas than receive antennas. This extension of the antenna selection, namely the postvoting vector selection (PVS), becomes a combinatorial problem. Using a low-complexity suboptimal detector (i.e., an LR-based detector or linear detector) for the subdetection, with the corresponding optimal PVS, it is shown that the near ML performance can be achieved. For slow fading MIMO channels, through simulations, we demonstrate that the computational complexity of the proposed PVC-MIMO detection with PVS is lower than that of TSD-CR.

6.1 Joint Detection for Underdetermined MIMO Systems

143

The rest of the chapter is organized as follows. The system model and the PVC-MIMO detection are presented in Sect. 6.1. The optimal PVS is discussed in Sect. 6.2. The performance of the proposed PVC-MIMO detectors is analyzed in Sect. 6.3. Simulation results and some further discussions are presented in Sect. 6.4. Finally, we conclude this chapter in Sect. 6.5 with some remarks. Throughout the chapter, vectors and matrices are represented by bold letters. For a matrix A, AT , AH , and A denote its transpose, Hermitian transpose, and pseudoinverse, respectively. EŒ denotes the statistical expectation. In addition, for a vector or matrix, k k denotes the two-norm. dˇc denotes the nearest integer to ˇ. Denote by n the set minus, by In an n n identity matrix, and by K D fk.1/ ; k.2/ ; : : : g the collection set of k.1/ ; k.2/ ; : : : . The .p; q/-th element of a matrix R is represented by ŒRp;q .

6.1 Joint Detection for Underdetermined MIMO Systems In this chapter, we consider underdetermined MIMO systems with receivers of limited complexity, where lower order modulation is employed as mentioned earlier. This would be the case for downlink channels in cellular systems, where the transmitter is a BS and the receiver is a mobile terminal that usually has a small number of receive antennas (e.g., N D 2) and limited computing power for detection. In this section, we present the system model for the underdetermined MIMO systems and introduce the PVC-MIMO detection after a brief description of some existing approaches.

6.1.1 System Model Consider a MIMO system with M transmit and N receive antennas. Let sm denote the data symbol to be transmitted by the mth transmit antenna. Assume that a common signal alphabet, denoted by S, is used for all sm . That is, sm 2 S, m D 1; 2; : : : ; M . Denote by yn the received signal at the nth receive antenna, and n D 1; 2; : : : ; N . Then, the received signal vector over a flat-fading MIMO channel is given by y D Œy1 ; y2 ; : : : ; yN T D Hs C n;

(6.3)

where s D Œs1 ; s2 ; : : : ; sM T and n D Œn1 ; n2 ; : : : ; nN T denote the transmit signal vector and noise vector, respectively. Note that n is assumed to be a zero-mean CSCG random vector with E nnH D N0 I. In addition, H is the channel matrix that can also be written as H D Œh1 ; h2 ; : : : ; hM ;

(6.4)

144


where hm denotes the mth column vector of H. Throughout this chapter, we assume that the CSI is perfectly known at the receiver, while the impact of channel estimation error on the performance will be discussed in Subsect. 6.4.2.

6.1.2 Existing Approaches Taking the equation in (6.3) as a linear system, various numerical algorithms can be used to estimate or detect s when the system is overdetermined (i.e., M N ). In particular, the LR-based detectors to provide the full receive diversity with low computational complexity have been proposed to detect s in Sect. 3.2. However, if the MIMO system is underdetermined (i.e., M > N ), few approaches (e.g., an exhaustive search for the ML detection, list-based detection, and GSD technique) could be applied to the MIMO detection. In this subsection, we briefly review two existing approaches, namely listbased Chase detection and GSD-based detection, which are comparable to our proposed PVC-MIMO detection. Then, the PVC-MIMO will be introduced in the next subsection.

6.1.2.1 List-Based Detection By dividing the symbols to be detected into two layers, the list-based Chase detection [14–19] is carried out and introduced in Sect. 3.1. Denote by P an M M permutation matrix which decides the detection order, and thus (6.3) becomes y D HPPT s C n N s C n; D HN

(6.5)

N D HP and sN D PT s. Let sN D sT sT T , where s1 and s2 denote .M N / 1 where H 1 2 and N 1 subvectors, respectively. Then, (6.5) is rewritten as y D H1 s1 C H2 s2 C n;

(6.6)

N D ŒH1 H2 for the N .M N / submatrix H1 and the N N submatrix where H H2 . The linear-list detection in Subsect. 3.1.1 can be employed to deal with the problem in (6.6), where a two-layer detection procedure is performed and briefly summarized as follows. With a proper P, the .M N / 1 subvector s1 is selected from s as the one with the smallest MSE (i.e., equivalently the highest SNR). After the linear filtering, a list of Q candidates for this subvector is constructed in the first layer. In the second layer, the contribution from the candidates in the list is treated as the interference and then canceled from the received signal. After that,

6.1 Joint Detection for Underdetermined MIMO Systems

145

the subdetection is employed with the corresponding square submatrix H2 to detect the residual N 1 subvector s2 . The final hard decision symbol vector is determined by MMSE over the concerned Q candidate vectors. With the linear-list detector, a low-complexity implementation can be obtained with a small list length Q. The performance of the detection scheme in above highly depends on the reliability of the detected subvector, s1 . Note that when H is square or tall, the SIC strategy can be carried out to estimate s1 without the impact of interference, which becomes the SIC-list detection in Subsect. 3.1.1. However, for underdetermined MIMO systems, since a linear filtering to obtain a list of candidate vectors for s1 suffers from the interference, good performance cannot be guaranteed, which is demonstrated by simulations in Sect. 6.4.

6.1.2.2 GSD-Based Detection In order to apply the well-known sphere decoding to underdetermined MIMO detection, the GSD-based detection is developed to provide the near ML performance [31–38]. Consider the real-valued matrix transformation in (6.3). Let yr D

R.y/ R.H/ J.H/ R.s/ R.n/ ; Hr D ; sr D ; nr D ; (6.7) J.y/ J.H/ R.H/ J.s/ J.n/

where R.:/ and J.:/ denote the real and imaginary parts operation, respectively, Then, (6.3) is rewritten as yr D Hr sr C nr ; (6.8) where the real-valued vectors yr D Œy1 ; y2 ; : : : ; y2N T , sr D Œs1 ; s2 ; : : : ; s2M T , and the size of matrix Hr becomes 2N 2M . By viewing S as a 2jSj -PAM signal set, where a mapping strategy is considered, the GSD-based detection is carried out to detect s by solving the underdetermined integer least squares (UILS) problem as sOr D arg min k yr Hr sr k2 : sr 2S 2M

(6.9)

With the QR factorization of Hr , we have Hr D Qr Rr ;

(6.10)

where the real-valued matrices Qr of size 2N 2N and Rr of size 2N 2M are unitary and upper trapezoidal, respectively. Letting

we have

yN r D QTr yr ;

(6.11)

yN r.A/ Rr.1/ Rr.2/ sr.A/ yN r D ; Rr D ; sr D ; yN 2N 0 0 rTr sr.B/

(6.12)

146


where yN r.A/ D ŒyN 1 ; yN 2 ; : : : ; yN 2N 1 T , rr D Œr1 ; r2 ; : : : ; r2M 2N C1 T , sr.A/ D Œs1 ; s2 ; : : : ; s2N 1 T , and sr.B/ D Œs2N ; s2N C1 ; : : : ; s2M T . Note that M > N . Then, the following two-layer detection is carried out to solve the problem in (6.9) as sOr D arg min k yN r Rr sr k2 sr 2S 2M

D arg

min

sr.B/ 2S 2M 2N C1

C arg

min

sr.A/ 2S 2N 1

2 yN 2N rTr sr.B/ k yN r.A/ Rr.2/ sr.B/ Rr.1/ sr.A/ k2 :

(6.13)

To apply a GSD-based detector, a radius is chosen such that k yN r Rr sr k2 < 2 :

(6.14)

From (6.13) and (6.14), we can also show that

yN 2N rTr sr.B/

2

< 2 :

(6.15)

With a fixed sr.B/ , a conventional sphere decoding algorithm is carried out to solve the problem in (6.13). The initial idea of GSD is proposed in [31], which takes all the possible candidates of sr.B/ to obtain the final decision. Note that the exhaustive search leads to high computational complexity. In order to reduce the complexity, different computationally efficient algorithms [32–35] are proposed. Using a column reordering strategy, the TSD-CR is introduced in [38] to further reduce the complexity. Although the GSD-based detection can provide the near ML performance, its complexity is still considerably high in some applications, which is illustrated through simulations in Sect. 6.4.

6.1.3 Prevoting Cancellation-Based MIMO Detection For underdetermined MIMO systems, since low complexity and good performance cannot be obtained by existing MIMO detectors (i.e., MMSE detector, ML detector, list-based detectors [14–19], and GSD-based detectors [31–35, 38]) at the same time, in this subsection, we introduce the PVC-MIMO detection to provide reasonably low complexity with the near optimal performance. Let R D M N and denote by P D fp1 ; p2 ; : : : ; pR g the index set for the prevoting signal vector (the selection of this vector will be discussed in Sect. 6.2), which is denoted by sP D Œsp1 ; : : : ; spR T . Then, (6.3) is rewritten as y D HP sP C HQ sQ C n;

(6.16)

6.2 Selection for Prevoting Vectors Depending on SubDetectors

147

where HP D Œhp1 ; : : : ; hpR is a submatrix of H associated with sP , sQ D Œsq1 ; : : : ; sqN T is the postvoting signal vector, and HQ D Œhq1 ; : : : ; hqN is a square submatrix of H associated with sQ . Here, the index set Q D f1; : : : ; M g n P, sP 2 S R , and sQ 2 S N . Define the finite set of all the possible candidate vectors for sP as R 2 fs1P ; s2P ; : : : ; sK P g, where K D jSj . For example, K D 4 D 16 if the size of sP is 2 1 and 4-QAM is used as the modulation method. Assuming that sP D skP for k 2 f1; : : : ; Kg, (6.16) is rewritten as rk D HQ sQ C n;

(6.17)

rk D y HP skP :

(6.18)

where After the PVC in (6.17), we can apply any conventional MIMO detector that works for a square MIMO channel to detect sQ . For convenience, denote by sOkQ the detected symbol vector of sQ (by any means) for given sP D skP . Let sk D

sP : sOkQ

(6.19)

With K candidates of sk , i.e., fs1 ; : : : ; sK g, the solution of the PVC-MIMO detection is obtained to best represent the observation vector y in the sense of SSE as sO D arg

min

sk 2fs1 ;:::;sK g

k y H0 sk k2 ;

(6.20)

where k 2 f1; : : : ; Kg and H0 D ŒHP HQ .

6.2 Selection for Prevoting Vectors Depending on SubDetectors In the PVC-MIMO detection, we can show that different postvoting vectors result in different HQ , which may lead to different performance of subdetection. For example, consider an original fat channel matrix H D ŒHP HQ and its column N Q , where HQ and H N D H NP H N Q are square submatrices. Then, reordered one H the performance of a suboptimal detector (e.g., an LR-based detector) employed N Q could vary depending on different submatrices HQ and H N Q . Note with HQ and H that column order of H equivalents to the pre-voting or postvoting vector selection. N Q ) in Since we are interested in the form of the square submatrix (i.e., HQ or H above only, in this section, we focus on the selection of the postvoting vector to exploit the performance of the PVC-MIMO detection. This is based on the fact that the prevoting vector is exhaustively selected for PVC, and thus there would

148


be no performance gain with different prevoting vectors. For the sub-detection, we consider a few low-complexity detectors including linear and LR-based detectors. Note that since a number of the subdetection operations are to be repeatedly performed, the complexity of subdetection should be low.

6.2.1 Selection Criterion with Linear Detector As a linear detector, we consider the MMSE detector in this section. Under the assumption that the detected prevoting vector is correct, from (6.17), the output of the MMSE detector is given by k sOkQ D WH mmse r ;

(6.21)

where Wmmse is the MMSE filter that is given by

N0 1 H C I : Wmmse D HQ HH Q Q Es

(6.22)

Here, Es represents the symbol energy with S. For the ZF detector, Wmmse is taken placed by the ZF filter as 1 Wzf D HQ HH : Q HQ

(6.23)

The detection performance depends on the channel matrix. For a given channel matrix, as will be discussed in Sect. 7.2, we can have the ME selection criterion for the selection of Q. Since Q f1; : : : ; M g, the optimal set Q can be found by using the ME criterion as (6.24) QME D arg max min HH Q HQ ; Q

where min .A/ denotes the minimum eigenvalue of A.

6.2.2 Selection Criteria with LR-Based Linear and SIC Detectors To determine Q for the PVS, we consider the case where LR-based MIMO detectors, which provide the near optimal performance with low complexity [21,24], are employed for the subdetection. Without loss of generality, we assume that the elements of s are complex integers [21, 24]. Then, the LR-based detection is carried out with HQ D GU;

(6.25)

6.2 Selection for Prevoting Vectors Depending on SubDetectors

149

where U is an integer unimodular matrix and G is an LR matrix of a nearly orthogonal basis. For the LR-based linear detection, from (6.17), the received signal vector can be rewritten as rk D Gc C n;

(6.26)

1

where G D HQ U and c D UsQ . Using the linear filter with rounding operation, the LR-based linear detection is carried out to detect c as Q k e; cO D bWr

(6.27)

where

1 Q D GH G C N0 UH U1 Q D GH G 1 GH and W GH W Es

(6.28)

denote the linear filter for the ZF detection and MMSE detection, respectively. For the LR-based MMSE-SIC detector, let " " # # k HQ nT r q q Hex D : (6.29) ; rex D ; and nex D N0 I 0 NE0s sQ Es Using the LR with Hex , the lattice reduced matrix Gex can be found as Hex D Gex Uex ;

(6.30)

where Gex is an LR matrix of a nearly orthogonal basis and Uex is an integer unimodular matrix. The LR-based MMSE-SIC detection is carried out using the QR factorization of Gex D QR; (6.31) where Q is a matrix whose column vectors are orthonormal and R is uppertriangular. Multiplying QH to y results in QH rex D RUex sQ C QH nex N D RNc C n;

(6.32)

where cN D Uex sQ and nN D QH nex . Note that the LR-based ZF-SIC can be carried out with another upper-triangular matrix R, which can be found in Subsect. 3.2.2. The SIC is performed with (6.32). With the upper-triangular matrix R, the last element of cN , i.e., the N th layer, is detected first. Then, in the detection of the .N 1/th layer, the contribution of the last element of cN is canceled and the signal of the .N 1/th layer is detected. This operation is terminated when all the layers are detected. With the LR-based linear and SIC detectors performed on HQ , where Q f1; : : : ; M g, the optimal set Q can be found by using the ME and the MD selection criteria [43], which are shown as QME D arg max min GH (6.33) Q GQ Q

150

and


n ˇ .Q/ ˇo ˇ ; QMD D arg max min ˇrr;r Q

r

(6.34)

respectively. In the above equations, GQ is the lattice reduced basis from HQ and .Q/ rr;r denotes the .r; r/th element of R from HQ in (6.17). Note that the use of ME and MD criteria will be further explained in Sect. 7.2.

6.3 Performance Analysis Assuming that the elements of H are independent CSCG random variables with mean zero and unit variance, i.e., Rayleigh MIMO channels, in this section, we analyze the diversity gain of the proposed PVC-MIMO detection through the error probability. Then, we discuss the complexity of the detection.

6.3.1 Diversity Analysis The error probability of the PVC-MIMO detection is characterized in this section. Denote by so the original transmitted vectors. Let S D fs1 ; : : : ; sK g represent the set of the candidate solutions provided by the PVC, where each sk is generated T OkT from (6.19), i.e., sk D skT , k D 1; 2; : : : ; K. In addition, denote by sO the final P s Q decision of the detector selected from the candidate solutions in S, which is obtained in (6.20). Then, we can define two error probabilities as follows: Definition 6.1. The probability that the transmitted symbol vector does not belong to the set of candidate solutions is defined as Pe;PV D Pr .so … S/ D 1 Pr .so 2 S/ 0 0 D 1 Pr 9sk 2 S W sk D so

(6.35)

for k 0 D 1; 2; : : : ; K, where the event of f9x W f .x/g denotes that there is at least one x such that a function of x, f .x/, is true. Definition 6.2. The probability that the final decision is not the transmitted one provided that the transmitted vector belongs to the set of candidate solutions is defined as Pe;SEL . In other words, Pe;SEL is the probability that the final decision is not correct conditioned on so 2 S, which is given by ˇ Pe;SEL D Pr sO ¤ so ˇso 2 S :

(6.36)

6.3 Performance Analysis

151

Using these two probabilities, the error probability of the PVC-MIMO detection becomes Pe D 1 .1 Pe;PV /.1 Pe;SEL / D Pe;PV C Pe;SEL Pe;PV Pe;SEL :

(6.37)

We will first discuss the error probability when an LR-based detector is employed for the subdetection of PVC-MIMO without PVS. Since an LR-based detector provides a full receive diversity order (as shown in Subsect. 3.2.6), the PVC-MIMO detection can provide reasonably good performance even without PVS. Next, we will consider the error probability when a linear detector is employed. In this case, the PVS plays a crucial role in order to achieve the good performance.

6.3.1.1 Error Probability with LR-Based Detectors Let us consider the case where LR-based detectors are used for the subdetection of PVC-MIMO without PVS. 0 0 A sufficient and necessary condition for so 2 S becomes f9sk 2 S W sk D so g. In the proposed PVC approach, since

s s D P sOQ k

we have

soP and s D o ; sQ o

(6.38)

0 0 0 Pr .so 2 S/ D Pr 9sk 2 S W skP D soP ; sOkQ D soQ :

(6.39) 0

0

That is, we have so 2 S if and only if there exists a candidate solution sk (sk 2 S h 0 iT 0 0 0 0 and sk D skP T sOkQT ), where the selected sP by the PVC approach, i.e., skP in sk , 0

satisfies skP D soP , and the detected postvoting vector (see (6.17)) after this PVC, 0 0 0 i.e., sOkQ in sk , also satisfies sOkQ D soQ . Since the exhaustive search approach is used for PVC, it is obtained that 0 0 Pr 9sk 2 S W skP D soP D 1: (6.40) Therefore, we have Pe;PV D 1 Pr.so 2 S/ 0 0 0 D 1 Pr 9sk 2 S W skP D soP ; sOkQ D soQ 0 ˇ 0 D 1 Pr sOkQ D soQ ˇskP D soP D EHQ PejHQ ;

(6.41)

152


where PejHQ denotes the error probability of the subdetection that detects sQ for a given HQ . That is, Pe;PV in (6.41) is equivalent to the (average) error probability of the subdetection performed on the square submatrix, HQ . Based on the principle of LR, we first consider Pe;PV for LR-based linear detectors. Since LR-based detection generates a nearly orthogonal basis for a given channel matrix [24] to mitigate the effect of (multiple antenna) interference, the full receive diversity with relatively low complexity can be obtained. In the LLL-LR [20] algorithm, HQ is transformed into a new basis denoted by G in (6.26), and thus we have L.G/ D L.HQ / ” G D HQ T; (6.42) where T is an integer unimodular matrix and L.A/ denotes a basis of lattice generated by A. Here, G is called LLL-reduced with parameter ı if G is QR factorized as G D QR; (6.43) where Q is unitary (QT Q D IN ), R is upper-triangular, and the elements of R satisfy the following inequalities: j ŒR`; j

1 j ŒR`;` j; 2

1` 0 and L D .M N /ŠN Š The relation between the PEP in (6.56) and the pdf of variable V can be deduced by Wang and Giannakis in [79]. According to [79], we can show that

2

0s

Pe;PV EV 4erfc @ Z

C1

13 h2 jjjj2 V A5 2N0

0s erfc @

0

1 h2 jjjj2 v A fV .v/dv 2N0

.LC1/ ; D c1 L C o

(6.61)

where D h2 jjjj2 =N0 and c1 > 0 is constant. Note that when M N C 1, we have LD

MŠ .M N /ŠN Š

MŠ .M N /ŠN.N 1/ 2

MŠ .M N /Š.M 1/.M 2/ .M N C 1/

D

MŠ .M 1/Š

D M > N;

(6.62)

that is, L > N . In addition, (6.52) and (6.53) also hold for linear detectors. Therefore, according to (6.61), a full diversity order N can be achieved by the proposed detectors when the ME criterion for index set Q selection is employed. Note that this result is based on the assumption that all the possible HQ ’s are statistically independent, which may not be true. In practice, different HQ ’s are not independent (i.e., XQ are correlated for different Q), and the minimum eigenvalues of HH Q HQ ’s are correlated in the proposed detection after PVS. Thus, (6.60) may not be valid (but just an approximation) and a full diversity order N cannot be achieved. However, for a small sized matrix HQ , a near full diversity order may be achieved due to the low correlation of the minimum eigenvalues of different HH Q HQ ’s, which is confirmed by the numerical results shown in the following section. Consequently, with the optimal PVS, the linear

6.3 Performance Analysis

157

detector-based PVC-MIMO detection can achieve higher diversity. In addition, for a small matrix HQ (e.g., a 2 2 matrix), a near full receive diversity order is achieved by the proposed detection.

6.3.2 Complexity Analysis Let CSub denote the complexity of the subdetection with a square channel matrix of N N . Excluding the complexity of the PVS, the complexity of the PVC-MIMO detection is given by CPVC D KCSub : (6.63) If an exhaustive search is employed to determine Q in (6.24), since there are LD

MŠ ; .M N /ŠN Š

(6.64)

possible index sets, the complexity for building Q is LCSel , where CSel denotes the computational complexity for each possible index set. On the other hand, if Q in (6.33) or (6.34) need to be determined by using the exhaustive search, we have LD

MŠ : .M N /Š

(6.65)

This is based on the fact that for the LR-based selection schemes, different order of candidates leads to different decision, which results in different performance of LR-based subdetection. Thus, the ordering of candidates is considered for the LRbased selection schemes (i.e., (6.33) and (6.34)). For example, if the MD selection criterion is used when M D 4 and N D 2, we need 4 3 D 12 LRs of 2 2 complex-valued channel matrices and CSel becomes the complexity for each LR. We will list the complexity of CSel with different PVS’s for their corresponding subdetectors in Sect. 6.4, empirically using the average number of flops. For a block fading channel, assume that the channel is not varying for a duration of W symbol vectors transmitted. Note that PVS is only performed once for a channel matrix. Then, including the complexity of PVS, the overall computational complexity of the PVC-MIMO detection per each symbol vector becomes CPVC D

LCSel C KCSub: W

(6.66)

For slow fading channels, as the coherence time is long, W will be large. In this case, the extra computational complexity required for PVS per each symbol detection

158


would be negligible, and thus we have CPVC KCSub . In Sect. 6.4, we will compare the complexity of our proposed PVC-MIMO detectors to other MIMO detectors based on the computational flops.

6.4 Simulation Results and Discussions 6.4.1 Simulation Results In this section, we present simulation results to compare our PVC-MIMO detectors with various conventional detectors for underdetermined MIMO systems. The conventional detectors (including the MMSE (linear) detector, ML detector, TSDCR [38], and the linear-list detector [14–19]1 ) and various PVC-MIMO detectors are considered and listed in Table 6.1. As we are interested in the case where the receivers’ computational complexity is limited, we only consider the cases of .M; N / 2 f.4; 2/; .4; 3/; .3; 2/g.2 Note that elements of MIMO channel matrices in simulations are generated as independent CSCG random variables with mean zero and unit variance. The SNR is defined as the energy per bit to the noise power spectral density ratio, Eb =N0 . We assume that 4-QAM and 16-QAM are used for signaling. Since the ML and TSD-CR share the same performance, we provide one curve for both in the performance simulation results, however, as will be shown in Table 6.3, their complexity can be different. When 4-QAM modulation is employed for signaling, in Figs. 6.1 and 6.2, for 2 4 and 3 4 channel matrices, respectively, we show simulation results of BER for different detectors in Table 6.1. In Figs. 6.3 and 6.4, with 16-QAM modulation, for channel matrices of size 2 3 and 3 4, respectively, simulation results of BER for various detectors in Table 6.1 are presented. From the simulation results, we can show that the full receive diversity is achieved by employing the PVC-MIMO detection approach with LR-based subdetectors. In Figs. 6.1 and 6.3, it is provided that Detectors VII and VIII have slight performance degradation from the ML performance and the SNR loss is a half dB at a broad range of BER. In all the simulation results, it is also shown that Detectors X and XI have negligible performance degradation compared to the ML performance. Furthermore, we note that Detectors IV and V cannot provide full diversity and good performance, especially when SNR is high. In Figs. 6.2 and 6.4, we show that Detector IX is able to provide reasonably good performance. For a 2 2 submatrix, we can observe that Detector IX can provide the near ML performance in Figs. 6.1 and 6.3, where the sizes of channel matrices 1

Two scenarios are considered for the linear-list detection: .1/ MMSE C linear-list (MMSE subdetector used in linear-list detection); .2/ LR-based MMSE-SIC C linear-list (LR-based MMSE-SIC subdetector used in linear-list detection). 2 The case of a large M N is discussed in Subsect. 6.4.2.

6.4 Simulation Results and Discussions

159

Table 6.1 Different detection methods for underdetermined MIMO systems Underdetermined MIMO systems with different detection methods I MMSE detector II ML detector III TSD-CR IV MMSE C linear-list: MMSE subdetector used in linear-list detection V LR-based MMSE-SIC C linear-list: LR-based MMSE-SIC subdetector used in linear-list detection VI MMSE C PVC-MIMO: MMSE subdetector used in PVC-MIMO VII LR-based MMSE C PVC-MIMO: LR-based MMSE subdetector used in PVC-MIMO VIII LR-based MMSE-SIC C PVC-MIMO: LR-based MMSE-SIC subdetector used in PVC-MIMO IX MMSE C PVC-MIMO C PVS: MMSE subdetector used in PVC-MIMO with optimal PVS (ME criterion) X LR-based MMSE C PVC-MIMO C PVS: LR-based MMSE subdetector used in PVC-MIMO with optimal PVS (ME criterion) XI LR-based MMSE-SIC C PVC-MIMO C PVS: LR-based MMSE-SIC subdetector used in PVC-MIMO with optimal PVS (MD criterion)

are 2 4 and 2 3, respectively. We also note that the performance of Detector IX is better than that with N D 3. Since a lower correlation of the minimum eigenvalue of HH Q HQ is obtained by employing a reduced-sized channel matrix HQ , the less error propagation is expected. It confirms that the PVC-MIMO detection with MMSE subdetector could be effective when N is sufficiently small. In Table 6.2, we list the complexity of CSel for Detectors IX, X, and XI, by using flops,3 for the case of N D 2 and N D 3, respectively. Since the computation for both LR and eigenvalue is considered with Detector X, the highest complexity is required. Since the TSD-CR approach [38] can be applied to underdetermined MIMO systems with reasonably low complexity and the optimal performance, it is worthy to compare its complexity with that of the PVC-MIMO detectors. In Table 6.3, we compare the complexity of our proposed PVC-MIMO detectors to that of other MIMO detectors, including the ML detector using an exhaustive search, MMSE detector, TSD-CR, and linear-list detectors, in terms of flops. Here, W D 1;000 is assumed for slow fading MIMO channels.4 Note that for PVCMIMO and TSD-CR, the PVS and Householder QR decomposition of channel

3

We simulate these systems using MATLAB-V5.3 on a PC. The MATLAB command “flops” is used to count the number of flops. 4 The complexity of PVC-MIMO with fast fading channels is discussed in Subsect. 6.4.2.

160


100

BER

10−1

10−2 Detector I Detector II/III Detector IV (Q = 8) Detector V (Q = 8) Detector VI Detector VII Detector VIII Detector IX Detector X Detector XI

10−3

10−4

6

8

10

12 14 Eb /N0 (dB)

16

18

20

Fig. 6.1 BER versus Eb =N0 of different detectors listed in Table 6.1 for 4-QAM, M D 4 and N D2

matrix with minimum column pivoting are carried out once for 1,000 symbol vectors transmitted, respectively, to make this comparison fair. The flops listed in Table 6.3 are obtained with Eb =N0 D 20 dB. Although the MMSE and linear-list detectors have low complexity, they do not suit for underdetermined MIMO systems. It is shown that the computational complexity of the PVC-MIMO detectors with optimal PVS for the case of fM; N g D f3; 2g, fM; N g D f4; 2g, and fM; N g D f4; 3g with 4-QAM is significantly lower than that of ML and TSD-CR. It is also shown that with 16-QAM, the proposed detectors can also provide relatively lower complexity for the cases of fM; N g D f3; 2g and fM; N g D f4; 3g. In addition, for different PVC-MIMO detectors in the same MIMO system, Detector IX has the lowest computational complexity among all, since no LR is required in PVS and subdetection. Overall, Detector XI is shown to be very attractive, because its performance is close to that of the ML detector and its complexity is low (the complexity is almost the same as that of IX, which is the lowest). From this, we can show that the combination of LR-based detectors and optimal PVS is the key ingredient to build low complexity, but near ML performance, detection schemes for underdetermined MIMO systems.


161

100

10−1

BER

10−2

10−3 Detector I Detector II/III Detector IV (Q = 2) Detector V (Q = 2) Detector VI Detector VII Detector VIII Detector IX Detector X Detector XI

10−4

10−5

4

6

8

10

12 Eb /N0 (dB)

14

16

18

20


6.4.2 Discussion In Subsect. 6.4.1, we have discussed the computational complexity of PVC-MIMO detectors with slow fading MIMO channels where M N is small (e.g., 1 or 2). In this subsection, we discuss the complexity of the PVC-MIMO detectors for fast fading channels and large M N . Moreover, the impact of channel estimation errors is considered.

6.4.2.1 Fast Fading Channels Previously, we have analyzed the complexity of the PVC-MIMO detection with PVS for slow fading MIMO channels, when W is large (e.g., W D 1;000). Note that fast fading channels lead to a small W . With the overall complexity per each symbol vector of the PVC-MIMO detection in (6.66), CPVC would be high due to the weight of CSel is high when W is small (i.e., the complexity of CSel is given in Table 6.2). Therefore, the PVC-MIMO detection with PVS could have high complexity with a small W .

162


100

BER

10−1

Detector I Detector II/III Detector IV (Q = 8) Detector V (Q = 8) Detector VI Detector VII Detector VIII Detector IX Detector X Detector XI

10−2

10−3

6

8

10

12 14 Eb /N0 (dB)

16

18

20


For the case of W D 10, where channel varies in the duration of every 10 symbol vectors transmitted (i.e., reasonably fast fading channels), with fN; M g D f2; 3g and fN; M g D f2; 4g, the average computational complexity per each symbol vector for PVS of Detector XI is 155 and 310, respectively, in terms of flops. In this case, compared to existing approaches (in Table 6.3), the complexity of the PVC-MIMO with PVS is still low. 6.4.2.2 Large M N Since there are underdetermined MIMO systems with a large M N , it is worthy to discuss the complexity of PVC-MIMO detectors employed in such MIMO systems. Considering a lower order modulation method (4-QAM), by using the same method that obtains the flops in Table 6.3, we compare the computational complexity of Detector XI and Detector III [38] for the cases of fM; N g D f5; 2g and f6; 2g, respectively, in terms of flops. For Detector XI, the flops of fM; N g D f5; 2g and


163

100

10−1

BER

10−2

Detector I Detector II/III Detector IV (Q = 8) Detector V (Q = 8) Detector VI Detector VII Detector VIII Detector IX Detector X Detector XI

10−3

10−4

10−5

6

8

10

12 14 Eb /N0 (dB)

16

18

20

Fig. 6.4 BER versus Eb =N0 of different detectors listed in Table 6.1 for 16-QAM, M D 4 and N D3 Table 6.2 Complexity comparison of CSel for Detectors IX, X, and XI, listed in Table 6.1

Average flops of CSel Detector N D2

N D3

IX X XI

1,608 3,070 1,587

258 678 473

f6; 2g are 3,106 and 12,263, respectively. For Detector III, the flops of fM; N g D f5; 2g and f6; 2g are 5,010 and 19,564, respectively. This shows that the PVC-MIMO detection has lower complexity than TSD-CR with a large M N and a lower order modulation method. We note that the PVC-MIMO detectors are not suitable for the case of a large M N and a high- order modulation method (16-QAM or 64-QAM) due to the exhaustive cancellation of prevoting vectors. However, it is noteworthy that the GSD-based detectors (e.g., TSD-CR) also have high complexity [31–38].

164

6 Detection for Underdetermined MIMO Systems Table 6.3 Complexity comparison of different detectors listed in Table 6.1 Average flops for each symbol vector detection 4-QAM

16-QAM

fM; N g D f3; 2g 78 4,484 753 168 170 193 201 197

Detector I II III IV V IX X XI

fM; N g D f4; 2g 109 22,021 1,296 623 626 770 783 778

fM; N g D f4; 3g 112 32,773 1,226 239 255 325 377 356

fM; N g D f3; 2g 302 286,724 3,467 1,671 1,673 3,056 3,074 3,060

fM; N g D f4; 3g 411 8,388,613 5,546 2,479 2,490 4,645 4,697 4,666

10−1

BER

10−2

Detector XI (ve = 0.05) Detector III (ve = 0.05) Detector XI (ve = 0.02) Detector III (ve = 0.02) Detector XI (ve = 0) Detector III (ve = 0)

10−3

10−4

6

8

10

12

14

16

18

20

Eb /N0 (dB)

Fig. 6.5 BER versus Eb =N0 of Detector III and Detector XI listed in Table 6.1 for ve D f0; 0:02; 0:05g with 4-QAM, M D 4 and N D 2

6.4.2.3 Imperfect CSI Estimation In practice, the channel matrix has to be estimated and there could be estimation errors. Consider an N M channel matrix H presented in (6.3), whose elements are


165

generated as independent CSCG random variables with mean zero and unit variance. When an imperfect CSI estimation is considered, the estimated channel matrix can be given by O D H C E: H (6.67) In (6.67), the N M matrix E represents errors in the CSI estimation, whose elements are generated as independent zero-mean CSCG random variables with variance v2e . With fN; M g D f2; 4g and 4-QAM modulation, in Fig. 6.5, we present simulation results of BER for Detector III and Detector XI with different CSI errors, where ve D 0, 0.02, and 0.05. Fig. 6.5 shows that the performance of Detectors III and XI degrades when ve increases in general. Nevertheless, it shows that our proposed PVC-MIMO detection with PVS (i.e., Detector XI) has a negligible performance gap from the ML performance (i.e., Detector III) with CSI estimation errors.

6.5 Conclusion and Remarks For underdetermined MIMO systems where a lower order modulation scheme is employed, we considered low-complexity MIMO detection approaches based on PVC in this chapter. It was shown that if an LR-based detector is used for the subdetection, the PVC-MIMO detection can achieve a full receive diversity order. We confirmed this through simulations. It was also shown that the complexity of the proposed PVC-MIMO detection is low and comparable to that of the MMSE detection when 4-QAM is used. Therefore, the proposed detection approach can be employed for underdetermined MIMO systems where the receivers’ computational complexity is limited such as mobile terminals. An extension of MIMO systems is multiuser MIMO systems, where the user selection plays a key role to exploit the diversity. In Part II of the book, we consider the user selection for multiuser MIMO systems with an actual employed MIMO detector.

Part II

Multiuser MIMO

In wireless communications, the spectral efficiency can be improved by exploiting the space domain when antenna arrays are employed. For example, space division multiple access (SDMA) [80–82] can be adopted with different beamforming techniques. If multiple antennas are equipped at both transmitter and receiver, we have MIMO channels formed by MIMO systems. Since a rich spatial diversity gain is obtained with MIMO channels, MIMO detection techniques have been well-discussed. Instead of performing an exhaustive search, tree search techniques (e.g., sphere decoding approaches [12, 83, 84]) are developed to provide the optimal performance in some cases with reduced complexity. Furthermore, using the properties of lattice, the LR-based low-complexity detectors [21–28, 85] are proposed which can provide a full receive diversity gain. In part I of the book, we have systematically introduced different low-complexity detection approaches for point to point MIMO systems. In multiuser systems, due to users’ different locations and channel conditions, it is possible to exploit another diversity gain, where the throughput can be maximized by choosing the user of the strongest channel gain at a time. The resulting diversity gain is called the multiuser diversity gain [40]. By extending multiuser systems to the case of MIMO systems [39], we have multiuser MIMO systems, where the multiuser MIMO user selection plays a key role in increasing the throughput or related SNR of channels. Although the achievable rate or related SNR can be used for the user selection criterion, it would be more practical to use a certain performance measure that is directly related to the performance of the actual detector or decoder employed. Therefore, in this part of the book, different user selection criteria are considered to maximize the performance of the actual MIMO detectors employed in multiuser MIMO systems.

Chapter 7

Selection Criteria of Single User

A rich spatial diversity gain can be obtained by employing various MIMO detectors (e.g., ML and LR-based detectors) to MIMO systems. Considering that multiple users are able to access the MIMO channel from different locations and under different channel conditions, it is possible to exploit another diversity gain, where the performance can be maximized by choosing the user of the best channel at a time. The resulting diversity gain is regarded as the multiuser diversity gain [40] in multiuser MIMO systems [39], where the multiuser MIMO user selection [41, 42] becomes an effective way to increase this diversity. In this chapter, for multiuser MIMO systems, we introduce the user selection criteria to select a single user from multiple users at a time, where different selection schemes are developed for different MIMO detection methods to be employed.

7.1 System Model We consider the multiuser MIMO system shown in Fig. 7.1 with K users in uplink channels to transmit signals to the BS, where each user is equipped with M transmit antennas. The BS is equipped with N receive antennas, where N M . Each user has an N M channel matrix and an M L signal matrix to be transmitted, which are denoted by Hk and Sk , respectively, and k 2 f1; 2; : : : ; Kg. The channel is assumed to be a quasi-static block flat-fading channel, which is not varying over a time slot duration of L symbols. Note that one user is selected to access the channel during one time slot. For uncoded signals, in this chapter, we can assume L D 1 (i.e., this assumption is used to simplify the derivation of user selection criteria, while the length of slot can be any number). Consider that the kth user is selected to transmit signal to the BS. Then, over a slot duration, the received signal at the BS is given by yk D Hk sk C nk ;


(7.1)

169

170

7 Selection Criteria of Single User

Fig. 7.1 Block diagram for multiuser MIMO uplink channels of K users equipped per user with M transmit antennas and the BS equipped with N receive antennas

where the background noise vector nk is an independent zero-mean CSCG random vector with E nk nH D N0 I. Furthermore, we assume that the CSI is perfectly k known at the receiver. In order to introduce different user selection criteria and derive the diversity gain, the following assumptions are used. (A1) A common signal alphabet, denoted by S, is used for all users, where sk 2 S M . Here, S A represents the A-dimensional Cartesian product of S. Furthermore, p let sk 2 Z C j Z, where Z denotes the set of integer numbers and j D 1. Note that the signal constellation or alphabet is a subset of Z C j Z. For example, the signal alphabet of 4-QAM is a subset of Z C j Z, where there are four lattice points. (A2) The elements of the channel matrix Hk are independent zero- mean CSCG random variables with variance h2 , where the signal power is absorbed for convenience. (A3) The transmitted signals are uncoded. This implies that the user selection criteria in this chapter are based on uncoded BER.

7.2 User Selection Criteria The SNR is usually considered for user selection criteria to exploit the multiuser diversity. With beamforming or antenna selection, a user selection criterion based on SNR can be easily derived [86]. In general, the SNR-based user selection criterion is directly related to the channel capacity-based user selection criterion

7.2 User Selection Criteria

171

since the channel capacity increases with the SNR. Thus, when MIMO systems are considered (without antenna selection), the user who has the highest channel capacity can be chosen. In this section, we first introduce the well-known maximum mutual information (MMI) criterion.

7.2.1 Maximum Mutual Information Criterion In [86], the MMI criterion is proposed to select the user who has the maximum mutual information between the transmitter and the receiver. The mutual information of the kth user is given by SNR H Hk Hk : Ik D log det IM C M

(7.2)

The MMI criterion is carried out to find the user index as kMMI D arg

max .Ik /:

kD1;2;:::;K

(7.3)

Although conventional user selection schemes [41, 42], including the MMI criterion, can be easily formulated to select user of the strongest channel gain at a time, the actual performance can be different from the expected one if nonideal or suboptimal MIMO detectors are employed for joint detection. Therefore, it is desirable to derive a user selection criterion which can maximize throughput over fading channels by exploiting multiuser diversity as well as multiple antenna diversity depending on the actual MIMO detector employed. Since a user can decide the transmission rate depending on applications, the actual transmission rate is not necessary to be close to the channel capacity. Denoting by Rk the transmission rate for user k, the throughput for user k can be expressed as Tk D Rk 1 Pe;.k/ , where Pe;.k/ is the error probability of packet or symbol of user k. It is noteworthy that the error probability depends on channel conditions as well as detection methods, where the performance of the actually employed MIMO detector need to be taken into account the error probability. Using the throughput expression, the throughput-based user selection criterion can be considered to choose the user with the maximum throughput. Alternatively, the user who maximizes the normalized throughput or throughput efficiency, denoted by Tk =Rk , can be chosen, which is equivalent to choosing the user who has the smallest error probability. Note that although the throughput has been considered to see the impact of multiuser diversity, the error probability can also be considered, especially when different suboptimal detection methods are employed. In [87], a geometricalbased criterion is developed for the LR-based linear detection to minimize the error probability. In [43], for the user selection in uplink channels, where a single user is selected to transmit signals to a BS at a time, the error probability is used for the user

172


selection criteria to choose the user who has the smallest error probability for given MIMO detectors (i.e., ML, linear, and LR-based detectors). In the following part of the chapter, we summarize the user selection criteria based on error probability with different employed detection methods.

7.2.2 User Selection Criteria for ML Detector 7.2.2.1 Selection Criteria Assume that user k is selected. From (7.1), using the exhaustive search, the ML detection is given by (7.4) sO ml D arg min ky Hsk2F ; s2S M

where the user index k is omitted and k:kF denotes the Frobenius norm. Considering the case that s.1/ is transmitted, while s.2/ is erroneously detected, the PEP [44] is given by 2 2

Pe s.1/ ! s.2/ D Pr y Hs.2/ F y Hs.1/ F 1 0s 2 kHdk A; D Q@ 2N0 where Q.x/ D

R1 x

(7.5)

2

z =2 ep dz 2

and d D s.1/ s.2/ . Letting dmin D arg min kHdk2 ; d2D;d¤0

(7.6)

the PEP is upper bounded by 1 0s 2 kHd min k A ; Pe s.1/ ! s.2/ Q @ 2N0

(7.7)

˚ where D D d D s s0 js; s0 2 S M ZM C j ZM . Denoting by D.H/ the length of the shortest nonzero vector of the lattice generated by H, we have D.H/ D kHdmin k :

(7.8)

According to (7.7), the user selection criterion to minimize the error probability becomes kMDist D arg max D .Hk / : (7.9) kD1;2;:::;K


173

The user selection criterion in above is referred to as the max–min distance (MDist) criterion with the ML detection as D.H/ is the minimum distance of the lattice generated by H. Note that the problem to find a nonzero shortest vector in a lattice is called the shortest vector problem (SVP) and known as an NP-hard problem [57]. By contrast, the LLL/CLLL algorithm (Subsect. 3.2.5) can be used as an approximation with polynomial time complexity. Alternatively, from (7.5), we have kHdk2 D dH HH Hd

kdk2 min HH H ;

(7.10)

where min .A/ represents the minimum eigenvalue of A. Thus, another approximation is carried out with the minimum eigenvalue of the channel matrix as kME D arg

max

kD1;2;:::;K

min HH k Hk :

(7.11)

Therefore, each user can feed back its minimum eigenvalue of the channel matrix and the user who has the maximum min HH can be selected to access the H k k channel. This user selection criterion is regarded as the ME criterion.

7.2.2.2 Diversity Analysis It is known that a full receive diversity gain is obtained by the ML detector with a fixed d. Theorem 7.1. Under (A1) and (A2), a full receive diversity gain is achieved by the ML detector, while the average PEP is given by E Pe s.1/ ! s.2/

X d2D;d¤0

N 2 det I C h ddH : 4N0

Proof. The proof can be found in Sect. 2.2.

(7.12)

t u

As the multiuser diversity is considered with MIMO systems, it is expected to have a higher diversity order by exploiting the multiuser diversity and receive diversity. The MDist and ME criteria have been introduced in (7.9) and (7.11), respectively, for the case that the ML detector is employed. With the MDist criterion, from (7.7) and (7.9), the PEP of the ML detector is bounded as 1 0s 2 .H / max D k k A : Pe s.1/ ! s.2/ Q @ 2N0

(7.13)

Furthermore, if the ME criterion is employed, by substituting (7.10) into (7.5), the PEP is bounded as

174


1 0s 2 maxk min HH H kdk k k A: Pe s.1/ ! s.2/ Q @ 2N0

(7.14)

Theorem 7.2. If the MDist criterion is employed, under A2), the diversity order with the ML detector is NK. Proof. Since max D2 .Hk / D max min dH HH k Hk d; k

k

(7.15)

d2D;d¤0

we can have that

Pe s.1/ ! s.2/

0s

X

Q@

d2D;d¤0

1 maxk dH HH k Hk d A : 2N0

(7.16)

Let wk D Hk d. Under (A2), it shows that wk is a CSCG random vector, where 2 2 2 E wk wH k D h kdk I. Then, Xk D kwk k becomes a chi-square random variable with 2N degrees of freedom, where its pdf is given by 2 2 xkN 1 exk=.h kdk / fX .xk / D

N h2 kdk2 .N 1/Š

(7.17)

and its cdf is given by 2 2 FX .xk / D 1 e.xk /=.h kdk /

N 1 X

q xk = h2 kdk2

qD0

qŠ

:

(7.18)

Since the wk ’s are independent, the pdf of V D max fX1 ; X2 ; : : : ; XK g is given by FV .v/ D KFXK1 .v/fX .v/ D c1 vNK1 C o vNK1C" ;

(7.19)

where c1 > 0 is constant. Therefore, according to [79], it can be derived as

E Pe s.1/ ! s.2/

X d2D;d¤0

2 0s E 4Q @

13 maxk dH HH H d k k A5 2N0

.NKC1/ D c2 dNK C o d ; 2

(7.20)

k 2 dk where d D Nh 0 and c2 > 0 is constant. It shows that the diversity order is NK. This completes the proof. t u


175

Theorem 7.2 shows that a full receive diversity gain of N together with a full multiuser diversity gain of K can be achieved by the ML detectors under the MDist user selection criterion. Theorem 7.3. If the ME criterion is employed and M D N , under (A2), the diversity order with the ML detector is K. When N > M , the diversity order becomes K.N M C 1/. Proof. Let X D min HH H =h2 . According to [48], as M D N , the pdf of the smallest eigenvalue is given by f .x/ D M eM x :

(7.21)

Let V D max fX1 ; X2 ; : : : ; XK g : Then, the pdf of V is derived as K1 M e fV ./ D KM 1 eM D KM K K1 C o K1C" ;

. ! 0C /;

(7.22)

where " > 0. Then, the upper bound on the PEP in (7.14) is rewritten as 1 0s Vh2 kdk2 A: Pe s.1/ ! s.2/ Q @ 2N0

(7.23)

Furthermore, according to [79], one can show that E

s.1/ ! s.2/

.KC1/ ; c3 dK C o d

(7.24)

where c3 > 0 is constant. As a result, we can see that the ME criterion cannot fully exploit the receive diversity gain, but can exploit the multiuser diversity gain, K. On the other hand, if N > M , the diversity order becomes K.N M C 1/ using the result in [48]. This completes the proof. t u

7.2.3 User Selection Criterion for Linear Detectors 7.2.3.1 Selection Criterion Let the user index k be omitted in (7.1). Using a linear transformation, s is estimated as sO D Wy; (7.25)

1 0 HH denotes a linear filter for the MMSE detector, where W D HH H C N Es I while the ZF detector is obtained if

N0 Es

D 0.

176


0 When the SNR is sufficiently large, we have N ! 0, and the MMSE detector Es becomes the ZF detector. Then, the PEP is given by [44]

0

1

B Pe s.1/ ! s.2/ D Q @ q

kdk 2N0

dH

2

.HH H/1

C A:

(7.26)

d

Furthermore, since 1 1

d max HH H dH HH H kdk2 D

kdk2 min .HH H/

(7.27)

and Q.:/ is a decreasing function, the PEP in (7.26) has the following upper bound: 1 0s 2 H min .H H/ kdk A: Pe s.1/ ! s.2/ Q @ 2N0

(7.28)

Hence, the ME criterion in (7.11) can be used for the user selection criterion.

7.2.3.2 Diversity Analysis Theorem 7.4. If the ME criterion is employed, under A2), the diversity order with the linear detector (ZF or MMSE detector) becomes K.N M C 1/. Proof. If the ME criterion is used with the linear detector, from (7.28), the PEP is bounded as 1 0s 2 maxk min HH H kdk k k A: Pe s.1/ ! s.2/ Q @ (7.29) 2N0 Applying the approach used to prove Theorem 7.3, we have Pe s.1/ ! s.2/ D c4 .N M C1/K C o ..N M C1/KC1/ ; where c4 is constant. This completes the proof.

(7.30) t u


177

7.2.4 User Selection Criteria for LR-Based Detectors 7.2.4.1 Selection Criterion To improve the performance of the linear and SIC detectors, the LR is performed in the LR-based detection. A complex-valued matrix is converted into a realvalued one for the LR as in [24] (i.e., LLL-LR). Alternatively, the LR can be directly performed with a complex-valued matrix as in [25] (i.e., CLLL-LR). In this subsection, we assume that the LR is performed with complex-valued matrices for convenience (i.e., CLLL-LR). For a given channel matrix H in (7.1), by omitting the user index k (for the sake of simplification), the LR basis is found as H D GU;

(7.31)

where U is a unimodular matrix whose elements are complex integers and G is a matrix whose column vectors are nearly orthogonal. Then, the received signal vector in (7.1) is rewritten as y D GUs C n D Gc C n;

(7.32)

where c D Us. In [87], the optimal decision region (ODR) criterion for the LR-based linear detection can be simplified to the min–max mean square error (MMMSE) criterion with the lattice reduced basis Gk from Hk , which is given by kODR=MMMSE D arg

min

kD1;2;:::;K

max

i D1;2;:::;M

k wk;.i /

k : 2

(7.33)

Here, wk;.i / denotes the i th row of the linear filter WH k from Gk and Wk D 1 Gk G k G H . k In [43], two selection criteria are proposed for the LR-based linear and LR-based SIC detectors, respectively. For the LR-based linear detectors, the ME criterion is also valid. Letting c.1/ D Us.1/ and c.2/ D Us.2/ , from (7.28), the PEP of LR-based linear detection is bounded as 1 0s H G/ kuk2 .G min A; (7.34) Pe s.1/ ! s.2/ Q @ 2N0 where u D c.1/ c.2/ D U s.1/ s.2/ :

(7.35)

178


According to (7.34), the selection criterion is given by kME D arg max min GH k Gk ; kD1;2;:::;K

(7.36)

where Gk is the reduced basis from Hk . It is noteworthy that this ME criterion is the same as that in (7.11) by replacing the channel matrix Hk with its lattice reduced one Gk . For the LR-based ZF-SIC detection, the QR factorization of G in (7.32) is given by G D QR; (7.37) where Q is a matrix whose column vectors are orthonormal and R is uppertriangular. By substituting (7.37) in (7.32), the LR-based ZF-SIC is carried out with the following signal vector: QH y D QH GUs C QH n Q D Rc C n;

(7.38)

where R D QH G, c D Us, and nQ D QH n. Since nQ and n have the same statistical property, nQ can be denoted by n. Alternatively, for the LR-based MMSE-SIC detection, define the extended channel matrix as (again we omit the user index k) " # H q : (7.39) Hex D N0 I Es The LR basis is found as Hex D Gex Uex ;

(7.40)

where Uex is a unimodular matrix whose elements are complex integers and Gex is a matrix whose column vectors are nearly orthogonal. Letting # " n y q yex D ; (7.41) and nex D 0 s 0 N Es we have yex D Hex s C nex :

(7.42)

Consider the QR factorization of Gex D QR, where Q is a matrix whose column vectors are orthonormal and R is upper-triangular. Then, the LR-based MMSE-SIC is carried out with the following signal: QH yex D QH Gex Uex s C QH nex Q D Rc C n;

(7.43)

where R D QH Gex , c D Uex s, and nQ D QH nex . Note that nQ can be denoted by n for convenience.


179

The SIC detection can be carried out with (7.38) or (7.43), respectively, for the LR-based ZF-SIC or the LR-based MMSE-SIC detector, where the elements of the last row, the M th layer, are detected first. Then, their contributions in the second last row are canceled and the signals of the .M 1/th row are detected. This operation is repeated up to the first row. Since the column vectors of G or Gex are nearly orthogonal after the LR operation, upper off-diagonal elements of R would be small. Therefore, the performance of the SIC detection would mainly depend on the diagonal elements of .k/ R. Denote by rq;q the .q; q/th element of R from the kth user’s channel Hk , where q D 1; 2; : : : ; M . Assuming that the detection of the lower layers is successfully carried out with no error, the SNR of qth layer of Hk is given by q.k/ D

1 ˇˇ .k/ ˇˇ2 ˇr ˇ : N0 q;q

(7.44)

From this, the user selection criterion for the LR-based SIC detection is given by kMD D arg

max

kD1;2;:::;K

min j q

.k/ rq;q

j ;

(7.45)

which is referred to as the MD criterion. Note that MD criterion can also be used with the SIC detection, by letting U; Uex D I (i.e., G D H or Gex D Hex ). In terms of the error probability, the MD criterion can also be derived. Letting x D QH yex , (7.43) is written as x D Rc C n:

(7.46)

Denoting by nq the qth element of n, there would be no error at the Mth layer of the LR-based SIC detection if rjnM j < 12 or 4 jnM j2 < jrM;M j2 . Therefore, there would j M;M j ˇ ˇ2 ˇ ˇ2 be no error across all the layers of the LR-based SIC detection if 4 ˇnq ˇ < ˇrq;q ˇ , for q D 1; 2; : : : ; M . From (3.179), the error probability of LR-based SIC detection becomes Pe 1

Q Y

1e

jrq;q j2

!

4N0

qD1

'e

minq

jrq;q j2 4N0

(7.47)

as N0 tends to 0. Therefore, ˇ ˇin order to minimize the error probability, the user who has the maximum minq ˇrq;q ˇ can be selected, which meets the MD criterion.

180


7.2.4.2 Diversity Analysis Theorem 7.5. If the CLLL-reduced basis is employed with ı D 1, under (A1), we have ˇ ˇ2 (7.48) min ˇrq;q ˇ ˇ M C1 D2 .H/; q

where ˇ > 43 is a constant. Moreover, under (A2), the LR-based SIC detector can achieve a full receive diversity gain. Proof. Consider the matrix G from H (i.e., H D GU, where U is a unimodular matrix) and the QR factorization of G D QR, where Q is unitary and R is upper-triangular. G is called CLLL-reduced if the elements of R satisfies (3.145) and (3.146). According to the proof of Theorem 3.6, (3.182) and (3.183) can be written as ˇ ˇ2 min ˇrq;q ˇ ˇ M C1 jr1;1 j2 (7.49) q

and kg1 k2

min kHdk2 d2D;d¤0

D D2 .H/;

(7.50) 1 respectively, where ˇ D ı 14 > 43 . Thus, from (7.49) and (7.50), we can obtain (7.48). Furthermore, applying the derivation used in Sect. 2.2, we have " ˇ ˇ2 !# N ˇrq;q ˇ X 2 ˇ M C1 H E exp min det I C h dd : (7.51) q 4N0 4N0 d2D;d¤0

From (7.47) and (7.51), the error probability is given by N X h2 ˇM C1 H Pe det I C dd ; 4N0

(7.52)

d2D;d¤0

where a full receive diversity order of N is achieved with the LR-based SIC detection. This completes the proof. t u Theorem 7.6. If the MD criterion is employed, under (A1) and (A2), the diversity order becomes NK, when the LR-based SIC detector is used with the CLLL-reduced basis. Proof. From (7.47) and (7.48), we have Pe exp ˇ M C1 D .Hk / X maxk dH HH k Hk d : exp ˇ M C1 2N0 d2D;d¤0

(7.53)

7.3 Simulation Results

181

Then, applying the approach used to prove Theorem 7.2, we can show that the average error probability becomes

.NKC1/ E ŒPe c5 dNK C o d ; (7.54) t u

where c5 is constant. This completes the proof.

7.3 Simulation Results In order to illustrate the impact of the multiuser diversity gain to multiuser MIMO systems, we present the BER simulation results of various multiuser MIMO systems in Figs. 7.2–7.4. Five multiuser MIMO systems are considered with M D N D 4 and K D 10, namely: 1. MMSE detection under ME criterion: MMSE (ME). 2. ML detection under MDist criterion: ML (MDist). 10−1

10−2

BER

10−3

10−4

10−5 MMSE (ME Criterion) LR−based MMSE−SIC (MMI Criterion) LR−based MMSE−SIC (ODR Criterion) LR−based MMSE−SIC (MD Criterion) ML (MDist Criterion)

10−6

10−7

0

1

2

3

4 Eb /N0 (dB)

5

6

7

8

Fig. 7.2 BER performance of various multiuser MIMO systems with 4-QAM, M D N D 4, and K D 10

182


100

10−1

BER

10−2

10−3


10−5

10−6

0

2

4

6 Eb /N0 (dB)

8

10

12


3. LR-based MMSE-SIC detection under MMI criterion: LR-based MMSE-SIC (MMI). 4. LR-based MMSE-SIC detection under ODR criterion: LR-based MMSE-SIC (ODR). 5. LR-based MMSE-SIC detection under MD criterion: LR-based MMSE-SIC (MD). In addition, 4, 16, and 64-QAM are used for signaling in Figs. 7.2, 7.3, and 7.4, respectively. From Figs. 7.2–7.4, we can show that the optimal performance is obtained by the ML detection under MDist criterion. The MMSE detection under ME criterion provides the worst simulation result as expected from the theoretical analysis, since it cannot fully exploit the spatial diversity. For the LR-based MMSE-SIC detection, the MMI criterion cannot provide a full diversity gain, although the performance can be improved by using the ODR criterion, there is still a BER gap compared to the one with the MD criterion. Overall, we can show that the best user selection criterion for the LR-based MMSE-SIC detection is the MD criterion, which can exploit a full diversity gain as the ML detection does with the MDist criterion.


183

100

10−1

BER

10−2

10−3


10−5

10−6

0

2

4

6

8 Eb /N0 (dB)

10

12

14

16


It is noteworthy that the LR basis needs to be generated when the LR-based detectors are used. For example, when the MD criterion is employed for the LRbased SIC detection, each user has to find the LR basis, which leads to increased computational complexity at both transmitter and receiver sides. Fortunately, the LR basis can be found in a polynomial time and the increased computational complexity would not be significant.

7.4 Conclusion and Remarks In this chapter, we introduced multiuser MIMO systems and the multiuser MIMO user selection criteria for different MIMO detectors, which select a single user among multiple users at a time. Using the error probability based selection criteria, we showed that the LR-based detection can exploit the same diversity as that of the ML detection in multiuser MIMO systems. Considering that multiple users can be selected to access the channel at a time, in the next chapter, selection schemes of multiple users for different MIMO detectors are discussed.

Chapter 8

Selection Criteria of Multiple Users

In the last chapter, different selection schemes have been introduced to choose a single user among multiple users to access the MIMO channel at a time. In some cases, as multiple users are able to access the MIMO channels at the same time, the selection of multiple users has to be considered. By viewing multiuser MIMO as virtual antennas in a single user MIMO system, various antenna selection techniques have been studied to determine the subset of multiple antennas that transmit or receive signals [86–88]. For example, a mutual information-based criterion is proposed in [86] to select the antenna subset that maximizes the mutual information. In addition, a geometrical-based criterion is developed with an LR-based linear detector to minimum the error probability in [87]. Note that those schemes can also be adopted to select multiple users in multiuser MIMO systems. Since the user selection problems that choose multiple users are combinatorial problems, the complexity required to solve these problems could be prohibitively high for large multiuser MIMO systems. For example, when 10 users are selected among 100 users, the number of possible subsets of selected users becomes U D

100Š D 1:731 1013 : .100 10/Š10Š

(8.1)

Furthermore, if the order of users in the subset is taken into account as different order may leads to different performance, we have U D

10Š D 6:2816 1019 : .10 4/Š

(8.2)

If a computational operation with certain complexity is carried out for each possible subset, the computational complexity required to determine the user subset becomes prohibitively high. Thus, low complexity suboptimal selection strategies are considered in [41,89–96] at the expense of degraded performance. In [41, 89–91], antennas are sequentially selected to maximize the throughput based on greedy selection schemes. L. Bai and J. Choi, Low Complexity MIMO Detection, DOI 10.1007/978-1-4419-8583-5 8, © Springer Science+Business Media, LLC 2012

185

186

8 Selection Criteria of Multiple Users

To be different from those throughput or SNR-based selection schemes, in Sect. 7.2, the error probability is used for the user selection criteria to choose the user who has the smallest error probability for a given MIMO detector. Various user selection criteria are derived with the ML detector as well as other low complexity suboptimal detectors. It has been shown that a full diversity gain (i.e., multiuser diversity and multiple antenna diversity) can be achieved using those user selection criteria presented in Subsect. 7.2.4 with LR-based detectors. In this chapter, we extend the user selection in Sect. 7.2 to support multiple users at a time. This extension of the user selection (i.e., multiple user selection) may not be straightforwardly employed as the multiple user selection problems become combinatorial problems. According to (8.2), for example, if an exhaustive search approach is used to select 10 users among 100 users when an LR-based MIMO detector is employed, LR needs to be performed on a huge number of 6:2816 1019 possible channel matrices composited by a group of subchannel matrices of the selected users (i.e., ME and MD criteria are employed on each possible channel matrix for LR-based linear and LR-based SIC detectors, respectively), which results in high computational complexity as the number of user combinations is large. Therefore, we introduce a greedy user selection approach in uplink channels for the complexity reduction when an LR-based detector is used. Moreover, an iterative LR updating algorithm is investigated to further reduce the computational complexity. From the theoretical analysis in this chapter, one can be shown that the LRbased detection can achieve the same diversity as the ML detector does, when the combinatorial user selection is carried out. Through simulations, we compare the BER performance of several MIMO detectors under different user selection criteria (i.e., combinatorial and greedy ones). When the LR-based detection is employed, simulation results confirm that the ME- or MD-based combinatorial user selection can provide the best performance, while the performance of the greedy user selection scheme could approach that of the combinatorial one as the correlation between possible composite channel matrices decreases. It also shows that the greedy user selection provides better performance and significantly reduced complexity compared to other approaches. This chapter is organized as follows. The system model for multiuser MIMO is presented in Sect. 8.1. In Sect. 8.2, different user selection criteria are discussed for given multiuser MIMO systems. The greedy user selection approach is derived in Sect. 8.3 with an iterative LR updating algorithm. The performance analysis and simulation results are presented in Sect. 8.4. Finally, we conclude this chapter with some remarks in Sect. 8.5. Throughout this chapter, the following notations are used and summarized as follows: 1. Vectors and matrices are represented by bold letters. 2. For a matrix A, AT , AH , and A denote its transpose, Hermitian transpose, and conjugate, respectively. 3. Denote by A.a W b; c W d / the submatrix of A with the elements obtained from rows a; : : : ; b and columns c; : : : ; d .

8.1 System Model

187

4. A.W; n/ and A.n; W/ denote the nth column and nth row vectors, respectively. 5. R.z/ and J.z/ denote the real and complex parts of a complex number z, respectively. 6. For a vector or matrix, k k denotes the 2-norm. 7. bˇc represents the closest integer which is smaller than ˇ, while dˇc denotes the nearest integer to ˇ. 8. “n” denotes the set minus. 9. In denotes an n n identity matrix. 10. K D fk.1/ ; k.2/ ; : : : g denotes the collection set of k.1/ ; k.2/ ; : : :

8.1 System Model Consider a multiuser MIMO system with K users in uplink channels to transmit signals to the BS, where each user is equipped with P transmit antennas and the BS is equipped with N receive antennas. Denote by L the number of symbols transmitted by a user. Then, each user has an N P channel matrix and a P L signal matrix, which are denoted by Hk and Sk , respectively, for k 2 f1; 2; : : : ; Kg. Various MIMO detection methods are considered at the BS to detect signal from users. It is assumed that all the users share a common uplink channel and M users can access the channel at a time, where M b N c. Note that the constraint P of M depends on the MIMO detectors employed at the BS, if an ML or linear detector is employed to detect signal, there could be more transmit antennas than receive antennas in multiuser MIMO systems, which results in M > b N c. However, P considering that the LR-based detection is used as the detection method, we have M bN c. In this chapter, for convenience, we assume M b N c as general cases P P are considered. The channel is defined to be a quasi-static block flat-fading channel, which is not varying over a time slot duration of L symbols. Here, a set of the M users who can access the channel could be updated for every time slot interval. For example, in a certain time slot interval, if K D 10 and M D 4, Fig. 8.1 presents the case that the 4th, 5th, 7th, and 9th users are selected to access the channel simultaneously. Note that this selection problem can also be regarded as that with virtual antennas in a single user MIMO system, where MP antennas are selected out of KP available antennas. Thus, the system shown in Fig. 8.1 is identical to the one in Fig. 8.2, where 4 antenna subsets (AS) (i.e., a sum of 4P antennas) are selected among an overall of 10P available antennas. In this chapter, channel conditions are the only metric to be considered to select users. However, this could be extended to include transmit optimization [97, 98], traffic conditions, and users’ priorities [99, 100], which are beyond the scope of the chapter. Let k.m/ be the mth selected user’s index. For convenience, define the set of the selected users’ indices as K D fk.1/ ; k.2/ ; : : : ; k.M / g (e.g., as shown in Fig. 8.1,

188


Fig. 8.1 Block diagram for multiuser MIMO uplink channels of K D 10 users equipped per user with P transmit antennas and the BS equipped with N receive antennas, while 4 users are selected to transmit signals to the BS during a time slot interval

we have K D fk.1/ ; k.2/ ; k.3/ ; k.4/ g f4; 5; 7; 9g and the number of possible permutation of K is 4!). Then, over a slot duration, the received signal at the BS can be expressed as YK D HK SK C N; (8.3) h i T where HK D Hk.1/ ; : : : ; Hk.M / , SK D STk.1/ ; : : : ; STk.M / , and N denote the N MP composite channel matrix, the MP L transmitted signal matrix, and the N L background noise matrix, respectively. It is assumed that each column vector of N is an independent zero mean CSCG random vector with EŒnl nH l D N0 I, where nl denotes the l-th column vector of N. Throughout this chapter, the following assumptions are used to derive user selection methods.


189

Fig. 8.2 Block diagram of virtual antennas in a single user MIMO system, while 4 AS are selected among 10 AS to transmit signals to the BS during a time slot interval

(A1) The CSI is perfectly known at the receiver. (A2) The elements of SK have a common signal alphabet, denoted p by S Z C j Z, where Z denotes the set of integer numbers with j D 1. Moreover, S A represents the A-dimensional Cartesian product of S. (A3) The transmitted signals are uncoded, which implies that the user selection criteria in this chapter are based on uncoded BER. Then, we assume L D 1 for uncoded signals. Note that this assumption is used to simplify the derivation of user selection criteria, while the length of slot can be any number. As a result, YK , SK , and N become vectors and will be denoted by yK , sK , and n, respectively.

8.2 User Selection Criteria When a single user is selected to access the channel during this time slot interval (i.e., M D 1), the user that provides the minimum BER can be chosen for a given MIMO detector to maximize the performance, which has been studied in Chap. 7 while different selection criteria have been developed depending on the types of actually employed MIMO detectors. It has been shown that the user selection criteria with the LR-based MMSE-SIC detector [2, 24] can provide good performance with reasonably low complexity, compared to that with the ML detector. Since only one user is selected (i.e., M D 1) in Chap. 7, in order to extend the user selection criteria to the case of M > 1, in this and next sections, we consider the combinatorial and greedy user selection criteria.

190


8.2.1 ML and Linear Selection Criteria According to (A3), (8.3) is rewritten as yK D HK sK C n:

(8.4)

For simplicity, we omit the user index set K. The estimated symbol vectors from the ML and linear detectors are given by sO D arg min k y Hs k2

(8.5)

sO D WH y;

(8.6)

s2S MP

and respectively. In (8.6), the linear filter 1 W D H HH H

(8.7)

is considered for the ZF detection and N0 1 H I WDH H HC Es

(8.8)

is used for the MMSE detection, where Es represents the symbol energy. Since the detection performance depends on the channel matrix for a certain MIMO detector, as discussed in Chap. 7, we can apply the MDist and the ME criteria for user selection. For a given M > 1, the set of the users who can access the channel is found by using the MDist or ME user selection criterion as follows: KMDist D arg max D .HK /

(8.9)

KME D arg max min HH K HK ;

(8.10)

K

or

K

respectively, where D.HK / denotes the length of the shortest non-zero vector of the lattice generated by HK and min .A/ represents the minimum eigenvalue of A. If the ML detector is employed, the MDist user selection criterion can be used to choose the M users who can provide the lowest BER, while the ME criterion is to choose the M users who have the highest worst SNR (i.e., max-min SNR). Note that if P D M D 1, both the criteria choose the user with the highest SNR. Although the MDist criterion in (8.9) has been derived to maximize the performance with the ML detector and the ME criterion in (8.10) suits for the MMSE detector, they can be used with any MIMO detector.


191

8.2.2 LR-Based Linear and SIC Selection Criteria In this subsection, the user selection criteria for LR-based detectors in Chap. 7 are extended to the case of M > 1. Although the LR can be performed with a complex-valued H [25–27] or a real-valued one converted from H [21, 24], there is no performance difference as shown in [25]. For convenience, in this chapter, we consider the LR with a complex-valued H as in [25]. Letting the user index set K in (8.4) be omitted, the LR is carried out with G D HU1

and c D Us;

(8.11)

where U is an integer unimodular matrix and G is a lattice basis reduced (LBR) matrix which has a nearly orthogonal basis. Note that the CLLL algorithm in Subsect. 3.2.5 can be employed to generate G from H. Using (8.11), the received signal vector in (8.4) can be rewritten as y D Gc C n:

(8.12)

Let W denote the linear filter of G, where 1 H G WH D G H G and

N0 H 1 1 H H W D G GC U U G Es H

(8.13)

(8.14)

are carried out for the LR-based ZF and the LR-based MMSE detection, respectively. Then, according to Subsect. 3.2.2, the LR-based linear detection is preformed to detect c as ˘ ˙ (8.15) cO D WH y : For the LR-based ZF-SIC detection, with the QR factorization of G in (8.12): G D QR;

(8.16)

where Q is unitary and R is upper triangular, (8.12) is rewritten as y D QRc C n:

(8.17)

From (8.17), we have QH y D Rc C QH n D Rc C n; since QH n and n share the same statistical properties.

(8.18)

192


For the LR-based MMSE-SIC detection, (8.4) is extended as yex D Hex s C nex ;

(8.19)

where y ; yex D 0

" Hex D

qH

#

" ;

N0 I Es

and nex D

# n q : 0 s N Es

(8.20)

Through LR (i.e., CLLL in Subsect. 3.2.5) with Hex , the LBR matrix Gex is found as Hex D Gex Uex ;

(8.21)

where again Uex is an integer unimodular matrix. Taking the QR factorization of Gex D Qex Rex

(8.22)

for a unitary Qex and an upper triangular Rex , (8.19) becomes yex D Qex Rex Uex s C nex :

(8.23)

Multiplying QH ex to yex in (8.23) leads to H QH ex yex D Rex Uex s C Qex nex

Q D Rex cQ C n;

(8.24)

where cQ D Uex s and nQ D QH ex nex . Using the SIC approach with (8.18) and (8.24) to detect c and cQ , the LR-based ZF-SIC and the LR-based MMSE-SIC detection are performed, respectively. Note that further particulars of SIC approach can be found in Sect. 2.4. Apparently, the ME criterion for the LR-based linear detection presented in Chap. 7 with M D 1 can be extended to the case of M > 1 as KME D arg max min GH (8.25) K GK : K

Similarly, the MD criterion for the LR-based SIC detection can also be modified as

.K/ j ; KMD D arg max min j rq;q K

.K/

q

(8.26)

where rq;q denotes the .q; q/-th element of R in (8.18) or Rex in (8.24). The user selection based on (8.9), (8.10), (8.26), and (8.25) is regarded as the combinatorial user selection since the users can be selected by combinatorial (or exhaustive) search.

8.3 LR-Based Greedy User Selection Using an Updating Method

193

8.3 LR-Based Greedy User Selection Using an Updating Method The computational complexity of the user selection under the criteria derived in Sect. 8.2 grows rapidly with M or K as they are all combinatorial optimization problems. For the LR-based combinatorial user selection schemes, we note that different order of user index leads to different decisions (based on ME in (8.25) or MD in (8.26)), which results in different performance. In order to maximize the performance with the combinatorial user selection, the order of user index is considered in this chapter. Therefore, there are U D

MŠ .M K/Š

(8.27)

possible user index set, while for each user index set, an LR of an N MP complex channel matrix is to be performed. For example, when K D 10, M D N D 4; and P D 1, 10 9 8 7 D 5;040 LRs of 4 4 complex-valued channel matrices have to be carried out to perform the selection in a time slot of L symbols. Since the complexity could be prohibitively high with U in (8.27), it is desirable to develop low complexity approaches for the user selection that can be adopted to practical systems. In this section, we introduce low complexity greedy approaches for the user selection. Note that we focus on the greedy user selection with LRbased MIMO detectors only as their performance is comparable to that of the ML detector and, more importantly, we can derive a computationally efficient LR updating method in conjunction with greedy user selection.

8.3.1 LR-Based Greedy User Selection To reduce the computational complexity of the LR-based combinatorial user selection schemes (i.e., ME in (8.25) and MD in (8.26)), in this subsection, a greedy approach is considered with an actually employed LR-based MIMO detector. The resulting approach is regarded as the LR-based greedy (LRG) user selection, which is of course suboptimal. Letting m D 1 and KN D f1; : : : ; Kg, the LRG user selection algorithm for the ME criterion with the LR-based MMSE detector is summarized within 3 steps as follows: (S1) The user index of the first user to be selected is obtained as k.1/ D arg max min GH k Gk ; N k2K

(8.28)

194


q iT h 0 where Gk is the LBR matrix of Hk or Hex;k D HTk N (i.e., for LR-based Es I ZF or LR-based MMSE, respectively). Note that as P D 1, no LR operation is required since Hk becomes a vector, and then (8.28) can be replaced by k.1/ D arg max min HH k Hk N k2K

(8.29)

for complexity reduction. Once the first user is chosen, we update as follows: Add k.1/ to˚ the index set of the selected users, K. KN ( KN n k.1/ . H.1/ D Hk.1/ . N The mth user can be (S2) Let m ( m C 1 and H.m/;k D H.m1/ Hk , k 2 K. chosen with the user index

k.m/ D arg max min GH (8.30) .m/;k G.m/;k ; (a) (b) (c)

N k2K

q iT h where G.m/;k is the LBR matrix of H.m/;k or Hex;.m/;k D HT.m/;k NE0s I (i.e., for LR-based ZF or LR-based MMSE, respectively). Once the mth user is found, we update as follows: (a) (b) (c)

Add k.m/ to the index set of the selected users, K. KN ( KN n k.m/ . H.m/ D H.m/;k.m/ .

(S3) If m D M , stop. Otherwise, go to (S2). On the other hand, with m D 1 and KN D f1; : : : ; Kg, we can conclude the LRG user selection algorithm for the MD criterion with the LR-based MMSE-SIC detector as follows: q iT h (S1) Denote by Gk the LBR matrix of Hk or Hex;k D HTk NE0s I (i.e., for LR-based ZF-SIC or LR-based MMSE-SIC, respectively). With the QR factorization of Gk D Qk Rk for a unitary matrix Qk and an upper triangular matrix Rk , the user index of the first user to be selected is obtained as

ˇ ˇ ˇ .k/ ˇ (8.31) k.1/ D arg max min ˇrq;q ˇ ; N k2K

.k/

q

where rq;q denotes the .q; q/-th element of Rk . Since P D 1 leads to a vector Hk , LR or QR operation in above becomes unnecessary, where (8.31) can be


195

replaced by (8.29) to reduce the complexity. Once the first user is chosen, we update as follows: (a) Add k.1/ to˚ the index set of the selected users, K. (b) KN ( KN n k.1/ . (c) H.1/ D Hk.1/ . N Let G.m/;k denote (S2) We update m ( m C 1 and H.m/;k D H.m1/ Hk , k 2 K. q iT h the LBR matrix of H.m/;k or Hex;.m/;k D HT.m/;k NE0s I (i.e., for LR-based ZF-SIC or LR-based MMSE-SIC, respectively), and then, the QR factorization of G.m/;k D Q.m/;k R.m/;k is employed, where Q.m/;k is unitary and R.m/;k is upper triangular. The mth user can be chosen with the user index

ˇ ˇ ˇ .m/;k ˇ k.m/ D arg max min ˇrq;q ˇ ; (8.32) N k2K

q

.m/;k

where rq;q represents the .q; q/-th element of R.m/;k . Once the mth user is found, we update as follows: (a) (b) (c)

Add k.m/ to the index set of the selected users, K. KN ( KN n k.m/ . H.m/ D H.m/;k.m/ .

(S3) If m D M , stop. Otherwise, go to (S2). Note that in both algorithms, the N mP complex-valued matrix H.m/ denotes the channel matrix for the first m selected users, while the N P complex-valued matrix Hk.m/ represents the channel matrix for the m-th selected user with the index ˚ k.m/ , where k.m/ 2 KN and KN D f1; : : : ; Kg n k.1/ ; : : : ; k.m1/ . Example 8.1. For example, consider the system shown in Fig. 8.1, where the LRG is employed to select users sequentially. If selection sequence of the 4 users becomes: User 7, User 5, User 9, User 4, we have k.1/ D 7, k.2/ D 5, k.3/ D 9, and k.4/ D 4. Furthermore, we can show that H.1/ D ŒH7 of size N P , H.2/ D ŒH7 H5 of size N 2P , H.3/ D ŒH7 H5 H9 of size N 3P , and H.4/ D ŒH7 H5 H9 H4 of size N 4P . In the LRG user selection, the number of required LR operations is given by ULRG D

M X

.K i C 1/;

(8.33)

i D1

where the matrix size for LR in selecting the mth user is N mP . Using the upper bound on the average complexity of LR analyzed in [25, 59, 60], the complexity of LRG can be upper bounded as CLRG D

M X .K i C 1/O .iP /3 N log.iP / : i D1

(8.34)

196 Table 8.1 Matrices with the LR at each user selection


1st user H1 H2 H3 H4 H5 H6 p H7 H8 H9 H10

2nd user

3rd user

ŒH7 ŒH7 ŒH7 ŒH7 ŒH7 ŒH7 ŒH7 ŒH7 ŒH7

ŒH7 ŒH7 ŒH7 ŒH7 ŒH7 ŒH7 ŒH7 ŒH7

H1 H2 H3 H4 p H5 H6 H8 H9 H10

H5 H5 H5 H5 H5 H5 H5 H5

H1 H2 H3 H4 H6 H8 p H9 H10

4th user ŒH7 ŒH7 ŒH7 ŒH7 ŒH7 ŒH7 ŒH7

H5 H9 H5 H9 H5 H9 H5 H9 H5 H9 H5 H9 H5 H9

H1 H2 H3 p H4 H6 H8 H10

Note that when P D 1, as no LR is required for the first user selection, (8.34) is reduced to M X .K i C 1/O .iP /3 N log.iP / : (8.35) CLRG D i D2

The number of required LR operations in the combinatorial user selection according to (8.26) or (8.25) is UCUS D D

KŠ .K M /Š M Y .K i C 1/;

(8.36)

i D1

where the matrix size for LR is always N MP and the overall complexity is upper bounded as M Y CCUS D .K i C 1/O .MP/3 N log.MP/ : (8.37) i D1

Comparing with (8.37), (8.34) or (8.35) shows a significant computational complexity reduction. However, since the LRG user selection does not jointly select M users, there will be performance loss. Example 8.2. According to Example 8.1, matrices to be performed with the LR at each user selection are presented in Table 8.1. In order to select the first user, User 7, a number of 10 matrices of size N P are carried out with the LR performed. At the second user selection, the LR is employed with 9 matrices of size N 2P , which results in User 5 is selected. After that, the LR is employed with 8 matrices of size N 3P to choose the third user, User 9. Finally, 7 matrices of size N 4P are computed with the LR to decide the forth one, User 4.


197

8.3.2 A Complexity Efficient Method for LR Updating We note that in the LRG user selection, the LR operation is repeatedly performed for each updated channel matrix. For example, at the mth user selection, an LR is carried out with the complex-valued channel matrix H.m/ D operation H.m1/ Hk as shown in (8.30), where Hk contains P newly added column vectors and the other .m1/P column vectors in H.m/ are already chosen and LBR. Instead of performing a new LR on all the mP column vectors in H.m/ , by utilizing the established .m 1/P LBR vectors, we can develop a computationally efficient LR updating method with new P column vectors, which is referred to as the updated basis LR (UBLR) in this chapter. The resulting user selection scheme is regarded as the UBLR-based greedy (UBLRG) user selection. Since UBLRG is a computationally efficient version of LRG, the performance of the LRG and UBLRG user selection schemes becomes the same. Thus, in order to study the performance of the LRG user selection, we can only consider UBLRG and assume that LRG and UBLRG are interchangeable. The UBLR algorithm is based on the CLLL algorithm in Subsect. 3.2.5. The CLLL algorithm is carried out to transform a given basis (i.e., a complex-valued channel matrix H.m/ of size N mP ) into a new basis consisting of nearly orthogonal basis vectors (i.e., a complex-valued matrix G.m/ of size N mP ), which can be further presented as L.G.m/ / D L.H.m/ / ” G.m/ D H.m/ U.m/

(8.38)

with a unimodular matrix U.m/ . The basis G.m/ in above is regarded as a reduced basis of a lattice with parameter ı if G.m/ is QR factorized as G.m/ D Q.m/ R.m/

(8.39)

for a unitary Q.m/ and an upper triangular R.m/ , while the elements of R.m/ satisfy the following inequalities [27]: j R.ŒR`; / j

1 1 j ŒR`;` j and j J.ŒR`; / j j ŒR`;` j; 2 2

1 ` < mP (8.40)

and ıjŒR1;1 j2 jŒR; j2 C jŒR1; j2 ;

D 2; : : : ; mP:

(8.41)

Here, ŒRp;q denotes the .p; q/-th element of R.m/ and the parameter ı is a factor selected to achieve a good quality-complexity trade-off [20]. In [27], it shows that ı can be chosen from . 14 ; 1/ and . 12 ; 1/ for the LLL and CLLL algorithms, respectively. In this chapter, we assume ı D 3=4 which is commonly considered for the complexity and performance trade-off.

198

8 Selection Criteria of Multiple Users Table 8.2 Size reduction in CLLL (15) for ` D 1 W 1 (16) dR.m/ . `; /=R.m/ . `; `/c (17) if ¤ 0 (18) R.m/ .1 W `; / R.m/ .1 W `; / R.m/ .1 W `; `/ U.m/ .W; / U.m/ .W; `/ (19) U.m/ .W; / (20) end if (21) end for

8.3.2.1 CLLL Algorithm Since the UBLR is based on the CLLL introduced in Subsect. 3.2.5, in order to build up the UBLR, the CLLL is characterized in details. With the CLLL, the matrix H.m/ is generated to a nearly orthogonal matrix G.m/ and a unitary matrix U.m/ . Let the QR factorization of H.m/ be carried out as H.m/ D Q0.m/ R0.m/ ;

(8.42)

where Q0.m/ is unitary and R0.m/ is upper triangular, let U0.m/ D ImP , and then let a set of matrices be n o A0.m/ D Q0.m/ ; R0.m/ ; U0.m/ : (8.43) Using A0.m/ as the input, the output of CLLL becomes ˚ A.m/ D Q.m/ ; R.m/ ; U.m/ ;

(8.44)

while the lattice reduced matrix G.m/ can be found as G.m/ D Q.m/ R.m/ D H.m/ U.m/ :

(8.45)

Since the CLLL algorithm is based on an iterative method, for the initialization, A.m/ is assigned by A0.m/ (i.e., Q.m/ Q0.m/, R.m/ R0.m/ , and U.m/ U0.m/ ) as the input and let D 2. Then, a version of CLLL algorithm is summarized as follows: (a) To fulfill (8.40), a size reduction is performed with the 1st to th columns of R.m/ and U.m/ , which is presented in Table 8.2. (b) As the basis of R.m/ is size reduced according to (8.40), let ( C 1 and go to Step (a) if (8.41) is fulfilled. Swap the . ˚ 1/th and th columns in R.m/ and U.m/ if (8.41) is not satisfied and update R.m/ ; Q.m/ . Let max. 1; 2/ and go to Step (a). Table 8.3 shows the algorithm in Step (b). Note that rows (23) and (26) in Table 8.3 are employed for the UBLR and will be discussed in the following part.


199

Table 8.3 Column swapping in CLLL (22) if ıj.R.m/ . 1; 1//j2 > jR.m/ .; /j2 C jR.m/ . 1; /j2 (23) .m/ .m/ C 1 (24) Swap the . 1/-th and th columns in R.m/ and U.m/ R.m/ .1;1/ ˛ D kR.m/ ˛ ˇ .1W;1/k (25) .m;.m/ / D with R .;1/ ˇ ˛ ˇ D kR.m/.m/ .1W;1/k (26) (27) (28) (29) (30) (31) (32)

”.m;.m/ / R.m/ . 1 W ; 1 W / .m;.m/ / R.m/ . 1 W ; 1 W / T Q.m/ .W; 1 W / Q.m/ .W; 1 W /.m; .m/ / maxf 1; 2g else C1 end if

(c) The algorithm is terminated if D mP . After the iterative˚operation, the lattice reduced matrix G.m/ is obtained with the updated A.m/ D Q.m/ ; R.m/ ; U.m/ as the output (i.e., refer to (8.45)). Note that in the LRG user selection at the mth user selection, we have H.m/ D H.m1/ Hk.m/ ;

(8.46)

where the CLLL has already been performed with H.m1/ at the .m 1/th user selection and of course its lattice-reduced matrix G.m1/ is available. Although the CLLL can be used to obtain the lattice reduced matrix G.m/ from H.m/ for the mth user selection, the knowledge of G.m1/ can be utilized to generate G.m/ with reduced complexity. This method is regarded as the UBLR.

8.3.2.2 UBLR Algorithm The UBLR algorithm is carried out to transform H.m/ into a reduced basis G.m/ by utilizing a given set of already available matrices, ˚ A.m1/ D Q.m1/ ; R.m1/ ; U.m1/ ;

(8.47)

associated with the lattice-reduced matrix G.m1/ in the previous .m 1/th user selection of the LRG, where G.m1/ D Q.m1/ R.m1/ D H.m1/ U.m1/ :

(8.48)

Here, the unimodular matrix U.m1/ is employed to present the column swapping in the CLLL, while R.m1/ satisfies (8.40) and (8.41). The transformation algorithm for generating G.m/ in UBLR is summarized as follows.

200


Table 8.4 Basis updating in UBLR (Part I)

U.m/ .1 W !; 1 W !/ U.m1/ Q.m/ Q.m1/ R.m/ .W; 1 W !/ R.m1/

(6) (7) (8)

Instead of starting the size reduction of R0.m/ with the first two columns (the 1st to th columns, where D 2 in Table 8.2), UBLR reduces the iteration by starting the size reduction with D .m 1/P C 1. In this case, the iteration of size reduction from that with D 2 to that with D .m 1/P C 1 needs to be obtained by updating A0.m/ from A.m1/ . Let the QR factorization of H.m1/ be H.m1/ D Q0.m1/ R0.m1/ ;

(8.49)

for a unitary Q0.m1/ and an upper triangular R0.m1/ . With U0.m1/ D I.m1/ , the set of matrices is given by n o A0.m1/ D Q0.m1/ ; R0.m1/ ; U0.m1/ :

(8.50)

Since R0.m1/ of size N P .m 1/ and R0.m/ of size N P m are both upper triangular, it is straightforward to show that R0.m1/ D R0.m/ .W; 1 W P .m 1//;

(8.51)

which results in that the size reduction and column swapping performed on the first P .m 1/ columns of R0.m/ are the same as those on R0.m1/ . Let A.m/ A0.m/ . By substituting the entries of R.m1/ into those of R.m/ as R.m/ .W; 1 W P .m 1//

R.m1/ ;

(8.52)

it is straightforward to show that the first to P .m 1/-th column vectors of R.m/ satisfying (8.40) and (8.41), while the ˚CLLL is partially performed on R.m/ can be updated with low with the basis updating approach. Similarly, Q ; U .m/ .m/ ˚ computational complexity from Q.m1/ ; U.m1/ as follows: Q.m/

Q.m1/ ;

U.m/ .1 W P .m 1/; 1 W P .m 1//

U.m1/ :

(8.53)

Letting ! D P .m 1/, the basis updating approach in above is presented in Table 8.4. Up to this point, matrices Q.m1/ and U.m1/ have been utilized to update Q.m/ and U.m/ , respectively, with low complexity. In addition, we note that R.m/ has been updated partially in row (8) of Table 8.4, since we do not consider updating R.m/ .1 W P .m 1/; P .m 1/ C 1 W P m/ in A.m/ . It can be observed that when a CLLL is performed on H.m/ with the same operations of the CLLL for previous user

8.3 LR-Based Greedy User Selection Using an Updating Method Table 8.5 Basis updating in UBLR (Part II)

201

(9) for ` D1 W .m1/ (10) R.m/ .m1;`/ 1 W .m1;`/ ; ! C 1 W .m1;`/ R.m/ .m1;`/ 1 W .m1;`/ ; ! C 1 W (11) end for

selections, R.m/ .1 W P .m 1/; P .m 1/ C 1 W P m/ will also be influenced. Thus, extra processing is necessary to recover R.m/ .1 W P .m 1/; P .m 1/ C 1 W P m/ in A.m/ . To this end, we define that ˚ B.m1/ D .m1/ ; .m1/ ; .m1/ ;

(8.54)

where ˚ .m1/ D .m1;1/ ; ; .m1;/ ; ˚ .m1/ D .m1;1/ ; ; .m1;/ ; .m1/ D :

(8.55)

The operations of swapping and updating R.m1/ and Q.m1/ are kept in .m1/ , .m1/ , and .m1;/ , where .m1/ keeps the number of swapping times, .m1/ keeps those columns involved in the swaps, and .m1;/ keeps the operations of column swaps. With the operation of row (27) in Table 8.3, we note that R.m/ .1 W P .m 1/; P .m 1/ C 1 W P m/ is generated by a transformation with .m/ . Thus, using the information kept in B.m1/, R.m/ .1 W P .m 1/; P .m 1/ C 1 W P m/ is updated in Table 8.5, where ` D P m. Consequently, using the basis updating approach in Tables 8.4 and 8.5, the UBLR is carried out to update the matrices in A.m/ with relatively low computational complexity. With an updated A.m/ , one CLLL can be carried out to generate the reduced basis G.m/ . The calculation of this new basis generation starts with D .m 1/P C 1. Hence, the computational complexity of UBLR is evidently reduced as compared to employing one CLLL starting with D 2. Note that since UBLR and CLLL generate the same reduced matrix G.m/ , they provide the same performance. In Sect. 8.4, this performance of UBLRG is validated by simulations. Using different segments provided in Tables 8.2–8.5, the UBLR algorithm of the mth user selection is summarized in Table 8.6. The input and output of the algorithm are given by ˚ Input W A.m1/ ; B.m1/ ; H.m1/ ; Hk.m/ (8.56) and

˚ Output W A.m/ ; B.m/ ;

(8.57)

respectively. Note that for the first user selection, taking its channel matrix Hk.1/ as the input, instead of using the UBLR, one CLLL is carried out to generate

202


Table 8.6 The UBLR (based on the CLLL) algorithm at the mth user selection

INPUT: fA.m1/ ; B.m1/ ; H.m1/ ; Hk.m/ g. OUTPUT: fA.m/ ; B.m/ g. (1) H.m/ H.m1/ Hk.m/ (2) ! size.H.m1/ ; 2/ (3) size.H.m/ ; 2/ (4) ŒQ.m/ R.m/ qr.H.m/ / I (5) U.m/ :: : Basis updating (Part I) in Table 8.4. :: : Basis updating (Part II) in Table 8.5. :: : (12) !C1 (13) .m/ 0 (14) while :: : Size reduction in Table 8.2. :: : Column swapping in Table 8.3. :: : (33) end while

Table 8.7 The average value of in a multiuser MIMO system when the CLLL-based MMSE-SIC detector is used with the LRG and UBLRG user selection, where K D 10, N D 8, and .M; P / D .8; 1/

Average value of mP LRG 2 0.2909 3 0.9029 4 1.8022 5 3.0633 6 4.7711 7 7.2925 8 12.1228 Sum 30.2457

UBLRG 0.2904 0.5851 0.8940 1.2708 1.7653 2.5620 4.7728 12.1404

fA.1/ ; B.1/ g as the output. Since the output of the mth user selection is regarded as the input at the .m C 1/-th user selection, the algorithm is recursively carried out from m D 2. The algorithm is terminated as m D M . The complexity of CLLL and UBLR algorithms highly depends on the number of column swapping, which is recorded by the parameter in the output. In Tables 8.7 and 8.8, the average values of per iteration are presented for the two possible cases of .M; P / D .8; 1/ and .M; P / D .4; 2/, respectively, in a multiuser MIMO system when the CLLL-based MMSE-SIC detector is used with the LRG and UBLRG user selection. Note that K D 10 and N D 8 are considered in the system. Based

8.4 Diversity Analysis and Numerical Results Table 8.8 The average value of in a multiuser MIMO system when the CLLL-based MMSE-SIC detector is used with the LRG and UBLRG user selection, where K D 10, N D 8, and .M; P / D .4; 2/

203

Average value of mP LRG 2 0.2926 4 1.7977 6 4.7663 8 12.0856 Sum 18.9422

UBLRG 0.2879 1.4952 3.0191 7.3761 12.1783

on these results, we can observe that the complexity is significantly reduced when UBLR is employed. We also note that with the LRG, the complexity for the case of .M; P / D .8; 1/ is higher than that of .M; P / D .4; 2/ as expected, where a large M implies higher complexity. We can also show that the complexity of UBLRG is upper bounded as 1 X M O .iP /3 N log.iP / : (8.58) CUBLRG D .K M C1/O .MP/3 N log.MP/ C i D1

Compared to the complexity of LRG which is upper bounded as CLRG D

M X

.K i C 1/O .iP /3 N log.iP / ;

(8.59)

i D1

the UBLRG scheme has lower complexity, especially when large K and M are considered. Using the real-valued LLL algorithm which is regarded as the LLL in Subsection 3.2.5, UBLR can also be performed with a real-valued channel matrix. Due to the derivation of such a UBLR is straightforward, we do not discuss it any further. Furthermore, the simulation results in Sect. 8.4 show the multiuser MIMO systems whose user selection using the LLL provide the same performance as those using the CLLL. In addition, as will be shown in Table 8.10, LLL-based UBLR algorithm can also be adopted to reduce the computational complexity of LLL-based LRG. Note that although the users are selected sequentially with the LRG or UBLRG, we need to detect all the signals from the selected users jointly for reasonably good performance, since the joint detection can be efficiently carried out with the LRbased detection [21, 24–26]. In addition, as will be shown by simulation results in Subsect. 8.4.2, the performance degradation due to the greedy user selection is not significant as long as joint detection is performed.

8.4 Diversity Analysis and Numerical Results In this section, we consider the diversity gain of the combinatorial user selection approaches with different detectors, including the ML, linear, and LR-based SIC detectors. We derive lower bounds on the diversity gain of those schemes. Since

204


the analysis of diversity gain with the proposed greedy user selection approach is difficult, we rely on simulations, from which we can show that our proposed LRG/UBLRG user selection approach has a similar diversity gain and comparable performance to the combinatorial one. Throughout this section, we assume that the elements of the channel matrix HK are independent zero-mean CSCG random variables with variance h2 .

8.4.1 Diversity Gain Analysis from Error Probability Through the following analysis of diversity gain, we can see the impact of different MIMO detectors with the corresponding user selection schemes on the performance of multiuser systems. Suppose that s.1/ is transmitted while s.2/ is erroneously detected, where s.i / 2 S MP and s.1/ ¤ s.2/ . Let d D s.1/ s.2/ . Using the PEP, we can show the diversity order obtained by multiple receive antennas as well as multiple user selection.

8.4.1.1 Diversity Gain of Combinatorial User Selection with ML Detector Theorem 8.1. The average PEP of the ML detector with the M selected users ml under the MDist user selection criterion in Sect. 8.2, denoted by Pe , is upper bounded as !

2 2 N b MK cC1

2 2 N b MK c ml k h dk k h dk Pe c1 Co ; (8.60) N0 N0 where c1 > 0 is constant. Proof. Consider M among K users are selected by the combinatorial user selection scheme under the MDist criterion. Suppose that signals from the selected M users are jointly detected with the N MP channel matrix HK using the ML detector. The PEP in detecting M users’ signals has the following upper bound [44]: 1 0s N 2 kH K dk A Pr s.1/ ! s.2/ erfc @ ; (8.61) 2N0 where dN D arg min kHK dk2

(8.62)

˚ D D d D s s0 j s ¤ s0 2 S MP ZMP C j ZMP :

(8.63)

d2D;d¤0

for

8.4 Diversity Analysis and Numerical Results

205

Note that erfc.x/ denotes the complementary error function of x, where Z

2 erfc.x/ D p

C1

2

ez dz:

(8.64)

x

Denote by D.HK / the length of the shortest nonzero vector of the lattice generated by HK . Then, we have

0s

Pr s.1/ ! s.2/ erfc @

1 D2 .HK / A; 2N0

(8.65)

where N D.HK / D kHK dk:

(8.66)

Since the MDist criterion in (8.9) is employed, it can be shown that

0s

Pr s.1/ ! s.2/ erfc @

1 maxK D2 .HK / A; 2N0

(8.67)

where max D 2 .HK / D max min dH HH K HK d: K

K

(8.68)

d2D;d¤0

Let wK D HK d. Note that wK is a zero-mean CSCG random vector and 2 2 E wK wH K D h kdk I:

(8.69)

Furthermore, we can show that XK D jjwK jj2 is a chi-square random variable with 2N degrees of freedom, while its pdf becomes fX .xK / D

1 . h2 kdk2 /N .N

2

1/Š

N 1 xK =. h kdk xK e

2/

(8.70)

and the cdf is given by 2

FX .xK / D 1 exK =. h kdk

2/

N 1 X qD0

.xK =. h2 kdk2 //q : qŠ

(8.71)

To obtain an upper bound on the error probability, we note that the number of alternative combinations of the channel matrices, which are statistically independent with each other, for selecting HK with the MDist selection in (8.9) is at K K least b M c. Let HK1 ; HK2 ; : : : ; HKb K c represent such b M c independent alternative M

206


K combinations of the channel matrices. As a result, there are at least b M c of wK , i.e., wK1 ; wK2 ; : : : ; wKb K c , which are independent. Let M

o n V D max X1 ; X2 ; : : : ; Xb K c ; M

(8.72)

K c. Using order statistics, the pdf of V where Xm D jjwKm jj2 for m D 1; 2; : : : ; b M is given by b K c1

fV .v/ D KFX M D c10 v

K NbM

.v/fX .v/

c1

K

C o.vN b M c1C" /;

(8.73)

where c10 > 0 is a constant and " > 0. Therefore, according to [79], it can be derived as 0s 2 13 X maxK dH HH H d ml K K A5 EV 4erfc @ Pe 2N0 d2D;d¤0

D c1

k h2 dk2 N0

N b MK c

0 1 2 2 N b MK cC1 k h dk A; Co@ N0

(8.74)

where c10 and c1 are proportional to each other [79]. This completes the proof.

t u

According to Theorem 8.1, we can show that a full receive diversity gain of N K together with a partial multiuser diversity gain of at least b M c can be achieved by the ML detectors under the MDist user selection criterion. This result is obtained under K the fact that there are at least b M c statistically independent alternative combinations of the composite channel matrix HK for M users. Thus, this result becomes a lower bound on the diversity gain. Actually, there could be more combinations for HK , which are not independent, that can increase the multiuser diversity gain. By simulations, we will further discuss the impact of these combinatorial matrices HK that are not independent. 8.4.1.2 Diversity Gain of Combinatorial User Selection with Linear Detector Theorem 8.2. The average PEP of the linear detector (i.e., the ZF or MMSE detector) with the selected M users under the ME user selection criterion in linear Sect. 8.2, denoted by Pe , is upper bounded as !

2 2 .N P C1/b MK c

2 2 .N P C1/b MK cC1 linear

h kdk

h kdk ; (8.75) Co Pe c2 N0 N0 where c2 > 0 is constant.


207

Proof. It can be shown that under the ME criterion, for a given HK , an upper bound on the error probability in detecting M users’ signals is given by [43] 0s erfc @

mmse

Pe

0s D erfc @ 0s D erfc @

1 2 maxK min .HH H /jjdjj K K A 2N0 1

h2 jjdjj2 maxK XQK A 2N0

1

h2 jjdjj2 V A; 2N0

(8.76)

2 Q where XQK D min .HH K HK /= h and V D maxK XK . According to [79], using the pdf of V (i.e., the approach used to prove Theorem 8.1), it can be deduced that linear

Pe

D EHK ŒPr s.1/ ! s.2/ 2 13 0s

h2 jjdjj2 V A5: EV 4erfc @ 2N0

(8.77)

Similar to the proof of Theorem 8.1, with independent alternative combinations of the channel matrices HK1 ; HK2 ; : : : ; HKb K c , we can follow the derivations in M [48, 79] and obtain that 0s 2 13

h2 jjdjj2 V linear A5 Pe EV 4erfc @ 2N0 Z

C1

0s erfc @

0

D c2

h2 kdk2 N0

1

h2 jjdjj2 v A 2N0

fV .v/d v

.N P C1/b MK c

0 1 2 .N P C1/b MK cC1

h kdk2 A; Co@ N0 (8.78)

where c2 > 0 is constant. This completes the proof.

t u

208


Theorem 8.2 shows that for the linear detector, the ME user selection criterion may not be able to exploit a full receive diversity gain. However, a partial multiuser K diversity gain of at least b M c can be achieved. 8.4.1.3 Diversity Gain of Combinatorial User Selection with LR-Based SIC Detector Theorem 8.3. The average PEP of the LR-based SIC detector with the selected M lr users under the MD user selection criterion in Sect. 8.2, denoted by Pe , is upperbounded as !

2 2 N b MK cC1

2 2 N b MK c lr k h dk k h dk Co ; (8.79) Pe c3 N0 N0 where c3 > 0 is constant. Proof. In the LR algorithm, the given channel matrix H is transformed into a new basis G. This transformation can also be presented as L.G/ D L.H/ ” G D HT;

(8.80)

where T is an integer unimodular matrix and L.A/ denotes the lattice generated by A. In (8.80), G is called LLL-reduced basis with parameter ı if G is QR factorized as G D QR for a unitary Q and an upper triangular R, while the elements of R satisfy (8.40) and (8.41) as m D M . Let (8.41) be rewritten as ı j r; j2 j r;C1 j2 C j rC1;C1 j2 ; and let

D 1; 2; : : : ; MP 1

1 1 4 > : ˇD ı 4 3

(8.81)

(8.82)

Then, we can obtain the following inequalities: j rC1;C1 j2 ˇ1 j r; j2

(8.83)

min j r; j2 ˇ MPC1 j r1;1 j2 :

(8.84)

and

Since G D QR, we can easily show j r1;1 j2 D kg1 k2 and kg1 k2 min kHdk2 D D2 .H/; d2D;d¤0

(8.85)


209

where g1 denotes the first column vector of G. Then, we have min j r; j2 ˇMPC1 D2 .H/:

(8.86)

Furthermore, in the proposed user selection for selecting M users with the LR-based SIC detector, (8.86) becomes min j r; j2 ˇ MPC1 D2 .HK /;

(8.87)

where K is the index set of the selected users. Note that the LR-based SIC detection is performed with (8.24). Let n denote the th element of nQ in (8.24). Then, the LR-based SIC detection does not have error across all the layers if we have jn j 1 < jr; j 2

or jn j2

Low Complexity MIMO Detection

Low power and low complexity constant multiplication using serial arithmetic

Low power and low complexity shift-and-add based computations

Detection of low-level optical signals

MIMO Signals and Systems

MIMO Radar Signal Processing

Complexity

MIMO Wireless Communications

Acoustic MIMO Signal Processing

Complexity

Ultra Wideband Systems with MIMO

MIMO Communication for Cellular Networks

Coding for MIMO Communication Systems

Wireless Communications over MIMO Channels

MIMO Systems, Theory and Applications

MIMO: From Theory to Implementation

Near-Capacity Multi-Functional MIMO Systems: Sphere-Packing, Iterative Detection and Cooperation (Wiley - IEE)

Detection Unlimited

Quickest detection

MIMO Antenna Technology for Wireless Communications

Multiuser Detection

Detection Unlimited

Communication complexity

Fault Detection

Detection Unlimited

Detection Unlimited

Malware Detection

Global Complexity

Communication Complexity

Low Complexity MIMO Detection

Low power and low complexity constant multiplication using serial arithmetic

Low power and low complexity shift-and-add based computations

Detection of low-level optical signals

MIMO Signals and Systems

MIMO Radar Signal Processing

Complexity

MIMO Wireless Communications

Acoustic MIMO Signal Processing

Complexity

Ultra Wideband Systems with MIMO

MIMO Communication for Cellular Networks

Coding for MIMO Communication Systems

Wireless Communications over MIMO Channels

MIMO Systems, Theory and Applications

MIMO: From Theory to Implementation

Near-Capacity Multi-Functional MIMO Systems: Sphere-Packing, Iterative Detection and Cooperation (Wiley - IEE)

Detection Unlimited

Quickest detection

MIMO Antenna Technology for Wireless Communications

Multiuser Detection

Detection Unlimited

Communication complexity

Fault Detection

Detection Unlimited

Detection Unlimited

Malware Detection

Global Complexity

Communication Complexity

Recommend Documents