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Random Matrix Theory and Wireless Communications Antonia M. Tulino Dept. Ingegneria Elettronica e delle Telecomunicazioni Universit´a degli Studi di Napoli ”Federico II” Naples 80125, Italy atulino@ee.princeton.edu Sergio Verd´u Dept. Electrical Engineering Princeton University Princeton, New Jersey 08544, USA verdu@princeton.edu

Foundations and Trends TM in Communications and Information Theory Published, sold and distributed by: PO Box 179 2600 AD Delft The Netherlands Tel: +31-6-51115274 www.nowpublishers.com sales@nowpublishers.com in North America: now Publishers Inc. PO Box 1024 Hanover, MA 02339 USA Tel. +1-781-985-4510 Printed on acid-free paper ISSNs: Paper version 1567-2190; Electronic version 1567-2328 c 2004 A.M. Tulino and S. Verd´u All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior written permission of the publishers. Now Publishers Inc. has an exclusive licence to publish this mate- rial worldwide. Permission to use this content must be obtained from the copyright licence holder. Please apply to now Publishers, PO Box 179, 2600 AD Delft, The Netherlands; www.nowpublishers.com; e-mail: sales@nowpublishers.com Printed in Great Britain by Antony Rowe Limited.

Foundations and Trends™ in Communications and Information Theory Vol 1, No 1 (2004) 1-182 © 2004 A.M. Tulino and S. Verd´u Random Matrix Theory and Wireless Communications Antonia M. Tulino1, Sergio Verd´u2 1 Dept. Ingegneria Elettronica e delle Telecomunicazion, i Universita degli Studi di Napoli “Federico II”, Naples 80125, Italy 2 Dept. Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA Abstract Random matrix theory has found many applications in physics, statis- tics and engineering since its inception. Although early developments were motivated by practical experimental problems, random matrices are now used in ﬁelds as diverse as Riemann hypothesis, stochastic diﬀerential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivari- ate statistics, information theory, signal processing and small-world networks. This article provides a tutorial on random matrices which provides an overview of the theory and brings together in one source the most signiﬁcant results recently obtained. Furthermore, the appli- cation of random matrix theory to the fundamental limits of wireless communication channels is described in depth.

Table of Contents Section 1 Introduction 1.1 Wireless Channels 1.2 The Role of the Singular Values 1.3 Random Matrices: A Brief Historical Account Section 2 Random Matrix Theory 2.1 Types of Matrices and Non-Asymptotic Results 2.2 Transforms 2.3 Asymptotic Spectrum Theorems 2.4 Free Probability 2.5 Convergence Rates and Asymptotic Normality Section 3 Applications to Wireless Communications 3.1 Direct-Sequence CDMA 3.2 Multi-Carrier CDMA 3.3 Single-User Multi-Antenna Channels 3.4 Other Applications Section 4 Appendices 4.1 Proof of Theorem 2.39 4.2 Proof of Theorem 2.42 4.3 Proof of Theorem 2.44 4.4 Proof of Theorem 2.49 4.5 Proof of Theorem 2.53 References 2 3 5 6 13 21 21 38 52 74 91 96 96 117 129 152 153 153 154 156 158 159 163

1 Introduction From its inception, random matrix theory has been heavily inﬂuenced by its applications in physics, statistics and engineering. The landmark contributions to the theory of random matrices of Wishart (1928) [311], Wigner (1955) [303], and Mar˘cenko and Pastur (1967) [170] were moti- vated to a large extent by practical experimental problems. Nowadays, random matrices ﬁnd applications in ﬁelds as diverse as the Riemann hypothesis, stochastic diﬀerential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivariate statistics, information theory, signal processing, and small-world networks. Despite the widespread applicability of the tools and results in random matrix theory, there is no tutorial reference that gives an accessible overview of the classical theory as well as the recent results, many of which have been obtained under the umbrella of free probability theory. In the last few years, a considerable body of work has emerged in the communications and information theory literature on the fundamental limits of communication channels that makes substantial use of results in random matrix theory. The purpose of this monograph is to give a tutorial overview of ran- 3

4 Introduction dom matrix theory with particular emphasis on asymptotic theorems on the distribution of eigenvalues and singular values under various as- sumptions on the joint distribution of the random matrix entries. While results for matrices with ﬁxed dimensions are often cumbersome and oﬀer limited insight, as the matrices grow large with a given aspect ratio (number of columns to number of rows), a number of powerful and appealing theorems ensure convergence of the empirical eigenvalue distributions to deterministic functions. The organization of this monograph is the following. Section 1.1 introduces the general class of vector channels of interest in wireless communications. These channels are characterized by random matrices that admit various statistical descriptions depending on the actual ap- plication. Section 1.2 motivates interest in large random matrix theory by focusing on two performance measures of engineering interest: Shan- non capacity and linear minimum mean-square error, which are deter- mined by the distribution of the singular values of the channel matrix. The power of random matrix results in the derivation of asymptotic closed-form expressions is illustrated for channels whose matrices have the simplest statistical structure: independent identically distributed (i.i.d.) entries. Section 1.3 gives a brief historical tour of the main re- sults in random matrix theory, from the work of Wishart on Gaussian matrices with ﬁxed dimension, to the recent results on asymptotic spec- tra. Section 2 gives a tutorial account of random matrix theory. Section 2.1 focuses on the major types of random matrices considered in the lit- erature, as well on the main ﬁxed-dimension theorems. Section 2.2 gives an account of the Stieltjes, η, Shannon, Mellin, R- and S-transforms. These transforms play key roles in describing the spectra of random matrices. Motivated by the intuition drawn from various applications in communications, the η and Shannon transforms turn out to be quite helpful at clarifying the exposition as well as the statement of many results. Considerable emphasis is placed on examples and closed-form expressions. Section 2.3 uses the transforms deﬁned in Section 2.2 to state the main asymptotic distribution theorems. Section 2.4 presents an overview of the application of Voiculescu’s free probability theory to random matrices. Recent results on the speed of convergence to the asymptotic limits are reviewed in Section 2.5. Section 3 applies the re-

1.1. Wireless Channels 5 sults in Section 2 to the fundamental limits of wireless communication channels described by random matrices. Section 3.1 deals with direct- sequence code-division multiple-access (DS-CDMA), with and without fading (both frequency-ﬂat and frequency-selective) and with single and multiple receive antennas. Section 3.2 deals with multi-carrier code- division multiple access (MC-CDMA), which is the time-frequency dual of the model considered in Section 3.1. Channels with multiple receive and transmit antennas are reviewed in Section 3.3 using models that incorporate nonideal eﬀects such as antenna correlation, polarization, and line-of-sight components. 1.1 Wireless Channels The last decade has witnessed a renaissance in the information theory of wireless communication channels. Two prime reasons for the strong level of activity in this ﬁeld can be identiﬁed. The ﬁrst is the grow- ing importance of the eﬃcient use of bandwidth and power in view of the ever-increasing demand for wireless services. The second is the fact that some of the main challenges in the study of the capacity of wireless channels have only been successfully tackled recently. Fading, wideband, multiuser and multi-antenna are some of the key features that characterize wireless channels of contemporary interest. Most of the information theoretic literature that studies the eﬀect of those fea- tures on channel capacity deals with linear vector memoryless channels of the form y = Hx + n (1.1) where x is the K-dimensional input vector, y is the N-dimensional output vector, and the N-dimensional vector n models the additive circularly symmetric Gaussian noise. All these quantities are, in gen- eral, complex-valued. In addition to input constraints, and the degree of knowledge of the channel at receiver and transmitter, (1.1) is char- acterized by the distribution of the N × K random channel matrix H whose entries are also complex-valued. The nature of the K and N dimensions depends on the actual ap- plication. For example, in the single-user narrowband channel with nT

6 Introduction and nR antennas at transmitter and receiver, respectively, we identify K = nT and N = nR; in the DS-CDMA channel, K is the number of users and N is the spreading gain. In the multi-antenna case, H models the propagation coeﬃcients between each pair of transmit-receive antennas. In the synchronous DS- CDMA channel, in contrast, the entries of H depend on the received signature vectors (usually pseudo-noise sequences) and the fading coef- ﬁcients seen by each user. For a channel with J users each transmitting with nT antennas using spread-spectrum with spreading gain G and a receiver with nR antennas, K = nTJ and N = nRG. Naturally, the simplest example is the one where H has i.i.d. entries, which constitutes the canonical model for the single-user narrowband multi-antenna channel. The same model applies to the randomly spread DS-CDMA channel not subject to fading. However, as we will see, in many cases of interest in wireless communications the entries of H are not i.i.d. 1.2 The Role of the Singular Values Assuming that the channel matrix H is completely known at the re- ceiver, the capacity of (1.1) under input power constraints depends on the distribution of the singular values of H. We focus in the simplest setting to illustrate this point as crisply as possible: suppose that the entries of the input vector x are i.i.d. For example, this is the case in a synchronous DS-CDMA multiaccess channel or for a single-user multi-antenna channel where the transmitter cannot track the channel. The empirical cumulative distribution function of the eigenvalues (also referred to as the spectrum or empirical distribution) of an n × n Hermitian matrix A is denoted by Fn A deﬁned as1 1{λi(A) ≤ x}, 1 n n i=1 Fn A(x) = (1.2) where λ1(A), . . . , λn(A) are the eigenvalues of A and 1{·} is the indi- cator function. A converges as n → ∞, then the corresponding limit (asymptotic empirical distribution 1 If Fn or asymptotic spectrum) is simply denoted by FA(x).

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