Random Matrix Theory and Wireless Communications Antonia M.Tulino Dept.Ingegneria Elettronica e delle Telecomunicazioni Universita degli Studi di Napoli"Federico II" Naples 80125,Italy atulino@ee.princeton.edu Sergio Verdu Dept.Electrical Engineering Princeton University Princeton,New Jersey 08544,USA verdu@princeton.edu now the essence of knowledge
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 TrendsTM in ∩Ow Communications and Information Theory the essence of knowledge Vol1,No1(2004)1-182 C 2004 A.M.Tulino and S.Verdui Random Matrix Theory and Wireless Communications Antonia M.Tulinol,Sergio Verdu2 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
Random Matrix Theory and Wireless Communications Antonia M. Tulino1, Sergio Verd´u2 Abstract Random matrix theory has found many applications in physics, statistics 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, multivariate 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 application of random matrix theory to the fundamental limits of wireless communication channels is described in depth. 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 Foundations and Trends™ in Communications and Information Theory Vol 1, No 1 (2004) 1-182 © 2004 A.M. Tulino and S. Verd´u
Table of Contents Section 1 Introduction 5 1.1 Wireless Channels 6 1.2 The Role of the Singular Values 6 1.3 Random Matrices:A Brief Historical Account 1 Section 2 Random Matrix Theory 21 2.1 Types of Matrices and Non-Asymptotic Results 21 2.2 Transforms 38 2.3 Asymptotic Spectrum Theorems 52 2.4 Free Probability 4 2.5 Convergence Rates and Asymptotic Normality 91 Section 3 Applications to Wireless Communications 96 3.1 Direct-Sequence CDMA 96 3.2 Multi-Carrier CDMA 117 3.3 Single-User Multi-Antenna Channels 129 3.4 Other Applications 152 Section 4 Appendices 153 4.1 Proof of Theorem 2.39 153 4.2 Proof of Theorem 2.42 154 4.3 Proof of Theorem 2.44 156 4.4 Proof of Theorem 2.49 158 4.5 Proof of Theorem 2.53 159 References 163 2
Table of Contents Section 1 Introduction 3 1.1 Wireless Channels 5 1.2 The Role of the Singular Values 6 1.3 Random Matrices: A Brief Historical Account 13 Section 2 Random Matrix Theory 21 2.1 Types of Matrices and Non-Asymptotic Results 21 2.2 Transforms 38 2.3 Asymptotic Spectrum Theorems 52 2.4 Free Probability 74 2.5 Convergence Rates and Asymptotic Normality 91 Section 3 Applications to Wireless Communications 96 3.1 Direct-Sequence CDMA 96 3.2 Multi-Carrier CDMA 117 3.3 Single-User Multi-Antenna Channels 129 3.4 Other Applications 152 Section 4 Appendices 153 4.1 Proof of Theorem 2.39 153 4.2 Proof of Theorem 2.42 154 4.3 Proof of Theorem 2.44 156 4.4 Proof of Theorem 2.49 158 4.5 Proof of Theorem 2.53 159 References 163 2
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 Marcenko 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
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 motivated 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,n,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 n 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-
4 Introduction dom matrix theory with particular emphasis on asymptotic theorems on the distribution of eigenvalues and singular values under various assumptions 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 application. Section 1.2 motivates interest in large random matrix theory by focusing on two performance measures of engineering interest: Shannon capacity and linear minimum mean-square error, which are determined 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 results in random matrix theory, from the work of Wishart on Gaussian matrices with fixed dimension, to the recent results on asymptotic spectra. Section 2 gives a tutorial account of random matrix theory. Section 2.1 focuses on the major types of random matrices considered in the literature, 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-