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-
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 x 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 nr
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 directsequence 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 codedivision 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 growing 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 features 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 general, complex-valued. In addition to input constraints, and the degree of knowledge of the channel at receiver and transmitter, (1.1) is characterized 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 application. For example, in the single-user narrowband channel with nT
6 Introduction and nr antennas at transmitter and receiver,respectively,we identify K=nr 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 nr antennas using spread-spectrum with spreading gain G and a receiver with nR antennas,K =nrJ 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 x n Hermitian matrix A is denoted by FA defined as! FA)=∑1A(A)≤， (1.2) i=1 where A1(A),...,An(A)are the eigenvalues of A and 1f.}is the indi- cator function 1IfF converges as n,then the corresponding limit(asymptotic empirical distribution or asymptotic spectrum)is simply denoted by FA(z)
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 DSCDMA 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 receiver, 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 Fn A(x) = 1 n n i=1 1{λi(A) ≤ x}, (1.2) where λ1(A),...,λn(A) are the eigenvalues of A and 1{·} is the indicator function. 1 If Fn A converges as n → ∞, then the corresponding limit (asymptotic empirical distribution or asymptotic spectrum) is simply denoted by FA(x)
1.2.The Role of the Singular Values 7 Now,consider an arbitrary N x K matrix H.Since the nonzero singular values of H and Ht are identical,we can write NF()-Nu(z)=KFRtH()-Ku(z) (1.3) where u(z)is the unit-step function (u(x)=0,x0). With an i.i.d.Gaussian input,the normalized input-output mutual information of (1.1)conditioned on H is2 ()logdet (I+SNR HH 1 1 (1.4) 片立s（1+sm) og（1+s5N到dF)回 (1.5) o with the transmitted signal-to-noise ratio (SNR) NE2] SNR= KE] (1.6) and with Ai(HHT)equal to the ith squared singular value of H. If the channel is known at the receiver and its variation over time is stationary and ergodic,then the expectation of(1.4)over the dis- tribution of H is the channel capacity (normalized to the number of receive antennas or the number of degrees of freedom per symbol in the CDMA channel).More generally,the distribution of the random variable (1.4)determines the outage capacity (e.g.[22]) Another important performance measure for (1.1)is the minimum mean-square-error (MMSE)achieved by a linear receiver,which deter- mines the maximum achievable output signal-to-interference-and-noise 2The celebrated log-det formula has a long history:In 1964,Pinsker [204]gave a general log-det formula for the mutual information between jointly Gaussian random vectors but did not particularize it to the linear model(1.1).Verdu [270]in 1986 gave the explicit form (1.4)as the capacity of the synchronous DS-CDMA channel as a function of the signature vectors.The 1991 textbook by Cover and Thomas [47]gives the log-det formula for the capacity of the power constrained vector Gaussian channel with arbitrary noise covariance matrix.In the mid 1990s,Foschini [77]and Telatar [250]gave (1.4)for the multi-antenna channel with i.i.d.Gaussian entries.Even prior to those works,the conventional analyses of Gaussian channels with memory via vector channels(e.g.[260,31])used the fact that the capacity can be expressed as the sum of the capacities of independent channels whose signal-to-noise ratios are governed by the singular values of the channel matrix
1.2. The Role of the Singular Values 7 Now, consider an arbitrary N × K matrix H. Since the nonzero singular values of H and H† are identical, we can write NFN HH† (x) − Nu(x) = KFK H†H(x) − Ku(x) (1.3) where u(x) is the unit-step function (u(x) = 0, x ≤ 0; u(x) = 1, x > 0). With an i.i.d. Gaussian input, the normalized input-output mutual information of (1.1) conditioned on H is2 1 N I(x; y|H) = 1 N log det I + SNR HH† (1.4) = 1 N N i=1 log 1 + SNR λi(HH† ) = ∞ 0 log (1 + SNR x) dFN HH† (x) (1.5) with the transmitted signal-to-noise ratio (SNR) SNR = NE[||x||2] KE[||n||2] , (1.6) and with λi(HH†) equal to the ith squared singular value of H. If the channel is known at the receiver and its variation over time is stationary and ergodic, then the expectation of (1.4) over the distribution of H is the channel capacity (normalized to the number of receive antennas or the number of degrees of freedom per symbol in the CDMA channel). More generally, the distribution of the random variable (1.4) determines the outage capacity (e.g. [22]). Another important performance measure for (1.1) is the minimum mean-square-error (MMSE) achieved by a linear receiver, which determines the maximum achievable output signal-to-interference-and-noise 2 The celebrated log-det formula has a long history: In 1964, Pinsker [204] gave a general log-det formula for the mutual information between jointly Gaussian random vectors but did not particularize it to the linear model (1.1). Verd´u [270] in 1986 gave the explicit form (1.4) as the capacity of the synchronous DS-CDMA channel as a function of the signature vectors. The 1991 textbook by Cover and Thomas [47] gives the log-det formula for the capacity of the power constrained vector Gaussian channel with arbitrary noise covariance matrix. In the mid 1990s, Foschini [77] and Telatar [250] gave (1.4) for the multi-antenna channel with i.i.d. Gaussian entries. Even prior to those works, the conventional analyses of Gaussian channels with memory via vector channels (e.g. [260, 31]) used the fact that the capacity can be expressed as the sum of the capacities of independent channels whose signal-to-noise ratios are governed by the singular values of the channel matrix.
8 Introduction ratio(SINR).For an i.i.d.input,the arithmetic mean over the users(or transmit antennas)of the MMSE is given,as function of H,by [271] (1.7) K 1 台I+SNR(H可 (1.8) 0 1+SNRT dF(z) N oo 1 dt(）- N-K K Jo 1+SNR K (1.9) where the expectation in(1.7)is over x and n while(1.9)follows from (1.3).Note,incidentally,that both performance measures as a function of sNR are coupled through d SNR- og et(+sHIr）=K-{在+H)} As we see in (1.5)and (1.9),both fundamental performance measures (capacity and MMSE)are dictated by the distribution of the empirical (squared)singular value distribution of the random channel matrix. In the simplest case of H having i.i.d.Gaussian entries,the density function corresponding to the expected value of Fcan be expressed explicitly in terms of the Laguerre polynomials.Although the integrals in (1.5)and (1.9)with respect to such a probability density function (p.d.f.)lead to explicit solutions,limited insight can be drawn from either the solutions or their numerical evaluation.Fortunately,much deeper insights can be obtained using the tools provided by asymptotic random matrix theory.Indeed,a rich body of results exists analyzing the asymptotic spectrum of H as the number of columns and rows goes to infinity while the aspect ratio of the matrix is kept constant. Before introducing the asymptotic spectrum results,some justifica- tion for their relevance to wireless communication problems is in order. In CDMA,channels with K and N between 32 and 64 would be fairly typical.In multi-antenna systems,arrays of 8 to 16 antennas would be
8 Introduction ratio (SINR). For an i.i.d. input, the arithmetic mean over the users (or transmit antennas) of the MMSE is given, as function of H, by [271] 1 K min M∈CK×N E ||x − My||2 = 1 K tr I + SNR H† H −1 (1.7) = 1 K K i=1 1 1 + SNR λi(H†H) (1.8) = ∞ 0 1 1 + SNR x dFK H†H(x) = N K ∞ 0 1 1 + SNR x dFN HH† (x) − N − K K (1.9) where the expectation in (1.7) is over x and n while (1.9) follows from (1.3). Note, incidentally, that both performance measures as a function of SNR are coupled through SNR d dSNR loge det I + SNR HH† = K − tr I + SNR H† H −1 . As we see in (1.5) and (1.9), both fundamental performance measures (capacity and MMSE) are dictated by the distribution of the empirical (squared) singular value distribution of the random channel matrix. In the simplest case of H having i.i.d. Gaussian entries, the density function corresponding to the expected value of FN HH† can be expressed explicitly in terms of the Laguerre polynomials. Although the integrals in (1.5) and (1.9) with respect to such a probability density function (p.d.f.) lead to explicit solutions, limited insight can be drawn from either the solutions or their numerical evaluation. Fortunately, much deeper insights can be obtained using the tools provided by asymptotic random matrix theory. Indeed, a rich body of results exists analyzing the asymptotic spectrum of H as the number of columns and rows goes to infinity while the aspect ratio of the matrix is kept constant. Before introducing the asymptotic spectrum results, some justification for their relevance to wireless communication problems is in order. In CDMA, channels with K and N between 32 and 64 would be fairly typical. In multi-antenna systems, arrays of 8 to 16 antennas would be
1.2.The Role of the Singular Values 9 at the forefront of what is envisioned to be feasible in the foreseeable fu- ture.Surprisingly,even quite smaller system sizes are large enough for the asymptotic limit to be an excellent approximation.Furthermore, not only do the averages of (1.4)and (1.9)converge to their limits surprisingly fast,but the randomness in those functionals due to the random outcome of H disappears extremely quickly.Naturally,such robustness has welcome consequences for the operational significance of the resulting formulas. 1.8 14 B=1 0.5 0 0.2 0.5 1.5 2.5 Fig.1.1 The Marcenko-Pastur density function (1.10)for B=1,0.5,0.2. As we will see in Section 2,a central result in random matrix theory states that when the entries of H are zero-mean i.i.d.with variance y, the empirical distribution of the eigenvalues of HH converges almost surely,asK,V一oo with→3，to the so-called Mar心enko-Pastur law whose density function is =(-动) x)+-Ot6-开 (1.10) 2π3x where (z)+=max(0,z)and a=(1-VE2 b=(1+V2. (1.11)
1.2. The Role of the Singular Values 9 at the forefront of what is envisioned to be feasible in the foreseeable future. Surprisingly, even quite smaller system sizes are large enough for the asymptotic limit to be an excellent approximation. Furthermore, not only do the averages of (1.4) and (1.9) converge to their limits surprisingly fast, but the randomness in those functionals due to the random outcome of H disappears extremely quickly. Naturally, such robustness has welcome consequences for the operational significance of the resulting formulas. 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 β= 0.2 0.5 1 Fig. 1.1 The Mar˘cenko-Pastur density function (1.10) for β = 1, 0.5, 0.2. As we will see in Section 2, a central result in random matrix theory states that when the entries of H are zero-mean i.i.d. with variance 1 N , the empirical distribution of the eigenvalues of H†H converges almost surely, as K, N → ∞ with K N → β, to the so-called Mar˘cenko-Pastur law whose density function is fβ(x) = 1 − 1 β + δ(x) + (x − a)+(b − x)+ 2πβx (1.10) where (z)+ = max (0, z) and a = (1 − β) 2 b = (1 + β) 2. (1.11)