10 Introduction =1 10 0.2 1012 14 16 Fig.1.2 The Marcenko-Pastur density function (1.12)for B=10,1,0.5,0.2.Note that the mass points at 0,present in some of them,are not shown. Analogously,the empirical distribution of the eigenvalues of HHt converges almost surely to a nonrandom limit whose density function is (cf.Fig.1.2) fg(x)=(1-3)6(x)+Bfg(x) =1-)+)+c-06- (1.12) 2πx Using the asymptotic spectrum,the following closed-form expres- sions for the limits of(1.4)[275]and (1.7)[271]can be obtained: (1.13) 1 odet(I+-SNR HH）→Blog(l+-)fa()dr =Blog 1+SNR-F (SNR,B) +og（1+s3-Fs, logeF(SNR, (1.14) 4SNR
10 Introduction 0 2 4 6 8 10 12 14 16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 β= 1 0.2 0.5 10 Fig. 1.2 The Mar˘cenko-Pastur density function (1.12) for β = 10, 1, 0.5, 0.2. Note that the mass points at 0, present in some of them, are not shown. Analogously, the empirical distribution of the eigenvalues of HH† converges almost surely to a nonrandom limit whose density function is (cf. Fig. 1.2) ˜fβ(x) = (1 − β) δ(x) + β fβ(x) = (1 − β) + δ(x) + (x − a)+(b − x)+ 2πx . (1.12) Using the asymptotic spectrum, the following closed-form expressions for the limits of (1.4) [275] and (1.7) [271] can be obtained: (1.13) 1 N log det I + SNR HH† → β b a log(1 + SNR x)fβ(x) dx = β log 1 + SNR − 1 4 F (SNR , β) + log 1 + SNR β − 1 4 F (SNR , β) − log e 4 SNR F (SNR , β) (1.14)
1.2.The Role of the Singular Values 11 表r{a+swH回}→厂1+ fe()dr (1.15) 1 F(SNR,B) 4B SNR (1.16) with Fe,)=（V(1+Va2+1-V(1-V2+1） (1.17) 4 6 8 10 2 46 8 10 N=15 SNR N=50 SNR Fig.1.3 Several realizations of the left-hand side of(1.13)are compared to the asymptotic limit in the right-hand side of (1.13)in the case of B=1 for sizes:N=3,5,15,50. The convergence of the singular values of H exhibits several key features with engineering significance: Insensitivity of the asymptotic eigenvalue distribution to the shape of the p.d.f.of the random matrix entries.This prop- erty implies,for example,that in the case of a single-user
1.2. The Role of the Singular Values 11 1 K tr I + SNR H† H −1 → b a 1 1 + SNR x fβ(x) dx (1.15) = 1 − F(SNR , β) 4 β SNR (1.16) with F(x, z) = x(1 + √z)2 + 1 − x(1 − √z)2 + 1 2 . (1.17) N = 50 SNR SNR SNR SNR N = 3 N = 5 N = 15 0 2 4 6 8 10 0 1 2 3 4 0 2 4 6 8 10 0 1 2 3 4 0 2 4 6 8 10 0 1 2 3 4 0 2 4 6 8 10 0 1 2 3 4 Fig. 1.3 Several realizations of the left-hand side of (1.13) are compared to the asymptotic limit in the right-hand side of (1.13) in the case of β = 1 for sizes: N = 3, 5, 15, 50. The convergence of the singular values of H exhibits several key features with engineering significance: • Insensitivity of the asymptotic eigenvalue distribution to the shape of the p.d.f. of the random matrix entries. This property implies, for example, that in the case of a single-user
12 Introduction multi-antenna link,the results obtained asymptotically hold for any type of fading statistics.It also implies that restrict- ing the CDMA waveforms to be binary-valued incurs no loss in capacity asymptotically.3 Ergodic behavior:it suffices to observe a single matrix realiza- tion in order to obtain convergence to a deterministic limit. In other words,the eigenvalue histogram of any matrix re- alization converges almost surely to the average asymptotic eigenvalue distribution.This "hardening"of the singular val- ues lends operational significance to the capacity formulas even in cases where the random channel parameters do not vary ergodically within the span of a codeword. Fast convergence of the empirical singular-value distribution to its asymptotic limit.Asymptotic analysis is especially use- ful when the convergence is so fast that,even for small values of the parameters,the asymptotic results come close to the finite-size results(cf.Fig.1.3).Recent works have shown that the convergence rate is of the order of the reciprocal of the number of entries in the random matrix [8,110] It is crucial for the explicit expressions of asymptotic capacity and MMSE shown in (1.14)and(1.16),respectively,that the channel matrix entries be i.i.d.Outside that model,explicit expressions for the asymp- totic singular value distribution such as (1.10)are exceedingly rare. Fortunately,in other random matrix models,the asymptotic singular value distribution can indeed be characterized.albeit not in explicit form,in ways that enable the analysis of capacity and MMSE through the numerical solution of nonlinear equations. The first applications of random matrix theory to wireless commu- nications were the works of Foschini [77]and Telatar [250]on narrow- band multi-antenna capacity;Verdu [271]and Tse-Hanly [256]on the optimum SINR achievable by linear multiuser detectors for CDMA; 3 The spacing between consecutive eigenvalues,when properly normalized,was conjectured in 65,66 to converge in distribution to a limit that does not depend on the shape of the p.d.f.of the entries.The universality of the level spacing distribution and other microscopic (local)spectral characteristics has been extensively discussed in recent theoretical physics and mathematical literature [174,106,200,52,54
12 Introduction multi-antenna link, the results obtained asymptotically hold for any type of fading statistics. It also implies that restricting the CDMA waveforms to be binary-valued incurs no loss in capacity asymptotically.3 • Ergodic behavior: it suffices to observe a single matrix realization in order to obtain convergence to a deterministic limit. In other words, the eigenvalue histogram of any matrix realization converges almost surely to the average asymptotic eigenvalue distribution. This “hardening” of the singular values lends operational significance to the capacity formulas even in cases where the random channel parameters do not vary ergodically within the span of a codeword. • Fast convergence of the empirical singular-value distribution to its asymptotic limit. Asymptotic analysis is especially useful when the convergence is so fast that, even for small values of the parameters, the asymptotic results come close to the finite-size results (cf. Fig. 1.3). Recent works have shown that the convergence rate is of the order of the reciprocal of the number of entries in the random matrix [8, 110]. It is crucial for the explicit expressions of asymptotic capacity and MMSE shown in (1.14) and (1.16), respectively, that the channel matrix entries be i.i.d. Outside that model, explicit expressions for the asymptotic singular value distribution such as (1.10) are exceedingly rare. Fortunately, in other random matrix models, the asymptotic singular value distribution can indeed be characterized, albeit not in explicit form, in ways that enable the analysis of capacity and MMSE through the numerical solution of nonlinear equations. The first applications of random matrix theory to wireless communications were the works of Foschini [77] and Telatar [250] on narrowband multi-antenna capacity; Verd´u [271] and Tse-Hanly [256] on the optimum SINR achievable by linear multiuser detectors for CDMA; 3 The spacing between consecutive eigenvalues, when properly normalized, was conjectured in [65, 66] to converge in distribution to a limit that does not depend on the shape of the p.d.f. of the entries. The universality of the level spacing distribution and other microscopic (local) spectral characteristics has been extensively discussed in recent theoretical physics and mathematical literature [174, 106, 200, 52, 54]
1.3.Random Matrices:A Brief Historical Account 13 Verdu [271]on optimum near-far resistance;Grant-Alexander [100], Verdu-Shamai [275,217],Rapajic-Popescu [206],and Miiller [185]on the capacity of CDMA.Subsequently,a number of works,surveyed in Section 3,have successfully applied random matrix theory to a vari- ety of problems in the design and analysis of wireless communication systems. Not every result of interest in the asymptotic analysis of channels of the form(1.1)has made use of the asymptotic eigenvalue tools that are of central interest in this paper.For example,the analysis of single-user matched filter receivers [275]and the analysis of the optimum asymp- totic multiuser efficiency [258 have used various versions of the central- limit theorem;the analysis of the asymptotic uncoded error probability as well as the rates achievable with suboptimal constellations have used tools from statistical physics such as the replica method [249,103]. 1.3 Random Matrices:A Brief Historical Account In this subsection,we provide a brief introduction to the main devel- opments in the theory of random matrices.A more detailed account of the theory itself,with particular emphasis on the results that are relevant for wireless communications,is given in Section 2. Random matrices have been a part of advanced multivariate statis- tical analysis since the end of the 1920s with the work of Wishart [311] on fixed-size matrices with Gaussian entries.The first asymptotic re- sults on the limiting spectrum of large random matrices were obtained by Wigner in the 1950s in a series of papers [303,305,306]motivated by nuclear physics.Replacing the self-adjoint Hamiltonian operator in an infinite-dimensional Hilbert space by an ensemble of very large Hermi- tian matrices,Wigner was able to bypass the Schrodinger equation and explain the statistics of experimentally measured atomic energy levels in terms of the limiting spectrum of those random matrices.Since then, research on the limiting spectral analysis of large-dimensional random matrices has continued to attract interest in probability,statistics and physics. Wigner [303]initially dealt with an nxn symmetric matrix A whose diagonal entries are 0 and whose upper-triangle entries are independent
1.3. Random Matrices: A Brief Historical Account 13 Verd´u [271] on optimum near-far resistance; Grant-Alexander [100], Verd´u-Shamai [275, 217], Rapajic-Popescu [206], and M¨uller [185] on the capacity of CDMA. Subsequently, a number of works, surveyed in Section 3, have successfully applied random matrix theory to a variety of problems in the design and analysis of wireless communication systems. Not every result of interest in the asymptotic analysis of channels of the form (1.1) has made use of the asymptotic eigenvalue tools that are of central interest in this paper. For example, the analysis of single-user matched filter receivers [275] and the analysis of the optimum asymptotic multiuser efficiency [258] have used various versions of the centrallimit theorem; the analysis of the asymptotic uncoded error probability as well as the rates achievable with suboptimal constellations have used tools from statistical physics such as the replica method [249, 103]. 1.3 Random Matrices: A Brief Historical Account In this subsection, we provide a brief introduction to the main developments in the theory of random matrices. A more detailed account of the theory itself, with particular emphasis on the results that are relevant for wireless communications, is given in Section 2. Random matrices have been a part of advanced multivariate statistical analysis since the end of the 1920s with the work of Wishart [311] on fixed-size matrices with Gaussian entries. The first asymptotic results on the limiting spectrum of large random matrices were obtained by Wigner in the 1950s in a series of papers [303, 305, 306] motivated by nuclear physics. Replacing the self-adjoint Hamiltonian operator in an infinite-dimensional Hilbert space by an ensemble of very large Hermitian matrices, Wigner was able to bypass the Schr¨odinger equation and explain the statistics of experimentally measured atomic energy levels in terms of the limiting spectrum of those random matrices. Since then, research on the limiting spectral analysis of large-dimensional random matrices has continued to attract interest in probability, statistics and physics. Wigner [303] initially dealt with an n×n symmetric matrix A whose diagonal entries are 0 and whose upper-triangle entries are independent
14 Introduction 0.2 0.15 01 Fig.1.4 The semicircle law density function (1.18)compared with the histogram of the average of 100 empirical density functions for a Wigner matrix of size n=100. and take the values 1 with equal probability.Through a combinatorial derivation of the asymptotic eigenvalue moments involving the Cata- lan numbers,Wigner showed that,as n-oo,the averaged empirical distribution of the eigenvalues ofA converges to the semicircle lw whose density is u()=了京V4-r2i证B z≤2 if >2 (1.18) Later,Wigner [305]realized that the same result would be obtained if the random selection was sampled from a zero-mean (real or complex) Gaussian distribution.In that case,it is even possible to find an exact formula for the joint distribution of the eigenvalues as a function of n [176].The matrices treated in [303]and [305]are special cases of Wigner matrices,defined as Hermitian matrices whose upper-triangle entries are zero-mean and independent.In [306],Wigner showed that the asymptotic distribution of any Wigner matrix is the semicircle law (1.18)even if only a unit second-moment condition is placed on its entries. Figure 1.4 compares the semicircle law density function (1.18)with the average of 100 empirical density functions of the eigenvalues of a 10 x 10 Wigner matrix whose diagonal entries are 0 and whose upper- triangle entries are independent and take the values +1 with equal probability. If no attempt is made to symmetrize the square matrix A and all
14 Introduction −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 0 0.05 0.1 0.15 0.2 0.25 0.3 Fig. 1.4 The semicircle law density function (1.18) compared with the histogram of the average of 100 empirical density functions for a Wigner matrix of size n = 100. and take the values ±1 with equal probability. Through a combinatorial derivation of the asymptotic eigenvalue moments involving the Catalan numbers, Wigner showed that, as n → ∞, the averaged empirical distribution of the eigenvalues of √ 1 nA converges to the semicircle law whose density is w(x) = 1 2π √ 4 − x2 if |x| ≤ 2 0 if |x| > 2 (1.18) Later, Wigner [305] realized that the same result would be obtained if the random selection was sampled from a zero-mean (real or complex) Gaussian distribution. In that case, it is even possible to find an exact formula for the joint distribution of the eigenvalues as a function of n [176]. The matrices treated in [303] and [305] are special cases of Wigner matrices, defined as Hermitian matrices whose upper-triangle entries are zero-mean and independent. In [306], Wigner showed that the asymptotic distribution of any Wigner matrix is the semicircle law (1.18) even if only a unit second-moment condition is placed on its entries. Figure 1.4 compares the semicircle law density function (1.18) with the average of 100 empirical density functions of the eigenvalues of a 10 × 10 Wigner matrix whose diagonal entries are 0 and whose uppertriangle entries are independent and take the values ±1 with equal probability. If no attempt is made to symmetrize the square matrix A and all