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Eigenvalue Dynamics of a Central Wishart Matrix With Application to MIMO Systems

F.Javier Lopez-Martinez,Member,IEEE,Eduardo Martos-Naya,Jose F.Paris,

and Andrea Goldsmith,Fellow,IEEE

Abstract—We investigate the dynamic behavior of the stationary random process de?ned by a central complex Wishart matrix W(t)as it varies along a certain dimension t. We characterize the second-order joint cumulative distribution function(cdf)of the largest eigenvalue,and the second-order joint cdf of the smallest eigenvalue of this matrix.We show that both cdfs can be expressed in exact closed-form in terms of a ?nite number of well-known special functions in the context of communication theory.As a direct application,we investigate the dynamic behavior of the parallel channels associated with multiple-input multiple-output(MIMO)systems in the presence of Rayleigh fading.Studying the complex random matrix that de?nes the MIMO channel,we characterize the second-order joint cdf of the signal-to-noise ratio(SNR)for the best and worst channels.We use these results to study the rate of change of MIMO parallel channels,using different performance metrics. For a given value of the MIMO channel correlation coef?cient, we observe how the SNR associated with the best parallel channel changes slower than the SNR of the worst channel.This different dynamic behavior is much more appreciable when the number of transmit(N T)and receive(N R)antennas is similar.However, as N T is increased while keeping N R?xed,we see how the best and worst channels tend to have a similar rate of change.

Index Terms—Complex Wishart matrix,cumulative distribution function,MIMO systems,mutual information, outage probability,random matrices,statistics.

I.I NTRODUCTION

A.Related Work

Since the seminal work by Wishart[2],random matrix theory has found application in very diverse?elds like physics[3],neuroscience[4]and many others[5].For Manuscript received June19,2014;revised January8,2015;accepted February24,2015.Date of publication March5,2015;date of current version April17,2015.This work was supported in part by Junta de Andalucia under Grant P11-TIC-7109,in part by the Spanish Government—FEDER under Grant TEC2010-18451,Grant TEC2011-25473,Grant TEC2013-44442-P,and Grant COFUND2013-40259,in part by the University of Malaga and Euro-pean Union through the Marie-Curie COFUND U-Mobility Program under Grant246550,in part by NEC,in part by Huawei,and in part by the NSF Center for Science of Information.This paper was presented at the2013IEEE Information Theory Workshop[1].

F.J.Lopez-Martinez was with the Wireless Systems Laboratory,Department of Electrical Engineering,Stanford University,Stanford,CA94305USA. He is now with the Departamento de Ingeniería de Comunicaciones, Universidad de Malaga,Málaga29071Spain(e-mail:fjlopezm@ic.uma.es).

E.Martos-Naya and J.

F.Paris are with the Departamento de Ingeniería de Comunicaciones,Universidad de Malaga,Málaga29071,Spain(e-mail: eduardo@ic.uma.es;paris@ic.uma.es).

A.Goldsmith is with the Wireless Systems Laboratory,Department of Electrical Engineering,Stanford University,Stanford,CA94305USA(e-mail: andrea@https://www.wendangku.net/doc/ab14917290.html,).

Communicated by R.Fischer,Associate Editor for Communications. Color versions of one or more of the?gures in this paper are available online at https://www.wendangku.net/doc/ab14917290.html,.

Digital Object Identi?er10.1109/TIT.2015.2409069instance,random matrix processes are useful in econometrics to study the stock volatility in portfolio management[6],[7]; in immunology,random matrix theory has been used to design immunogens targeted for rapidly mutating viruses[8].

In the context of information and communication the-ory,random matrices have been used to characterize the performance of communication systems that make use of multiple antennas at the transmitter and the receiver sides of a communication link,referred to as a multiple-input multiple-output(MIMO)systems.This technique has become the standard transmission mechanism for current wireless communication systems[9]due to its increased capacity and reliability.In this scenario,the channel is described as a random matrix H whose size is determined by the numbers of transmit and receive antennas.

The characterization of the eigenvalues of the matrix W HH H has been used to study the fundamental performance limits of MIMO systems[10],[11];speci?cally, the ordered eigenvalues of W characterize the parallel eigenchannels used to achieve multiplexing gain,and,in particular the largest eigenvalue of W determines the diversity gain of the system.

When the entries of H are distributed as complex Gaussian random variables,then W is said to follow a complex Wishart(CW)distribution[2].The eigenvalue statistics of CW matrices have been studied in depth in the literature, both for central[12]–[15]and non-central[16]–[22]Wishart distributions.These results can be seen as a?rst-order charac-terization of a CW random process,and can be used to derive useful performance metrics such as the outage probability or the channel capacity.

However,wireless communication systems are in general non-static and hence the stochastic process associated with H exhibits a variation along different dimensions due to mobility of users or objects in the propagation environment. For example,temporal variation of wireless communication channels due to user mobility has an impact on the ability to estimate channel state information and hence limits the achievable performance.Equivalently,the frequency variation due to the effect of multipath has a similar effect on the equivalent channel gain in the frequency domain.

If we consider two samples of a stationary CW random process W(t),namely W1 W(t)and W2 W(t+τ),the dynamics of W(t)are captured by the joint distribution of W1and W2.More precisely,the dynamics of the MIMO parallel channels(or eigenchannels)can be studied sepa-rately by studying the joint distribution of the eigenvalues

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of W1and W2.Along these lines,the statistical analysis of two correlated CW matrices was tackled in[23]and[24], deriving the2s-dimensional joint pdf of the s eigenvalues of a CW matrix and a perturbed version of it.Rather than the joint distribution of all the eigenvalues,we consider the marginal joint distribution of a particular eigenvalue as our metric to capture the dynamic behavior of a CW random process,since this distribution allows for the separate statistical characteriza-tion of all s eigenvalues.Therefore,we will focus our attention on this set of second-order(or bivariate)distributions.

This problem was addressed in[25]when studying the mutual information distribution in orthogonal frequency division multiplexing(OFDM)systems operating under frequency-selective MIMO channels;speci?cally,a closed-form expression for the joint second-order pdf was given for arbitrarily-selected eigenvalues of the equivalent frequency-domain Wishart matrix.However,in order to obtain the joint bivariate cdf or the correlation coef?cient for a particular eigenvalue,a two-fold numerical integration with in?nite limits was required.In[26],an expression for this bivariate cdf was derived for the extreme eigenvalues(i.e.the largest and the smallest)in terms of the determinant of a matrix whose entries are expressed as in?nite series of products of incomplete gamma functions;hence,its evaluation is highly impractical as the number of antennas is increased.

B.Contributions

In this paper,we show that the joint second-order cdf of the random process given by the largest eigenvalue of a complex Wishart matrix can be expressed in closed-form; this also holds when considering the smallest eigenvalue. Speci?cally,we provide exact closed-form analytical results for these distributions in terms of the determinant of a matrix, whose entries are expressed as a?nite number of Marcum Q-functions[27]and modi?ed Bessel functions of the?rst kind.Therefore,they can be ef?ciently computed with commercial software mathematical packages.

These results complete the current landscape of closed-form bivariate cdfs for the most common fading distributions such as Rayleigh[28],Weibull[29]and Nakagami-m[30],[31]. Interestingly,they are expressed in terms of the Marcum Q-function,and hence the results for Nakagami-m and Rayleigh fading can be seen as particular cases of the expres-sions derived herein when particularized for N T=m,N R=1 and N T=1,N R=1respectively.

We also show how our results can be used to directly eval-uate many performance metrics that allow us to quantify the rates of change of MIMO parallel channels in different ways: 1)We study the behavior of a mutual information met-

ric associated with two different observations of the eigenvalue of interest.We quantify the loss in mutual information as the CW process changes,in order to determine the impact of increasing N T or N R in the rate of change of the best and worst parallel channels.

2)We evaluate the probability of having two outages in two

different instants.This metric applies to two different transmissions within a given time windowτin?at fading channels,as well as to two different transmissions

with a frequency separation f in MIMO OFDM systems affected by multipath fading.

3)The level crossing rate(LCR)and the average fade

duration(AFD)are often used to characterize the dynamics of fading channels.In the context of time-varying MIMO channels,this problem was tackled in[32]using Rice’s original framework for LCR analy-sis[33]of continuous random processes.However,the inherent sampling of the equivalent frequency-domain channel in OFDM systems makes the LCR obtained using Rice’s approach to be an overestimation of the actual LCR as observed in[34].Here,we use our closed-form results for the bivariate cdfs of interest to study the LCR and the AFD of the extreme eigenvalues in MIMO OFDM systems,using the method described in[35]for sampled fading channels.

The remainder of the paper is structured as follows:The main mathematical contributions are presented in Section II, i.e.the bivariate cdfs of the extreme eigenvalues of two corre-lated CW matrices.As an application,the performance metrics that characterize the dynamics of MIMO parallel channels are introduced in Section III,and then used in Section IV to pro-vide some numerical results in practical scenarios of interest. Finally,our main conclusions are discussed in Section V.The proofs for the results in Section II are included as appendices.

II.S TATISTICAL A NALYSIS

A.Notation and Preliminaries

Throughout this paper,vectors and matrices are denoted in bold lowercase a and bold uppercase A,respectively.We use |a|to indicate the modulus of a complex number a,whereas |A|indicates the determinant of a matrix A.The symbol～

means statistically distributed as,whereas E{·}represents the expectation operation,and the super-index H denotes the Hermitian transpose.For the sake of coherence with the scenario where the general results of this paper are used, we denote the numbers of rows and columns of the random matrices with Gaussian entries as N R and N T,respectively. The correlation coef?cient of two random variables X and Y is de?ned as

ρX,Y cov{X,Y}

σ2Xσ2Y

,(1)

where cov{}denotes covariance operation,andσ2X,σ2Y represent the variances of the random variables X and Y respectively.

The cdf of X is de?ned as F X(x) Pr{X≤x}= x

?∞f X

(z)dz,where f X(x)is the pdf of X.Similarly,the joint cdf of two correlated random variables X and Y is de?ned as

F X,Y(x,y) Pr{X≤x,Y≤y}

?∞

f X,Y(z1,z2)dz1dz2,(2)

where f X,Y(x,y)is the joint cdf of X and Y.The joint complementary cdf(ccdf),de?ned asˉF X,Y(x,y) Pr{X>x, Y>y}and the joint cdf of two correlated identically distributed random variables X and Y are related through ˉF

X,Y(x,y)=F X,Y(x,y)?F X(x)?F Y(y)+1.(3)

LOPEZ-MARTINEZ et al.:EIGENV ALUE DYNAMICS OF A CENTRAL WISHART MATRIX 2695

B.De?nitions

Before presenting the main analytical results,it is necessary to introduce the following de?nition of interest related with central CW matrices.

De?nition 1(Central Complex Wishart Matrix):Let us consider the complex random matrix H ∈C N R ×N T with zero-mean i.i.d.Gaussian entries ～CN 0,σ2

.If we de?ne s =min (N R ,N T ),t =max (N R ,N T )and the matrix W as

W =

HH H N R ≤N T

H H H N R >N T (4)

then W ∈C s ×s follows a complex central Wishart distribution,i.e.W ～CW t ,σ2I s ,0s ,where I s and 0s are the identity and the null s ×s matrices,respectively.

It is also necessary to introduce an auxiliary integral function.

De?nition 2(Incomplete Integral of Nuttall Q-Function):

J α,β,γ

k ,c ,j (u )=

∞

z 2j +k ?1e ?αz 2

Q 2c +k +1,k (γz ,β)dz ,

(5)

where Q m ,n (x ,y )is the Nuttall Q-function [36],k ,c ,j ∈Z +,u ∈[0,∞)and α,β,γ∈R .

The function J α,β,γ

k ,c ,j (u )is an incomplete integral of Nuttall Q -function (IINQ),and generalizes a class of incomplete integrals of Marcum Q -functions which appear in different problems in communication theory [30],[36],[37].

We also restate previous results for the ?rst-order statistics of the extreme eigenvalues of CW matrices that will be used in our later derivations.

Proposition 1:Let λ1be the largest eigenvalue

of a com-plex central Wishart matrix W ～CW t ,σ2I s ,0s .The cdf of λ1is given in closed-form as [12],[16]

F λ1(λ)=

|M (λ)|

|K |

,(6)

where

M i ,j =γ(t ?s +i +j ?1,λσ2

),(7)K i ,j

= (t ?s +i +j ?1),

(8)γ(m ,x )is the lower incomplete Gamma function,and (·)denotes the Gamma function.

For the sake of compactness in the following derivations,we will use the ccdf of the smallest eigenvalue.

Proposition 2:Let λs be the smallest eigenvalue

of a com-plex central Wishart matrix W ～CW t ,σ2I s ,0s .The ccdf of λs is given in closed-form as [12],[16]

ˉF

λS (λ)= ?M (λ)

|K |,(9)where

i ,j = (t ?s +i +j ?1,λσ2

),(10)

and (m ,x )is the upper incomplete Gamma function.

C.Problem Statement

We are interested in the study of a stationary CW random process W (t ).Hence,we consider two realizations of this random process at two different instants,i.e.W (t ) W 1and W (t +τ) W 2.The correlation between the underlying Gaussian processes H (t ) H 1and H (t +τ) H 2corre-sponding to the two realizations of the Gaussian matrix can be modelled as

H 2=ρH 1+

1?ρ2 ,(11)where ρis the correlation coef?cient 1between the {i ,j }

entries of H 1and H 2,and is an auxiliary N R ×N T matrix with i.i.d.entries ～CN 0,σ2

,which is independent of H 1.In virtue of the spectral theorem,the matrices W 1and W 2are orthogonally diagonalizable,i.e.there exist s ×s matrices

V 1and V 2such as W 1=V 1 V H 1and W 2=V 2 V H 2

.The diagonal matrices formed by the ordered eigenvalues of W 1and W 2are then given by diag {λ1,...,λs }and diag {?1,...,?s },where λk and ?k represent the k th eigen-value of the W 1and W 2matrices,respectively.

Throughout this paper,we will focus our attention on the extreme eigenvalues,i.e.the largest (k =1)and the smallest (k =s ).Speci?cally,we are interested in the characterization of the dynamics of the random process given by the 1st and s th eigenvalues of a complex Wishart matrix.Therefore,we aim to study the joint distributions of the random processes {λi ,?i }for i ={1,s }.

With the previous de?nitions,it is clear that the random matrices H 2|H 1and W 2|W 1are distributed as

H 2|H 1～CN N R ,N T (ρH 1,σ2(1?ρ2)I N R ?I N T )

(12)W 2|W 1～CW s t ,σ2(1?ρ2

)I s ,ρ2σ2

(1?ρ2)W 1

.(13)Hence,W 2|W 1follows a non-central Wishart distribution with

non-centrality parameter matrix given by ρ2

σ2(1

?ρ2)W 1

.The distribution of the k th eigenvalue of a non-central CW matrix was calculated in [18];here,we use the distribution of the eigenvalues for the conditioned random matrix W 2|W 1to obtain exact closed-form expressions for the marginal joint distributions of the largest and the smallest eigenvalues of two correlated CW matrices.D.Main Results

In the following theorem,we obtain the joint distribution of the largest eigenvalue of two correlated Wishart matrices.

1In our case,we assume that each of the elements of this channel matrix

h i ,j (t )evolves as an independent random process.This is the natural extension of the i.i.d.case to include the evolution of the random matrix H as it varies along a certain dimension t .We also assume that the correlation coef?cient

for each of the channel matrix elements is the same,i.e.H 2=RH 1+ˉR

,where R and ˉR are N R ×N R diagonal matrices with elements r i ,i =ρ

and ˉr i ,i = 1?|ρ|2;therefore,both R and ˉR

have the form of the scaled identity matrix.As we will later see,this model is useful in a number of scenarios involving MIMO communications.Assuming a more general correlation model is indeed possible,although a different approach would be required in order to analyze the dynamics of the extreme eigenvalues,and a closed-form characterization is probably unattainable in such situation.

2696IEEE TRANSACTIONS ON INFORMATION THEORY,VOL.61,NO.5,MAY 2015

Theorem 1:Let λ1and ?1be the largest eigenvalues of two complex central Wishart matrices W 1and W 2,respectively,where W 1and W 2are identically distributed W ～CW t ,σ2I s ,0s ,with underlying Gaussian matrices correlated according to (11).The exact joint cdf of λ1and ?1can be expressed as

F λ1,?1(λ,?)=C |?(λ,?)|,

(14)

where

C =

2s σ?2ts (?1)s (s ?1)2εs (s ?t )

s i =1ε1?i

(s ?i )!(t ?i )!

,(15)

κ=1(1?ρ2)σ2,ε=κρ2,the entries of the s ×s matrix ?are

given by

?i ,j =2(2i ?s ?t )/2

J σ?2,0,√2εt ?s ,s ?i ,j (0)?J σ?2,√

2κ?,√2εt ?s ,s ?i ,j (0)

?J σ?2,0,√2ε

t ?s ,s ?i ,j

√λ +J σ?2,√

2κ?,√2εt ?s ,s ?i ,j √λ .(16)

Proof:See Appendix A.

Following a similar approach,we derive a closed-form expression for the joint distribution of the smallest eigenvalue in the following theorem.In this case,and for the sake of compactness,we present an expression for the bivariate ccdf,being the cdf obtained in a straightforward manner using (3).Theorem 2:Let λs and ?s be the smallest eigenvalues of two complex central Wishart matrices W 1and W 2,respectively,where W 1and W 2are identically distributed W ～CW t ,σ2I s ,0s ,with underlying Gaussian matrices correlated according to (11).The exact joint ccdf of λs and ?s can be expressed as

ˉF λs ,?s (λ,?)=C ??(λ,?) ,

(17)where the entries of the s ×s matrix ??

are given by ??

i ,j =2(2i ?s ?t )/2·J σ?2,√2κ?,√2εt ?s ,s ?i ,j √λ ,(18)

Proof:See Appendix B.

Expressions (14)and (17)are given in terms of the IINQ de?ned in (5).As we show in the following theorem,this IINQ can be expressed in closed-form in terms of a ?nite sum of Marcum Q -functions.

Theorem 3:The IINQ de?ned in (5)admits a closed-form solution in terms of a ?nite number of Marcum Q-functions given in (19)–(23),as shown at the bottom of this page.

Proof:See Appendix C.

Note that in (19),we have de?ned some auxiliary para-meters δ=α+γ2/2and θ=γ2/(γ2+2α),whereas the coef?cients P c ,l ,k and ωl ,k are detailed in Appendix C.In order to obtain this result,two auxiliary integrals (20)and (21)are solved;the proof for these results are given in Appendix D.We observe that the solutions for (20)and (21)are given in terms of the Nuttall Q -function,and the regularized con?u-ent hypergeometric function of two variables ?

3(b ,c ,w,z ),de?ned as the classic 3function in [38]normalized to (c ).However,for the set of indices in (19),the Nuttall Q -function can be computed in terms of the Marcum Q -function and the modi?ed Bessel function of the ?rst kind,using the relation de?ned in [39]and restated in Appendix C in eq.(59).

With regard to the ?

3function,it can also be expressed as a ?nite number of Marcum Q -functions using the relationship recently derived in [31]and restated in (22).Hence,(19)is given in closed-form in terms of a ?nite number of Marcum Q -functions,which are included as built-in functions in most commercial mathematical packages.2

2Even though the generalized Marcum Q -function Q m

(a ,b )is mostly used

for m >0,there exists a simple relation when m <0given in [40]as Q m (a ,b )=1?Q 1?m (b ,a ).

J α,β,γ

k ,c ,j (u )=e

?β22

1?γ22α

c l =1

P c ,l ,k (β2

)γl ?1βk +l +1√2δ

2j +(k +l ?1)Q 2(j ?1)+(k +l ?1)+1,k +l ?1

γβ

√2δ

,u √2δ

c +1

l =1

ωl ,k (c )γk +2(l ?1)K α,β,γ

k +l ,j +k +l ?1(u ),

(19)K α,β,γ

m ,n (u )

∞

z 2n ?1e

?αz 2

Q m (γz ,β)dz =δ?n 2(n ?1)!e ?β22e ?u 2δ

n ?1 k =0

δu 2 k k !

I β,δ,θ

m ,n ,k (u ),(20)

I β,δ,θm ,n ,k (u )=(1?θ)k ?n e β22e δθu 2?(1?θ)k ?n

β22 k +m ?n

? 3 1,k +m ?n +1;β22,δθu 2β22

+n ?k l =1

(1?θ)k +l ?n ?1

β22 k +m ?n +l ?1? 3 l ,k +m ?n +l ;θβ

22,δθu 2β22

,(21)? 3(b ,c ;w,z )= z w b ?12(b ?1) i =0

A i (b ,c ;z )

w c ?i ?1z i exp w +z w Q 2?c +i √2w, 2

z w

(22)

A i (b ,c ;z )=

(?1)b ?1(b ?1)!

i /2

k =0

(?1)k (b ?i +k )i ?k (c ?i ?1+k )i ?2k (i ?2k )!k !

(23)

LOPEZ-MARTINEZ et al.:EIGENV ALUE DYNAMICS OF A CENTRAL WISHART MATRIX 2697

This closed-form result for the integral (19)is new in the literature to the best of our knowledge,and allows us to directly obtain a closed-form expression for the bivariate distribution of the extreme eigenvalues of a central complex Wishart matrix.These closed-form results for the cdfs have similar form to existing results in the literature for related dis-tributions;speci?cally,they reduce to the recently calculated expression for the bivariate Nakagami-m cdf [30],[31]when N T =m and N R =1,as well as to the well-known expression for the bivariate Rayleigh cdf [28]letting N T =N R =1.For the readers’convenience,a brief description of the Q -functions used in this paper is included in Appendix E.

III.A PPLICATION TO MIMO S YSTEMS

The dynamic behavior of the random processes of interest,namely the largest and the smallest eigenvalues of a CW matrix,is fully characterized by their joint distribution.In this section,we illustrate how these bivariate cdfs can be used to derive practical metrics for the performance evaluation of MIMO systems.

Let us consider a MIMO system with N T transmit and N R receive antennas.In this scenario,the received vector y ∈C N R ×1is given by

y =Hx +n ,

(24)

where x ∈C N T ×1is the transmitted vector,n ∈C N R ×1is the noise vector with i.i.d.entries modeled as complex Gaussian

RV’s ～CN (0,1),and H ∈C N R ×N T represents the Rayleigh fading channel matrix with i.i.d.entries ～CN 0,σ2

The MIMO channel can be decomposed into up to s parallel scalar subchannels,where the power gain of the k th eigen-channel depends on the k th eigenvalue of the matrix W .The best and worst achievable performance will be attained using the channels given by the largest and smallest eigenvalues,for which the SNR γis known to be proportional to the magnitude of their respective eigenvalues [9],i.e.γ(t )∝λand γ(t +τ)∝?according to the notation in Section II-C.We now de?ne a set of performance metrics that will allow for studying the rate of change of the random processes of interest.A.Normalized Mutual Information of SNR Values

The joint distributions characterized in this paper incorpo-rate the dynamics of the CW random process through the correlation coef?cient ρof the underlying Gaussian channel matrix,according to the correlation model in eq.(11).How-ever,the relation between ρand the correlation coef?cient ρi of each one of the ordered eigenvalues of the CW matrix is not fully understood.Analytical results for this correlation coef?cient are hard to obtain,as they require a two-fold numerical integration over the joint bivariate pdf (or the joint ccdf)of the eigenvalue of interest [25].

Simulations in [34]show that the rate of change of the largest eigenvalue is much slower than the rate of change of the smaller eigenvalue in a 4×4MIMO setup,i.e.worse eigenchannels decorrelate faster;however,little is known about how the dynamics of MIMO parallel channels change as the number of antennas is increased.

For this reason,we propose an alternative metric to quantify the rate of change of MIMO parallel channels.Instead of deriving the mutual information of λand φ(i.e.the mutual information of the equivalent SNR γat two different instants),which is still an open problem in the literature [25],[41]–[44],we study the mutual information of the discrete and identically distributed random variables de?ned as X {γi (t )<γth }and Y {γi (t +τ)<γth },where i =1and i =s indicate the SNR of the best and worst eigenchannels,respectively.

The mutual information is a measure of the amount of information shared by two random sets X and Y .Thus,it is useful to determine how by knowing one of them,the uncertainty about the other one is reduced.In the case of X and Y being two different samples of a random process,this metric gives information about how spacing these two samples affects their independence.The use of normalized mutual information metrics as a measure of similarity is a well investigated subject in statistics [45],[46].Speci?cally,we have chosen for our analysis the mutual information metric introduced by [45]as

MI X ,Y I (X ,Y )=I (X ,Y )

√

H (X )H (Y )

(25)

Because they are two samples of a stationary random process,X and Y have the same marginal distribution and therefore H (X )=H (Y ).Thus,this metric also reduces to other conventional mutual information metrics [46]as I (X ,Y )=

I (X ,Y )

max {H (X ),H (Y )}

2I (X ,Y )

H (X )+H (Y )=I (X ,Y )H (X )=I (X ,Y )H (Y )

;(26)

which takes values in the range [0,1].Speci?cally,if

ρ(τ)=1then X and Y are the same random variable;thus,I (X ,X )=H (X )and therefore MI (γth ,τ)=1.Conversely,letting ρ(τ)=0,then X and Y are independent and we have I (X ,Y )=0?MI (γth ,τ)=0.

Mutual information metrics are more general than correlation measures,in the sense that they capture dependence other than linear.Besides,the normalized mutual informa-tion of discrete random variables can be easily computed,provided that their joint distribution is known.Hence,we use the closed-form expressions for these joint distributions to calculate normalized mutual information of the discrete and identically distributed random variables prevously de?ned as X {γi (t )<γth }and Y {γi (t +τ)<γth }.This metric gives a normalized measure of the rate of change of the random process γi (t ),in terms of the amount of information that is kept between two different samples of the process.Thus,this metric will give information on the similarity between X and Y ,providing a quantitative mechanism to evaluate the dynamics of the SNR.

B.Probability of Two Outage Events

The outage probability is de?ned as the probability of the instantaneous SNR γi to be below a certain threshold,i.e.P out (γth ) Pr {γi <γth }.This metric does not incorporate any information related with the dynamic variation of γi ;

2698IEEE TRANSACTIONS ON INFORMATION THEORY,VOL.61,NO.5,MAY 2015

however,we can equivalently de?ne the probability of two outage events occurring in two different instants as

P out (γth ,τ) Pr {γi (t )<γth ,γi (t +τ)<γth }

(27)

where τis de?ned as the separation between two transmissions.Note that this separation has dimensions of time when applied to a time-varying ?at-fading MIMO channel,whereas it can take dimensions of frequency when analyzing the equivalent MIMO channel in the frequency domain for MIMO OFDM systems affected by multipath fading.

Since the instantaneous SNR for the eigenchannel associ-ated with a certain eigenvalue is proportional to λ[9],we can easily see that P out (γth ,τ)=F λi ,?i (γth ,γth ).In the limiting case of τ→∞the process γi (t )decorrelates and we have F λi ,?i (γth ,γth )|τ→∞=F λ(γth ,γth )2,whereas for τ=0we have F λi ,?i (γth ,γth )|τ=0=F λi (γth ).Intuitively,as the only dependence on ρin (14)and (17)is through |ρ|2,then P out (γth ,τ)is to be bounded above the outage probability and below the square of the outage probability.C.Level Crossing Statistics

The level crossing rate (LCR)is used to determine how often a random process crosses a threshold value.In his seminal work [33],Rice introduced a way to compute this metric for continuous processes,using the joint statistics of the random process and its ?rst derivative.This has been the preferred approach for analyzing the LCR in fading channels,as it allows for obtaining compact expressions when the underlying processes are Gaussian.

In some scenarios,the random process of interest is not necessarily continuous.In OFDM systems affected by multipath Rayleigh fading,a discretized equivalent model in the frequency domain is usually assumed,where the chan-nel frequency response at a ?nite number of subcarriers is de?ned as a sampled Gaussian random process.In this scenario,the LCR obtained using Rice’s approach is only approximated [26]and in general overestimates the number of crossings,especially when the condition f ·τ<<1does not hold,for f the subcarrier spacing and τthe rms delay spread.This has an intuitive explanation:when studying the number of crossings of the discretized version of a continuous random process,the possibility of missing a crossing is nonzero,and grows as the sampling period is increased.

An alternative formulation for the LCR analysis of sampled random processes was proposed in [35];interestingly,the LCR can be directly calculated from the bivariate cdf of the random process of interest.Hence,the LCR of the extreme eigenvalues in a MIMO OFDM system can be calculated as

N f (u )=F λi

(u )?F λi ,?i (u ,u )

.(28)Equivalently,the average fade duration (AFD)gives infor-mation about how long the random process remains below a certain threshold level.This metric can be computed as

A f (u )=Pr {λi ≤u }

N f (u )

.(29)

In this case,the AFD has dimension of Hertz,and measures the average number of subcarriers undergoing a

fade.

Fig.SNR and of We i.e.the two P out γth how note the of we 3We must note that the results here derived correspond to a Gaussian channel

matrix with i.i.d.entries.Even though this assumption does not hold when the number of transmit antennas is very large,it is usually considered as a reference case and in some scenarios it is a reasonably good approximation for the massive linear array case [47].

LOPEZ-MARTINEZ et al.:EIGENV ALUE DYNAMICS OF A CENTRAL WISHART MATRIX2699 Fig.

bers

The

and

the

i.e.

the

because of the inherent discretization there is a need to de?ne a value of γfor which the probabilities are computed.In our case,we decided to chose a value ofγthat satis?es a certain value of outage probability(OP). Hence,we have computed this value ofγby inverting the corresponding cdf (i.e.,the OP)for values at which communication systems operate.of transmit antennas is increased,we observe again how the worst channel tends to become more stable.

In Fig.4,we illustrate how the information provided by the metric MI does not strongly depend on the value ofγ.

2700IEEE TRANSACTIONS ON INFORMATION THEORY,VOL.61,NO.5,MAY

2015

Fig.We the N T is that a and In the =u =antennas are considered,and Monte Carlo simulations are included with markers.When the number of transmit antennas is increased,we observe how the value of u for which the maximum number of crossings occurs also grows;

however,

Fig.6.this that for rate as N four Tx the For of We a are for in two of how is the i.e.an the

2701

Fig.8.Normalized AFD vs threshold level u for different numbers of transmit antennas and different values of normalized delay spread .Dashed lines indicate =0.1,solid lines indicate =0.2.Parameter values are N R=4. the AFD?gures are now more abrupt when4receive antennas are used,compared to the case of2receive antennas.

V.C ONCLUSIONS

We obtained exact closed-form expressions for the joint cdfs of the extreme eigenvalues of complex central Wishart matrices.Despite the inherent complexity of the analyzed distributions,the results here derived have a similar form as other related distributions in the context of communication theory,and can be computed with well-know special functions included in commercial mathematical packages.

We have used these results to characterize the dynamics of the parallel channels associated with MIMO transmission. The analytical results here obtained for the best and worst channels are useful to study the evolution of the SNRs in MIMO Rayleigh fading channels.We illustrated how different performance metrics can be easily evaluated:In addition

metrics such as the LCR,the AFD or the of two outage events,we used a normalized mutual

metric to characterize the rate of change of MIMO channels directly from the derived joint cdf.Other metrics like the transition probabilities for a?rst-

chain model[48]can be calculated from these

observed that when the number of transmit and antennas is similar,the worst channel has a much

than the best channel.While the dynamics of are barely affected by using more transmit antennas,

that the worst channel tends to have a more stable

as N T is increased.This suggests that the user

variation in massive MIMO systems would be similar with the same mobility.

derivation of asymptotic limit results for large

N R would be of general interest in random matrix

Unfortunately,even though our results for the bivariate analyzed are given in closed-form for the?rst the literature,they are still quite involved and not easily to the asymptotic regime.In fact,a different than the one taken herein might be required for this Therefore,characterizing the dynamics of the eigenvalues in the limit of large N T and N R is left

for future research.

A PPENDIX A

B IVARIATE CDF OF THE L ARGEST E IGENVALUE

random matrix W2|W1follows a non-central CW according to(13).Hence,the cdf of the largest

of W2|W1is given by[18]

?1| (?,λ1,...,λs)=|C||G(0, )?G(?, )|(31) entries of matrix G(?, )are given by

?,λj

=ˉλs?t2j eˉλj22i?s?t2Q a,b

2ˉλj,

2ˉ?

,(32) for{i,j}=1...s,a=s+t?2i+1,b=t?s,Q m,n(·,·) is the Nuttall Q-function,the determinant|C|is given by

|C|=e

j=1

ˉλj

λi?ˉλj

(33)

andˉλj andˉ?are de?ned as scaled versions ofλj and?respectively

ˉλj =ρ2

1?ρ2

σ2

λj=ελj,(34)

ˉ? 1

1?ρ2

σ2

?=κ?.(35) For convenience of calculation,we re-express

|C|=c s|E|/|V|,(36) where E is a diagonal matrix whose entries are given by E j,j=e?ˉλj,V is a Vandermonde matrix with entries

V i,j=λj?1

,and

c s=

(?1)s(s?1)/2

s?1

i=0

εi

.(37)

2702IEEE TRANSACTIONS ON INFORMATION THEORY,VOL.61,NO.5,MAY 2015

Note that both E and V matrices depend on the eigenvalues λ=[λ1...λs ]T ,although this dependence is omitted for the sake of compactness.Analogously,we split (31)into three terms as

G (0, )?G (?, )=AYB ,

(38)

where

B i ,j (?)=22i ?s ?t

2 Q a ,b

2ˉλj ,0 ?Q a ,b

2ˉλj , 2ˉ? ,(39)

and A ,Y are diagonal matrices whose entries are given by A j ,j =ε(s ?t )

2e ˉ

λj ,and Y j ,j =λs ?t

2j .Again,the dependence on

λis omitted in these matrices.With these de?nitions,we can express (31)as

F ?1| (?,λ1,...,λs )=c s ε

s (s ?t )2

|Y ||B (?)|/|V |.(40)

In order to obtain the joint cdf F λ1,?1(λ,?)of the largest eigenvalues {λ1,?1},we average (40)using the joint pdf of the eigenvalues {λ1,...,λs }given by [13]

f (λ1,...,λs )=K |Z ||V |2,λ1>λ2>...>λs >0,

(41)

where Z is a diagonal matrix with entries given by

Z j ,j =λt ?s i e

?λi /σ

,(42)

V is the Vandermonde matrix de?ned in (36),and

K = σ

2 ?t ·s s

i =1

(s ?i +1) (t ?i +1)

(43)

is a normalization factor.Thus,the joint cdf can be expressed as

F λ1,?1(λ,?)=K c s εs (s ?t )

... λ>λ1>...λs >0

s ?f old

| ||B (?)|d λ1...d λs ,(44)

where =YVZ .Using [49,eq.(51)]we can express

the s ?fold integral of a determinant,as

F λ1,?1(λ,?)=K c s εs (s ?t )2

λ 0

(x )dx

,(45)where the entries of the s ×s matrix are given by i ,j =2(2i ?t ?s )/2e ?

σ2x j +t ?s

2?1

× Q ι,t ?s √2εx ,0 ?Q ι,t ?s √2εx , 2κ? ,

(46)

and ι=2(s ?i )+(t ?s )+https://www.wendangku.net/doc/ab14917290.html,ing a change of variables

x =z 2,we can express the integral (45)as

F λ1,?1(λ,?)=C |?|,

(47)

where

C =2s σ?2ts (?1)

s (s ?1)2

ρ2

(1?ρ2)σ2

s (s ?t )2

×s i =1

ρ2(1?ρ2)σ2

1?i

(s ?i )!(t ?i )!

(48)

and the entries of the s ×s matrix ?are given by

?i ,j =2(2i ?s ?t )/2

J σ?2,0,√2εt ?s ,s ?i ,j (0)?J σ?2,√

2κ?,√2εt ?s ,s ?i ,j (0)

?J σ?2,0,√2εt ?s ,s ?i ,j √λ

+J σ?2,√

2κ?,√2εt ?s ,s ?i ,j √

λ ,(49)using the de?nition given in (19)for the IINQ function.Since

a closed-form expression for J α,β,γ

k ,c ,j (y )is given in (19),this yields a closed-form expression for the joint cdf F λ1,?1(λ,?)in (45).

A PPENDIX B

J OINT B IVARIATE CCDF OF THE S MALLEST E IGENVALUE This proof is similar to the one detailed in the previous appendix.The random matrix W 2|W 1follows a non-central CW distribution according to (13).Hence,the ccdf of the smallest eigenvalue of W 2|W 1is given by [18]

ˉF

?s | (?,λ1,...,λs )=|C ||G (?, )|(50)

where the entries of matrix G (?, )are given in (32),and

the determinant |C |is detailed in (33).Following a similar reasoning as in Appendix A,we can re-express (50)as

ˉF ?s | (?,λ1,...,λs )=c s εs (s ?t )2|Y | ?B (?) /|V |,(51)where

i ,j (?)=22i ?s ?t

Q a ,b 2ˉλj ,

2ˉ?

(52)

and the rest of parameters were de?ned in the previous proof.

In order to obtain the joint ccdf F λs ,?s (λ,?)of the smallest eigenvalues {λs ,?s },we average (51)using the joint pdf of the eigenvalues {λ1,...,λs }given in (41).After some manipulations,the joint ccdf can be expressed as

F λs ,?s (λ,?)=K c s εs (s ?t )2 ... ∞>λ>λ1>...λs

s ?f old

| | ?B (?) d λ1...d λs ,

(53)

Using [49,eq.(51)]we can express the s ?fold integral of a determinant,as

F λs ,?s (λ,?)=K c s εs (s ?t )2

∞ λ

? (x )dx

,(54)where the entries of the s ×s matrix ?

are given by ? i ,j =2

(2i ?t ?s )/2e ?x σ2x j +t ?s 2?1Q ι,t ?s √2εx ,√2κ? ,(55)

LOPEZ-MARTINEZ et al.:EIGENV ALUE DYNAMICS OF A CENTRAL WISHART MATRIX 2703

and ι=2(s ?i )+(t ?s )+https://www.wendangku.net/doc/ab14917290.html,ing a change of variables x =z 2,we can express the integral (54)as

F λs ,?s (λ,?)=C ??

,(56)where C is given in (48)and the entries of the s ×s matrix

are given by ??i ,j =2(2i ?s ?t )/2J σ?2,√2κ?,√2εt ?s ,s ?i ,j √λ ,(57)using the de?nition given in (19)for the IINQ function.Since a closed-form expression for J α,β,γ

k ,c ,j (y )is given in (19),this yields a closed-form expression for the joint ccdf F λs ,?s (λ,?)in (54).

A PPENDIX C

A PPENDIX :S OLUTION FOR J α,β,γ

k ,c ,j (u )

Let us consider the incomplete integral de?ned in (19)as

J α,β,γ

k ,c ,j (u )=∞

z 2j +k ?1e ?αz 2Q 2c +k +1,k (γz ,β)dz ,(58)

where k ,c ,j ∈N and α,β,γ,u ∈R +.The Nuttall Q m ,n function can be expressed in terms of a ?nite sum of Marcum Q k (·,·)and modi?ed Bessel functions of the ?rst kind I n (·),when the sum of indices m +n is an odd number [39,eq.(8)]as

Q 2c +k +1,k (γz ,β)

=c +1 l =1

ωl ,k (c )(γz )k +2(l ?1)Q k +l (γz ,β)

?(γz )2+β

l =1

P c ,l ,k (β2)(γz )l ?1βk +l +1

I k +l ?1(γz β) ,(59)

where

ωl ,k (c )=2

c ?l +1c !

(l ?1)!

c +k

c ?l +1

,(60)P c ,l ,k (β2)=c ?l r =0

2c ?l ?r (c ?1?r )!(l ?1)!

c +k

c ?l ?r β2r .

(61)

Substituting (59)into (58),we split the IINQ into a ?nite

sum of Nuttall Q -functions and incomplete integrals of the Marcum Q -function as in (19).The integral denoted as K α,β,γ

m ,n (u )in (20)can be seen as a generalization of that given in [30];the solution for this integral is given in Appendix D.

A PPENDIX D

A PPENDIX :S OLUTION FOR K α,β,γ

m ,n (u )

We aim to ?nd a closed-form expression for the integral

K α,β,γ

m ,n (u )= ∞

x 2n ?1e ?αx 2Q m (γx ,β)dx ,(62)

where n ,m ∈N and α,β,γ,u ∈R +,which is a generalization

of that given in [35].The Marcum Q -function can be expressed in terms of a contour integral as [50]

Q m (γx ,β)=e ?γ2x 2+β

22 1p m

11?p

e γ2x 22p +β2p 2dp ,(63)where is a circular contour o

f radius less than unity enclosin

g the origin.Thus,we can express

K α,β,γ

m ,n (u )= ∞

u x 2n ?1e ?αx 2× e ?γ2x 2+β22

1p m 11?p e γ2x 22p

+β2p 2dp dx .(64)After some manipulations,and letting δ=α+γ2/2we have K α,β,γ

m ,n (u )=e

?β22

∞

x 2n ?1e

?δx 2

1p m 1

1?p

e γ2x 22p e β2p 2dp

dx ,(65)

and changing the integration order K α,β,γ

m ,n (u )=e

?β22

1p m 1

1?p

e β2p 2

∞

x 2n ?1e

?δx 2

γ2x 22p

dx dp .

(66)

Let ε=2δ?γ2/p ;the inner integral in (66)is given by

∞u x 2n ?1e ?εx 22dx =ε?n 2n ?1 n ,

2ε2 (67)where (n ,w)is the upper incomplete Gamma function.Since

n is a positive integer,we can use the following relationship

(n ,w)=(n ?1)!exp (?w)

n ?1 k =0

w k k !

,(68)

to express (66)as K α,β,γ

m ,n (u )

?β2

1p m 1

1?p

e β2p 2

ε?n 2n ?1(n ?1)!e

?u 2

2n ?1 k =0

u 2ε2 k 1

k !

dp ,

(69)

which can be conveniently rearranged as

K α,β,γ

m ,n (u )

n ?1

(n ?1)!e

?β2

×n ?1 k =0

u 22 k 1k ! ε?(n ?k )p m 11?p

e β2p 2e ?u 2ε2dp .

(70)

If we de?ne θ=γ2/2δ,we have

K α,β,γ

m ,n (u )

=δ?n 2

(n ?1)!e ?β22e ?u 2

×n ?1 k =0 u 2δ k k !

1?θp ?1 ?(n ?k )p m 1

1?p e β2p 2e δθu 2p dp .(71)

2704IEEE TRANSACTIONS ON INFORMATION THEORY,VOL.61,NO.5,MAY

2015

Fig.9.Contour integration for integral I m,n,k.

After some algebra,we obtain

Kα,β,γm,n(u)=δ?n

(n?1)!e?β22e?u2δ

n?1

k=0

u2δ

Iβ,δ,θ

m,n,k

(u),

(72)

where

Iβ,δ,θm,n,k (u) =?

p k?(n?m)

p?1

(p?θ)n?k

eβ

δθu2

p dp.

(73)

Let us de?ne the function G(p)as

G(p) =

p k?(n?m)

p?1

(p?θ)n?k

eδθu

p.(74)

Thus,we have

Iβ,δ,θm,n,k (u)=?

G(p)eβ

2dp.(75)

Interestingly,the integrand is in the form of an inverse Laplace transform.Hence,we aim to?nd a connection between this integral de?ned in the contour and the general Bromwich integral

g(t)=

2πj

c+j∞

c?j∞

G(p)exp(tp)dp=L?1{G(p);t}

(76)

Let us de?ne the contour given in Fig.9,where the value of c is chosen to be at the right of all the singularities in G(p).

Note thatθ=γ2

γ2+2α∈(0,1)by de?nition.Hence,the contour

integral along C is given by

C G(p)eβ

2dp=

2πj

c+j∞

c?j∞

G(p)eβ

2dp

Cβ

G(p)eβ

2dp.(77)

Using Cauchy-Goursat theorem,we can equivalently express

G(p)eβ

2dp=

G(p)eβ

2dp+

G(p)eβ

2dp,

(78)

where 0and 1are closed contours which enclose the

singularities at{p=0,p=θ}and p=1,respectively.

Combining(77)and(78),we have

2πj

c+j∞

c?j∞

G(p)eβ

2dp

L?1{G(p);t}

Cβ

G(p)eβ

2dp

G(p)eβ

2dp+

G(p)eβ

2dp.(79)

Since|G(p)|≤M R?l for some l>0on Cβas R→∞,the

integral

Cβ

equals zero.Hence,choosing a contour 0≡ ,

we have

Iβ,δ,θ

m,n,k

(u)=?

G(p)eβ

2dp

G(p)eβ

2dp?L?1{G(p)}t=β2

.(80)

Using the residue theorem,we can express

Iβ,δ,θ

m,n,k

(u)=Res

G(p)eβ

p=1

?L?1{G(p)}

t=b2

,(81)

where Res{F(p)}p=x denotes the residue of F(p)at p=x.

The calculation of this residue yields

Res

G(p)eβ

p=1

(1?θ)n?k

eδθu2eβ

2.(82)

To calculate the inverse Laplace transform of G(p),we use

partial fraction expansion in(74)to identify

G(p)=G1(p)+G2(p)

p k?(n?m)

eδθu

p?1

n?k

l=1

A l

(p?θ)l

,(83)

where the constants A0and A l are given by

A0=(1?θ)k?n(84)

A l=(1?θ)k+l?n?1.(85)

Using[51],we?nd an expression for the inverse Laplace

transforms as

L?1{G1(p)}

t=β22

=A0

β2

k+m?n

? 3

1,k+m?n+1;

β2

δθu2β2

(86)

and

L?1{G2(p)}

t=β2

n?k

l=1

A l

β2

k+m?n+l?1

×? 3

l,k+m?n+l;

θβ2

δθu2β2

(87)

LOPEZ-MARTINEZ et al.:EIGENV ALUE DYNAMICS OF A CENTRAL WISHART MATRIX 2705

where ?

3(b ,c ,w,z )is the regularized con?uent hypergeometric function of two variables,de?ned as a normalized version of the con?uent hypergeometric function of two variables given in [38,9.261.3],i.e.

3(b ,c ,w,z ) 3(b ,c ,w,z )

(c )

.(88)

Finally,combining (82),(86)and (87)yields the desired expression for (62).

A PPENDIX E

G ENERALIZED Q -F UNCTIONS IN

C OMMUNICATION T HEORY

In this appendix,we introduce some fundamentals on the generalized Q -functions used in this paper,which have played a key role in communication theory for almost 70years.The Marcum Q -function is named after Jess Ira Marcum (born Marcovitch [52]),who ?rst introduced this nomenclature in 1948when developing a statistical theory of target detection by pulsed radar [53,eq.(16)]as

Q (α,β)= ∞

βx exp ?x 2+α2

2 I 0(αx )dx ,

(89)where I 0(·)is the modi?ed Bessel function of the ?rst kind

and order zero.A more general form of this integral was also included in Marcum’s original report [53,eq.(49)],which is usually referred to as generalized Marcum Q -function.This function can be expressed as

Q M (α,β)=1αM ?1 ∞β

x M exp ?x

2+α22 I M ?1(αx )dx ,

(90)where I M ?1(·)is the modi?ed Bessel function of the ?rst kind and order M ?1.We see that (90)reduces to (89)for M =1,and hence it is usual to include the subindex in Q 1(α,β)when using the function in (89).Interestingly,the generalized Marcum Q -function can be expressed in terms of the standard Marcum Q 1function,and a ?nite number of modi?ed Bessel functions as [37,eq.(4.81)]Q M (α,β)=Q 1(α,β)

+exp ?

α2

+β22 M ?1 n =1

βα

I n (αβ).(91)

This family of functions is relevant not only in communi-cation theory but in general statistics,since the generalized Marcum Q -function represents the ccdf of the non-central chi-squared distribution (i.e.,the squared norm of a Gaussian vector)[54].

Later in the 70’s,Albert H.Nuttall introduced a more general form of this function [36,eq.(86)]as

Q M ,N (α,β)=

∞βx M exp ?x

2+α22 I N (αx )dx ,(92)which is usually referred to as Nuttall Q -function [39].Letting

M =N +1,the Nuttall Q -function relates to the generalized Marcum Q -function as

Q N +1,N (α,β)=αN Q N +1(α,β),

(93)

whereas the standard Marcum Q -function is given by

Q 1,0(α,β)=Q 1(α,β).As previously stated in (59),the Nuttall Q -function can be expressed in terms of a ?nite number of modi?ed Bessel functions and generalized Marcum Q -functions if the sum of the indices M +N is an odd number.

A CKNOWLEDGMENT

The authors would like to thank Dr.David Morales-Jimenez for insightful discussions and for his assistance on elaborating Fig.9.

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F.Javier Lopez-Martinez(S’05–M’10)received the M.Sc.and Ph.D.degrees in Telecommunication Engineering in2005and2010,respectively,from the University of Malaga(Spain).He joined the Communication Engineering Department at the University of Malaga in2005,as an associate researcher. In2010he stays for3months as a visitor researcher at University College London.He is the recipient of a Marie Curie fellowship from the UE under the U-mobility program at University of Malaga.Within this project, between August2012-2014he held a postdoc position in the Wireless Systems Lab(WSL)at Stanford University,under the supervision of Prof.Andrea J.Goldsmith.He’s now a Postdoctoral researcher at the Communication Engineering Department,Universidad de Malaga.

He has received several research awards,including the best paper award in the Communication Theory symposium at IEEE Globecom2013,and the IEEE Communications Letters Exemplary Reviewer certi?cate in2014. His research interests span a diverse set of topics in the wide areas of Communication Theory and Wireless Communications:stochastic processes, random matrix theory,statistical characterization of fading channels,physical layer security,massive MIMO and mmWave for5G.

Eduardo Martos-Naya received the M.Sc.and Ph.D.degrees in telecommunication engineering from the University of Malaga,Malaga,Spain, in1996and2005,respectively.In1997,he joined the Department of Communication Engineering,University of Malaga,where he is currently an Associate Professor.His research activity includes digital signal processing for communications,synchronization and channel estimation,and performance analysis of wireless systems.Currently,he is the leader of a project supported by the Andalusian regional Government on cooperative and adaptive wireless communications systems.

Jose F.Paris received his MSc and PhD degrees in Telecommunication Engineering from the University of Málaga,Spain,in1996and2004, respectively.From1994to1996he worked at Alcatel,mainly in the development of wireless telephones.In1997,he joined the University of Málaga where he is now an associate professor in the Communication Engi-neering Department.His teaching activities include several courses on signal processing,digital communications and acoustic engineering.His research interests are related to wireless communications,especially channel modeling and performance analysis.Currently,he is the leader of several projects supported by the Spanish Government and FEDER on wireless acoustic and electromagnetic communications over underwater channels.In2005,he spent?ve months as a visitor associate professor at Stanford University with Prof.Andrea J.Goldsmith.

LOPEZ-MARTINEZ et al.:EIGENV ALUE DYNAMICS OF A CENTRAL WISHART MATRIX2707

Andrea Goldsmith(S’90–M’93–SM’99–F’05)is the Stephen Harris professor in the School of Engineering and a professor of Electrical Engi-neering at Stanford University.She was previously on the faculty of Electrical Engineering at Caltech.Dr.Goldsmith co-founded and served as CTO for two wireless companies:Accelera,Inc.,which develops software-de?ned wireless network technology for cloud-based management of WiFi access points, and Quantenna Communications,Inc.,which develops high-performance WiFi chipsets.She has previously held industry positions at Maxim Tech-nologies,Memorylink Corporation,and AT&T Bell Laboratories.She is a Fellow of the IEEE and of Stanford,and has received several awards for her work,including the IEEE ComSoc Armstrong Technical Achievement Award,the National Academy of Engineering Gilbreth Lecture Award,the IEEE ComSoc and Information Theory Society joint paper award,the IEEE ComSoc Best Tutorial Paper Award,the Alfred P.Sloan Fellowship,and the Silicon Valley/San Jose Business Journals Women of In?uence Award.She is author of the book“Wireless Communications”and co-author of the books “MIMO Wireless Communications”and Principles of Cognitive Radio,all published by Cambridge University Press,as well as inventor on25patents.She received the B.S.,M.S.and Ph.D.degrees in Electrical Engineering from U.C.Berkeley.

Dr.Goldsmith has served as editor for the IEEE T RANSACTIONS ON

I NFORMATION T HEORY,the Journal on Foundations and Trends in Communi-cations and Information Theory and in Networks,the IEEE T RANSACTIONS ON C OMMUNICATIONS,and the IEEE Wireless Communications Magazine as well as on the Steering Committee for the IEEE T RANSACTIONS ON

W IRELESS C OMMUNICATIONS.She participates actively in committees and conference organization for the IEEE Information Theory and Communica-tions Societies and has served on the Board of Governors for both societies. She has also been a Distinguished Lecturer for both societies,served as President of the IEEE Information Theory Society in2009,founded and chaired the Student Committee of the IEEE Information Theory Society,and chaired the Emerging Technology Committee of the IEEE Communications Society.At Stanford she received the inaugural University Postdoc Mentoring Award,served as Chair of Stanfords Faculty Senate in2009,and currently serves on its Faculty Senate,Budget Group,and Task Force on Women and Leadership.