Complex singular Wishart matrices and applications

Transcription

1 Complex singular Wishart matrices and applications T Ratnarajah R Vaillancourt CRM-3185 May 2005 Thans are due to the Natural Sciences and Engineering Council of Canada and the Centre de recherches mathématiques of the Université de Montréal for their support Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave, Ottawa ON K1N 6N5 Canada tharm@mathstatuottawaca remi@uottawaca

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3 Abstract In this paper, complex singular Wishart matrices and their applications are investigated In particular, a volume element on the space of positive semidefinite m m complex matrices of ran n < m is introduced and some transformation properties are established The Jacobian for the change of variables in the singular value decomposition of general m n complex matrices is derived Then the density functions are formulated for all ran n complex singular Wishart distributions From this, the joint eigenvalue density of low ran complex Wishart matrices are derived Finally, application of these densities in information theory is given Keywords and Phrases Complex singular matrix distribution, complex Wishart distribution, singular value decomposition, channel capacity, Rayleigh distributed MIMO channel AMSMOS subject classification 6210, 60E05, 94A15, 94A05 Résumé On étudie les matrices de Wishart complexes singulières On introduit un élément volumétrique sur l espace des matrices m m complexes semi-définies de rang n < m et l on en établit certaines propriétés, telles la décomposition selon les valeurs singulières et leurs fonctions de densité, d où l on obtient la densité conjointe des valeurs propres de ces matrices de bas rang On appliques ces densités à la théorie de l information

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5 1 Introduction The present communictions revolution is due, in part, by advances in wireless communications, such as wireless internet and multimedia communications As the wireless industry becomes ubiquitous and polular, the need for a high-data rate and large user capacity on a wireless platform is the ey driver in developing robust communications techniques, which offer a substantially increased information capacity Recently, the theory of complex random matrices has attracted a lot of attention as a powerful mathematical tool that enables the computation of MIMO capacities for correlated channels Multiple input, multiple output MIMO wireless communication systems, where the number of inputs or transmitters n t is greater than the number of outputs or receivers n r n t > n r and the channel is correlated at the transmitter end The capacity of a communication channel expresses the maximum rate at which information can be reliably conveyed by the channel [1] A MIMO channel can be represented by an n r n t complex random matrix CN0, Σ r Σ t, where Σ t and Σ r represent the channel correlation at the transmitter and receiver ends, respectively If Σ r = σ 2 I nr or I nr and Σ t = I nt or σ 2 I nt then the channel is said to be an uncorrelated Rayleigh distributed channel, and the complex nonsingular Wishart matrix theory can be used to compute the capacities for both cases n r n t and n t > n r, see [2] owever, if Σ r = I nr and Σ t = Σ then the complex nonsingular and singular Wishart matrix theories are needed to compute the capacities for the cases n r n t and n t > n r, respectively See [3] for the case n r n t In this paper we assume that Σ r = I nr, Σ t = Σ and n t > n r, which leads us to represent the channel capacity in the form of a complex singular Wishart matrix This is the motivation behind this study Let an n m complex Gaussian or normal random matrix A be distributed as A CNM, I n Σ with mean E{A} = M and covariance cov{a} = I n Σ ere we read the symbol as is distributed as, CN denotes the complex normal distribution and denotes the Kronecer product Then the matrix W = A A is called a complex noncentral Wishart matrix If M = 0, then W is called a complex central Wishart matrix The complex central and noncentral Wishart distributions are denoted by CW m n, Σ and CW m n, Σ, Ω, respectively, where Ω = Σ 1 M M The complex Wishart matrices are well studied in the literature only for n m, for example, see [4], [5] and [6] In this paper, we extend the study of complex central Wishart distributions to the singular case, where 0 < n < m and n, m Z Thus the ran of W C m m is n provided the ran of A C n m is n A volume element on the space of positive semidefinite m m ermitian matrices of ran n < m is introduced see Theorem 1 The Jacobian of the change of variables in the singular value decomposition of general m n complex matrices is derived see Theorem 2 The density is derived for ran-n complex central Wishart distributions for all integers n, 0 < n < m see Theorem 3 The joint eigenvalue density of low ran complex Wishart matrices are derived see Theorem 4 It should be noted that singular Wishart and beta distributions are studied in [7] for real random matrices The singular value distribution of Gaussian random matrices is given in [8] This paper is organized as follows Section 2 provides the necessary tools for deriving the complex singular Wishart distribution theory Complex singular Wishart matrices are studied in Section 3 The capacity of a MIMO channel and the computational method are given in Section 4 2 Necessary tools In this section, we derive necessary tools for studying the singular Wishart distribution theory and MIMO channel capacity If 0 < n < m, then the density does not exist for W CW m n, Σ on the space of ermitian m m matrices, because W is singular and of ran n almost surely It can be shown that the density does exist on the 2mn n 2 -dimensional manifold, CS m,n, of ran n of positive semidefinite m m ermitian matrices W with n distinct positive eigenvalues Moreover, the set of all m n matrices E 1 with orthonormal columns is called the Stiefel manifold, denoted by CV n,m Thus, CV n,m = {E 1 m n; E 1 E 1 = I n } 1 The elements of E 1 can be regarded as the coordinates of a point on a 2mn n 2 -dimensional surface in the 2mn-dimensional Euclidean space Theorems 1 and 2 below are derived by means of the exterior product approach See [9], p 57, for the definition of the exterior product for real matrices, such as symmetric, sew-symmetric, upper-triangular and arbitrary matrices On the other hand, if X = X r + ix c is a complex matrix, then the exterior product is dx = dx r dx c Jacobian formulas for some important complex matrix factorizations are given in [10] The volume element dw in the reduced spectral decomposition W = E 1 ΛE1 is given by the following theorem Theorem 1 Let m and n be two positive integers such that 0 < n < m and consider an m m positive semidefinite ermitian matrix W CS m,n of ran n with decomposition W = E 1 ΛE 1, where the diagonal elements of Λ = diagλ 1,, λ n are positive real eigenvalues in decreasing order, λ 1 > > λ n > 0, and E 1 CV n,m Then the 1

6 volume element is where dw = 2π n n dλ = λ 2m 2n n λ λ l 2 dλ E1 de 1, 2 dλ, E1 de 1 = =l l= m e l de, and the matrix E 1 is appended with an m m n matrix E 2 such that the compound m m matrix, E = [E 1 : E 2 ] = [e 1,, e n : e n+1,, e m ] is unitary and Proof First, we note that Therefore, we have E dw E = dw = de 1 ΛE 1 + E 1 dλe 1 + E 1 Λ de 1, E 1 E 1 = I n, E 2 E 1 = 0 [ E 1 de 1 Λ + dλ + Λ de1 E 1 Λ de1 E 2 E2 de 1 Λ 0 The exterior product on the left side of equation 3 is equal to ] 3 E dw E = det E 2m dw = dw The lth row of E 2 de 1 Λ is [ e l de 1 e l de n ] Λ, n + 1 l m, and the exterior product of its elements is equal to det Λ 2 n e l de = n λ 2 n e l de Therefore, the exterior product of all the elements of E 2 de 1 Λ is m l=n+1 [ n λ 2 n ] n e l de = λ 2m 2n n m l=n+1 e l de 4 Second, we consider E 1 de 1 Λ + dλ + ΛdE 1 E 1 Since the columns of E 1 are orthonormal, we have E 1 E 1 = I n = E 1 de 1 = de 1 E 1 = E 1 de 1 This implies that the real and the imaginary parts of E1 de 1 are sew-symmetric and symmetric, respectively Moreover, we have E1 de 1 Λ + dλ + ΛdE1 E 1 = E1 de 1 Λ ΛE1 de 1 + dλ 5 The exterior product of the diagonal elements of the right side of equation 5 is given by dλ = dλ 6 Note that the diagonal elements of E1 de 1 Λ ΛE1 de 1 are zeros Now, we consider the upper diagonal elements of E1 de 1 Λ ΛE1 de 1 Since the matrix Re E1 de 1 is sewsymmetric, it can be written as 0 Re e 2 de 1 Re e n de 1 Re E1 Re e 2 de 1 0 Re e n de 2 de 1 = Re e 3 de 1 Re e 3 de 2 0 Re e n de 3 Re e n de 1 Re e n de 2 0 For < l, the, lth element of Re E 1 de 1 Λ Λ Re E 1 de 1 is given by Re e l de λ λ l Therefore, the exterior product of the upper diagonal elements of Re E 1 de 1 Λ Λ Re E 1 de 1 2

7 is given by Similarly, since the matrix Im E 1 Im E1 de 1 = n Ree l de λ λ l 7 de 1 is symmetric, it can be written as Im e 1 de 1 Im e 2 de 1 Im e 3 de 1 Im e n de 1 and the exterior product of the upper diagonal elements of is given by Im e 2 de 1 Im e 2 de 2 Im e 3 de 2 Im e n de 2 Im E 1 de 1 Λ Λ Im E 1 de 1 Im e n de 1 Im e n de 2 Im e n de 3 Im e n de n n Ime l de λ λ l 8 Therefore, the exterior product of the elements of the right side of equation 3 is obtained by multiplying equations 4, 6, 7, and 8, ie, n dw = 2π n n λ 2m 2n λ λ l 2 dλ E 1 de 1 9 Note that we must divide the volume element by 2π n to normalize the arbitrary phases of the n elements in the first row of E 1 The volume element dw or Jacobian in the singular value decomposition is given by the following theorem Theorem 2 Let Z be an m n complex matrix and Z = E 1 Υ the nonsingular part of the singular value decomposition, where E 1 CV n,m, Un and the diagonal elements of Υ = diagυ 1,, υ n are positive real singular values with υ 1 > > υ n > 0 Then we have where dυ = dz = 2π n n υ 2m 2n+1 n υ 2 υl 2 2 dυ E1 de 1 d 10 dυ, d = =l l= h l dh, E1 de 1 = l= m e l de, and the matrix E 1 is appended with an m m n matrix E 2 such that the compound m m matrix, E = [E 1 : E 2 ] = [e 1,, e n : e n+1,, e m ] is unitary and Proof First, we note that Therefore, we have dz = de 1 Υ + E 1 dυ + E 1 Υ d E 1 E 1 = I n, E 2 E 1 = 0 [ E E dz = 1 de 1 Υ + dυ + Υ d de 1 Υ The exterior product of the left side of equation 11 is equal to E 2 E dz = dz ] 11 3

8 As in the proof of Theorem1 see equation 4, the exterior product of all the elements of E2 de 1 Υ is n n m E2 de 1 Υ = e l de 12 υ 2m 2n l=n+1 Second, we consider E 1 de 1 Υ + dυ + Υd Since is unitary we have = I n = d = d = d This implies that the real and imaginary parts of d are sew-symmetric and symmetric, respectively Similarly, for E1 de 1 Thus the n n matrix of differential forms is with entries The exterior product of the elements of T is T = E 1 de 1 Υ + dυ + Υd = E 1 de 1 Υ Υ d + dυ, 13 dυ + υ Ime de Imh dh = l, T l = e de lυ l υ h dh l > l, e l de υ l υ h l dh < l T = T n T l T l, where T l T l = υ 2 υl 2 Re e l de h l dh υ 2 υl 2 Im e l de h l dh = υ 2 υl 2 2 e l de h l dh 14 Therefore, the exterior product of equation 11 is dz = 2π n E2 de 1 Υ T n n = 2π n l= = 2π n n n υ 2m 2n e l de υ 2m 2n+1 m l=n+1 l= e l h l dh n n de υ υ 2 υ 2 2 n l dυ υ 2 υl 2 2 dυ E 1 de 1 d 15 Note that we must divide the volume element by 2π n to normalize the arbitrary phases of the n elements in the first row of E 1 The volume of the Stiefel manifold CV n,m is given by where the complex multivariate gamma function is VolCV n,m = E CV 1 de 1 = 2n π mn n,m CΓ n m, 16 CΓ n m = π nn 1/2 n Γm + 1, m > n 1 The differential form de 1 1 Vol[CV n,m ] E 1 de 1 = CΓ nm 2 n π mn E 1 de

9 has the property that CV n,m de 1 = 1, and it represents the aar invariant probability measure on CV n,m If m = n, then we get a special case of Stiefel manifold, the so-called unitary manifold, defined by CV n,n Un = {En n; E E = I n }, that is, the set of unitary n n matrices The volume of Un is given by Vol [Un] = Un E de = 2n π n2 CΓ n n 18 The probability distributions of random matrices are often derived in terms of hypergeometric functions of matrix arguments The following complex hypergeometric function of two matrix arguments is required in the sequel, ie, 0F n 0 X, Y = =0 κ C κ XC κ Y,!C κ I m where X C m m, Y C n n and 0 < n < m Moreover, κ = 1,, n denotes a partition of the integer with 1 n 0 and = n and κ denotes summation over all partitions κ of The complex zonal polynomial also called Schur polynomial[11] of a complex matrix Y defined in [5] is C κ Y = χ [κ] 1χ [κ] Y, 19 where χ [κ] 1 is the dimension of the representation [κ] of the symmetric group, n i<j χ [κ] 1 =! i j i + j n i=1, 20 i + n i! and χ [κ] Y is the character of the representation [κ] of the linear group given as a symmetric function of the eigenvalues, λ 1,, λ n, of Y by [ ] det λ j+n j i χ [κ] Y = [ ] 21 det Note that both the real and complex zonal polynomials are particular cases of the general α Jac polynomials C κ α Y, where α = 1 for complex and α = 2 for real zonal polynomials, respectively See[12] for details In this paper we only consider the complex case; therefore, for notational simplicity we drop the superscript of Jac polynomials, as was done in equation 19, ie, C κ Y := C κ 1 Y Finally, we have [ ] r 1 C κ I n = 2 2! 2 n i<j 2 i 2 j i + j r κ i=1 2 i + r i! where [ ] 1 2 n = κ r i=1 λ n j i 1 n i i for a partition κ of with r nonzero parts and a = aa + 1 a Complex Singular Wishart Matrices In this section, we derive the complex singular Wishart density and the joint eigenvalue density of the complex singular Wishart matrix Theorem 3 Let m and n be two positive integers such that 0 < n < m The density of W CW m n, Σ on the space CS m,n of m m positive semidefinite ermitian matrices of ran n is given by fw = π nn m CΓ n ndet Σ n etr Σ 1 W det Λ n m, 22 where W = E 1 ΛE 1, E 1 CV n,m, Λ = diagλ 1,, λ n, and etr denotes the exponential of the trace, etr = exptr 5

10 Proof Let W = A A, where the n m matrix A is distributed as A CN0, I n Σ and Σ is an m m positive definite ermitian matrix The density of A is fa = π nm det Σ n etr AΣ 1 A da = π nm det Σ n etr Σ 1 A A da, 23 where the volume element da n m l=1 da l is included to facilitate the calculation of Jacobians when we transform A Transforming A = E 1 Υ as in Theorem 2 results in the desired parameterization W = E 1 ΛE 1, where Λ = Υ 2 and det Λ = det Υ 2 Thus we have dλ = n n n dλ = 2 n υ dυ = 2 n υ dυ 24 From Theorems 1 and 2 and equation 24 we can write da as n da = 2 n dw d Since n λ = det Λ, the joint density of W and is λ n m π nm det Σ n etr Σ 1 W 2 n det Λ n m dw d Integrating with respect to over the Stiefel manifold CV n,m and using equation 16, we obtain the marginal density function of W, given by equation 22 The following theorem gives the joint density of the eigenvalues of a complex singular Wishart matrix Theorem 4 Let m and n be two positive integers such that 0 < n < m and consider the m m positive semidefinite ermitian matrix W CW m n, Σ The joint density of the positive eigenvalues, λ 1,, λ n, of W is fλ = πnn 1 det Σ n n n λ m n λ λ l 2 CΓ n ncγ n m etr Σ 1 E 1 ΛE 1 de1, 25 E 1 CV n,m where W = E 1 ΛE1, E 1 CV n,m, and Λ = diagλ 1,, λ n Moreover etr Σ 1 E 1 ΛE 1 de1 = 0 F n 0 Σ 1, Λ 26 E 1 CV n,m Proof Substituting 2 into the Wishart density 22 and integrating with respect to E 1 over the Stiefel manifold CV n,m, we have fλ = πnn 1 2π n n n CΓ n ndet Σ n λ m n λ λ l 2 etr Σ 1 E 1 ΛE 1 E 1 de 1 27 E 1 CV n,m Now using 17 we obtain the desired results 25 Note that, if Σ = σ 2 I m, then the joint density of the eigenvalues λ 1,, λ n has a simple form which does not require a complex hypergeometric function representation Corollary 1 Let W CW m n, σ 2 I m with 0 < n < m Then the joint density of the eigenvalues, λ 1,, λ n, of W is gλ = πnn 1 σ 2 nm n n λ m n λ λ l 2 exp 1 n CΓ n ncγ n m σ 2 λ, 28 where W = E 1 ΛE 1, E 1 CV n,m, and Λ = diagλ 1,, λ n 6

11 Proof Putting Σ = σ 2 I m in Theorem 4 and noting that complete the proof E 1 CV n,m etr 1 σ 2 E 1ΛE1 de 1 = etr 1σ 2 Λ de 1 E 1 CV n,m = exp 1 n σ 2 λ i i= The MIMO Channel Capacity Recently, in response to the demand for higher bit rates in wireless communications, industrial researchers have exploited the use of multiple input, multiple output systems, as shown in Fig 1 below These studies show that MIMO systems increase capacity significantly over single input, single output SISO systems For example, if n = min{n t, n r }, a MIMO uncorrelated Rayleigh distributed channel achieves almost n more bits per hertz for every 3-dB increase in signal-to-noise ratio SNR compared to a SISO system, which achieves only one additional bit per hertz for every 3-dB increase in SNR [2] But the channel coefficients from different transmitter antennas to a single receiver antenna can be correlated This channel correlation, which degrades the channel capacity [13], depends on the physical parameters of a MIMO system and the scatterer characteristics The physical parameters include the antenna arrangement and spacing, the angle spread, the angle of arrival, etc One of the objectives of this paper is to evaluate this capacity degradation for the channel matrix CN0, I nr Σ with n t > n r This will be done by deriving closed form ergodic capacity formulas for correlated channels and their numerical evaluation The complex signal received at the jth output can be written as n t y j = h ij x i + v j, 30 i=1 where h ij is the complex channel coefficient between input i and output j, x i is the complex signal at the ith input and v j is complex Gaussian noise The signal vector received at the output can be written as y 1 = h 11 h nt1 x 1 + v 1, y nr h 1nr h ntn r x nt v nr ie, in vector notation, y = x + v, 31 where y, v C nr, C nr nt, and x C nt see Fig 1 The total power of the input is constrained to ρ, E{x x} ρ or tr E{xx } ρ In this section, we shall deal exclusively with the linear model 31 and compute the capacity of MIMO channel models for n t > n r We assume that is a complex Gaussian random matrix whose realization is nown to the receiver, or equivalently, the channel output consists of the pair y, Note that the transmitter does not now the channel and the input power is distributed equally over all transmitting antennas Moreover, if we assume a bloc-fading model and coding over many independent fading intervals, then the Shannon or ergodic capacity of the random MIMO channel [2] is given by C = E { log det Int + ρ/n t }, 32 where the expectation is evaluated using a complex Gaussian density If CN0, I nr Σ then the channel is Rayleigh distributed and correlated at the transmitter end This is typical of fixed or mobile communication environments ere the covariance matrix of the rows of is denoted by Σ, which is an n t n t positive definite ermitian matrix Let W = CW nt n r, Σ and n t > n r Then the channel capacity can be written as C = E W {log det I nt + ρ/n t W}, 33 7

12 v 1 x 1 h 11 + y 1 h 12 h 21 v 2 x 2 h 22 + y 2 h 2nr xn t h n 1 t h n 2 t h n t nr h 1nr + v nr y nr Figure 1: A MIMO communication system where the expectation is evaluated using a complex singular Wishart density given in Theorem 3 Let λ 1 > > λ nr be the eigenvalues of W and Λ = diagλ 1,, λ nr Then the capacity can also be computed using a joint eigenvalue density, fλ, or the single unordered eigenvalue density, fλ, ie, C = E Λ {log = n r nr [1 + ρ/n t λ ] E λ {log1 + ρ/n t λ } } = n r E λ {log1 + ρ/n t λ} 34 The joint eigenvalue density of a complex singular Wishart matrix is given in Theorem 4 From this joint eigenvalue density we can obtain a single unordered eigenvalue density, fλ, by dividing fλ by n r! and integrating with respect to λ 2,, λ nr As a numerical example, we compute the channel capacity of a correlated 2 4 channel matrix n r = 2 and n t = 4, ie, CN0, I 2 Σ, where we assume the eigenvalues of the positive definite ermitian matrix Σ are 18090, 13090, 06910, In this case W is 4 4 complex singular Wishart matrix with two nonzero eigenvalues λ 1 and λ 2 The joint eigenvalue distribution is given by 1 fλ 1, λ 2 = 12det Σ 2 λ 1λ 2 2 λ 1 λ 2 2 0F 2 0 Σ 1, Λ, 35 where Λ = diagλ 1, λ 2 The capacity of the correlated channel, CN0, I 2 Σ, is given by C c = λ1 0 0 [log1 + ρ/4λ 1 + log1 + ρ/4λ 2 ] fλ 1, λ 2 dλ 2 dλ 1 36 For comparison purpose we also compute the capacity of the uncorrelated 2 4 channel matrix, ie, CN0, I 2 σ 2 I 4 In this case, the joint eigenvalue density of the complex singular Wishart matrix is given by gλ 1, λ 2 = 1 12σ 16 λ 1λ 2 2 λ 1 λ 2 2 e λ1+λ2/σ 2 37 From equation 37, we can evaluate the single unordered eigenvalue density, fλ, which is given by gλ = λ2 e λ/σ2 12σ 6 λ 2 σ 4 6 λ σ

13 The capacity of this uncorrelated channel CN0, I 2 σ 2 I 4 is given by C u = 2 0 log1 + ρ/4λ gλ dλ 39 Figure 2 shows the capacity in nats 5 vs signal-to-noise ratio for a 2 4 correlated/uncorrelated Rayleigh fading channel matrix From this figure we note the following: i the capacity is decreasing with channel correlation, ii the capacity is increasing with SNR Capacity in nats 8 6 C u 4 C c Signal to noise ratio SNR Figure 2: Capacity vs SNR for n t = 4 and n r = 2, ie, 2 4 channel matrix C u and C c denote the capacity of uncorrelated and correlated Rayleigh channels, respectively 5 Conclusion In this paper, we studied the complex singular Wishart distribution and its application In particular, we derived the complex singular Wishart density and joint eigenvalue density of a complex singular Wishart matrix Using these distributions, both correlated and uncorrelated MIMO Rayleigh channel capacity formulas were obtained The capacities of 2 4 MIMO Rayleigh channel matrices were computed for b correlated and uncorrelated channels It was also shown how channel correlation degrades the capacity of the communication systems Appendix The real singular Wishart density is derived in [7, Theorem 6] In this appendix we give the joint eigenvalue density of a real singular Wishart matrix Note that we follow the notation of [7] and [9] Theorem 5 Let m and n be two positive integers such that 0 < n < m and consider an m m positive semidefinite symmetric Wishart matrix W W m n, Σ Then the joint density of the positive eigenvalues, λ 1,, λ n, of W is π n2 /2 2 mn/2 det Σ n/2 n n λ m n 1/2 λ λ l Γ n n/2γ n m/2 1 2 Σ 1 E 1 ΛE1 de 1, E 1 V n,m etr 5 In equation 34, if we use log e then the capacity is measured in nats If we use log 2 then the capacity is measured in bits Thus, one nat is equal to e bits/sec/z e =

14 where W = E 1 ΛE1, E 1 V n,m and Λ = diagλ 1,, λ n Moreover 1 2 Σ 1 E 1 ΛE1 de 1 = 0 F n Σ 1, Λ References E 1 V n,m etr [1] R B Ash, Information Theory, Dover, New Yor, 1965 [2] I E Telatar, Capacity of multi-antenna Gaussian channels, Eur Trans Telecom, [3] T Ratnarajah, R Vaillancourt and M Alvo, Complex random matrices and Rayleigh channel capacity, Communications in Information & Systems, 3 Oct [4] N R Goodman, Statistical analysis based on a certain multivariate complex Gaussian distribution An introduction, Ann Math Statist, [5] A T James, Distributions of matrix variate and latent roots derived from normal samples, Ann Math Statist, [6] T Ratnarajah, R Vaillancourt and M Alvo, Eigenvalues and condition numbers of complex random matrices, SIAM Journal on Matrix Analysis and Applications, to appear [7] Uhlig, On singular Wishart and singular multivariate beta distributions, Ann of Statist, [8] J Shen, On the singular values of Gaussian random matrices, Linear Alg Appl, [9] R J Muirhead, Aspects of Multivariate Statistical Theory, Wiley, New Yor, 1982 [10] T Ratnarajah, R Vaillancourt and M Alvo, Jacobians and hypergeometric functions in complex multivariate analysis, Canadian Applied Mathematics Quarterly, to appear [11] I G Macdonald, Symmetric Functions and all Polynomials, Oxford University Press, New Yor, 1995 [12] T Baer and P Forrester, The Calogero-Sutherland model and generalized classical polynomials, Commun Math Phys, [13] C N Chuah, D Tse, J M Kahn and R A Valenzuela, Capacity scaling in MIMO wireless systems under correlated fading, IEEE Trans on Information Theory, 48 March