Eigenvalue Statistics of Finite-Dimensional Random Matrices for MIMO Wireless Communications

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1 Eigenvalue Statistics of Finite-Dimensional Random Matrices for MMO Wireless Communications Giuseppa Alfano Universitá del Sannio, Benevento, taly Antonia M. Tulino Universitá Federico, Napoli, taly Angel Lozano Bell Labs Lucent Technologies, Holmdel, Nj, USA Sergio Verdú Princeton University, Princeton, NJ, USA Abstract This paper characterizes the marginal probability density function of an unordered eigenvalue of finite-dimensional random matrices of particular interest in MMO multiple-input multiple-output wireless communications. Specifically, a technique is presented for deriving the eigenvalue statistics in one-side correlated Rayleigh-faded channels and iicean-faded channels, or out cochannel interferers. The exact expressions found turn out to be extremely useful in calculating information-theoretic quantities. As an application, we calculate the ergodic mutual information for all the abovementioned channel fading conditions, obtaining a closed form formula for the Rayleigh case and, in turn, a series expression for the Ricean faded one.. NTRODUCTON Much of the early analytical characterizations of MMO channels were asymptotic, either in the number of antennas or in the SNR signal-to-noise ratio. More recently, strides have been made in finding exact nonasymptotic expressions. The ergodic mutual information, specifically, has been expressed for: Channels D independent identically distributed Rayleigh-faded entries [1]. Channels D Ricean-faded entries [2], [3]. Channels various forms of antenna correlation [4], [5], [6], [7] and/or interference [8]. n [1], the ergodic capacity of channels D Rayleighfaded entries was obtained by integrating over the marginal distribution of an unordered eigenvalue of a central Wishart matrix. Posterior works, however, mostly abandoned this approach due to the lack of explicit expressions for such marginal distribution in more general channels. The aim of this paper is, precisely, to present a unified procedure for calculating the marginal density distribution of an unordered eigenvalue for the most popular MMO channel models. Via this distribution, one can then calculate the ergodic mutual information and/or the channel capacity via a simple, onefolded integral. n addition, this distribution is also relevant to other MMO problems such as the study of adaptive transmission schemes spatial diversity at the receiver [9] and/or at the transmitter [10] and the performance evaluation of differential MMO transmission [11].. SGNAL AND CHANNEL MODEL Denoting by n T and the number of transmit and receive antennas, we consider the complex frequency-flat linear model y = ghx+ n where x and y are the input and output vectors while n is white Gaussian noise. The channel is represented by the n T zero-mean random matrix H normalized such that E[Tr{HH }]= n T. 1 The noise is composed of both thermal noise and out-of-cell interference such that L n = gl H l x l + n th 2 where n th is the thermal noise, L the number of interferers, x l the signal transmitted by the l-th interferer, modeled as a Gaussian vector unit covariance matrix, and g l H l the channel from such interferer. The number of transmit antennas at the l-th interferer is denoted by m l while the entries of H l are modeled as zero-mean jointly Gaussian random variables distribution and such that e Tr{Θ 1 H lh l } fh l = π m ldetθ m l E[Tr{H l H l }]=m l. The single-sided spectral density of the noise is denoted by = E[ n 2 ] /06/$20.00 c 2006 EEE

2 and its normalized conditional covariance is defined as Φ n {H l } E[nn {H l }]. 4 Under these assumptions, the received SNR can be computed as [12] SNR = g E[ x 2 ] E[Tr{HH Φ 1 n {H l }}] n T 5 expectation over x, H and the set {H l }.fφ n =, 5 reduces to SNR = g E[ x 2 ]. 6 Since the fading encountered by wireless systems tends to be either Rayleigh or Ricean, the entries of H can be modeled as jointly Gaussian. With that, the characterization of H entails simply determining the mean and correlation between its entries. A. MMO channel Rayleigh-faded nterferers We model the out-of-cell interferers as Rayleigh-faded. By conveniently defining [12] l g l E[ x l 2 ] m l E[Tr{H l H l }] = g l E[ x l 2 ] 7 as the average energy per antenna received from the l-th interferer expectation over both x l and H l, we can write L = l + ρ th, 8 ρ th = E[ n th 2 ]/. n the remainder we seek some insight by concentrating on the interference-limited scenario that is [12] 1 ρ th l. For this scenario we can reasonably assume that Φ n can be modeled as a complex Wishart matrix L = L m l degrees of freedom and covariance matrix Θ, whose distribution can be written as [13] fφ n = detφ n L nr exp [ TrΘ 1 Φ n ] Γ nr LdetΘ L, Γ p q, p q, the complex multivariate Gamma function [13] p Γ p q =π pp 1 2 q l! We will also assume that L. Under these assumptions, we can define the positive-definite matrix S = HH Φ n 1 whose marginal eigenvalues distribution we shall characterize in the following. Herein, the law of H depends on the statistical characterization of the fading. n the remainder we focus for H on two different scenarios: MMO Rayleigh-faded channels Rayleigh-faded interferers, MMO Ricean-faded channels Rayleigh-faded interferers. 1 Rayleigh-faded Channels: n the Rayleigh case, we adhere to the so-called separable model whereby H = Θ 1/2 R WΘ1/2 T 9 where W is a n T matrix D zero-mean unit-variance complex Gaussian random entries while Θ T and Θ R are deterministic transmit and receive correlation matrices. n the following, we further assume that Θ T = nt and n T, and for ease of notation define τ = n T. The entries of H are jointly Gaussian distribution [14] e Tr{Θ 1 R HH } fh = π ntnr detθ R nt hence the distribution of S is [13] fs = τ+l l! ω n T l n T l!l l! dets τ det + Ω 1 S nt+l 10 ω l the l-th eigenvalue of Ω = Θ 1 Θ R. 2 Ricean-faded Channels: The Rice case can be modeled by incorporating an additional deterministic matrix H 0 containing unit-magnitude entries 1 K H = K +1 Θ1/2 R WΘ1/2 T + K +1 H 0 11 so that K E[H] = K +1 H 0 12 the Ricean K-factor quantifying the ratio between the deterministic unfaded and the random faded energies. For this channel, still assuming uncorrelated transmit antennas, namely Θ T = nt, the distribution of S can be written as [13] fs = K+1e Tr{M} dets τ det+k+1s n T +L 1F 1 n T +L, n T, M τ+l l! n T l!l l! + 1 K+1 S

3 M = KH 0 H 0 Θ 1 and 1 F 1 a, b, the confluent hypergeometric function of matrix argument [13].. CHARACTERZATON OF THE UNORDERED EGENVALUES This main section is devoted to the formulation of a general strategy to obtain the marginal density distribution of an unordered eigenvalue of finite-dimensional random matrices conforming to the models described in the previous section. The method is based on two main results that are stated as Lemmas in the following. Lemma 1 [15], [16] A hypergeometric function of two square matrix arguments A and B of the same 1 dimension m and all distinct eigenvalues can be expressed as a ratio of determinants pf q x1 + m,...,x p + m y 1 + m,...,y q + m A, B = c m detf m m a k a l b k b l a k resp. b k thek-th eigenvalue of A resp. B, [ m 1 p ] t m i=1 c m = x i + t t q j=1 y j + t t=1 and the i, j-th entry of F given by x1 +1,...,x F i,j = p F p +1 q y 1 +1,...,y q +1 ; a ib j 14 Lemma 2 [17] Let F and G be two n n matrices whose i, j-th entries are, respectively, F i,j = f j w i and G i,j = g j w i where f j and g j, j =1,...,n,are functions defined o +. Then, for b>a>0, n detf detg hw k dw 1...dw n = n! deta [a,b] n k=1 where h is a function defined o + and A is another n n matrix whose i, j-th entry is A i,j = b a f i wg j whw dw Note that, in [17], the factor n! does not appear because the variables w 1,...,w n are ordered. As claimed in Lemma 2, in contrast, they are unordered. Using these lemmas, we can now state the main result of the paper. 1 For a generalization of this formula to square matrices of different dimensions, or nondistinct eigenvalues, see [16]. Theorem 1 Let ψ i be a function defined o + and consider a random matrix whose unordered eigenvalues admit a joint distribution of the form m m fλ =K ζλ k λ k λ l detψλ 15 k=1 Λ = diag{λ 1,...,λ m }, K a normalization constant and ΨΛ a matrix whose i, j-th entry is ψ i λ j. The marginal probability density function of the unordered eigenvalues can be written as fλ = K m m λ j 1 ζλψ i λdi, j 16 where K is a proper normalizing constant and Di, j is the i, j-cofactor of the m m matrixa whose l, k-th entry equals 0 λ l 1 ζλψ k λdλ Proof: Using the Laplace determinant expansion, 15 becomes m m fλ = K 1 i+j λ j 1 1 ζλ 1 ψ i λ 1 m k=2 ζλ k detṽλ det ΨΛ 17 ΨΛ and ṼΛ the m 1 m 1 matrices obtained deleting the first row and column from the matrix ΨΛ and VΛ, respectively. Here, VΛ is the matrix whose i, j-th entry is λ j 1 i. Eq. 16 follows from integration of 17 over m 1 eigenvalues via Lemma 2. Note that the choice of λ 1 in 17 has no effect on the final result since we started from an unordered distribution. Based on this general result, we can express the marginal density distributions of the unordered eigenvalues for two cases of interest in MMO for which no analytical solutions existed: One-side correlated Rayleigh-faded channel oneside correlated Rayleigh-faded interferers. One-side correlated Ricean-faded channel one-side correlated Rayleigh-faded interferers. These expressions will be stated as corollaries in the remainder 2. n the absence of interference, the corresponding characterizations can be found in [7] and [3], respectively. Our formulation in this paper unifies and generalizes those 2 For sake of brevity the proofs are relegated to the full version of the paper [18]. 4127

4 previous results and may lead to further expressions of interest. Corollary 1 Consider a MMO communication where both the desired user and the interferers are affected by Rayleigh fading. Let ω 1,...,ω nr be the distinct 3 singular values of Ω. Then, the marginal density distribution of an unordered singular value of S is fλ =K K = nr 1 nt +L+l l nr 1 ω l 1 ω k λ τ+j 1 Di, j λ/ω i τ+l+1 l nr ω n T l τ+l l! n T l! l!l l! 19 and Di, j the i, j-cofactor of the matrix whose l, k-th entry equals τ+l 1!L l! τ+l! ω τ+l k. Corollary 2 Consider a MMO communication where the desired user experiences a Rice distributed fading, while the interferers are affected by Rayleigh fading. Let us assume Θ = Θ R and let µ 1,...,µ nr be the distinct singular values of M = KH 0 H 0 Θ 1. The marginal density distribution of an arbitrary singular value of S is K = fλ =K 1F 1 τ+l+1,τ+1, e i µ i nr µ k µ l K+1 τ+j λ τ+j 1 1+K+1λ τ+l+1 Di, j K+1µ iλ 1+K+1λ n T+L l! n T l! l!l l! 1 20 l nr τ+l+l lτ+l 21 and Di, j the i, j-cofactor of the matrix whose l, k-th entry equals L L µ L l! t k δ+l 1! t [δ+1] tδ+l! 1 F 1 δ + l, δ + L +1,µ k, t=0 δ = τ + t and [a] b = a+b 1! a 1!. 3 The assumption of distinct eigenvalues can be relaxed resorting to the results in [16]. This allows, in particular, to deal the case Θ = Θ R. V. ERGODC MUTUAL NFORMATON n this section, we capitalize on the unordered eigenvalue characterizations in order to calculate the ergodic mutual information coherent reception, which can be expressed as a function of the SNR as [1] [ SNR = E log 2 det + = E [log 2 1+ SNR n ] R ge[tr{s}] S ] SNR nr ge[tr{s}] λs 22 where λ denotes an arbitrary nonzero eigenvalue of S and the expectation is over its distribution. n the remaining of the Section we exploit 22 together the newly obtained expressions to evaluate the mutual information. Proposition 1 Under the hypotheses of Corollary 1, the ergodic mutual information is given by SNR= K log e 2 τ+j 1 ν=0 Di,jω τ+j i τ+j 1!L j! L+τ! 23 j=1 i=1 2F 1 1,τ + L ν;τ + L + 1;1 1 γω i L + τ ν γ = SNR L, gn T Tr{Ω} and K and Di, j as in Corollary 1. Example 1 Consider a mobile terminal radiating from n T =4 transmit antennas, loosely correlated, i.e., Θ T =. Then, let the receiving base station have =2 antennas Θ R i,j = e 0.05 d2 i j 2 24 which corresponds to a d-wavelength antenna separation and a broadside truncated Gaussian power azimuth spectrum 2 root-mean-square spread. Consider further the presence of an interferer, equipped m 1 =5uncorrelated transmit antennas, and assume the channel from this interferer to be H 1 = Θ 1/2 W, Θ i,j = e 0.03 d2 i j For d=2, the ergodic mutual information conveyed by the channel, given by 23, is depicted in Fig 1. Also shown is the result of 2000 random realizations obtained via Montecarlo, which validates the analytical function. The ergodic mutual information for the case of Ricean faded desired user impaired by cochannel Rayleigh interference can be encompassed by expanding the confluent 4128

5 SNR hypergeometric function in 20 and following the same steps as for Proposition 1. Proposition 2 Under the hypotheses of Corollary 2, the ergodic mutual information is given by SNR = nr K log e 2 where γ = SNR j=1 i=1 δ+j 1 ν=0 Di,jL j! [τ+1] L t=0 µ t i δ+j 1! δ! t! 2F 11,δ+L ν;δ+l+1;1 K+1 γ L+δ ν L K+1 gn T[+KTr{H 0H 0 Θ 1 }], while δ, K and Di, j are as in Corollary 2. V. CONCLUSONS Analytical expressions for the marginal density distribution of an unordered eigenvalue of finite-dimensional random matrices of particular interest in wireless communications are given and their validity is confirmed by matching Montecarlo simulated statistics. The newly obtained formulas are especially useful in evaluating information theoretic performance measures of MMO channels whose communication is impaired by cochannel interference nr=2 nt=4 L= SNR Fig. 1. Ergodic mutual information given in 23 n T =4, =2, L=5, Θ T =, Θ R as in 24 and Θ as in 25. REFERENCES [2] M. Kang and M. S. Alouini, Capacity of MMO Rician channels, EEE Transactions on Wireless Communications, Vol. 5, No. 1, pp , Januray [3] G. Alfano, A. Lozano, A. Tulino, S. Verdú, Mutual nformation and Eigenvalues Distribution of MMO Ricean Channels, Proc. of STA 2004, Parma, taly, October 10-13, [4] M. Chiani, M. Z. Win, and A. Zanella, On the capacity of Spatially Correlated MMO Rayleigh-Fading Channels, EEE Transactions on nformation Theory, Vol.49, No.10, pp , October [5]P.J.Smith,S.Roy,M.Shafi CapacityofMMOSystems Semicorrelated Flat-Fading, EEE Transactions on nformation Theory, Vol.49, No.10, pp , Oct [6] M. Kiessling, Unifying Analysis of Ergodic MMO Capacity in Correlated Rayleigh Fading Environments, Eur. Trans. Telecom., Vol. 16, No.1, pp , January [7] G. Alfano, A. Lozano, A. Tulino, S. Verdú, Capacity of MMO Channels One-sided Correlation, Proc. of SSSTA 2004, Sydney, AU, August 30-September 3, [8] M. Kang, L. Yang and M. S. Alouini, Capacity of MMO channels in the presence of co-channel interference, EEE Transactions on Wireless Communications, submitted. [9] M. S. Alouini and A. J. Goldsmith, Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques, EEE Transactions on Vehicular Technology, Vol. 48, No. 4, pp , July [10] S. K. Jayaweera, and H. V. Poor, Capacity of Multi-antenna Systems both Receiver and Transmitter Channel State nformation, EEE Transactions on nformation Theory, Vol.49, No.10, pp , Oct [11] A. L. Moustakas, S. H. Simon, T. L. Marzetta Capacity of Differential versus non-differential Space-Time Unitary Modulation for MMO Channels, EEE Transactions on nformation Theory, submitted. [12] A. Lozano, A. Tulino, S. Verdú, Multiple Antenna Capacity in the Low Power Regime, EEE Transactions on nformation Theory, Vol.49, No.10, pp , October [13] A. T. James, Distribution of Matrix Variates and Latent Roots Derived from Normal Samples, The Annals of Mathematical Statistics, Vol. 35, No.2, pp , June [14] C. G. Khatri, On Certain Distribution Problems Based on Positive Definite Quadratic Functions in Normal Vectors, The Annals of Mathematical Statistics, Vol.37, No.2, pp , April [15] K.. Gross, D. St. P. Richards, Total positivity, spherical series, and hypergeometric functions of matrix argument, Journal of Approximation Theory, Vol. 59, No. 2, pp , February [16] A. Y. Orlov New solvable matrix integrals, nt. Journ. of Modern Physics A, Vol. 19, pp , [17] C. G. Khatri, On the moments of traces of two matrices in three situations for complex multivariate normal populations, Sankhya, The ndian J. Statist., Ser. A, Vol. 32, Pt.1, pp , February [18] G. Alfano, A. Tulino, A. Lozano, S. Verdú, Eigenvalues Statistics of Finite-Dimensional Random Matrices for MMO Channels, in preparation. [1] E. Telatar, Capacity of Multi-antenna Gaussian Channels, Eur. Trans. Telecom., Vol. 10, No.6, pp , Nov.-Dec