Complex Wishart Matrix Research Articles

Expected Values of Scalar-Valued Functions of a Complex Wishart Matrix

The complex Wishart distribution has ample applications in science and engineering. In this paper, we give explicit expressions for E(tr(Wh))g(tr(Wj))i and E(tr(W−h))g(tr(W−j))i, respectively, for particular values of g, h, i, j, g+h+i+j≤5, where W follows a complex Wishart distribution. For specific values of g, h, i, j, we first write (tr(Wh))g(tr(Wj))i and (tr(W−h))g(tr(W−j))i in terms of zonal polynomials and then by using results on integration evaluate resulting expressions. Several expected values of matrix-valued functions of a complex Wishart matrix have also been derived.

Block-modified Wishart matrices: the easy case

Associated to any complex Wishart matrix $W$ of parameters $(dn,dm)$ and any linear map $\varphi:M_n(\mathbb C)\to M_n(\mathbb C)$ is the block-modified matrix $\tilde{W}=(id\otimes\varphi)W$. Following some previous work with Nechita, we study here the asymptotic $*$-distribution of $\tilde{W}$, in the $d\to\infty$ limit, in the case where the modification map $\varphi$ is easy, or more generally super-easy, in the quantum algebra/representation theory sense. Under suitable assumptions on $\varphi$ we obtain in this way a compound free Poisson law.

A unified complex noncentral Wishart type distribution inspired by massive MIMO systems

The eigenvalue distributions from a complex noncentral Wishart matrix S=XHX has been the subject of interest in various real world applications, where X is assumed to be complex matrix variate normally distributed with nonzero mean M and covariance Σ. This paper focuses on a weighted analytical representation of S to alleviate the restriction of normality; thereby allowing the choice of X to be complex matrix variate elliptically distributed for the practitioner. New results for eigenvalue distributions of more generalised forms are derived under this elliptical assumption, and investigated for certain members of the complex elliptical class. The distribution of the minimum eigenvalue enjoys particular attention. This theoretical investigation has proposed impact in communications systems (where massive datasets can be conveniently formulated in matrix terms), in particular the case where the noncentral matrix has rank one which is useful in practice.

Interference Coordination for 3-D Beamforming-Based HetNet Exploiting Statistical Channel-State Information

In this paper, we investigate the inter-tier interference coordination for a heterogeneous network (HetNet), where both macro and small cells work in the frequency-division duplexing mode and share the same spectrum. A 2-D large-scale antenna array is deployed at a macro-base station (BS), while conventional multiple antenna array is deployed at each small cell. First, we derive an approximation of macro-users’ signal-to-interference-plus-noise ratio (SINR), under the assumption of only statistical channel-state information (CSI) at macro-BS, and a 3-D beamforming transmission scheme is applied for macro-users. A lower-bound of small-cell users’ expected SINR is also derived using the property of complex Wishart matrix. Based on these, simple metrics are derived to measure the inter-tier interference. Then, new interference coordination algorithms are proposed to achieve a good tradeoff for the performance and traffic between macro-cells and small cells. The proposed algorithms require only a few additional statistical CSI. Moreover, we derive tractable ergodic rate approximation of both the macro-cell and small-cell users which are shown to match well with the Monte Carlo results.

NON-CENTRAL MULTIVARIATE CHI-SQUARE AND GAMMA DISTRIBUTIONS

A (p-1)-variate integral representation is given for the cumulative distribution function of the general p-variate non-central gamma distribution with a non-centrality matrix of any admissible rank. The real part of products of well known analytical functions is integrated over arguments from (-pi,pi). To facilitate the computation, these formulas are given more detailed for p=2 and p=3. These (p-1)-variate integrals are also derived for the diagonal of a non-central complex Wishart Matrix. Furthermore, some alternative formulas are given for the cases with an associated one-factorial (pxp)-correlation matrix R, i.e. R differs from a suitable diagonal matrix only by a matrix of rank 1, which holds in particular for all (3x3)-R with no vanishing correlation.

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

We investigate the dynamic behavior of the stationary random process defined by a central complex Wishart (CW) matrix ${\bf{W}}(t)$ as it varies along a certain dimension $t$. We characterize the second-order joint 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 finite 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 defines 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 coefficient, 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 $N_R$ fixed, we see how the best and worst channels tend to have a similar rate of change.

Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations

Let $$\lambda _1, \ldots , \lambda _n$$ be random eigenvalues coming from the beta-Laguerre ensemble with parameter $$p$$ , which is a generalization of the real, complex and quaternion Wishart matrices of parameter $$(n,p).$$ In the case that the sample size $$n$$ is much smaller than the dimension of the population distribution $$p$$ , a common situation in modern data, we approximate the beta-Laguerre ensemble by a beta-Hermite ensemble, which is a generalization of the real, complex and quaternion Wigner matrices. As corollaries, when $$n$$ is much smaller than $$p,$$ we show that the largest and smallest eigenvalues of the complex Wishart matrix are asymptotically independent; we obtain the limiting distribution of the condition numbers as a sum of two i.i.d. random variables with a Tracy–Widom distribution, which is much different from the exact square case that $$n=p$$ by Edelman (SIAM J Matrix Anal Appl 9:543–560, 1988); we propose a test procedure for a spherical hypothesis test. By the same approximation tool, we obtain the asymptotic distribution of the smallest eigenvalue of the beta-Laguerre ensemble. In the second part of the paper, under the assumption that $$n$$ is much smaller than $$p$$ in a certain scale, we prove the large deviation principles for three basic statistics: the largest eigenvalue, the smallest eigenvalue and the empirical distribution of $$\lambda _1, \ldots , \lambda _n$$ , where the last large deviation is derived by using a non-standard method.

The Exact Distribution of the Condition Number of Complex Random Matrices

Let G m×n (m ≥ n) be a complex random matrix and W = G m×n H G m×n which is the complex Wishart matrix. Let λ 1 > λ 2 > …>λ n > 0 and σ 1 > σ 2 > …>σ n > 0 denote the eigenvalues of the W and singular values of G m×n, respectively. The 2-norm condition number of G m×n is . In this paper, the exact distribution of the condition number of the complex Wishart matrices is derived. The distribution is expressed in terms of complex zonal polynomials.

Distribution of Diagonal Elements of a General Central Complex Wishart Matrix

We consider a general central complex Wishart matrix derived from non-circular complex Gaussian vectors. We present a technique for evaluation of the probability density, cumulative distribution, and moment generating functions of the joint distribution of the diagonal elements of the Wishart matrix. We give examples of application of the obtained results.

On the Diagonal Distribution of a Complex Wishart Matrix and its Application to the Analysis of MIMO Systems

The statistical properties of Wishart matrices have been extensively used to analyze the performance of multiple-input multiple-output (MIMO) systems. In particular, the signal-to-noise ratio (SNR) output statistics of several MIMO systems depends on the diagonal distribution of a complex Wishart matrix. In this paper, we derive the joint density of the diagonal elements of a complex Wishart matrix, which follows a multivariate chi-square distribution. The density expression is in the form of an infinite series representation which converges rapidly and is easy to compute. This expression is used to obtain the distribution of the maximum of the diagonal elements, which allows analyzing the performance of two different MIMO systems under practical conditions. First, our statistical results are applied to the outage probability characterization of MIMO systems with receive antenna selection in spatially correlated Rayleigh fading. Then, the same results are used to analyze the outage probability of transmit beamforming systems under limited-rate feedback.

Capacity of multiuser multiple input multiple output-orthogonal frequency division multiplexing systems with doubly correlated channels for various fading distributions

In this article, the capacity of multiuser multiple input multiple output (MIMO) orthogonal frequency division multiplexing (OFDM) system is analysed under Rayleigh, Rician, Lognormal and Weibull fading channels, and compared with each other under both semi- and doubly correlated channels. The distribution of the complex Wishart matrix is utilised to derive closed-form expressions of ergodic capacity of MIMO-OFDM systems under doubly correlated channels when the transmitter has no channel state information (CSI) and the receiver has perfect CSI. The capacity of MIMO-OFDM systems is plotted for various fading channels under different system specifications, such as number of transmit and receive antennas, number of subcarriers and different power levels. The obtained results are used to study the impact of channel correlation on capacity.

Capacity of single user MIMO-OFDM systems with doubly-correlated channels under various fading distributions

In this paper, the capacity of Single User (SU) Multiple Input Multiple Output (MIMO) Orthogonal Frequency Division Multiplexing (OFDM) system is analyzed under Rayleigh, Rician, Lognormal and Weibull fading channels, and compared with each other under both singly- and doubly-correlated channel conditions. The distribution of the complex Wishart matrix is utilized to derive closed form expressions of ergodic capacity of MIMO-OFDM systems under doubly-correlated channels when the transmitter has no Channel State Information (CSI) and the receiver has perfect CSI. The capacity of MIMO-OFDM systems is plotted for various channels under different system specifications, such as, number of transmit and receive antennas, number of subcarriers, and different power levels. The obtained results are used to study the impact of channel correlation on capacity.

On the Performance of Eigenvalue-Based Cooperative Spectrum Sensing for Cognitive Radio

In this paper, the distribution of the ratio of extreme eigenvalues of a complex Wishart matrix is studied in order to calculate the exact decision threshold as a function of the desired probability of false alarm for the maximum-minimum eigenvalue (MME) detector. In contrast to the asymptotic analysis reported in the literature, we consider a finite number of cooperative receivers and a finite number of samples and derive the exact decision threshold for the probability of false alarm. The proposed exact formulation is further reduced to the case of two receiver-based cooperative spectrum sensing. In addition, an approximate closed-form formula of the exact threshold is derived in terms of a desired probability of false alarm for a special case having equal number of receive antennas and signal samples. Finally, the derived analytical exact decision thresholds are verified with Monte-Carlo simulations. We show that the probability of detection performance using the proposed exact decision thresholds achieves significant performance gains compared to the performance of the asymptotic decision threshold.

Expectations of Functions of Complex Wishart Matrix

In this article, we study lower and upper triangular factorizations of the complex Wishart matrix. Further, using these factorizations, we obtain several expected values of scalar and matrix valued functions of the complex Wishart matrix. We also generalize Muirhead's identity for the complex case which gives a number of interesting special cases.

The Tracy–Widom limit for the largest eigenvalues of singular complex Wishart matrices

This paper extends the work of El Karoui [Ann. Probab. 35 (2007) 663–714] which finds the Tracy–Widom limit for the largest eigenvalue of a nonsingular p-dimensional complex Wishart matrix Wℂ(Ωp, n) to the case of several of the largest eigenvalues of the possibly singular (n<p) matrix Wℂ(Ωp, n). As a byproduct, we extend all results of Baik, Ben Arous and Peche [Ann. Probab. 33 (2005) 1643–1697] to the singular Wishart matrix case. We apply our findings to obtain a 95% confidence set for the number of common risk factors in excess stock returns.

Joint Density for Eigenvalues of Two Correlated Complex Wishart Matrices: Characterization of MIMO Systems

In this letter, the joint probability density function (PDF) for the eigenvalues of a complex Wishart matrix and a perturbed version of it are derived. The latter version can be used to model channel estimation errors and variations over time or frequency. As an example, the joint PDF is used to calculate the transition probabilities between modulation states in an adaptive MIMO system. This leads to a Markov model for the system. We then use the model to investigate the modulation state entering rates (MSER), the average stay duration (ASD), and the effects of feedback delay on the accuracy of modulation state selection in mobile radio systems. Other applications of this PDF are also discussed.

A trivariate chi-squared distribution derived from the complex Wishart distribution

The joint density for a particular trivariate chi-squared distribution given by the diagonal elements of a complex Wishart matrix is derived. This distribution has applications in the processing of multilook synthetic aperture radar data. The expression for the density is in the form of an infinite series that converges rapidly and is simple and fast to compute. The expression is shown to reduce to known forms for a number of special cases and is validated by simulation. The characteristic function is also derived and used to relate joint moments of the trivariate distribution to the parameters of the density function.

Complex random matrices and Rician channel capacity

Eigenvalue densities of complex noncentral Wishart matrices are investigated to study an open problem in information theory. Specifically, the largest, smallest, and joint eigenvalue densities of complex noncentral Wishart matrices are derived. These densities are expressed in terms of complex zonal polynomials and invariant polynomials. A connection between the complex Wishart matrix theory and information theory is given. This facilitates evaluation of the most important information-theoretic measure, the so-called ergodic channel capacity. In particular, the capacity of multiple-input multiple-output (MIMO) Rician distributed channels is investigated. We consider both spatially correlated and uncorrelated MIMO Rician channels and derive exact and easily computable tight upper bound formulas for ergodic capacities. Numerical results are also given, which show how the channel correlation degrades the capacity of the communication system.

Complex random matrices and Rayleigh channel capacity

The eigenvalue densities of complex central Wishart matrices are investigated with the objective of studying an open problem in channel capacity. These densities are represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials. The connection between the complex Wishart matrix theory and information theory is given. This facilitates the evaluation of the most important information-theoretic measure, the so-called channel capacity. In particular, the capacity of multiple input, multiple output (MIMO) Rayleigh distributed channels are fully investigated. We consider both correlated and uncorrelated channels and derive the corresponding channel capacity formulas. It is shown how the channel correlation degrades the capacity of the communication system. To appear in Communications in Information & Systems

A Note on Universality of the Distribution of the Largest Eigenvalues in Certain Sample Covariance Matrices

Recently Johansson and Johnstone proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X*X(XtX) converges to the Tracy–Widom law as n,p (the dimensions of X) tend to ∞ in some ratio n/p→γ>0. We extend these results in two directions. First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy–Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner random matrices in refs. 27, 38, and 39 allows to extend the results by Johansson and Johnstone to the case of X with non-Gaussian entries, provided n−p=O(p1/3). We also prove that λmax≤(n1/2+p1/2)2+O(p1/2 log(p)) (a.e.) for general γ>0.