Complex Wishart Matrices Research Articles

Second order cumulants: Second order even elements and $R$-diagonal elements

We introduce R -diagonal and even operators of second order. We give a formula for the second order free cumulants of the square x^2 of a second order even element in terms of the second order free cumulants of x . Similar formulas are proved for the second order free cumulants of aa^* , when a is a second order R -diagonal operator. We also show that if r is second order R -diagonal and b is second order free from r , then rb is also second order R -diagonal. We present a large number of examples, in particular, the limit distribution of products of Ginibre matrices. We prove the conjectured formula of Dartois and Forrester for the fluctuations moments of the product of two independent complex Wishart matrices and generalize it to any number of factors.

The Eigenvectors of Single-Spiked Complex Wishart Matrices: Finite and Asymptotic Analyses

Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {W}\in \mathbb {C}^{n\times n}$ </tex-math></inline-formula> be a single-spiked Wishart matrix in the class <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {W}\sim \mathcal {CW}_{n}(m,\mathrm {I}_{n}+ \theta \mathrm {v}\mathrm {v}^{\dagger}) $ </tex-math></inline-formula> with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\geq n$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {I}}_{n}$ </tex-math></inline-formula> is the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\times n$ </tex-math></inline-formula> identity matrix, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {v}\in \mathbb {C}^{n\times 1}$ </tex-math></inline-formula> is an arbitrary vector with unit Euclidean norm, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\theta \geq 0$ </tex-math></inline-formula> is a non-random parameter, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\cdot)^{\dagger} $ </tex-math></inline-formula> represents the conjugate-transpose operator. Let u1 and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {u}}_{n}$ </tex-math></inline-formula> denote the eigenvectors corresponding to the smallest and the largest eigenvalues of W, respectively. This paper investigates the probability density function (p.d.f.) of the random quantity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Z_{\ell }^{(n)}=\left |{\mathrm {v}^{\dagger} \mathrm {u}_\ell }\right |^{2}\in (0,1)$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell =1,n$ </tex-math></inline-formula> . In particular, we derive a finite dimensional closed-form p.d.f. for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Z_{1}^{(n)}$ </tex-math></inline-formula> which is amenable to asymptotic analysis as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m,n$ </tex-math></inline-formula> diverges with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m-n$ </tex-math></inline-formula> fixed. It turns out that, in this asymptotic regime, the scaled random variable <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$nZ_{1}^{(n)}$ </tex-math></inline-formula> converges in distribution to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi ^{2}_{2}/2(1+\theta)$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi _{2}^{2}$ </tex-math></inline-formula> denotes a chi-squared random variable with two degrees of freedom. This reveals that u1 can be used to infer information about the spike. On the other hand, the finite dimensional p.d.f. of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Z_{n}^{(n)}$ </tex-math></inline-formula> is expressed as a double integral in which the integrand contains a determinant of a square matrix of dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n-2)$ </tex-math></inline-formula> . Although a simple solution to this double integral seems intractable, for special configurations of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n=2,3$ </tex-math></inline-formula> , and 4, we obtain closed-form expressions.

Distribution of the Scaled Condition Number of Single-Spiked Complex Wishart Matrices

Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {X}\in \mathbb {C}^{n\times m}$ </tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\geq n$ </tex-math></inline-formula> ) be a random matrix with independent columns each distributed as complex multivariate Gaussian with zero mean and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">single-spiked</i> covariance matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {I}_{n}+ \eta \mathbf {u}\mathbf {u}^{*}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {I}_{n}$ </tex-math></inline-formula> is the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\times n$ </tex-math></inline-formula> identity matrix, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {u}\in \mathbb {C}^{n\times 1}$ </tex-math></inline-formula> is an arbitrary vector with unit Euclidean norm, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\eta \geq 0$ </tex-math></inline-formula> is a non-random parameter, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\cdot)^{*}$ </tex-math></inline-formula> represents the conjugate-transpose. This paper investigates the distribution of the random quantity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{2}(\mathbf {X})=\sum _{k=1}^{n} \lambda _{k}/\lambda _{1}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0\le \lambda _{1}\le \lambda _{2}\le \ldots \leq \lambda _{n} &lt; \infty $ </tex-math></inline-formula> are the ordered eigenvalues of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {X}\mathbf {X}^{*}$ </tex-math></inline-formula> (i.e., single-spiked Wishart matrix). This random quantity is intimately related to the so called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">scaled condition number</i> or the Demmel condition number (i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}(\mathbf {X})$ </tex-math></inline-formula> ) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{-2}(\mathbf {X})$ </tex-math></inline-formula> ). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{2}(\mathbf {X})$ </tex-math></inline-formula> which is amenable to asymptotic analysis as matrix dimensions grow large. Our asymptotic results reveal that, as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m,n\to \infty $ </tex-math></inline-formula> such that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m-n$ </tex-math></inline-formula> is fixed and when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula> scales on the order of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1/n$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{2}(\mathbf {X})$ </tex-math></inline-formula> scales on the order of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n^{3}$ </tex-math></inline-formula> . In this respect we establish simple closed-form expressions for the limiting distributions. It turns out that, as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m,n\to \infty $ </tex-math></inline-formula> such that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n/m\to c\in (0,1)$ </tex-math></inline-formula> , properly centered <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{2}(\mathbf {X})$ </tex-math></inline-formula> fluctuates on the scale <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m^{\frac {1}{3}}$ </tex-math></inline-formula> .

Noncentral complex Wishart matrices: Moments and correlation of minors

Complex Wishart matrices are a class of random matrices with numerous emerging applications. In particular, the statistical characterization of such class of random matrices is essential for solving problems in various fields, including statistics, finance, physics and engineering. This paper establishes a new way to solve such problems based on the statistical moments and correlation of the minors. The first two minors’ moments and correlation for noncentral complex Wishart matrices are derived as a function of size and degrees of freedom, as well as of covariance and noncentrality matrices. As a case study, the findings are applied to the statistical characterization of the capacity of wireless multiple-input–multiple-output systems.

Global fluctuations for Multiple Orthogonal Polynomial Ensembles

We study the fluctuations of linear statistics with polynomial test functions for Multiple Orthogonal Polynomial Ensembles. Multiple Orthogonal Polynomial Ensembles form an important class of determinantal point processes that include random matrix models such as the GUE with external source, complex Wishart matrices, multi-matrix models and others.Our analysis is based on the recurrence matrix for the multiple orthogonal polynomials, that is constructed out of the nearest neighbor recurrences. If the coefficients for the nearest neighbor recurrences have limits, then we show that the right-limit of this recurrence matrix is a matrix that can be viewed as representation of a Toeplitz operator with respect to a non-standard basis. This will allow us to prove Central Limit Theorems for linear statistics of Multiple Orthogonal Polynomial Ensembles. A particular novelty is the use of the Baker–Campbell–Hausdorff formula to prove that the higher cumulants of the linear statistics converge to zero.We illustrate the main results by discussing Central Limit Theorems for the Gaussian Unitary Ensembles with external source, complex Wishart matrices and specializations of the Schur measure related to multiple Charlier, multiple Krawtchouk and multiple Meixner polynomials.

Integer moments of complex Wishart matrices and Hurwitz numbers

We give formulae for the cumulants of complex Wishart (LUE) and inverse Wishart matrices (inverse LUE). Their large- N expansions are generating functions of double (strictly and weakly) monotone Hurwitz numbers which count constrained factorisations in the symmetric group. The two expansions can be compared and combined with a duality relation proved in [F. D. Cunden, F. Mezzadri, N. O’Connell, and N. J. Simm, Moments of random matrices and hypergeometric orthogonal polynomials, Comm. Math. Phys. 369 (2019), no. 3, 1091–1145] to obtain: i) a combinatorial proof of the reflection formula between moments of LUE and inverse LUE at genus zero and, ii) a new functional relation between the generating functions of monotone and strictly monotone Hurwitz numbers. The main result resolves the integrality conjecture formulated in [F. D. Cunden, F. Mezzadri, N. J. Simm, and P. Vivo, Correlators for the Wigner–Smith time-delay matrix of chaotic cavities, J. Phys. A 49 (2016), no. 18, 18LT01, 20 pp] on the time-delay cumulants in quantum chaotic transport. The precise combinatorial description of the cumulants given here may cast new light on the concordance between random matrix and semiclassical theories.

Spectral statistics for the difference of two Wishart matrices

In this work, we consider the weighted difference of two independent complex Wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite-dimension scenario using two distinct approaches. The first derivation involves the use of unitary group integral, while the second one relies on applying the derivative principle. The latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements. Exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with Monte Carlo simulation results. Analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived. Additionally, we provide a large-dimension asymptotic result for the spectral density using the Stieltjes transform approach for algebraic random matrices. Finally, we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean.

On the distribution of an arbitrary subset of the eigenvalues for some finite dimensional random matrices

We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Specifically, we derive compact expressions for the joint probability distribution function of the eigenvalues and the expectation of functions of the eigenvalues, including joint moments, for the case of both ordered and unordered eigenvalues.

Non‐crossing annular pairings and the infinitesimal distribution of thegoe

We present a combinatorial approach to the infinitesimal distribution of the Gaussian orthogonal ensemble (GOE). In particular we show how the infinitesimal moments are described by non-crossing partitions, but not of type B. We demonstrate the asymptotic infinitesimal freeness of independent complex Wishart matrices. With the combinatorial picture we can easily compute the infinitesimal cumulants of the GOE and demonstrate the lack of asymptotic infinitesimal freeness of independent GOE ensembles. I have added a new figure and some additional explanatory remarks. This update corrects a number of typographical errors; I am grateful to the readers who brought these to my attention.

Cooperative secrecy transmission in multi‐hop relay networks with interference alignment

In this study, the authors consider the secrecy communication of one source-destination pair in a wireless multi-hop relay network, where each node is equipped with multiple antennas. They use the method of interference alignment for secrecy transmission in the presence of an eavesdropper. Firstly, they propose a decode-and-forward (DF) relaying protocol and give a new result on the joint probability density function of the first, second, horizontal ellipsis , kth ($k=1, 2,\ldots $k=1,2, horizontal ellipsis ) largest eigenvalues of the complex Wishart matrices $\boldsymbol{H} \boldsymbol{H}H$HHH, where $\boldsymbol{H}$H is an matrix whose elements are independent identically distributed complex Gaussian variables with zero mean and unit variance. They then utilise this result to characterise the legitimate outage probability and the diversity order of the proposed DF protocol. The case where the relay works in the amplify-and-forward (AF) mode is considered at last, and the diversity order of the proposed AF protocol is given.

Standard Condition Number Distributions of Finite Wishart Matrices for Cognitive Radio Networks

The standard condition number (SCN) in random matrix theory (RMT) has widely been employed in cognitive radio networks (CRNs). The paper studies the SCN distributions of central and complex Wishart matrices with finite dimensions. The exact and closed-form expressions of SCN distributions are first proposed in sum of multiple polynomials. The proposed formulations provide an efficient way to determine probability density function and cumulative distribution function of the SCN in finite RMT. In particular, the specific SCN distributions of triple Wishart matrix are initially presented. The theoretical and precise sensing thresholds for cooperative spectrum sensing (CSS) systems in CRNs are calculated using the proposed SCN distributions. Numerical results indicate that the proposed theoretical SCN distributions match the empirical SCN distributions very well. Furthermore, the exact SCN-based CSS systems are able to obtain higher sensing performance than the asymptotic SCN-based schemes because precise sensing thresholds can be determined.

Estimating tail probabilities of the ratio of the largest eigenvalue to the trace of a Wishart matrix

This paper develops an efficient Monte Carlo method to estimate the tail probabilities of the ratio of the largest eigenvalue to the trace of the Wishart matrix, which plays an important role in multivariate data analysis. The estimator is constructed based on a change-of-measure technique and it is proved to be asymptotically efficient for both the real and complex Wishart matrices. Simulation studies further show the improved performance of the proposed method over existing approaches based on asymptotic approximations, especially when estimating probabilities of rare events.

Sample Mean Versus Sample Fréchet Mean for Combining Complex Wishart Matrices: A Statistical Study

The space of covariance matrices is a non-Euclidean space. The matrices form a manifold which if equipped with a Riemannian metric becomes a Riemannian manifold, and recently this idea has been used for comparison and clustering of complex-valued spectral matrices, which at a given frequency are typically modeled as complex Wishart-distributed random matrices. Identically distributed sample complex Wishart matrices can be combined via a standard sample mean to derive a more stable overall estimator. However, using the Riemannian geometry, their so-called sample Frechet mean can also be found. We derive the expected value of the determinant of the sample Frechet mean and the expected value of the sample Frechet mean itself. The population Frechet mean is shown to be a scaled version of the true covariance matrix. The risk under convex loss functions for the standard sample mean is never larger than for the Frechet mean. In simulations, the sample mean also performs better for the estimation of an important functional derived from the estimated covariance matrix, namely partial coherence.

A note on spiked Wishart matrices

Consider the spiked complex Wishart matrices with sample size M and population size N. As N,M→∞ such that N/M→γ∈(0,1], Baik et al. (2005) established a phase transition of the largest eigenvalue. In this paper we show that some of their main results also hold true in the case where N/M→0. More precisely, we prove that limiting distribution of the largest eigenvalue is the finite GUE distribution.

On the Scaled Eigenvalue Distributions of Complex Wishart Matrices

The eigenvalue distributions of complex Wishart matrices are critical research issues in random matrix theory (RMT). The scaled eigenvalue (SE) distributions of complex Wishart matrices with finite dimensions are deduced in this paper. The probability density function (PDF) and cumulative distribution function (CDF) of the SE are formulated in the closed-form and coefficient-based expressions. Moreover, the derivative of SE PDF is provided in an exact formulation utilizing the same coefficient vectors. The numerical results verify that the newly proposed SE distributions fit the empirical distributions very well and the dimensions of Wishart matrix can be identified by the derivative of SE PDF.

Some inferences on the distribution of the Demmel condition number of complex Wishart matrices

Abstract In recent years, many researchers have studied the distributions of the Demmel (or the scaled) condition numbers (DCN) of complex Wishart matrices. In this paper, several new distributional properties of the distribution of the Demmel condition number of complex Wishart matrices are presented. The limiting distributions of the standardized extreme order statistics are given. Since the truncated distributions arise in practical statistics where the ability of record observations is limited to a given threshold or within a specified range, there has been a great interest, in recent years, in the characterizations of probability distributions by truncated moments. Before a particular probability distribution model is applied to fit the realworld data, it is necessary to confirm whether the given continuous probability distribution satisfies the underlying requirements by its characterization. Therefore, in this paper, some characterizations of the distribution of DCN are also shown. We hope that the findings of this paper will be quite useful in many fields of pure and applied sciences, such as, probability, statistics, multivariate statistics, linear algebra, operator algebra theory, actuarial science, physics, wireless communications, and polarimetric synthetic aperture radar (PolSAR), among others.

Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances

We compute the spectral statistics of the sum H of two independent complex Wishart matrices, each of which is correlated with a different covariance matrix. Random matrix theory enjoys many applications including sums and products of random matrices. Typically ensembles with correlations among the matrix elements are much more difficult to solve. Using a combination of supersymmetry, superbosonisation and bi-orthogonal functions we are able to determine all spectral k-point density correlation functions of H for arbitrary matrix size N. In the half-degenerate case, when one of the covariance matrices is proportional to the identity, the recent results by Kumar for the joint eigenvalue distribution of H serve as our starting point. In this case the ensemble has a bi-orthogonal structure and we explicitly determine its kernel, providing its exact solution for finite N. The kernel follows from computing the expectation value of a single characteristic polynomial. In the general non-degenerate case the generating function for the k-point resolvent is determined from a supersymmetric evaluation of the expectation value of k ratios of characteristic polynomials. Numerical simulations illustrate our findings for the spectral density at finite N and we also give indications how to do the asymptotic large-N analysis.

Classification of complex Wishart matrices with a diffusion-reaction system guided by stochastic distances.

We propose a new method for polarimetric synthetic aperture radar (PolSAR) imagery classification based on stochastic distances in the space of random matrices obeying complex Wishart distributions. Given a collection of prototypes [Formula: see text] and a stochastic distance d(.,.), we classify any random matrix X using two criteria in an iterative set-up. First, we associate X with the class which minimizes the weighted stochastic distance w(m)d(X,Z(m)), where the positive weights w(m) are computed to max- imize the class discrimination power. Second, we improve the result by embedding the classification problem into a diffusion-reaction partial differential system where the diffusion term smooths the patches within the image, and the reaction term tends to move the pixel values towards the closest class prototype. In particular, the method inherits the benefits of speckle reduction by diffusion-like methods. Results on synthetic and real PolSAR data show the performance of the method.

Error probability analysis of a novel adaptive beamforming receiver for large‐scale multiple‐input–multiple‐output communication system

A novel adaptive beamforming (BF) receiver is proposed for a multiple-input–multiple-output communication system over multipath fading channels. By dividing a large number of receive antennas into many adaptive beamformers (ABFs), the proposed system reduces the array dimension in order to simplify the signal detection process and enjoys the benefit of BF techniques to improve the symbol-error-rate (SER) performance. The effectiveness of the proposed system depends on the number of ABFs and the number of antennas per ABF. The statistical properties of complex Wishart matrices are applied to analyse the SER performance of the proposed system with both maximum-likelihood (ML) detection and zero-forcing detection. Furthermore, the authors have derived an upper bound for SER of the ML detection based on the smallest eigenvalue distribution. Through the authors analysis, it has been shown that the proposed system using ML detection provides comparable SER performance, but with a lower complexity as compared with that of a conventional ML receiver because of reduced array dimensions.

Outage Probability for MIMO Relay Channel

The channel capacity for the multiple-input multiple-output (MIMO) relay channel is still an open problem. In view of this gap, upper and lower bounds were found by Wang et al., 2005. This paper presents an extensive study of mutual information and outage probability of Rayleigh fading MIMO relay channels based on the eigenvalues distribution for the sum of complex Wishart matrices. Interestingly, for certain scenarios, upper and lower bounds are coincident providing the real channel capacity. To compute outage probability, a Gaussian approximation for mutual information has been used. Furthermore, it was proposed an equivalent distribution for the sum of Wishart matrices, which proves to be excellent in all scenarios. Closed-form expressions for probability density function and outage probability have been found for the general case of any number of antennas and any SNR values. Analytical expressions have been validated by means of Monte Carlo simulations.