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Let $\mathrm {W}\in \mathbb {C}^{n\times n}$ be a single-spiked Wishart matrix in the class $\mathrm {W}\sim \mathcal {CW}_{n}(m,\mathrm {I}_{n}+ \theta \mathrm {v}\mathrm {v}^{\dagger}) $ with $m\geq n$ , where ${\mathrm {I}}_{n}$ is the $n\times n$ identity matrix, $\mathrm {v}\in \mathbb {C}^{n\times 1}$ is an arbitrary vector with unit Euclidean norm, $\theta \geq 0$ is a non-random parameter, and $(\cdot)^{\dagger} $ represents the conjugate-transpose operator. Let u1 and ${\mathrm {u}}_{n}$ 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 $Z_{\ell }^{(n)}=\left |{\mathrm {v}^{\dagger} \mathrm {u}_\ell }\right |^{2}\in (0,1)$ for $\ell =1,n$. In particular, we derive a finite dimensional closed-form p.d.f. for $Z_{1}^{(n)}$ which is amenable to asymptotic analysis as $m,n$ diverges with $m-n$ fixed. It turns out that, in this asymptotic regime, the scaled random variable $nZ_{1}^{(n)}$ converges in distribution to $\chi ^{2}_{2}/2(1+\theta)$ , where $\chi _{2}^{2}$ 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 $Z_{n}^{(n)}$ is expressed as a double integral in which the integrand contains a determinant of a square matrix of dimension $(n-2)$. Although a simple solution to this double integral seems intractable, for special configurations of $n=2,3$ , and 4, we obtain closed-form expressions. [ABSTRACT FROM AUTHOR]

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

One of the key issues in the acquisition of sparse data by means of compressed sensing is the design of the measurement matrix. Gaussian matrices have been proven to be information-theoretically optimal in terms of minimizing the required number of measurements for sparse recovery. In this paper, we provide a new approach for the analysis of the restricted isometry constant (RIC) of finite dimensional Gaussian measurement matrices. The proposed method relies on the exact distributions of the extreme eigenvalues for Wishart matrices. First, we derive the probability that the restricted isometry property is satisfied for a given sufficient recovery condition on the RIC, and propose a probabilistic framework to study both the symmetric and asymmetric RICs. Then, we analyze the recovery of compressible signals in noise through the statistical characterization of stability and robustness. The presented framework determines limits on various sparse recovery algorithms for finite size problems. In particular, it provides a tight lower bound on the maximum sparsity order of the acquired data allowing signal recovery with a given target probability. Also, we derive simple approximations for the RICs based on the Tracy–Widom distribution. [ABSTRACT FROM AUTHOR]

We derive the probability that all eigenvalues of a random matrix M lie within an arbitrary interval [a,b] , \psi (a,b)\triangleq \Pr \{a\leq \lambda _{\min }({\text{M}}), \lambda _{\max }({\text{M}})\leq b\} , when M is a real or complex finite-dimensional Wishart, double Wishart, or Gaussian symmetric/Hermitian matrix. We give efficient recursive formulas allowing the exact evaluation of \psi (a,b)$ for Wishart matrices, even with a large number of variates and degrees of freedom. We also prove that the probability that all eigenvalues are within the limiting spectral support (given by the Marčenko-Pastur or the semicircle laws) tends for large dimensions to the universal values 0.6921 and 0.9397 for the real and complex cases, respectively. Applications include improved bounds for the probability that a Gaussian measurement matrix has a given restricted isometry constant in compressed sensing. [ABSTRACT FROM PUBLISHER]

This paper presents a new connection between the generalized Marcum-Q function and the confluent hypergeometric function of two variables, \Phi3. This result is then applied to the closed-form characterization of the bivariate Nakagami-m distribution and of the distribution of the minimum eigenvalue of correlated noncentral Wishart matrices, both important in communication theory. New expressions for the corresponding cumulative distributions are obtained and a number of communication-theoretic problems involving them are pointed out. [ABSTRACT FROM AUTHOR]

The fundamental diversity-multiplexing tradeoff of the three-node, multi-input, multi-output (MIMO), quasi-static, Rayleigh faded, half-duplex relay channel is characterized for an arbitrary number of antennas at each node and in which opportunistic scheduling (or dynamic operation) of the relay is allowed, i.e., the relay can switch between receive and transmit modes at a channel dependent time. In this most general case, the diversity-multiplexing tradeoff is characterized as a solution to a simple, two-variable optimization problem. This problem is then solved in closed form for special classes of channels defined by certain restrictions on the numbers of antennas at the three nodes. The key mathematical tool developed here that enables the explicit characterization of the diversity-multiplexing tradeoff is the joint eigenvalue distribution of three mutually correlated random Wishart matrices. Besides being relevant here, this distribution result is interesting in its own right. Previously, without actually characterizing the diversity-multiplexing tradeoff, the optimality in this tradeoff metric of the dynamic compress-and-forward protocol based on the classical compress-and-forward scheme of Cover and El Gamal was shown by Yuksel and Erkip. However, this scheme requires global channel state information at the relay. In this paper, the so-called quantize-map and forward (QMF) coding scheme is adopted as the achievability scheme with the added benefit that it achieves optimal tradeoff with only the knowledge of the (channel dependent) switching time at the relay node. Moreover, in special classes of the MIMO half-duplex relay channel, the optimal tradeoff is shown to be attainable even without this knowledge. Such a result was previously known only for the half-duplex relay channel with a single antenna at each node, also via the QMF scheme. More generally, the explicit characterization of the tradeoff curve in this study enables the in-depth comparisons herein of full-duplex versus half-duplex relaying as well as static versus dynamic relaying, both as a function of the numbers of antennas at the three nodes. [ABSTRACT FROM AUTHOR]

Multiple-input multiple-output (MIMO) systems are being considered as one of the key enabling technologies for future wireless networks. However, the decrease in capacity due to the presence of interferers in MIMO networks is not well understood. In this paper, we develop an analytical framework to characterize the capacity of MIMO communication systems in the presence of multiple MIMO co-channel interferers and noise. We consider the situation in which transmitters have no channel state information, and all links undergo Rayleigh fading. We first generalize the determinant representation of hypergeometric functions with matrix arguments to the case when the argument matrices have eigenvalues of arbitrary multiplicity. This enables the derivation of the distribution of the eigenvalues of Gaussian quadratic forms and Wishart matrices with arbitrary correlation, with application to both single-user and multiuser MIMO systems. In particular, we derive the ergodic mutual information for MIMO systems in the presence of multiple MIMO interferers. Our analysis is valid for any number of interferers, each with arbitrary number of antennas having possibly unequal power levels. This framework, therefore, accommodates the study of distributed MIMO systems and accounts for different spatial positions of the MIMO interferers. [ABSTRACT FROM AUTHOR]

Following the seminal work of Zheng and Tse, this paper investigates the fundamental diversity and multiplexing tradeoff of multiple-input--multiple-output (MIMO) systems in which knowledge of the channel state at both sides of the link is employed to transmit independent data streams through the channel eigenmodes. First, the fundamental diversity and multiplexing tradeoff of each of the individual substreams is obtained and this result is then used to derive a tradeoff optimal scheme for rate allocation along channel eigenmodes. The tradeoff of spatial multiplexing is finally compared to the fundamental tradeoff of the MIMO channel and to the one of both space only codes and V-BLAST which do not require channel state information (CSI) at the transmit side. [ABSTRACT FROM AUTHOR]