Almost sure localization of the eigenvalues in a gaussian information plus noise model. Applications to the spiked models

Let $\boldsymbol{\Sigma}_N$ be a $M \times N$ random matrix defined by $\boldsymbol{\Sigma}_N = \mathbf{B}_N + \sigma \mathbf{W}_N$ where $\mathbf{B}_N$ is a uniformly bounded deterministic matrix and where $\mathbf{W}_N$ is an independent identically distributed complex Gaussian matrix with zero mean and variance $\frac{1}{N}$ entries. The purpose of this paper is to study the almost sure location of the eigenvalues $\hat{\lambda}_{1,N} \geq ... \geq \hat{\lambda}_{M,N}$ of the Gram matrix ${\boldsymbol \Sigma}_N {\boldsymbol \Sigma}_N^*$ when $M$ and $N$ converge to $+\infty$ such that the ratio $c_N = \frac{M}{N}$ converges towards a constant $c > 0$. The results are used in order to derive, using an alernative approach, known results concerning the behaviour of the largest eigenvalues of ${\boldsymbol \Sigma}_N {\boldsymbol \Sigma}_N^*$ when the rank of $\mathbf{B}_N$ remains fixed when $M$ and $N$ converge to $+\infty$.Comment: 19 pages, 1 figure, Accepted for publication in Electronic Journal of Probabilit

On the Capacity Achieving Covariance Matrix for Frequency Selective MIMO Channels Using the Asymptotic Approach

In this contribution, an algorithm for evaluating the capacity-achieving input covariance matrices for frequency selective Rayleigh MIMO channels is proposed. In contrast with the flat fading Rayleigh cases, no closed-form expressions for the eigenvectors of the optimum input covariance matrix are available. Classically, both the eigenvectors and eigenvalues are computed numerically and the corresponding optimization algorithms remain computationally very demanding. In this paper, it is proposed to optimize (w.r.t. the input covariance matrix) a large system approximation of the average mutual information derived by Moustakas and Simon. An algorithm based on an iterative water filling scheme is proposed, and its convergence is studied. Numerical simulation results show that, even for a moderate number of transmit and receive antennas, the new approach provides the same results as direct maximization approaches of the average mutual information.Comment: presented at ISIT 2010 Conference, Austin, Texas, June 13-18, 2010 (5 pages, 1 figure, 2 tables

On the precoder design of flat fading MIMO systems equipped with MMSE receivers: a large system approach

This paper is devoted to the design of precoders maximizing the ergodic mutual information (EMI) of bi-correlated flat fading MIMO systems equiped with MMSE receivers. The channel state information and the second order statistics of the channel are assumed available at the receiver side and at the transmitter side respectively. As the direct maximization of the EMI needs the use of non attractive algorithms, it is proposed to optimize an approximation of the EMI, introduced recently, obtained when the number of transmit and receive antennas $t$ and $r$ converge to $\infty$ at the same rate. It is established that the relative error between the actual EMI and its approximation is a $O(\frac{1}{t^{2}})$ term. It is shown that the left singular eigenvectors of the optimum precoder coincide with the eigenvectors of the transmit covariance matrix, and its singular values are solution of a certain maximization problem. Numerical experiments show that the mutual information provided by this precoder is close from what is obtained by maximizing the true EMI, but that the algorithm maximizing the approximation is much less computationally intensive.Comment: Submitted to IEEE Transactions on Information Theor

On the almost sure location of the singular values of certain Gaussian block-Hankel large random matrices

This paper studies the almost sure location of the eigenvalues of matrices ${\bf W}_N {\bf W}_N^{*}$ where ${\bf W}_N = ({\bf W}_N^{(1)T}, ..., {\bf W}_N^{(M)T})^{T}$ is a $ML \times N$ block-line matrix whose block-lines $({\bf W}_N^{(m)})_{m=1, ..., M}$ are independent identically distributed $L \times N$ Hankel matrices built from i.i.d. standard complex Gaussian sequences. It is shown that if $M \rightarrow +\infty$ and $\frac{ML}{N} \rightarrow c_*$ ($c_* \in (0, \infty)$), then the empirical eigenvalue distribution of ${\bf W}_N {\bf W}_N^{*}$ converges almost surely towards the Marcenko-Pastur distribution. More importantly, it is established that if $L = \mathcal{O}(N^{\alpha})$ with $\alpha < 2/3$, then, almost surely, for $N$ large enough, the eigenvalues of ${\bf W}_N {\bf W}_N^{*}$ are located in the neighbourhood of the Marcenko-Pastur distribution.Comment: 67 pages. Revised version, to appear in Journal of Theoretical Probability, published on line at http://link.springer.com/article/10.1007/s10959-015-0614-

A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile

Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $$Y_{ij}^{n}=\frac{\sigma_{ij}(n)}{\sqrt{n}} X_{ij}^{n}$$ the $X_{ij}^{n}$ being centered, independent and identically distributed random variables with unit variance and $(\sigma_{ij}(n); 1\le i\le N, 1\le j\le n)$ being an array of numbers we shall refer to as a variance profile. We study in this article the fluctuations of the random variable $$\log\det(Y_n Y_n^* + \rho I_N)$$ where $Y^*$ is the Hermitian adjoint of $Y$ and $\rho > 0$ is an additional parameter. We prove that when centered and properly rescaled, this random variable satisfies a Central Limit Theorem (CLT) and has a Gaussian limit whose parameters are identified. A complete description of the scaling parameter is given; in particular it is shown that an additional term appears in this parameter in the case where the 4$^\textrm{th}$ moment of the $X_{ij}$'s differs from the 4$^{\textrm{th}}$ moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications

Improved subspace estimation for multivariate observations of high dimension: the deterministic signals case

We consider the problem of subspace estimation in situations where the number of available snapshots and the observation dimension are comparable in magnitude. In this context, traditional subspace methods tend to fail because the eigenvectors of the sample correlation matrix are heavily biased with respect to the true ones. It has recently been suggested that this situation (where the sample size is small compared to the observation dimension) can be very accurately modeled by considering the asymptotic regime where the observation dimension $M$ and the number of snapshots $N$ converge to $+\infty$ at the same rate. Using large random matrix theory results, it can be shown that traditional subspace estimates are not consistent in this asymptotic regime. Furthermore, new consistent subspace estimate can be proposed, which outperform the standard subspace methods for realistic values of $M$ and $N$. The work carried out so far in this area has always been based on the assumption that the observations are random, independent and identically distributed in the time domain. The goal of this paper is to propose new consistent subspace estimators for the case where the source signals are modelled as unknown deterministic signals. In practice, this allows to use the proposed approach regardless of the statistical properties of the source signals. In order to construct the proposed estimators, new technical results concerning the almost sure location of the eigenvalues of sample covariance matrices of Information plus Noise complex Gaussian models are established. These results are believed to be of independent interest.Comment: New version with minor corrections. The present paper is an extended version of a paper (same title) to appear in IEEE Trans. on Information Theor

Deterministic equivalents for certain functionals of large random matrices

Consider an $N\times n$ random matrix $Y_n=(Y^n_{ij})$ where the entries are given by $Y^n_{ij}=\frac{\sigma_{ij}(n)}{\sqrt{n}}X^n_{ij}$, the $X^n_{ij}$ being independent and identically distributed, centered with unit variance and satisfying some mild moment assumption. Consider now a deterministic $N\times n$ matrix A_n whose columns and rows are uniformly bounded in the Euclidean norm. Let $\Sigma_n=Y_n+A_n$. We prove in this article that there exists a deterministic $N\times N$ matrix-valued function T_n(z) analytic in $\mathbb{C}-\mathbb{R}^+$ such that, almost surely, $$\lim_{n\to+\infty,N/n\to c}\biggl(\frac{1}{N}\operatorname {Trace}(\Sigma_n\Sigma_n^T-zI_N)^{-1}-\frac{1}{N}\operatorname {Trace}T_n(z)\biggr)=0.$$ Otherwise stated, there exists a deterministic equivalent to the empirical Stieltjes transform of the distribution of the eigenvalues of $\Sigma_n\Sigma_n^T$. For each n, the entries of matrix T_n(z) are defined as the unique solutions of a certain system of nonlinear functional equations. It is also proved that $\frac{1}{N}\operatorname {Trace} T_n(z)$ is the Stieltjes transform of a probability measure $\pi_n(d\lambda)$, and that for every bounded continuous function f, the following convergence holds almost surely $$\frac{1}{N}\sum_{k=1}^Nf(\lambda_k)-\int_0^{\infty}f(\lambda)\pi _n(d\lambda)\mathop {\longrightarrow}_{n\to\infty}0,$$ where the $(\lambda_k)_{1\le k\le N}$ are the eigenvalues of $\Sigma_n\Sigma_n^T$. This work is motivated by the context of performance evaluation of multiple inputs/multiple output (MIMO) wireless digital communication channels. As an application, we derive a deterministic equivalent to the mutual information: $$C_n(\sigma^2)=\frac{1}{N}\mathbb{E}\log \det\biggl(I_N+\frac{\Sigma_n\Sigma_n^T}{\sigma^2}\biggr),$$ where $\sigma^2$ is a known parameter.Comment: Published at http://dx.doi.org/10.1214/105051606000000925 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Performance analysis of an improved MUSIC DoA estimator

This paper adresses the statistical performance of subspace DoA estimation using a sensor array, in the asymptotic regime where the number of samples and sensors both converge to infinity at the same rate. Improved subspace DoA estimators were derived (termed as G-MUSIC) in previous works, and were shown to be consistent and asymptotically Gaussian distributed in the case where the number of sources and their DoA remain fixed. In this case, which models widely spaced DoA scenarios, it is proved in the present paper that the traditional MUSIC method also provides DoA consistent estimates having the same asymptotic variances as the G-MUSIC estimates. The case of DoA that are spaced of the order of a beamwidth, which models closely spaced sources, is also considered. It is shown that G-MUSIC estimates are still able to consistently separate the sources, while it is no longer the case for the MUSIC ones. The asymptotic variances of G-MUSIC estimates are also evaluated.Comment: Revised versio