arXiv subjects

Jamal Najim

Publications and source records attributed to Jamal Najim.

33 records · Page 2Linked to original sources

arXiv2012

Performance of mutual information inference methods under unknown interference

The problem of fast point-to-point MIMO channel mutual information estimation is addressed, in the situation where the receiver undergoes unknown colored interference, whereas the channel with the transmitter is perfectly known. The considered scenario assumes that the estimation is based on a few channel use observations during a short sensing period. Using large dimensional random matrix theory, an estimator referred to as {\em G-estimator} is derived. This estimator is proved to be consistent as the number of antennas and observations grow large and its asymptotic performance is analyzed. In particular, the G-estimator satisfies a central limit theorem with asymptotic Gaussian fluctuations. Simulations are provided which strongly support the theoretical results, even for small system dimensions.

math.PR↗

arXiv2012

Estimation of the Covariance Matrix of Large Dimensional Data

This paper deals with the problem of estimating the covariance matrix of a series of independent multivariate observations, in the case where the dimension of each observation is of the same order as the number of observations. Although such a regime is of interest for many current statistical signal processing and wireless communication issues, traditional methods fail to produce consistent estimators and only recently results relying on large random matrix theory have been unveiled. In this paper, we develop the parametric framework proposed by Mestre, and consider a model where the covariance matrix to be estimated has a (known) finite number of eigenvalues, each of it with an unknown multiplicity. The main contributions of this work are essentially threefold with respect to existing results, and in particular to Mestre's work: To relax the (restrictive) separability assumption, to provide joint consistent estimates for the eigenvalues and their multiplicities, and to study the variance error by means of a Central Limit theorem.

cs.IT↗

arXiv2011

Fluctuations of an improved population eigenvalue estimator in sample covariance matrix models

This article provides a central limit theorem for a consistent estimator of population eigenvalues with large multiplicities based on sample covariance matrices. The focus is on limited sample size situations, whereby the number of available observations is known and comparable in magnitude to the observation dimension. An exact expression as well as an empirical, asymptotically accurate, approximation of the limiting variance is derived. Simulations are performed that corroborate the theoretical claims. A specific application to wireless sensor networks is developed.

math.PR↗

arXiv2011

On bilinear forms based on the resolvent of large random matrices

Consider a matrix $Σ_n$ with random independent entries, each non-centered with a separable variance profile. In this article, we study the limiting behavior of the random bilinear form $u_n^* Q_n(z) v_n$, where $u_n$ and $v_n$ are deterministic vectors, and Q_n(z) is the resolvent associated to $Σ_n Σ_n^*$ as the dimensions of matrix $Σ_n$ go to infinity at the same pace. Such quantities arise in the study of functionals of $Σ_n Σ_n^*$ which do not only depend on the eigenvalues of $Σ_n Σ_n^*$, and are pivotal in the study of problems related to non-centered Gram matrices such as central limit theorems, individual entries of the resolvent, and eigenvalue separation.

math.PR↗

arXiv2011

A CLT for Information-theoretic statistics of Non-centered Gram random matrices

In this article, we study the fluctuations of the random variable: $$ {\mathcal I}_n(ρ) = \frac 1N \log\det(Σ_n Σ_n^* + ρI_N),\quad (ρ>0) $$ where $Σ_n= n^{-1/2} D_n^{1/2} X_n\tilde D_n^{1/2} +A_n$, as the dimensions of the matrices go to infinity at the same pace. Matrices $X_n$ and $A_n$ are respectively random and deterministic $N\times n$ matrices; matrices $D_n$ and $\tilde D_n$ are deterministic and diagonal, with respective dimensions $N\times N$ and $n\times n$; matrix $X_n=(X_{ij})$ has centered, independent and identically distributed entries with unit variance, either real or complex. We prove that when centered and properly rescaled, the random variable ${\mathcal I}_n(ρ)$ satisfies a Central Limit Theorem and has a Gaussian limit. The variance of ${\mathcal I}_n(ρ)$ depends on the moment $\E X_{ij}^2$ of the variables $X_{ij}$ and also on its fourth cumulant $κ= \E|X_{ij}|^4 - 2 - |\E X_{ij}^2|^2$. The main motivation comes from the field of wireless communications, where ${\mathcal I}_n(ρ)$ represents the mutual information of a multiple antenna radio channel. This article closely follows the companion article "A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile", {\em Ann. Appl. Probab. (2008)} by Hachem et al., however the study of the fluctuations associated to non-centered large random matrices raises specific issues, which are addressed here.

math.PR↗

arXiv2011

Large information plus noise random matrix models and consistent subspace estimation in large sensor networks

In array processing, a common problem is to estimate the angles of arrival of $K$ deterministic sources impinging on an array of $M$ antennas, from $N$ observations of the source signal, corrupted by gaussian noise. The problem reduces to estimate a quadratic form (called "localization function") of a certain projection matrix related to the source signal empirical covariance matrix. Recently, a new subspace estimation method (called "G-MUSIC") has been proposed, in the context where the number of available samples $N$ is of the same order of magnitude than the number of sensors $M$. In this context, the traditional subspace methods tend to fail because the empirical covariance matrix of the observations is a poor estimate of the source signal covariance matrix. The G-MUSIC method is based on a new consistent estimator of the localization function in the regime where $M$ and $N$ tend to $+\infty$ at the same rate. However, the consistency of the angles estimator was not adressed. The purpose of this paper is to prove the consistency of the angles of arrival estimator in the previous asymptotic regime. To prove this result, we show the property that the singular values of M x N Gaussian information plus noise matrix escape from certain intervals is an event of probability decreasing at rate O(1/N^p) for all p. A regularization trick is also introduced, which allows to confine these singular values into certain intervals and to use standard tools as Poincaré inequality to characterize any moments of the estimator. These results are believed to be of independent interest.

math.PR↗

arXiv2010

On the capacity achieving covariance matrix for Rician MIMO channels: an asymptotic approach

The capacity-achieving input covariance matrices for coherent block-fading correlated MIMO Rician channels are determined. In this case, no closed-form expressions for the eigenvectors of the optimum input covariance matrix are available. An approximation of the average mutual information is evaluated in this paper in the asymptotic regime where the number of transmit and receive antennas converge to $+\infty$. New results related to the accuracy of the corresponding large system approximation are provided. An attractive optimization algorithm of this approximation is proposed and we establish that it yields an effective way to compute the capacity achieving covariance matrix for the average mutual information. Finally, 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, while being much more computationally attractive.

math.PR↗

arXiv2010

Performance of Statistical Tests for Single Source Detection using Random Matrix Theory

This paper introduces a unified framework for the detection of a source with a sensor array in the context where the noise variance and the channel between the source and the sensors are unknown at the receiver. The Generalized Maximum Likelihood Test is studied and yields the analysis of the ratio between the maximum eigenvalue of the sampled covariance matrix and its normalized trace. Using recent results of random matrix theory, a practical way to evaluate the threshold and the $p$-value of the test is provided in the asymptotic regime where the number $K$ of sensors and the number $N$ of observations per sensor are large but have the same order of magnitude. The theoretical performance of the test is then analyzed in terms of Receiver Operating Characteristic (ROC) curve. It is in particular proved that both Type I and Type II error probabilities converge to zero exponentially as the dimensions increase at the same rate, and closed-form expressions are provided for the error exponents. These theoretical results rely on a precise description of the large deviations of the largest eigenvalue of spiked random matrix models, and establish that the presented test asymptotically outperforms the popular test based on the condition number of the sampled covariance matrix.

math.PR↗

arXiv2008

BER and Outage Probability Approximations for LMMSE Detectors on Correlated MIMO Channels

This paper is devoted to the study of the performance of the Linear Minimum Mean-Square Error receiver for (receive) correlated Multiple-Input Multiple-Output systems. By the random matrix theory, it is well-known that the Signal-to-Noise Ratio (SNR) at the output of this receiver behaves asymptotically like a Gaussian random variable as the number of receive and transmit antennas converge to +$\infty$ at the same rate. However, this approximation being inaccurate for the estimation of some performance metrics such as the Bit Error Rate and the outage probability, especially for small system dimensions, Li et al. proposed convincingly to assume that the SNR follows a generalized Gamma distribution which parameters are tuned by computing the first three asymptotic moments of the SNR. In this article, this technique is generalized to (receive) correlated channels, and closed-form expressions for the first three asymptotic moments of the SNR are provided. To obtain these results, a random matrix theory technique adapted to matrices with Gaussian elements is used. This technique is believed to be simple, efficient, and of broad interest in wireless communications. Simulations are provided, and show that the proposed technique yields in general a good accuracy, even for small system dimensions.

cs.IT↗

arXiv2008

A Central Limit Theorem for the SINR at the LMMSE Estimator Output for Large Dimensional Signals

This paper is devoted to the performance study of the Linear Minimum Mean Squared Error estimator for multidimensional signals in the large dimension regime. Such an estimator is frequently encountered in wireless communications and in array processing, and the Signal to Interference and Noise Ratio (SINR) at its output is a popular performance index. The SINR can be modeled as a random quadratic form which can be studied with the help of large random matrix theory, if one assumes that the dimension of the received and transmitted signals go to infinity at the same pace. This paper considers the asymptotic behavior of the SINR for a wide class of multidimensional signal models that includes general multi-antenna as well as spread spectrum transmission models. The expression of the deterministic approximation of the SINR in the large dimension regime is recalled and the SINR fluctuations around this deterministic approximation are studied. These fluctuations are shown to converge in distribution to the Gaussian law in the large dimension regime, and their variance is shown to decrease as the inverse of the signal dimension.

cs.IT↗

arXiv2008

A Central Limit Theorem for the SNR at the Wiener Filter Output for Large Dimensional Signals

Consider the quadratic form $β= {\bf y}^* ({\bf YY}^* + ρ{\bf I})^{-1} {\bf y}$ where $ρ$ is a positive number, where ${\bf y}$ is a random vector and ${\bf Y}$ is a $N \times K$ random matrix both having independent elements with different variances, and where ${\bf y}$ and ${\bf Y}$ are independent. Such quadratic forms represent the Signal to Noise Ratio at the output of the linear Wiener receiver for multi dimensional signals frequently encountered in wireless communications and in array processing. Using well known results of Random Matrix Theory, the quadratic form $β$ can be approximated with a known deterministic real number $\barβ_K$ in the asymptotic regime where $K\to\infty$ and $K/N \to α> 0$. This paper addresses the problem of convergence of $β$. More specifically, it is shown here that $\sqrt{K}(β- \barβ_K)$ behaves for large $K$ like a Gaussian random variable which variance is provided.

cs.IT↗

arXiv2007

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 $Y^n_{ij}=\frac{σ_{ij}(n)}{\sqrt{n}}X^n_{ij}$, 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 $Σ_n=Y_n+A_n$. 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, \[\lim_{n\to+\infty,N/n\to c}\biggl(\frac{1}{N}\operatorname {Trace}(Σ_nΣ_n^T-zI_N)^{-1}-\frac{1}{N}\operatorname {Trace}T_n(z)\biggr)=0.\] Otherwise stated, there exists a deterministic equivalent to the empirical Stieltjes transform of the distribution of the eigenvalues of $Σ_nΣ_n^T$. 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 $π_n(dλ)$, and that for every bounded continuous function f, the following convergence holds almost surely \[\frac{1}{N}\sum_{k=1}^Nf(λ_k)-\int_0^{\infty}f(λ)π_n(dλ)\mathop {\longrightarrow}_{n\to\infty}0,\] where the $(λ_k)_{1\le k\le N}$ are the eigenvalues of $Σ_nΣ_n^T$. 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: \[C_n(σ^2)=\frac{1}{N}\mathbb{E}\log \det\biggl(I_N+\frac{Σ_nΣ_n^T}{σ^2}\biggr),\] where $σ^2$ is a known parameter.

math.PR↗

arXiv2007

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 $$ Y_{ij}^{n}=\frac{σ_{ij}(n)}{\sqrt{n}} X_{ij}^{n} $$ the $X_{ij}^{n}$ being centered, independent and identically distributed random variables with unit variance and $(σ_{ij}(n); 1\le i\le N, 1\le j\le n)$ being an array of numbers we shall refer to as a variance profile. We study in this article the fluctuations of the random variable $$ \log\det(Y_n Y_n^* + ρI_N) $$ where $Y^*$ is the Hermitian adjoint of $Y$ and $ρ> 0$ 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.

math.PR↗

arXiv2007

Large deviations for weighted empirical mean with outliers

We study in this article large deviations for the empirical mean of iid random vectors with some deterministic weights, whose empirical measure weakly converges to some compactly support probability distribution. The scope of this paper is to study the effect on the LDP of outliers, that is sequences of weights that remain far from the support of the limiting measure.

math.PR↗

arXiv2006

A New Approach for Capacity Analysis of Large Dimensional Multi-Antenna Channels

This paper adresses the behaviour of the mutual information of correlated MIMO Rayleigh channels when the numbers of transmit and receive antennas converge to infinity at the same rate. Using a new and simple approach based on Poincaré-Nash inequality and on an integration by parts formula, it is rigorously established that the mutual information converges to a Gaussian random variable whose mean and variance are evaluated. These results confirm previous evaluations based on the powerful but non rigorous replica method. It is believed that the tools that are used in this paper are simple, robust, and of interest for the communications engineering community.

cs.IT↗