arXiv subjects

Walid Hachem

Publications and source records attributed to Walid Hachem.

52 records · Page 3Linked to original sources

arXiv2012

A Subspace Estimator for Fixed Rank Perturbations of Large Random Matrices

This paper deals with the problem of parameter estimation based on certain eigenspaces of the empirical covariance matrix of an observed multidimensional time series, in the case where the time series dimension and the observation window grow to infinity at the same pace. In the area of large random matrix theory, recent contributions studied the behavior of the extreme eigenvalues of a random matrix and their associated eigenspaces when this matrix is subject to a fixed-rank perturbation. The present work is concerned with the situation where the parameters to be estimated determine the eigenspace structure of a certain fixed-rank perturbation of the empirical covariance matrix. An estimation algorithm in the spirit of the well-known MUSIC algorithm for parameter estimation is developed. It relies on an approach recently developed by Benaych-Georges and Nadakuditi, relating the eigenspaces of extreme eigenvalues of the empirical covariance matrix with eigenspaces of the perturbation matrix. First and second order analyses of the new algorithm are performed.

math.PR↗

arXiv2012

A CLT on the SNR of Diagonally Loaded MVDR Filters

This paper studies the fluctuations of the signal-to-noise ratio (SNR) of minimum variance distorsionless response (MVDR) filters implementing diagonal loading in the estimation of the covariance matrix. Previous results in the signal processing literature are generalized and extended by considering both spatially as well as temporarily correlated samples. Specifically, a central limit theorem (CLT) is established for the fluctuations of the SNR of the diagonally loaded MVDR filter, under both supervised and unsupervised training settings in adaptive filtering applications. Our second-order analysis is based on the Nash-Poincaré inequality and the integration by parts formula for Gaussian functionals, as well as classical tools from statistical asymptotic theory. Numerical evaluations validating the accuracy of the CLT confirm the asymptotic Gaussianity of the fluctuations of the SNR of the MVDR filter.

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arXiv2012

Fluctuations of spiked random matrix models and failure diagnosis in sensor networks

In this article, the joint fluctuations of the extreme eigenvalues and eigenvectors of a large dimensional sample covariance matrix are analyzed when the associated population covariance matrix is a finite-rank perturbation of the identity matrix, corresponding to the so-called spiked model in random matrix theory. The asymptotic fluctuations, as the matrix size grows large, are shown to be intimately linked with matrices from the Gaussian unitary ensemble (GUE). When the spiked population eigenvalues have unit multiplicity, the fluctuations follow a central limit theorem. This result is used to develop an original framework for the detection and diagnosis of local failures in large sensor networks, for known or unknown failure magnitude.

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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↗

arXiv2009

Error exponents for Neyman-Pearson detection of a continuous-time Gaussian Markov process from noisy irregular samples

This paper addresses the detection of a stochastic process in noise from irregular samples. We consider two hypotheses. The \emph{noise only} hypothesis amounts to model the observations as a sample of a i.i.d. Gaussian random variables (noise only). The \emph{signal plus noise} hypothesis models the observations as the samples of a continuous time stationary Gaussian process (the signal) taken at known but random time-instants corrupted with an additive noise. Two binary tests are considered, depending on which assumptions is retained as the null hypothesis. Assuming that the signal is a linear combination of the solution of a multidimensional stochastic differential equation (SDE), it is shown that the minimum Type II error probability decreases exponentially in the number of samples when the False Alarm probability is fixed. This behavior is described by \emph{error exponents} that are completely characterized. It turns out that they are related with the asymptotic behavior of the Kalman Filter in random stationary environment, which is studied in this paper. Finally, numerical illustrations of our claims are provided in the context of sensor networks.

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arXiv2009

Resource Allocation for Downlink Cellular OFDMA Systems: Part I - Optimal Allocation

In this pair of papers (Part I and Part II in this issue), we investigate the issue of power control and subcarrier assignment in a sectorized two-cell downlink OFDMA system impaired by multicell interference. As recommended for WiMAX, we assume that the first part of the available bandwidth is likely to be reused by different base stations (and is thus subject to multicell interference) and that the second part of the bandwidth is shared in an orthogonal way between the different base stations (and is thus protected from multicell interference). Although the problem of multicell resource allocation is nonconvex in this scenario, we provide in Part I the general form of the global solution. In particular, the optimal resource allocation turns out to be "binary" in the sense that, except for at most one pivot-user in each cell, any user receives data either in the reused bandwidth or in the protected bandwidth, but not in both. The determination of the optimal resource allocation essentially reduces to the determination of the latter pivot-position.

cs.IT↗

arXiv2009

Resource Allocation for Downlink Cellular OFDMA Systems: Part II - Practical Algorithms and Optimal Reuse Factor

In a companion paper, we characterized the optimal resource allocation in terms of power control and subcarrier assignment, for a downlink sectorized OFDMA system. In our model, the network is assumed to be one dimensional for the sake of analysis. We also assume that a certain part of the available bandwidth is likely to be reused by different base stations while that the other part of the bandwidth is shared in an orthogonal way between these base stations. The optimal resource allocation characterized in Part I is obtained by minimizing the total power spent by the network under the constraint that all users rate requirements are satisfied. When optimal resource allocation is used, any user receives data either in the reused bandwidth or in the protected bandwidth, but not in both (except for at most one pivot-user in each cell). We also proposed an algorithm that determines the optimal values of users resource allocation parameters. The optimal allocation algorithm proposed in Part I requires a large number of operations. In the present paper, we propose a distributed practical resource allocation algorithm with low complexity. We study the asymptotic behavior of both this simplified resource allocation algorithm and the optimal resource allocation algorithm of Part I as the number of users in each cell tends to infinity. Our analysis allows to prove that the proposed simplified algorithm is asymptotically optimal. As a byproduct of our analysis, we characterize the optimal value of the frequency reuse factor.

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arXiv2009

Performance Analysis over Slow Fading Channels of a Half-Duplex Single-Relay Protocol: Decode or Quantize and Forward

In this work, a new static relaying protocol is introduced for half duplex single-relay networks, and its performance is studied in the context of communications over slow fading wireless channels. The proposed protocol is based on a Decode or Quantize and Forward (DoQF) approach. In slow fading scenarios, two performance metrics are relevant and complementary, namely the outage probability gain and the Diversity-Multiplexing Tradeoff (DMT). First, we analyze the behavior of the outage probability P_o associated with the proposed protocol as the SNR tends to infinity. In this case, we prove that SNR^2 P_o converges to a constant. We refer to this constant as the outage gain and we derive its closed-form expression for a general class of wireless channels that includes the Rayleigh and the Rice channels as particular cases. We furthermore prove that the DoQF protocol has the best achievable outage gain in the wide class of half-duplex static relaying protocols. A method for minimizing the outage gain with respect to the power distribution between the source and the relay, and with respect to the durations of the slots is also provided. Next, we focus on Rayleigh distributed fading channels to derive the DMT associated with the proposed DoQF protocol. Our results show that the DMT of DoQF achieves the 2 by 1 MISO upper-bound for multiplexing gains r < 0.25.

cs.IT↗

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.

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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.

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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↗

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.

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