Asymptotic properties of robust complex covariance matrix estimates

In many statistical signal processing applications, the estimation of nuisance parameters and parameters of interest is strongly linked to the resulting performance. Generally, these applications deal with complex data. This paper focuses on covariance matrix estimation problems in non-Gaussian environments and particularly, the M-estimators in the context of elliptical distributions. Firstly, this paper extends to the complex case the results of Tyler in [1]. More precisely, the asymptotic distribution of these estimators as well as the asymptotic distribution of any homogeneous function of degree 0 of the M-estimates are derived. On the other hand, we show the improvement of such results on two applications: DOA (directions of arrival) estimation using the MUSIC (MUltiple SIgnal Classification) algorithm and adaptive radar detection based on the ANMF (Adaptive Normalized Matched Filter) test

Exploiting persymmetry for low-rank Space Time Adaptive Processing

International audienceReducing the number of secondary data used to estimate the Covariance Matrix (CM) for Space Time Adaptive Processing (STAP) techniques is still an active research topic. Within this framework, the Low-Rank (LR) structure of the clutter is well-known and the corresponding LR STAP filters have been shown to exhibit a smaller Signal Interference plus Noise Ratio (SINR) loss than classical STAP filters, only 2r secondary data (where r is the clutter rank) instead of 2m (where m is the data size) are required to reach the classical 3 dB SNR loss. By using other features of the radar system, other properties of the CM can be exploited to further reduce the number of secondary data; this is the case for active systems using a symmetrically spaced linear array with constant pulse repetition interval, which results in a persymmetric structure of the noise CM. In this context, we propose to combine this property of the CM and the LR structure of the clutter to perform CM estimation. In this paper, the resulting STAP filter is shown, both theoretically and experimentally, to exhibit good performance with fewer secondary data; 3 dB SINR Loss is achieved with only r secondary data

Performance of two Low-Rank STAP Filters in a Heterogeneous Noise

International audienceThis paper considers the Space Time Adaptive Processing (STAP) problem where the disturbance is modeled as the sum of a Low-Rank (LR) Spherically Invariant Random Vector (SIRV) clutter and a zero-mean white Gaussian noise. To derive our adaptive LR-STAP filters, the estimation of the projector onto the clutter subspace is performed from the Sample Covariance Matrix (SCM) and the Normalized Sample Covari-ance Matrix (NSCM). We compute the theoretical performance of both corresponding LR-STAP filters through the analysis of the Signal to Interference plus Noise Ratio (SINR) Loss, based on a perturbation analysis. Numerical simulations validate the theoretical formula and allow to show that the LR-STAP filter built from the SCM performance does not depend on the heterogeneity of the SIRV clutter whereas the LR-STAP filter built from the NSCM performance does

A NEW DERIVATION OF THE BAYESIAN BOUNDS FOR PARAMETER ESTIMATION

International audienceThis paper deals with minimal bounds in the Bayesian context. We express the minimum mean square error of the conditional mean estimator as the solution of a continuum constrained optimization problem. And, by relaxing these constraints, we obtain the bounds of the Weiss-Weinstein family. Moreover, this method enables us to derive new bounds as the Bayesian version of the deterministic Abel bound

NON ASYMPTOTIC EFFICIENCY OF A MAXIMUM LIKELIHOOD ESTIMATOR AT FINITE NUMBER OF SAMPLES

International audienceIn estimation theory, the asymptotic (in the number of samples) efficiency of the Maximum Likelihood (ML) estimator is a well known result [1]. Nevertheless, in some scenarios, the number of snapshots may be small. We recently investigated the asymptotic behavior of the Stochastic ML (SML) estimator at high Signal to Noise Ratio (SNR) and finite number of samples [2] in the array processing framework: we proved the non-Gaussiannity of the SML estimator and we obtained the analytical expression of the variance for the single source case. In this paper, we generalize these results to multiple sources, and we obtain variance expressions which demonstrate the non-efficiency of SML estimates

The Bayesian ABEL Bound on the Mean Square Error

International audienceThis paper deals with lower bound on the Mean Square Error (MSE). In the Bayesian framework, we present a new bound which is derived from a constrained optimization problem. This bound is found to be tighter than the Bayesian Bhattacharyya bound, the Reuven-Messer bound, the Bobrovsky-Zakai bound, and the Bayesian Cramér-Rao bound

Clutter Subspace Estimation in Low Rank Heterogeneous Noise Context

International audienceThis paper addresses the problem of the Clutter Subspace Projector (CSP) estimation in the context of a disturbance composed of a Low Rank (LR) heterogeneous clutter , modeled here by a Spherically Invariant Random Vector (SIRV), plus a white Gaussian noise (WGN). In such context, the corresponding LR adaptive filters and detectors require less training vectors than classical methods to reach equivalent performance. Unlike classical adaptive processes, which are based on an estimate of the noise Covariance Matrix (CM), the LR processes are based on a CSP estimate. This CSP estimate is usually derived from a Singular Value Decomposition (SVD) of the CM estimate. However, no Maximum Likelihood Estimator (MLE) of the CM has been derived for the considered disturbance model. In this paper, we introduce the fixed point equation that MLE of the CSP satisfies for a disturbance composed of a LR-SIRV clutter plus a zero mean WGN. A recursive algorithm is proposed to compute this solution. Numerical simulations validate the introduced estimator and illustrate its interest compared to the current state of art

Robust estimation of the clutter subspace for a Low Rank heterogeneous noise under high Clutter to Noise Ratio assumption

International audienceIn the context of an heterogeneous disturbance with a Low Rank (LR) structure (called clutter), one may use the LR approximation for filtering and detection process. These methods are based on the projector onto the clutter subspace instead of the noise covariance matrix. In such context, adaptive LR schemes have been shown to require less secondary data to reach equivalent performances as classical ones. The main problem is then the estimation of the clutter subspace instead of the noise covariance matrix itself. Maximum Likelihood estimator (MLE) of the clutter subspace has been recently studied for a noise composed of a LR Spherically Invariant Random Vector (SIRV) plus a white Gaussian Noise (WGN). This paper focuses on environments with a high Clutter to Noise Ratio (CNR). An original MLE of the clutter subspace is proposed in this context. A cross-interpretation of this new result and previous ones is provided. Validity and interest - in terms of performance and robustness - of the different approaches are illustrated through simulation results

CRLB under K-distributed observation with parameterized mean

International audienceA semi closed-form expression of the Fisher information matrix in the context of K-distributed observations with parameterized mean is given and related to the classical, i.e. Gaussian case. This connection is done via a simple multiplicative factor, which only depends on the intrinsic parameters of the texture and the size of the observation vector. Finally, numerical simulation is provided to corroborate the theoretical analysi

Non-efficacité et non-gaussianité asymptotiques d'un estimateur du maximum de vraisemblance à fort rapport signal sur bruit

National audienceEn théorie de l'estimation, dans le cas d'observations indépendantes de mêmes densités de probabilité, l'efficacité asymptotique en le nombre T d'observations de la méthode du Maximum de Vraisemblance (MV) est un résultat bien connu qui permet d'appréhender ses performances lorsque T est grand. Dans certaines situations, le nombre d'observations peut être faible et ce résultat ne s'applique plus. Dans le cadre du traitement d'antenne et d'une modélisation stochastique des signaux émis par les sources, nous remédions à cette lacune lorsque le Rapport Signal sur Bruit (RSB) est grand. Nous montrons que dans cette situation, l'estimateur du MV est asymptotiquement (en RSB) non-efficace et non-gaussien