Generalized robust shrinkage estimator and its application to STAP detection problem

Recently, in the context of covariance matrix estimation, in order to improve as well as to regularize the performance of the Tyler's estimator [1] also called the Fixed-Point Estimator (FPE) [2], a "shrinkage" fixed-point estimator has been introduced in [3]. First, this work extends the results of [3,4] by giving the general solution of the "shrinkage" fixed-point algorithm. Secondly, by analyzing this solution, called the generalized robust shrinkage estimator, we prove that this solution converges to a unique solution when the shrinkage parameter $\beta$ (losing factor) tends to 0. This solution is exactly the FPE with the trace of its inverse equal to the dimension of the problem. This general result allows one to give another interpretation of the FPE and more generally, on the Maximum Likelihood approach for covariance matrix estimation when constraints are added. Then, some simulations illustrate our theoretical results as well as the way to choose an optimal shrinkage factor. Finally, this work is applied to a Space-Time Adaptive Processing (STAP) detection problem on real STAP data

Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 1: The Over-Sampled Case

In \cite{Abramovich04}, it was demonstrated that the likelihood ratio (LR) for multivariate complex Gaussian distribution has the invariance property that can be exploited in many applications. Specifically, the probability density function (p.d.f.) of this LR for the (unknown) actual covariance matrix $\R_{0}$ does not depend on this matrix and is fully specified by the matrix dimension $M$ and the number of independent training samples $T$. Since this p.d.f. could therefore be pre-calculated for any a priori known $(M,T)$, one gets a possibility to compare the LR of any derived covariance matrix estimate against this p.d.f., and eventually get an estimate that is statistically ``as likely'' as the a priori unknown actual covariance matrix. This ``expected likelihood'' (EL) quality assessment allows for significant improvement of MUSIC DOA estimation performance in the so-called ``threshold area'' \cite{Abramovich04,Abramovich07d}, and for diagonal loading and TVAR model order selection in adaptive detectors \cite{Abramovich07,Abramovich07b}. Recently, a broad class of the so-called complex elliptically symmetric (CES) distributions has been introduced for description of highly in-homogeneous clutter returns. The aim of this series of two papers is to extend the EL approach to this class of CES distributions as well as to a particularly important derivative of CES, namely the complex angular central distribution (ACG). For both cases, we demonstrate a similar invariance property for the LR associated with the true scatter matrix \mSigma_{0}. Furthermore, we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology. This first part is devoted to the conventional scenario ($T \geq M$) while Part 2 deals with the under-sampled scenario ($T \leq M$)

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

Numerical performances of low rank stap based on different heterogeneous clutter subspace estimators

International audienceSpace time Adaptive Processing (STAP) for airborne RADAR fits the context of a disturbance composed of a Low Rank (LR) clutter, here modeled by a Compound Gaussian (CG) process, plus a white Gaussian noise (WGN). In such context, the corresponding LR adaptive filters used to detect a target require less training vectors than classical methods to reach equivalent performance. Unlike the classical filter which is based on the Covariance Matrix (CM) of the noise, the LR filter is based on the clutter subspace projector, which is usually derived from a Singular Value Decomposition (SVD) of a noise CM estimate. Regarding to the considered model of LR-CG plus WGN, recent results are providing both direct estimators of the clutter subspace [1][2] and an exact MLE of the noise CM [3]. To promote the use of these new estimation methods, this paper proposes to apply them to realistic STAP simulations