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

摘要

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 ($T leq M$).