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

In Abramovich [“Bounds on Maximum Likelihood Ratio-Part I: Application to Antenna Array Detection-Estimation With Perfect Wavefront Coherence,” IEEE Trans. Signal Process., vol. 52, pp. 1524-1536, June 2004], it was demonstrated, for multivariate complex Gaussian distribution, that the probability density function (p.d.f.) of the likelihood ratio (LR) for the (unknown) actual covariance matrix R₀ does not depend on this matrix and is fully specified by the matrix dimension M and the number of independent training samples T. This invariance property hence enables one 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 R₀. This “expected likelihood” quality assessment allowed significant improvement of MUSIC DOA estimation performance in the so-called “threshold area,” and for diagonal loading and TVAR model order selection in adaptive detectors. Recently, the so-called complex elliptically symmetric (CES) distributions have 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, 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 Σ₀. 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 ≥ M) while Part II deals with the undersampled scenario (T ≤ M).