THEORETICAL ANALYSIS OF AN IMPROVED COVARIANCEMATRIX ESTIMATOR IN NON-GAUSSIAN NOISE

This paper presents a detailed theoretical analysis of a recently introduced covariance matrix estimator, called the Fixed Point Es-timate (FPE). It plays a significant role in radar detection applica-tions. This estimate is provided by the Maximum Likelihood Es-timation (MLE) theory when the non-Gaussian noise is modelled as a Spherically Invariant Random Process (SIRP). We study in details its properties: existence, uniqueness, unbiasedness, consis-tency and asymptotic distribution. We propose also an algorithm for its computation and prove the convergence of this numerical procedure. These results will allow to study the performance anal-ysis of the adaptive CFAR radar detectors (GLRT-LQ, BORD,...). 1. PROBLEM STATEMENT AND BACKGROUND Non-Gaussian noise characterization has gained many interests sin-ce experimental radar clutter measurements showed that these data are correctly described by non-Gaussian statistical models. One of the most tractable and elegant non-Gaussian model comes from the so-called Spherically Invariant Random Process (SIRP) theory. A SIRP is the product of a Gaussian random process- called speckle- with a non-negative random variable- called texture. This model leads to many results [1, 2, 3, 4]. The basic problem of detecting a complex signal corrupted by an additive SIRP noise c in a m-dimensional complex vector y al-lowed to build several Generalized Likelihood Ratio Tests like the GLRT-Linear Quadratic (GLRT-LQ) in [1, 2] or the Bayesian Op-timum Radar Detector (BORD) in [3, 4]. Let us recall some SIRP theory results. A noise modelled as a SIRP is a non-homogeneous Gaussian process with random power. More precisely, a SIRP [5] is the product of a positive ran-dom variable τ (texture) and a m-dimensional independent com-plex Gaussian vector x (speckle) with zero mean covariance matrix M = E(xx†)with normalization tr (M) = m, where † denotes the conjugate transpose operator: c = τ x. The SIRP PDF expression is: pm(c)