A Geometric Approach to Covariance Matrix Estimation and its Applications to Radar Problems

A new class of disturbance covariance matrix estimators for radar signal processing applications is introduced following a geometric paradigm. Each estimator is associated with a given unitary invariant norm and performs the sample covariance matrix projection into a specific set of structured covariance matrices. Regardless of the considered norm, an efficient solution technique to handle the resulting constrained optimization problem is developed. Specifically, it is shown that the new family of distribution-free estimators shares a shrinkage-type form; besides, the eigenvalues estimate just requires the solution of a one-dimensional convex problem whose objective function depends on the considered unitary norm. For the two most common norm instances, i.e., Frobenius and spectral, very efficient algorithms are developed to solve the aforementioned one-dimensional optimization leading to almost closed-form covariance estimates. At the analysis stage, the performance of the new estimators is assessed in terms of achievable signal-to-interference-plus-noise ratio (SINR) both for spatial and Doppler processing scenarios assuming different data statistical characterizations. The results show that interesting SINR improvements with respect to some counterparts available in the open literature can be achieved especially in training starved regimes.

I. Introduction

Interference covariance matrix estimation is a longstanding and basic problem in adaptive radar signal processing and naturally arises in several areas such as target detection, direction of arrival estimation, sidelobe cancelling, and secondary data selection [1]– [4] (just to list a few). Conventional adaptive architectures (such as Sample Matrix Inversion (SMI) Doppler filter [1], Kelly's receiver [4], and spatial beamformers [5]) resort to the Sample Covariance Matrix (SCM) of a secondary data set collected from range gates spatially close to the one under test to estimate the interference covariance. These algorithms are often very prohibitive because they lean on the assumption that the environment remains stationary and homogeneous during the adaptation process. Precisely, they provide satisfactory performance when the secondary vectors share the same spectral properties of the interference in the test cell, are statistical independent, and their number is higher than twice the useful signal dimension [1]. These requisites however may represent important limitations since in real environments the number of data where the disturbance is homogeneous (often referred to as sample support) is very limited. Besides, poor training data selection, in such adaptive algorithms, can result in severe radar performance degradation [6] and [7] .