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

## TL;DR

This paper introduces a geometric class of covariance matrix estimators for radar signal processing that improve SINR performance, especially in data-scarce scenarios, by projecting sample covariance matrices into structured sets using efficient algorithms.

## Contribution

It proposes a novel geometric framework for covariance estimation using unitary invariant norms, with efficient algorithms for Frobenius and spectral cases, enhancing radar processing performance.

## Key findings

- Significant SINR improvements in training-starved regimes.
- Efficient algorithms for near closed-form covariance estimates.
- Performance validated in spatial and Doppler radar scenarios.

## Abstract

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 shrinkagetype 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 a spatial and a Doppler processing 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.