Approximate MLE of High-Dimensional STAP Covariance Matrices with Banded & Spiked Structure -- A Convex Relaxation Approach

TL;DR

This paper introduces a convex relaxation method for estimating high-dimensional clutter covariance matrices in STAP, effectively handling ICM and high noise floors with limited samples, outperforming existing techniques.

Contribution

It proposes a novel convex relaxation approach for the non-convex MLE problem, enabling accurate covariance estimation in high-dimensional, low-sample regimes with theoretical guarantees.

Findings

Achieves higher SCNR than state-of-the-art methods.

Effectively estimates covariance with more dimensions than samples.

Provides theoretical bounds and conditions for positive definiteness.

Abstract

Estimating the clutter-plus-noise covariance matrix in high-dimensional STAP is challenging in the presence of Internal Clutter Motion (ICM) and a high noise floor. The problem becomes more difficult in low-sample regimes, where the Sample Covariance Matrix (SCM) becomes ill-conditioned. To capture the ICM and high noise floor, we model the covariance matrix using a ``Banded+Spiked'' structure. Since the Maximum Likelihood Estimation (MLE) for this model is non-convex, we propose a convex relaxation which is formulated as a Frobenius norm minimization with non-smooth convex constraints enforcing banded sparsity. This relaxation serves as a provable upper bound for the non-convex likelihood maximization and extends to cases where the covariance matrix dimension exceeds the number of samples. We derive a variational inequality-based bound to assess its quality. We introduce a novel algorithm to jointly estimate the banded clutter covariance and noise power. Additionally, we establish conditions ensuring the estimated covariance matrix remains positive definite and the bandsize is accurately recovered. Numerical results using the high-fidelity RFView radar simulation environment demonstrate that our algorithm achieves a higher Signal-to-Clutter-plus-Noise Ratio (SCNR) than state-of-the-art methods, including TABASCO, Spiked Covariance Stein Shrinkage, and Diagonal Loading, particularly when the covariance matrix dimension exceeds the number of samples.