Bosung Kang

Rank-Constrained Maximum Likelihood Estimation of Structured Covariance Matrices

This paper develops and analyzes the performance of a structured covariance matrix estimate for the important practical problem of radar space-time adaptive processing in the face of severely limited training data. Traditional maximum likelihood (ML) estimators are effective when training data are abundant, but they lead to poor estimates, degraded false alarm rates, and detection loss in the realistic regime of limited training. The problem is exacerbated by recent advances, which have led to high-dimensional N of the observations arising from increased antenna elements, as well as higher temporal resolution (P time epochs and finally N = JP). This work addresses the problem by incorporating constraints in the ML estimation problem obtained from the geometry and physics of the airborne phased array radar scenario. In particular, we exploit the structure of the disturbance covariance and, importantly, knowledge of the clutter rank to derive a new rank-constrained maximum likelihood (RCML) estimator of clutter and disturbance covariance. We demonstrate that despite the presence of the challenging rank constraint, the estimation can be transformed to a convex problem and derive closed-form expressions for the estimated covariance matrix. Performance analysis using the knowledge-aided sensor signal processing and expert reasoning data set (where ground truth covariance is made available) shows that the proposed estimator outperforms state-of-the-art alternatives in the sense of a higher normalized signal-to-interference and noise ratio. Crucially, the RCML estimator excels for low training, including the notoriously difficult regime of K ≤ N training samples.

Computationally efficient toeplitz approximation of structured covariance under a rank constraint

Disturbance covariance estimation is a centrally important problem in radar space-time adaptive processing (STAP). Because training is invariably scarce, estimators that exploit inherent structure and physical radar constraints are needed in practice. This paper develops a new computationally efficient estimator that obtains a Toeplitz approximation of the structured interference covariance under a rank constraint. Previous work has shown that exact maximum likelihood (ML) estimation of Toeplitz covariance matrix has no closed-form solution, and most versions of this problem result in iterative estimators that are computationally expensive. Our proposed solution focuses on a computationally efficient approximation and involves a cascade of two closed-form solutions. First, we obtain the rank-constrained ML estimator whose merits have recently been established firmly for radar STAP. The central contribution of this paper is the rank-preserving Toeplitz approximation, which we demonstrate can be modeled as an equality-constrained quadratic program and also admits a closed form. Extensive performance evaluation on both simulated and knowledge-aided sensor signal processing and expert reasoning data confirms that the proposed estimator yields unbeatable performance for radar STAP under the previously stated conditions of rank and Toeplitz constraints.

On the practical merits of rank constrained ML estimator of structured covariance matrices

Estimation of the disturbance or interference covariance matrix plays a central role on radar target detection in the presence of clutter, noise and jammer. The disturbance covariance matrix should be inferred from training sample observations in practice. Traditional maximum likelihood (ML) estimators lead degraded false alarm and detection performance in the realistic regime of limited training. For this reason, informed estimators have been actively researched. Recently, a new estimator [1] that explicitly incorporates rank information of the clutter subspace was proposed. This paper reports significant new analytical and experimental investigations on the rank-constrained maximum likelihood (RCML) estimator. First, we show that the RCML estimation problem formulated in [1] has a closed form. Next, we perform new and rigorous experimental evaluation in the form of reporting: 1.) probability of detection versus signal to noise ratio (SNR), and 2.) SINR performance under heterogeneous (target corrupted) training data. In each case, we compare against widely used existing estimators and show that exploiting the rank information has significant practical merits in robust estimation.

Automatic rank estimation for practical STAP covariance estimation via an expected likelihood approach

We address the problem of estimation of structured covariance matrices for radar space-time adaptive processing (STAP)1. The knowledge of the interference environment has been exploited in many previous works to accurately estimate a structured disturbance covariance matrix. In particular, it has been shown that employing the rank of clutter subspace, i.e. rank constrained maximum likelihood (RCML) estimation, leads to a practically powerful estimator as well as a closed form solution. While the rank is a very effective constraint, often practical non-idealities make it difficult to be known precisely using physical models. We propose an automatic rank estimation method in STAP via an expected likelihood (EL) approach. We formulate rank estimation as an optimization problem with the expected likelihood criterion and formally prove that the proposed optimization has a unique solution. Through experimental results from a simulation model and KASSPER dataset, we show the RCML estimator with the rank obtained via the EL approach outperforms RCML estimators with the other rank selection methods in the sense of a normalized signal-to-interference and noise ratio (SINR) and the probability of detection.

Estimation of structured covariance matrices for radar STAP under practical constraints

Estimation of the disturbance or interference covariance matrix plays a central role in radar target detection. Traditional maximum likelihood (ML) estimators lead to degraded false alarm and detection performance in the realistic regime of limited training. For this reason, structured covariance estimators have been actively researched. This paper reviews as well as proposes new structured covariance estimation methods which exploit physically motivated practical constraints. We first review the rank constrained maximum likelihood (RCML) estimator which explicitly incorporates the rank of the clutter subspace as a constraint in the ML problem. Next, we introduce an efficient approximation of structured covariance under joint Toeplitz and rank constraint (EASTR). In particular, we propose new quadratic optimization problems that enforce Toeplitz structure while preserving rank. Crucially, both the RCML estimator and the EASTR admit closed form solutions and hence facilitate real time implementation. We perform experimental evaluation in the form of normalized SINR, probability of detection, and whiteness tests. In each case, we compare against widely used existing estimators and show that exploiting the practical constraints has significant merits in covariance estimation.

Constrained ML estimation of structured covariance matrices with applications in radar STAP

The disturbance covariance matrix in radar space time adaptive processing (STAP) must be estimated from training sample observations. Traditional maximum likelihood (ML) estimators are effective when training is generous but lead to degraded false alarm rates and detection performance in the realistic regime of limited training. We exploit physically motivated constraints such as 1.) rank of the clutter subspace which can be inferred using existing physics based models such as the Brennan rule, and 2.) the Toeplitz constraint that applies to covariance matrices obtained from stationary random processes. We first provide a closed form solution of the rank constrained maximum likelihood (RCML) estimator and then subsequently develop an efficient approximation under joint Toeplitz and rank constraints (EASTR). Experimental results confirm that the proposed EASTR estimators outperform state-of-the-art alternatives in the sense of widely used measures such as the signal to interference and noise ratio (SINR) and probability of detection - particularly when training support is limited.

Expected likelihood approach for determining constraints in covariance estimation

Recent covariance estimation methods for radar space-time adaptive processing exploit practical constraints such as the rank of clutter subspace and the condition number of disturbance covariance to estimate accurate covariance even when training is not generous. While rank and condition number are very effective constraints, often practical nonidealities make it difficult to know them precisely using physical models. Therefore, we propose a method to determine constraints in covariance estimation for radar space-time adaptive processing via an expected likelihood approach. We analyze three cases of constraints: 1) a rank constraint, 2) both rank and noise power constraints, and 3) a condition number constraint. In each case, we formulate precise constraint determination as an optimization problem. For each of the three cases, we derive new analytical results which allow for computationally efficient, practical ways of determining these constraints with formal proofs. Through experimental results from a simulation model and the KASSPER data set, we show that the estimator with optimal constraints obtained by the expected likelihood approach outperforms state-of-the-art alternatives.

Efficient approximation of structured covariance under joint Toeplitz and rank constraints

The disturbance (clutter plus noise and jamming) covariance matrix which plays a central role in radar space time adaptive processing (STAP) should be estimated from sample training observations in practice. Traditional maximum likelihood (ML) estimators lead to degraded false alarm and detection performance in the realistic regime of limited training. Therefore constrained ML estimation has received much attention which exploits structure and other properties that a disturbance covariance matrix exhibits. In this paper1, we derive a new covariance estimator for STAP that jointly considers a Toeplitz structure and a rank constraint on the clutter component. Past work has shown that in the regime of low training, even handling each constraint individually is hard and techniques often resort to slow numerically based solutions. Our proposed solution leverages a recent advance called rank constrained ML estimator (RCML) of structured covariances to build a computationally friendly approximation that involves a cascade of two closed form solutions. Experimental investigation shows that the proposed estimator outperforms state of the art alternatives in the sense of: normalized signal to interference and noise ratio (SINR), and probability of detection versus signal to noise ratio (SNR).