Normalized signal-to-noise ratio for covariance estimation in target detection

We address the Normalized Signal to Noise Ratio (NSNR) metric defined in the seminal paper by Reed, Mallett, and Brennan on adaptive detection. NSNR is the ratio between the SNR of a linear detector which uses an estimated noise covariance and the SNR of a clairvoyant detector based on the exact unknown covariance. It is not obvious how to evaluate NSNR since it is a function of the target vector. To close this gap, we consider the NSNR associated with the worst target. Using the Kantorovich Inequality, we provide a closed-form solution for the worst-case NSNR. Then, we prove that the classical Gaussian Kullback Leibler (KL) divergence bounds it. Motivated by these results, we derive a simple variant of a classic norm based estimator by incorporating KL in a leave-one-out cross-validation (LOOCV) framework. Numerical experiments with different true covariances and various estimates suggest that the KL metric is more correlated with the NSNR metric than competing norm-based metrics and simply changing the metric in the LOOCV estimator improves KL and NSNR performance.