A Matrix Extension Built Under-Sampled Likelihood Ratio Test with Application to MUSIC Breakdown Prediction and Cure

Likelihood ratio tests are used in a range of detection-estimation problems, but normally cannot be extended to cases where training data volume T is smaller than the dimension M of the observations. We propose a non-degenerate normalized LR test that can be used for detection-estimation in such under-sampled training conditions. The LR is formed based on non-degenerate band extension of the original degenerate sample covariance matrix. This LR is then applied within a generalized likelihood ratio test framework to an array processing problem where the presence of closely spaced signal can be robustly detected, but their individual directions of arrival cannot be fully resolved by subspace-based DOA techniques such as MUSIC. In that case, MUSIC produces direction of arrival estimates for some sources with very large errors (outliers). We use the under-sampled likelihood ratio to detect the presence of such MUSIC outliers and provide corrected DOA estimates.