The surprising sample covariance matrix: Unexpected characteristics and understanding them

Abstract

The Sample Covariance Matrix (SCM) is important to many problems in acoustic signal processing in both time-domain and frequency-domain processing. However, some careful analysis suggests that the SCM contains a variety of surprises. For instance, not all elements of the SCM have the same error—the error of a particular element depends on the location of the element in the matrix and is different for SCMs of real-valued processes and complex-valued processes. Moreover, when the samples of the process used to compute the SCM are correlated, the sample covariance matrix behaves differently yet again. In particular, for frequency-domain sample covariance matrices, which are common in underwater acoustic communication and sonar signal processing, the SCM obtained using a tapped delay line is a better estimate of the covariance matrix than that obtained using independent samples—a most counterintuitive result. In this talk, we present a unified analysis technique using ergodic theory that predicts a variety of unexpected characteristics of the SCM, and explain the reasons behind some of them. Given the wide variety of algorithms that rely on the SCM, understanding this behavior is crucial to designing robust and accurate algorithms.