This paper considers the problem of covariance matrix estimation from the viewpoint of statistical signal processing for high-dimensional or wideband random processes. Due to limited sensing resources, it is often desired to accurately estimate the covariance matrix from a small number of sample observations. To make up for the lack of observations, this paper leverages the structural characteristics of the random processes by considering the interplay of three widely-available signal structures: stationarity, sparsity and the underlying probability distribution of the observed random signal. New problem formulations are developed that incorporate both compressive sampling and sparse covariance estimation strategies. Tradeoff study is provided to illustrate the design choices when estimating the covariance matrices using a handful of sample observations.
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