On the second-order statistics of the weighted sample covariance matrix

The second-order statistics of the sample covariance are encountered in many covariance based processing algorithms. This paper derives closed-form expressions for the covariance of the weighted sample covariance matrix with an arbitrary weight for both a real system and complex system. Given a system model, the results explicitly rely on the second-order and fourth-order statistics of the channel noise and inputs. They are shown to coincide with the existing results when the channel inputs and noise are Gaussian distributed. Our results can be directly applied to analyze the statistical properties of subspace-based channel estimation methods for single-input multiple-output (SIMO) systems and code-division multiple access (CDMA) systems. Numerical examples are provided to further verify analyses.