On the second-order statistics of the sample covariance

The second-order statistics of the sample covariance are encountered in many covariance based processing algorithms such as subspace based and covariance-matching based channel estimation methods. This paper provides closed-form expressions for the covariance of either the weighted sample covariance matrix or vectored sample covariance. Given a system model, the results explicitly rely on the second-order and fourth-order statistics of the channel noise and inputs. They can be directly applied to study the statistical properties of covariance based channel estimators and detectors. Various numerical examples are provided to further verify derived results.