Covariance estimation in non-stationary interference

We consider the problem of estimating the covariance matrix of a 0-mean Gaussian random vector, when the observations consist of i.i.d. samples corrupted by additive noise. The noise vectors are independent 0-mean Gaussian random vectors with known covariance which is varying across observations. Such a problem could arise in an adaptive radar system operating in a non-stationary interference environment. We derive the log-likelihood function and state necessary conditions which a maximizer of the log-likelihood must satisfy. We derive an EM algorithm for numerical maximization of the log-likelihood. Some interesting convergence properties of the EM algorithm can be shown analytically (for special cases) and via simulation