Cramér-Rao Bound and Maximum Likelihood Estimation of Covariance Matrices With Non-Homogeneous Snapshots

We consider the problem of estimating the covariance matrix R_T of an observation vector, using K groups of snapshots Z_k = [z_k.1 ... z_k.L_k], of respective size L_k, whose covariance matrices R_k are randomly distributed around R_t, and hence are different from R_t. The Cramer-Rao bound (CRB) for estimation of Rt is derived as well as its maximum likelihood estimator (MLE). We illustrate the behavior of the CRB in the two opposite cases, namely K = 1 where all snapshots share a common covariance matrix, and L_k = 1 where each snapshot has a different covariance matrix. We also discuss the influence of the degree of heterogeneity on the estimation performance.