Maximum likelihood estimation of Toeplitz-block-Toeplitz covariances in the presence of subspace interference

The EM algorithm is a commonly cited solution in the literature for the problem of maximum likelihood estimation of covariance matrices under a Toeplitz constraint. In this paper, the solution is extended to the case of twodimensional signals, where spatial stationarity enforces a Toeplitz-block-Toeplitz structure on the covariance matrix. A further generalisation which is presented involves the estimation of the covariance when the observations are subject to subspace interference. It is shown that this situation is amenable to a missing data interpretation, and can be incorporated into the EM iteration with moderate ease. The solution shares all the characteristics of the 1-D Toeplitz estimate. The need to solve this problem arises in many invariance applications, where it is required to fit a stationary multivariate normal model to data which is subject to a certain type of interference. The case of unknown DC offset is included in this class.