Progress in structured covariance estimation

The author summarizes recent work on the problem of computing maximum-likelihood estimates of structured covariance matrices, as it applies to problems in array processing and spectrum estimation. Two areas are discussed; the existence of positive definite solutions when the number of observations is less than the dimension of the matrix, and the efficient implementation of the algorithm expectation-maximization for estimating Toeplitz matrices. For the Toeplitz case, it is shown that, for a single observation vector, the probability of generating a positive definite solution can be very small, whereas when the Toeplitz covariance matrix is constrained to have a nonnegative definite circulant extension, a positive definite solution will exist with probability