Locally optimal maximum-likelihood estimate of a Toeplitz matrix of given rank

We derive an algorithm to compute a maximum-likelihood (ML) estimate of a Toeplitz covariance matrix T whose rank is known a priori that is "locally optimal", by maximising the likelihood ratio in the neighborhood of T. This problem arises, for example, in the detection and/or direction-of-arrival (DOA) estimation of m uncorrelated plane-wave sources using a nonuniform (sparse) M-sensor linear antenna array, where m⩾M (the "superior" case), but the problem is important in its own right, and has application to other areas of signal processing and communications. The algorithm relies upon the solution of a convex linear programming (LP) problem, whose feasibility is guaranteed