Locally optimal maximum-likelihood completion of a partially specified Toeplitz covariance matrix

The problem of maximum-likelihood (ML) completion of a partially specified Toeplitz covariance matrix is crucial in several applications, such as the detection and estimation of more independent Gaussian sources than sensors (m>M) in minimum-redundancy sparse linear antenna arrays. Given the sufficient statistic in the form of the M-variate direct data covariance matrix Rˆ, we describe an algorithm that finds a positive-definite completed M_α-variate Toeplitz matrix (M_α≫M) with (locally) maximal likelihood ratio (LR). Simulations demonstrate that a statistically high LR is achieved, compared with L₂ optimisation