On Toeplitz-constrained weights for spatially distributed source localization

The paper offers two contributions. Firstly, we provide a relationship between two well-known methods - redundancy averaging (RA) and weighted covariance-matching (WCM) - for estimating any Toeplitz Hermitian covariance matrix. The analysis presented enables us to perform their connection by means of optimal weights. By applying both Toeplitzifications in asymptotic weight determination, another virtue is to propose an improvement on the asymptotic best consistent (ABC) estimator for direction finding in the presence of a spatially distributed source. Since the Toeplitz structure of the array covariance matrix is held itself in a symmetric angle deviation model, it is thus advantageous to incorporate such a priori knowledge of the matrix structure into a consistent weight estimation step. Based on weighted least squares (WLS) criteria, our enforcement upon the Toeplitz structure is pictorially illustrated that, in a small, or even moderate, number of temporal snapshots, the estimations of nominal direction and their associated angular spread with the additional constraint can outperform the conventional approach without preprocessing of the structured covariance estimation.