Positive-definite Toeplitz completion in DOA estimation for nonuniform linear antenna arrays. I. Fully augmentable arrays

This paper considers the problem of direction-of arrival (DOA) estimation for multiple uncorrelated plane waves incident on so-called “fully augmentable” sparse linear arrays. In situations where a decision is made on the number of existing signal sources (m) prior to the estimation stage, we investigate the conditions under which DOA estimation accuracy is effective (in the maximum-likelihood sense). In the case where m is less than the number of antenna sensors (M), a new approach called “MUSIC-maximum-entropy equalization” is proposed to improve DOA estimation performance in the “preasymptotic region” of finite sample size (N) and signal-to-noise ratio. A full-sized positive definite (p.d.) Toeplitz matrix is constructed from the M×M direct data covariance matrix, and then, alternating projections are applied to find a p.d. Toeplitz matrix with m-variate signal eigensubspace (“signal subspace truncations”). When m⩾M, Cramer-Rao bound analysis suggests that the minimal useful sample size N is rather large, even for arbitrarily strong signals. It is demonstrated that the well-known direct augmentation approach (DAA) cannot approach the accuracy of the corresponding Cramer-Rao bound, even asymptotically (as N→∞) and, therefore, needs to be improved. We present a new estimation method whereby signal subspace truncation of the DAA augmented matrix is used for initialization and is followed by a local maximum-likelihood optimization routine. The accuracy of this method is demonstrated to be asymptotically optimal for the various superior scenarios (m⩾M) presented