Cramér-Rao Bound for DOA Estimators Under the Partial Relaxation Framework: Derivation and Comparison

Abstract

A class of computationally efficient DOA estimators under the Partial Relaxation (PR) framework has recently been proposed. Conceptually different from conventional DOA estimation methods in the literature, the estimators under the PR framework rely on the non-complete relaxation of the array manifold while performing a spectral-search in the field of view. This particular type of relaxation essentially implies a modified signal model with partial information loss due to the relaxation. The information loss and its impact on the DOA estimation performance have not yet been analytically quantified in the literature. In this paper, the information loss induced by the relaxation of the array manifold is investigated through the Cramér-Rao Bound (CRB). The closed-form expression of the CRB for DOA estimation under the PR model, on the one hand, provides insight on the information loss in the asymptotic region where the number of snapshots tends to infinity. On the other hand, the proposed CRB characterizes the lower bound for the DOA estimation performance of all PR estimators. We prove that, under the assumptions of Gaussian source signal and noise, the CRB of the PR signal model is lower-bounded by the conventional stochastic CRB. We also prove that the previously proposed Weighted Subspace Fitting estimator under the PR framework asymptotically achieves the CRB of the PR signal model. Furthermore, it is shown that the asymptotic mean-squared errors of all Weighted Subspace Fitting estimators under the PR framework for any positive definite weighting matrix are identical.