Cramér-Rao Bound for Convolutional Beamspace Method

The Cramér-Ran Bound (CRB) for a recently proposed beamspace method, called convolutional beamspace (CBS), is studied. CBS offers the same advantages as classical beamspace methods, including lower computational complexity and higher DOA resolution. CBS also yields smaller estimation errors than element-space methods for correlated sources. The good MSE performance of CBS was demonstrated by simulations and theoretical analysis in previous works. However, they were compared to element-space CRB but not to CBS CRB. In this paper, the CRB for CBS is derived so that more insight can be further obtained. Conventionally, the CRB is a lower bound for unbiased estimators. Yet, a lower bound on the variances of the biased CBS estimator is obtained and shown to be well approximated by the classical CRB for unbiased estimators. Two forms of CRB expressions are derived, and they offer different insights as explained in the paper. All the results also apply to element-space since it is a special case of CBS. The insights obtained from the CRB expressions are verified by simulations.