Finite data performance analysis of MVDR beamformer with and without spatial smoothing

Recently, the performance of a minimum variance distortionless response (MVDR) beamformer has been extensively studied for the case when a true or asymptotic covariance matrix is available. In practical situations, however, we only have a finite number of snapshots of data from which the array covariance matrix can be estimated. In this paper, we analyze the finite-data performance of this beamformer with and without spatial smoothing, using first-order perturbation theory. In particular, we develop expressions for the mean values of the power gain in any direction of interest, the output power and the norm of the weight-error vector, as a function of the number of snapshots and the number of smoothing steps. We show that, in general, the smoothing, in addition to decorrelating the sources, can also dlleviate the effects of finite-data perturbations. Next, we reduce the above expressions to the case when no spatial smoothing is used. These expressions are valid for an arbitrary array and for arbitrarily correlated signals. For this case, we also develop an expression for the variance of the power gain. We simplify these expressions for a single interference case to show explicitly how the SNR, spacing of the interference from the desired signal and the correlation between them influence the beamformer performance. Simulations are used to verify the usefulness of the theoretical expressions and the results show an excellent agreement with predicted results.