Influence Function and Asymptotic Efficiency of Scatter Matrix Based Array Processors: Case MVDR Beamformer

Abstract

In this paper, we consider array processors that are scale-invariant functions of the array covariance matrix. The emphasis is on Capon's MVDR beamformer. We call such an array processor as scatter matrix based (SMB) array processor since the covariance matrix is required only up to a constant scalar and thus a scatter matrix (proportional to covariance under finite covariance assumption) provides sufficient information. In order to establish interesting statistical robustness and large sample properties, we derive a general expression for the influence function and the asymptotic covariance structure of SMB-MVDR beamformer weights. Our results apply under the class of complex elliptically symmetric distributions, which includes the commonly used complex normal distribution as a special case. We illustrate the theory by deriving the IF and asymptotic relative efficiencies of the conventional SMB-MVDR beamformer that employs the sample covariance matrix and beamformers that employ robust M -estimators of scatter. Theoretical findings are confirmed by simulations. Our findings favor beamformers based upon M-estimators of scatter, since they combine a high efficiency with appealing robustness properties.