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

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.