Statistical analysis of the finite sample forward-backward MVDR beamformer

A statistical analysis is provided for the sample co-variance matrix of the forward-backward (FB) minimum variance distortionless response (MVDR) beamformer. Unlike the forward-only case, its covariance matrix does not follow the complex-valued Wishart distribution. However, we can prove that after a proposed unitary transformation, the new covariance matrix has a real-valued Wishart distribution. Based on this result, a close-form expression of the output SINR (signal-to-interference-plus-noise ratio) for the finite sample FB MVDR beamformer is then derived. Our analysis shows that the error of the output SINR with FB precessing is reduced approximately by half compared with the case without it, as also verified by simulations.