Jointly optimal dereverberation and beamforming

We previously proposed an optimal (in the maximum likelihood sense) convolutional beamformer that can perform simultaneous denoising and dereverberation, and showed its superiority over the widely used cascade of a Weighted Prediction Error (WPE) dereverberation filter and a conventional Minimum-Power Distortionless Response (MPDR) beamformer. However, it has not been fully investigated which components in the convolutional beamformer yield such superiority. To this end, this paper presents a new derivation of the convolutional beamformer that allows us to factorize it into a WPE dereverberation filter, and a special type of a (non-convolutional) beamformer, referred to as a weighted MPDR (wM-PDR) beamformer, without loss of optimality. With experiments, we show that the superiority of the convolutional beamformer in fact comes from its wMPDR part.