Theoretical analysis of binaural transfer function MVDR beamformers with interference cue preservation constraints

The objective of binaural noise reduction algorithms is not only to selectively extract the desired speaker and to suppress interfering sources (e.g., competing speakers) and ambient background noise, but also to preserve the auditory impression of the complete acoustic scene. For directional sources this can be achieved by preserving the relative transfer function (RTF) which is defined as the ratio of the acoustical transfer functions relating the source and the two ears and corresponds to the binaural cues. In this paper, we theoretically analyze the performance of three algorithms that are based on the binaural minimum variance distortionless response (BMVDR) beamformer, and hence, process the desired source without distortion. The BMVDR beamformer preserves the binaural cues of the desired source but distorts the binaural cues of the interfering source. By adding an interference reduction (IR) constraint, the recently proposed BMVDR-IR beamformer is able to preserve the binaural cues of both the desired source and the interfering source. We further propose a novel algorithm for preserving the binaural cues of both the desired source and the interfering source by adding a constraint preserving the RTF of the interfering source, which will be referred to as the BMVDR-RTF beamformer. We analytically evaluate the performance in terms of binaural signal-to-interference- and-noise ratio (SINR), signal-to-interference ratio (SIR), and signal-to-noise ratio (SNR) of the three considered beamformers. It can be shown that the BMVDR-RTF beamformer outperforms the BMVDR-IR beamformer in terms of SINR and outperforms the BMVDR beamformer in terms of SIR. Among all beamformers which are distortionless with respect to the desired source and preserve the binaural cues of the interfering source, the newly proposed BMVDR-RTF beamformer is optimal in terms of SINR. Simulations using acoustic transfer functions measured on a binaural hearing aid validate our theoretical results.