New insights into non-causal multichannel linear filtering for noise reduction

We investigate a general framework for noise reduction which consists in controlling the level of signal distortion while reducing the level of noise. A parameterized non-causal filter that allows for tuning the signal distortion and noise reduction inversely is obtained and is referred to as parameterized multichannel non-causal Wiener filter (PMWF) herein. The same optimization problem leads to the minimum variance distortionless response (MVDR) as a particular case of the PMWF. In contrast to earlier works, the proposed expressions of the PMWF and MVDR are simplified and require the knowledge of the speech and noise statistics only. To rigorously quantify the gains and losses when using these filters, we establish simplified closed-form expressions for three measures, namely, the signal distortion index, the noise reduction factor, and the output signal-to-noise ratio (SNR), and highlight the tradeoff between noise reduction and speech distortion in the multichannel case.