On single-channel noise reduction in the time domain

In this paper, we revisit the noise-reduction problem in the time domain and present a way to decompose the filtered speech into two uncorrelated (orthogonal) components: the desired speech and the interference. Based on this new decomposition, we discuss how to form different optimization cost functions and address the issue of how to design different noise-reduction filters by optimizing these new cost functions. Particularly, we cover the design of the maximum signal-to-noise-ratio (SNR), the Wiener, the minimum variance distortionless response (MVDR), and the tradeoff filters. It is interesting that with this new decomposition, we can now design the MVDR filter that can achieve noise reduction without adding speech distortion in the single-channel case, which has never been seen before. We also demonstrate that the maximum SNR, Wiener, and tradeoff filters are identical to the MVDR filter up to a scaling factor. From a theoretical point of view, this scaling factor is not significant and should not affect the output SNR at any processing time. But from a practical viewpoint, the scaling factor can be time-varying due to the nonstationarity of the speech and possibly the noise and can cause discontinuity in the residual noise level, which is unpleasant to listen to. As a result, it is essential to have the scaling factor right from one processing sample (or frame) to another in order to avoid large distortions and for this reason, it is recommended to use the MVDR filter in speech enhancement applications.