On the optimal linear filtering techniques for noise reduction

Noise reduction, which aims at extracting the clean speech from noisy observations, has plenty of applications. It has attracted a considerable amount of research attention over the past several decades. Although many methods have been developed, the most widely used one, by far, is the optimal linear filtering technique, which achieves clean speech estimate by passing the noisy observation through an optimal linear filter/transformation. The representative algorithms of this include Wiener filtering, spectral restoration, subspace method, etc. Many experiments have been carried out, from various points of view, to show that the optimal filtering technique can reduce the level of noise that is present in the speech signal and improve the corresponding signal-to-noise ratio (SNR). However, there is not much theoretical justification so far for the noise reduction and SNR improvement. This paper attempts to provide a theoretical analysis on the performance (including noise reduction, speech distortion, and SNR improvement) of the optimal filtering noise-reduction techniques including the time-domain causal Wiener filter, the subspace method, and the frequency-domain subband Wiener filter. We show that the optimal linear filter, regardless of how we delineate it, can indeed reduce the level of noise (but at a price of attenuating the desired speech signal). Most importantly, we prove that the a posteriori SNR (defined after the optimal filtering) is always greater than, or at least equal to the a priori SNR, which reveals that the optimal linear filtering technique is indeed able to make noisy speech signals cleaner. We will also discuss the bounds for noise reduction, speech distortion, and SNR improvement. 2007 Elsevier B.V. All rights reserved.