A widely linear distortionless filter for single-channel noise reduction

Traditionally in the single-channel noise-reduction problem, speech distortion is inevitable since the desired signal is also filtered while filtering the noise. In fact, the more the noise is reduced, the more the speech distortion is added into the desired signal, as proved in the literature. So, if we require no speech distortion, we either end up with no noise reduction at all or have to use multiple sensors. In this paper, we attempt to apply the widely linear (WL) estimation theory to noise reduction. Unlike the traditional approaches that only filter the short-time Fourier transform (STFT) of the noisy signal, the method developed in this paper applies the noise-reduction filter to both the STFT of the noisy signal and its conjugate. With the constraint of no speech distortion, a WL distortionless filter is derived. We show that this new optimal filter can fully take advantage of the noncircularity property of speech signals to achieve up to 3-dB signal-to-noise-ratio (SNR) improvement without introducing any speech distortion, which can only be obtained with the traditional approaches if two or more microphones are used.