On Optimal Frequency-Domain Multichannel Linear Filtering for Noise Reduction

Abstract

Several contributions have been made so far to develop optimal <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">multichannel</i> linear filtering approaches and show their ability to reduce the acoustic noise. However, there has not been a clear <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unifying theoretical</i> analysis of their performance in terms of both noise reduction and speech distortion. To fill this gap, we analyze the frequency-domain (non-causal) multichannel linear filtering for noise reduction in this paper. For completeness, we consider the noise reduction constrained optimization problem that leads to the parameterized multichannel non-causal Wiener filter (PMWF). Our contribution is fivefold. First, we formally show that the minimum variance distortionless response (MVDR) filter is a particular case of the PMWF by properly formulating the constrained optimization problem of noise reduction. Second, we propose new simplified expressions for the PMWF, the MVDR, and the generalized sidelobe canceller (GSC) that depend on the signals' statistics only. In contrast to earlier works, these expressions are explicitly independent of the channel transfer function ratios. Third, we quantify the theoretical gains and losses in terms of speech distortion and noise reduction when using the PWMF by establishing new simplified closed-form expressions for three performance measures, namely, the signal distortion index, the noise reduction factor (originally proposed in the paper titled ldquoNew insights into the noise reduction Wiener filter,rdquo by J. Chen ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IEEE Transactions on Audio, Speech, and Language Processing</i> , Vol. 15, no. 4, pp. 1218-1234, Jul. 2006) to analyze the single channel time-domain Wiener filter), and the output signal-to-noise ratio (SNR). Fourth, we analyze the effects of coherent and incoherent noise in addition to the benefits of utilizing multiple microphones. Fifth, we propose a new proof for the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> SNR improvement achieved by the PMWF. Finally, we provide some simulations results to corroborate the findings of this work.