Musical-Noise-Free Blind Speech Extraction Based on Higher-Order Statistics Analysis

Abstract

In this chapter, we introduce a musical-noise-free blind speech extraction method using a microphone array for application to nonstationary noise. In the recent noise reduction study, it was found that optimized iterative spectral subtraction (SS) results in speech enhancement with almost no musical noise generation, but this method is valid only for stationary noise. The method presented in this chapter consists of iterative blind dynamic noise estimation by, e.g., independent component analysis (ICA) or multichannel Wiener filtering, and musical-noise-free speech extraction by modified iterative SS, where multiple iterative SS is applied to each channel while maintaining the multichannel property reused for the dynamic noise estimators. Also, in relation to the method, we discuss the justification of applying ICA to signals nonlinearly distorted by SS. From objective and subjective evaluations simulating a real-world hands-free speech communication system, we reveal that the method outperforms the conventional speech enhancement methods.

Appendix

This appendix provides a brief review of the time-variant nonlinear noise estimator. For more detailed information, Ref. [24] is available.

Let $X_{1}(f,\tau )$ and $X_{2}(f,\tau )$ be noisy signals received at the microphones in the time-frequency domain, defined as

$$\begin{aligned} X_{1}(f,\tau ) = H_{1}(f)S(f,\tau )+N_{1}(f,\tau ),\end{aligned}$$

(13.55)

$$\begin{aligned} X_{2}(f,\tau ) = H_{2}(f)S(f,\tau )+N_{2}(f,\tau ), \end{aligned}$$

(13.56)

where $H_{1}(f)$ and $H_{2}(f)$ are the transfer functions from the target signal position to each microphone. Next, the auto-power PSDs in each microphone, $\varGamma _{11}(f)$ and $\varGamma _{22}(f)$, can be expressed as follows:

$$\begin{aligned} \varGamma _{11}(f,\tau ) = |H_{1}(f)|^2 \varGamma _\mathrm{SS}(f,\tau )+\varGamma _\mathrm{NN}(f,\tau ) ,\end{aligned}$$

(13.57)

$$\begin{aligned} \varGamma _{22}(f,\tau ) = |H_{2}(f)|^2 \varGamma _\mathrm{SS}(f,\tau )+\varGamma _\mathrm{NN}(f,\tau ) , \end{aligned}$$

(13.58)

where $\varGamma _\mathrm{SS}(f,\tau )$ is the PSD of the target speech signal and $\varGamma _\mathrm{NN}(f,\tau )$ is the PSD of the noise signal. In this chapter, we assume that the left and right noise PSDs are approximately the same, i.e., $\varGamma _\mathrm{N_1 N_1}(f,\tau ) \simeq \varGamma _\mathrm{N_2 N_2}(f,\tau ) \simeq \varGamma _\mathrm{NN}(f,\tau )$.

Next, we consider the Wiener solution between the left and right transfer functions, which is defined as

$$\begin{aligned} H_\mathrm{W}(f,\tau )=\frac{\varGamma _\mathrm{12}(f,\tau )}{\varGamma _\mathrm{22}(f,\tau )}, \end{aligned}$$

(13.59)

where $\varGamma _\mathrm{12}(f)$ is the cross-PSD between the left and right noisy signals. The cross-PSD expression then becomes

$$\begin{aligned} \varGamma _\mathrm{12}(f,\tau )= \varGamma _\mathrm{SS}(f,\tau )H_\mathrm{1}(f)H^*_\mathrm{2}(f). \end{aligned}$$

(13.60)

Therefore, substituting (13.60) into (13.59) yields

$$\begin{aligned} H_\mathrm{W}(f,\tau )=\frac{\varGamma _\mathrm{SS}(f,\tau )H_\mathrm{1}(f)H^*_\mathrm{2}(f)}{\varGamma _\mathrm{22}(f,\tau )}. \end{aligned}$$

(13.61)

Furthermore, using (13.57) and (13.58), the squared magnitude response of the Wiener solution in (13.61) can also be expressed as

$$\begin{aligned} |H_\mathrm{W}(f,\tau )|^2\!=\!\frac{(\varGamma _\mathrm{11}(f,\tau )\!-\!\varGamma _\mathrm{NN}(f,\tau ))(\varGamma _\mathrm{22}(f,\tau )\!-\!\varGamma _\mathrm{NN}(f,\tau ))}{\varGamma ^2_\mathrm{22}(f,\tau )}. \end{aligned}$$

(13.62)

Equation (13.62) is rearranged into the following quadratic equation:

$$\begin{aligned}&\varGamma ^2_\mathrm{NN}(f,\tau ) - \varGamma _\mathrm{NN}(f,\tau )\left( \varGamma _\mathrm{11}(f,\tau )+\varGamma _\mathrm{22}(f,\tau )\right) \nonumber \\&+\varGamma _\mathrm{EE}(f,\tau )\varGamma _\mathrm{22}(f,\tau )=0, \end{aligned}$$

(13.63)

where

$$\begin{aligned} \varGamma _\mathrm{EE}(f,\tau )=\varGamma _\mathrm{11}(f,\tau ) - \varGamma _\mathrm{22}(f,\tau )|H_\mathrm{W}(f)|^2. \end{aligned}$$

(13.64)

Consequently, the noise PSD $\varGamma _\mathrm{NN} (f)$ can be estimated by solving the quadratic equation in (13.63) as follows:

$$\begin{aligned} \varGamma _\mathrm{NN}(f,\tau )&=\frac{1}{2}\left( \varGamma _\mathrm{11}(f,\tau )+\varGamma _\mathrm{22}(f,\tau )\right) - \varGamma _\mathrm{avg}(f,\tau ), \end{aligned}$$

(13.65)

$$\begin{aligned} \varGamma _\mathrm{avg}(f,\tau )&= \frac{1}{2}\{(\varGamma _\mathrm{11}(f,\tau )+\varGamma _\mathrm{22}(f,\tau ))^2\nonumber \\&-4\varGamma _\mathrm{EE}(f,\tau )\varGamma _\mathrm{22}(f,\tau )\}^{0.5}. \end{aligned}$$

(13.66)