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Minimum mean square error estimator for speech enhancement in additive noise assuming Weibull speech priors and speech presence uncertainty

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Abstract

A novel single-channel technique was proposed based on a minimum mean square error (MMSE) estimator to enhance short-time spectral amplitude (STSA) in the Discrete Fourier Transform (DFT) domain. In the present contribution, a Weibull distribution was used to model DFT magnitudes of clean speech signals under the additive Gaussian noise assumption. Moreover, the speech enhancement procedure was conducted with (WSPU) and without speech presence uncertainty (WoSPU). The theoretical spectral gain function was obtained as a weighted geometric mean of hypothetical gains associated with signal presence and absence. Extensive experiments were conducted with clean speech signals taken from the TIMIT database, which had been degraded by various additive non-stationary noise sources, and then enhanced signals were evaluated. The evaluation results demonstrated the outperformance of the proposed method compared to the probability density functions (PDF) of Rayleigh and Gamma distributions in terms of segmental signal-to-noise ratio (segSNR), general SNR, and perceptual evaluation of speech quality (PESQ). The performance in the WSPU case was also significantly improved compared to WoSPU, assuming Weibull speech priors in the MMSE-STSA based speech enhancement algorithm.

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Authors and Affiliations

  1. Department of Electrical Engineering, Imam Khomeini International University, Qazvin, Iran

    Mojtaba Bahrami & Neda Faraji

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  1. Mojtaba Bahrami
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  2. Neda Faraji
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Correspondence to Neda Faraji.

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Appendices

Appendix 1

According to Papoulis and Pillai (2002) by integration of Eq. (23) in polar coordinate the univariate distribution is obtained

$$ p(R) = \frac{\partial }{\partial R}\int\limits_{0}^{R} {\int\limits_{0}^{2\pi } {\frac{1}{2\pi }} } p(Y \equiv X)rd\varphi dr, $$
(36)

where \( R = \left| Y \right| = \sqrt {Y_{\text{Re}}^{2} + Y_{\text{Im}}^{2} } , \) and \( Y_{\text{Re}} \) and \( Y_{\text{Im}} \) denote the real and imaginary parts of noisy speech spectrum, respectively. We have the Weibull PDF in Eq. (1) in order to solve the integral in Eq. (36), we utilize Eq. (37)

$$ \frac{\partial }{\partial w}\int\limits_{0}^{w} {z(v)dv} = z(w). $$
(37)

Appendix 2

The 2-D convolution formula in polar coordinate is defined as

$$ f(R,\theta ) = g(R,\theta )\, * \,h(R,\theta ) = \int\limits_{0}^{\infty } {\int\limits_{0}^{2\pi } {g(R - t,\theta - \alpha )h(t,\alpha )td\alpha dt} } $$
(38)

where r and θ are amplitude and phase variables.

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Bahrami, M., Faraji, N. Minimum mean square error estimator for speech enhancement in additive noise assuming Weibull speech priors and speech presence uncertainty. Int J Speech Technol 24, 97–108 (2021). https://doi.org/10.1007/s10772-020-09767-y

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