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Bayesian estimation for speech enhancement given a priori knowledge of clean speech phase

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Abstract

In this paper, STFT based speech enhancement algorithms based on estimation of short time spectral amplitudes are proposed. These algorithms use maximum likelihood, maximum a posterior and minimum mean square error (MMSE) estimators which respectively uses Laplace, Gamma and Exponential probability density functions as noise spectral amplitude priors and Nakagami distribution as speech spectral amplitude priors. The phase of noisy speech carries significant information to be retrieved and utilized. However, the undesired artifacts which are the resultant of the process do create many challenges. In this paper, the reconstructed phase is treated as an uncertain prior knowledge when deriving a joint MMSE estimate of the (C)omplex speech coefficients given (U)ncertain (P)hase information is proposed. The proposed phase reconstruction algorithm assists in generating a clean speech phase. The proposed estimator reduces undesired artifacts and also gives satisfactory values between noisy phase signal and estimate of prior phase and hence yields superior performance in the instrument measures, informal listening and speech quality.

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Author information

Authors and Affiliations

  1. Electronics and Communication Engineering Department, National Institute of Technology Warangal, Warangal, Telangana, 506 004, India

    V. Sunnydayal & T. Kishore Kumar

Authors
  1. V. Sunnydayal
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  2. T. Kishore Kumar
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Corresponding author

Correspondence to V. Sunnydayal.

Appendix: MMSE CUP Estimator (Speech as Nakagami PDF and noise as exponential PDF)

Appendix: MMSE CUP Estimator (Speech as Nakagami PDF and noise as exponential PDF)

$$ \hat{S}^{\left( \beta \right)} = E\left( {A^{\beta } e^{{j\varPhi_{S} }} \left| {y,\tilde{\phi }_{S} } \right.} \right) = \int\limits_{0}^{\infty } {\int\limits_{0}^{2\pi } {a^{\beta } e^{{j\phi_{S} }} p_{{A,\varPhi_{S} \left| {y,\tilde{\phi }_{S} } \right.}} \left( {a,\varPhi_{S} \left| {y,\tilde{\phi }_{S} } \right.} \right)d\phi_{S} da} } $$
(42)
$$ \begin{aligned} p_{{A,\varPhi_{S} \left| {y,\tilde{\phi }_{S} } \right.}} \left( {a,\varPhi_{S} \left| {y,\tilde{\phi }_{S} } \right.} \right) & = \frac{{p_{{Y,A,\varPhi_{S} ,\tilde{\varPhi }_{S} \left| {y,\tilde{\phi }_{S} } \right.}} \left( {y,a,\phi_{S} ,\tilde{\phi }_{S} } \right)}}{{p_{{Y,\tilde{\varPhi }_{S} }} \left( {y,\tilde{\phi }_{S} } \right)}} \\ & = \frac{{p_{{Y\left| {s,\tilde{\phi }_{S} } \right.}} \left( {y\left| {a,\phi_{S} ,\tilde{\phi }_{S} } \right.} \right)p_{A} \left( a \right)p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)}}{{\iint {p_{{Y\left| {s,\tilde{\phi }_{S} } \right.}} \left( {y\left| {a,\phi_{S} ,\tilde{\phi }_{S} } \right.} \right)p_{A} \left( a \right)p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)dad\phi_{S} }}} \\ \end{aligned} $$
(43)

Assuming that speech coefficients as Nakagami

$$ p_{A} \left( a \right) = \frac{2}{\varGamma \left( \mu \right)}\left( {\frac{\mu }{{\sigma_{S}^{2} }}} \right)^{\mu } a^{2\mu - 1} \exp \left( { - \frac{\mu }{{\sigma_{S}^{2} }}a^{2} } \right) $$
(44)

Assuming that noise coefficients as exponential

$$ P_{Y\left| S \right.} \left( {y\left| s \right.} \right) = \frac{1}{{\sigma_{N} }}\exp \left( {\frac{y - s}{{\sigma_{N} }}} \right) $$
(45)

von Mises distribution with concentration

$$ p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right) = \exp \left( {\kappa \cos \left( {\phi_{S} - \tilde{\phi }_{S} } \right)} \right)/2\pi I_{0} \left( \kappa \right) $$
(46)

To determine posterior, substitute (44), (45) and (46) in (43),

$$ \begin{aligned} \hat{S}^{\left( \beta \right)} & = E\left( {A^{\beta } e^{{j\varPhi_{S} }} \left| {y,\tilde{\phi }_{S} } \right.} \right) \\ & = \frac{{a^{\beta } e^{{j\phi_{S} }} \int\limits_{0}^{2\pi } {\int\limits_{0}^{\infty } {\frac{2}{\varGamma \left( \mu \right)}\left( {\frac{\mu }{{\sigma_{S}^{2} }}} \right)^{\mu } a^{2\mu - 1} \exp \left( { - \frac{\mu }{{\sigma_{S}^{2} }}a^{2} } \right)} \frac{1}{{\sigma_{N} }}\exp \left( {\frac{y - s}{{\sigma_{N} }}} \right)p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)dad\phi_{S} } }}{{\int\limits_{0}^{2\pi } {\int\limits_{0}^{\infty } {\frac{2}{\varGamma \left( \mu \right)}\left( {\frac{\mu }{{\sigma_{S}^{2} }}} \right)^{\mu } a^{2\mu - 1} \exp \left( { - \frac{\mu }{{\sigma_{S}^{2} }}a^{2} } \right)} \frac{1}{{\sigma_{N} }}\exp \left( {\frac{y - s}{{\sigma_{N} }}} \right)p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)dad\phi_{S} } }} \\ & = \frac{{\frac{{e^{{j\phi_{S} }} }}{{\sigma_{N} }}\frac{2}{\varGamma \left( \mu \right)}\left( {\frac{\mu }{{\sigma_{S}^{2} }}} \right)^{\mu } \exp \left( {\frac{y}{{\sigma_{N} }}} \right)\int\limits_{0}^{2\pi } {\int\limits_{0}^{\infty } {a^{\beta + 2\mu - 1} \exp \left( { - \frac{\mu }{{\sigma_{S}^{2} }}a^{2} - \frac{{e^{{j\phi_{S} }} }}{{\sigma_{N} }}a} \right)} p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)dad\phi_{S} } }}{{\frac{1}{{\sigma_{N} }}\frac{2}{\varGamma \left( \mu \right)}\left( {\frac{\mu }{{\sigma_{S}^{2} }}} \right)^{\mu } \exp \left( {\frac{y}{{\sigma_{N} }}} \right)\int\limits_{0}^{2\pi } {\int\limits_{0}^{\infty } {a^{2\mu - 1} \exp \left( { - \frac{\mu }{{\sigma_{S}^{2} }}a^{2} - \frac{{e^{{j\phi_{S} }} }}{{\sigma_{N} }}a} \right)} p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)dad\phi_{S} } }} \\ & = \frac{{e^{{j\phi_{S} }} \int\limits_{0}^{2\pi } {\int\limits_{0}^{\infty } {a^{\beta + 2\mu - 1} \exp \left( { - \frac{\mu }{{\sigma_{S}^{2} }}a^{2} - \frac{{e^{{j\phi_{S} }} }}{{\sigma_{N} }}a} \right)} p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)dad\phi_{S} } }}{{\int\limits_{0}^{2\pi } {\int\limits_{0}^{\infty } {a^{2\mu - 1} \exp \left( { - \frac{\mu }{{\sigma_{S}^{2} }}a^{2} - \frac{{e^{{j\phi_{S} }} }}{{\sigma_{N} }}a} \right)} p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)dad\phi_{S} } }} \\ \end{aligned} $$
(47)

From Gradshteyn and Ryzhik (2007), Eq. 3.462.1

$$ \int_{0}^{\infty } {x^{v - 1} e^{{ - \beta x^{2} - \gamma x}} } = \left( {2\beta } \right)^{{\frac{ - v}{2}}} \varGamma \left( v \right)\exp \left( {\frac{{\gamma^{2} }}{8\beta }} \right)D_{ - v} \left( {\frac{\gamma }{2\sqrt \beta }} \right) $$
(48)

Compare (47) and (48)

$$ v = \beta + 2\mu ,\quad \beta = \frac{\mu }{{\sigma_{S}^{2} }},\quad \gamma = \frac{{e^{j\phi } }}{{\sigma_{N} }}\;{\text{in}}\;{\text{numerator}} $$
$$ v = 2\mu ,\quad \beta = \frac{\mu }{{\sigma_{S}^{2} }},\quad \gamma = \frac{{e^{j\phi } }}{{\sigma_{N} }}\;{\text{in}}\;{\text{denominator}} $$
$$ = \frac{{\int\limits_{0}^{2\pi } {e^{{j\phi_{S} }} \left( {\frac{2\mu }{{\sigma_{s}^{2} }}} \right)^{{ - \left( {\frac{\beta + 2\mu }{2}} \right)}} \varGamma \left( {\beta + 2\mu } \right)\exp \left( {\frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N}^{2} }$}}}}{{{\raise0.7ex\hbox{${8\mu }$} \!\mathord{\left/ {\vphantom {{8\mu } {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}}}} \right)D_{{ - \left( {\beta + 2\mu } \right)}} \left( {\frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N} }$}}}}{{2\sqrt {{\raise0.7ex\hbox{$\mu $} \!\mathord{\left/ {\vphantom {\mu {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}} }}} \right)} p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)d\phi_{S} }}{{\int\limits_{0}^{2\pi } {\left( {\frac{2\mu }{{\sigma_{s}^{2} }}} \right)^{{ - \left( {\frac{2\mu }{2}} \right)}} \varGamma \left( {2\mu } \right)\exp \left( {\frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N}^{2} }$}}}}{{{\raise0.7ex\hbox{${8\mu }$} \!\mathord{\left/ {\vphantom {{8\mu } {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}}}} \right)D_{{ - \left( {2\mu } \right)}} \left( {\frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N} }$}}}}{{2\sqrt {{\raise0.7ex\hbox{$\mu $} \!\mathord{\left/ {\vphantom {\mu {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}} }}} \right)} p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)d\phi_{S} }} $$

Multiply and divide with \( \sigma_{N}^{2} \) with in the integration

$$ = \frac{{\left( {\frac{2\mu }{{\sigma_{s}^{2} }}\frac{{\sigma_{N}^{2} }}{{\sigma_{N}^{2} }}} \right)^{{ - \left( {\frac{\beta + 2\mu }{2}} \right)}} \varGamma \left( {\beta + 2\mu } \right)\int\limits_{0}^{2\pi } {e^{{j\phi_{S} }} \exp \left( {\frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N}^{2} }$}}}}{{{\raise0.7ex\hbox{${8\mu }$} \!\mathord{\left/ {\vphantom {{8\mu } {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}}}} \right)D_{{ - \left( {\beta + 2\mu } \right)}} \left( {\frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N} }$}}}}{{2\sqrt {{\raise0.7ex\hbox{$\mu $} \!\mathord{\left/ {\vphantom {\mu {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}} }}} \right)} p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)d\phi_{S} }}{{\left( {\frac{2\mu }{{\sigma_{s}^{2} }}\frac{{\sigma_{N}^{2} }}{{\sigma_{N}^{2} }}} \right)^{{ - \left( {\frac{2\mu }{2}} \right)}} \varGamma \left( {2\mu } \right)\int\limits_{0}^{2\pi } {\exp \left( {\frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N}^{2} }$}}}}{{{\raise0.7ex\hbox{${8\mu }$} \!\mathord{\left/ {\vphantom {{8\mu } {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}}}} \right)D_{{ - \left( {2\mu } \right)}} \left( {\frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N} }$}}}}{{2\sqrt {{\raise0.7ex\hbox{$\mu $} \!\mathord{\left/ {\vphantom {\mu {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}} }}} \right)} p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)d\phi_{S} }} $$
$$ = \frac{{\left( {\sqrt {\left( {\frac{{\xi \sigma_{N}^{2} }}{2\mu }} \right)} } \right)^{\left( \beta \right)} \varGamma \left( {\beta + 2\mu } \right)\int\limits_{0}^{2\pi } {e^{{j\phi_{S} }} \exp \left( {\frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N}^{2} }$}}}}{{{\raise0.7ex\hbox{${8\mu }$} \!\mathord{\left/ {\vphantom {{8\mu } {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}}}} \right)D_{{ - \left( {\beta + 2\mu } \right)}} \left( {\frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N} }$}}}}{{2\sqrt {{\raise0.7ex\hbox{$\mu $} \!\mathord{\left/ {\vphantom {\mu {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}} }}} \right)} p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)d\phi_{S} }}{{\varGamma \left( {2\mu } \right)\int\limits_{0}^{2\pi } {\exp \left( {\frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N}^{2} }$}}}}{{{\raise0.7ex\hbox{${8\mu }$} \!\mathord{\left/ {\vphantom {{8\mu } {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}}}} \right)D_{{ - \left( {2\mu } \right)}} \left( {\frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N} }$}}}}{{2\sqrt {{\raise0.7ex\hbox{$\mu $} \!\mathord{\left/ {\vphantom {\mu {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}} }}} \right)} p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)d\phi_{S} }} $$
$$ \hat{S}^{\left( \beta \right)} = \frac{{\left( {\sqrt {\left( {\frac{{\xi \sigma_{N}^{2} }}{2\mu }} \right)} } \right)^{\left( \beta \right)} \varGamma \left( {\beta + 2\mu } \right)\int\limits_{0}^{2\pi } {e^{{j\phi_{S} }} \exp \left( {\frac{{\upsilon^{2} }}{2}} \right)D_{{ - \left( {\beta + 2\mu } \right)}} \left( \upsilon \right)} p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)d\phi_{S} }}{{\varGamma \left( {2\mu } \right)\int\limits_{0}^{2\pi } {\exp \left( {\frac{{\upsilon^{2} }}{2}} \right)D_{{ - \left( {2\mu } \right)}} \left( \upsilon \right)} p_{{\varPhi_{S} \left| {\tilde{\phi }_{S} } \right.}} \left( {\phi_{S} \left| {\tilde{\phi }_{S} } \right.} \right)d\phi_{S} }} $$

where \( \xi = {\raise0.7ex\hbox{${\sigma_{S}^{2} }$} \!\mathord{\left/ {\vphantom {{\sigma_{S}^{2} } {\sigma_{N}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N}^{2} }$}} \), \( \upsilon = \sqrt {\frac{\xi }{4\mu }} \cos \left( {\phi_{S} - \phi_{Y} } \right) \)

$$ \upsilon = \left( {\frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N} }$}}}}{{2\sqrt {{\raise0.7ex\hbox{$\mu $} \!\mathord{\left/ {\vphantom {\mu {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}} }}} \right) = \frac{{\sqrt {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sigma_{N}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N}^{2} }$}}} }}{{2\sqrt {{\raise0.7ex\hbox{$\mu $} \!\mathord{\left/ {\vphantom {\mu {\sigma_{S}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{S}^{2} }$}}} }} = \frac{{\sqrt {{\raise0.7ex\hbox{${\sigma_{S}^{2} }$} \!\mathord{\left/ {\vphantom {{\sigma_{S}^{2} } {\sigma_{N}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma_{N}^{2} }$}}} }}{{\sqrt {4\mu } }} = \sqrt {\frac{\xi }{4\mu }} \cos \left( {\phi_{S} - \phi_{Y} } \right) $$

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Sunnydayal, V., Kumar, T.K. Bayesian estimation for speech enhancement given a priori knowledge of clean speech phase. Int J Speech Technol 18, 593–607 (2015). https://doi.org/10.1007/s10772-015-9306-4

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