Speech Enhancement - Digital Speech: Coding for Low Bit Rate Communication Systems - page 382

Assuming that each spectral component of speech and noise has complex
Gaussian distribution, and that the noise is additive to and uncorrelated with
the speech signal, the conditional probability density functions observing a
noisy spectral component Y k ,given H 0 and H 1 ,are

2 ) exp

2

1 | Y k |

p(Y k | H 0 ) = − (11.37)

2 )

πE( | D k | E( | D k |

exp

(11.38)

2

1 | Y k |

p(Y k | H 1 ) = −

π(E( | D k |

2 ) +

E( | X k |

2 ))

E( | D k |

2 ) +

E( | X k |

2 )

2 ) and E( 2 )

where k is the spectral bin index, 0 ≤ k ≤ K/ 2, and E( | D k | | X k |

denote the variances of the k th

spectral components of noise and speech,

respectively.
The probability of speech presence can be given by Bayes' rule,

p(Y k | H 1 )p(H 1 )

p(H 1 | Y k ) =

| + |

p(Y k H 0 )p(H 0 ) p(Y k H 1 )p(H 1 )

µ

= (11.39)

1 + µ k

where,

p(H 1 )
p(H 0 )

µ = (11.40)

in which p(H 1 ) and p(H 0 ) denote the apriori probability of speech presence
and absence, respectively. The likelihood ratio of the k th spectral bin k can
be defined from the above two likelihood ratios,

p(Y k | H 1 )

k =

p(Y k | H 0 )

exp ( 1

1 + γ k )ξ k

=

(11.41)

1 + ξ k 1 + ξ k

The enhanced spectrum based on the probability of speech presence is
written as,

X k

= E(X k | Y k ,H 0 )p(H 0 | Y k ) + E(X k | Y k ,H 1 )p(H 1 | Y k ) (11.42)