Speech Enhancement - Digital Speech: Coding for Low Bit Rate Communication Systems - page 375

11.2 Review of STSA-based Speech Enhancement

Assuming that the noise d(n) is additive to the speech signal x(n) ,thenoisy
speech y(n) can be written as,

y(n) = x(n) + d(n), for 0 ≤ n ≤ K − 1 (11.1)

where n is the time index. The objective of speech enhancement is to find
the enhanced speech

ˆ

x(n) given y(n) , with the assumption that d(n) is
uncorrelated with x(n) . The time-domain signals can be transformed to the
frequency domain as,

Y k = X k + D k , for 0 ≤ k ≤ K − 1 (11.2)

where Y k , X k ,and D k denote the short-time DFT of y(n) , x(n) ,and d(n) ,
respectively. The STSA-based speech enhancement filters out the noise by
modifying the spectral amplitudes of Y k in equation (11.2). Therefore, the
enhanced spectrum X k can be written in terms of the modification factor
(gain) G k and the noisy spectrum Y k as,

X k =

G k Y k , for 0 ≤ G k ≤ 1 (11.3)

The gain G k is a function of a posteriori SNR,

2

| Y k |

≡

γ k (11.4)

E( | D k |

2 )

and apriori SNR,

2 )

E( | X k |

ξ k ≡ (11.5)

E( | D k |

2 )

2 ) are the statistical variances of the k th spectral
components of the noise and the speech, respectively. The function definition
of the gain G k depends on specific enhancement methods. The a posteriori SNR
γ k in equation (11.4) can be obtained easily as Y k is the input noisy spectrum
and E(

2 ) and E( |

where E( | D k | X k |

2 ) can be obtained through a noise adaptation procedure discussed
in Section 11.3. However, the speech variance E(

| D k |

2 ) for the estimation of
ξ k in equation (11.5) is not available. As a solution, Ephraim and Malah [10]
proposed the decision-directed (DD) method given by,

| X k |

α | X (t − 1 )

2

|

ξ (t)

α)MAX(γ (t)

=

2 ) +

( 1 − − 1 , 0 ) (11.6)

k k

D (t)

E( | |

k

where 0 ≤ α< 1and t is the frame index.