Digital Signal Processing Reference
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in which
α
k
and
θ
k
are dummy variables for the spectral amplitude and phase,
respectively, of
X
k
. The amplitude has the Rayleigh distribution given by,
exp
α
k
2
α
k
=
−
p(α
k
)
(11.29)
|
|
2
)
|
|
2
)
E(
X
k
E(
X
k
and the phase has the uniform distribution given by,
1
2
π
p(θ
k
)
=
(11.30)
Through derivation given in [10], equation (11.26) can be rewritten as,
(
1
.
5
)
√
v
k
γ
k
exp
2
(
1
v
k
)I
0
v
k
2
v
k
I
1
v
k
2
v
k
|
X
k
|=
−
+
+
|
Y
k
|
(11.31)
=
√
π/
2,
I
0
(
where
(
)
denote
the modified Bessel functions of zero and first order, respectively, and
v
k
≡
·
)
is the gamma function with
(
1
.
5
)
·
)
and
I
1
(
·
ξ
k
ξ
k
γ
k
.
As a variant, Ephraim and Malah [15] proposed an MMSE log spectral
amplitude (MMSE-LSA) estimator, based on the well-known fact that a
distortionmeasurewith the log spectral amplitudes ismore suitable for speech
processing. The MMSE-LSA estimator minimizes the following distortion
measure,
1
+
log
2
|
X
k
ε
=
|
X
k
|−
log
|
(11.32)
with
exp
E
Y
k
}
|
X
k
|=
{
log
(
|
X
k
|
)
|
(11.33)
From [15], the final estimate becomes,
ξ
k
exp
1
dt
∞
e
−
t
t
ξ
k
|
X
k
|=
|
Y
k
|
(11.34)
1
+
2
v
k
11.2.5 Spectral EstimationBasedontheUncertaintyofSpeech
Presence
The conventional speech enhancement methods can be extended by incorpo-
rating the uncertainty of speech presence [14, 15]. The absence and presence
of speech,
H
0
and
H
1
, respectively, can be defined as,
H
0
:
Y
k
=
D
k
(11.35)
H
1
:
Y
k
=
X
k
+
D
k
(11.36)
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