Digital Signal Processing Reference
In-Depth Information
11.2.3 Wiener Filtering
The Wiener filter (WF) is a minimum mean square error (MMSE) estimate
of a desired signal in the time domain [1, 4]. Given a noisy signal
y(n)
,for
0
≤
n
≤
N
−
1, the Wiener filter produces the MMSE estimate
x(n)
of speech
ˆ
x(n)
as,
x(
0
)
ˆ
ˆ
w
0
w
1
.
w
P
−
1
y(
0
)
y(
−
1
)
···
y(
1
−
P)
x(
1
)
.
y(
1
)
y(
0
)
···
y(
2
−
P)
=
···
···
···
···
y(N
−
1
) y(N
−
2
)
···
y(N
−
P)
x(N
ˆ
−
1
)
w
Y
x
ˆ
(11.16)
where
w
k
are the filter coefficients for 0
1 with the filter order
P
.
Equation (11.16) can be rewritten in the algebraic form as,
≤
k
≤
P
−
=
(11.17)
x
ˆ
Yw
The Wiener filter error signal
e
is the difference between the desired and
estimated speech signals given by,
=
−
ˆ
e
x
x
(11.18)
The error metric
ε
is defined as,
e
T
e
ε
=
(11.19)
Yw
)
T
(
x
=
(
x
−
−
Yw
)
x
T
x
w
T
Y
T
x
x
T
Yw
w
T
Y
T
Yw
=
−
−
−
The filter coefficients
w
are derived by setting the derivative of
ε
to zero with
respect to
w
,
∂ε
∂
w
=−
2
(
x
T
Y
w
T
y
T
Y
)
−
=
0
(11.20)
Then, the optimal
w
is given by,
(
Y
T
Y
)
−
1
Y
T
x
=
(11.21)
w
in which
Y
T
Y
and
Y
T
x
are the autocorrelation matrix
R
yy
of
y(n)
and
the cross-correlation vector
r
yx
between
y(n)
and
x(n)
, respectively. Thus,
equation (11.21) can be written as,
R
−
1
=
yy
r
yx
(11.22)
w
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