Biomedical Engineering Reference
In-Depth Information
2.1.4 Wiener filter
cific frequency response. It is not the case for a Wiener filter, introduced by Norbert
Wiener [Wiener, 1949].
Let's assume that a certain system generates signal
u
. We record the signal with an
apparatus which has a known impulse response
r
. We assume that the measurement
is corrupted by the additive Gaussian white noise
v
. This can be expressed as:
]=
∑
i
x
[
n
]=
y
[
n
]+
v
[
n
r
[
n
−
i
]
u
[
i
]+
v
[
n
]
(2.11)
In such a system
u
cannot be accessed directly. However, minimizing the square error
we can find
u
which estimates
u
:
arg min
E
∑
n
|
u
[
n
]
−
u
[
n
]
|
2
u
[
n
]=
(2.12)
From the Parseval theorem it follows, that the above relation will also hold for the
Fourier transforms of the respective signals:
arg min
E
2
U
∑
f
|
U
[
f
]
−
U
[
f
]
|
[
f
]=
(2.13)
Taking advantage of the convolution theorem (Sect. 1.4.4) for the signal
y
[
n
]=
∑
i
r
[
n
−
i
]
u
[
i
]
we obtain:
Y
[
f
]=
U
[
f
]
R
[
f
]
(2.14)
Assuming that the estimator has the form:
X
[
f
]
Φ
[
f
]
U
[
f
]=
(2.15)
R
[
f
]
It can be shown that condition (2.13) is satisfied for:
2
|
Y
[
f
]
|
Φ
[
f
]=
(2.16)
|
Y
[
f
]
|
2
+
|
V
[
f
]
|
2
Φ
[
f
]
is called a Wiener filter. Thus:
2
Φ
[
f
]
1
|
Y
[
f
]
|
H
[
f
]=
]
=
(2.17)
2
2
R
[
f
R
[
f
]
|
Y
[
f
]
|
+
|
V
[
f
]
|
gives the frequency domain representation of the transfer function of the optimal
filter.
In practice neither
Y
[
f
]
nor
V
[
f
]
is known. The empirical estimate of Φ
[
f
]
may be
computed as:
S
x
[
f
]
−
S
v
[
]
f
S
v
for S
x
[
f
]
>
[
f
]
Φ
[
f
]=
S
x
[
f
]
(2.18)
S
v
0
for S
x
[
f
]
≤
[
f
]
is the power spectrum of
x
and
S
v
where
S
x
[
f
]
[
f
]
is the approximation of the noise
power spectrum.
Search WWH ::
Custom Search