Biomedical Engineering Reference
In-Depth Information
The operation of the filter is much easier to understand in the frequency domain.
First let's reorder the equation (2.1):
y
[
n
]+
a
(
2
)
∗
y
[
n
−
1
]+
...
+
a
(
n
a
+
1
)
∗
y
[
n
−
n
a
]
=
b
(
1
)
∗
x
[
n
]+
b
(
2
)
∗
x
[
n
−
1
]+
...
+
b
(
n
b
+
1
)
∗
x
[
n
−
n
b
]
(2.2)
Application of
Z
transform (Sect. 1.4.6) to both sides of (2.2) yields:
A
(
z
)
Y
(
z
)=
B
(
z
)
X
(
z
)
(2.3)
From this we get:
)
−
1
B
Y
(
z
)=
A
(
z
(
z
)
X
(
z
)=
H
(
z
)
X
(
z
)
(2.4)
Function
H
in (2.4) is called the frequency response function, and it has the form:
z
−
1
z
−
n
b
b
(
1
)+
b
(
2
)
+
···
+
b
(
n
b
+
1
)
H
(
z
)=
(2.5)
a
(
1
)+
a
(
2
)
z
−
1
+
···
+
a
(
n
a
+
1
)
z
−
n
a
This function is a ratio of two polynomials. We can factor the numerator and denom-
inator to obtain:
q
1
z
−
1
q
2
z
−
1
q
n
b
z
−
1
g
(
1
−
)(
1
−
)
...
(
1
−
)
H
(
z
)=
(2.6)
p
1
z
−
1
p
2
z
−
1
p
n
a
z
−
1
(
1
−
)(
1
−
)
...
(
1
−
)
The numbers
are zeros of the numerator and are called zeros of the
transfer function. The numbers
{
q
1
,
q
2
,...,
q
n
b
}
are zeros of the denominator and
are called poles of the transfer function. The filter order equals the number of poles
or zeros, whichever is greater. We can obtain frequency dependent transfer function
H
{
p
1
,
p
2
,...,
p
n
a
}
e
i
2π
f
. The function assigns to each frequency
f
a complex
number with magnitude
M
and phase φ:
H
(
f
)
substituting
z
=
e
i
φ
(
f
)
. From equation (2.4) we
see that the operation of a filter is multiplication of each of the Fourier components
of the signal by the complex number
H
(
f
)=
M
(
f
)
;thatis,filter changes the magnitude and
phase of the component. A very useful measure can be derived from the phase of the
frequency response function—the group delay. The group delay, defined as:
(
f
)
d
φ
(
f
)
τ
g
(
f
)=
−
(2.7)
df
is a measure of the time delay introduced by the filter to the signal component of
frequency
f
. From the formula (2.7) we see that if the phase φ depends linearly on
the frequency then τ
g
const
. This means that all frequency components are
equally delayed. The phase structure of the signal is not distorted, which is essential,
if the filtering is used as a preprocessing step for more advanced methods of signal
processing that are phase sensitive. Such a linear phase delay is the property of the
FIR filters. In case of off-line applications the delays introduced by the filter, in any
form (not only the linear ones), can be corrected. The technique relies on applying
the same filter twice. After filtering the signal in the forward direction, the filtered
(
f
)=
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