Biomedical Engineering Reference
In-Depth Information
H
(
s
)
Ideal low-pass f ilter
A
0
Practical low-pass f ilter
0.707
A
0
w
w
c
FIGURE 11.1
:
Butterworth low-pass characteristic transfer function response curves. The ideal
low-pass filter is shown as the red trace, and the normal practical response trace is shown in black
magnitude monotonically decreases as frequency increases in both the band-pass and
band-stop regions. The flattest magnitude of the filter is in the vicinity of
ω
=
0, or DC,
with its greatest error about the cutoff frequency.
The “Transfer Function” or describing equation for the Butterworth low-pass filter
is given by (11.1) for the s-domain and rewritten in (11.2) in terms of frequency.
A
lp
K
H
(
s
)
=
1
=
(11.1)
s
+
s
+
j
ω
A
lp
H
(
j
ω
)
=
1
(11.2)
ω
c
2
+
Low-pass gain, and
ω
c
where
A
lp
=
=
ω
n
The second-order transfer function for a low-pass Butterworth filter is given by
(11.3):
2
n
A
lp
ω
H
(
s
)
=
s
2
+
2
ζω
n
s
+
b
ω
n
(11.3)
1
R
1
R
2
C
1
C
2
=
1
T
1
T
2
2
c
where
b
ω
=
11.1.2 Chebyshev low-pass Filter
The Chebyshev low-pass filter has an underdamped response, meaning that a filter of
second order or higher order may oscillate and become unstable if the gain is too high
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