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410 Chapter 8 IIR FILTER DESIGN
Magnitude Response
Magnitude in dB
|H|
1
0.4467
decibels
0
7
16
0.1585
0
0 0.2 0.3 0.5
Analog frequency in π units
30
0 0.2 0.3 0.5
Analog frequency in π units
1
Phase Response
0.2
Impulse Response
0.5
0.15
radians
0
−0.5
ha(t)
0.1
0.05
−1
0 0.2 0.3 0.5
Analog frequency in π units
0
0 10 20 30
time in seconds
FIGURE 8.15 Butterworth analog filter in Example 8.4
8.3.5 CHEBYSHEV LOWPASS FILTERS
There are two types of Chebyshev filters. The Chebyshev-I filters have
equiripple response in the passband, while the Chebyshev-II filters have
equiripple response in the stopband. Butterworth filters have monotonic
response in both bands. Recall our discussions regarding equiripple FIR
filters. We noted that by choosing a filter that has an equiripple rather
than a monotonic behavior, we can obtain a lower-order filter. Therefore
Chebyshev filters provide lower order than Butterworth filters for the
same specifications.
The magnitude-squared response of a Chebyshev-I filter is
|H a (jΩ)| 2 =
1
( ) (8.52)
Ω
Ω c
1+ɛ 2 T 2 N
where N is the order of the filter, ɛ is the passband ripple factor, which is
related to R p , and T N (x)istheNth-order Chebyshev polynomial given by
{ (
cos N cos −1 (x) ) , 0 ≤ x ≤ 1
T N (x) =
cosh ( cosh −1 (x) ) , 1