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Characteristics of Prototype Analog Filters 419
Magnitude Response
Magnitude in dB
1
0.8913
0
1
|H|
decibels
16
0.1585
radians
0
0 0.2 0.3 0.5
Analog frequency in π units
1
0.5
0
−0.5
Phase Response
−1
0 0.2 0.3 0.5
Analog frequency in π units
ha(t)
30
0 0.2 0.3 0.5
Analog frequency in π units
0.2
0.1
0
−0.1
−0.2
−0.3
Impulse Response
0 10 20 30
time in seconds
FIGURE 8.17 Chebyshev-II analog filter in Example 8.7
8.3.10 ELLIPTIC LOWPASS FILTERS
These filters exhibit equiripple behavior in the passband as well as in
the stopband. They are similar in magnitude response characteristics to
the FIR equiripple filters. Therefore elliptic filters are optimum filters
in that they achieve the minimum order N for the given specifications
(or alternately, achieve the sharpest transition band for the given order
N). These filters, for obvious reasons, are very difficult to analyze and,
therefore, to design. It is not possible to design them using simple tools,
and often programs or tables are needed to design them.
The magnitude-squared response of elliptic filters is given by
|H a (jΩ)| 2 =
1
1+ɛ 2 U 2 N
( Ω
Ω c
) (8.61)
where N is the order, ɛ is the passband ripple (which is related to R p ),
and U N (·) istheNth-order Jacobian elliptic function. The analysis of
this function, even on a superficial level, is beyond the scope of this book.
Note the similarity between the preceding response (8.61) and that of the
Chebyshev filters given by (8.52). Typical responses for odd and even N
are as follows.
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