DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

A Signal Processing Perspective 247
Design Theorem, it is necessary that Itcan also be shown that to insure the
zeros of are within and that monotonically
increases for it is necessary that
An explicit formula for the ultraspherical filter polynomials, where, for
convenience, has been normalized to unity, is as follows:
An important property readily follows from (8.11): for all N and
Therefore, the required value for is as follows:
where in (8.12) is the attenuation in dB at the passband edge, i.e., at
in general).
A recursion may also be used:
where and It can readily be seen from (8.13) that,
if which would make (8.10) a Butterworth response.
It can also be shown that if that (8.10) is a Chebyshev Type I
response. If where m = 0, 1, · · · , any non-negative integer, then (8.10)
is a Legendre response: if m = 0 the response would be the standard Legendre
response, otherwise the modified m-th associated Legendre response.
Values of other than those mentioned immediately above yield responses
unique to the ultraspherical response. One will be chosen here for display of response
characteristics: let and This value of is selected
because it is an interesting compromise between the Butterworth and the Chebyshev
Type I response. The ultraspherical magnitude response has a Shaping Factor and
Filter Selectivity considerably better than a Butterworth response of the same order,
but not as good as a Chebyshev Type I with 1 dB of ripple, but the ripple in the
passband of the ultraspherical response is much less than in the Chebyshev response.
The other responses are also compromises between that of the Butterworth and the
Chebyshev Type I. See Figure 8.22 for plots of (8.10) for the ultraspherical filter,
for several values of N. See Figure 8.23 for detailed plots across the passband: the
passband ripple is less than 0.2 dB for orders greater than two and becomes less and
less as the order becomes greater. The phase response is shown in Figure 8.24. The
phase delay is shown in Figure 8.25 and the group delay in Figure 8.26. Note that
the group delay is significantly more constant across the passband, and that the peak
delay near the normalized passband edge frequency of unity is significantly less than
Section 8.4 Ultraspherical Filters