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352 Other Direct Filter Design Methods
9.2.4. Summary of Introduction to eauer Elliptic Filters. The lowpass response
functions of Butterworth, Chebyshev, inverse Chebyshev, and Cauer
elliptic filters have certain similarities and increasing complexity in the order
given. The rational elliptic function may be written as the ratio of two even
polynomials of the same even degree. Their roots are easily computed using
the doubly periodic Jacobian elliptic sine function.
Selectivity estimates related to choice of filter degree can be accomplished
using a nomogram or a computer program. Program B9-1 does that; it also
computes the trap frequencies and the attenuation at any given frequency.
Symmetric elliptic filters have odd-degree N, have (N - 1)/2 traps, and
have equal terminating resistances. Even-degree elliptic filters are called
antimetric (from synthesis terminology, which is irrelevant to these purposes).
They have N/2 traps, have unequal terminating resistances, and require either
ideal transformers or one negative element. The even-degree filter just described
has the exact response shape and is called a type-a filter. By arbitrarily
moving two traps to infinity, the negative element or the ideal transformers
are eliminated; this filter, with an approximate elliptic response, is called a
type-b filter. A further aberration moves two passband zeros to the origin, to
obtain equal terminating resistances; this is called a type-c filter. Both type-b
and type-c filters have slightly reduced selectivity cutoff rates, but usually this
is not a serious departure from predictions based on the nomogram.
9.3. Doubly Terminated Elliptic Filters
Amstutz (1978) published a procedure and two FORTRAN computer programs
for calculating the elements of doubly terminated elliptic filters. The
basis of his method will be described, and BASIC language translations of his
programs, with examples, will be furnished. As seen in Program B9-1, there is
little difficulty in computing the doubly periodic Jacobian elliptic functions.
However, it was also noted that round-off error can be a problem. Another
Amstutz contribution was a more accurate computation of these functions,
especially the elliptic sine function; his method utilizes infinite products
instead of infinite summations.
The Amstutz method is quite straightforward, although it incorporates one
subtle step. He computes the poles and zeros of elliptic filter transducer
function H and characteristic function K, as described in Section 3.2.4. As in
(3.52), he obtains the rational polynomial of the reflection coefficient in factor
form. Attention is then restricted to the trap frequencies. It is clear from
Figure 9.12 that the input coefficient magnitude at these frequencies must be
unity (complete reflection). Thus Amstutz finds very simple expressions for
both the input impedance and its derivative with respect to frequency Zin (i.e.,
group delay) at the trap frequencies. Again referring to Figure 9.12a, the input
impedance at/the frequency where M I and K 1 are resonant is clearly Zin = sL I .
Also, it can be shown that M 1 and K 1 can be found from Z'n'