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348 Other Direct Filter Design Methods
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Figure 9.13. Low-order Chebyshev rational functions. [From Daniels, 1974. Copyright 1974 Bell
Telephone Laboratories. Reprinted by permission.]
the Jacobian elliptic sine function, which is doubly periodic and only slightly
harder to evaluate than the trigonometric sine function. This calculation is
included in BASIC Program B9- I. It requires less than II lines of code. The
rational functions so obtained behave as indicated in Figure 9.13. It is easy to
imagine how the square of these functions produces the frequency response in
Figure 9.1 I b.
9.2.2. Eltiptic Filter Degree, Attenuation, and Pole Frequencies. Zverev
(1967) published the well-known nomogram in Figure 9.14, which relates the
elliptic filter degree and attenuation to the bandwidth parameters. Figures
9.I1b and 9.14 show that the passband ripple (Am,,=O.1 dB) and the
stopband shelf attenuation (Amin =43 dB) require N = 5 for a stopband/passband
transition ratio (WH/WB) of 1.5. Program B9-1 has been translated to
BASIC from a program by Daniels (1974). It duplicates the calculations
illustrated in Figure 9.14 and computes the attenuation pole (trap) frequencies.
A word of caution: Program B9-l is inaccurate for the solution in the
upper-left corner of Figure 9.14 because of the round-off error in the complete
elliptic integral calculation (another Jacobian function). A more accurate