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FOIl' Important Possband Shopes 309
dB
fifo
Figure 8.25. Maximally flat breakpoint plot: N "" 4, L p = 1 dB, Fp = 0.1.
(8.72)
To locate other points at F = F, on the curve in Figure 8.25, given the loss
value Lx, use
where q was given in (8.66).
F = __F..:.,~
P (q/e)'/N'
(8.73)
Example 8.6, Suppose that the Butterworth defined passband edge is L p
= 0.1
dB at bandwidth F p
=0.15, and the stopband loss is L,=30 dB at F,=0.195.
Find N and, using the next higher integer, find the frequencies where L = 3
dB. Solving (8.72) yields N=20.3. Using N=21 in (8.73) yields F,=0.1640.
Summarizing, for N =21 the frequency response curve is 15% wide at the
O.I-dB points, 16.4% wide at the 3 dB points, and 19.5% wide at the 30 dB
points. Note that the N = 8 Chebyshev filter in Example 8.5 met the same
performance specifications.
The 2N zeros of the Butterworth function in (8.71) are shown on the circle
in Figure 8.23, where the dimensioning parameter is
cosh -1 [(sinh0.8814N)/e]
b=cosh N. .
(8.74)