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214 Impedance Matching
repeated here using identities (2.34) and (2.35). The lowpass prototype frequency
variable will be w' and the degree will be n':
for w'" I, and
IH(w')1 2 = I +K 2 + ,2 cos'(n' cos- I w') (6.95)
IH(w')1 2 = I + K 2 + ,2 cosh'(n' cosh - I w') (6.96)
for w' > 1. These equations define the passband and stopband, respectively, in
Figure 6.21. Substituting the following frequency-mapping function into (6.95)
and (6.96) produces the response in Figure 6.36:
",2_",2
w'= 0
A
where the defined constants are
(6.97)
(6.98)
w o = Jw;;w~ . (6.99)
(See Figure 6.36, which is plotted in terms of the frequency variable w.)
Although the defined constant W
o is shown, the band-center frequency is taken
as the arithmetic average,
(6.100)
and is scaled so that W m = 1. The relative bandwidth is defined with respect to
Wm:
(6.101)
Note that both Cottee parameters, W o and w, differ from the parameters with
the same names discussed earlier in this chapter.
With (6.97) substituted, the transducer function of w, defined by (6.95) and
(6.96), is a double mapping of the conventional (w') function shown in Figure
6.21; it maps into Figure 6.36 from Wo to 0 and from Wo to infinity. It is easy to
confirm the mappings of w~w' for passband edge frequencies Wb~ I and
wa~-l, the w' image of w'~+ 1. The nature of this mapping is such that the
conventional lowpass prototype filter having n' reactive elements corresponds
to a new filter "having n=2n' elements, giving the response in Figure 6.36. It
will be important to keep track of the complex frequency domains s' and s,
corresponding to degrees n' and n and frequency domains U)' and w, respectively.
6.6.2. Evaluation ofGain-Bandwidth Integrals. Fano's gain-bandwidth integrals
were given in (6.45) and (6.46) for one- and two-reactance lowpass loads,