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48 Some Tools and Examples ofFilter Syntlu!sis
The transducer magnitude function in (3.49) implies the existence of H(s),
and it will be apparent later as to the convenience of defining a companion
function,
K(s) = pH(s), (3.52)
\
called the characteristic function. Using (3.49) and (3.50), an important energy
relationship between the transducer and characteristic functions is obtained:
H(s)H( - s) = I + K(s)K( - s). (3.53)
This shows that IHUw)I;;> = 1, as required. Both H(s) and K(s) are rational
functions with numerators and denominators identified as
e(s)
H(s)=pes)
,
f(s)
K(s)=-.
pes)
(3.54)
(3.55)
A concise statement can be made about the nature of the individual
polynomials e, f, and p. The roots of e(s) and f(s) are real or in conjugate
pairs. The roots of e(s) lie in the open (not on jw axis) left-half plane and are
the natural modes of the LC network; the roots of f(s) are called reflection
zeros or zero-loss frequencies. Polynomial f(s) is either even or odd, with
degree no greater than that of e(s). As in Figure 3.2, the roots of pes) are
conjugate by pairs, are purely imaginary (on the jw axis) for ladder networks,
and are called the loss poles (peaks) or transmission zeros. Polynomial pes) is
either even or odd.
Using (3.53)-(3.55), the fundamental polynomial relationship in doubly
terminated network synthesis is
e(s)e( - s) = p(s)p( - s) + f(s)f( - s). (3.56)
Either H or K is given, so that either f or e must be found from (3.56),
respectively. The latter is illustrated in the example from Ternes and Mitra
(1973).
Example 3.10.
Find H(s) given
K(s)= -76s 4 + Ils 3 -33s'+ 12s-4. (3.57)
4{3(s'+4)
Compare (3.57) with (3.55) to identify f(s) and pes); e.g., f(s) is shown in (3.51).
Use Program B3-3 to calculate p(s)p( - s) and f(s)f( - s). Adding these manually
(a program to do this will be described in Section 3.3.1), (3.56) yields
e(s)e( - s) = 5776s 8 + 48955· + 14815 4 + 504s' + 784. (3.58)
The eight roots of (3.58) are found easily using Program B3-1:
± 0.226127 ±jO.828392; ± 0.596242 ±j0.379658. (3.59)