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62 Some Tools and Examples ofFilter Synthesis
Utilization of the continued fraction expansion of reactance functions from
Section 3.4 requires the conversion of ZRLC to its corresponding ZLC" This is
obtained by inspection of ZRLC behavior at s=O and reference to a standard
table, which was derived. The worked example relates to the looking-back
impedance at the output of a resistively driven LC two-port network. Starting
from a given resistance function polynomial, which could have been obtained
by the fitting procedure of Section 2.5, pi-network element values that realized
this behavior versus frequency were obtained. This example will be of cen tral
importance as the final operation in a relatively new broadband impedancematching
procedure considered in Section 6.7.
3.6. Long Division and Partial Fraction Expansion
The last section of Chapter Three describes an important design tool that is
useful for network synthesis in the frequency domain as well as for Laplace
analysis in the time domain. The former is illustrated by Bode's alternative to
the Gewertz procedure (see Guillemin, 1957). The time domain application of
partial fraction expansions will be illustrated next (from Blinchikoff and
Zverev, 1976).
Suppose that a given system transfer function is
H,(s)= s4+6s'+22s'+30s+ 14. (3.109)
S4 + 6s' + 22s' + 30s + 13
As will be demonstrated, it can also be expressed in the form
H,(s)=I+ 0.1 0.02 + 0.02s+0.04 (3.110)
(s+ I)' (s+ I) (S-SI)(S-Sf)'
where s is the Laplace complex frequency variable, and root SI is S, = -2+j3.
Using a standard table of Laplace transforms for time and frequency functions,
it is easy to show that the time response corresponding to (3.110) is
h,(t) = 8(t) + [O.lte-'-0.02e-'+ 2e-'(0.01 cos 3t - si~t) ]U(t), (3.111)
where 8(t) is an impulse function, and u(t) is a unit-step function.
The algorithm to be described operates on proper rational functions, i.e.,
those whose numerator degree is lower than the denominator degree. Clearly,
(3.109) is not proper, but would be if one long-division step were accomplished.
The first subject treated in this section will be a compact long-division
algorithm, botb for obtaining proper fractions and to convince the reader that
it is not complicated to program. This is important, because long division is
one of two main features of the partial fraction expansion algorithm to follow.
3.6.1. Long Division. Vlach (1969) gives a brief FORTRAN program for
long division; it is adapted to BASIC language in Appendix-B Program B3-6.
The calculation occurs in the last 10 lines of the program.