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44 Some Tools and Examples ofFilter Synthesis
3.1.6. Root-Finder Summary. Moore"s root finder is a practical tool that is
accurate and robust except for repeated roots, when accuracy is reduced. It is
based on the Cauchy-Riemann condition and the Mitrovic method for
evaluation of the polynomial and its derivative. Computation is reduced by
more than half when all polynomial coefficients are real. which is usually the
case in modern network synthesis. There are many other applications for this
fast root finder, such as in root locus plotting versus gain factors and in
z-transform calculations in sampled data system design. The structure of the
particular problem may result in roots being far from the origin of the
complex plane; in these cases, where the root finder may be slow or may fail
to converge, scaling of polynomial coefficients can reduce each root by either
a factor or a fixed amount. Roots thus found closer to the origin can then be
moved back to their original location by shifting in the opposite fashion.
The following sections will employ this root finder for network synthesis
steps and partial fraction expansions.
3.2. Polynomials From Complex Zeros and Products
The next two sections describe the composition of polynomials by multiplication
and addition, respectively. The computer programs provided will continue
to be in BASIC language, although these calculations are just as feasible
in hand-held computers. This section begins with composition of polynomials
from known root factors as available in the preceding root-finder section.
Complex factors will be multiplied to find the generally complex coefficients
of the resulting polynomial. Then a program will be given that multiplies a
sequence of polynomials having real coefficients.
The last half of this section includes the beginning steps in doubly terminated
network synthesis; both the ideas and the use of the computing aids are
important in what follows. Power transfer from a complex source to a complex
load will be introduced and then specialized to the real-source impedance
case. The generalized reflection coefficient will be defined, and the Feldtkeller
energy equation will be discussed for a given steady-state frequency of
excitation. Finally, polynomials used in network synthesis will be described,
and the fundamental polynomial relationship will be derived from power
transfer considerations of a lossless two-port network.
3.2.1. Polynomials From Complex Zeros. Only polynomials with real coefficients
are considered, so that their roots must be real or occur in conjugate
pairs. A conjugate pair of complex numbers can always be expressed as a
quadratic factor, as previously described by (3.18). Program B3-2 in appendix
B asks for a set of complex zeros in rectangular components, then outputs the
resulting polynomial coefficients, also in their rectangular components. It is
interesting to confirm some of the previously described characteristics of
polynomials by use of this program.