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Chapter Three
Some Tools and Examples
ofFilter Synthesis
This chapter provides the nec.essary computing aids for manipulating polynomials
in Laplace complex frequency s. These programs are explained and then
applied to a meaningful sequence of modern network synthesis steps by way
of example. The result is a sense of confidence, ease, and insight that is
difficult to obtain by a purely academic approach to either computing
methods or synthesis.
A reliable root finder based on useful, important principles begins the
chapter. The synthesis process involves assembly as well as disassembly
(factoring) of polynomials; so, programs that form polynomials from factors
and by the polynomial four functions (add, subtract, multiply, and divide) are
. considered next. Also, programs for continued and partial fraction expansion
are presented with some applications.
By the end of Chapter Three, those who have used the programs, tried the
examples, and followed the fairly routine mathematical steps should be able to
appreciate more detailed explanations of synthesis methods, for example,
those of Ternes and Mitra (1973).
3.1. Complex Zeros of Complex Polynomials
Finding complex zeros of polynomials ranks, along with solution of linear
systems of equations, as a fundamental tool in engineering analysis. Textbooks
usually give examples that factor by the quadratic formula or inspection,
leaving the serious student to do his own numerical root finding by some
system routine on a large and perhaps inconvenient computer. Moore (1967)
described a conceptually interesting root finder that works well and fits easily
into small computers. This is the time to eliminate the frustration or missed
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