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Polynomial Addition and Subtraction ofParts 49
The last step is to associate the left-half-plane roots with e(s) as required
above. Using the four left-half-plane roots from (3.59) and Program B3-2, the
e(s) polynomial is obtained, except for a constant. By (3.58), that constant
must be J5776 = 76, so that
e(s) = 76s 4 + 125s 3 + 135s 2 +84s+28. (3.60)
Using the denominator of (3.57) as pes) and (3.60), the rational polynomial
H(s) is thus found according to (3.54).
3.2.5. SUtnmlJry of Polynomials From Zeros and Products. Programs that
calculate polynomial coefficients given complex zeros or given a sequence of
polynomials to multiply are easy to program and require very little computer
memory. QuadratiC factors, magnitude functions, and polynomial factors
having roots in the left-half plane are important parts of the mathematics of
network synthesis.
The basis of doubly terminated network selectivity behavior is the Feldtkeller
energy equation (3.49), which describes the power transfer from a
source, relative to maximum available power, in terms of the reflection
coefficient at that interface. This leads to the transducer and characteristic
functions that are polynomials in complex frequency s= a +jw. There is a free
exchange of sand jw in the magnitude-function relationships (the interested
reader is referred to Van Valkenburg, 1960, for details of the underlying
analytic continuation principle). There is a straightforward procedure for
finding the transducer numerator polynomial given the characteristic function
numerator and denominator, and vice versa. The programs in this chapter
make these computations relatively easy.
33. Polynomial Addition and Subtraction of Parts
The transducer and characteristic functions Hand K have been introduced by
way of the Feldtkeller energy equation. The chain (or ABCD) parameters for
two-port networks are commonly encountered as complex numbers at a
frequency, but also may be rational functions of complex frequency s. This
section will introduce a simple program for adding and subtracting polynomials.
the main step required to use H(s) and K(s) to find the polynomials A, B,
C, and D prior to finding an LC network that corresponds to the given data.
The program and the synthesis steps will be described.
3.3.1. Program for Addition and Subtraction of Parts. Program B3-4 in
Appendix B adds or subtracts coefficients of like powers of s in two given
polynomials, or just those coefficients of even powers or of odd powers. It is
written in BASIC, but the single-subscript array R(·) is the basis of the
memory assignment; this makes its translation to hand-held calculators especially
elementary. The computation occurs in lines 200-370.