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42 Some Tools and Examples of Filter Synthesis
Repeated roots are usually determined to within two or three significant
figures; this may be adequate for most but certainly not all engineering work.
3.1.5. Polynomial Scaling. By the initial guess z=O.1 +jl for the root location,
there is an assumption that the roots are not too far from the origin.
Some polynomials may require scaling of coefficients to obtain the assumed
condition, and the roots will require subsequent rescaling to correspond to the
original problem. Two methods will be described, as given by Turnbull (1952):
(1) decreasing all roots by the factor 10; (2) decreasing all roots by subtracting
some fixed amount. The choice of method and amount depends on the
problem being solved; there is usually adequate information to make those
choices.
To reduce all root real and imaginary components by a factor of 10, reduce
all polynomial coefficients of the kth-power terms by lOexp(n - k), where the
polynomial degree is n. The following example clarifies the procedure.
Example 3.5.
Consider the polynomial
f(z) = 19404-394z+2z', (3.33)
which has roots 98 + jO and 99 + jO (available from the root-finder program).
Rewrite the polynomial with revised coefficients using the rule given above:
f,(z)= 194.04-39.4z+2z'. (3.34)
The root-finder program will show that the roots of (3.34) are 9.8 + jO and
9.9 + jO. Similarly, the roots of
are 0.98 + jO and 0.99 + jO.
f,(z)=1.9404-3.94z+2z' (3.35)
The method for shifting the roots by a given amount is somewhat more
involved but uses synthetic division in an interesting way. Again, consider the
degree-3 polynomial in (3.12) without loss of generality. Suppose that variable
z is decreased by amount h:
z=s+h or s=z- h. (3.36)
Making that substitution in (3.12), there must be an equivalent polynomial,
F(s), in the new variable s:
This is rewritten two more ways:
f(z) = F(s)=bo+ b,s+b,s' + bJs'.
(3.37)
F(s) = bo+ s(co+ c,z + c,z'), (3.38)
F(s)=bo+s[b,+s(co+zc;>J. (3.39)
Note that (3.39) is nested in the same fashion as (3.25). Now (3.13) is written
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