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38 Some Tools and Examples ofFilter Synthesis
Example 3.1.
Consider the factors
which are equal to the polynomial
fez) = (z+ 2)(3 + 2z+ z'), (3.15)
f(z)=6+7z+4z'+z'. (3.16)
The algorithm in (3.14) will be used to find the quadratic factor in (3.15),
which is the unknown in real problems. Proceeding with (3.14):
k=2:
k= I:
k=O:
c,=I+(-2)XO=I,
c, = 4+(- 2) X I "" 2,
co=7+(-2)X2=3.
(3.17)
There is no change in the algebra when coefficients a k
in (3.12) and c k
in
(3.13) are complex; complex arithmetic is employed in (3.14) instead of the
real arithmetic previously indicated. However, when all b k
in (3.1) are zero, so
that coefficients ai in (3.12) are known to be real, then there may be one or
more real roots and any complex roots will occur in conjugate pairs. This will
be the case in ordinary filter synthesis, so that computing effort can be
reduced substantially in both synthetic division and evaluation of the polynomial
and its derivatives. Assuming real coefficients, real roots are removed, as
in (3.14), using only real arithmetic. When a root's imaginary part is not
essentially zero, then the quadratic factor containing the root and its conjugate
is removed.
Consider the identity
(z - Zi)(Z - zi') = Z2 + p,z + qi, (3.18)
where Pi= -2x" qi=X?+Y?, and z*=x-jy (see (3.2». Ralston (1965, p. 372)
described removal of quadratic factors; no complex arithmetic is involved.
Without loss of generality, consider the polynomial
fez) = ao + a,z + a,z' + a,z' + a 4 z 4 + asz s
(3.19)
and its equivalent product form
(3.20)
where the quadratic term corresponds to (3.18) with the one discovered root Zi'
The recursion is
(3.21 )
Example 3.2.
Consider the factors
fez) = (z' + 3z+2)(60+ 47z+ 12z' + z'), (3.22)