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64 Some Tools and Examples ofFilter Synthesis
Figure 3.9 shows that two successive long divisions by the factor x + I were
accomplished with remainder numerators - 24 and 23 - 24/(x + I), respectively.
It is helpful to follow this proce$.s by doing the division either manually
or with Program B3-6 and writing the results of each separate division. Then a
division step by the factor x - I occurs, leaving the constant - II plus the
prior rational remainder. The process is fairly clear up to the point where the
second main operation occurs. However, the next (second) division by factor
x - I leaves the constant 3 plus the expression
_1_(_Il+2L_ 24 )= -11+8 +--=.§...+ 12 . (3.Il4)
x-I x+1 (x+I)2 x-I x+1 (x+I)2
In this identity the right side preserves the form of the preceding collection of
terms, and thus preserves the algorithm as different root factors are encountered.
This illustrates the general scheme; interested readers are referred to
Chin and Steiglitz (1977) for further detail.
Two more comments are appropriate. Some ill-conditioned roots may cause
rounding errors to accumulate unless the roots are processed in order of
ascending magnitude. Note that the example in Figure 3.9 employs real roots;
the roots may be complex and therefore in conjugate complex pairs. They are
processed separately in Program B3-7 using complex arithmetic. As in the
root-finder Program B3-1, this can be avoided by dealing only with quadratic
factors, as mentioned by Chin and Steiglitz (1977).
Example 3.18. First run the example in Figure 3.9 to be sure that the output
sequence of residues is understood. Then perform a partial fraction expansion
of (3.109) by first obtaining the proper fraction in (3.113) by one long-division
step (Program B3-6). Use root-finder Program B3-1 to find denominator roots
- I + jO, - I + jO, - 2 + j3, and - 2 - j3. Enter these roots, in that order, into
partial fraction expansion Program B3-7 to find the residues of each term.
These are shown in (3.110), except for the combined conjugate roots term.
This is obtained with the .useful identity
K, Kf (K,+Kns-(K,sf+K~s;)
-+--= (3.115)
S-Sj S-Sj* (S-Si)(S-st) l
where K, is a residue. Note that residues of complex conjugate roots also
occur as complex conjugates.
3.6.3. Summary of Partial Fraction Expansion. A long-division algorithm
that is simple enough for even hand-held computers is furnished in BASIC
language. It is useful in reducing rational polynomials to proper form, i.e.,
numerator degree less than denominator degree. Long division is also one of
the two main features of an efficient algorithm that is also especially suitable
for small computers.
The input to the partial fraction expansion algorithm consists of the
numerator real coefficients and the denominator roots in order of ascending