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Summary
of the Partial Fraction Expansion
In summary, the partial fraction expansion can be used to expand
any rational z transform
as a sum of firstorder terms

(7.17) 
for
, and

(7.18) 
for
, where the term
is optional, but often
preferred. For real filters, the complex onepole terms may be paired
up to obtain secondorder terms with real coefficients.
The PFE procedure occurs in two or three steps:
 When
, perform a step of long division to obtain
an FIR part
and a strictly proper IIR part
.
 Find the
poles
,
(roots of
).
 If the poles are distinct, find the
coefficients (residues)
,
from
 If there are repeated poles, find the additional coefficients via
the method of §6.8.5, and the general form of the PFE is

(7.19) 
where
denotes the number of distinct poles, and
denotes the multiplicity of the
th pole.
In step 2, the poles are typically found by factoring the
denominator polynomial
. This is a dangerous step numerically
which may fail when there are many poles, especially when many poles
are clustered close together in the
plane.
The following matlab code illustrates factoring
to
obtain the three roots,
,
:
A = [1 0 0 1]; % Filter denominator polynomial
poles = roots(A) % Filter poles
See Chapter 9 for additional discussion regarding digital filters
implemented as parallel sections (especially §9.2.2).
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