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

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(7.17) |
for

for


- 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
residues
,
from
- If there are repeated poles, find the additional residues via
the method of §6.8.5, and the general form of the PFE is
wheredenotes 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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