In summary, the partial fraction expansion can be used to expand
any rational z transform
as a sum of first-order terms
for , and
for , where the term
is optional, but often
preferred. For real filters, the complex one-pole terms may be paired
up to obtain second-order 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 residues ,
If there are repeated poles, find the additional residues via
the method of §6.8.5, and the general form of the PFE is
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
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).