Summary of the Partial Fraction Expansion
In summary, the partial fraction expansion can be used to expand any rational z transform
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 to obtain the three roots, , :
A = [1 0 0 -1]; % Filter denominator polynomial poles = roots(A) % Filter poles
Software for Partial Fraction Expansion
Alternate Stability Criterion