### 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 first-order terms

 (7.17)

for , and

 (7.18)

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:
1. When , perform a step of long division to obtain an FIR part and a strictly proper IIR part .
2. Find the poles , (roots of ).
3. If the poles are distinct, find the residues , from

4. If there are repeated poles, find the additional residues 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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