Partial Fraction Expansion

An important tool for inverting the *z* transform and converting among digital
filter implementation structures is the *partial fraction
expansion* (PFE). The term ``partial fraction expansion'' refers to the
expansion of a rational transfer function into a sum of first and/or
second-order terms. The case of first-order terms is the simplest and
most fundamental:

where

and . (The case is addressed in the next section.)
The denominator coefficients are called the *poles* of the
transfer function, and each numerators is called the
*residue* of pole . Equation (6.7) is general only if the poles
are *distinct*. (Repeated poles are addressed in
§6.8.5 below.) Both the poles and their residues may be complex.
The poles may be found by factoring the polynomial into
first-order terms,^{7.2} *e.g.*,
using the `roots` function in matlab.
The residue corresponding to pole may be found
analytically as

when the poles are distinct. In the Matlab Signal Processing Tool Box, the function

Note that in Eq. (6.8), there is always a pole-zero cancellation at
. That is, the term
is always cancelled by an
identical term in the denominator of , which must exist because
has a pole at . The residue is simply the
*coefficient* of the one-pole term
in the partial
fraction expansion of at . The transfer function
*is*
, in the limit, as .

- Example
- Complex Example
- PFE to Real, Second-Order Sections
- Inverting the Z Transform
- FIR Part of a PFE

- Alternate PFE Methods
- Repeated Poles
- Dealing with Repeated Poles Analytically
- Example
- Impulse Response of Repeated Poles
- So What's Up with Repeated Poles?

- Alternate Stability Criterion
- Summary of the Partial Fraction Expansion
- Software for Partial Fraction Expansion

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