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 numerator
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,^{7.4}*e.g.*, using the `roots` function in matlab.
The residue
corresponding to pole
may be found
analytically as

when the poles are distinct. Thus, it is the ``residue'' left over after multiplying by the pole term and letting approach the pole . In a partial fraction expansion, the th residue can be thought of as simply the

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University