Summary of the Partial Fraction Expansion

In summary, the partial fraction expansion can be used to expand any rational z transform

$\displaystyle H(z) \eqsp \frac{B(z)}{A(z)} \eqsp \frac{b_0 + b_1 z^{-1}+ \cdots + b_M z^{-M}}{1 + a_1 z^{-1}+ \cdots + a_N z^{-N}}$

as a sum of first-order terms

$\displaystyle H(z) \isdefs \frac{B(z)}{A(z)} \eqsp \sum_{i=1}^{N} \frac{r_i}{1-p_iz^{-1}} \protect$

(7.17)

for

, and

$\displaystyle H(z) \eqsp F(z) + z^{-(K+1)}\sum_{i=1}^{N}\frac{r_i}{1-p_iz^{-1}} \protect$

(7.18)

for $M\geq N$ , where the term $z^{-(K+1)}$ 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 $M\geq N$ , perform a step of long division to obtain an FIR part and a strictly proper IIR part $B^\prime(z)/A(z)$ .
Find the poles , $i=1,\ldots,N$ (roots of ).
If the poles are distinct, find the residues , $i=1,\ldots,N$ from

$\displaystyle r_i = \left.(1-p_iz^{-1})\frac{B(z)}{A(z)}\right\vert _{z=p_i}$
If there are repeated poles, find the additional residues via the method of §6.8.5, and the general form of the PFE is

$\displaystyle H(z) \eqsp F(z) + z^{-(K+1)}\sum_{i=1}^{N_p}\sum_{k=1}^{m_i}\frac{r_{i,k}}{(1-p_iz^{-1})^k} \protect$ (7.19)

where denotes the number of distinct poles, and $m_i\ge 1$ 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 $A(z) = 1 - z^{-3}$ to obtain the three roots, $p_k=e^{jk2\pi/3}$ , :

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).

Order a Hardcopy of Introduction to Digital Filters

Previous: Alternate Stability Criterion
Next: Software for Partial Fraction Expansion

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

Sign in

Search Online Books

Free Online Books

Ads

Chapters

Introduction to Digital Filters
    Transfer Function Analysis
       Partial Fraction Expansion
          Summary of the Partial Fraction Expansion