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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:

$\displaystyle H(z) \isdefs \frac{B(z)}{A(z)} \eqsp \sum_{i=1}^{N} \frac{r_i}{1-p_iz^{-1}} \protect$ (7.7)


B(z) &=& b_0 + b_1 z^{-1}+ b_2z^{-2}+ \cdots + b_M z^{-M}\\
A(z) &=& 1 + a_1 z^{-1}+ a_2z^{-2}+ \cdots + a_N z^{-N}

and $ M<N$ . (The case $ M\geq N$ is addressed in the next section.) The denominator coefficients $ p_i$ are called the poles of the transfer function, and each numerator $ r_i$ is called the residue of pole $ p_i$ . Equation (6.7) is general only if the poles $ p_i$ 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 $ A(z)$ into first-order terms,7.4e.g., using the roots function in matlab. The residue $ r_i$ corresponding to pole $ p_i$ may be found analytically as

$\displaystyle r_i \eqsp \left.(1-p_iz^{-1})H(z)\right\vert _{z=p_i} \protect$ (7.8)

when the poles $ p_i$ are distinct. Thus, it is the ``residue'' left over after multiplying $ H(z)$ by the pole term $ (1-p_iz^{-1})$ and letting $ z$ approach the pole $ p_i$ . In a partial fraction expansion, the $ i$ th residue $ r_i$ can be thought of as simply the coefficient of the $ i$ th one-pole term $ r_i/(1-p_iz^{-1})$ in the PFE. The matlab function residuez7.5 will find poles and residues computationally, given the difference-equation (transfer-function) coefficients.

Note that in Eq.(6.8), there is always a pole-zero cancellation at $ z=p_i$ . That is, the term $ (1-p_iz^{-1})$ is always cancelled by an identical term in the denominator of $ H(z)$ , which must exist because $ H(z)$ has a pole at $ z=p_i$ . The residue $ r_i$ is simply the coefficient of the one-pole term $ 1/(1-p_i z^{-1})$ in the partial fraction expansion of $ H(z)$ at $ z=p_i$ . The transfer function is $ r_i/(1-p_iz^{-1})$ , in the limit, as $ z\to p_i$ .

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University