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Inverting the Z Transform

The partial fraction expansion (PFE) provides a simple means for inverting the z transform of rational transfer functions. The PFE provides a sum of first-order terms of the form

$\displaystyle H_i(z) \eqsp \frac{r_i}{1-p_iz^{-1}}.
$

It is easily verified that such a term is the z transform of

$\displaystyle h_i(n) \eqsp r_i p_i^n, \quad n=0,1,2,\ldots\,.
$

Thus, the inverse z transform of $ H(z)$ is simply

$\displaystyle h(n) \eqsp \sum_{i=1}^N h_i(n) \eqsp \sum_{i=1}^N r_i p_i^n,
\quad n=0,1,2,\ldots\,.
$

Thus, the impulse response of every strictly proper LTI filter (with distinct poles) can be interpreted as a linear combination of sampled complex exponentials. Recall that a uniformly sampled exponential is the same thing as a geometric sequence. Thus, $ h$ is a linear combination of $ N$ geometric sequences. The term ratio of the $ i$ th geometric sequence is the $ i$ th pole, $ p_i$ , and the coefficient of the $ i$ th sequence is the $ i$ th residue, $ r_i$ .

In the improper case, discussed in the next section, we additionally obtain an FIR part in the z transform to be inverted:

$\displaystyle F(z) \eqsp f_0 + f_1z^{-1}+ f_2z^{-2}+ \cdots + f_K z^{-K} \;\longleftrightarrow\;
[f_0,f_1,\ldots,f_K,0,0,\ldots].
$

The FIR part (a finite-order polynomial in $ z^{-1}$ ) is also easily inverted by inspection.

The case of repeated poles is addressed in §6.8.5 below.


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA