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Repeated Poles

When poles are repeated, an interesting new phenomenon emerges. To see what's going on, let's consider two identical poles arranged in parallel and in series. In the parallel case, we have

$\displaystyle H_1(z) \eqsp \frac{r_1}{1-pz^{-1}} + \frac{r_2}{1-pz^{-1}}
\eqsp \frac{r_1+r_2}{1-pz^{-1}}
\isdefs \frac{r_3}{1-pz^{-1}}.
$

In the series case, we get

$\displaystyle H_2(z) \eqsp \frac{r_1}{1-pz^{-1}} \cdot \frac{r_2}{1-pz^{-1}}
\eqsp \frac{r_1r_2}{(1-pz^{-1})^2}
\isdefs \frac{r_3}{(1-pz^{-1})^2}.
$

Thus, two one-pole filters in parallel are equivalent to a new one-pole filter7.8 (when the poles are identical), while the same two filters in series give a two-pole filter with a repeated pole. To accommodate both possibilities, the general partial fraction expansion must include the terms

$\displaystyle \frac{r_{1,1}}{(1-pz^{-1})^2} + \frac{r_{1,2}}{(1-pz^{-1})}
$

for a pole $ p$ having multiplicity 2.



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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
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