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Complex Example

To illustrate an example involving complex poles, consider the filter

$\displaystyle H(z) \eqsp \frac{g}{1+z^{-2}},
$

where $ g$ can be any real or complex value. (When $ g$ is real, the filter as a whole is real also.) The poles are then $ p_1=j$ and $ p_2=-j$ (or vice versa), and the factored form can be written as

$\displaystyle H(z) \eqsp \frac{g}{(1-jz^{-1})(1+jz^{-1})}.
$

Using Eq.$ \,$ (6.8), the residues are found to be

\begin{eqnarray*}
r_1 &=& \left.(1-jz^{-1})H(z)\right\vert _{z=j}
\eqsp \left.\frac{g}{1+jz^{-1}}\right\vert _{z=j}
\eqsp \frac{g}{2}\,,\mbox{ and}\\
r_2 &=& \left.(1+jz^{-1})H(z)\right\vert _{z=-j}
\eqsp \left.\frac{g}{1-jz^{-1}}\right\vert _{z=-j}
\eqsp \frac{g}{2}\,.
\end{eqnarray*}

Thus,

$\displaystyle H(z) \eqsp \frac{g/2}{1-jz^{-1}} + \frac{g/2}{1+jz^{-1}}.
$

A more elaborate example of a partial fraction expansion into complex one-pole sections is given in §3.12.1.


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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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