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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}}. $

More examples are given in §3.12.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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