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Introduction to Digital Filters
    Transfer Function Analysis
       Partial Fraction Expansion
          Complex Example

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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.\... ...eft.\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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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