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Embedded SystemsFPGAElectronics

Introduction to Digital Filters
    Transfer Function Analysis
       Partial Fraction Expansion
          Repeated Poles
             Dealing with Repeated Poles Analytically

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Dealing with Repeated Poles Analytically

A pole of multiplicity $ m_i$ has $ m_i$ residues associated with it. For example,

$\displaystyle H(z)$ $\displaystyle \isdef$ $\displaystyle \frac{7 - 5z^{-1}+ z^{-2}}{\left(1-\frac{1}{2}z^{-1}\right)^3}$  
  $\displaystyle =$ $\displaystyle \frac{1}{\left(1-\frac{1}{2}z^{-1}\right)^3} + \frac{2}{\left(1-\frac{1}{2}z^{-1}\right)^2} + \frac{4}{\left(1-\frac{1}{2}z^{-1}\right)} \protect$ (7.12)

and the three residues associated with the pole $ z=1/2$ are 1, 2, and 4.

Let $ r_{ij}$ denote the $ j$th residue associated with the pole $ p_i$, $ j=1,\ldots,m_i$. Successively differentiating $ (1-p_iz^{-1})^{m_i}H(z)$ $ k-1$ times with respect to $ z^{-1}$ and setting $ z=p_i$ isolates the residue $ r_{ik}$:

\begin{eqnarray*} r_{i1} &=& \left.(1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i}\\... ...ac{d^3}{d(z^{-1})^3} (1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i} \end{eqnarray*}

or

$\displaystyle \zbox {r_{ik} = \left.\frac{1}{(k-1)!(-p_i)^{k-1}}\frac{d^{k-1}}{d(z^{-1})^{k-1}} (1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i}} $

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About the Author: 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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