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Dealing with Repeated Poles Analytically

A pole of multiplicity $ m_i$ has $ m_i$ coefficients7.9 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 coefficients associated with the pole $ z=1/2$ are 1, 2, and 4.

Let $ r_{ij}$ denote the $ j$ th coefficient 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 coefficient $ r_{ik}$ :

\begin{eqnarray*}
r_{i1} &=& \left.(1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i}\\ [5pt]
r_{i2} &=& \left.\frac{1}{-p_i}\frac{d}{dz^{-1}} (1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i}\\ [5pt]
r_{i3} &=& \left.\frac{1}{2(-p_i)^2}\frac{d^2}{d(z^{-1})^2} (1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i}\\ [5pt]
r_{i4} &=& \left.\frac{1}{3\cdot 2(-p_i)^3}\frac{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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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2015-04-22 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA