Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Example

For the example of Eq.$ \,$ (6.12), we obtain

\begin{eqnarray*}
r_{11} &=& \left.\left(1-\frac{1}{2}z^{-1}\right)^3H(z)\right\vert _{z=\frac{1}{2}}\\
&=& \left.7 - 5z^{-1}+ z^{-2}\right\vert _{z=\frac{1}{2}}
= 7 - 5\cdot 2 + 2^2 = 1\\ [10pt]
r_{12} &=& \left.-2\frac{d}{dz^{-1}} (7 - 5z^{-1}+ z^{-2})\right\vert _{z^{-1}=2}\\
&=& \left.-2(- 5 + 2z^{-1})\right\vert _{z^{-1}=2} = 2\\ [10pt]
r_{13} &=& \left.\frac{1}{2\left(-\frac{1}{2}\right)^2}\frac{d}{dz^{-1}} (-5 + 2z^{-1})\right\vert _{z^{-1}=2} = 2\cdot 2 = 4.
\end{eqnarray*}


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``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
CCRMA