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Frequency Domain Analysis

FDA, force-drive mass $ m$:

$\displaystyle v_n=v_{n-1} + \frac{T}{m}f_n, \quad n=0,1,2,\ldots $

$ z$ transform ($ v_{-1}=0$):

$\displaystyle V(z) = z^{-1}V(z) + \frac{T}{m} F(z) $

Driving Point Impedance (digital):

$\displaystyle \zbox{R(z) \isdef \frac{F(z)}{V(z)} = m \frac{1-z^{-1}}{T}} $

Continuous-time driving point impedance:

$\displaystyle f(t)=m\frac{dv}{dt} \quad\longleftrightarrow\quad F(s)=msV(s) $

$\displaystyle \Rightarrow\quad \zbox{R(s) \isdef \frac{F(s)}{V(s)} = ms} $

Thus, the FDA maps $ s$ plane to the $ z$ plane as follows:

$\displaystyle \zbox{s \leftarrow \frac{1-z^{-1}}{T}} $

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Download SMAC03S.pdf
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``Recent Developments in Musical Sound Synthesis Based on a Physical Model'', by Julius O. Smith III, (Stockholm Musical Acoustics Conference (SMAC-03), August 6--9, 2003).
Copyright © 2006-02-19 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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