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Digitization of the Damped-Spring Plectrum

Applying the bilinear transformation7.3.2) to the reflectance $ \hat{\rho}_f(s)$ in Eq.$ \,$ (9.23) (including damping) yields the following first-order digital force-reflectance filter:

$\displaystyle \hat{\rho}_f(z) \;=\; \frac{\mu}{\mu+2R}
\cdot
\frac{
\frac{2}{T}\frac{1 - z^{-1}}{1 + z^{-1}}
+
\frac{K}{\mu}}{
\frac{2}{T}\frac{1 - z^{-1}}{1 + z^{-1}}
+\frac{K}{\mu+2R}}
\;=\; g\frac{1-\zeta z^{-1}}{1-pz^{-1}}
$

where
$\displaystyle p$ $\displaystyle =$ $\displaystyle \frac{1-\frac{KT}{2(\mu+2R)}}{1+\frac{KT}{2(\mu+2R)}}$   (digital pole)$\displaystyle \protect$ (10.27)
$\displaystyle \zeta$ $\displaystyle =$ $\displaystyle \frac{1-\frac{KT}{2\mu}}{1+\frac{KT}{2\mu}}$   (digital zero)$\displaystyle \protect$ (10.28)
$\displaystyle g$ $\displaystyle =$ $\displaystyle \frac{1-p}{1-\zeta}$   (gain term)$\displaystyle \protect$ (10.29)

The transmittance filter is again $ 1+\hat{\rho}_f(z)$ , and there is a one-filter form for the scattering junction as usual.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA