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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}
...z^{-1}}{1 + z^{-1}}
\;=\; g\frac{1-\zeta z^{-1}}{1-pz^{-1}}

$\displaystyle p$ $\displaystyle =$ $\displaystyle \frac{1-\frac{KT}{2(\mu+2R)}}{1+\frac{KT}{2(\mu+2R)}}$   (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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Plectrum Damping