Digitization of the Damped-Spring Plectrum

Applying the bilinear transformation (§7.3.2) to the reflectance $\hat{\rho}_f(s)$ in Eq.(9.23) (including damping) yields the following first-order digital force-reflectance filter:

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

where

$$p = \frac{1-\frac{KT}{2(\mu+2R)}}{1+\frac{KT}{2(\mu+2R)}} \protect$$ (10.27)

$$\zeta = \frac{1-\frac{KT}{2\mu}}{1+\frac{KT}{2\mu}} \protect$$ (10.28)

$$g = \frac{1-p}{1-\zeta} \protect$$ (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.