DSPRelated.com
Free Books
   Free Books    Physical Audio Signal Processing  

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:

$\displaystyle \hat{\rho}_f(z) \;=\; \frac{\mu}{\mu+2R} \cdot \frac{ \frac{2}{T... ...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.

Next Section:
Feathering
Previous Section:
Plectrum Damping