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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Virtual Musical Instruments
       String Excitation
          Pluck Modeling
             Digitization of the Damped-Spring Plectrum

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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.

Previous: Plectrum Damping
Next: Feathering

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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