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Physical Audio Signal Processing
    Elementary String Instruments
       Ideal String Struck by a Mass
          Digital Waveguide Model

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Digital Waveguide Model

To obtain a force-wave digital waveguide model of the string-mass assembly after the mass has struck the string, it only remains to digitize the model of Fig.4.23. The delays are obviously to be implemented using digital delay lines. For the mass, we must digitize the force reflectance

$\displaystyle \hat{\rho}(s) = \frac{ms}{ms+2R}. \protect$ (5.23)

A common choice of digitization procedure is the bilinear transform (§L.4). This will effectively yield a wave digital filter model for the mass in this context (see Appendix Q for a tutorial on wave digital filters).

The bilinear transform is typically scaled as

$\displaystyle s = \frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}} $

where $ T$ denotes the sampling interval. This choice optimizes the low-frequency match between the digital and analog mass frequency responses. Rewriting Eq.$ \,$(4.23) as

$\displaystyle \hat{\rho}(s) = \frac{1}{1+\frac{2R}{ms}} $

the bilinear transform gives

\begin{eqnarray*} \hat{\rho}_d(z) &=& \frac{1}{1+\frac{2R}{m}\frac{T}{2}\frac{1+z^{-1}}{1-z^{-1}}}\\ [5pt] &=& g\frac{1-z^{-1}}{1-pz^{-1}} \end{eqnarray*}

where the gain coefficient $ g$ and pole $ p$ are given by

\begin{eqnarray*} g&\isdef &\frac{1}{1+\frac{RT}{m}}\;<\;1\\ [5pt] p&\isdef &\frac{1-\frac{RT}{m}}{1+\frac{RT}{m}}\;<\;1. \end{eqnarray*}

Thus, the reflectance of the mass is a one-pole, one-zero filter. The zero is exactly at dc, the real pole is close to dc, and the gain at half the sampling rate is $ 1$. We may recognize this as the classic dc-blocking filter [460]. Comparing with Eq.$ \,$(4.23), we see that the behavior at dc is correct, and that the behavior at infinite frequency ( $ \hat{\rho}(\infty)=1$) is now the behavior at half the sampling rate (