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Physical Audio Signal Processing
    Equivalence of Digital Waveguide and Finite Difference Schemes
       State Space Formulation
          Boundary Conditions
             Reactive Terminations

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Reactive Terminations

In typical string models for virtual musical instruments, the ``nut end'' of the string is rigidly clamped while the ``bridge end'' is terminated in a passive reflectance $ S(z)$. The condition for passivity of the reflectance is simply that its gain be bounded by 1 at all frequencies [458]:

$\displaystyle \left\vert S(e^{j\omega T})\right\vert\leq 1, \quad \forall\, \omega T\in[-\pi,\pi). \protect$ (P.42)

A very simple case, used, for example, in the Karplus-Strong plucked-string algorithm, is the two-point-average filter:

$\displaystyle S(z) = -\frac{1+z^{-1}}{2} $

To impose this lowpass-filtered reflectance on the right in the chosen subgrid, we may form

   $\displaystyle \mbox{$\stackrel{{\scriptscriptstyle \vdash\!\!\dashv}}{\mathbf{A}}$}$$\displaystyle _W=$   $\displaystyle \mbox{$\stackrel{{\scriptscriptstyle \vdash}}{\mathbf{A}}$}$$\displaystyle _W- \frac{1}{2}{\bm \Delta}_{8,5} - \frac{1}{2}{\bm \Delta}_{8,7} $

which results in the FDTD transition matrix

\begin{eqnarray*} \mbox{$\stackrel{{\scriptscriptstyle \vdash\!\!\dashv}}{\mathb... ... \\ 0 & 0 & 0 & 0 & -1/2 & 1/2 & -1 & -1 \end{array}\!\right]. \end{eqnarray*}

This gives the desired filter in a half-rate, staggered grid case. In the full-rate case, the termination filter is really

$\displaystyle S(z) = -\frac{1+z^{-2}}{2} $

which is still passive, since it obeys Eq.$ \,$(P.42), but it does not have the desired amplitude response: Instead, it has a notch (gain of 0) at one-fourth the sampling rate, and the gain comes back up to 1 at half the sampling rate. In a full-rate scheme, the two-point-average filter must straddle