Dispersive Waveguide Modeling

In a digital waveguide model, the effects of dispersion can be accounted for by considering the relationship between temporal and spatial sampling intervals:
$\displaystyle X = c(\omega) T_{s} \hspace{0.2in} \Longrightarrow \hspace{0.2in} T_{s}(\omega) = \frac{X}{c(\omega)} = \frac{c_0T_{s0}}{c(\omega)},$
where $T_{s0}$ is the unit delay time without dispersion.
As a result, unit delays ( $z^{-1}$ ) must be replaced by
$\displaystyle z^{-1} \longrightarrow z^{-c_0/c(\omega)}.$
We can interpret this new delay unit as an allpass filter that approximates the corresponding frequency-dependent delay.
In a digital waveguide model, dispersive wave propagation can thus be simulated by replacing unit delays with allpass filters, as illustrated in Fig. 3.

Figure 3: Discrete-time simulation of dispersive wave propagation, where digital allpass filters replace unit delay elements.
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Because allpass filters are linear and time invariant, they can be commuted and implemented at discrete spatial locations in the model, in exactly the same way as previously discussed for other linear, time invariant gain factors or filters.
In general, good approximations to dispersion require fairly high-order allpass filters. Contrary to the case with commuted frequency-dependent loss filters, the consolidation of allpass units does not necessarily lead to computational savings by way of lower-order filters.
A number of approaches have been reported for dispersion filter design.
To model stiff strings, the allpass response must exhibit a decreasing phase delay with increasing frequency.