Wave propagation in distributed physical systems can be efficiently simulated using DWNs. The simplest case is a single bidirectional delay line which simulates wave propagation in a lossless cylindrical acoustic tube or ideal vibrating string [97] (see Fig. 1). A slightly more complex case is the cascade connection of two or more bidirectional delay lines, interconnected by two-port lossless scattering junctions, as in the Kelly-Lochbaum vocal-tract model (see Fig. 5, allpass section).

Terminating a cascade of bidirectional delay lines with an infinite or
zero impedance yields total reflection from the termination (far left or
far right in Fig. 5). In this
case, delay elements for sampled signals traveling toward the
termination can be *commuted* through the scattering junctions and
combined with the delay elements for signals traveling away from the
termination, and the sampling rate can be lowered by a factor of two
[92]. This derives the ladder digital filter structure used in
more modern speech coding via linear prediction [57].
Extending this procedure, the multivariable digital lattice
filters synthesized according to [112] can be transformed via
elementary delay manipulations to a multivariable DWN.

Wave propagation in *membranes* and *volumes* can be
efficiently simulated using a *waveguide mesh* which is a grid
of bidirectional delay lines interconnected by -port lossless
scattering junctions
[93,122,84,125,30].
Highly efficient, multiply-free, lossless meshes can be obtained when
the number of waveguides intersecting at each junction is a power of
two [91]; this happens in the simple rectilinear 2D mesh, and
also in the tetrahedral 3D mesh (analogous to the diamond crystal
lattice)
[125]. The common feature of DWNs is sampled unidirectional traveling waves
in distributed wave-propagation media.

Digital waveguide networks (as well as WDFs or any digital filter
with a physical interpretation) can be regarded as a special case of
*finite difference methods* (FDM) [109]. Normally,
FDMs are derived directly from differential equations by replacing
differentials (, , etc.) by finite steps (, , etc.). A major distinction of DWN-induced FDMs is their inherent
*stability*. Since DWNs for lossless media are by construction
lossless networks of lossless sampled transmission lines, the
stability problems normally associated with FDMs are avoided at the
outset. Moreover, *dispersion* is often avoided completely in the
1D case, and it can be controlled from a different perspective in
higher dimensions [124]. These advantages are quite
useful in acoustic modeling applications which involve large-order
systems which are very close to lossless (such as a vibrating string,
plate, or reverberating chamber).

To incorporate linear propagation distortions (frequency-dependent attenuation and dispersion), recursive digital filters are typically embedded in the DWN at specific points. Rather than implement attenuation and dispersion in a uniformly sampled, distributed fashion (i.e., a small amount of filtering between each pair of unit-delay elements), one digital filter will normally implement the attenuation and dispersion associated with a much larger section of medium [97]. In principle, the required filter order increases with the size of the section being ``summarized,'' but in nearly lossless media such as strings and acoustic tubes, the filtering per unit length of medium is so weak that very low-order filters give very good approximations to many samples worth of wave propagation. This ``lumping'' of losses and dispersion in the discrete-time simulation can yield enormous complexity reductions, but it is also a source of approximation error which must be considered in the context of the application. In acoustic modeling over audio bandwidths, approximations of this kind are normally inaudible so long as the overall decay rates and tunings of the modes of the structure being modeled are preserved.

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