Physical Room Modeling with FDNs

Applications

Recent developments in physical modeling using digital waveguides have
included the use of a *waveguide mesh* to model 2D membranes and 3D
rooms [32,33,24]. In the membrane,
for example, a rectilinear mesh of digital waveguides can be interconnected
via four-port scattering junctions to provide lossless prototypes for
``plate reverberators'' and the like. A single dispersive waveguide (made
dispersive using embedded allpass filters) can be used to model ``spring
reverberators.'' Savioja et al. [24] have found that the rectilinear 3D
waveguide mesh has good room simulation properties at low frequencies.

Since reverberation quality generally increases with the number dimensions (from spring to plate to acoustic space), it is plausible to expect that higher dimensional waveguide meshes will provide better reverberation than we have ever known. Generalizing (35) to higher dimensions, one can see that the higher the dimensionality, the more rapidly the mode density increases with frequency. However, above the ``Schroeder limit'' at which the modes are so densely packed that the ear cannot resolve them, increasing the density should have no audible effect. Nevertheless, it is an interesting direction to pursue.

The waveguide mesh is structurally lossless so that there is no attenuation
error in the sampled wave propagation. However, the grid quantization does
give rise to *dispersion* error: The speed of sound effectively
varies somewhat as a function of frequency and propagation direction on the
mesh. Generally, results are very accurate at low frequencies, but sound
speed decreases gradually as frequency increases in all but certain
directions which tend to be diagonals along the mesh
[32]. The choice of mesh geometry has a strong
effect on the dispersion behavior [33]. It also strongly
affects computational complexity. As an example, whenever an isotropic mesh
utilizes -port scattering junctions in which is a power of 2, the
scattering matrices require no multiplies [26]. For rectilinear
meshes, membranes are multiply free, as are solids in 4D (since the number
of ports is , in dimensional space). The tetrahedral mesh,
analogous to the diamond crystal, requires no multiplies to fill 3D space.
Multiply-free waveguide meshes can be integrated very densely in VLSI.

A final word about waveguide meshes is that they, like any other LTI system, can be expressed in a sparse state-space form which yields an FDN that can be interpreted as a physical model.

Practical FDN Design

Physical Room Modeling with FDNs

Applications

``Circulant and Elliptic Feedback Delay Networks for Artificial Reverberation'', by Davide Rocchesso and Julius O. Smith III, preprint of version in

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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