Conclusions

A variety of topics related to digital waveguide networks (DWN) were discussed, within a generalized formulation which arises naturally from sampled traveling-wave solutions to physical wave equations. The idealized elements of DWNs are sampled lossless waveguides, or ideal transmission lines, intersecting at lossless scattering junctions. Such networks are a generalization of well known lattice and ladder digital filter structures, and many desirable properties of such filters are carried over. In physical modeling applications, lossy and dispersive waveguides and junctions are utilized, as are nonlinear and time-varying extensions.

Lossless junctions of normalized and unnormalized waveguides were investigated, and losslessness was characterized for both physical and non-physical junctions, in both energy-based and algebraic approaches. Depending on the application at hand, the energy-based viewpoint may be more advantageous over the algebraic viewpoint, or vice versa. For example, we introduced three new three-multiply, normalized, lattice-filter sections, each derived by adopting a different approach.

The main applications of digital waveguide networks so far seem to be in the fields of artificial reverberation, musical sound synthesis, and physical simulation of nearly lossless propagation in media such as air, plates, membranes, and vibrating strings. We expect future applications to be plausible in any field needing numerically robust and computationally efficient simulation of largely linear propagation media, especially when the wave propagation is close to lossless, and when a fixed upper frequency limit makes sense in the application, as it does in most audio applications. In other terms, DWNs provide an effective ``bandlimited medium'' simulation, being especially efficient in the lossless case. Also, in many audio applications, it has proved sufficient to implement losses only sparsely in the DWN, using a single digital filter, e.g., to ``summarize'' the losses and dispersion over a fairly large section of medium, so that most of the efficiency of the lossless case is retained in the lossy, dispersive case. The time-varying case can be made free of ``parametric amplification'' effects by means of waveguide normalization. Finally, the nonlinear case, while not being bandlimited and therefore prone to aliasing, is greatly facilitated and made more robust by the use of a simulation structure which is precisely understood from a physical point of view.