Properties of DWNs

Each delay element of a DWN can be interpreted precisely as a sampled traveling-wave component in a physical system, unlike the delay elements in ladder and lattice digital filters. Due to the particular bilinear-transform frequency warping used in typical WDFs, the delay elements in WDFs can be precisely interpreted as containing samples of physical traveling waves at dc, $f_s/4$, and $f_s/2$, where $f_s$ denotes the sampling rate.

Because simple sampling of traveling waves is used to define DWNs, aliasing can occur if the bandwidths of the physical signals become too large, or if nonlinearities or time-varying parameters create signal components at frequencies above the Nyquist limit. (The bilinear transform, on the other hand, does not alias.) An advantage of simple sampling is that the frequency axis is preserved exactly up to half the sampling rate, while in the case of the bilinear transform, the frequency axis is warped so that only dc, $f_s/2$ and one other frequency can be mapped across exactly.

Due to the precise physical interpretation of DWNs, nonlinear and time-varying extensions are well behaved and tend to remain ``physical'', provided aliasing is controlled. (See Section 6.)

Because delay elements appear in physically meaningful locations in both the forward and reverse signal paths of a DWN, there is no restriction to a reflectively terminated cascade chain of scattering junctions as is normal in the ladder/lattice filter context. Digital waveguides can be coupled at junctions, cascaded, looped, or branched, to any degree of network complexity. As a result, much more general network topologies are available, corresponding to arbitrary physical constructions.

Lumped elements can be integrated into DWNs and results from WDF theory can be used to model both linear and nonlinear lumped circuit elements [58,127,21].

The instantaneous power anywhere in a DWN can be made invariant with respect to time-varying filter coefficients, as discussed in Sections 4 and 5. This can be seen as generalizing the normalized ladder filter [35,92,95].

As a result of the strict passivity which follows directly from the physical interpretation, no instability, limit cycles, or overflow oscillations can occur, even in the time-varying case, as long as ``passive scattering'' is used at all waveguide junctions [113,92]. As explained in Section 6, passive scattering may be trivially obtained simply by using extended internal precision in each junction followed by magnitude truncation of all outgoing waves leaving the junction. However, in scattering intensive applications such as the 2D and 3D mesh, magnitude truncation often yields too much damping due to round-off, and more refined schemes must be used.

The basic characteristics of DWNs can be summarized as follows [95]:

• DWNs are derived by sampling traveling-wave descriptions of distributed physical wave-propagation systems such as strings, acoustic tubes, plates, gases, and solids.

• Each delay element of a DWN has a precise physical interpretation as a sample of a unidirectional traveling wave.

• The frequency axis is preserved up to half the sampling rate (i.e., it is not warped according to the bilinear transform).

• Physically meaningful nonlinear, time-varying extensions are straightforward.

• Aliasing can occur due to nonlinearities, time variation, or inadequate bandlimiting of initial conditions and/or input signals.

• Fully general modeling geometries are available (e.g., in contrast to ladder/lattice filters).

• Lumped models can be simply interfaced to DWNs.

• Overall signal energy can be simply controlled.

• Instability, limit cycles, can overflow oscillations can be suppressed by using ``passive scattering.''

• Sensitivity to coefficient quantization can be minimized.

• A synthesis procedure exists for constructing any single-input, single-output (SISO) transfer function by means of a DWN [111], and any $m$-input, $k$-output transfer function by means of a multivariable DWN [112].