DIGITAL WAVEGUIDE MODELING OF ACOUSTIC SYSTEMS

Julius O. Smith III Associate Professor of Music

Digital Waveguide Filters (DWF) have proven useful for building computational models of acoustic systems which are both physically meaningful and efficient for applications such as digital synthesis. The physical interpretation opens the way to capturing valued aspects of real instruments which have been difficult to obtain by more abstract synthesis techniques.

Waveguide filters were initially derived to construct digital reverberators out of energy-preserving building blocks, but any linear acoustic system can be approximated using waveguide networks. For example, the bore of a wind instrument can be modeled very inexpensively as a digital waveguide. Similarly, a violin string can be modeled as a digital waveguide with a nonlinear coupling to the bow. When the computational form is physically meaningful, it is often obvious how to introduce nonlinearities correctly, thus leading to realistic behaviors far beyond the reach of purely analytical methods.

A basic feature of DWF building blocks is the exact physical interpretation of the contained digital signals as samples of traveling pressure waves, velocity waves, or the like. A by-product of this formulation is the availability of signal power defined instantaneously with respect to both space and time. This instantaneous handle on signal power yields a simple picture of the effects of round-off error on the growth or decay of the signal energy within the DWF system.

Another nice property of waveguide filters is that they can be reduced in special cases to standard lattice/ladder digital filters which have been extensively developed in recent years. One immediate benefit of this connection is a body of techniques for realizing any digital filter transfer function as a DWF. Waveguide filters are also related to "wave digital filters'' (WDF) which have been developed primarily by Fettweis. Waveguide filters can be viewed as providing a discrete-time "building material'' incorporating aspects of lattice and ladder digital filters, wave digital filters, one-dimensional waveguide acoustics, and classical network theory. Using a "mesh'' of one-dimensional waveguides, modeling can be carried out in two and higher dimensions. (See Van Duyne, Smith, p. 23.)

In this context, a waveguide can be defined as any medium in which wave motion can be characterized by the one-dimensional wave equation. In the lossless case, all solutions can be expressed in terms of left-going and right-going traveling waves in the medium. The traveling waves propagate unchanged as long as the wave impedance of the medium is constant. The wave impedance is the square root of the "massiness'' times the "stiffness'' of the medium; that is, it is the geometric mean of the two sources of resistance to motion: the inertial resistance of the medium due to its mass, and the spring-force on the displaced medium due to its elasticity.

Digital waveguide filters are obtained (conceptually) by sampling the unidirectional traveling waves which occur in a system of ideal, lossless waveguides. Sampling is across time and space. Thus, variables in a DWF structure are equal exactly (at the sampling times and positions, to within numerical precision) to variables propagating in the corresponding physical system. In other applications, the propagation in the waveguide is extended to include frequency dependent losses and dispersion. In still more advanced applications, nonlinear effects are introduced as a function of instantaneous signal level.

When the wave impedance, signal scattering occurs, i.e., a traveling wave impinging on an impedance discontinuity will partially reflect and partially transmit at the junction in an energy-preserving way. Real-world examples of waveguides include the bore of a clarinet, a violin string, horns, organ pipes, the vocal tract in speech, microwave antennas, electric transmission lines, and optical fibers.

References

J. O. Smith, "Physical Modeling Using Digital Waveguides,'' Computer Music Journal, special issue: Physical Modeling of Musical Instruments, Part I, vol. 16, no. 4, pp. 74-91, Winter, 1992.

Music 220B Course Description, J. O. Smith, Stanford University.

Music 421 (EE 266) Course Description, J. O. Smith, Stanford University.

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