Introduction

Digital Waveguide Filters (DWF) have proven useful for building computational models of acoustic systems which are both physically meaningful and efficient for 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 derived for the purpose of building reverberators using lossless building blocks [6], 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 [7]. Similarly, a violin string can be modeled as a digital waveguide with a nonlinear coupling to the bow [7]. When the basic model 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 traveling pressure waves or velocity waves. A byproduct 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 [8]. 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 [4]. 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 very closely related to Wave Digital Filters (WDF) which have been developed primarily by Fettweis [2]. Waveguide filters can be viewed as a generalized framework incorporating aspects of lattice and ladder digital filters, wave digital filters, one-dimensional waveguide acoustics, and classical network theory [1].

A waveguide for our purposes is any medium in which wave motion can be characterized by the one-dimensional wave equation [5]. 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 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. For example, the wave impedance $R$ of a vibrating string is $R= \sqrt{T\rho}=\rho c$, where $\rho$ is string density (mass per unit length) and $T$ is the tension of the string.

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