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 of a
vibrating string is
, where is string
density (mass per unit length) and 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.

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