The mainstream literature in digital signal processing does not
routinely include
methodology for using digital filters as physical modeling elements.
On the other hand, literature on general numerical simulation techniques for
physical dynamic systems does not generally cover relevant good practice
in digital signal processing. Recent applications
[91,56,64,122,134] have shown that
many benefits can be derived from taking a physical point of view with
respect to digital filtering computations. For example, robust, rapidly
time-varying, nearly lossless digital filters (such as needed, e.g., for a
guitar string simulation) can be developed more easily using a
physical approach, and conditions for the absence of unnatural
artifacts become more clear. Moreover, *nonlinear* extensions
are much more straightforward when there is a physical interpretation
for each of the processing elements. While some benefits of the physical
modeling perspective have been realized in the form of general-purpose
digital filter structures with good numerical properties, such as
ladder and lattice filters,^{1} more general use of signal processing elements and
analysis/optimization methods in physical modeling applications
appears to be rare.

The original Kelly-Lochbaum (KL) speech model [39] employed a ladder filter with delay elements in physically meaningful locations, allowing it to be interpreted as a discrete-time, traveling-wave model of the vocal tract (see the allpass portion of Fig. 5 for a similar diagram). Assuming a reflecting termination at the lips, the KL model can be transformed via elementary manipulations to the ladder/lattice filters used in linear predictive coding (LPC) of speech [92]. The early work of Kelly and Lochbaum appears to have been followed by two main lines of development: (1) ``articulatory'' speech synthesis which utilizes increasingly sophisticated physical simulations [50,22], and (2) linear predictive coding of speech [57]. The all-pole synthesis filter used in LPC, when implemented as a ladder filter, can be loosely interpreted as a transformation of the KL model [57]. There have been ongoing efforts to develop low bit-rate speech coders based on simplified articulatory models [89,29]. The main barrier to obtaining practical speech coding methods has been the difficulty of estimating vocal-tract shape given only the speech waveform.

There have been a few developments toward higher quality
speech models retaining the simplicity of a source-filter model such
as LPC while
building on a true physical model interpretation: An extended
derivative of the Kelly-Lochbaum (KL) model, adding a nasal tract,
neck radiation, and internal damping, has been used to synthesize
high-quality female *singing* voice [14]. *Sparse*
acoustic tubes, in which many reflection coefficients are constrained
to be zero, have been proposed
[54,32]. *Conical* (rather than
the usual cylindrical) tube segments and sparsely distributed *interpolating* scattering junctions have been proposed as further
refinements [117,115].

In musical sound synthesis and delay-effects applications, *digital waveguide models* have been used for *distributed*
media, such as vibrating strings, bores, horns, plates, solids,
acoustic spaces, and the like
[93,94,97,14,15,16,18,17,,37,103,44,46,47,8,,98,123,119,,116,,115,118,120,130,6,,49,82,121,41,,68,100,45,77,,87,107,,74,101,102,30].
A digital waveguide is typically defined as a *bidirectional delay
line* containing *sampled traveling waves*.
This paper is concerned with general properties of digital waveguide
models.

Digital waveguide models often include *lumped* elements such as
masses and springs. Lumped element modeling is the main focus of *wave digital filters* (WDF) as developed principally by Fettweis
[26,27]. Wave digital filters derive from a
scattering-theoretic formulation [4] and bilinear
transformation [65] of *lumped* RLC elements. The
formal traveling-wave signals in the scattering-theoretic description
of lumped circuit elements are known as *wave variables*.
Interfacing to digital waveguide models is simple since both use
scattering-theoretic formulations. In acoustic modeling applications,
WDFs can provide digital filters which serve as explicit physical
models of mass, spring, and dashpot elements, as well as more exotic
elements from classical network theory such as transformers, gyrators,
and circulators. An example of using a WDF to model a nonlinear
mass-spring system is the ``wave digital piano hammer''
[127]. For realizability of lumped models
with feedback, wave digital filters also incorporate short waveguide
sections called ``unit elements,'' but these are ancillary to the main
development. The digital waveguide formulation is actually more
closely related to ``unit element filters'' which were developed much
earlier than WDFs in microwave engineering [72].

Vaidyanathan and Mitra [113] developed a class of digital
filters containing both the WDFs and the Gray-Markel normalized ladder
structures [35], and proved low passband sensitivity to
coefficient quantization and the absence of limit cycles, even under
time-varying conditions. The generalized filter structure in
[113] can itself be interpreted as a DWN consisting of a single
*normalized* scattering junction (see Section 4)
joining an input/output port and waveguides which are each
sample long and reflectively terminated at their other end.

Vaidyanathan and Mitra also defined a general family of *multivariable* digital ladder filters [112] and derived a
synthesis procedure for -input -output transfer-function
matrices based on recursive extraction of generalized scattering
junctions from an allpass transfer-function matrix in which the
desired transfer function is embedded. Each ``vector'' scattering
junction is an elegant generalization of that in the Gray-Markel
normalized ladder filter [57]. The extraction approach
starts from the specification of a matrix fraction
description [43] of the desired allpass matrix and proceeds
directly in the domain.

Digital waveguide models, in contrast to WDFs and generalized
Gray-Markel ladder filters, are typically derived directly from the
geometry and physical properties of a desired acoustic system. They
are often used to simulate nearly lossless distributed vibrating
structures such as strings, tubes, rods, membranes, plates, and so on, but
only up to a certain bandwidth (the limit of human hearing in audio
applications). In many cases we do not want simply to
implement a transfer function; we need instead a complete *model*
with which we can interact in a physically consistent way, even when
parameters are time-varying and nonlinearities are introduced. In
such cases, digital waveguide networks can provide a solid
foundation.

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