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Digital Waveguide Networks (DWN) have been widely used to develop efficient discrete-time physical models for sound synthesis, particularly for woodwind, string, and brass musical instruments [1,2,3,4,5,6,7]. They were initially developed for artificial reverberation [8,9,10], and more recently they have been applied to robust numerical simulation of 2D and 3D vibrating systems [11,12,13,14,15,16,17].

A digital waveguide may be thought of as a sampled transmission line--or acoustic waveguide--in which sampled, unidirectional traveling waves are explicitly simulated. Simulating traveling-wave components in place of physical variables such as pressure and velocity can lead to significant computational reductions, particularly in sound synthesis applications, since the models for most traditional musical instruments (in the string, wind, and brass families), can be efficiently simulated using one or two long delay lines together with sparsely distributed scattering junctions and filters [2,18,19]. Moreover, desirable numerical properties are more easily ensured in this framework, such as stability [20], ``passivity'' of round-off errors, and minimized sensitivity to coefficient quantization [21,22,23,14].

In [24], a multivariable formulation of digital waveguides was proposed in which the real, positive, characteristic impedance of the waveguide medium (be it an electric transmission line or an acoustic tube) is generalized to any $q\times q$ para-Hermitian matrix. The associated wave variables were generalized to a $q\times
m$ matrix of $z$ transforms. From fundamental constraints assumed at a junction of two or more waveguides (pressure continuity, conservation of flow), associated multivariable scattering relations were derived, and various properties were noted.

In this paper, partially based on [25], we pursue a different path to vectorized DWNs, starting with a multivariable generalization of the well known telegrapher's equations [26]. This formulation provides a more detailed physical interpretation of generalized quantities, and new potential applications are indicated.

The paper is organized as follows. Section II introduces the generalized DWN formulation, starting with the scalar case and proceeding to the multivariable case. The generalized wave impedance and complex signal power appropriate to this formulation are derived, and conditions for ``passive'' computation are given. In Section III, losses are introduced, and some example applications are considered. Finally, Section IV presents a derivation of the general form of the physical scattering junctions induced intersecting multivariable digital waveguides.

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``Generalized Digital Waveguide Networks'', by Julius O. Smith III and Davide Rocchesso, preprint submitted for publication, Summer 2001.
Copyright © 2008-03-12 by Julius O. Smith III and Davide Rocchesso
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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