ï~~VLSI for a Physical Model
of Musical Instrument Oscillations.
David Rivas, Steve Watkins, Paul M. Chau
Department of ECE R-007
University of California at San Diego,
La Jolla, California, 92093
david@polanski.ucsd.edu
watkins@cs.ucsd.edu, chau@celece.ucesd.edu
ABSTRACT
A VLSI implementation of a physical model for musical instrument oscillation based
on the Mclntrye, Schumacher, and Woodhouse model is presented. This implementation utilizes a generalization of Smith's wave digital filter formulations, and is suitable
for a variety of musical instrument simulations, reverberation constructions, as well as
traditional digital filtering. The Woodhouse et al. model is described including
Smiths additions. Salient features of the model are pointed out and a simple generalization is provided. Following is a discussion of the generalized architecture required
to implement the different models including bow-string and reed-bore simulations as
well as traditional digital filter operations. A discussion of the the device implementation is considered and a short discussion on implementation trade-offs is provided.
Results of the design choices are noted including the results of using a bit-serial
design.
1. The Model and Previous Work.
McIntyre, Schumacher, and Woodhouse [McShWo83] present a model for oscillations in musical instruments that corresponds to an energy source driving a network
composed of a non-linear section and a linear section. The linear section is effectively
a simple lumped transmission line of length related to the period of oscillation.
The McIntyre, Shumacher, and Woodhouse model generally requires solving
simultaneously a pair of equations introducing the non-linearity. By modeling the
linear portion of the;network as a transmission line, the interconnection between the
linear and non-linear portions of the network can be modeled as impedance
mismatches at ends of the transmission line. Thus a single value in the form of a
reflection coefficient can be used to parameterize the effect of the non-linear section of
the system [RaWhVa84]. Smith and others [Smith86] [Garnet87] have produced results
in software by generating the non-linearity via simple table look-up.
In a digital implementation, the lossless transmission line becomes a simple delay
loop representing the negatively and positively traveling waves in the line. Since we
are modeling physical systems the connection of the two lossless lines must obey conservation properties. Modeling the connection as a standard 2-port, and using traditional current and voltage variables wb-,n obtain following relations:
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