Transfer Function Models

Rather than build an explicit model for every mass, spring, and dashpot in a lumped system, we may instead choose to model only the transfer function between selected inputs and outputs of the physical system. This is can be considered a kind of ``large-scale'' physical modeling in which the physical system is modeled as a transfer function relating specific inputs and outputs. Such models are used extensively in the field of control systems design [152].<sup>R.1</sup> Transfer-function modeling is often the best way to incorporate lumped elements in an otherwise physical computational model. Maximum computational efficiency is typically obtained by deciding which portions of a physical model can be ``frozen'' as ``black boxes'' characterized only by their transfer functions. This is normally possible only for linear, time-invariant model components for which there is no need to ever ``look inside the black box.''

An example where such ``macroscopic'' transfer-function modeling is normally applied is the trumpet bell (§8.2). A fine-grained model might use a piecewise cylindrical or piecewise conical approximation to the flaring bell [74]. However, there is normally no need for an explicit bell model in a practical virtual instrument, and its transmittance and reflectance can be perfectly well summarized by digital filters having frequency responses thar are optimal approximations to the measured (or theoretical) bell response. A disadvantage to having done this is that it is no longer possible to ``stick a mute'' into the bell. This is an example of the general tradeoff between physical extensibilty and computational efficiency/parsimony when designing computational models based on physics.

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Subsections

• Outline
• Sampling the Impulse Response
  • Impulse Invariant Method
  
  
• Pole mapping
• Filter Design Methods
• Modal Expansion
  • Parallel Second-Order Expansion
  • Delay Loop Expansion

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.