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 [150].<sup>Q.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.