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 system. While this is not really physical modeling,^P.1 we include some discussion of it because it is often the best way to incorporate lumped elements in an otherwise physical computational model. Often we begin with a physical model, and decide which portions of the model can be ``frozen'' as ``black boxes'' characterized only by their transfer functions. This is normally done 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 often applied is the trumpet bell (see §8.2). A fine-grained model might use a piecewise cylindrical or piecewise conical approximation to the flaring bell. However, there is normally no need for an explicit bell model, and its transmittance and reflectance can be perfectly well summarized by digital filters having frequency responses which are optimal approximations to the measured or theoretical bell response. The disadvantage to having done this is that it is no longer possible to ``stick a mute'' into the bell (which normally is not done, but could potentially be called for). This is an example of the general tradeoff that exists between model flexibility and model economy.