Frequency-Response Matching Using Digital Filter Design Methods

The lumped modeling methods discussed in §L.3 and §L.4 are order preserving. As a result, they suffer from severe approximation error such as frequency warping and perhaps artificial damping as well. By allowing the order to increase in the digital model, it is possible to obtain arbitrarily accurate models of individual masses and springs insofar as their frequency response is concerned. A possible drawback is that the precise physical interpretation is lost for the internal state of the filter.

Given force inputs and velocity outputs, the frequency response of an ideal mass was given in Eq.(L.1.2) as

and the frequency response for a spring was given by Eq.(L.1.3) as

Thus, an ideal mass is an integrator and an ideal spring is a differentiator. The modeling problem for masses and springs can thus be posed as a problem in digital filter design given the above desired frequency responses.

Consider the case of a spring model (differentiator). In audio applications, it is usually desirable to evaluate numerical approximations in the frequency domain. This is because the ear acts to a first approximation like a spectrum analyzer [551]. In the plane, an ideal differentiator can be characterized as a linear filter with transfer function . A transfer function