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Frequency-Response Matching Using
Digital Filter Design Methods

Given force inputs and velocity outputs, the frequency response of an ideal mass was given in Eq.$ \,$ (7.1.2) as

$\displaystyle \Gamma_m(j\omega) \eqsp \frac{1}{m j\omega},

and the frequency response for a spring was given by Eq.$ \,$ (7.1.3) as

$\displaystyle \Gamma_k(j\omega) \eqsp \frac{j\omega}{k}.

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. More generally, the admittance frequency response ``seen'' at the port of a general $ N$ th-order LTI system is, from Eq.$ \,$ (8.3),

$\displaystyle H(s) \isdefs \frac{B(s)}{A(s)} \isdefs \frac{b_M s^M + \cdots b_1 s + b_0}{a_N s^N + \cdots a_1 s + a_0} \protect$ (9.14)

where we assume $ M<N$ . Similarly, point-to-point ``trans-admittances'' can be defined as the velocity Laplace transform at one point on the physical object divided by the driving-force Laplace transform at some other point. There is also of course no requirement to always use driving force and observed velocity as the physical variables; velocity-to-force, force-to-force, velocity-to-velocity, force-to-acceleration, etc., can all be used to define transfer functions from one point to another in the system. For simplicity, however, we will prefer admittance transfer functions here.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2015-05-22 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University