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Linearly Growing State Variables

It may seem disturbing that such a simple, passive, physically rigorous simulation of a mass-spring oscillator should have to make use of state variables which grow without bound for the limiting cases of simple harmonic motion at frequencies zero and half the sampling rate. This is obviously a valid concern in practice as well. However, it is easy to show that this only happens at dc and $ f_s/2$, and that there is a true degeneracy at these frequencies, even in the physics. For all frequencies in the audio range (e.g., for typical sampling rates), such state variable growth cannot occur. Let's take closer look at this phenomenon, first from a signal processing point of view, and second from a physical point of view.


A signal processing perspective.

Going back to the poles of the mass-spring system in Eq. (N.39), we see that, as the imaginary part of the two poles, $ \pm\omega_0 =
\pm\sqrt{k/m}$, approach zero, they come together at $ s=0$ to create a repeated pole. The same thing happens at $ \omega_0=\infty$ since both poles go to ``the point at infinity''.

It is a well known fact from linear systems theory that two poles at the same point $ s=s_0=\sigma_0$ in the $ s$ plane can correspond to an impulse-response component of the form $ te^{\sigma_0 t}$, in addition to the component $ e^{\sigma_0 t}$ produced by a single pole at $ s=\sigma_0$. In the discrete-time case, a double pole at $ z=r_0$ can give rise to an impulse-response component of the form $ n r_0^n$. This is the fundamental source of the linearly growing internal states of the wave digital sine oscillator at dc and $ f_s/2$. It is interesting to note, however, that such modes are always unobservable at any physical output such as the mass force or spring force that is not actually linearly growing.


A physics perspective.

In the physical system, dc and infinite frequency are in fact strange cases. In the case of dc, for example, a nonzero constant force implies that the mass $ m$ is under constant acceleration. It is therefore the case that its velocity is linearly growing. Our simulation predicts this, since, using Eq. (N.43) and Eq. (N.42),

\begin{eqnarray*}
v_m(n) &=& \frac{f^{{+}}_m(n)}{m} - \frac{f^{{-}}_m(n)}{m}
=...
...m} \left[2(n+1) + 2n\right]x_0
= \frac{1}{m} (4 n x_0 + 2 x_0).
\end{eqnarray*}

The dc term $ 2x_0/m$ is therefore accompanied by a linearly growing term $ 2nx_0/m$ in the physical mass velocity. It is therefore unavoidable that we have some means of producing an unbounded, linearly growing output variable.


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[How to cite and copy this work] 
``Physical Audio Signal Processing for Virtual Musical Instruments and Digital Audio Effects'', by Julius O. Smith III, (December 2005 Edition).
Copyright © 2006-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]