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A Signal Processing Perspective on Repeated Mass-Spring Poles

Going back to the poles of the mass-spring system in Eq.$ \,$(F.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.
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Physical Perspective on Repeated Poles in the Mass-Spring System
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Linearly Growing State Variables in WD Mass-Spring Oscillator