A Signal Processing Perspective on Repeated Mass-Spring Poles

Going back to the poles of the mass-spring system in Eq.(F.56), 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.