#### 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, , approach zero, they come together at to create a*repeated pole*. The same thing happens at 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 in the plane can correspond to an impulse-response component of the form , in addition to the component produced by a single pole at . In the discrete-time case, a double pole at can give rise to an impulse-response component of the form . This is the fundamental source of the linearly growing internal states of the wave digital sine oscillator at dc and . 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