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
