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Impulse Response of Repeated Poles
In the time domain, repeated poles give rise to polynomial
amplitude envelopes on the decaying exponentials corresponding to the
(stable) poles. For example, in the case of a single pole repeated
twice, we have
Proof:
First note that
Therefore,
Note that
is a firstorder polynomial in
. Similarly, a pole
repeated three times corresponds to an impulseresponse component that
is an exponential decay multiplied by a quadratic polynomial in
, and so on. As long as
, the impulse response will
eventually decay to zero, because exponential decay always overtakes
polynomial growth in the limit as
goes to infinity.
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