#### 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

(7.13) |

Note that is a first-order polynomial in . Similarly, a pole repeated three times corresponds to an impulse-response 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.

**Next Section:**

So What's Up with Repeated Poles?

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Example