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,

(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

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University