#### So What's Up with Repeated Poles?

In the previous section, we found that repeated poles give rise to polynomial amplitude-envelopes multiplying the exponential decay due to the pole. On the other hand, two*different*poles can only yield a convolution (or sum) of two different exponential decays, with no polynomial envelope allowed. This is true no matter how closely the poles come together; the polynomial envelope can occur only when the poles merge exactly. This might violate one's intuitive expectation of a continuous change when passing from two closely spaced poles to a repeated pole.

To study this phenomenon further, consider the convolution of two one-pole impulse-responses and :

The finite limits on the summation result from the fact that both and are causal. Recall the closed-form sum of a truncated geometric series:

(7.15) |

Setting yields

(7.16) |

which is the first-order polynomial amplitude-envelope case for a repeated pole. We can see that the transition from ``two convolved exponentials'' to ``single exponential with a polynomial amplitude envelope'' is perfectly continuous, as we would expect. We also see that the polynomial amplitude-envelopes fundamentally arise from

*iterated convolutions*. This corresponds to the repeated poles being arranged in

*series*, rather than in parallel. The simplest case is when the repeated pole is at , in which case its impulse response is a constant:

^{7.9}

**Next Section:**

Example 2

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Impulse Response of Repeated Poles