#### 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 :

 (7.14)

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:

Applying this to Eq.(6.14) yields

Note that the result is symmetric in and . If , then becomes proportional to for large , while if , it becomes instead proportional to .

Going back to Eq.(6.14), we have

 (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:

The convolution of a constant with itself is a ramp:

The convolution of a constant and a ramp is a quadratic, and so on:7.9

Next Section:
Example 2
Previous Section:
Impulse Response of Repeated Poles