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:

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:

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