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 :
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
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