### Repeated Poles

When poles are repeated, an interesting new phenomenon emerges. To see what's going on, let's consider two identical poles arranged in parallel and in series. In the parallel case, we have

*parallel*are equivalent to a new one-pole filter

^{7.8}(when the poles are identical), while the same two filters in

*series*give a

*two-pole*filter with a repeated pole. To accommodate both possibilities, the general partial fraction expansion must include the terms

#### Dealing with Repeated Poles Analytically

A pole of *multiplicity* has
residues associated with it. For example,

and the three residues associated with the pole are 1, 2, and 4.

Let denote the th residue associated with the pole , . Successively differentiating times with respect to and setting isolates the residue :

or

#### Example

For the example of Eq.(6.12), we obtain

#### 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

(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

*quadratic*polynomial in , and so on. As long as , the impulse response will eventually decay to zero, because exponential decay always overtakes polynomial growth in the limit as goes to infinity.

#### 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:

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:

^{7.9}

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Alternate Stability Criterion

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Alternate PFE Methods