### Prony's Method

There are several variations on equation-error minimization, and some
confusion in terminology exists. We use the definition of *Prony's
method* given by Markel and Gray [48]. It is equivalent to ``Shank's
method'' [9]. In this method, one first computes the
denominator
by minimizing

This step is equivalent to minimization of *ratio error*
(as used in *linear prediction*) for the
all-pole part
, with the first terms of the time-domain
error sum discarded (to get past the influence of the zeros
on the impulse response). When
, it coincides with the
covariance method of linear prediction [48,47]. This idea for
finding the poles by ``skipping'' the influence of the zeros on the
impulse-response shows up in the stochastic case under the name of *modified Yule-Walker equations* [11].

Now, Prony's method consists of next minimizing output error with the pre-assigned poles given by . In other words, the numerator is found by minimizing

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The Padé-Prony Method

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An FFT-Based Equation-Error Method