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
where is now known. This hybrid method is not as sensitive to the time distribution of as is the pure equation-error method. In particular, the degenerate equation-error example above (in which was obtained) does not fare so badly using Prony's method.