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.