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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 $ \hat{A}^\ast (z)$ by minimizing

J_S^2(\hat{\theta}) &= \sum_{n={{n}_b}+1}^\infty\left(\hat{a}\ast h(n) - \hat{b}(n)\right)^2 \\
&= \sum_{n={{n}_b}+1}^\infty\left(\hat{a}\ast h(n) \right)^2. \\

This step is equivalent to minimization of ratio error (as used in linear prediction) for the all-pole part $ \hat{A}(z)$ , with the first $ {{n}_b}+1$ terms of the time-domain error sum discarded (to get past the influence of the zeros on the impulse response). When $ {{n}_b}={{n}_a}-1$ , 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 $ \ensuremath{L_2}$ output error with the pre-assigned poles given by $ \hat{A}^\ast (z)$ . In other words, the numerator $ \hat{B}(z)$ is found by minimizing

$\displaystyle \left\Vert\,H(\ejo ) - \frac{\hat{B}(\ejo )}{\hat{A}^\ast (\ejo )}\,\right\Vert _2,

where $ \hat{A}^\ast (\ejo )$ is now known. This hybrid method is not as sensitive to the time distribution of $ h(n)$ as is the pure equation-error method. In particular, the degenerate equation-error example above (in which $ \hat{H}\equiv 0$ was obtained) does not fare so badly using Prony's method.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University