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

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

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

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

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.