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The Padé-Prony Method

Another variation of Prony's method, described by Burrus and Parks [9] consists of using Padé approximation to find the numerator $ \hat{B}^\ast $ after the denominator $ \hat{A}^\ast $ has been found as before. Thus, $ \hat{B}^\ast $ is found by matching the first $ {{n}_b}+1$ samples of $ h(n)$ , viz., $ \hat{b}^\ast _n = \hat{a}^\ast \ast h (n),
n=0\ldots\,,{{n}_b}$ . This method is faster, but does not generally give as good results as the previous version. In particular, the degenerate example $ h(n)=0, n\leq {{n}_b}$ gives $ \hat{H}^\ast (z)\equiv 0$ here as did pure equation error. This method has been applied also in the stochastic case [11].

On the whole, when $ H(\ejo )$ is causal and minimum phase (the ideal situation for just about any stable filter-design method), the variants on equation-error minimization described in this section perform very similarly. They are all quite fast, relative to algorithms which iteratively minimize output error, and the equation-error method based on the FFT above is generally fastest.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-04-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University