The Padé-Prony Method

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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 , 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 $H(e^{j\omega})$ 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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Chapters

Introduction to Digital Filters
    Recursive Digital Filter Design
       Filter Design by Minimizing the L2 Equation-Error Norm
          The Padé-Prony Method