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 $H(e^{j\omega})$ 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.