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I want to use Prony's method to fit a signal given by the sum of $p$ complex exponentials

$x[n]=\displaystyle{\sum_{k=1}^p} A_k{\rm e}^{\,{\rm j}\theta_k} {\rm e}^{2\pi\, {\rm j} f_k (n-1)}$.

The problem is that Prony's standard formulation (Hildebrand, Introduction to Numerical Analysis, pp. 378-382) assumes that all the samples $x[n]$ with $n=1,\ldots,N$ are available, however in some applications $x[1]$ can't be measured. It seems that in this case Prony's method can't be applied. Am I missing something?

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Why can't you observe $x[1]$? Wouldn't you then just fit the exponentials to the portion of the signal that you can observe? –  Jason R Jan 3 '13 at 14:12
    
Jason, the main reason why I can't observe $x[1]$ is that I'm using Prony's method in a non-standard way. The $x[n]$ are not samples in the time domain, but in the frequency domain (thinks about applications like MRI or SAR imaging where you acquire complex data at different modulation frequencies). So in my case when $n=1$ the exponent is 0 and that is a problem. –  Arrigo Jan 3 '13 at 17:45
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2 Answers 2

Drop the first sample - apply the formulation as if x[2] was x[1].

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Tha's not possible if you use Prony's algorithm as presented in Hildebrand, Introduction to Numerical Analysis, pp. 378-382, where $x[1]$ is explicitly needed. See the answer below for a different implementation that does not need this term. –  Arrigo Jan 4 '13 at 18:16
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An variant of Prony's method that does not require $x[1]$ is presented in
Kahn et al. "On the consistency of Prony's method and related algorithms." Journal of Computational and Graphical Statistics 1.4 (1992): 329-349. MATLAB code is available at http://www.statsci.org/other/prony.html The signal model in this implementation is

$x[n]=\displaystyle{\sum_{k=1}^p} A_k{\rm e}^{\,{\rm j}\theta_k} {\rm e}^{2\pi\, {\rm j} f_k t_n}$

where $n=1,\ldots,N$ and the only restriction is that the $t_n$ are equispaced.

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