Question about Prony's method

Question

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?

Answer 1

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

Answer 2

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.