Prony method extracts damped complex exponential functions (or sinusoids)
from a given signal by solving a set of linear equations
(Lobos et al., 2003; Mitrofanov and Priimenko, 2015; Peter and Plonka, 2013; Prony, 1795).
The Prony method allows for estimation of frequencies, amplitudes and phases
of a signal (For details see Appendix). Assume we want to solve the problem:
, we derive the concise form
The above M equations can be written in a matrix form:
of equation 4
can be computed by solving a polynomial of the form:
equation 5 can also be written in a form:
The coefficients of the polynomial can be computed by solving the following equation:
We use the method proposed by Toh and Trefethen (1994) to compute the roots of equation 6.
If the roots are solved,
the can be computed using equation 3.
Finally, the components are
computed based on the equation below(For details see Appendix):