Prony analysis (Prony's method) was developed by Gaspard Riche de Prony in 1795. However, practical use of the method awaited the digital computer. Similar to the Fourier transform, Prony's method extracts valuable information from a uniformly sampled signal and builds a series of damped complex exponentials or damped sinusoids. This allows the estimation of frequency, amplitude, phase and damping components of a signal.
The method Let f ( t ) {\displaystyle f(t)} be a signal consisting of N {\displaystyle N} evenly spaced samples. Prony's method fits a function
f ^ ( t ) = ∑ i = 1 N A i e σ i t cos ( ω i t + ϕ i ) {\displaystyle {\hat {f}}(t)=\sum _{i=1}^{N}A_{i}e^{\sigma _{i}t}\cos(\omega _{i}t+\phi _{i})}
to the observed f ( t ) {\displaystyle f(t)} . After some manipulation utilizing Euler's formula, the following result is obtained, which allows more direct computation of terms:
f ^ ( t ) = ∑ i = 1 N 1 2 A i ( e j ϕ i e λ i + t + e − j ϕ i e λ i − t ) , {\displaystyle {\begin{aligned}{\hat {f}}(t)=\sum _{i=1}^{N}{\tfrac {1}{2}}A_{i}\left(e^{j\phi _{i}}e^{\lambda _{i}^{+}t}+e^{-j\phi _{i}}e^{\lambda _{i}^{-}t}\right),\end{aligned}}}
where
λ i ± = σ i ± j ω i {\displaystyle \lambda _{i}^{\pm }=\sigma _{i}\pm j\omega _{i}} are the eigenvalues of the system,
σ i = − ω 0 , i ξ i {\displaystyle \sigma _{i}=-\omega _{0,i}\xi _{i}} are the damping components,
ω i = ω 0 , i 1 − ξ i 2 {\displaystyle \omega _{i}=\omega _{0,i}{\sqrt {1-\xi _{i}^{2}}}} are the angular-frequency components,
ϕ i {\displaystyle \phi _{i}} are the phase components,
A i {\displaystyle A_{i}} are the amplitude components of the series,
j {\displaystyle j} is the imaginary unit ( j 2 = − 1 {\displaystyle j^{2}=-1} ).
… excerpt ends here. Continue reading the full article.
