Prony's method

Prony analysis (Prony's method) was developed by Gaspard Riche de Prony in 1795.

However, practical use of the method awaited the digital computer.

[1] 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.

be a signal consisting of

evenly spaced samples.

Prony's method fits a function to the observed

After some manipulation utilizing Euler's formula, the following result is obtained, which allows more direct computation of terms: where Prony's method is essentially a decomposition of a signal with

complex exponentials via the following process: Regularly sample

samples may be written as If

happens to consist of damped sinusoids, then there will be pairs of complex exponentials such that where Because the summation of complex exponentials is the homogeneous solution to a linear difference equation, the following difference equation will exist: The key to Prony's Method is that the coefficients in the difference equation are related to the following polynomial: These facts lead to the following three steps within Prony's method: 1) Construct and solve the matrix equation for the

, a generalized matrix inverse may be needed to find the values

values, find the roots (numerically if necessary) of the polynomial The

-th root of this polynomial will be equal to

values are part of a system of linear equations that may be used to solve for the

It is possible to use a generalized matrix inverse if more than

Note that solving for

will yield ambiguities, since only

This leads to the same Nyquist sampling criteria that discrete Fourier transforms are subject to