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.

Prony analysis of a time-domain signal

The method

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Let   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

  are the eigenvalues of the system,
  are the damping components,
  are the angular-frequency components,
  are the phase components,
  are the amplitude components of the series,
  is the imaginary unit ( ).

Representations

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Prony's method is essentially a decomposition of a signal with   complex exponentials via the following process:

Regularly sample   so that the  -th of   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   values:

 

Note that if  , a generalized matrix inverse may be needed to find the values  .

2) After finding the   values, find the roots (numerically if necessary) of the polynomial

 

The  -th root of this polynomial will be equal to  .

3) With the   values, the   values are part of a system of linear equations that may be used to solve for the   values:

 

where   unique values   are used. It is possible to use a generalized matrix inverse if more than   samples are used.

Note that solving for   will yield ambiguities, since only   was solved for, and   for an integer  . This leads to the same Nyquist sampling criteria that discrete Fourier transforms are subject to

 

See also

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Notes

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  1. ^ Hauer, J. F.; Demeure, C. J.; Scharf, L. L. (1990). "Initial results in Prony analysis of power system response signals". IEEE Transactions on Power Systems. 5 (1): 80–89. Bibcode:1990ITPSy...5...80H. doi:10.1109/59.49090. hdl:10217/753.

References

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