Prony analysis (Prony's method) was developed by
Gaspard Riche de Prony
Baron Gaspard Clair François Marie Riche de Prony (22 July 1755 – 29 July 1839) was a French mathematician and engineer, who worked on hydraulics. He was born at Chamelet, Beaujolais, France and died in Asnières-sur-Seine, France.
Educati ...
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 sinusoid
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
s. This allows for the estimation of frequency, amplitude, phase and damping components of a signal.
The method
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
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...
, the following result is obtained. This 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, and
*
is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
(
).
Representations
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 to 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
*
Generalized pencil-of-function method Generalized pencil-of-function method (GPOF), also known as matrix pencil method, is a signal processing technique for estimating a signal or extracting information with complex exponentials. Being similar to Prony and original pencil-of-function m ...
*Computation of Prony decomposition using
Autoregression analysis
*Application of Prony decomposition in
Time-frequency analysis
Notes
References
*
*
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Signal processing