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.

The method

Let $f(t)$ be a signal consisting of $N$ evenly spaced samples. Prony's method fits a function

{\hat {f}}(t)=\sum _{i=1}^{N}A_{i}e^{\sigma _{i}t}\cos(\omega _{i}t+\phi _{i})

to the observed $f(t)$ . After some manipulation utilizing Euler's formula, the following result is obtained, which allows more direct computation of terms:

{\begin{aligned}{\hat {f}}(t)&=\sum _{i=1}^{N}A_{i}e^{\sigma _{i}t}\cos(\omega _{i}t+\phi _{i})\\&=\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

\lambda _{i}^{\pm }=\sigma _{i}\pm j\omega _{i}

are the eigenvalues of the system,

\sigma _{i}=-\omega _{0,i}\xi _{i}

are the damping components,

\omega _{i}=\omega _{0,i}{\sqrt {1-\xi _{i}^{2}}}

are the angular-frequency components,

\phi _{i}

are the phase components,

A_{i}

are the amplitude components of the series,

j

is the imaginary unit (

j^{2}=-1

).

Representations

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

Regularly sample ${\hat {f}}(t)$ so that the $n$ -th of $N$ samples may be written as

F_{n}={\hat {f}}(\Delta _{t}n)=\sum _{m=1}^{M}\mathrm {B} _{m}e^{\lambda _{m}\Delta _{t}n},\quad n=0,\dots ,N-1.

If ${\hat {f}}(t)$ happens to consist of damped sinusoids, then there will be pairs of complex exponentials such that

{\begin{aligned}\mathrm {B} _{a}&={\tfrac {1}{2}}A_{i}e^{\phi _{i}j},\\\mathrm {B} _{b}&={\tfrac {1}{2}}A_{i}e^{-\phi _{i}j},\\\lambda _{a}&=\sigma _{i}+j\omega _{i},\\\lambda _{b}&=\sigma _{i}-j\omega _{i},\end{aligned}}

where

{\begin{aligned}\mathrm {B} _{a}e^{\lambda _{a}t}+\mathrm {B} _{b}e^{\lambda _{b}t}&={\tfrac {1}{2}}A_{i}e^{\phi _{i}j}e^{(\sigma _{i}+j\omega _{i})t}+{\tfrac {1}{2}}A_{i}e^{-\phi _{i}j}e^{(\sigma _{i}-j\omega _{i})t}\\&=A_{i}e^{\sigma _{i}t}\cos(\omega _{i}t+\phi _{i}).\end{aligned}}

Because the summation of complex exponentials is the homogeneous solution to a linear difference equation, the following difference equation will exist:

{\hat {f}}(\Delta _{t}n)=\sum _{m=1}^{M}{\hat {f}}[\Delta _{t}(n-m)]P_{m},\quad n=M,\dots ,N-1.

The key to Prony's Method is that the coefficients in the difference equation are related to the following polynomial:

z^{M}-P_{1}z^{M-1}-\dots -P_{M}=\prod _{m=1}^{M}\left(z-e^{\lambda _{m}}\right).

These facts lead to the following three steps within Prony's method:

1) Construct and solve the matrix equation for the $P_{m}$ values:

{\begin{bmatrix}F_{M}\\\vdots \\F_{N-1}\end{bmatrix}}={\begin{bmatrix}F_{M-1}&\dots &F_{0}\\\vdots &\ddots &\vdots \\F_{N-2}&\dots &F_{N-M-1}\end{bmatrix}}{\begin{bmatrix}P_{1}\\\vdots \\P_{M}\end{bmatrix}}.

Note that if $N\neq 2M$ , a generalized matrix inverse may be needed to find the values $P_{m}$ .

2) After finding the $P_{m}$ values, find the roots (numerically if necessary) of the polynomial

z^{M}-P_{1}z^{M-1}-\dots -P_{M}.

The $m$ -th root of this polynomial will be equal to $e^{\lambda _{m}}$ .

3) With the $e^{\lambda _{m}}$ values, the $F_{n}$ values are part of a system of linear equations that may be used to solve for the $\mathrm {B} _{m}$ values:

{\begin{bmatrix}F_{k_{1}}\\\vdots \\F_{k_{M}}\end{bmatrix}}={\begin{bmatrix}(e^{\lambda _{1}})^{k_{1}}&\dots &(e^{\lambda _{M}})^{k_{1}}\\\vdots &\ddots &\vdots \\(e^{\lambda _{1}})^{k_{M}}&\dots &(e^{\lambda _{M}})^{k_{M}}\end{bmatrix}}{\begin{bmatrix}\mathrm {B} _{1}\\\vdots \\\mathrm {B} _{M}\end{bmatrix}},

where $M$ unique values $k_{i}$ are used. It is possible to use a generalized matrix inverse if more than $M$ samples are used.

Note that solving for $\lambda _{m}$ will yield ambiguities, since only $e^{\lambda _{m}}$ was solved for, and $e^{\lambda _{m}}=e^{\lambda _{m}\,+\,q2\pi j}$ for an integer $q$ . This leads to the same Nyquist sampling criteria that discrete Fourier transforms are subject to

\left|\operatorname {Im} (\lambda _{m})\right|=\left|\omega _{m}\right|<{\frac {\pi }{\Delta _{t}}}.

Notes

^ 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

Carriere, R.; Moses, R. L. (1992). "High resolution radar target modeling using a modified Prony estimator". IEEE Transactions on Antennas and Propagation. 40: 13–18. doi:10.1109/8.123348.
Slyusar, V. I. (1998). "Interpretation of the Proni method for solving long-range problems" (PDF). Radioelectronics and Communications Systems. 41 (1): 35–39.

[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.

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