- Prony's method
Prony analysis (Prony's method) was developed by
Gaspard Riche de Prony in1795 . However, practical use of the method awaited the digital computer [1] . Similar to theFourier transform , Prony's method extracts valuable information from a uniformly sampled signal and builds a series of damped complex exponentials or sinusoids. This allows for 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(2pi f_i t + phi_i) to the observed f(t). After some manipulation utilizing
Euler's formula , the following result is obtained. This allows for more direct computation of terms.:hat{f}(t) = sum_{i=1}^{N} A_i e^{sigma_i t} cos(2pi f_i t + phi_i)= sum_{i=1}^{N} frac{1}{2} A_i e^{phi_i j}e^{lambda_i t} where::lambda_i = (-sigma_i pm j omega_i )t are the eigenvalues of the system, sigma_i are the damping components, phi_i are the phase components, f_i are the frequency components, A_i are the amplitude components of the series, and j=sqrt{-1}.Example
References
[1] Hauer, J.F. et al (1990). "Initial Results in Prony Analysis of Power System Response Signals". "IEEE Transactions on Power Systems", 5, 1, 80-89.
How to
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 follows::F_n=hat{f}(Delta_t n) = sum_{m=1}^{M} Beta_m e^{Lambda_m t}
If hat{f}(t) happens to be consist of dampened sinusoids then there will be pairs of complex exponentials such that :Beta_a = frac{1}{2} A_i e^{phi_i j} :Beta_b = frac{1}{2} A_i e^{-phi_i j} :Lambda_a = sigma_i + j omega_i :Lambda_b = sigma_i - j omega_i where:Beta_a e^{Lambda_a t} + Beta_b e^{Lambda_b t} = frac{1}{2} A_i e^{phi_i j} e^{(sigma_i + j omega_i) t} + frac{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)
??Because the sumation of complex exponentials is the homogeneous solution to a linear differential equation the following difference equation will exist??::hat{f}(Delta_t n) = -sum_{m=1}^{M} hat{f}(Delta_t (n-m)) P_m The key to Prony's Method is that the coefficients in the difference equation are related to the following polynomial:sum_{m=1}^{M+1} P_m x^{m-1} = prod_{m=1}^{M} ( x - e^{Lambda_m} )
These facts lead to the following three steps to Prony's Method
1) Construct and solve the matrix equation for the P_m values:
egin{bmatrix}F_N \: \F_{2N-1} end{bmatrix}=-egin{bmatrix}F_{N-1} & .. & F_{0} \: & . & : \F_{2N-2} & .. & F_{N-1} end{bmatrix}egin{bmatrix}P_1 \: \P_Mend{bmatrix}
Note that if N ≠ M 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 :sum_{m=1}^{M+1} P_m x^{m-1}
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 which may be used to solve for the Beta_m values:
egin{bmatrix}F_{k_1} \: \F_{k_M} end{bmatrix}=egin{bmatrix}(e^{Lambda_1})^{k_1} & .. & (e^{Lambda_M})^{k_1} \: & . & : \(e^{Lambda_1})^{k_M} & .. & (e^{Lambda_M})^{k_M} end{bmatrix}egin{bmatrix}Beta_1 \: \Beta_Mend{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+q 2 pi j} for and integer q . This leads to the same nyquist sampling criteria that discrete fourier transforms are subject to: :Im(Lambda_m)|=|omega_m|
reference:Rob Carriere and Randolph L. Moses, “High Resolution Radar Target Modeling Using a Modified Prony Estimator,” IEEE Trans. Antennas Propogat., vol.40, pp. 13-18, January 1992.
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