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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Time-frequency characterization of random systems

Lorenzo Galleani, Leon W. Cohen

Published Dec 31, 2003• 4 citations
Publisher

Abstract

We have previously presented a method to write equations for the Wigner distribution corresponding to the solution of a linear differential equation. We extend this method to arbitrary time-frequency distributions, the Short Time Fourier Transform, and the wavelet transform. Also, we apply the method to stochastic signals and address a number of applications including Brownian motion. We obtain the Wigner distribution for Brownian motion by solving the Langevin equation and also by using our method. We obtain exact solutions.

DOI
10.1117/12.513900

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We have previously presented a method to write equations for the Wigner distribution corresponding to the solution of a linear differential equation. We extend this method to arbitrary time-frequency distributions, the Short Time Fourier Transform, and the wavelet transform. Also, we apply the method to stochastic signals and address a number of applications including Brownian motion. We obtain the Wigner distribution for Brownian motion by solving the Langevin equation and also by using our method. We obtain exact solutions.
We use the Wigner distribution to study systems subjected to random forces. We define the instantaneous spectrum as the ensemble average of the Wigner distribution, and we write the differential equation whose solution gives us the time-varying spectrum of the state variable. We consider the cases of both constant and time-varying coefficients. We apply the method to study the instantaneous spectrum of a harmonic oscillator driven by Gaussian noise, with both constant and time-varying coefficients. In the latter case our method clearly reveals the nonstationarity of the power spectrum and we confirm our result by numerical simulations.
We present a general procedure for obtaining equations of motion for the Wigner distribution of functions that are governed by ordinary and partial differential equations. For the case of fields we show that in general one must consider Wigner distribution of the four variables, position, momentum, time and frequency. We also show that in general one cannot write an equation of motion for position and momentum however it can be done in some cases, the Schrodinger equation being one such case. Our method leads to an equation of motion for the Schrodinger equation with time dependent potentials in contrast to the result obtained by Wigner and Moyal which was for time independent potentials.
We present a general method for calculating relevant moments of the wavelet transform.Explicit results are given for the time, frequency, and scale moments.Using the results obtained, we show that the wavelet transform has unique characteristics that are not possessed by other methods.We discuss whether these unusual characteristics are physically or mathematically important.Exactly solvable examples are given, and the results are contrasted to those of the standard methods such as the spectrogram and the Wigner distribution.
There is a mathematical method for finding frequency components and their amplitudes, or spectra from periodic patterns. It is called Fourier transform (FT) spectral analysis. In this chapter, how we apply Fourier transform to the following subjects will be explored, emphasizing theoretical discussions: electronics circuits and telecommunication; spectroscopy—magnetic resonance, and IR-spectroscopy; optics—one- and two-dimensional diffractions, lens, hologram, and x-ray diffraction; stochastic process—central limit theorem, Brownian motion, fluctuation–dissipation theorem, and motional narrowing; spectral method for solving diffusion equation and Green’s function for wave equation; and quantum mechanics—wave packet, group velocity, imaginary time Schrödinger equation, Heisenberg’s uncertainty principle, and the Born approximation of scattering of a particle of potential.