Theoretical and experimental results for the distribution of a certain nonlinear functional of the Ornstein-Uhlenbeck process
- 1 September 1969
- journal article
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Information Theory
- Vol. 15 (5) , 532-535
- https://doi.org/10.1109/tit.1969.1054364
Abstract
Letx(t)be the Ornstein-Uhlenbeck process andy(t)the result of low-passRCfiltering of sgnx(t). This paper considers the problem of determining the first-order probability density function ofy(t). The approach is to apply the\nuth-order Fokker-Planck-Kolmogorov type equations. Based upon an assumption as to the linearity of a coefficient of the resulting differential equation, a closed-form solution is obtained forp(y). The result agrees with the previous work of Doyle, McFadden and Marx who solved the special case when the bandwidth of theRCfilter is twice the bandwidth of the input noise. The result also agrees, to within experimental error, with a Monte Carlo simulation over four orders of magnitude of variation of the ratio of the bandwidths of theRCfilter and the input process.Keywords
This publication has 8 references indexed in Scilit:
- On the response of a class of self-excited oscillators to stochastic excitationInternational Journal of Non-Linear Mechanics, 1967
- Generalizations and extensions of the Fokker- Planck-Kolmogorov equationsIEEE Transactions on Information Theory, 1967
- Some analytical and experimental phase-locked loop results for low signal-to-noise ratiosProceedings of the IEEE, 1966
- On the Relaxation of the Hard—Sphere Rayleigh and Lorentz GasThe Journal of Chemical Physics, 1964
- Phase-locked loop dynamics in the presence of noise by Fokker-Planck techniquesProceedings of the IEEE, 1963
- The Distribution of a Certain Nonlinear Functional of an Ornstein-Uhlenbeck ProcessJournal of the Society for Industrial and Applied Mathematics, 1962
- On the probability density of the output of a low-pass system when the input is a Markov step processIEEE Transactions on Information Theory, 1960
- A systematic approach to a class of problems in the theory of noise and other random phenomena--IIEEE Transactions on Information Theory, 1957