DocumentCode :
943769
Title :
A systematic approach to a class of problems in the theory of noise and other random phenomena--II: Examples
Author :
Siegert, A.
Volume :
3
Issue :
1
fYear :
1957
fDate :
3/1/1957 12:00:00 AM
Firstpage :
38
Lastpage :
43
Abstract :
The method of Part I is applied to the problem of finding the probability distribution of u \\equiv \\int_0^t K(\\tau )x^2(\\tau ) d\\tau , where K(\\tau ) is a given function and x(\\tau ) is the Uhlenbeck process. The earlier methods of Kac and the author yielded the characteristic function of this distribution as the reciprocal square root of the Fredholm determinant D of an integral equation. The present method yields a second-order linear differential equation with initial condition only for D as function of t . For the special cases K(\\tau ) = 1 and K(\\tau ) = e^{-\\alpha \\tau } the characteristic function is obtained in closed form. In Section III, we have verified directly from the integral equation the differential equation for D and some relations between D and the initial and end point values of the Volterra reciprocal kernel which appear in the joint characteristic function for u, x(0) and x(t) .
Keywords :
Markov processes; Noise; Probability functions; Autocorrelation; Detectors; Differential equations; Gaussian processes; Information theory; Integral equations; Kernel; Noise reduction; Nonlinear equations; Partial differential equations; Physics; Probability distribution; Receivers;
fLanguage :
English
Journal_Title :
Information Theory, IRE Transactions on
Publisher :
ieee
ISSN :
0096-1000
Type :
jour
DOI :
10.1109/TIT.1957.1057391
Filename :
1057391
Link To Document :
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