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First and second passage times of Rayleigh processes (Corresp.)

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The first and second passage times of a stationary Rayleigh processR(t,a)are discussed.R(t,a)represents the envelope of a stationary random process consisting of a sinusoidal signal of amplitude and frequencyf_{0}plus stationary Gaussian noise of unit variance having a narrow-band power spectral density which is symmetrical aboutf_{0}. Approximate integral equations are developed whose solutions yield approximate probability densities concerning the first and second passage times ofR(t,a). The resulting probability functions are presented in graphs for the case when the power spectral density of the noise is Gaussian. Related results concerning the approximate distribution function of the absolute minimum or absolute maximum ofR(t,a)in the closed interval[0,tau]are also presented. The exact probability densities are expressed in the form of an infinite series of multiple integrals.

Published in:

Information Theory, IEEE Transactions on  (Volume:33 ,  Issue: 3 )

Date of Publication:

May 1987

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