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

The first and second passage times of a stationary Rayleigh process
R(t,a)
are discussed.
R(t,a)
represents the envelope of a stationary random process consisting of a sinusoidal signal of amplitude and frequency
f_{0}
plus stationary Gaussian noise of unit variance having a narrow-band power spectral density which is symmetrical about
f_{0}
. Approximate integral equations are developed whose solutions yield approximate probability densities concerning the first and second passage times of
R(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 of
R(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.