B.S.T.J. Brief: Axis-Crossing Intervals of Rayleigh Processes

Let R(t.,a) denote the envelope of a stationary random process consisting of a sinusoidal signal of amplitude / 2 a and frequency / 0 plus Gaussian noise of unit variance having a narrow-band power spectral density which is symmetrical about f0. When a = 0 Rice 1 presented some theoretical results which are very useful for studying statistical properties of the axis-crossing intervals of R(l,0). The axis-crossing points and the axis-crossing intervals of the Rayleigh process R(t,a) are defined in Fig. 1. Some recent work concerning the axis-crossing in tervals of R(t,a) was reported by Levin and Fomin 2 , Goryainov, 3 and Rainal. 4,6 The purpose of this brief is to present some theoretical results when a ^ 0. These results stem from a straightforward extension of Rice's analysis. The Rayleigh process R(t,a) occurs at the output of a typical radio or radar receiver during the reception of a sinusoidal signal immersed in Gaussian noise.