B.S.T.J. Briefs: Axis-Crossing Intervals of Sine Wave Plus Noise

Axis-Crossing Intervals of Sine Wave Plus Noise By I. INTRODUCTION A. J. RAINAL Let I(t, a) denote the stationary random process consisting of a sinusoidal signal of amplitude s/2a and angular frequency q plus Gaussian noise, I.x(t), of zero mean and unit variance. Thus, / ( / , a) = V 2 a c o s (ql + 80) + / , ( / ) . (1) 0(, denotes a random phase angle which is distributed uniformly in the interval ( -- r, 7r). "a" denotes the signal-to-noise power ratio. When a = 0 Rice1 presented some theoretical results which are very useful for studying statistical properties of the axis-crossing intervals and the axis-crossing points of I(t, 0) at an arbitrary level I. The axis-crossing intervals and the axis-crossing points of I(t, a) are defined in Fig. 1. In recent work Cobb 2 presented some theoretical results concerning the zero-crossing intervals, the axis-crossing intervals defined by the level I = 0, of I(t, a). Some experimental and theoretical results concerning (he zero-crossing intervals of I(t, a) were reported by Rainal. 3 For the case when the power spectral density of I N (t) is narrow-band and symmetrical about the sine wave frequency,