First and Second Passage Times of Sine Wave Plus Noise

Exact, explicit, results concerning the first passage times of a Markov or "Markov-like" random process have been given by many authors. 1-7 But very little is known about the first passage times of a random process consisting of a sinusoidal signal plus stationary gaussian noise. This random process is of interest because it serves as a realistic model for the output of the IF amplifier of a typical radio or radar receiver during the reception of a sinusoidal signal immersed in Gaussian noise. Let I(t, a) denote the stationary random process consisting of a sinusoidal signal of amplitude (2a) * and angular frequency q plus stationary gaussian noise, IN(t), of zero mean and unit variance. Thus, I(t, a) = (2a)1 cos (qt + d0) + IN(t). (1) d0 denotes a random phase angle which is distributed uniformly in the interval ( -- ir, T). a denotes the signal-to-noise power ratio. 2239 2240 T H E BELL SYSTEM TECHNICAL J O U R N A L , DECEMBER 1968 The first and second passage times of I(t, a) are indicated in Fig. 1 and are defined as: (t) r + represents the time I(t, a) takes in going from an upcrossing of the level 7X to the first crossing of the level I2 Ii · (it) T~ represents the time I(t, a) takes in going from a downcrossing of the level Ii to the first crossing of the level I2 h . (iii) T represents the time I(t, a) takes in going from an upcrossing of the level to the second crossing of the level I2 Ii · (iv) T represents the time I(t, a) takes in going from a downcrossing of the level Ix to the second crossing of the level I2 I .