Axis-Crossings of the Phase of Sine Wave Plus Noise
A. J. Rainal · Bell System Technical Journal · 1967
This paper is concerned with the axis-crossings of the resultant phase$-\pi \leqq \theta(t,a)\leqq π$of a sinusoidal signal of amplitude$\sqrt{2a}$and frequency ∫0plus Gaussian noise of unit variance having a narrow-band power spectral density which is symmetrical about ∫0. The discontinuous phase process θ(t,a) is present at the output of the IF amplifier of a radio or radar receiver during the reception of a sinusoidal signal immersed in Gaussian noise. Also, the phase process θ(t,a) is basic in Rice's recent analysis of noise in FM receivers. The following theoretical results are presented concerning the axis-crossings (level-crossings) of θ(t,a) at an arbitrary level θ (i) The average number of upward (or downward) axis-crossings per second. (ii) The conditional probability that an upward axis-crossing occurs between t + τ and t + τ + dr given a downward axis-crossing at t. (iii) The conditional probability that a downward axis-crossing occurs between t + τ and t + τ + dr given an upward axis-crossing at t. (iv) The conditional probability that an upward axis-crossing occurs between t + τ and t + τ + dr given an upward axis-crossing at t. (v) The variance of the number of axis-crossings observed in a time τ. The theoretical probability functions are presented in graphs as a continuous function of τ for various values of θ and “a” for the case when the Gaussian noise has a Gaussian power spectral density.