Level Crossings of Phase of Sine Wave Plus Gaussian Noise

Abstract

Theoretical results concerning the level crossings of the phase -π≤θ(t)≤π of the process consisting of a sine wave disturbed by additive Gaussian noise are presented, along with the experimental results supporting the theory. The average number of crossings of the phase angle θ(t), as well as the conditional probability densities for R(t), the envelope of the considered process, and for θ′(t), the time derivative of θ(t), are derived for an arbitrary crossing level θ. These derivations are a generalization of the existing theory to the case of a nonsymmetric (about the sine wave frequency) noise spectrum and an arbitrary crossing level -π≤θ≤π. To verify the derivations, experimental measurement of the considered quantities is performed for the case of a Gaussian noise having the 7th-order Butterworth spectrum. The experimental results are compared with the corresponding theoretical ones, and are found to be in good agreement.

Generalized Nakagami-m Phase Crossing Rate

“…In this sense, Rice obtained the phase crossing rate at the particular phase levels θ = 0 and θ = π for the envelope lying within an arbitrary range. In [3], the work by Rice was extended to consider asymmetrical noise spectrum as well as arbitrary phase levels, i.e., −π ≤ θ ≤ π. More recently, [4] and [5] investigated the phase crossing statistics, respectively, for the Hoyt (Nakagami-q) and Weibull processes.…”