Generalized Phase-Crossing Rate and Random FM Noise for $\alpha{-}\mu$ Fading Channels

In this paper, the crossing statistics of phase processes and random frequency-modulation (FM) noise are studied for $\alpha{-}\mu$ fading channels. Specifically, simple closed-form expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of random FM noise as well as the generalized phase-crossing rate (GPCR) are derived. In addition, simple closed-form expressions for the conditional pdfs of the envelope and the random FM noise, which are conditioned on an arbitrary phase upward crossing level, are obtained and investigated. The formulations developed here are validated by means of Monte Carlo simulation and verified by comparison with results that are already known in the literature. These results correspond to special cases of the $\alpha{-}\mu$ fading channels, such as Nakagami-$m$ and Weibull channels.