On the crossing statistics of phase processes and random FM noise in Nakagami-q mobile fading channels

The crossing statistics of phase processes and random frequency modulation (FM) noise are studied for Nakagami-q fading channels. Closed-form expressions are first derived for the probability density function (PDF) and the cumulative distribution function (CDF) of random FM noise. The crossing rate of the phase process is then obtained for any crossing level of the phase. Moreover, the conditional PDF of random FM noise and envelope processes-conditioned on the crossings of an arbitrary level of the phase-are investigated. Since the Rayleigh fading channel is a special case of the Nakagami-q fading channel, the derived expressions are verified by comparison with results known for Rayleigh fading channels. In addition, it is shown that the derived analytical results are in excellent agreement with those obtained by computer simulations. The presented results are useful, for example, for studying the statistics of noise spikes occurring in limiter-discriminator FM receivers and for investigating the cycle slipping phenomenon in phase-locked-loop schemes when considering the transmission over Nakagami-q mobile fading channels.