A Study on the Phase Crossing Statistics and Random FM Noise in M2M Double Rayleigh Fading Channels

In this paper, we study the statistics of phase processes and random frequency modulation (FM) noise encountered in mobile-to-mobile (M2M) double Rayleigh fading channels. The Rayleigh processes making up the double Rayleigh channel are assumed to be independent but not necessarily identically distributed. The Doppler power spectral densities of these processes are assumed to be symmetric about the carrier frequency. Under these fading conditions, we derive at first an expression for the joint probability density function (JPDF) of the phase process and its rate of change, i.e., random FM noise. Second, and based on this JPDF formula, we investigate the probability density function (PDF) and cumulative distribution function (CDF) of random FM noise. Moreover, the average crossing rate of phase processes as well as that of the Laplace distributed inphase and quadrature components of the channel complex gain are obtained. It is found that the presented analytical expressions encompass known results corresponding to the case of single Rayleigh fading channels (unlike the case of the amplitude fading where the statistics of single scattering channels cannot, in general, be attained from those of double fading processes). For the verification of the validity and accuracy of the presented theoretical expressions, we compare them with those ones obtained by means of computer simulations. Excellent agreements are shown between theoretical and simulation quantities.