In this paper, we study the statistics of phase processes and random frequency modulation (FM) noise encountered in 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 supposed to be symmetric about the carrier frequency. Under these fading conditions, we derive first an expression for the joint probability density function (jpdf) of the phase process and its rate of change. Capitalizing on this jpdf formula, we then 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 Laplacian distributed in-phase and quadrature components of the complex channel gain, are obtained. Aside from the aforementioned statistics, we also provide exact expressions for the pdf, cdf, and second-order moment of the random FM noise at the output of a selection combiner, operating on independent and identically distributed (i.i.d.) branches. The validity of the analysis has been carried out by reducing the derived expressions to already-known results for single Rayleigh channels and by comparing our analytical results with the corresponding simulation results.
['Nazih Hajri', 'Neji Youssef', 'Matthias Patzold']
On the Statistical Properties of Phase Crossings and Random FM Noise in Double Rayleigh Fading Channels