Envelope PDF in Multipath Fading Channels with Random Number of Paths and Nonuniform Phase Distributions

In a multipath fading channel, the transmitted signal travels through several different paths to the receiver. In each path, amplitude and phase of the signal vary in a random manner. It is common to consider the number of paths as a large constant and to model random fluctuations of the phase by the uniform probability density function (PDF). However, these assumptions are not realistic in many cases. In this paper, a general multipath fading channel with random number of paths (with negative binomial distribution) and nonuniform phase distributions (with von Mises PDFs) is considered and it is shown that the envelope fluctuates according to a gamma PDF. It is also shown that the parameters of this gamma PDF are directly related to the physical parameters of the channel. Due to the realistic assumptions made in the derivation, the gamma PDF is a promising candidate for accurate modeling of envelope statistics in multipath fading channels.