Envelope Probability Density Functions for Fading Model in Wireless Communications

An analytical model that describes small-scale local area fading experienced by narrowband radio receivers consisting of N dominant specular waves and an arbitrary number of additional weaker waves lumped into a Gaussian distributed diffuse component was recently proposed. The exact envelope pdf of this model is in a definite integral form involving a product of N zeroth-order Bessel functions. A closed form formula for this envelope pdf is known only for N = 2. For analytic studies, a closed-form expression for this pdf for an arbitrary N = 3 is desirable. In this paper, we derived a new expression for this pdf in terms of a series of orthogonal Laguerre polynomials that can be efficiently computed for any arbitrary value of TV. In the course of our derivation, we gained much physical insight and were able to show, from our computations and analysis, that for the sum of N = 6 specular waves, the envelope pdf can be approximated closely by a Rayleigh distribution.