Unified Laguerre Polynomial-Series-Based Distribution of Small-Scale Fading Envelopes

We derive new unified probability density function (pdf) formulas based on the generalized Laguerre polynomial series expansion that cover a wide range of small-scale fading distributions in wireless communications. The unified envelope pdf covers many known Laguerre polynomial-series-based pdf´s and small-scale fading distributions, which include the multiple-waves-plus-diffuse-power (MWDP) fading, kappa-mu, Nakagami-m, Rician (Nakagami-n), Nakagami-q (Hoyt), Rayleigh, Weibull, and alpha-mu (Stacy) distributions as special cases. We highlight that the unified pdf formulas are general and applicable to characterize the distribution of any random variable defined on the positive range. We then propose two truncated forms of the pdf and prove that the original pdf and both truncated forms satisfy the properties of a valid pdf. A new cumulative distribution function and a moment-generating function of this pdf are derived. We also analyze the truncation error for the case when only a finite number of terms are kept in the series expansion.