On the Distribution of the Sum of Gamma-Gamma Variates and Applications in RF and Optical Wireless Communications

The Gamma-Gamma (Γ Γ ) distribution has recently attracted the interest of the research community due to its involvement in various communication systems. In the context of RF wireless communications, Γ Γ distribution accurately models the power statistics in composite shadowing/fading channels as well as in cascade multipath fading channels, while in optical wireless (OW) systems, it describes the fluctuations of the irradiance of optical signals distorted by atmospheric turbulence. Although Γ Γ channel model offers analytical tractability in the analysis of single input single output (SISO) wireless systems, difficulties arise when studying multiple input multiple output (MIMO) systems, where the distribution of the sum of independent Γ Γ variates is required. In this paper, we present a novel and simple closed-form approximation for the distribution of the sum of independent, but not necessarily identically distributed Γ Γ variates. It is shown that the probability density function (PDF) of the Γ Γ sum can be efficiently approximated either by the PDF of a single Γ Γ distribution, or by a finite weighted sum of PDFs of Γ Γ distributions. To reveal the importance of the proposed approximation, the performance of RF wireless systems in the presence of composite fading, as well as MIMO OW systems impaired by atmospheric turbulence, are investigated. Numerical results and simulations illustrate the accuracy of the proposed approach.