New Accurate Approximation for the Sum of Generalized Gamma Distributions and its Applications

By considering the probability density function (PDF) of a generalized Gamma (GG) random variable (RV) evaluated in terms of a proper subset H1,0 class of Fox’s H-function (FHF) and the moment-based approximation to estimate the H-function 1,1 parameters, a closed-form tight approximate expression for the distribution of the sum of independent and not necessarily identical GG distributed RVs is presented as well as a sufficient condition for the convergence is verified. Such proposed approximate PDF is useful analytical tool for analyzing the performance of L-branch maximal-ratio combining receivers subject to such a fading model. This result can be also of paramount importance when dealing with an intelligent reflecting surface subject to GG fading channels. Furthermore, various closed-form approximate expressions, such as the cumulative distribution function (CDF), moment-generating function, outage probability (OP), average channel capacity, nth moment of the signal-to-noise ratio (SNR), amount of fading, and average symbol error probability (ASEP) for numerous coherent digital modulation schemes, are derived and examined in terms of FHF. To gain further insight into the system performance, asymptotic closed-form expressions for the ASEP and OP are derived and interesting observations are made. Particularly, our asymptotic analysis reveals that the achievable diversity order for high SNR values depends essentially on the branches’ number. The proposed mathematical analysis is assessed and corroborated by Monte-Carlo simulations using computer algebra systems, while the PDF and CDF are validated further with the aid of the Kullback-Leibler divergence criterion and Kolmogorov-Smirnov test, respectively