Applications of Meijer's factorization theorems in performance analyses of all-optical multi-hop FSO systems

The use of bivariate Fox H-functions (BFHFs) in performance analyses of wireless communication systems has gained considerable attention in past few decades. However, the non-existence of robust built-in routines for evaluating such functions in standard computing systems poses numerous challenges in numerical experiments and simulations. Motivated by the apparent need to circumvent these difficulties in performance analyses of cooperative wireless communications, this work presents an alternative method for obtaining the exact, approximate and asymptotic BFHF-free cumulative distribution function (CDF) of the end-to-end (e2e) signal-to-noise ratio (SNR) of multi-hop amplify-and-forward (AF) relaying wireless communication systems. As an illustration, the e2e performance analysis of an all-optical dual-hop free-space optical (FSO) transmission system over Gamma-Gamma turbulence in the presence of pointing errors is revisited. Specifically, new mathematical formulae for the statistical characteristics of the e2e SNR for systems with AF fixed-gain relaying as well as channel-state-information(CSI)-assisted using heterodyne detection (HD) or intensity modulation with direct detection (IM/DD) are derived in terms of mathematically malleable and uniformly convergent infinite series of weighted Meijer G-functions. The usefulness of the derived CDFs is illustrated through derivation of traditional system performance metrics. The accuracy of the derived analytical formulae is verified via Monte Carlo simulations in MATLAB®. Finally, based on results observed in this paper, useful expansions of common BFHFs in terms of easily computable univariate hypergeometric functions are proposed