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.

中文翻译：

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Meijer 因子分解定理在全光多跳 FSO 系统性能分析中的应用

在过去的几十年中，在无线通信系统的性能分析中使用二元 Fox H 函数 (BFHF) 已经引起了相当大的关注。然而，在标准计算系统中不存在用于评估此类功能的强大内置例程，这给数值实验和模拟带来了许多挑战。由于明显需要规避协作无线通信性能分析中的这些困难，这项工作提出了一种替代方法，用于获得端到端 (e2e) 的精确、近似和渐近无 BFHF 累积分布函数 (CDF)多跳放大转发 (AF) 中继无线通信系统的信噪比 (SNR)。作为例证，重新审视了在存在指向错误的情况下通过 Gamma-Gamma 湍流对全光双跳自由空间光 (FSO) 传输系统的端到端性能分析。具体来说，对于具有 AF 固定增益中继以及信道状态信息 (CSI) 辅助的系统 e2e SNR 统计特性的新数学公式，使用外差检测 (HD) 或直接检测强度调制 (IM/DD) ) 是根据加权梅杰 G 函数的数学可延展性和一致收敛的无穷级数推导出来的。派生的 CDF 的有用性通过传统系统性能指标的派生来说明。通过 MATLAB® 中的 Monte Carlo 模拟验证了导出分析公式的准确性。最后，根据本文观察到的结果，