Second Order Statistics of -Fisher-Snedecor Distribution and Their Application to Burst Error Rate Analysis of Multi-Hop Communications

An advantage of using the composite fading models (CFMs) is their ability to concurrently address the impact of multi-path and shadowing phenomena on the system performance in wireless communications. A Fisher-Snedecor (FS) $\mathcal{F}$ CFM has been recently proposed as an experimentally verified and tractable fading model that can be efficiently applied for 5G and beyond 5G wireless communication systems. This paper provides second-order (s-order) performance analysis of the product of $N$ independent but not identically distributed (i.n.i.d) FS $\mathcal{F}$ random variables (RVs). In particular, accurate and closedform approximations for level crossing rate (LCR) and average fade duration (AFD) of the product of $N$ i.n.i.d FS $\mathcal{F}(N-{\mathrm {FS}}~ \mathcal{F})$ RVs are successfully derived by exploiting a general property of a Laplace approximation method for evaluation of the $N$ -folded integral-form LCR expression. Based on the obtained s-order statistical results, the burst error rate and maximum symbol rate of the $N$ -FS $\mathcal{F}$ distribution are addressed and thoroughly examined. The numerical results of the considered performance measures are discussed in relation to the $N-\mathrm{FS}~ \mathcal{F}$ multi-path and shadowing severity parameters. Moreover, the impact of the number of hops $(N)$ of the $N$ -FS $\mathcal{F}$ CFM on the s-order metrics, the burst error rate and maximum symbol rate are numerically evaluated and investigated. The derived s-order statistical results can be used to address the cooperative relay-assisted (RA) communications for vehicular systems. Monte-Carlo $(\mathrm{M}-\mathrm{C})$ simulations for the addressed statistical measures are developed in order to confirm the provided theoretical results.