FIRST ORDER OUTAGE STATISTICS OF ASYMMETRICAL RF- OW DUAL-HOP RELAY COMMUNICATIONS

This paper addresses the first-order outage statistics of asymmetrical radio frequency (RF)-optical wireless (OW) relay systems over non turbulent-induced-fading (nTIF) and turbulent-induced-fading (TIF) channels. We rely on dual-hop amplify-and-forward relay (AFR) scheme and provide detailed mathematical development for derivation of novel exact analytical as well as novel closed form approximative expressions for: i). cumulative distribution function, ii.) outage probability, and iii.) average bit-error-rate. The system under consideration is modeled as the product of independent Nakagmi-m and double squared Nakagami-m (also known as Gamma-Gamma) random processes. The obtained results of the proposed system are graphically presented for RF -OW TIF and nTIF channel sets of parameters. Moreover, the detailed comparisons of exact and approximated numerical results whose derivation resorts on exponential Laplace approximation method (LAM) are provided and thoroughly examined for the considered RF-OW statistical measures.


INTRODUCTION
Optical  wireless (OW) communicationsas well as asymmetrical radio frequency (RF)-OW communications are relevant research topic within academia and industry for future 5G and even beyond 5G (B5G) network deployments (Hamza et al., 2018;Khalighi & Uysal, 2014;Illi et al., 2017;Douik et al., 2016). The OW communications are primarily intended to speed up the transmission rate and ensure higher capacity and wider bandwidth compared to RF links. Moreover, OW communications, especially free space optical (FSO) communications which operates at near infrared part of the spectrum are i.) cost effective, ii.) spectrum license free, iii.) channel interference free. On the other hand, one of the main FSO system performance impairments is the impact of atmospheric turbulence due to the small-and large-scales atmospheric cells. Weather conditions as well as misalignment of the system's transmitter-receiver apparatus can cause further degradation of the system performance stability.
Namely, Nakagami-m (Nm) RP can address RF links over non turbulence induced fading (nTIF) channels (Nakagami, 1960) while double squared Nakagami-m (d-sNm), also known as gamma-gamma RP can address FSO links over moderate to strong turbulence induced fading (TIF) channels (Andrews & Phillips, 2005;Vetelino et al., 2007;Al-Ahmadi, 2014). The papers (Zedini et al., 2014;Anees & Bhatnagar, 2015;Petkovic et al., 2017) address mixed RF-FSO relay systems over Nm nTIF channels and d-sNm TIF channels and provide closed form analytical results for the first order statistical measures expressed through Meijer's G function. In paper , Nm nTIF and d-sNm TIF are used to address cooperative mixed RF-FSO relay link and obtained analytical results are given in terms of H-fox and Meijer's G functions. Moreover, in (Stefanovic et al., 2019a) the closed form analytical expressions for second order statistics of the products of Nm, d-sNm and Nm RPs are derived by Laplace approximation method (LAM) and efficiently applied to address TIF and nTIF channels of mixed triple-hop RF-FSO-RF vehicle-to-vehicle (V2V) AFR communications.

MATHEMATICS, COMPUTER SCIENCE AND MECHANICS
approximations by solving complex many-folded integrals and significantly decrease computational time of complex analytical expressions. The proofs that LAM is able to provide precise results even under sub-asymptotic conditions is provided in (Butler & Wood, 2002). Another remark of the LAM is its generality and simplicity of application, which in many cases can provide fast computing closed form accurate approximations derived from latent Gaussian models (Wang, 2010).
In this paper we provide comprehensive mathematical development for computing novel exact expressions as well as novel closed form approximative expressions for i.) cumulative distribution function, ii.) outage probability, iii.) and average bit error rate of the product of Nm and d-sNm RPs. Moreover, we rely on exponential LAM for derivation of closed form analytical expressions for the first order statistical measures. The obtained results are further used to address asymmetrical RF-FSO dualhop AFR system in the case when nTIF (fading over RF channel) and TIF (turbulence induced fading over FSO channel) are the main cause of the system model performance degradation.
To the best of author's knowledge there is no paper in open technical literature that applies exponential LAM for derivation of the first order outage statistics of RF-FSO AFR system over Nm nTIF and d-sNm TIF channels.

SYSTEM MODEL OF RF-FSO DUAL-HOP AFR COMMUNICATIONS
We model the mixed RF-FSO dual-hop AFR system as the product of the independent Nm random process (RP),  Further, 2 n C is Refractive index, k=2π/λ is wave-number (λwavelength), D is receiver aperture diameter and L is propagation distance.

Probability Density Function (PDF)
The PDF of x out can be expressed through joint and conditional probabilities as:

Cumulative Distribution Function (CDF)
The CDF of x out can be calculated by using (Simon & Alouini, 2000;Gradshteyn & Ryzhik, 2000 It has been shown in (Butler & Wood, 2002) that accurate results can be obtained for real value parameter, T=1. Further, the LAM, also known as exponential LAM for constant multivariate function f 1 and variable multivariate function f 2 is considered in 58 MATHEMATICS, COMPUTER SCIENCE AND MECHANICS (Harding & Hausman, 2007). Accordingly, the arguments in Eq. where p and q denote parameters for different binary modulation such as:  coherent binary frequency shift keying (CBFSK) for p=0.5 and q=0.5,  coherent binary phase shift keying (CBPSK) for p=0.5 and q=1,  non-coherent binary frequency shift keying (NBFSK) for p=1, q=0.5,  differential binary phase shift keying (DBPSK) for p=1, q=1. The closed form approximation for BER RF-FSO can be obtained by evaluating J 2 in Eq. (13) by using exponential LAM for three folded integrals (Zlatanov et al., 2008, Equation I.

NUMERICAL RESULTS
In numerical results we provide performance analysis as well as comparison of exact and approximated results of the first order outage statistics under weak, moderate and strong nTIF and TIF conditions of the RF-FSO dual-hop AFR system.
Moreover, it can be seen that nTIF and TIF severity parameters (α, β, m RF ) have stronger impact on BER RF-FSO than considered binary modulations. In most of the observed range and under considered nTIF and TIF channel conditions the best performance results relating to BER RF-FSO can be achieved for CBPSK. The comparison of exact analytical expression and approximated closed form expression for BER RF-FSO of DBPSK modulated signal are provided in Figure 3. It can be noticed that exponential LAM fails to match approximation with exact analytical results for higher Ω RF dB values.