# Performance Analysis of Dual-Hop DF Multi-Relay FSO System with Adaptive Modulation
## 1 Introduction  
Free space optical (FSO) communication systems are gaining substantial attention as a viable solution to fulfill the demanding requirements of upcoming 6G networks. These systems can easily integrate with or complement other technologies, providing numerous potential applications, such as high-speed transmission in backhaul and fronthaul links, satellite communications, and urban and metropolitan areas for last-mile connectivity. Although the FSO system can offer a solution for the increasing demand for high-speed and reliable communication, it can be affected by atmospheric turbulence (AT), which may reduce signal quality and transmission range [1; 2]. Numerous models have been proposed in the literature to describe the fading that occurs in the channel of the FSO system due to AT. The most commonly used models are the Log-normal (\(\mathcal{LN}\)), Gamma-Gamma (\(\mathcal{GG}\)), and Malaga (\(\mathcal{M}\)) distribution models. It has been observed that \(\mathcal{GG}\) and \(\mathcal{M}\) are the most appropriate distributions to represent conditions ranging from weak to strong turbulence, whereas the \(\mathcal{LN}\) distribution is primarily useful for weak turbulence conditions. In a recent work, the authors of [3] conducted a study to assess the feasibility of utilizing the Fisher-Snedecor \(\mathcal{F}\) distribution for AT in the FSO channel. The findings of their study revealed that the \(\mathcal{F}\) distribution offers a fit to experimental data that is as good as or superior to that of the \(\mathcal{GG}\) distribution. In order to address the effect of AT along with other issues, such as long-distance links and no direct link between the transmitter and the receiver, relaying techniques have been introduced in FSO systems as an efficient solution. These techniques also aim toincrease spatial diversity. The most popular relaying strategies, which have been discussed in the literature are the decode-and-forward (DF) and amplify-and-forward (AF). The single relaying technique (dual-hop) has been widely studied in FSO networks, including those integrated with other technologies. A dual-hop FSO system was considered with an AF scheme in [4] and with a DF scheme in [5] over \(\mathcal{F}\) and \(\mathcal{M}\) channels, respectively. In [6], the authors presented an AF-based high-altitude platform station (HAPS) FSO system over a \(\mathcal{GG}\) channel. Using unmanned aerial vehicles (UAVs) as a relay in a dual-hop FSO system was considered in [7]. Furthermore, many works have studied the integration of FSO with radio frequency (RF) communication, for various applications, such as terrestrial, airborne, and satellite communication [8; 9; 10; 11]. Other studies have investigated the integration of FSO with underwater wireless optical communication [12; 13; 14] and power line communication (PLC) [15]. In a multi-relay system, parallel relays are employed between the transmitter and receiver, either with or without an existing direct link. This configuration was investigated in [16; 17; 18; 19]. In these works, all the relays were active during transmission, which required precise synchronization between these relays. Relay selection (RS), which is based on channel condition, is a method that chooses the relay with the best signal-to-noise ratio (SNR) to forward the data to destination. It was proposed in cooperative FSO systems to enhance resource utilization and eliminate the need for complex relay synchronization [20]. In [21], the average bit error rate (ABER) expression of the DF cooperative FSO system was derived. The cooperative system was further investigated in [22], where the exact expressions of outage probability (OP) and ABER were derived as well as the asymptotic expressions. In these works, the optimal relay selection (ORS) scheme was used to select the best relay. This scheme improves the system's performance but requires the channel state information (CSI) of all hops.