# Performance Analysis of Dual-Hop DF Multi-Relay FSO System with Adaptive Modulation
## 1 Introduction
To reduce this complexity as well as the consuming power, a proposed solution known as the partial relay selection (PRS) scheme was introduced. Within this scheme, RS is dependent on CSI of either the source-to-relay or relay-to-destination links. The most comprehensive work in RS cooperative FSO systems was introduced in [23], where the ABER of the cooperative system with ORS and PRS schemes have been investigated over \(\mathcal{LN}\) and \(\mathcal{GG}\) channel models. In [24], the PRS scheme was also proposed for a two-way relay FSO system, and its performance was investigated in terms of OP. Another technique to mitigate the impact of AT in FSO systems is adaptive transmission. This approach involves dynamically adjusting the transmission parameters, such as modulation order and transmitted power, based on the channel conditions. Adaptive modulation (AM), for instance, adapts the modulation order depending on the channel conditions. This adaptation improves spectral efficiency (SE) of the system while achieving the required target performance in terms of ABER or OP. On the other hand, transmitted power control can be employed to regulate transmitted power according to received signal levels, thereby improving the SNR and reducing the effect of AT. In [25], the authors investigated the channel capacity of an FSO system over an \(\mathcal{M}\) channel with truncated channel inversion with fixed rate adaptive transmission. They extended the analysis based on optimal rate, optimal power and rate, and channel inversion with fixed rate adaptive transmission in [26]. The authors in [27] proposed a link adaptation algorithm using M-ary pulse position modulation (PPM) over an \(\mathcal{LN}\) fading channel to improve the bandwidth efficiency by varying the code rate and modulation order. Enhancement of the SE of the FSO system while maintaining the target ABER was achieved using the AM technique in [28; 29; 30; 31; 32]. The performance analysis of the AM for a subcarrier intensity modulation (SIM) system utilizing phase shift keying (PSK) was presented in [28]. The analysis specifically focused on single-input single-output (SISO) systems over the \(\mathcal{LN}\) and \(\mathcal{GG}\) channels, and subsequently extended to multiple-input multiple-output (MIMO) systems. The analysis in [28] was extended to include PSK and quadrature amplitude modulation (QAM) over a \(\mathcal{GG}\) channel in [29]. The performance of the AM for SIM systems utilizing PSK modulation and M-ary differential PSK modulation was studied over an \(\mathcal{M}\) channel, respectively, in [30; 31], where a SISO configuration was considered in [30] and an MIMO configuration was considered in [31]. Recently, in [32], we investigated the performance of a coherent parallel-branch maximum ratio combining system with adaptive M-PSK modulation over \(\mathcal{G}\mathcal{G}\) channels. The authors in [33] proposed a method to improve the SE with maintaining the target OP. They investigated multiple-input single-output (MISO) and single-input multiple-output (SIMO) adaptive coherent systems employing PSK or QAM over an \(\mathcal{LN}\) channel. The authors extended the work and investigated AM for the intensity modulation/direct detection (IM/DD) system with PPM and pulse amplitude modulation (PAM) in [34]. On the other hand, the integration of adaptive transmission with relay-assisted FSO systems has received little attention. In [35], the authors proposed the use of AM in a multi-hop coherent FSO system (i.e., multi-relays are serially placed between the source and the destination) with an AF scheme. The OP, SE, and ABER were derived over an \(\mathcal{M}\) channel with pointing errors. Additionally, [36] suggested adjusting the transmitted power over an \(\mathcal{LN}\) channel to enhance bandwidth efficiency in a multi-hop FSO system with a DF scheme. Motivated by these observations, the performance of a dual-hop DF multi-relay system that employs PSK modulation is investigated under SIM detection.