# Performance Analysis of Dual-Hop DF Multi-Relay FSO System with Adaptive Modulation
## 6 Numerical Results  
This section presents selected numerical results for adaptive system performance along with MC simulations. To gain insights into the impact of fading parameters on system performance, different cases are assumed under various turbulence conditions, as shown in Table 1. We also consider various pointing errors, set \(P_{o}=10^{-5}\), and assume that \(\gamma_{s,l}=\gamma_{l,d}=\gamma\). The performance of the adaptive dual-hop single-relay (DH-SR) system is depicted in Figure 1, showcasing the SE across varying turbulence conditions, including strong, moderate, and weak turbulence. These analytical results are obtained using (17), and are \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Turbulence** & **Case** & \(\mathbf{a_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{b_{1}}\) & \(\mathbf{b_{2}}\) \\ \hline \multirow{3}{*}{Strong turbulence} & Balanced & 1.6795 & 1.6795 & 3.0746 & 3.0746 \\  & Unbalanced-1 & 1.6102 & 2.0945 & 3.1128 & 3.3851 \\  & Unbalanced-2 & 2.0945 & 1.6102 & 3.3851 & 3.1128 \\ \hline \multirow{3}{*}{Moderate turbulence} & Balanced & 2.452 & 2.452 & 4.0912 & 4.0912 \\  & Unbalanced-1 & 1.8573 & 4.2554 & 3.5841 & 5.6848 \\  & Unbalanced-2 & 4.2554 & 1.8573 & 5.6848 & 3.5841 \\ \hline \multirow{3}{*}{Weak turbulence} & Balanced & 3.4542 & 3.4542 & 5.1751 & 5.1751 \\  & Unbalanced-1 & 2.3882 & 6.582 & 4.2057 & 7.9741 \\ \cline{1-1}  & Unbalanced-2 & 6.582 & 2.3882 & 7.9741 & 4.2057 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Turbulence conditions. plotted for the balanced case vs. the ASNR, \(\tilde{\gamma}\), with two different numbers of modulation levels, \(N=3\), and \(N=5\). It can be seen that the SE improves with increasing \(\tilde{\gamma}\). Furthermore, the SE improves with the enhancement of the channel conditions (i.e., the turbulence condition becomes lower) and with an increase in the number of modulation levels. Moreover, Figure 1 illustrates a comparison of the SE for the adaptive DH-SR system under various pointing errors, where \(\xi=1.1\) refers to the strong pointing errors, while \(\xi=6.7\) refers to the weak pointing errors. The results clearly show that the adaptive DH-SR system achieves higher SE under weak pointing errors. This indicates that both turbulence conditions and pointing errors significantly impact SE performance and must be considered in the analysis of adaptive systems. The modulation level selection probability of the adaptive DH-SR system, provided in (14), is illustrated in Figure 2. It is investigated under strong turbulence with two pointing errors and number of modulation levels, \(N=5\). It can be observed that at the lowest ASNR values, the modulation level selection probability is low where the OP of the adaptive DH-SR system is high. With increasing the values of the ASNR, a higher modulation level selection probability increases. For the higher values of the ASNR, the largest modulation level, \(M_{5}\), dominates the selection process. This observation explains how the SE improves with increasing the values of the ASNR until it reaches its maximum at higher values. In same figure, the impact of the pointing errors on the modulation level selection probability is also illustrated. It is depicted that the modulation level selection probability of all levels decreases significantly with strong pointing errors, \(\xi=1.1\), which leads also to a reduction in the SE. This behavior is expected since the strong pointing errors reduce the received SNR, and this, in turn, decreases the modulation level selection probability and the SE. Figure 1: SE of the DH-SR system under diverse turbulence conditions and different pointing errors with \(N=3\), and \(N=5\). (**a**) Strong turbulence, (**b**) Moderate turbulence, (**c**) Weak turbulence. The SE of the adaptive DH-MR system under three RS schemes given in (23) is presented in Figures 3 and 4.