# Performance Analysis of Dual-Hop DF Multi-Relay FSO System with Adaptive Modulation
## 6 Numerical Results
The SE for the balanced case of the adaptive DH-MR system improves as the number of relays, \(L\), increases as shown in Figure 3. This improvement can be attributed to the fact that a larger value of \(L\) increases the chance of selecting the relay with better channel gain. However, it is worth noting that the improvement in SE is not linear with the increase in \(L\), indicating that larger values of \(L\) may not necessarily lead to further enhancements. This observation underscores the significance of our investigation in determining the optimal number of relays for such system. The comparison of the SE of the adaptive DH-MR system for three selection schemes is shown in Figure 4. The comparison is provided for three cases: balanced, unbalanced-1, and unbalanced-2 under strong turbulence conditions. Additionally, \(\xi_{1}=\xi_{2}=1.1\) for both hops of the balanced case, \(\xi_{1}=1.1\), \(\xi_{2}=6.7\) for the first and the second hop of the unbalanced-1 case, and \(\xi_{1}=6.7\), \(\xi_{2}=1.1\) for the first and the second hop of the unbalanced-2 case. It is observed that the SE of the adaptive system under the ORS scheme is the best in all cases. In the balanced case, both PRS schemes, PRS1 and PRS2, exhibit the same performance due to identical channel gains in both hops. Moreover, the PRS1 scheme outperforms the PRS2 scheme in the unbalanced-1 case, while the opposite occurs in the unbalanced-2 case. This difference in the performance can be attributed to the nature of the channel gains and the selection schemes employed in these cases. In the unbalanced-1 case, the turbulence conditions are stronger in the first hops compared to the second hops. Therefore, the PRS1 scheme, in which RS depends on the CSI of the first hop, performs better than the PRS2 scheme, in which RS depends on the CSI of the second hop, as both schemes rely on the minimum capacity of the hops of the selected relay. Conversely, in the unbalanced-2 case, the turbulence conditions are weaker in the first hops compared to the second hops. Consequently, the PRS2 scheme outperforms the PRS1 scheme in this scenario. Figure 5 shows the modulation level selection probability for the balanced cases of the adaptive DH-MR system under moderate turbulence conditions and strong pointing errors, \(\xi=1.1\). The modulation level selection probability of both adaptive DH-MR and adaptive DH-SR systems is compared under the ORS scheme, which gives us more insights into the impact of increasing the number of relays, \(L\), on the SE. For all modulation levels, \(M_{i}\), we can see that the modulation level selection probability of the adaptive DH-MR system is higher than that of the adaptive DH-SR system and its maximum values are shifted to the left compared to the adaptive DH-SR system. This is because increasing \(L\) improves the e2e-received SNR,  
Figure 2: Modulation level selection probability of the DH-SR system under strong turbulence with various pointing errors. In Figure 6, we present the OP of the adaptive DH-MR system under three RS schemes along with the OP of the adaptive DH-SR system. The OP is investigated under different cases with strong turbulence conditions. The results indicate improvements in the OP of the adaptive DH-MR system compared to the adaptive DH-SR system, especially for the ORS scheme in all cases, the PRS1 scheme in the unbalanced-1 case, and the PRS2 scheme in the unbalanced-2 case. For other cases, the PRS2 scheme in the unbalanced-1 case and the PRS1 scheme in the unbalanced-2 case, there is only a little improvement in low ASNR values, which confirms the discussion of Figure 4.  
Figure 5: Modulation level selection probability of the DH-MR system under moderate turbulence with \(N=5\) and \(\xi=1.1\). Figure 6: OP of the DH-MR system under strong turbulence with \(N=5\) and \(L=3\). **(a)** Balanced case, **(b)** unbalanced-1 case, **(c)** unbalanced-2 case.