## 4 Performance Analysis
### C. Average Ergodic Channel Capacity
According to Figure 2, the overall trend of the OP curves of the CSI-assisted relay and the FG relay is consistent. Therefore, in the following analysis, in order to avoid repeated descriptions, we only describe the case of CSI-assisted relay in detail. Figure 3 demonstrates the effects of light fog (LF) and medium fog (MF) on the OP of \(N\)-hop FSOC system, which considers CSI-assisted relays. The system OP of the MF channel is higher than that of the LF channel. For example, when \(N=3\) and \(P_{t}=30\) dBm, the system OPs of LF channel and MF channel are \(2.95\times 10^{-4}\) and \(1.06\times 10^{-1}\), respectively. The reason is that the optical power attenuation of the MF channel is greater than that of the LF channel. Therefore, in the case of fixed system noise, the SNR of the MF channel is lower than that of the LF channel and the OP of the MF channel is higher than that of the LF channel. This also means that the MF channel needs more hops to achieve the same communication performance as the LF channel. Figure 2: Outage probability (OP) for \(N\)-hop FSOC system with CSI\(-\)assisted relays and FG relays under light fog. Figure 4 demonstrates the effects of different atmospheric refractive index structure constants (\(C_{n}^{2}\)) on the OP of \(N\)-hop FSOC system, which considers CSI-assisted relays and light fog weather. Table 3 shows the intensity of atmospheric turbulence corresponding to different \(C_{n}^{2}\). When \(N=2\), the Rytov variances of each hop link corresponding to \(C_{n}^{2}=2\times 10^{-14}\), \(C_{n}^{2}=6\times 10^{-14}\), \(C_{n}^{2}=2\times 10^{-13}\), and \(C_{n}^{2}=6\times 10^{-13}\) are \(\sigma_{R}^{2}=0.2350\), \(\sigma_{R}^{2}=0.7049\), \(\sigma_{R}^{2}=2.3498\), and \(\sigma_{R}^{2}=7.0495\), respectively. Since \(\sigma_{R}^{2}\leq 0.3\), \(0.3\leq\sigma_{R}^{2}<5\) and \(\sigma_{R}^{2}\geq 5\) represent weak, moderate and strong turbulence, respectively [43,44], \(C_{n}^{2}=2\times 10^{-14}\), \(C_{n}^{2}=6\times 10^{-14}\), \(C_{n}^{2}=2\times 10^{-13}\), and \(C_{n}^{2}=6\times 10^{-13}\) represent weak, moderate, moderate, and strong turbulence, respectively. In Figure 4, in the case of fixed transmitted power, when \(C_{n}^{2}\) is increased, the OP of the single-hop FSOC system is almost unchanged, while with the increase in \(C_{n}^{2}\), the OP of the multi-hop FSOC system increases. For example, when \(P_{t}=30\) dBm and \(N=1\), the OPs with \(C_{n}^{2}=6\times 10^{-14}\) (Moderate) and \(C_{n}^{2}=6\times 10^{-13}\)(Strong) are \(4.28\times 10^{-1}\) and \(4.37\times 10^{-1}\), respectively, and the system OP is almost unchanged. When \(P_{t}=30\) dBm and \(N=3\), the OPs with \(C_{n}^{2}=6\times 10^{-14}\) (Moderate) and \(C_{n}^{2}=6\times 10^{-13}\) (Moderate) are \(2.95\times 10^{-4}\) and \(7.20\times 10^{-4}\), respectively, and the system OP increases by about two times. The reason is that when the total link length is fixed, on a single-hop link, the optical power attenuation caused by fog is more serious than that caused by atmospheric turbulence. Therefore, increasing the atmospheric turbulence intensity has little effect on the system SNR, and the system OP is basically unchanged. Increasing the number of hops shortens the length of each hop link, which leads to a significant reduction in the optical power attenuation caused by fog at each hop link. As a result, the optical power attenuation caused by atmospheric turbulence at each hop link cannot be ignored. Increasing the atmospheric turbulence intensity leads to an increase in the optical power attenuation of each hop link, which reduces the SNR of each hop link and increases the OP of the \(N\)-hop FSOC system. In short, on a single-hop link, the atmospheric turbulence intensity has little effect on the system performance. When the number of hops is increased, the influence of atmospheric turbulence intensity on the multi-hop FSOC cannot be ignored. Increasing the atmospheric turbulence intensity reduces multi-hop FSOC system performance.