## 4 Performance Analysis
### C. Average Ergodic Channel Capacity
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 normalized jitter standard deviation. \(5.25\times 10^{-1}\), respectively, and the system OP is basically unchanged. When \(N=3\) and \(P_{t}=30\) dBm, the OPs with \(\mu_{H}/r=\mu_{V}/r=3\) and \(\mu_{H}/r=\mu_{V}/r=7\) are \(2.95\times 10^{-4}\) and \(3.1\times 10^{-3}\), respectively, and the system OP increases by about one order. The reason for this phenomenon is similar to Figure 5a, and we do not repeat the description here. In the multi-hop FSOC system, from the comparison of Figure 5a,b, the variation in the system OP caused by increasing the normalized jitter standard deviation is greater than that caused by increasing the normalized boresight error. Therefore, in the multi-hop FSOC system, compared with correcting the boresight error of each hop link, compensating for the platform jitter error of each hop link can significantly improve the system communication performance. Figure 6a demonstrates the OP versus normalized beamwidth for the \(N\)-hop FSOC system under LF and MF weather conditions, which considers CSI-assisted relays. The total link length is 1.2 km, and the transmitted optical power is 10 dBm. According to Figure 6a, the normalized beam width has an optimum value that minimizes the OP of an \(N\)-hop FSOC system. On one hand, in the case that the normalized beam width is lower than the critical value, an excessively narrow beam width leads to misalignment, which increases the OP of the \(N\)-hop FSOC system. On the other hand, above the critical value, the excessive beam width will reduce the optical power received by the optical antenna at each hop link, which reduces the SNR and increases the system OP. In Figure 6a, as the number of hops increases, the optimal beam width shows an increasing trend, and the optimal normalized beam width of LF is wider than that of MF. For example, under the LF condition, the optimal normalized beam widths for each hop link corresponding to \(N=3\) and \(N=5\) are 9.7 and 10.9, respectively. When \(N=5\), the optimal normalized beam widths for each hop link corresponding to LF and MF are 10.9 and 9.7, respectively. In the previous analysis, we have illustrated that as the number of hops increases, the impact of the pointing error on the system OP becomes severe. To minimize the system OP, a wider normalized beam width needs to be used to suppress pointing errors. Similarly, the received optical power per hop at the LF channel is higher than that at the MF channel. Therefore, compared with the MF channel, the pointing error at the LF channel has a more severe impact on the system OP. To minimize the system OP, a wider normalized beam width needs to be used in the LF channel to suppress pointing errors. Figure 6b demonstrates the OP versus normalized beamwidth for the \(N\)-hop FSOC system under different total link length, which considers CSI-assisted relays. As shown in Figure 6b, reducing the total link length widens the optimal normalized beam width. For example, when \(N=3\), the optimal normalized beam widths for each hop link corresponding to \(d=0.6\) km and \(d=1.8\) km are 11.6 and 8.9, respectively. The reason is similar to that mentioned above, and we do not repeat here. To sum up, increasing the number of hops or decreasing the total link length results in an increase in the optimal normalized beam width at per hop link. The optimal normalized beam width in LF is wider than that in MF. In an actual multi-hop FSOC system, it is very important to select an appropriate normalized beam width for different link states. Figure 5: Outage probability (OP) under light fog for \(N\)-hop FSOC system with different (**a**) normalized jitter standard deviation and (**b**) normalized boresight errors. Figure 7a demonstrates the effects of different total link length on the OP of \(N\)-hop FSOC system, which considers CSI-assisted relays and light fog weather.