## 4 Performance Analysis
### C. Average Ergodic Channel Capacity
Increasing the total link length increases the system OP. For example, when \(N=3\), \(P_{t}=20\) dBm, the system OPs corresponding to \(d=1\) km and \(d=2\) km are \(3.14\times 10^{-5}\) and \(6.36\times 10^{-2}\), respectively. The reason is that increasing the total link length reduces the optical power received at the receiving end, which reduces the system SNR and increases the system OP. In addition, even if the total link length is 1 km, the OP of the single-hop link is still high, and multi-hop link can significantly reduce the OP. The reason is that foggy weather greatly attenuates the optical power of the link, which causes the single-hop link to be unable to communicate at a length of 1 km. Therefore, more relay nodes need to be set up between the source node and the target node to ensure communication quality. Figure 7b demonstrates the effects of different SNR threshold on the \(N\)-hop FSOC OP, which considers CSI-assisted relays and light fog weather. As the SNR threshold increases, the system OP in both the LF and MF channels decreases. For example, in the LF channel, when \(N=3\) and \(P_{t}=20\) dBm, the system OPs corresponding to \(\gamma_{\text{th}}=2\)dB and \(\gamma_{\text{th}}=10\)dB are \(3.6\times 10^{-3}\) and \(1.14\times 10^{-2}\), respectively. The reason is that as the SNR threshold increases, \(\Pr(\gamma_{\text{ub}}\leq\gamma_{\text{th}})\) will increase, so the OP will increase. To sum up, by reducing the total link length and SNR threshold, the stability of multi-hop FSOC systems can be improved. Figure 8 demonstrates the effects of different modulation orders on the ABER of \(N\)-hop FSOC system, which considers CSI-assisted relays and light fog weather. The system ABER increases as the modulation order increases. The reason is that as the modulation order increases, the voltage interval of the M-PAM signal becomes smaller and more susceptible to link noise. Therefore, as the modulation order increases, the end-to-end link ABER increases. Moreover, as the modulation order increases, the variation in the ABER in the single-hop FSOC system is smaller than that in the multi-hop FSOC system. For example, when \(N=1\) and \(P_{t}=30\) dBm, the system ABERs corresponding to OOK and 64 PAM are \(1.89\times 10^{-1}\) and \(2.97\times 10^{-1}\), respectively, and the system ABER of 64 PAM nearly doubles  
Figure 6: Outage probability (OP) under light fog for \(N\)-hop FSOC system with different (**a**) fog conditions and (**b**) total link length. Figure 7: Outage probability (OP) under light fog for \(N\)-hop FSOC system with different (**a**) total link length and (**b**) SNR threshold. that of OOK. When \(N=3\) and \(P_{t}=30\) dBm, the system ABERs corresponding to OOK and 64 PAM are \(7.6\times 10^{-5}\) and \(8.1\times 10^{-3}\), respectively, and the system ABER increases by about two orders. The reason is that increasing the number of hops significantly increases the SNR of the \(N\)-hop FSOC system. When the modulation order increases, the variation in SNR in the multi-hop FSOC system is larger than that in the single-hop FSOC system. This results in a larger ABER variation for multi-hop FSOC systems than for single-hop FSOC systems. Therefore, increasing the modulation order seriously deteriorates the multi-hop FSOC system performance. To sum up, in a multi-hop FSOC system, a low-order modulation scheme should be selected to ensure the system communication performance. In addition, we take the OOK modulation method as an example to analyze the relationship between ABER and average SNR of the multi-hop AF FSOC link. According to the instantaneous SNR expression of the \(i\)-th hop link shown in Equation (18): \(\gamma_{i}=\left|h_{i}\right|^{2}P_{t}^{2}/\sigma_{n}^{2}\), we can obtain the average SNR, \(u_{i}\), of the \(i\)-th hop link: [23; 45]  
\[u_{i}=E[\gamma_{i}]=\frac{P_{t}^{2}}{\sigma_{n}^{2}}E^{2}[h_{i}] \tag{50}\]  
where \(E[\cdot]\) represents expectation, \(E[h_{i}]=E\left[h_{fj}\right]E[h_{pj}]E[h_{a\downarrow}]\).