## 7 Conclusions  
In this work, we studied the performance of an IM/DD multi-hop FSOC system with CSI-assisted relay or FG relay under a composite channel of Gamma-Gamma distribution atmospheric turbulence, Gamma distribution random fog, and Beckmann distribution generalized pointing errors. Specifically, we derive closed expressions for the PDF of the end-to-end SNR upper bound by using the \(I\)-function. Based on these formulas, the effects of system parameters on OP, ABER, and average ergodic channel capacity of multi-hop FSOC system are studied. The results show that, firstly, both CSI-assisted relay and FG relay can significantly improve multi-hop FSOC system performance. Under low transmitted power, FG relay outperforms CSI-assisted relay. Under high transmitted power, CSI-assisted relay outperforms FG relay. Second, compared to the LF channel, the MF channel significantly reduces the multi-hop FSOC system performance. Increasing the number of hops and transmitted optical power or reducing the total link length can significantly improve the multi-hop FSOC system performance. In addition, on a single-hop link, the influence of fog channel on system performance is dominant, while atmospheric turbulence intensity,  
Figure 10: Average ergodic channel capacity for \(N\)-hop FSOC system with different fog conditions.  
normalized jitter standard deviation and normalized boresight error have little effect on the system performance. With the increase in hops number, atmospheric turbulence intensity, normalized jitter standard deviation and normalized boresight error have serious effects on multi-hop FSOC systems. Compared with reducing the normalized boresight error, reducing the normalized jitter standard deviation greatly improves the multi-hop FSOC system performance. Therefore, in a multi-hop FSOC system, compared with correcting the boresight error of each hop link, compensating for the platform vibration error of each hop link can significantly improve the system performance. Finally, when the number of hops increases or the severity of random fog decreases or the transmission distance decreases, a wider beam width is required to minimize the OP of a multi-hop FSOC system. Therefore, in the actual system design, selecting the optimal beam width for each hop link can optimize multi-hop FSOC system performance. We also found that increasing the modulation order seriously deteriorates the ABER of the multi-hop FSOC system. Therefore, in a multi-hop FSOC system, a low-order modulation scheme should be selected to ensure the system communication performance. This work contributes to the system design and performance optimization of multi-hop FSOC systems under composite channels.  
[PERSON] and [PERSON] proposed the project. [PERSON] conducted the equation derivation and simulation and wrote the manuscript. [PERSON], [PERSON] and [PERSON] helped to modify the manuscript. [PERSON] helped to translate the paper and conduct relevant literature search. All authors have read and agreed to the published version of the manuscript.  
This work is financially supported by National Natural Science Foundation of China (62105029, U2141231); China Postdoctoral Science Foundation (2021M700415, 2021 TQ0035); State Key Laboratory of Applied Optics (SKLA02022001A11).  
Not applicable.  
Data presented in this paper is not publicly available, but may be obtained from the authors upon reasonable request.  
The authors declare no conflict of interest.