## 4 Performance Analysis
### B. Average Bit Error Rate  
ABER is a key indicator to evaluate the communication quality of FSOC system. The ABER of various binary and nonbinary modulations can be uniformly expressed as [29; 39]:  
\[P_{b}=\frac{\delta}{2\Gamma(p)}\sum\limits_{l=1}^{n}q_{l}^{p}\int_{0}^{\infty} \gamma^{p-1}e^{-q_{l}\gamma}F_{\gamma_{\text{ub}}}(\gamma)d\gamma \tag{38}\]  
where the values of \(\delta\), \(p\), \(q_{l}\), and \(n\) represent different modulation schemes. Table 1 shows the specific parameters under the two modulation types of OOK and PAM. In IM/DD FSOC links, OOK and PAM are two typical modulation methods, and both methods have been effectively used in field experiments [40; 41; 42]. Both OOK and PAM modulation types have their own advantages and disadvantages: OOK has the advantage of strong resistance to atmospheric turbulence, but the disadvantage is low spectral efficiency; M-PAM has the advantage of high spectral efficiency, and the theoretical maximum spectral efficiency of M-PAM is \(\log_{2}M\) times higher than that of OOK. However, the disadvantage of M-PAM is its weak antiturbulence ability. In summary, it is very meaningful to carry out ABER performance research under two modulation methods: OOK and PAM. Substitute Equation (36) into Equation (38). We obtain the expression of ABER for \(N\)-hop links:  
\[ \begin{array}{ll}P_{b}&=\frac{\delta}{2\Gamma(p)}\prod\limits_{i=1}^{N}\frac {\varepsilon_{\text{mod},i}^{2}z_{i}^{k_{i}}}{\Gamma(\alpha_{i})\Gamma(\beta_{i })}\sum\limits_{l=1}^{n}q_{l}^{p}\int_{0}^{\infty}\gamma^{p-s-1}e^{-q_{l} \gamma}d\gamma\\ &\times\frac{1}{2\pi l}\int\limits_{l}\prod\limits_{i=1}^{N}\left(\frac{ \varepsilon_{\text{mod},i}^{2}+\frac{\gamma_{i}}{N}s}{\Gamma\left(\alpha_{i} +\frac{2l}{N}s\right)\Gamma\left(\beta_{i}+\frac{2l}{N}s\right)\Gamma^{k_{i}} \left(z_{i}+\frac{2l}{N}s\right)\Gamma\left(-s\right)}{\prod\limits_{i=1}^{N} \Gamma\left(z_{\text{mod},i}^{2}+\frac{2l}{N}s\right)\Gamma^{k_{i}}\left(z_{i }+1+\frac{2l}{N}s\right)\Gamma\left(1-s\right)}\right.\\ &\left.\left(N\prod\limits_{i=1}^{N}\psi_{i}\left(\frac{a_{i}\beta_{i}}{A_{ \text{mod},i}\sqrt{\gamma_{i}}}\right)^{\frac{2l}{N}}\right)^{-s}ds\right.\end{array}  \tag{39}\]  
where the internal integral with respect to \(\gamma\) is:  
\[\int_{0}^{\infty}\gamma^{p-s-1}e^{-q_{l}\gamma}d\gamma=q_{l}^{-p+s}\Gamma(-s+p) \tag{40}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Modulation** & \(\delta\) & \(p\) & \(q_{l}\) & \(n\) \\ \hline OOK & 1 & 1/2 & 1/4 & 1 \\ M-PAM & 1 & 1/2 & \(\frac{\log_{2}M}{8(M-1)^{2}}\) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for different modulation types. Substituting the above equation into Equation (39). After simplification, we obtain:  
\[ \begin{array}{ll}P_{b}&=\frac{\delta}{2\Gamma(p)}\prod\limits_{i=1}^{N}\frac{ \varepsilon_{\text{mod},i}^{2}\varepsilon_{i}^{k_{i}}}{\Gamma(\alpha_{i}) \Gamma(\beta_{i})}\\ &\times\sum\limits_{l=1}^{n}\frac{1}{2\pi j}\int_{l}^{l}\frac{\sum\limits_{i=1 }^{N}\Gamma\left(\varepsilon_{\text{mod},i}^{2}+\frac{2l_{i}}{N}s\right) \Gamma\left(\alpha_{i}+\frac{2l_{i}}{N}s\right)\Gamma\left(\beta_{i}+\frac{2l_ {i}}{N}s\right)\Gamma^{k_{i}}\left(z_{i}+\frac{2l_{i}}{N}s\right)\Gamma(-s+p )}{\sum\limits_{i=1}^{N}\Gamma\left(\varepsilon_{\text{mod},i}^{2}+1+\frac{2 l_{i}}{N}s\right)\Gamma^{k_{i}}\left(z_{i}+1+\frac{2l_{i}}{N}s\right)\Gamma(1-s)}\\ &\left(\frac{N}{q_{l}}\prod\limits_{i=1}^{N}\psi_{i}\left(\frac{\alpha_{i}\beta _{i}}{A_{\text{mod},i}\sqrt{\gamma_{i}}}\right)^{\frac{2l_{i}}{N}}\right)^{- s}ds\end{array}