## 4 Performance Analysis
### C. Average Ergodic Channel Capacity  
The average ergodic channel capacity is used to describe the maximum amount of information that the channel can transmit, which is defined as the highest transmission rate when the bit error rate tends to zero. The average ergodic channel capacity of an IM/DD system, \(C_{\text{erg}}\), is expressed as [35]:  
\[C_{\text{erg}}=\frac{1}{\ln(2)}\int_{0}^{\infty}\ln(1+\theta\gamma)f_{\gamma}( \gamma)d\gamma \tag{44}\]  
where \(\theta=e/2\pi\). Taking Equation (32) into Equation (44) with \(\ln(1+\theta\gamma)=G_{2,2}^{1,2}\bigg{[}\theta\gamma\big{|} \begin{matrix}1,1 \\ 1,0\end{matrix} \bigg{]}\). We can obtain the average ergodic channel capacity of \(N\)-hop links:  
\[ \begin{array}{ll}C_{\text{erg}}&=\frac{1}{\ln(2)}\sum\limits_{i=1}^{N}\frac{ \varepsilon_{\text{mod},i}^{2}\varepsilon_{i}^{k_{i}}}{\Gamma(\alpha_{i}) \Gamma(\beta_{i})}\int_{0}^{\infty}\gamma^{-s-1}G_{2,2}^{1,2}\bigg{[}\theta \gamma\big{|}\begin{matrix}1,1\\ 1,0\end{matrix}\bigg{]}d\gamma\\ &\times\frac{1}{2\pi j}\int_{l}^{l}\frac{\sum\limits_{i=1}^{N}\Gamma\left( \varepsilon_{\text{mod},i}^{2}+\frac{2l_{i}}{N}s\right)\Gamma\left(\alpha_{i} +\frac{2l_{i}}{N}s\right)\Gamma\left(\beta_{i}+\frac{2l_{i}}{N}s\right)\Gamma ^{k_{i}}\left(z_{i}+\frac{2l_{i}}{N}s\right)}{\sum\limits_{i=1}^{N}\Gamma \left(\varepsilon_{\text{mod},i}^{2}+1+\frac{2l_{i}}{N}s\right)\Gamma^{k_{i}} \left(z_{i}+1+\frac{2l_{i}}{N}s\right)}\\ &\left(N\prod\limits_{i=1}^{N}\psi_{i}\left(\frac{\alpha_{i}\beta_{i}}{A_{\text {mod},i}\sqrt{\gamma_{i}}}\right)^{\frac{2l_{i}}{N}}\right)^{-s}ds\end{array}  \tag{45}\]  
We use the identity [38] (Equation (2.24.2.1)). Then the internal integral with respect to \(\gamma\) is expressed as:  
\[\int_{0}^{\infty}\gamma^{-s-1}G_{2,2}^{1,2}\bigg{[}\theta\gamma\big{|} \begin{matrix}1, 1\\ 1,0\end{matrix} \bigg{]}d\gamma=\frac{\Gamma(-s+1)\Gamma^{2}(s)}{\Gamma(s+1)} \theta^{s} \tag{46}\]  
For example, when \(N=1\), the OPs of \(P_{t}=10\) dBm and \(P_{t}=30\) dBm are 0.73 and 0.43, respectively. For the CSI-assisted relay system, when \(P_{t}=30\) dBm, the OPs of \(N=2\) and \(N=3\) are \(4.32\times 10^{-2}\) and \(2.95\times 10^{-4}\), respectively. The reason is that increasing the transmitted power means increasing the SNR of each hop link, which reduces the system OP. Since the total link length is fixed, increasing the number of hops shortens the distance between two adjacent nodes, which greatly reduces the probability that the channel of each hop link is in deep fading. Therefore, the SNR of each hop link is significantly increased, and the OP of the system is significantly reduced. According to Figure 2, when the number of hops is fixed, under low transmitted power, the OP of the FG relay system is lower than that of the CSI-assisted relay system. However, under high transmitted power, the OP of the CSI-assisted relay system is lower than that of the FG relay system. For example, when \(N=3\) and \(P_{t}=10\) dBm, the OPs of the CSI-assisted relay system and the FG relay system are 0.1086 and \(8.9\times 10^{-2}\), respectively. When \(N=3\) and \(P_{t}=30\) dBm, the OPs of the CSI-assisted relay system and the FG relay system are \(2.95\times 10^{-4}\) and \(5.27\times 10^{-4}\), respectively. The reason is that CSI-assisted relay assumes that the first transmitter sends a reference signal with known characteristics, and the receiver at the next relay point can determine the distortion caused by composite channel fading. In principle, it is possible to correct such distortion, but the CSI-assisted relay is rarely perfect because the received signal is polluted by noise. This implies that CSI-assisted relay works best when the transmitted power is very high (high SNR), the algorithms used to determine the CSI of link may \"ignore\" noise. However, they give incorrect results when the transmitted power is very low (low SNR). Consequently, FG relay links can outperform CSI-assisted relay links when the transmitted power is low.