## 3 Performance Analysis: Ergodic Capacity  
Based on [17], we can express the ergodic capacity \(\overline{C}\) as  
\[\overline{C}=\mathbb{E}\big{[}\log_{2}(1+\gamma^{\prime})\big{]}, \tag{7}\]  
where \(\gamma^{\prime}=\frac{(\eta I^{\prime})^{r}}{N_{0}}\) is the instantaneous SNR and \(r\) is \(r=1\) for HD and \(r=2\) for IM/DD detection techniques commonly used in wireless optical communication systems. The average electrical SNR \(\mu^{\prime}\) can be expressed as  
\[\mu^{\prime}=\mathbb{E}\big{[}\gamma^{\prime}\big{]}=\frac{\eta^{\prime}( \mathbb{E}[I^{\prime}])^{r}}{N_{0}}. \tag{8}\]  
Since \(I_{l}\) is deterministic and \(I_{a}\) and \(I_{p}\) are independent random variables, the power of expectation of \(I^{\prime}\) can be obtained as \((\mathbb{E}[I^{\prime}])^{r}=\big{(}\mathbb{E}\big{[}I_{l}a_{I}p\big{]}\big{)} ^{r}=I_{l}^{r}(\mathbb{E}[I_{a}])^{r}(\mathbb{E}\big{[}I_{p}\big{]})^{r}\). With \((\mathbb{E}[I_{a}])^{r}=\big{(}\frac{z}{z+1}\big{)}^{kr}\) and \(\big{(}\mathbb{E}\big{[}I_{p}\big{]}\big{)}^{r}=\big{(}\frac{\xi^{2}A_{0}}{ \xi^{2}+1}\big{)}^{r}\), derived in the Appendices A-D, the resulting average electrical SNR can be written as  
\[\mu^{\prime}=\frac{\Big{(}\eta I_{l}\frac{A_{0}\xi^{2}}{1+\xi^{2}}\big{(} \frac{z}{z+1}\big{)}^{k}\Big{)}^{r}}{N_{0}}. \tag{9}\]  
For \(\xi^{2}\gg 1\), it can be assumed that the pointing error is negligible. Therefore, this can be simplified as  
\[\mu^{\prime}=\frac{\Big{(}\eta I_{l}A_{0}\big{(}\frac{z}{z+1}\big{)}^{k}\Big{)} ^{r}}{N_{0}}. \tag{10}\]  
In our case, it is impossible or extremely difficult to obtain the ergodic capacity in a closed-form expression. Therefore, the ergodic capacity can be analyzed using the moment expressions in the following section. Based on [17], asymptotic ergodic capacity can be written as  
\[\overline{C}\simeq\frac{1}{\log 2}\bigg{(}\frac{\partial}{\partial n}\mathbb{E} \Big{[}\gamma^{\prime n}\Big{]}\bigg{|}_{n=0}\bigg{)}, \tag{11}\]  
where, with the definition of the average electrical SNR, the \(n\)th moment of \(\gamma^{\prime}\) in (11) can be obtained using irradiance as [9]  
\[\mathbb{E}\Big{[}\gamma^{\prime n}\Big{]}=\bigg{(}\frac{\eta^{\prime}}{N_{0} }\bigg{)}^{n}\mathbb{E}\Big{[}I^{\prime m}\Big{]}=\frac{\mathbb{E}\big{[}I^{ \prime m}\big{]}}{(\mathbb{E}[I^{\prime}])^{m}}\mu^{\prime n}=\frac{\mathbb{E }\big{[}I_{a}^{m}\big{]}\mathbb{E}\big{[}I_{p}^{m}\big{]}}{\big{(}\mathbb{E}[ I_{a}]\big{)}^{m}\big{(}\mathbb{E}[I_{p}]\big{)}^{m}}\mu^{\prime n}. \tag{12}\]  
Subsequently, by applying the quotient rule for differentiation, the derivative of (12) can be obtained as given in (13). As a result, after some manipulation, we can obtain asymptotic ergodic capacity in a closed-form expression, as given in (14). \[ \begin{split}&\frac{\partial}{\partial n}\bigg{[}\frac{\mathbb{E}[I _{a}^{m}]\mathbb{E}\big{[}I_{p}^{m}\big{]}}{\big{(}\mathbb{E}[I_{a}]\big{)}^{ m}\big{(}\mathbb{E}[I_{p}]\big{)}^{m}}\mu^{\prime n}\bigg{]}\bigg{|}_{n=0}\\ &=\frac{\Big{(}\frac{\partial}{\partial n}\mathbb{E}[I_{a}^{m}] \mathbb{E}\big{[}I_{p}^{m}\big{]}\Big{)}(\mathbb{E}[I_{a}])^{m}(\mathbb{E}[ I_{p}])^{m}-\Big{(}\frac{\partial}{\partial n}(\mathbb{E}[I_{a}])^{m}(\mathbb{E}[ I_{p}])^{m}\Big{)}\mathbb{E}[I_{a}^{m}]\mathbb{E}\big{[}I_{p}^{m}\Big{]}}{ \Big{(}(\mathbb{E}[I_{a}])^{m}(\mathbb{E}[I_{p}])^{m}\Big{)}^{2}}\mu^{\prime n }\bigg{|}_{n=0}+\Big{(}\frac{\partial}{\partial n}\mu^{\prime n}\Big{)}\frac{ \mathbb{E}[I_{a}^{m}]\mathbb{E}\big{[}I_{p}^{m}\big{]}}{(\mathbb{E}[I_{a}])^{ m}(\mathbb{E}[I_{p}])^{m}}\bigg{|}_{n=0},\end{split}