## 3 Performance Analysis
### Error Performance with Phase Noise Effects  
If we express the instantaneous loop SNR as a multiple \(\psi\) of the instantaneous SNR, i.e., \(v=\psi\,\gamma\) as suggested by [PERSON] et al. in [3], then under phase noise impairments, the total angular deviation, which is denoted by \(\zeta\)= \(\varphi+\theta\), where \(\theta\) represents the contribution introduced by the phase noise process, follows a zero-mean Gaussian PDF with total variance [5]:  
\[\sigma_{\zeta}^{2}=\sigma_{\varphi}^{2}+\sigma_{\theta}^{2}=\frac{1}{2\gamma}+ \frac{1}{v}=\frac{1}{2\gamma}\left(1+\frac{2}{\psi}\right). \tag{25}\]  
By considering also that \(\gamma=\mu_{NZB}I^{2}/A_{q}^{2}\), the SEP conditioned on \(I\) is obtained as [5]:  
\[p_{t}(I)\cong 1-\frac{1}{\sqrt{2\pi\sigma_{\zeta}^{2}}-\pi/M}\int\limits_{\zeta }^{\pi/M}\text{e}^{-\zeta^{2}/(2\sigma_{\zeta}^{2})}d\zeta=\text{erfc}\left( \frac{\eta_{M}I\sqrt{\mu_{NZB}}}{A_{q}}\right), \tag{26}\]where \(\eta_{M}=\pi\left(M\sqrt{1+2/\psi}\right)^{-1}\) is introduced for simplification reasons. Consequently, the corresponding ASEP is given by averaging Equation (26) with respect to the PDF of \(l\) through the following integral:  
\[p_{t}\cong\int\limits_{0}^{\infty}\text{erfc}\left(\frac{\eta_{M}l\sqrt{\eta_{ NZB}}}{A_{q}}\right)f_{l}(1)\,dI. \tag{27}\]  
By replacing the complementary error function in Equation (27) by its Meijer-G function equivalent [20], and the PDF of \(l\) by Equation (13) as well as considering the similarity between Equations (19) and (26), the ASEP is eventually derived as:  
\[p_{t}\cong\frac{\frac{\gamma^{2}A_{\text{(Nor\,R)}}}{2\pi^{3/2}}}{\sum\limits_{ \text{(Nor\,R)}}}\sum\limits_{\text{2}^{a+k-3}}a_{k\text{(Nor\,R)}}B_{\text{(Nor \,R)}}^{-\frac{a+k}{2}}\,G_{6,3}^{2.5}\left( \begin{array}{c}\frac{16\,\eta_{M }^{2}\eta_{NZB}(1+q^{-2})^{2}}{B_{\text{(Nor\,R)}}^{2}}&\frac{2-q^{2}}{2}, \frac{1-k}{2},\frac{2-k}{2},\frac{1}{2},\frac{2-k}{2},\,1\\ \frac{1}{2}&\frac{-q^{2}}{2}\end{array} \right). \tag{28}\]  
Similarly, the corresponding ASEP of the multi-hop FSO system under consideration is given by [24]:  
\[ \begin{array}{l}p_{t,tot}&\cong\sum\limits_{i=1}^{N}\left[p_{t,i},\prod \limits_{j=i+1}^{N}\left(1-2p_{t,j}\right)\right]\\ &=\frac{1}{2\pi^{3/2}}\sum\limits_{i=1}^{N}\left\{\Psi_{i}\,\psi_{i}(\mu_{i, NZB})\prod\limits_{j=i+1}^{N}\left[1-\frac{1}{\pi^{3/2}}\Psi_{j}\,\psi_{j}(\mu_{j, NZB})\right]\,\right\},\end{array}  \tag{29}\]  
where:  
\[\Psi_{i}=q_{i}^{2}A_{i\text{(Nor\,R)}}\sum\limits_{\text{(Nor\,R)}}2^{a_{i}+k _{i}-3}a_{k,i\text{(Nor\,R)}}B_{i\text{(Nor\,R)}}^{-\frac{a_{i}+k_{i}}{2}} \tag{30}\]  
and:  
\[\psi_{i}(\mu_{i,NZB})=G_{6,3}^{2.5}\left(\frac{16\,\eta_{i,}^{2}\,M\mu_{i, NZB}(1+q_{i}^{-2})^{2}}{\left(c_{i}+\Omega_{i}\right)^{2}B_{i\text{(Nor\,R)}}^{2}} \right|\,\,\frac{2-q_{i}^{2}}{2},\frac{1-a_{i}}{2},\frac{2-a_{i}}{2},\frac{1- k_{i}}{2},\frac{2-k_{i}}{2},\,1\\ \frac{0}{2},\frac{1}{2},-\frac{q_{i}^{2}}{2}\end{array}\right). \tag{31}\]  
Additionally, when \(p_{t,i}=p_{t,i}\), \(i\
eq j\), which means that \(\psi_{1}(\mu_{1,NZB})=\psi_{2}(\mu_{2,NZB})=\ldots=\psi_{i}(\mu_{i,NZB})\) = \(\psi(\mu_{NZB})\) and \(\Psi_{1}=\Psi_{2}=\ldots=\Psi_{i}=\Psi\), the mathematical expression of Equation (29) takes the following form:  
\[p_{t,tot}\cong\frac{\Psi}{2\pi^{3/2}}\,\psi(\mu_{NZB})\sum\limits_{i=1}^{N} \prod\limits_{j=i+1}^{N}\left[1-\frac{\Psi}{\pi^{3/2}}\,\psi(\mu_{NZB})\right]= \frac{1}{2}\left[1-\left(1-\frac{\Psi}{\pi^{3/2}}\,\psi(\mu_{NZB})\right)^{N}\right] \tag{32}\]  
Finally, by ignoring the presence of pointing errors, but taking account of the phase noise effect, it can be shown that the ASEP for the single-hop FSO configuration reduces to:  
\[p_{t,\rho}\cong\frac{A_{\text{(Nor\,R)}}}{2\pi^{3/2}}\sum\limits_{\text{(Nor \,R)}}2^{a+k-2}a_{k\text{(Nor\,R)}}B_{\text{(Nor\,R)}}^{-\frac{a_{k}+k}{2}}\,G_ {5,2}^{2.4}\left( \begin{array}{c}\frac{16\,\eta_{M}^{2}\eta_{NZB}}{B_{\text{(Nor \,R)}}^{2}}&\frac{1-a}{2},\frac{2-a}{2},\frac{1-k}{2},\frac{2-k}{2},\,1\\ 0,\frac{1}{2}\end{array}