## 2 Materials and Methods
### B. Channel Model
Therefore, the PDF of the Beckmann distribution generalized pointing error in the \(i\)-th hop link is as follows:  
\[f_{h_{p,i}}(h_{p,i})=\frac{\varepsilon_{\text{mod},i}^{2}}{A_{\text{mod},i}^{ \varepsilon_{\text{mod},i}^{2}}}h_{p_{i}}^{2\text{mod},i},0\leq h_{p,i}\leq A _{\text{mod},i}, \tag{8}\]  
where \(\varepsilon_{\text{mod},i}=\omega_{zeq,i}/2\sigma_{\text{mod},i}\) represents the ratio of the equivalent beam radius at the \(i\)-th receiving node to the modified Rayleigh distribution parameter. \(\omega_{zeq,i}^{2}=\omega_{d_{i}}^{2}\sqrt{\pi erf}(v_{i})/\left(2v_{i}\exp(- v_{i}^{2})\right)\), with \(v_{i}=\sqrt{\pi/2}(r_{i}/\omega_{d_{i}})\). \(erf(\bullet)\) represents the error function. \(r_{i}\) and \(\omega_{d_{i}}\) represent the receiving aperture radius and beam width at the \(i\)-th node, respectively. The modified Rayleigh distribution parameter, \(\sigma_{\text{mod},i}\), is as follows:  
\[\sigma_{\text{mod},i}=\left(\frac{3\mu_{H,i}^{2}\sigma_{H,i}^{4}+3\mu_{V,i}^{2 }\sigma_{V,i}^{4}+\sigma_{H,i}^{6}+\sigma_{V,i}^{6}}{2}\right)^{\frac{1}{6}} \tag{9}\]  
where \(\mu_{H,i}^{2}\) and \(\mu_{V,i}^{2}\) represent the boresight variance of the horizontal and vertical displacements at the \(i\)-th receiving node, respectively. \(\sigma_{H,i}^{2}\) and \(\sigma_{V,i}^{2}\) represent the jitter variance of the horizontal and vertical displacements at the \(i\)-th receiving node, respectively. Another parameter \(A_{\text{mod},i}\) in Formula (8) can be expressed as:  
\[A_{\text{mod},i}=A_{0,i}\exp\left(\frac{1}{\varepsilon_{\text{mod},i}^{2}}-\frac{ 1}{2\varepsilon_{H,i}^{2}}-\frac{1}{2\varepsilon_{V,i}^{2}}-\frac{1}{2 \varepsilon_{H,i}^{2}\sigma_{H,i}^{2}}-\frac{1}{2\varepsilon_{V,i}^{2}\sigma_ {V,i}^{2}}\right) \tag{10}\]  
where \(A_{0,i}=\left[erf(v_{i})\right]^{2}\) represents the maximum power fraction collected at the \(i\)-th node. \(\varepsilon_{H,i}=\omega_{2 eq,i}/2\sigma_{H,i}\) and \(\varepsilon_{V,i}=\omega_{2 eq,i}/2\sigma_{V,i}\) represent the ratio of the equivalent beam radius to the jitter standard deviation of the horizontal or vertical direction at the \(i\)-th receiving node, respectively. (4) Composite Channel Statistical Model  
The product of atmospheric turbulence and generalized pointing error in the \(i\)-th hop link is defined as \(h_{\text{ap},i}=h_{a,i}h_{p,i}\), and its PDF is as follows [33]:  
\[f_{h_{\text{ap},i}}\big{(}h_{\text{ap},i}\big{)}=\frac{\alpha_{i}\beta_{i} \varepsilon_{\text{mod},i}^{2}}{A_{\text{mod},i}\Gamma(\alpha_{i})\Gamma( \beta_{i})}\times G_{1,3}^{3,0}\Bigg{[}\left.\frac{\alpha_{i}\beta_{i}}{A_{ \text{mod},i}}h_{\text{ap},i}\right|\varepsilon_{\text{mod},i}^{2}-1,\alpha_{ i}-1,\beta_{i}-1\right] \tag{11}\]  
where \(G_{p,q}^{m,n}[\bullet]\) is the Meijer-G function. Applying the theory of product distribution, the PDF of \(h_{i}=h_{\text{ap},i}h_{f,i}\) is as follows:  
\[f_{h_{f,i}}(h_{f,i})=\frac{z_{i}^{k_{i}}}{\Gamma(k_{i})}\left[\ln\left(\frac{1 }{h_{f,i}}\right)\right]^{k_{i}-1}h_{f,i}^{z_{i}-1},0<h_{f,i}<1, \tag{4}\]0  
Putting Equations (4) and (11) into Equation (12), we obtain:  
\[ \begin{array}{ll}f_{h_{i}}(h_{i})&=\frac{\alpha_{i}\beta_{i} \varepsilon_{\text{mod},i}^{2}\varepsilon_{i}^{k_{i}}h_{i}^{z_{i}-1}}{A_{\text {mod},i}\Gamma(\alpha_{i})\Gamma(\beta_{i})\Gamma(k_{i})}\\ &\int_{h_{i}}^{\infty}\Big{[}\ln\Big{(}\frac{h_{\text{ap},i}}{h_{i}}\Big{)} \Big{]}^{k_{i}-1}h_{\text{ap},i}^{-z_{i}}G_{1,3}^{3,0}\Bigg{[}\left.\frac{ \alpha_{i}\beta_{i}}{A_{\text{mod},i}}h_{\text{ap},i}\right|\begin{array}{c} \varepsilon_{\text{mod},i}^{2}\\ \varepsilon_{\text{mod},i}^{2}-1,\alpha_{i}-1,\beta_{i}-1\end{array}\Bigg{]}dh _{\text{ap},i}\end{array}  \tag{13}\]  
According to the definition of the Meijer-G, the above formula is expanded as:  
\[