## 2 System and Channel Models
### RF Link Statistical Distributions  
It is assumed that the RF link between the HAP and terrestrial users link follows Shadowed-Rician fading distribution and the PDF of instantaneous SNR \(\gamma_{2}\) can be expressed as [17; 25; 37]:  
\[f_{\gamma_{2}}(\gamma_{2})=\Omega exp(-a\gamma_{2})_{1}F_{1}(m_{h};1;\delta \gamma_{2}),\ \gamma_{2}>0 \tag{4}\]  
where \(\Omega=1/2\Big{(}\frac{2 bm_{h}}{2 bm_{h}+\Omega}\Big{)}^{m_{h}}\), \(a=1/2b\) and \(\delta=\Omega/2b(2 bm_{h}+\Omega)\), with \(\Omega\), \(m_{h}\), and \(2b\) being the average power of LOS, Nakagami fading severity parameter with \(0<m_{h}<\infty\), and the multipath component, respectively, and \(\ {}_{1}F_{1}(x;y;z)\) denotes the confluent hypergeometric function. By using the identity detail in Reference [38], the confluent hypergeometric function in (4) can be expressed as:  
\[{}_{1}F_{1}(m_{h};1;\delta\gamma_{2})=exp(\delta\gamma_{2})\sum_{q=0}^{m_{h}-1 }\frac{(1-m_{h})_{q}(-\delta\gamma_{SR})^{q}}{(q!)^{2}}, \tag{5}\]  
where \((.)_{q}\) denotes the Pochhammer symbol.  
By invoking (5) into (4), the Shadowed-Rician distribution of the RF link can be further simplified as:  
\[f_{\gamma_{2}}(\gamma_{2})=\sum_{q=0}^{m_{h}-1}\frac{(\Omega 1-m_{h})_{p}(-\delta)^{q}}{(p!)^{2}(\overline{\gamma}_{1})^{q+1}}\gamma_{2}^{ q}\exp(-\Delta_{H}\gamma_{SR}), \tag{6}\]  
where \(\Delta_{H}=(a-\delta)/\overline{\gamma}_{2}\).  
Since the sum of SNR at the terrestrial users can be expressed as \(\gamma_{2}=\sum_{q=1}^{N_{u}}\abs{h_{2,q}}^{2}\), then, by following the same approach detailed in Reference [17], the PDF of instantaneous SNR \(\gamma_{2}\) Shadowed-Rician distribution can be expressed as:  
\[f_{\gamma_{2}}(\gamma_{2})=\sum_{q=1}^{m_{h}-1}\cdots\sum_{q_{N_{u}}=0}^{m_{h} -1}\Xi(N_{u})\gamma_{2}^{\lambda-1}\exp(-\Delta_{H}\gamma_{2}), \tag{7}\]where  
\[\left\{\begin{array}{c}\Xi(N_{u})=\prod_{j=1}^{N_{u}}\rho\big{(}q_{j}\big{)} \Omega^{N_{u}}\prod_{l=1}^{N_{u}-1}B(\sum_{k=1}^{\varepsilon}q_{k}+\varepsilon, q_{\varepsilon+1}+1)\\ \rho\big{(}q_{j}\big{)}=\frac{(1-m_{h})_{q_{j}}(-\delta)^{j}}{(q_{j}!)^{2}( \gamma_{2})^{\delta^{j+1}}}\\ \lambda=\sum_{i=1}^{N_{u}}q_{j}+N_{u}\end{array},\right. \tag{8}\]  
with \(B(.,.)\) denoting Beta function.  
By integration (7) using the integral identity detailed in Reference [38] (Equation (3.351(1), the CDF of the link can be obtained as:  
\[F_{\gamma_{2}}(\gamma_{2})=\sum_{q_{1}=0}^{m_{h}-1}\ldots\sum_{q_{N_{u}}=0}^{ m_{h}-1}\Xi(N_{u})\Delta_{H}^{-\lambda}\gamma(\lambda,\Delta_{H}\gamma_{2}), \tag{9}\]  
where \(\gamma(.,.)\) is the lower incomplete Gamma function.  
By converting the incomplete Gamma function to Meijer-G function using the identity detailed in Reference [39] (Equation (8.4.16(1))), then, (9) can be further expressed as:  
\[F_{\gamma_{2}}(\gamma_{2})=\sum_{q_{1}=0}^{m_{h}-1}\ldots\sum_{q_{N_{u}}=0}^{ m_{h}-1}\Xi(N_{u})\Delta_{H}^{-\lambda}G_{1,2}^{1,1}\bigg{(}\Delta_{H}\gamma_{2} \bigg{|}\begin{array}{c}1\\ \lambda,\end{array}\bigg{)}. \tag{10}\]  
Owing to broadcast nature of RF link, the eavesdropper attempts to intercept the secret information of the ground users from the HAP relay node. As a result of this, the eavesdropper link is assumed to follow Rican distribution with the PDF of instantaneous SNR \(\gamma_{e}\) defined as [40; 41]:  
\[f_{e}(\gamma_{e})=\frac{(K+1)\exp(-K)}{\overline{\gamma}_{e}}exp\left(-\frac{ K+1}{\overline{\gamma}_{e}}\right)I_{o}\left(2\sqrt{\frac{K(K+1)}{\overline{ \gamma}_{e}}}\gamma_{e}\right), \tag{11}\]  
where \(K\) is the Rician fading factor which is defined as the ratio of the power of the line-of-sight component to the scattered components, and \(I_{o}(.)\) denotes the zero-order modified Bessel function of the first kind. By using the identity detailed in Reference [38], \(I_{o}=\sum_{l=0}^{\infty}x^{l}/2^{2l}(l!)^{2}\),  
\[f_{e}(\gamma_{e})=\eta exp(-K)\sum_{l=0}^{\infty}\frac{(Kq)^{l}}{\left(l! \right)^{2}}\gamma_{e}^{\,l}\exp(-\eta\gamma_{e}), \tag{12}\]  
where \(\eta=\frac{K+1}{\overline{\gamma}_{e}}\).