## 2 System Model
### Channel Model  
During propagation, the wireless signal is normally influenced by path loss, shadows, and multipath fading. The path loss is the mean of the wireless signal transmission attenuation in large scale fading (usually across tens to hundreds of wavelengths). In the HAPs communication scenario, it is generally believed that the Rician fading characteristic is appropriate, which includes the LOS and NLOS components [12]. The LOS component is modeled as a shadowed free space propagation loss. If \(\alpha^{2}\) and \(2\sigma^{2}\) denote the power of the LOS component and NLOS component respectively, the normal PDF of the signal can be represented as given in [25] by:  
\[f\left(s|\alpha,\sigma\right)=\frac{s}{\sigma^{2}}\exp\left\{-\frac{s^{2}+ \alpha^{2}}{2\sigma^{2}}\right\}K_{0}\left(\frac{s\alpha}{\sigma^{2}}\right),\quad s\geq 0 \tag{5}\]  
where \(K_{0}(\cdot)\) denotes the zero order modified Bessel function of the first kind. However, the LOS component is frequently shadowed by trees, buildings, and other obstacles, therefore if we further take the shadows into consideration, the LOS component in Equation (5) obeys a lognormal distribution, and its distribution function is:  
\[f_{L_{s}}(l)=\frac{1}{\sqrt{2\pi}\delta l}\exp(-\frac{\left(\ln l-\ln L_{0} \right)^{2}}{2\delta^{2}}) \tag{6}\]  
where \(L_{s}\) denotes the shadowed LOS, \(\delta\) is the severity of the shadow fading, typically in the range 1.5-7 dB and \(L_{0}\) denotes the average path loss. A larger value of \(\delta\) means more serious shadows in the LOS signal. In the proposed model, a typical value of \(L_{0}\) is 129.1 dB when the HAPs are at an altitude of 22 km. For mathematical analysis convenience, Equation (6) can be considered to be equivalent to the Gamma distribution given by [26]:  
\[f_{L_{s}}(l)=\frac{l^{m_{s}-1}}{\Gamma(m_{s})(\Omega_{s}/m_{s})^{m_{s}}}\exp(-l/( \Omega_{s}/m_{s})) \tag{7}\]  
where \(\Gamma(m_{s})\) is the Gamma function, \(m_{s}=1/\exp(\delta^{2}-1)\), and \(\Omega_{s}=L_{0}\exp(\delta^{2}/2)\). The NLOS component refers to the rapid fluctuation of the received signal amplitude based on the shadow slow fading. Generally, the Nakagami distribution is used, and its distribution function is given by:  
\[f_{L_{m}}(l)=\frac{1}{\Gamma(m)}\bigg{(}\frac{m}{\bar{L}_{s}}\bigg{)}^{m}l^{m-1 }\exp(-\frac{m}{L_{s}}l) \tag{8}\]  
where \(L_{m}\) denotes the loss of multipath fading (NLOS) and \(m\geq 0\) is the fading factor. Typically \(L_{m}\) is in the rang 7-12 dB and a larger \(L_{m}\) means more serious NLOS signal loss. A larger \(m\) means better channel conditions, and it is typically in the rang 1.2-10 dB. \(\bar{L}_{s}=\mathrm{E}(r^{2})\) represents the average power of multipath fading, which obeys the Gamma distribution. Therefore, the comprehensive channel fading model \(L_{f}\) can be represented as given in [25] by:  
\[f_{L_{f}}(l)=\int_{0}^{\infty}f_{L_{m}}(l|x)f_{L_{s}}(x|L_{0})dx=\frac{2}{ \Gamma(m)\Gamma(m_{s})}\Big{(}\frac{mm_{s}}{(4\pi d/\lambda)^{2}}\Big{)}^{ \frac{m+ms}{2}}l^{\frac{m+ms-2}{2}}K_{m_{s}-m}\left(2\sqrt{\frac{mm_{s}}{(4\pi d /\lambda)^{2}}}l\right) \tag{9}\]  
where \(K_{m_{s}-m}(\cdot)\) denotes the \(m_{s}-m\) order modified Bessel function of the second kind. Due to the far distance between HAPs, we assume that each sub-channel obeys the same distribution given in Equation (9). Therefore, the channel model for the multiple HAP V-MIMO network is given by:  
\[\mathbf{H}(f)=\sqrt{\frac{\kappa}{\kappa+1}}\mathbf{H}_{LOS}(f)+\sqrt{\frac{ 1}{\kappa+1}}\mathbf{H}_{NLOS}(f) \tag{10}\]  
where \(\kappa=\alpha^{2}/\left(2\sigma^{2}\right)\) is the Rician factor, \(\mathbf{H}_{LOS}\left(f\right)\) represents the channel matrix of the shadowed LOS path, and \(\mathbf{H}_{NLOS}\left(f\right)\) denotes the scattering and NLOS path. Note that the channel model is remarkably different from that in [16; 17; 18], in which the system model is a true MIMO-HAP architecture.