## 3 System Model
### Channel Model  
The transmit power and geometric gain of the respective nodes and links are denoted as \(P_{i}^{t}\) and \(G_{ij}\) (\(i\in\text{S},\text{UR}_{1},\text{UR}_{2},j\in\text{UR}_{1},\text{UR}_{2}, \text{D},i\
eq j\)). The RF signals received by \(\text{UR}_{1}\) and \(\text{UR}_{2}\) are given as follows:  
\[y_{\text{UR}_{1}}[n]= \sqrt{P_{\text{S}}^{t}G_{\text{S-UR}_{1}}}h_{\text{S-UR}_{1}}x_{ \text{S}}[n]+n_{\text{UR}_{1}}[n] \tag{1}\]  
and  
\[y_{\text{UR}_{2}}[n]= \sqrt{P_{\text{S}}^{t}G_{\text{S-UR}_{2}}}h_{\text{S-UR}_{2}}x_{ \text{S}}[n]+n_{\text{UR}_{2}}[n]. \tag{2}\]  
The signals received at D from the relays are described as follows:  
\[y_{\text{D}_{1}}[n]=\sqrt{P_{\text{UR}_{1}}^{t}G_{\text{UR}_{1} \text{-D}}}h_{\text{UR}_{1}\text{-D}}x_{\text{UR}_{1}}[n]+n_{\text{D}}[n] \tag{3}\]  
and  
\[y_{\text{D}_{2}}[n]=\sqrt{P_{\text{UR}_{2}}^{t}G_{\text{UR}_{2} \text{-D}}}h_{\text{UR}_{2}\text{-D}}x_{\text{UR}_{2}}[n]+n_{\text{D}}[n], \tag{4}\]  
where \(h_{ij}\) represents the complex channel gain, influenced by various factors such as multipath effects, weather conditions, and obstructions in air-to-ground communication. Noise is denoted by \(n_{j}\), assumed to be additive white Gaussian noise with zero mean. The assumption \(\mathbb{E}[|h_{ij}|^{2}]=1\) holds, assuming block-fading over one block duration.  
The geometric gain \(G_{ij}\) for link \(i\)-\(j\) is characterized by a two-ray transmission model [35], Section 2.4.1. The receive power \(P_{j}^{r}\) (\(i\in\text{S},\text{UR}_{1},\text{UR}_{2},j\in\text{UR}_{1},\text{UR}_{2}, \text{D}\)) is expressed as  
\[P_{j}^{r}= \left(\frac{\sqrt{M_{ij}}l_{i}l_{j}}{d_{ij}^{2}}\right)^{2}P_{i}^ {t}, \tag{5}\]  
where \(M_{ij}\) and \(d_{ij}\) denote the pattern of radiation and the corresponding link distance, respectively. The terms \(l_{i}\) and \(l_{j}\) refer to the heights of the source and destination.  
The average and instantaneous receive signal-to-noise power ratio (SNR) at \(\text{UR}_{1}\), \(\text{UR}_{2}\), and \(\text{D}\) are expressed as \(\overline{\gamma}_{ij}=P_{j}^{T}\frac{E_{s}}{N_{0}}\) and \(\gamma_{ij}=|h_{ij}|^{2}\overline{\gamma}_{ij}\) (\(i\in\{\text{S},\text{UR}_{1},\text{UR}_{2},\text{D}\},j\in\{\text{UR}_{1}, \text{UR}_{2},\text{D}\}\)), where \(E_{s}\) represents the symbol-level transmit power and \(N_{0}\) being noise variance. \(h_{ij}\) represents the complex channel gain. We assume that all the wireless channels experience Nakagami-m fading with the probability density function (PDF) of instantaneous SNR \(f_{\gamma}(\gamma_{ij})\) of link \(ij\), where \(\gamma_{ij}\), is given by  
\[f_{\gamma}(\gamma_{ij})=\frac{m^{m}}{\Gamma(m)\overline{\gamma}_{ij}^{m}} \gamma_{ij}^{m-1}\exp\left(-\frac{m\gamma_{ij}}{\overline{\gamma}_{ij}}\right), \tag{6}\]  
where  
* \(\gamma_{ij}\) is the amplitude of the received signal;
* \(m\) is the shape factor of the Nakagami-\(m\) distribution, typically \(m\geq 0.5\);
* \(\Gamma_{ij}\) is the average SNR of the link \(ij\);
* \(\Gamma(m)\) is the Gamma function, defined as \(\Gamma(m)=\int_{0}^{\infty}t^{m-1}e^{-t}\,dt\).  
The cumulative distribution function (CDF) of the Nakagami-m distribution is as follows:  
\[F_{\gamma}(\gamma_{ij})=\frac{\gamma\left(m,\frac{m\gamma_{ij}}{\overline{ \gamma}_{ij}}\right)}{\Gamma(m)}, \tag{7}\]  
where  
* \(\gamma\left(m,\frac{m\gamma_{ij}}{\overline{\gamma}_{ij}}\right)\) is the lower incomplete gamma function, defined as \(\gamma(a,x)=\int_{0}^{x}t^{a-1}e^{-t}\,dt\).  
All links are assumed to be mutually independent and identically distributed.