# Maximization of Average Achievable Rate for NOMA-UAV Dual-User Communication System Assisted by RIS
## 2 System Model
These signals will experience attenuation or interference during propagation. A commonly used model to describe this phenomenon is Rayleigh fading. The values of \(h_{ug_{2}}[k]\) and \(h_{ug_{1}}[k]\) in the \(k\)th time slot can get  
\[h_{ug_{1}}[k]=\sqrt{c_{0}d_{ug_{1}}^{-\alpha}[k]\tilde{h}_{1}}, \tag{2}\]  
\[h_{ug_{2}}[k]=\sqrt{c_{0}d_{ug_{2}}^{-\alpha}[k]\tilde{h}_{2}}, \tag{3}\]  
\(d_{ug_{1}}[k]=\sqrt{Z_{UAV}^{2}+\left\|q[k]-L_{\text{G1}}\right\|^{2}}\) and \(d_{ug_{2}}[k]=\sqrt{Z_{UAV}^{2}+\left\|q[k]-L_{\text{G2}}\right\|^{2}}\) are the distance from UAV to User 1 and User 2 respectively. The \(c_{0}\) is the path loss, and \(\alpha\) is the path loss index of UAV-user. \(\tilde{h}_{1}\sim\mathbb{CN}(0,1)\) and \(\tilde{h}_{2}\sim\mathbb{CN}(0,1)\). Assuming that the channel between  
Figure 1: The RIS-assisted NOMA-UAV system model. UAV and RIS is a LoS channel and is set to \(h_{ur}\), the value of \(h_{ur}[k]\) at the \(k\)th time slot can obtain  
\[ \begin{split} h_{ur}[k]=\sqrt{c_{0}d_{ur}^{-\tau}[k]}\{1,...,e^{-j \frac{2\pi}{\lambda}d_{ur}(h^{(m,n)}[k]}\\...,e^{-j\frac{2\pi}{\lambda}d_{ur}(h^{(m-k,\lambda t-1)}[k]} \}^{T},\end{split}  \tag{4}\]  
the \(d\) represents the interval between the antennas, \(\lambda\) represents the wavelength, \(\kappa\) is the signal loss index of the UAV-RIS, and \(d_{ur}[k]=\sqrt{(Z_{\lambda UAV}-Z_{RIS})^{2}+\left\|q[k]-L_{R}\right\|^{2}}\) is the distance of the UAV-RIS in the time slot. \(\chi_{ur}^{(m,n)}[k]=m\sin\phi_{ur}[k]\cos\phi_{ur}[k]+n\sin\phi_{ur}[k]\), \(\sin\phi_{ur}[k]\cos\phi_{ur}[k]=\frac{x_{R}-x_{M}[k]}{d_{ur}[k]}\), \(\sin\phi_{ur}[k]=\frac{Z_{\lambda UAV}-Z_{RIS}}{d_{ur}[k]}\), \(\phi_{ur}[k]\) and \(\phi_{ur}[k]\) respectively denote the azimuth and pitch angle of the signal from UAV to RIS within time slot \(k\). We define the distance between the user and the RIS as the distance from the user to the first array element of the RIS [25]. The channel between RIS users is modeled using Rician fading effect. Unlike Rayleigh fading, the Rician model accounts for the reflected signals from the RIS, denoted as \(h_{rg_{1}}\) and \(h_{rg_{2}}\). Then \(h_{rg_{1}}\) and \(h_{rg_{2}}\) can get  
\[h_{rg_{1}}= \sqrt{c_{0}d_{rg_{1}}^{-\tau}}\Bigg{(}\sqrt{\frac{\sigma}{1+\sigma }}h_{rg_{1}}^{\text{LOS}}+\sqrt{\frac{1}{1+\sigma}}h_{rg_{1}}^{\text{NLOS}} \Bigg{)}, \tag{5}\]  
\[h_{rg_{1}}^{\text{LOS}}=\Bigg{\{}1,...,e^{-j\frac{2\pi}{\lambda}d_{rg_{1}}^{ \text{(m,n)}}[k]},...,e^{-j\frac{2\pi}{\lambda}d_{rg_{1}}^{\text{(M_{R}-1)}[ k]}}\Bigg{\}}^{T}, \tag{6}\]  
\[h_{rg_{2}}= \sqrt{c_{0}d_{rg_{2}}^{-\tau}}\Bigg{(}\sqrt{\frac{\sigma}{1+\sigma }}h_{rg_{2}}^{\text{LOS}}+\sqrt{\frac{1}{1+\sigma}}h_{rg_{2}}^{\text{NLOS}} \Bigg{)}, \tag{7}\]  
\[h_{rg_{2}}^{\text{LOS}}=\Bigg{\{}1,...,e^{-j\frac{2\pi}{\lambda}d_{rg_{2}}^{ \text{(m,n)}}[k]},...,e^{-j\frac{2\pi}{\lambda}d_{rg_{2}}^{\text{(M_{R}-1)}[ k]}}\Bigg{\}}^{T}, \tag{8}\]  
we define parameters \(d_{rg_{1}}=\sqrt{Z_{RIS}^{2}+\left\|L_{R}-L_{G1}\right\|^{2}}\) and \(d_{rg_{2}}=\sqrt{Z_{RIS}^{2}+\left\|L_{R}-W_{G2}\right\|^{2}}\) as the distances from the RIS to User 1 and User 2, which facilitates subsequent performance analysis. \(h_{rg_{1}}^{\text{NLOS}}\sim\mathbb{CN}(0,1)\) and \(h_{rg_{2}}^{\text{NLOS}}\sim\mathbb{CN}(0,1)\) represent the circular symmetric complex Gaussian distribution. \(\sigma\) is the Rician factor; \(\tau\) is the path loss index corresponding to RIS-ground users. And \(\chi_{rg_{1}}^{\text{(m,n)}}[k]=m\cos\phi_{rg_{1}}\sin\varphi_{rg_{1}}+n\cos \varphi_{rg_{1}}\), where \(\cos\phi_{rg_{1}}\sin\varphi_{rg_{1}}=\frac{x_{R}-x_{G1}}{d_{We assume that the signals in the system take the form of linear combinations, with each signal affected by different noises and disturbances. The signal in the system can be expressed as: \(y=H*s+n\), where \(y\) is the received signal, \(H\) is the channel matrix, \(s\) is the transmitted signal, and \(n\) is the noise.