# Maximization of Average Achievable Rate for NOMA-UAV Dual-User Communication System Assisted by RIS
## 2 System Model  
This paper discusses a RIS-assisted NOMA-UAV communication system architecture. The system consists of a UAV in flight, a RIS deployed on a wall, and two ground terminal users utilizing NOMA communication. The UAV maintains a line-of-sight (LoS) wireless connection with the RIS installed on the building's side wall to ensure stable signal propagation. Even if the wireless connection between the UAV and the terrestrial user is interrupted, the system can still use RIS to enable effective signal reception by the ground user. All nodes involved in the communication process are positioned in a three-dimensional rectangular coordinate system, as shown in Figure 1. The location of the User 1 is set to \(L_{\text{G1}}=[x_{\text{G1}},y_{\text{G1}}]^{T}\), and the location of the User 2 is set to \(L_{\text{G2}}=[x_{\text{G2}},y_{\text{G2}}]^{T}\). The complete flight path of the UAV is expanded in the X-Y-Z three-dimensional coordinate system. During the whole flight cycle, the flight height of UAV remains unchanged, which is \(Z_{UAV}\)[9]. In addition, the height of RIS is set to \(Z_{RIS}\), the location of the RIS is set to \(L_{R}=[x_{R},y_{R}]^{T}\). The maximum speed during flight is limited to \(V_{max}\). The UAV's flight \begin{table}
\begin{tabular}{c c c} \hline \hline
**Reference** & **Application Scene** & **Solution Method** \\ \hline
[8] & RIS-assisted UAV communication system & Joint design iterative optimization \\ \hline
[9] & New RIS-assisted UAV communication system & Closed phase shift solution, SCA \\ \hline
[10] & Hybrid air full-duplex relay mode & Riemann gradient method \\ \hline
[11] & STAR-RIS-assisted UAV & LRCPPO algorithm \\ \hline
[12] & RIS-assisted multi-UAV communication system & Double deep Q network \\ \hline
[13] & RIS combined with NOMA multi-UAV & Quadratic programming, SCA \\ \hline
[15] & RIS mobile edge computing framework & Semidefinite relaxation, SCA \\ \hline
[16] & STAR-RIS-assisted UAV & Joint design iterative optimization \\ \hline
[17] & STAR-RIS assisted NOMA-UAV network & Joint optimization algorithm \\ \hline
[19] & T-RIS assisted UAV & D3 RL framework \\ \hline
[20] & Aerial RIS-assisted wireless network & MADDPG method \\ \hline
[21] & RIS-assisted UAV communication system & Diversity combining techniques \\ \hline
[23] & STAR-RIS-assisted UAV & Double deep Q network, SCA \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of some contributions of existing papers. cycle is denoted as \(J\), which is divided into \(K\) time intervals, each of duration \(\delta\). The UAV can cover a maximum distance in a single time interval, as calculated by \(D=V_{m}\delta\). Based on this, the position of the UAV at time \(k\) can be represented by \(Q[k]=[x_{l}[k],y_{l}[k]]^{T}\), \(k\in\{1,2,...,K\}\). To better align with the actual flight trajectory, the UAV follows the constraints outlined below. \[\left\|q[k+1]-q[k]\right\|^{2}\leq D^{2},k\in\{1,2,...,K-1\} \tag{1a}\] \[\left\|q[K]-q_{e}\right\|^{2}\leq D^{2},\] (1b) \[q[1]=q_{s}. \tag{1c}\]  
Here \(q_{s}\) and \(q_{e}\) denote the UAV's starting and terminal points. The constraint achieves a high-precision dynamic approximation of the real flight trajectory by discretizing the continuous path into a series of differential segments. The RIS is assumed to consist of a uniform planar array with \(M=M_{x}\times M_{z}\) elements. \(\Theta[k]=diag\{e^{i\theta_{(0,m)}[k]},...,e^{i\theta_{(m,m)}[k]},...,e^{i \theta_{(M_{x}-1,M_{x}-1)}[k]}\}\) is defined as a diagonal phase matrix, where \(\theta_{(m,m)}[k]\in[0,2\pi),k\in\{1,2,...,K\},m\in\{0,1,...,M_{x}\},n\in\{0,1,...,M_{z}\}\) denotes the stage value of the \((m,n)\)th reflection unit in the \(k\) slot. \(h_{ug_{1}}\) and \(h_{ug_{2}}\) are set as the direct links from UAV to ground User 1 and User 2, respectively. Although the direct signal from the transmitting source to the receiver may be blocked by obstacles, some signals can bypass these obstructions through reflection or scattering.