# NOMA-Based Rate Optimization for Multi-UAV-Assisted D2D Communication Networks
## 2 System Model
### Propagation Model  
The path loss depends on the \(LoS\) and \(NLoS\) link states, where the probabilities under the \(LoS\) and \(NLoS\) conditions are denoted as \(P_{LoS}\) and \(P_{NLoS}\), respectively, and \(P_{LoS}+P_{NLoS}=1\) is satisfied. The path losses under conditions \(LoS\) and \(NLoS\) are defined as \(L_{LoS}\) and \(L_{NLoS}\), respectively. Therefore, the average path loss \(L_{j}^{i}(t)\) between any two points \(i\) and \(j\) in the model between time slot \(t\) is denoted as  
\[L_{j}^{i}(t)=P_{\text{LoS}}\cdot L_{\text{LoS}}+P_{\text{NLoS}}\cdot L_{\text {NLoS}} \tag{1}\]  
The channels between the cellular user and the UAV and between the D2D transmitter and the UAV are modeled as air-to-ground channels. The air-to-ground channel model adopted in this paper is provided by the 3 GPP-36.777 standard [26]. For the UAV movement model, \(\{x_{u}(t),y_{u}(t),h_{u}(t)\}\) denotes the position of the UAV in time slot \(t\), \(v(t)\) denotes the UAV flight speed, and the 3D distance between UAV \(u\) and cellular user \(C_{i}\) at the moment \(t\) is denoted as \(d_{i}^{u}\big{(}t\big{)}\). \[d_{i}^{u}(t)=\sqrt{h_{u}^{2}(t)+\left[x_{u}(t)-x_{i}(t)\right]^{2}+\left[y_{u} (t)-y_{i}(t)\right]^{2}} \tag{2}\]  
where \(\{x_{i}(t),y_{i}(t)\}\) denotes the location of the cellular user of the UAV service in time slot \(t\). Let \(f_{c}\) be the carrier frequency. The path loss \(L_{i}^{u}(t)\) and the corresponding conditional probability \(P_{LoS}(t)\) between cellular user \(C_{i}\) and UAV \(u\) can be expressed by Equations (3) and (4). \[L_{i}^{u}(t)=\begin{cases}30.9+(22.25-0.5\text{log}_{10}h_{u}(t))\log_{10}d_{i} ^{u}(t)(t)+20\text{log}_{10}f_{c}\text{, if LoS link}\\ \\ \max\big{\{}L_{\text{Loss}},32.4+(43.2-7.6\text{log}_{10}h_{u}(t))\log_{10}d_{i} ^{u}(t)+20\text{log}_{10}f_{c}\big{\}}\text{, if NLoS link}\\ \\ P_{\text{LoS}}(t)=\left\{ \begin{array}{cc}1,&\text{if}\ \sqrt{\left(d_{i}^{u}(t) \right)^{2}-\left(h_{u}(t)\right)^{2}}\leqslant d_{0}\\ \frac{d_{0}}{\sqrt{\left(d_{i}^{u}(t)\right)^{2}-\left(h_{u}(t)\right)^{2}}}+ \exp\Bigg{\{}\frac{-\sqrt{\left(d_{i}^{u}(t)\right)^{2}-\left(h_{u}(t)\right)^ {2}}}{p_{1}}+\frac{d_{0}}{p_{1}}\Bigg{\}},&\text{if}\ \sqrt{\left(d_{i}^{u}(t)\right)^{2}-\left(h_{u}(t)\right)^{2}}>d_{0}\\ \end{array} \right.\\ \text{where}\ d_{0}=\max[294.05\cdot\text{log}_{10}\ h_{u}(t)-432.94,18]\), \(p_{1}=233.98\cdot\text{log}_{10}h_{u}(t)-0.95\). Figure 1: Model diagram of the UAV-assisted D2D-NOMA system. Considering the small-scale fading, the channel gain from UAV \(u\) to cellular user \(C_{i}\) at \(t\) time slot can be calculated as in Equation (5):  
\[g_{i}^{u}(t)=H_{i}^{u}(t)\cdot 10^{-L_{i}^{u}(t)/10} \tag{5}\]  
where \(H_{i}^{u}(t)\) denotes the attenuation coefficient for UAV \(u\) and cellular user \(C_{i}\) at time slot \(t\). Similarly, the channel gain of D2D transmitter \(D_{j}\) and UAV \(u\) in a D2D cluster at time slot \(t\) can be calculated as in Equation (6):  
\[g_{DT_{j}}^{u}(t)=H_{DT_{j}}^{u}(t)\cdot 10^{-L_{DT_{j}}^{u}(t)/10} \tag{6}\]  
where \(=H_{DT_{j}}^{u}(t)\) denotes the attenuation coefficient of UAV \(u\) and D2D transmitter \(DT_{j}\) at \(t\) time slot. The channels between the cellular user and the D2D transmitter and between the D2D receiver and the D2D transmitter are modeled as ground-to-ground channels. The distance between cellular user \(C_{i}\) and D2D transmitter \(DT_{j}\) at the moment \(t\) is denoted as \(d_{i,DT_{j}}^{u}(t)\). \[d_{i,DT_{j}}^{u}(t)=\sqrt{\left[x_{i}(t)-x_{DT_{j}}(t)\right]^{2}+\left[y_{i} (t)-y_{DT_{j}}(t)\right]^{2}} \tag{7}\]  
where \(\{x_{DT_{j}}(t),y_{DT_{j}}(t)\}\) denotes the position of D2D transmitter \(DT_{j}\) in time slot \(t\).