# Optimal Position and Target Rate for Covert Communication in UAV-Assisted Uplink RSMA Systems
## 2 System Model
### Proposed Transmission Scheme
Correspondingly, the achievable rates of \(\mathbf{s}_{11}\), \(\mathbf{s}_{2}\), and \(\mathbf{s}_{12}\) are expressed as \(R_{11}=\log_{2}(1+\gamma_{11})\), \(R_{2}=\log_{2}(1+\gamma_{2})\) and \(R_{12}=\log_{2}(1+\gamma_{12})\), respectively. To maximize the sum rate for the RS strategy, U1 needs to allocate the maximum possible power to \(\mathbf{s}_{12}\). As per \(\gamma_{12}\), \(\mathbf{s}_{12}\) is free from interference, hence allocating power to \(\mathbf{s}_{12}\) would be more efficient compared to \(\mathbf{s}_{11}\). Considering that the interference received by U2 should be no large than \(\tau\) to keep U2 uninterrupted, we have \((1-\alpha)g_{11}P_{1}\leq\tau\). Obviously, the allowed maximum power of \(\mathbf{s}_{12}\) is \(\tau/g_{1b}\) with \(\alpha=1-\frac{\tau}{g_{11}P_{1}}\). Meanwhile, U2's messages are not supposed to be decoded firstly for covertness, which results in \(\alpha P_{1}\geq 0\), i.e., \(P_{1}\geq\tau/g_{1b}\). The target rates of \(\mathbf{s}_{12}\) and \(\mathbf{s}_{11}\) are set as \(\hat{R}_{12}=\log_{2}(1+\frac{\tau}{\sigma_{b}^{2}})\) and \(\hat{R}_{11}=\hat{R}_{1}-\hat{R}_{12}\), respectively. The power allocation scheme and decoding order are shown in Figure 2b. Figure 2: (a) A sample for splitting \(\mathbf{s}_{1}\) into \(\mathbf{s}_{11}\) and \(\mathbf{s}_{12}\). (b) An illustration of power allocation and decoding order. where \(\rho_{1}=P_{1}^{\text{min}}+\frac{S_{2w}}{S_{1w}}P_{2},\rho_{2}=P_{1}^{\text{max}}+ \frac{S_{2w}}{S_{1w}}P_{2}\). Then, the DEP at Willie is given by  
\[\mathbb{P}_{E}= \begin{cases}1,&\frac{\lambda-\sigma_{w}^{2}}{S_{1w}}<P_{1}^{ \text{min}},\\ \frac{S_{1w}P_{1}^{\text{max}}+\sigma_{2}^{2}-\lambda}{S_{1w}(P_{1}^{\text{max }}-P_{1}^{\text{min}})},&P_{1}^{\text{min}}\leq\frac{\lambda-\sigma_{w}^{2}}{S_ {1w}}<\rho_{1},\\ 1-\frac{S_{2w}P_{2}}{S_{1w}(P_{1}^{\text{max}}-P_{1}^{\text{min}})},&\rho_{1} \leq\frac{\lambda-\sigma_{w}^{2}}{S_{1w}}\leq P_{1}^{\text{max}},\\ \frac{\lambda-S_{1w}P_{1}^{\text{min}}-S_{2w}P_{2}-\sigma_{w}^{2}}{S_{1w}(P_{ 1}^{\text{max}}-P_{1}^{\text{min}})},&\rho_{1}^{\text{max}}<\frac{\lambda- \sigma_{w}^{2}}{S_{1w}}\leq\rho_{2},\\ 1,&\frac{\lambda-\sigma_{w}^{2}}{S_{1w}}>\rho_{2}.\end{cases}  \tag{11}\]  
Note that the condition \(g_{1w}(P_{1}^{\text{max}}-P_{1}^{\text{min}})\geq g_{2w}P_{2}\) needs to be satisfied; otherwise, [PERSON] has zero probability of making detection errors. According to the monotonicity of (11), the minimum DEP is given by  
\[\mathbb{P}_{E}^{\ddagger}=1-\frac{g_{2w}P_{2}}{g_{1w}(P_{1}^{\text{max}}-P_{1} ^{\text{min}})}, \tag{12}\]  
and the corresponding detection threshold satisfies \(\rho_{1}\leq\frac{\lambda-\sigma_{w}^{2}}{S_{1w}}\leq P_{1}^{\text{max}}\). Since legitimate users don't have instantaneous CSI between Willie and U2, we consider the expected minimum DEP \(\overline{\mathbb{P}_{E}^{\ddagger}}\) over all possible realization of \(h_{2w}\) as the measurement of covertness from the perspective of legitimate users. \(\overline{\mathbb{P}_{E}^{\ddagger}}\) is given by  
\[ \begin{split}\overline{\mathbb{P}_{E}^{\ddagger}}& =\int_{0}^{\frac{S_{1w}(P_{1}^{\text{max}}-P_{1}^{\text{min}})}{P_ {2}}}\mathbb{P}_{E}^{\ddagger}f_{2w}(x)dx\\ &=\int_{0}^{\frac{S_{1w}(P_{1}^{\text{max}}-P_{1}^{\text{min}})} {P_{2}}}\left[1-\frac{xP_{2}}{g_{1w}(P_{1}^{\text{max}}-P_{1}^{\text{min}})} \right]\frac{1}{L_{2w}}e^{-\frac{x}{L_{2w}}}dx\\ &=1+\frac{L_{2w}P_{2}}{g_{1w}(P_{1}^{\text{max}}-P_{1}^{\text{min }})}\Bigg{[}e^{-\frac{S_{1w}(P_{1}^{\text{max}}-P_{1}^{\text{min}})}{L_{2w}P_{ 2}^{\ddagger}}}-1\Bigg{]},\end{split}  \tag{13}\]  
where \(f_{g_{2w}}(x)\) is the probability density function of \(g_{2w}\). The results indicate that as the variation interval of \(P_{1}\) increases, there is a corresponding rise in DEP, leading to a larger value of \(\hat{R}_{2}\). In addition, U1 can modify its channel to Willie by repositioning to meet the covertness requirement.