# Optimal Position and Target Rate for Covert Communication in UAV-Assisted Uplink RSMA Systems
## 2 System Model
### Sum Rate Analysis  
In the proposed scheme, the power allocation for \(\mathbf{s}_{12}\) is designed to prevent connection outages during the decoding of both \(\mathbf{s}_{2}\) and \(\mathbf{s}_{12}\). However, an outage may still occur during the decoding of \(\mathbf{s}_{11}\) due to the randomness of \(P_{1}\). Therefore, the outage probability of the system is determined by that of decoding \(\mathbf{s}_{11}\). We respectively analyze the outage probabilities and ESR under \(\mathcal{H}_{0}\) and \(\mathcal{H}_{1}\) in the following.  
1. Under \(\mathcal{H}_{1}\) The achievable rate under \(\mathcal{H}_{1}\) of \(\mathbf{s}_{11}\) is given by \[R_{11}^{1}=\log_{2}\Bigg{(}1+\frac{g_{1b}P_{1}-\tau}{g_{2b}P_{2}+\tau+\sigma_ {b}^{2}}\Bigg{)}.\] (14)Thus, the outage probability of \(\mathbf{s}_{11}\) under \(\mathcal{H}_{1}\) is expressed as  
\[\begin{split}\mathbb{O}_{11}^{1}&=\mathbb{P}\{R_{11}^ {1}<\hat{R}_{11}\}\\ &=\mathbb{P}\bigg{\{}P_{1}<\frac{\mu_{11}(g_{2b}P_{2}+\tau+\sigma _{b}^{2})+\tau}{g_{1b}}\bigg{\}}\\ &=\max\bigg{\{}\frac{\mu_{11}(g_{2b}P_{2}+\tau+\sigma_{b}^{2})+ \tau-g_{1b}P_{1}^{\min}}{g_{1b}(P_{1}^{\max}-P_{1}^{\min})},0\bigg{\}},\end{split} \tag{15}\]  
where \(\mu_{11}=2^{\hat{R}_{11}}-1\). Eventually, the ESR under \(\mathcal{H}_{1}\) is given by  
\[\overline{R}_{\text{sum}}^{1}=(\hat{R}_{11}+\hat{R}_{2}+\hat{R}_{12})(1- \mathbb{O}_{11}^{1}). \tag{16}\]
2. Under \(\mathcal{H}_{0}\) Similarly, the achievable rate under \(\mathcal{H}_{0}\) of \(\mathbf{s}_{11}\) is given by \[R_{11}^{1}=\log_{2}\Bigg{(}1+\frac{g_{1b}P_{1}-\tau}{\tau+\sigma_{b}^{2}}\Bigg{)}.\] (17) And the outage probability of \(\mathbf{s}_{11}\) under \(\mathcal{H}_{0}\) is expressed as \[\begin{split}\mathbb{O}_{11}^{0}&=\mathbb{P}\{R_{11 }^{0}<\hat{R}_{11}\}\\ &=\mathbb{P}\bigg{\{}P_{1}<\frac{\mu_{11}(\tau+\sigma_{b}^{2})+ \tau}{g_{1b}}\bigg{\}}\\ &=\max\bigg{\{}\frac{\mu_{11}(\tau+\sigma_{b}^{2})+\tau-g_{1b}P_{ 1}^{\min}}{g_{1b}(P_{1}^{\max}-P_{1}^{\min})},0\bigg{\}}.\end{split}\] (18) Since fixed power is allocated to \(\mathbf{s}_{12}\) to satisfy \(\hat{R}_{12}\), \(\mathbb{P}\{R_{11}^{0}<\hat{R}_{11}\}=\mathbb{P}\{R_{1}^{0}<\hat{R}_{1}\}\). The ESR under \(\mathcal{H}_{0}\) is given by \[\overline{R}_{\text{sum}}^{0}=(\hat{R}_{11}+\hat{R}_{12})(1-\mathbb{O}_{11}^ {0}).\] (19) Finally, the ESR of the system is expressed as \[\begin{split}\overline{R}_{\text{sum}}&=\frac{1}{ 2}\Big{(}\overline{R}_{\text{sum}}^{0}+\overline{R}_{\text{sum}}^{1}\Big{)}\\ &\triangleq f\Big{[}\Big{(}a-b2^{\hat{R}_{11}}\Big{)}\hat{R}_{11} -c2^{\hat{R}_{11}}+d\Big{]},\end{split}\] (20) where \(a=2g_{1b}P_{1}^{\max}+g_{2b}P_{2}+2\sigma_{b}^{2}\), \(b=g_{2b}P_{2}+2\tau+2\sigma_{b}^{2}\), \(c=(2\hat{R}_{12}+\hat{R}_{2})(\tau+\sigma_{b}^{2})+(\hat{R}_{12}+\hat{R}_{2})g _{2b}P_{2}\), \(d=(2\hat{R}_{12}+\hat{R}_{2})(g_{1b}P_{1}^{\max}+\sigma_{b}^{2})+(\hat{R}_{12}+ \hat{R}_{2})g_{2b}P_{2}\), and \(f=\frac{1}{2g_{1b}(P_{1}^{\max}-P_{1}^{\min})}\). Component \(1/2\) is due to \(\mathbb{P}(\mathcal{H}_{0})=\mathbb{P}(\mathcal{H}_{1})=1/2\). Equation (20) demonstrates that as \(\hat{R}_{11}\) increases, the outage probabilities also increase, whereas the change in ESR is uncertain. Therefore, to maximize ESR, it is necessary to consider how to set \(\hat{R}_{11}\).  
Lemma 1: _The optimal \(\hat{R}_{11}\) to maximize the ESR is given by_  
\[\hat{R}_{11}^{\star}=\max\bigg{\{}0,\log_{2}e\Bigg{[}W\Bigg{(}\frac{ae2^{c/b}}{ b}\Bigg{)}-1\Bigg{]}-\frac{c}{b}\bigg{\}}, \tag{21}\]  
_where \(e\) is [PERSON]'s number, and \(W(\cdot)\) denotes Lambert \(W\) Function [22]. The corresponding \(\overline{R}_{\text{sum}}\) is denoted as \(\overline{R}_{\text{sum}}(\hat{R}_{11}^{\star})\)._Proof.: See Appendix A.  
We notice that as \(g_{1w}\) increases, \(\hat{R}_{11}^{\dagger}\) increases, resulting in a higher ESR. Therefore, ESR can be increased by placing U1 closer to the BS.