# Optimal Position and Target Rate for Covert Communication in UAV-Assisted Uplink RSMA Systems
## 2 System Model
### Proposed Transmission Scheme  
U1 adopts the RS strategy with random transmit power to assist U2's covert and reliable transmission. U2 conveys secret messages at a fixed rate and probability of 0.5, while U1 transmits public information. To confuse [PERSON]'s detection, U1 adopts the random transmit power. Furthermore, U1 shares the same resource block with U2 while preventing U2's outage probability from deteriorating through an RS strategy. Denoting \(P_{i}\), \(\mathbf{s}_{i}\), and \(\hat{R}_{i}\) as the transmit power, messages, and target rate of U\(i\), respectively. For simplicity, we adopt \(\mu_{i}\) to represent \(2^{\hat{R}_{i}}-1\). U2 transmits with a fixed power \(P_{2}\) and rate \(\hat{R}_{2}\). Then, the interference threshold of U1 is given by  
\[\tau=\frac{g_{2b}P_{2}}{\mu_{2}}-\sigma_{b}^{2}, \tag{2}\]  
where \(\sigma_{b}^{2}\) represents the received noise power at BS. When the interference received by U2 is lower than \(\tau\), there is no outage. Conversely, U2's connection is always interrupted when the interference is large than \(\tau\). Note that to ensure \(\tau\geq 0\), we have \(\mu_{2}\leq\frac{g_{2b}P_{2}}{\sigma_{b}^{2}}\). Figure 1: The uplink RSMA covert communication system model. To enhance the covertness of U2's transmission, \(P_{1}\) changes from slot to slot, following a continuous uniform distribution within \([P_{1}^{\text{min}},P_{1}^{\text{max}}]\). The probability density function of \(P_{1}\) is given by  
\[f_{P_{1}}(x)= \begin{cases}\frac{1}{P_{1}^{\text{max}}-P_{1}^{\text{min}}},& \text{if }P_{1}^{\text{min}}\leq x\leq P_{1}^{\text{max}},\\ 0,&\text{otherwise}.\end{cases}  \tag{3}\]  
In order to obfuscate [PERSON]'s detection and transmit more information, U1 continuously sends messages to BS. The random transmit power of U1 is designed to create ambiguity in [PERSON]'s received power. Consequently, it becomes challenging for [PERSON] to determine whether the increase in received power is due to U2's transmission or simply a variation in U1's transmit power. To increase the total throughput without causing interruptions to U2, U1 applies RS in each time slot. In uplink RSMA systems, U1 needs to split its messages \(\mathbf{s}_{1}\) into two parts \(\mathbf{s}_{11}\) and \(\mathbf{s}_{12}\), as shown in Figure 2a. Note that Figure 2a depicts only one possible splitting scheme. There are also alternative approaches that can be considered. And the received signals at node \(j\) can be expressed as  
\[\mathbf{y}_{j}[n]= \begin{cases}\sqrt{P_{1}}h_{j}(\sqrt{\alpha}\mathbf{s}_{11}[ n]+\sqrt{1-\alpha}\mathbf{s}_{12}[n])+\mathbf{n}_{j}[n],&\mathcal{H}_{0},\\ \sqrt{P_{1}}h_{j}(\sqrt{\alpha}\mathbf{s}_{11}[n]+\sqrt{1-\alpha}\mathbf{s}_{ 12}[n])+\sqrt{P_{2}}h_{2j}\mathbf{s}_{2}[n]+\mathbf{n}_{j}[n],&\mathcal{H}_{1},\end{cases}  \tag{4}\]  
where \(n=1\ldots N\) is the index of channel use, \(\mathbf{n}_{j}[n]\) is the received AWGN at \(j\) with the variance of \(\sigma_{j}^{2}\), and \(\alpha\) is the power allocation factor satisfying \(0\leq\alpha\leq 1\). The hypotheses \(\mathcal{H}_{1}\) and \(\mathcal{H}_{0}\) represent the existence and non-existence of U2's secret transmission, respectively. It is assumed that \(\mathbf{s}_{k}\), \(k\in\{1,2,11,12\}\), is independently coded with the Gaussian codebook satisfying \(\mathbb{E}\{\mathbf{s}_{k}[n]\mathbf{s}_{k}^{*}[n]\}=1\), where \(\mathbb{E}\{\cdot\}\) and \((\cdot)^{*}\) represent the expectation and conjugate transpose operators, respectively. The decoding order for uplink RSMA is \(\mathbf{s}_{11}\rightarrow\mathbf{s}_{2}\rightarrow\mathbf{s}_{12}\)[11]. Thus, the signal-to-interference-plus-noise ratios (SINRs) for BS decoding \(\mathbf{s}_{11}\), \(\mathbf{s}_{2}\), and \(\mathbf{s}_{12}\) are given by \(\gamma_{11}=\frac{\alpha\mathbf{s}_{11}P_{1}}{g_{22}P_{2}+(1-\alpha)\mathbf{s }_{11}P_{1}+\sigma_{b}^{2}}\), \(\gamma_{2}=\frac{g_{22}P_{2}}{(1-\alpha)\delta\mathbf{s}_{11}P_{1}+\sigma_{b} ^{2}}\), and \(\gamma_{12}=\frac{(1-\alpha)\mathbf{s}_{11}P_{1}}{\sigma_{b}^{2}}\), respectively.