# Optimal Position and Target Rate for Covert Communication in UAV-Assisted Uplink RSMA Systems

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## Abstract

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## 1. Introduction

- We investigate a novel application of RSMA systems, where a UAV splits its rate to avoid deteriorating the covert transmission of a ground user while increasing the ESR. To the best of the authors’ knowledge, this is the first work that studied the covet communication in UAV-assisted uplink RSMA system.
- We derive the closed-form expressions of the ESR and obtain the optimal target rate of UAV which maximizes the ESR of the system. Subjected to minimum detection error probability (DEP) and expected covert rate (ECR) constraints, a joint position and target rate optimization problem is formulated for maximizing the ESR of uplink RSMA systems.
- The numerical results show that the proposed scheme outperforms NOMA systems in terms of ESR with the same DEP and ECR and illustrate the effect of constraints on the ESR.

## 2. System Model

#### 2.1. Communication Scenario

#### 2.2. Proposed Transmission Scheme

#### 2.3. Detection Metrics at Willie

## 3. Performance Analysis

#### 3.1. Covertness Analysis

#### 3.2. Sum Rate Analysis

- (1)
- Under ${\mathcal{H}}_{1}$The achievable rate under ${\mathcal{H}}_{1}$ of ${\mathbf{s}}_{11}$ is given by$${R}_{11}^{1}={log}_{2}\left(1+\frac{{g}_{1b}{P}_{1}-\tau}{{g}_{2b}{P}_{2}+\tau +{\sigma}_{b}^{2}}\right).$$Thus, the outage probability of ${\mathbf{s}}_{11}$ under ${\mathcal{H}}_{1}$ is expressed as$$\begin{array}{cc}\hfill {\mathbb{O}}_{11}^{1}& =\mathbb{P}\{{R}_{11}^{1}<{\widehat{R}}_{11}\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\mathbb{P}\left\{{P}_{1}<\frac{{\mu}_{11}({g}_{2b}{P}_{2}+\tau +{\sigma}_{b}^{2})+\tau}{{g}_{1b}}\right\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =max\left\{\frac{{\mu}_{11}({g}_{2b}{P}_{2}+\tau +{\sigma}_{b}^{2})+\tau -{g}_{1b}{P}_{1}^{\mathrm{min}}}{{g}_{1b}({P}_{1}^{\mathrm{max}}-{P}_{1}^{\mathrm{min}})},0\right\},\hfill \end{array}$$$$\begin{array}{c}\hfill {\overline{R}}_{\mathrm{sum}}^{1}=({\widehat{R}}_{11}+{\widehat{R}}_{2}+{\widehat{R}}_{12})(1-{\mathbb{O}}_{11}^{1}).\end{array}$$
- (2)
- Under ${\mathcal{H}}_{0}$Similarly, the achievable rate under ${\mathcal{H}}_{0}$ of ${\mathbf{s}}_{11}$ is given by$${R}_{11}^{1}={log}_{2}\left(1+\frac{{g}_{1b}{P}_{1}-\tau}{\tau +{\sigma}_{b}^{2}}\right).$$And the outage probability of ${\mathbf{s}}_{11}$ under ${\mathcal{H}}_{0}$ is expressed as$$\begin{array}{cc}\hfill {\mathbb{O}}_{11}^{0}& =\mathbb{P}\{{R}_{11}^{0}<{\widehat{R}}_{11}\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\mathbb{P}\left\{{P}_{1}<\frac{{\mu}_{11}(\tau +{\sigma}_{b}^{2})+\tau}{{g}_{1b}}\right\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =max\left\{\frac{{\mu}_{11}(\tau +{\sigma}_{b}^{2})+\tau -{g}_{1b}{P}_{1}^{\mathrm{min}}}{{g}_{1b}({P}_{1}^{\mathrm{max}}-{P}_{1}^{\mathrm{min}})},0\right\}.\hfill \end{array}$$Since fixed power is allocated to ${\mathbf{s}}_{12}$ to satisfy ${\widehat{R}}_{12}$, $\mathbb{P}\{{R}_{11}^{0}<{\widehat{R}}_{11}\}=\mathbb{P}\{{R}_{1}^{0}<{\widehat{R}}_{1}\}$. The ESR under ${\mathcal{H}}_{0}$ is given by$$\begin{array}{c}\hfill {\overline{R}}_{\mathrm{sum}}^{0}=({\widehat{R}}_{11}+{\widehat{R}}_{12})(1-{\mathbb{O}}_{11}^{0}).\end{array}$$Finally, the ESR of the system is expressed as$$\begin{array}{cc}\hfill {\overline{R}}_{\mathrm{sum}}& =\frac{1}{2}\left({\overline{R}}_{\mathrm{sum}}^{0}+{\overline{R}}_{\mathrm{sum}}^{1}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \triangleq f\left[\left(a-b{2}^{{\widehat{R}}_{11}}\right){\widehat{R}}_{11}-c{2}^{{\widehat{R}}_{11}}+d\right],\hfill \end{array}$$

**Lemma 1.**

**Proof.**

## 4. Optimization Problem

**Lemma 2.**

**Proof.**

## 5. Numerical Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PLS | Physical layer security |

AWGN | Additive white Gaussian noise |

UAV | Unmanned aerial vehicle |

RSMA | Rate-splitting multiple access |

NOMA | Non-orthogonal multiple access |

RS | Rate-splitting |

BS | Base station |

ESR | Expected sum rate |

DEP | Detection error probability |

3D | Three-dimensional |

CSI | Channel state information |

LOS | Line-of-sight |

SINR | Signal-to-interference-plus-noise ratio |

LRT | Likelihood ratio test |

FAP | False alarm probability |

MDP | Miss detection probability |

KKT | Karush–Kuhn–Tucker |

## Appendix A. Proof of Lemma 1

## Appendix B. Proof of Lemma 2

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**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.

**Figure 3.**Procedure for solving optimization problem (Section 4).

**Figure 4.**The maximum ESR versus the expected minimum DEP for different ECR constraints in RSMA and NOMA systems, where ${\widehat{R}}_{2}=3$ bpcu.

**Figure 5.**The outage probability of ${\widehat{R}}_{11}^{\u2020}$ under ${\mathcal{H}}_{1}$ for different covertness constraints in RSMA and NOMA systems, where $\u03f5=0.8{\widehat{R}}_{2}$.

**Figure 6.**The maximum ESR versus the target rate of U2 ${\widehat{R}}_{2}$ for different covertness constraints in RSMA and NOMA systems, where $\u03f5=0.8{\widehat{R}}_{2}$.

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**MDPI and ACS Style**

Duan, Z.; Yang, X.; Zhang, T.; Wang, L. Optimal Position and Target Rate for Covert Communication in UAV-Assisted Uplink RSMA Systems. *Drones* **2023**, *7*, 237.
https://doi.org/10.3390/drones7040237

**AMA Style**

Duan Z, Yang X, Zhang T, Wang L. Optimal Position and Target Rate for Covert Communication in UAV-Assisted Uplink RSMA Systems. *Drones*. 2023; 7(4):237.
https://doi.org/10.3390/drones7040237

**Chicago/Turabian Style**

Duan, Zhengxiang, Xin Yang, Tao Zhang, and Ling Wang. 2023. "Optimal Position and Target Rate for Covert Communication in UAV-Assisted Uplink RSMA Systems" *Drones* 7, no. 4: 237.
https://doi.org/10.3390/drones7040237