Joint Hovering Height, Power, and Rate Optimization for Air-to-Ground UAV-RSMA Covert Communications

Abstract

In this paper, we investigate covert communications for an unmanned aerial vehicle (UAV)-aided rate-splitting multiple access (RSMA) system in which the UAV transmits to the covert and public users separately while shielding covert transmissions from a warden. Under the RSMA principles, the messages of the covert and public users are converted to common and private streams for air-to-ground transmissions. Subject to the quality of service (QoS) requirements of the public user and covertness constraint of the UAV-aid RSMA (UAV-RSMA) system, the aim of covert communication design is to maximize the covert rate by jointly optimizing the transmit power allocation, common rate allocation, and UAV hovering height, which each contribute to the uncertainty of the warden’s binary decision and attempts to enhance communication performance. To address the non-convex covert rate maximization problem in addition to the highly coupled system parameters, we decouple the original problem into three subproblems of transmit power allocation, common rate allocation, and UAV hovering height. We derive the optimal solutions for each of the subproblems of the transmit power and rate allocations and formulate a signomial programming problem to tackle UAV hovering height optimization. The simulation results indicate the superior covert rate performance achieved by the proposed AO algorithm and demonstrate that the proposed UAV-RSMA scheme achieves a higher covert rate than the benchmark schemes.

1. Introduction

Rate-splitting multiple access (RSMA) has been recognized as a promising multiple access technology for future wireless communications due to its high spectral efficiency, high energy efficiency, and flexibility on inter-user interference (IUI) management [1,2,3]. In downlink RSMA, each user’s message is split into a common part and a private part to generate the overall transmit signal, which consists of a common steam and multiple private streams, respectively. Specifically, the common stream is obtained by merging the common messages of all users via encoding with a shared codebook. The private streams are independently generated from all users’ private messages with each private stream being intended for a distinctive receiver. At the receiver side, the common stream is decoded by all the receivers, whereas each receiver decodes its own private stream by using successive interference cancellation (SIC) and treating other private streams as noise [4]. Because of its flexibility in partially decoding interference and treating partial interference as noise, RSMA has drawn enormous attention from industry and academia to tackle various challenges that have emerged for the upcoming 6G wireless communications [4,5,6,7].

With the unprecedented proliferation of confidential and sensitive data over wireless communications, privacy protections have rendered massive critical security issues in 6G wireless communication systems. A notable challenge of RSMA is encountered in security and privacy protection, due to open propagation of wireless signals. To prevent information leakage in RSMA systems from the perspective of physical layer security (PLS), common message and cooperative jamming have been exploited to address eavesdropping issues [8,9]. It has been shown in [8] that common messages can not only convey information signals but also act as artificial noise (AN) to interfere with eavesdroppers such that cooperative RSMA can achieve a higher secrecy sum rate than conventional non-orthogonal multiple access (NOMA) and space-division multiple access (SDMA). Due to imperfect channel state information at transmitter (CSIT), the secrecy performance of the RSMA system is decreased. Assuming the worst-case uncertainty channel model, the authors of [9] jointly designed the IUI mitigation and secrecy sum-rate maximization for the RSMA system, which significantly enhanced the secrecy rate performance compared to that of NOMA. In the presence of an internal eavesdropper, a secure beamforming design was proposed that takes into account all users’ secrecy rate constraints, which resulted in RSMA outperforming the benchmark schemes in terms of weighted sum rate [10]. To prevent untrusted relays from eavesdropping on private messages owned by other users, a two-slot cooperative rate-splitting scheme was proposed in which the precoding, slot allocation, and rate allocation were jointly optimized to effectively guarantee users are safe from eavesdropping [11]. However, PLS approaches cannot prevent adversaries from receiving and detecting legitimate signals, which results in novel threats to the actual hiding of legitimate transmissions.

Covert communications aim to provide a higher level of security protections than conventional PLS approaches by ensuring adversaries have a large detection error probability (DEP) of monitoring behaviors of legitimate communications [11,12]. Since leaking a small amount of data or exposing existence information—such as those that occur in military operations, wearable medical devices, or wireless financial transactions—can result in unexpected harms to legitimate systems, covert communications have emerged and are desired in the context of RSMA. To maximize the sum rate and maintain fairness, transmit power allocation and rate allocation in RSMA systems need to smartly cooperate with each other to realize covert communications, and a policy gradient method-based learning algorithm was proposed in [13]. It has been shown that under the stochastic optimization framework, the learning algorithm can achieve non-saturated transmission rate at high signal-to-noise (SNR) regions with infinite block length. Furthermore, jamming signal was superimposed to common and private messages to maximize the minimum data rate among all users subject to the covertness constraint [14]. By using AN signal to effectively prevent adversary from detecting legitimate RSMA transmissions, up to three times the minimum data rate can be achieved over that of the NOMA scheme [14].

Recently, unmanned aerial vehicle (UAV)-aided wireless communications have gained much attention from academia and the industry due to the high mobility, enhanced coverage, and flexible deployment of UAVs. PLS techniques were applied to prevent the exchange of confidential information to and from the UAVs, but they cannot hide the UAV’s communication behaviors and prevent them from being detected by wardens. To overcome the challenges raised in UAV-aided military and financial fields, covert communication techniques were exploited [15]. In [16], both the PLS and covert communication performance were analyzed for a multi-hop network against the UAV surveillance. By using beam sweeping, an optimal detection strategy was proposed to detect the UAV’s covert transmissions [17]. Also, the beam sweeping was adopted at the UAV transmitter to maximize the average throughput of the covert transmissions from the UAV to the ground terminal [18]. For long-distance covert communications, the UAV was deployed as a relay to maximize the transmission rate between the legitimate transmitter and receiver against a flying warden [19]. In [20], an UAV full-duplex relay was applied to maximize the effective covert throughput in the practical Rician channels. In [21], an UAV-aided uplink covert communication system was investigated, where a multi-antenna jammer was deployed to maximize the transmission rate between the ground transmitter and the UAV receiver against several randomly distributed wardens. Under the assumption that block length is finite, the downlink transmissions from the UAV to mutliple ground users were hidden by maximizing the minimum average covert rate [22]. In [23], smart contracts and blockchain techniques were used to ensure the security and guarantee the privacy of UAV collaborations. The UAVs can also be used to covertly collect data from ground users [24]. By using the UAV as a jammer, the covert communication performance can be improved as shown in [25]. To clearly show the considered UAV security and privacy protection techniques, we plotted the tree diagram in Figure 1 for related works in the research field.

In the aforementioned works, UAV maneuverability was not fully exploited to enhance the covert communication performance. By exploiting their trajectory design along with wireless resource optimization, UAVs have played an important role in wireless covert communications, which hide the UAV communication behavior and protect the UAV from being exposed in various civilian and military applications. So far, a notable research direction is to jointly optimize UAV placement and transmit power with the aim of enhancing the covert communication performance. In [26], the UAV placement and transmit power were jointly optimized to improve the received signal-to-noise (SNR) at the covert user by using the heuristic approaches. For a single-user scenario, the optimal UAV horizontal location was shown to be on the line segment between the covert user and the warden [27]. To minimize the probability of UAV exposure, the hovering height and transmit power were jointly optimized [28]. Moreover, the UAV movements were taken into account, and the system throughput was improved by jointly optimizing the UAV trajectory and transmit power [29]. However, most of the existing works on the UAV-aided covert communications have focused on the single-user scenario, whereas UAV-aided covert communications with multiple access, including NOMA and RSMA, have rarely been studied [30]. From the perspective of privacy protections, early stage research on covert communications for the UAV-aided RSMA (UAV-RSMA) systems are urgently demanded.

Figure 1. Diagram showing research on UAV security and privacy protection techniques (Wang, H.M. 2020 [16], Hu, J. 2020 [17], Zhou, X. 2019 [24], Liang, W. 2020 [25], Yan, S. 2021 [26], Zhou, X. 2019 [29]).

Figure 1. Diagram showing research on UAV security and privacy protection techniques (Wang, H.M. 2020 [16], Hu, J. 2020 [17], Zhou, X. 2019 [24], Liang, W. 2020 [25], Yan, S. 2021 [26], Zhou, X. 2019 [29]).

As outlined above, the existing works on the security and privacy protections of UAV-aided wireless covert communication systems have mainly focused on single-user scenarios and cannot be applied to wireless multiple access systems. Furthermore, current works on UAV hovering height optimization are confined to the NOMA-based covert communication systems, whose implemented in UAV-RSMA systems is, however, non-trivial. The reason for this issue is that the UAV hovering height is highly coupled with common and private messages, which uniquely belong to the UAV-RSMA systems in contrast to the UAV-NOMA systems. Therefore, it is important to consider covert communications in UAV-RSMA systems. To realize covert communication between the UAV and RSMA users, the common and private messages, transmit power, and hovering height of the UAV need to be jointly optimized. To fully exploit the uncertainties of the common and private messages, transmit power, and UAV hovering height to realize these covert communications, we focus on an air-to-ground UAV-RSMA system and develop an optimal covert communication design. A comparison between the work in paper and the existing related works is summarized in

Table 1. Comparison of the proposed scheme with the existing works on air-to-ground UAV covert communications.

References	Multiple Access Scheme	Transmit Power Optimization	Rate Allocation Optimization	Hovering Height Optimization
[26]	–	✓	–	–
[27]	–	✓	–	✓
[28]	–	✓	–	✓
[29]	–	✓	–	–
[30]	NOMA	✓	–	✓
Proposed Scheme	RSMA	✓	✓	✓

. The main contributions of this paper are summarized as follows:

• We exploit covert communications in an air-to-ground UAV-RSMA system, where a covert user and a public user can receive the RSMA signal transmitted from the UAV in the presence of a ground warden. The transmit power, common and private messages, and UAV hovering height are jointly utilized as new uncertainty medium to protect the private message transmissions of the covert user.
• Subject to the covertness constraint of the UAV-RSMA system and the QoS requirements of the public user, the transmit power allocation, common rate allocation, and UAV hovering height are jointly optimized to maximize the covert rate. To solve the non-convex covert rate maximization problem, we propose an efficient alternating optimization (AO) algorithm to deal with three subproblems of optimizing the transmit power allocation, common rate allocation, and UAV hovering height. The optimal solution to the subproblem of transmit power optimization is elaborately determined in a novel AO manner, the optimal solution to the subproblem of optimizing common rate allocation is derived, and a signomial programming problem is formulated to tackle hovering height optimization.
• The simulation results validate the effectiveness of the proposed AO algorithm. By jointly optimizing the UAV hovering height, power allocation, and rate allocation, the simulation results also verify that the proposed UAV-RSMA scheme achieves a higher covert rate than the benchmark schemes.

The rest of the paper is organized as follows. The system model of the air-to-ground UAV-RSMA covert communication system is presented in Section 2. Section 3 formulates the covert rate maximization problem and develops the optimization method. The simulation results are presented in Section 4. The conclusions of this work are given in Section 5.

2. System Model

We consider an air-to-ground UAV-RSMA system as shown in Figure 2, in which a public user (Grace) and a covert user (Bob) are located in a geographical area, and a UAV is hovering at a certain location to provide downlink transmissions to Bob and Grace by using the RSMA, while a warden (Willie) operates within the vicinity of Bob to monitor the covert transmissions from the UAV to Bob. The horizontal locations of Bob, Grace, and Willie are denoted according to the Cartesian coordinate system by ${\mathbf{q}}_b=[x_b,y_b]$, ${\mathbf{q}}_g=[x_g,y_g]$, and ${\mathbf{q}}_w=[x_w,y_w]$, respectively, while the UAV’s horizontal location is ${\mathbf{q}}_w=[x_w,y_w]$ and its hovering height is H satisfying $H_{min}\le H\le H_{max}$, where $H_{min}$ and $H_{max}$ are the minimum and maximum hovering altitudes, respectively.

In this work, we apply the following assumptions for the considered UAV-RSMA system: (1) All nodes are respectively equipped with a single antenna for reception or transmission. (2) Alice knows the locations of Bob and grace and also knows the instantaneous CSI of the channels from UAV to Bob and Grace. (3) The UAV knows the statistical CSI of the channels from Alice to Willie, which can be obtained by observing the location information of Willie, whereas Alice does not know the corresponding instantaneous CSI because Willie tries to hide its existence from the legitimate system. (4) From the perspective of covert communication design, we consider the worst-case scenario, i.e., Willie knows the instantaneous CSI of the channels from Alice to Willie [31,32]. (5) Since the strength of the line of sight (LoS) component is much stronger than the non-line of sight (NLoS), the air-to-ground channel is simply modeled by the LoS component. Then, the channel from the UAV to the node k ($k\in \{b,g,w\}$) can be expressed as:

$$\begin{array}{c}\hfill h_k=\sqrt{\frac{{\mathcal{L}}_0}{\parallel {\mathbf{q}}_u-{\mathbf{q}}_k{\parallel }^2+H^2}},\end{array}$$

(1)

where ${\mathcal{L}}_0$ is the path-loss measured at the reference distance of 1 m and $\parallel ·\parallel$ denotes the Euclidean norm.

2.1. RSMA Transmissions

In the considered UAV-RSMA system, the UAV adopts RSMA to transmit the messages $S_b\left(k\right)$ and $S_g\left(k\right)$ to Bob and Grace, respectively, where $k=1,\cdots ,K$ denotes the index of the channel usage and K is the total number of the channel uses. Before the UAV’s transmission, $S_b\left(k\right)$ and $S_g\left(k\right)$ are each split into the two parts $\{S_{b_1}\left(k\right),S_{b_2}\left(k\right)\}$ and $\{S_{g_1}\left(k\right),S_{g_2}\left(k\right)\}$, respectively. Then, the messages $S_{b_1}\left(k\right)$ and $S_{g_1}\left(k\right)$ are merged and encoded to a common stream $s_0\left(k\right)$, while $S_{b_2}\left(k\right)$ and $S_{g_2}\left(k\right)$ are encoded to two private streams $s_1\left(k\right)$ and $s_2\left(k\right)$, respectively. Then, the transmitted signal of the UAV can be expressed as $z_1=\left(\sqrt{{\alpha }_0{P}_u}{s}_0\left(k\right)+\sqrt{{\alpha }_1{P}_u}{s}_1\left(k\right)+\sqrt{{\alpha }_2{P}_u}{s}_2\left(k\right)\right)$, where ${\alpha }_0,{\alpha }_1$, and ${\alpha }_2$ are transmit power allocation factors satisfying $0\le {\alpha }_0,{\alpha }_1,{\alpha }_2\le 1$ and $P_u$ is the peak transmit power at the UAV. We assume that both common and private streams satisfy $\mathbb{E}(|s_0{|}^2)=\mathbb{E}(|s_1{|}^2)=\mathbb{E}(|s_2{|}^2)=1$. For each transmission block, the signals received at Bob and Grace can be respectively expressed as:

$$\begin{array}{c}\hfill y_b\left(k\right)=\left(\sqrt{{\alpha }_0{P}_u}{s}_0\left(k\right)+\sqrt{{\alpha }_1{P}_u}{s}_1\left(k\right)+\sqrt{{\alpha }_2{P}_u}{s}_2\left(k\right)\right)h_b+n_b\left(k\right)\end{array}$$

(2)

and

$$\begin{array}{c}\hfill y_g\left(k\right)=\left(\sqrt{{\alpha }_0{P}_u}{s}_0\left(k\right)+\sqrt{{\alpha }_1{P}_u}{s}_1\left(k\right)+\sqrt{{\alpha }_2{P}_u}{s}_2\left(k\right)\right)h_g+n_g\left(k\right),\end{array}$$

(3)

where $n_b\left(k\right)\sim \mathcal{CN}(0,{\sigma }_b^2)$ ($n_g\left(k\right)\backsim \mathcal{CN}(0,{\sigma }_g^2)$) is the additive white Gaussian noise (AWGN) at Bob (Grace). When Bob receives the signal, the common stream $s_0\left(k\right)$ is first detected and the private stream $s_1\left(k\right)$ and $s_2\left(k\right)$ regarded as noise [1]. Thus, the achievable rate of detecting $s_0\left(k\right)$ at Bob is given by $R_{b,s_0\left(k\right)}={log}_2(1+{\gamma }_{b,s_0\left(k\right)})$, where the received signal-to-interference-plus-noise ratio (SINR) ${\gamma }_{b,s_0\left(k\right)}$ is given by

$$\begin{array}{c}\hfill {\gamma }_{b,s_0\left(k\right)}=\frac{{\alpha }_0{P}_u{\left|h_b\right|}^2}{({\alpha }_1+{\alpha }_2)P_u{\left|h_b\right|}^2+{\sigma }_b^2}.\end{array}$$

(4)

After successfully detecting and eliminating the common stream from the received signal, Bob detects its own private stream $s_1\left(k\right)$. Consequently, the achievable rate of detecting $s_1\left(k\right)$ at Bob can be expressed as $R_{b,s_1\left(k\right)}={log}_2(1+{\gamma }_{b,s_1\left(k\right)})$, where the received SINR ${\gamma }_{b,s_1\left(k\right)}$ is given by

$$\begin{array}{c}\hfill {\gamma }_{b,s_1\left(k\right)}=\frac{{\alpha }_1{P}_u{\left|h_b\right|}^2}{{\alpha }_2{P}_u{\left|h_b\right|}^2+{\sigma }_b^2}.\end{array}$$

(5)

Similarly, the achievable rates for decoding the common stream $s_0\left(k\right)$ and private stream $s_2\left(k\right)$ at Grace are given by $R_{g,s_0\left(k\right)}={log}_2(1+{\gamma }_{g,s_0\left(k\right)})$ and $R_{g,s_2\left(k\right)}={log}_2(1+{\gamma }_{g,s_2\left(k\right)})$, respectively, where

$$\begin{array}{c}\hfill {\gamma }_{g,s_0\left(k\right)}=\frac{{\alpha }_0{P}_u{\left|h_g\right|}^2}{({\alpha }_1+{\alpha }_2)P_u{\left|h_g\right|}^2+{\sigma }_g^2}\end{array}$$

(6)

and

$$\begin{array}{c}\hfill {\gamma }_{g,s_2\left(k\right)}=\frac{a_2{P}_u{\left|h_g\right|}^2}{{\alpha }_1{P}_u{\left|h_g\right|}^2+{\sigma }_g^2},\end{array}$$

(7)

respectively. In this work, we assume $|h_b{|}^2>{|h_g|}^2$ without loss of generality. Consequently, we have $R_{g,s_0\left(k\right)}<R_{b,s_0\left(k\right)}$. Shared by Bob and Grace, the transmission rate of the common message can be characterized by

$$\begin{array}{ccc}\hfill R_c& =& min\left\{R_{b,s_0\left(k\right)},R_{g,s_0\left(k\right)}\right\}\hfill \\ & =& R_b^c+R_g^c,\hfill \end{array}$$

(8)

where $R_b^c\triangleq {\beta }_1{R}_{g,s_0\left(k\right)}$ and $R_g^c\triangleq {\beta }_2{R}_{g,s_0\left(k\right)}$ denote the achievable rates of the common message transmissions of Bob and Grace, respectively, ${\beta }_1$ and ${\beta }_2$ are the common rate allocation factors satisfying $0\le {\beta }_1,{\beta }_2\le 1$ and ${\beta }_1+{\beta }_2\le 1$ [1,33]. Then, the achievable rates of Bob and Grace can be expressed as $R_b={\beta }_1{R}_{g,s_0\left(k\right)}+R_{b,s_1\left(k\right)}$ and $R_g={\beta }_2{R}_{g,s_0\left(k\right)}+R_{g,s_2\left(k\right)}$, respectively.

2.2. Willie’s Detection

As the warden, Willie faces two hypotheses in monitoring the covert transmissions from Alice to Bob: (1) The null hypothesis ${\mathcal{H}}_0$ corresponding to the case in which Alice does not transmit a private message to Bob. (2) The alternate hypothesis ${\mathcal{H}}_1$ corresponding to the case in which Alice transmits a private message to Bob. Under the two hypotheses, the signal received at Willie can be respectively expressed as:

$$\begin{array}{c}\hfill {\mathcal{H}}_0:y_w\left(k\right)=\left(\sqrt{{\alpha }_0{P}_u}{s}_0\left(k\right)+\sqrt{{\alpha }_2{P}_u}{s}_2\left(k\right)\right)h_w+n_w\end{array}$$

(9)

and

$$\begin{array}{c}\hfill {\mathcal{H}}_1:y_w\left(k\right)=\left(\sqrt{{\alpha }_0{P}_u}{s}_0\left(k\right)+\sqrt{{\alpha }_1{P}_u}{s}_1\left(k\right)+\sqrt{{\alpha }_2{P}_u}{s}_2\left(k\right)\right)h_w+n_w,\end{array}$$

(10)

where $n_w\left(k\right)\backsim \mathcal{CN}(0,{\sigma }_w^2)$ denotes the AWGN at Willie. Under ${\mathcal{H}}_0$ and ${\mathcal{H}}_1$, the likelihood functions of the received signals can be evaluated as $p_0\left(y_w\right)=e^{-\frac{{|y_w|}^2}{{\sigma }_0^2}}/\pi {\sigma }_0^2$ and $p_1\left(y_w\right)=e^{-\frac{{|y_w|}^2}{{\sigma }_1^2}}/\pi {\sigma }_1^2$, respectively, with

$$\begin{array}{c}\hfill {\sigma }_0^2=({\alpha }_0+{\alpha }_2)P_u{\left|h_w\right|}^2+{\sigma }_w^2\end{array}$$

(11)

and

$$\begin{array}{c}\hfill {\sigma }_1^2=P_u{\left|h_w\right|}^2+{\sigma }_w^2.\end{array}$$

(12)

To minimize the DEP, Willie conducts the likelihood ratio test $\frac{p_1\left(y_w\right)}{p_0\left(y_w\right)}\underset{{}_{{\mathcal{D}}_0}}{\overset{{}_{{\mathcal{D}}_1}}{\gtrless }}1$, where ${\mathcal{D}}_1$ and ${\mathcal{D}}_0$ are the binary decisions that infer whether Alice transmits or not. Equivalently, Willie adopts its average received power $P_w\stackrel{K\to \infty }{=}\frac{1}{K}{\sum }_{k=1}^K{\left|y_w\left(k\right)\right|}^2$ as the decision statistic. Then, the optimal decision rule can be expressed as $P_w\underset{{}_{{\mathcal{D}}_0}}{\overset{{}_{{\mathcal{D}}_1}}{\gtrless }}{\varphi }^{*}$, where ${\varphi }^{*}$ is the optimal detection threshold for Willie’s decision, with its form being given by:

$$\begin{array}{c}\hfill {\varphi }^{*}=\frac{{\sigma }_0^2{\sigma }_1^2}{{\sigma }_1^2-{\sigma }_0^2}\ln \frac{{\sigma }_1^2}{{\sigma }_0^2}.\end{array}$$

(13)

By using the optimal detection threshold, the achieved minimum DEP can be expressed as:

$$\begin{array}{ccc}\hfill {\xi }^{*}& =& Pr\left({\mathcal{D}}_1|{\mathcal{H}}_0\right)+Pr\left({\mathcal{D}}_0|{\mathcal{H}}_1\right)\hfill \\ & =& Pr\left(P_w\ge {\varphi }^{*}|{\mathcal{H}}_0\right)+Pr\left(P_w\le {\varphi }^{*}|{\mathcal{H}}_1\right)\hfill \\ & =& 1+{\left(\frac{{\sigma }_1^2}{{\sigma }_0^2}\right)}^{-\frac{{\sigma }_1^2}{{\sigma }_1^2-{\sigma }_0^2}}-{\left(\frac{{\sigma }_1^2}{{\sigma }_0^2}\right)}^{-\frac{{\sigma }_0^2}{{\sigma }_1^2-{\sigma }_0^2}},\hfill \end{array}$$

(14)

where $Pr\left({\mathcal{D}}_1|{\mathcal{H}}_0\right)$ and $Pr\left({\mathcal{D}}_0|{\mathcal{H}}_1\right)$ denote the false alarm probability and missed detection probability, respectively. Unfortunately, the above expression for ${\xi }^{*}$ is intractable in further analysis of corresponding optimizations. To formulate a feasible covert rate maximization problem, we introduce a lower bound on ${\xi }^{*}$ as:

$$\begin{array}{c}\hfill {\xi }^{*}\ge 1-\sqrt{\frac{1}{2}\mathcal{D}\left(p_0\left(y_w\right)\Vert p_1\left(y_w\right)\right)},\end{array}$$

(15)

where $\mathcal{D}\left(p_0\left(y_w\right)\Vert p_1\left(y_w\right)\right)$ is the Kullback–Leibler (KL) divergence, which can be evaluated as:

$$\begin{array}{ccc}\hfill \mathcal{D}\left(p_0\left(y_w\right)\Vert p_1\left(y_w\right)\right)& =& \int p_0\left(y_w\right)ln\frac{p_0\left(y_w\right)}{p_1\left(y_w\right)}d{y}_w\hfill \\ & =& ln\left(\frac{{\sigma }_1^2}{{\sigma }_0^2}\right)+\left(\frac{{\sigma }_0^2}{{\sigma }_1^2}\right)-1.\hfill \end{array}$$

(16)

To guarantee the covert transmissions from Alice to Bob, the covert constraint

$$\begin{array}{c}\hfill {\xi }^{*}\ge 1-\epsilon ,\end{array}$$

(17)

needs to be satisfied with $\epsilon >0$ being a small value to represent the required covertness level. By using the KL divergence and the lower bound on ${\xi }^{*}$ in (15), a stricter covert constraint over (17) can be formulated as:

$$\begin{array}{c}\hfill \mathcal{D}\left(p_0\left(y_w\right)\Vert p_1\left(y_w\right)\right)\le 2{\epsilon }^2.\end{array}$$

(18)

Considering that $f\left(\lambda \right)=ln\lambda +\frac{1}{\lambda }-1$ is a monotonically increasing function with respect to $\lambda \in \left[1,\infty \right)$ with $f\left(1\right)=0$ and $f(\infty )=\infty$, the covert constraint (18) can be rewritten as:

$$\begin{array}{ccc}\hfill & & \left(1-\kappa ({\alpha }_0+{\alpha }_2)\right)P_u|h_w{|}^2\le (\kappa -1){\sigma }_w^2,\hfill \end{array}$$

(19)

where $\kappa$ is the unique solution of $f\left(\lambda \right)=2{\epsilon }^2$ in the interval $\left[1,\infty \right)$.

3. Joint Optimization of Hovering Height, Power, and Rate Allocation

3.1. Problem Formulation and Parameters Optimization

Subject to the covertness constraint and QoS requirements of the considered UAV-RSMA system, the aim of this work is to maximize Bob’s covert rate (equivalently the achievable rate of Bob) by jointly optimizing the transmit power allocation, common rate allocation, and UAV hovering height. Specifically, the achievable rate of Grace needs to be no less than the target rate $R_g^{\min }$, which represents the QoS requirements of Grace. Then, the problem of maximizing the covert rate can be formulated as:

$$\begin{array}{}\mathrm{(20a)}& \begin{array}{ccc}\hfill \left(\mathrm{P}1\right):& \underset{{\alpha }_0,{\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2,H}{\max }{R}_b\hfill & \hfill \end{array}\mathrm{(20b)}& \begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& {\alpha }_0+{\alpha }_1+{\alpha }_2\le 1,0\le {\alpha }_1,{\alpha }_2,{\alpha }_3\le 1,\hfill & \hfill \end{array}\mathrm{(20c)}& \begin{array}{ccc}\hfill & {\beta }_1+{\beta }_2\le 1,0\le {\beta }_1,{\beta }_2\le 1,\hfill & \hfill \end{array}\mathrm{(20d)}& \begin{array}{ccc}\hfill & R_b^c\ge 0,R_g^c\ge 0,\hfill & \hfill \end{array}\mathrm{(20e)}& \begin{array}{ccc}\hfill & R_g\ge R_g^{\min },\hfill & \hfill \end{array}\mathrm{(20f)}& \begin{array}{ccc}\hfill & \left(1-\kappa ({\alpha }_0+{\alpha }_2)\right)P_u|h_w{|}^2\le (\kappa -1){\sigma }_w^2,\hfill & \hfill \end{array}\mathrm{(20g)}& \begin{array}{ccc}\hfill & H_{min}\le H\le H_{max}.\hfill & \hfill \end{array}\end{array}$$

In problem (P1), (20b) is the constraint on the transmit power allocation factors, (20c) represents the constraint on the common rate allocation factors, (20d) are constraints placed on the common rates of Bob and Grace, respectively, (20e) is the QoS constraint required by Grace, (20f) guarantees the preset covertness level is satisfied, and (20g) is the hovering height constraint.

Obtaining the optimal solution to problem (P1) is extremely challenging. The reason is that the optimization variables in the objective function and non-convex constraints (20d)–(20f) are highly coupled. Consequently, it is impossible to directly solve problem (P1) by jointly optimizing the system parameters. To tackle this challenge, we decouple the covert rate maximization problem (P1) into three subproblems of optimizing the transmit power allocation, common rate allocation, and UAV hovering height and propose an AO algorithm to obtain the optimized system parameters. In the following, the subproblems of optimizing the transmit power allocation, common rate allocation, and UAV hovering height are separately presented.

3.2. Subproblem of Transmit Power Allocation Optimization

By observing the objective function in problem (P1), once ${\beta }_1$, ${\beta }_2$, and H are given, the value of the objective function is only determined by the transmit power allocation, i.e., the variables ${\alpha }_0$, ${\alpha }_1$, and ${\alpha }_3$. Similarly, constraints (20d) and (20e) are determined by the transmit power allocation once ${\beta }_1$, ${\beta }_2$ and H are given. Moreover, constraints (20c) and (20g) are independent of the transmit power allocation. Thus, for any given ${\beta }_1$, ${\beta }_2$, and H, the covert rate maximization problem (P1) is reduced to optimize the transmit power allocation only, which can be expressed as follows:

$$\begin{array}{}\mathrm{(21a)}& \begin{array}{ccc}\hfill \left(\mathrm{P}2\right):& \underset{{\alpha }_0,{\alpha }_1,{\alpha }_2}{\max }{R}_b\hfill & \hfill \end{array}\mathrm{(21b)}& \begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& {\alpha }_0+{\alpha }_1+{\alpha }_2\le 1,0\le {\alpha }_0,{\alpha }_1,{\alpha }_2\le 1,\hfill & \hfill \end{array}\mathrm{(21c)}& \begin{array}{ccc}\hfill & {\beta }_2{log}_2\left(1+\frac{{\alpha }_0{P}_T{\left|h_g\right|}^2}{({\alpha }_1+{\alpha }_2)P_T{\left|h_g\right|}^2+{\sigma }_g^2}\right)+{log}_2\left(1+\frac{{\alpha }_2{P}_T{\left|h_g\right|}^2}{{\alpha }_1{P}_T{\left|h_g\right|}^2+{\sigma }_g^2}\right)\ge R_g^{\min },\hfill & \hfill \end{array}\mathrm{(21d)}& \begin{array}{ccc}\hfill & \left(1-\kappa ({\alpha }_0+{\alpha }_2)\right)P_u|h_w{|}^2\le (\kappa -1){\sigma }_w^2.\hfill & \hfill \end{array}\end{array}$$

Since ${\alpha }_0$, ${\alpha }_1$, and ${\alpha }_2$ are coupled with each other in the objective function and all the constraints, it is challenging to obtain the optimal solution of problem (P2) directly. However, for any given ${\alpha }_0$, the optimal solution of ${\alpha }_1^{*}$ and ${\alpha }_2^{*}$ can be determined for problem (P2). As such, for any given ${\alpha }_1$ and ${\alpha }_2$, the optimal ${\alpha }_0^{*}$ can also be determined. Thus, the optimal solution of ${\alpha }_0^{*}$, ${\alpha }_1^{*}$, and ${\alpha }_2^{*}$ to problem (P2) can be obtained in the manner of the AO. In the following, we present the corresponding solution optimized from the perspective of the AO.

Proposition 1.

The optimal solution to problem (P2) can be achieved only when ${\alpha }_0+{\alpha }_1+{\alpha }_2=1$.

Proof. 

The proof is given in Appendix A. □

Based on the results in Proposition 1, we have ${\alpha }_1+{\alpha }_2=1-{\alpha }_0$ for any given ${\alpha }_0$. Consequently, constraint (21c) can be simplified as $R_g^c+{log}_2\left(1+\frac{(1-{\alpha }_0-{\alpha }_1)P_u{\left|h_g\right|}^2}{{\alpha }_1{P}_u{\left|h_g\right|}^2+{\sigma }_g^2}\right)\ge R_g^{\min }$. For constraint (21c), we have the following observations: (1) The case of $R_g^c\ge R_g^{\min }$. In this case, Grace’s common rate already satisfies the corresponding QoS requirements. (2) The case of $R_g^c<R_g^{\min }$. In this case, Grace’s common rate cannot satisfy the corresponding QoS requirements. Based on the above observations and taking into account that $R_b$ is a monotonically increasing function with respect to ${\alpha }_1$ with any given ${\alpha }_0$, the optimal ${\alpha }_1^{*}$ and ${\alpha }_2^{*}$ can be respectively given by:

$$\begin{array}{c}\hfill {\alpha }_1^{*}=\left\{\begin{array}{cc}\min \left\{1-{\alpha }_0,{\Xi }_1\right\},& \mathrm{if}{R}_g^c\ge R_g^{\min }\\ \min \left\{1-{\alpha }_0,{\Xi }_1,{\Xi }_2\right\},& \mathrm{otherwise}\end{array}\right.\end{array}$$

(22)

and

$$\begin{array}{c}\hfill {\alpha }_2^{*}=\left\{\begin{array}{cc}0,& \mathrm{if}{R}_g^c\ge R_g^{\min }\\ {\Xi }_3,& \mathrm{otherwise}\end{array}\right.,\end{array}$$

(23)

where ${\Xi }_1=\frac{\kappa -1}{\kappa }\left(\frac{{\sigma }_w^2}{P_u{\left|h_w\right|}^2}+1\right)$, ${\Xi }_2=\frac{(1-{\alpha }_0)P_u{\left|h_g\right|}^2-\left(2^{R_g^{\min }-R_g^c}-1\right){\sigma }_g^2}{P_u{\left|h_g\right|}^2{2}^{R_g^{\min }-R_g^c}}$, and

${\Xi }_3=\frac{\left(2^{R_g^{\min }-R_g^c}-1\right)\left({\alpha }_1{P}_u{\left|h_g\right|}^2+{\sigma }_g^2\right)}{P_u{\left|h_g\right|}^2}$. On the other hand, for any given ${\alpha }_1$ and ${\alpha }_2$, the optimal ${\alpha }_0^{*}$ is given by ${\alpha }_0^{*}=1-{\alpha }_1^{*}-{\alpha }_2^{*}$ based on the results of Proposition 1. As a result, the AO approach to obtain the optimal solution to problem (P2) is summarized in Algorithm 1.

Algorithm 1: AO Algorithm of Optimizing Transmit Power

1: Initialize a feasible ${\alpha }_0$ and $\iota =0$ 2: repeat 3:  Given ${\alpha }_0^{\iota }$, update ${\alpha }_1^{\iota +1}$ and ${\alpha }_2^{\iota +1}$ according to (22) and (23) 4:  Given ${\alpha }_1^{\iota +1}$ and ${\alpha }_2^{\iota +1}$, update ${\alpha }_0^{\iota +1}$ according to ${\alpha }_0^{\iota +1}=1-{\alpha }_1^{\iota +1}-{\alpha }_2^{\iota +1}$ 5:  Update $\iota =\iota +1$ 6: until $R_b^{\iota +1}-R_b^{\iota }<{\zeta }_1$

3.3. Subproblem of Common Rate Allocation Optimization

By observing the objective function in problem (P1), once ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, and H are given, the value of the objective function is only determined by the common rate allocation, i.e., the variables ${\beta }_1$ and ${\beta }_2$. Similarly, constraint (20e) is determined by the common rate allocation once ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, and H are given. Moreover, constraints (20b), (20d), (20f), and (20g) are independent of the common rate allocation. Thus, for any given ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, and H, problem (P1) is simplified to optimize the common rate allocation and the reduced problem is given by:

$$\begin{array}{}\mathrm{(24a)}& \begin{array}{ccc}\hfill \left(\mathrm{P}3\right):& \underset{{\beta }_1,{\beta }_2}{\max }{R}_b\hfill & \hfill \end{array}\mathrm{(24b)}& \begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& {\beta }_2{log}_2\left(1+\frac{{\alpha }_0{P}_u{\left|h_g\right|}^2}{({\alpha }_1+{\alpha }_2)P_u{\left|h_g\right|}^2+{\sigma }_g^2}\right)+{log}_2\left(1+\frac{{\alpha }_2{P}_u{\left|h_g\right|}^2}{{\alpha }_1{P}_u{\left|h_g\right|}^2+{\sigma }_g^2}\right)\ge R_g^{\min }.\hfill & \hfill \end{array}\end{array}$$

In problem (P3), constraint (24b) can be rewritten as ${\beta }_2\ge {\Xi }_4$ with

${\Xi }_4=\frac{R_g^{\min }-{log}_2\left(1+\frac{{\alpha }_2{P}_u{\left|h_g\right|}^2}{{\alpha }_1{P}_u{\left|h_g\right|}^2+{\sigma }_g^2}\right)}{{log}_2\left(1+\frac{{\alpha }_0{P}_u{\left|h_b\right|}^2}{({\alpha }_1+{\alpha }_2)P_u{\left|h_b\right|}^2+{\sigma }_b^2}\right)}$. Taking into account that $R_b$ is a monotonically decreasing function with respect to ${\beta }_2$, the optimal solution to problem (P3) can be written as ${\beta }_1^{*}=1-{\beta }_2^{*}$ and ${\beta }_2^{*}={\Xi }_4$.

3.4. Subproblem of Hovering Height Optimization

By observing the objective function in problem (P1), once ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, ${\beta }_1$, and ${\beta }_2$ are given, the value of the objective function is only determined by the hovering height, i.e., the variable H. Similarly, constraint (20e) is determined by H once ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, ${\beta }_1$, and ${\beta }_2$ are given. Moreover, constraints (20b), (20d), (20f), and (20g) are independent of the feasible H. Thus, for any given ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, ${\beta }_1$, and ${\beta }_2$, problem (P1) turns to be the covert rate maximization problem with respect to the hovering height, which is expressed as:

$$\begin{array}{}\mathrm{(25a)}& \begin{array}{ccc}\hfill \left(\mathrm{P}4\right):& \underset{H}{\max }{R}_b\hfill & \hfill \end{array}\mathrm{(25b)}& \begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& R_g\ge R_g^{\min },\hfill & \hfill \end{array}\mathrm{(25c)}& \begin{array}{ccc}\hfill & H_{min}\le H\le H_{max},\hfill & \hfill \end{array}\mathrm{(25d)}& \begin{array}{ccc}\hfill & \left(1-\kappa ({\alpha }_0+{\alpha }_2)\right)P_u|h_w{|}^2\le (\kappa -1){\sigma }_w^2.\hfill & \hfill \end{array}\end{array}$$

To ensure the corresponding optimization problem feasible, we first rewrite the covert constraint (25d) as follows:

$$\begin{array}{c}\hfill \frac{P_u{\mathcal{L}}_0}{d_w^2+H^2}\le \frac{(\kappa -1){\sigma }_w^2}{1-\kappa ({\alpha }_0+{\alpha }_2)},\end{array}$$

(26)

where $d_w$ is the horizontal distance from the UAV to Willie. Since maximizing $R_b={\beta }_1{R}_{g,s_0\left(k\right)}+R_{b,s_1\left(k\right)}$ is equivalent to maximizing ${\beta }_1{R}_{g,s_0\left(k\right)}$ and $R_{b,s_1\left(k\right)}$, respectively, while maximizing $R_{g,s_0\left(k\right)}$ and $R_{b,s_1\left(k\right)}$ are equivalent to maximize the SINR ${\gamma }_{g,s_0\left(k\right)}$ and ${\gamma }_{b,s_1\left(k\right)}$, respectively, the maximum $R_b$ can be obtained by maximizing ${\gamma }_{g,s_0\left(k\right)}+{\gamma }_{b,s_1\left(k\right)}$. Note that the covert constraint (25d) has been transformed into a signomial function (26), and we replace maximization in problem (P4) by minimization, which generates the SP problem as:

$$\begin{array}{}\mathrm{(27a)}& \begin{array}{ccc}\hfill \left(\mathrm{P}5\right):& \underset{H}{\min }\frac{1}{{\gamma }_{g,s_0\left(k\right)}}+\frac{1}{{\gamma }_{b,s_1\left(k\right)}}\hfill & \hfill \end{array}\mathrm{(27b)}& \begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& \left(1-\kappa ({\alpha }_0+{\alpha }_2)\right)P_u|h_w{|}^2\le (\kappa -1){\sigma }_w^2.\hfill & \hfill \end{array}\mathrm{(27c)}& \begin{array}{cc}\hfill & \left(\text{25b}\right),\left(\text{25c}\right),\left(\text{26}\right).\hfill \end{array}\end{array}$$

To solve the SP problem, the left-hand-side of (26) needs to be transformed into a posynomial. To this end, we approximate the posyomial denominator $\theta =d_w^2+H^2$ in (26) by the geometric mean [34]. Then, the approximation can be written as:

$$\begin{array}{c}\hfill \widehat{\theta }={\left(\frac{d_w^2}{{\lambda }_0}\right)}^{{\lambda }_0}{\left(\frac{H^2}{{\lambda }_1}\right)}^{{\lambda }_1},\end{array}$$

(28)

where we have ${\lambda }_0=\frac{d_w^2}{\theta }$ and ${\lambda }_1=\frac{H^2}{\theta }$. Through approximating the arithmetic mean by the geometric mean based on the classic arithmetic–geometric mean inequality [35], the SP problem (P5) is approximated by the following geometric programming problem:

$$\begin{array}{}\mathrm{(29a)}& \begin{array}{ccc}\hfill \left(\mathrm{P}6\right):& \underset{H}{\min }\frac{1}{{\gamma }_{g,s_0\left(k\right)}}+\frac{1}{{\gamma }_{b,s_1\left(k\right)}}\hfill & \hfill \end{array}\mathrm{(29b)}& \begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& \frac{P_u{\mathcal{L}}_0}{\widehat{\theta }}\le \frac{(\kappa -1){\sigma }_w^2}{1-\kappa ({\alpha }_0+{\alpha }_2)},\hfill & \hfill \end{array}\mathrm{(29c)}& \begin{array}{cc}\hfill & \left(\text{25b}\right),\left(\text{25c}\right).\hfill \end{array}\end{array}$$

Then, problem (P6) can be solved by using a convex solver such as CVX. The whole procedure of the AO approach is given in Algorithm 2. The convergence of Algorithm 2 is characterized by the following proposition.    

Algorithm 2: AO Algorithm of Maximizing the Covert Rate

1: Initialize the feasible ${\beta }_1^0$, ${\beta }_2^0$, and $H^0$, set $\ell =0$ 2: repeat 3:  Given ${\beta }_1^{\ell },{\beta }_2^{\ell }$ and $H^{\ell }$, solve the problem (P2) by Algorithm 1 to obtain ${\alpha }_0^{\ell +1},{\alpha }_1^{\ell +1}$, and ${\alpha }_2^{\ell +1}$ 4:  Given ${\alpha }_0^{\ell +1},{\alpha }_1^{\ell +1}$, ${\alpha }_2^{\ell +1}$, and $H^{\ell }$, solve the problem (P3) to obtain ${\beta }_1^{\ell +1}$ and ${\beta }_2^{\ell +1}$ 5:  Given $H^{\ell }$, compute ${\widehat{\theta }}^{\ell }$ according to (28) 6:  Given ${\alpha }_0^{\ell +1},{\alpha }_1^{\ell +1}$, ${\alpha }_2^{\ell +1}$, ${\beta }_1^{\ell +1}$ and ${\beta }_2^{\ell +1}$, solve the problem (P6) to obtain $H^{\ell +1}$ 7:  Update $\ell =\ell +1$ 8: until  $R_b^{\ell +1}-R_b^{\ell }<{\zeta }_2$

Proposition 2.

The developed Algorithm 2 is guaranteed to converge.

Proof. 

The proof is given in Appendix B. □

In Algorithm 2, the convex optimizations are involved in the iterations. It is noted that the complexity of Algorithm 2 mainly originates from solving the GP problem (P6). Since problem (P6) only contains a single variable, the complexity of solving problem (P6) is characterized by $\mathcal{O}\left(I_1\right)$ [36], where $I_1$ is the number of iterations of the AO approach.

3.5. Benchmark Scheme

In this section, we investigate the covert performance of the UAV-assisted NOMA (UAV-NOMA) system, where the conventional NOMA is applied for the transmissions from Alice to Bob and Grace [37,38]. Based on the assumption of channel ordering $|h_b{|}^2>{|h_g|}^2$, the SIC decoding order at Bob is $S_g\left(k\right)\to S_b\left(k\right)$, i.e., decoding $S_g\left(k\right)$ first before decoding $S_b\left(k\right)$. At Grace, $S_g\left(k\right)$ is decoded by treating the signal related to $S_b\left(k\right)$ as noise. Thus, more transmit power is allocated to transmit $S_g\left(k\right)$ at the UAV to ensure the successful SIC at the receiver [37,39], which also provides helps to shield Bob’s covert communications [38]. Accordingly, the achievable rate of Bob obtained from the transmission of $S_b\left(k\right)$ is given by $R_{b,s_g\left(k\right)}^{{}^{\mathrm{NOMA}}}={log}_2\left(1+{\gamma }_{b,s_g\left(k\right)}^{{}^{\mathrm{NOMA}}}\right)$, where

$$\begin{array}{c}\hfill {\gamma }_{b,s_g\left(k\right)}^{{}^{\mathrm{NOMA}}}=\frac{{\overline{\alpha }}_2{P}_u{\left|h_b\right|}^2}{{\overline{\alpha }}_1{P}_u{\left|h_b\right|}^2+{\sigma }_b^2}\end{array}$$

(30)

with ${\overline{\alpha }}_1$ and ${\overline{\alpha }}_2$ denoting the transmit power allocation factors for $x_b\left(k\right)$ and $x_g\left(k\right)$, respectively, and satisfying ${\overline{\alpha }}_1+{\overline{\alpha }}_2\le 1$ and $0\le {\overline{\alpha }}_1,{\overline{\alpha }}_2\le 1$. Furthermore, Bob’s covert rate obtained from the transmission of $x_b\left(g\right)$ can be expressed as $R_{b,s_b\left(k\right)}^{{}^{\mathrm{NOMA}}}={log}_2\left(1+{\gamma }_{b,s_b\left(k\right)}^{{}^{\mathrm{NOMA}}}\right)$, where

$$\begin{array}{c}\hfill {\gamma }_{b,s_b\left(k\right)}^{{}^{\mathrm{NOMA}}}=\frac{{\overline{\alpha }}_1{P}_u{\left|h_b\right|}^2}{{\sigma }_b^2}.\end{array}$$

(31)

Similarly, the achievable rate of Grace is given by $R_{g,s_g\left(k\right)}^{{}^{\mathrm{NOMA}}}={log}_2\left(1+{\gamma }_{g,s_g\left(k\right)}^{{}^{\mathrm{NOMA}}}\right)$, where

$$\begin{array}{c}\hfill {\gamma }_{g,s_g\left(k\right)}^{{}^{\mathrm{NOMA}}}=\frac{{\overline{\alpha }}_2{P}_u{\left|h_g\right|}^2}{{\overline{\alpha }}_1{P}_u{\left|h_g\right|}^2+{\sigma }_g^2}.\end{array}$$

(32)

We assume that Willie also adopts the Neyman–Pearson criterion to make the binary decision in the UAV-NOMA system. By replacing ${\overline{\alpha }}_2={\alpha }_0+{\alpha }_2$, where ${\alpha }_0$ and ${\alpha }_2$ are the transmit power allocation factor defined in the UAV-RSMA system, it is readily shown that the covertness constraint is given by

$$\begin{array}{ccc}\hfill & & (1-\kappa {\overline{\alpha }}_2)P_u|h_w{|}^2\le (\kappa -1){\sigma }_w^2.\hfill \end{array}$$

(33)

To maximize Bob’s covert rate, the transmit power allocation and UAV hovering height should be jointly optimized and the corresponding optimization problem is formulated as:

$$\begin{array}{}\mathrm{(34a)}& \begin{array}{ccc}\hfill \left(\mathrm{P}7\right):& \underset{{\overline{\alpha }}_1,{\overline{\alpha }}_2,H}{\max }{R}_{b,s_b\left(k\right)}^{{}^{\mathrm{NOMA}}}\hfill & \hfill \end{array}\mathrm{(34b)}& \begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& {\overline{\alpha }}_1+{\overline{\alpha }}_2\le 1,{\overline{\alpha }}_1\le {\overline{\alpha }}_2,\hfill & \hfill \end{array}\mathrm{(34c)}& \begin{array}{ccc}\hfill & R_{b,s_b\left(k\right)}^{{}^{\mathrm{NOMA}}}>R_{g,s_g\left(k\right)}^{{}^{\mathrm{NOMA}}},\hfill & \hfill \end{array}\mathrm{(34d)}& \begin{array}{ccc}\hfill & R_{g,s_g\left(k\right)}^{{}^{\mathrm{NOMA}}}\ge R_g^{\min },\hfill & \hfill \end{array}\mathrm{(34e)}& \begin{array}{cc}\hfill & (1-\kappa {\overline{\alpha }}_2)P_u|h_w{|}^2\le (\kappa -1){\sigma }_w^2.\hfill \end{array}\end{array}$$

In problem (P7), (34b) is the transmit power allocation constraint, constraint (34c) guarantees the successful SIC at Bob, constraint (34d) represents Grace’s QoS requirements, which can be further written as ${\overline{\alpha }}_2{P}_u{\left|h_g\right|}^2\ge {\gamma }_{\mathrm{th}}\left({\overline{\alpha }}_1{P}_u{\left|h_g\right|}^2+{\sigma }_g^2\right)$ with ${\gamma }_{\mathrm{th}}=2^{R_g^{\min }}-1$. Similarly to the UAV-RSMA system, the joint optimization of the transmit power allocation and UAV hovering height is impossible in the considered UAV-NOMA system. Fortunately, the AO approach can be applied to obtain the optimal $\{{\overline{\alpha }}_1^{*},{\overline{\alpha }}_2^{*}\}$ with the given H and $h^{*}$ and $\{{\overline{\alpha }}_1,{\overline{\alpha }}_2\}$. The subproblems of optimizing the transmit power allocation and UAV hovering height are separately presented in the following.

For any given H, the covert rate maximization problem of optimizing the transmit power allocation can be formulated as:

$$\begin{array}{}\mathrm{(35a)}& \begin{array}{ccc}\hfill \left(\mathrm{P}8\right):& \underset{{\overline{\alpha }}_1,{\overline{\alpha }}_2}{\max }{R}_{b,s_b\left(k\right)}^{{}^{\mathrm{NOMA}}}\hfill & \hfill \end{array}\mathrm{(35b)}& \begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& {\overline{\alpha }}_1+{\overline{\alpha }}_2\le 1,{\overline{\alpha }}_1\le {\overline{\alpha }}_2,\hfill & \hfill \end{array}\mathrm{(35c)}& \begin{array}{ccc}\hfill & {\overline{\alpha }}_2{P}_u{\left|h_g\right|}^2\ge {\gamma }_{\mathrm{th}}({\overline{\alpha }}_1{P}_u{\left|h_g\right|}^2+{\sigma }_g^2),\hfill & \hfill \end{array}\mathrm{(35d)}& \begin{array}{ccc}\hfill & (1-\kappa {\overline{\alpha }}_2)P_u|h_w{|}^2\le (\kappa -1){\sigma }_w^2.\hfill & \hfill \end{array}\end{array}$$

Proposition 3.

Problem (P8) can be optimized only when ${\overline{\alpha }}_1+{\overline{\alpha }}_2=1$.

Proof. 

The proof is similar to that in Appendix A. □

Based on the results in Proposition 3, we let ${\overline{\alpha }}_2=1-{\overline{\alpha }}_1$ and transform (35c) equivalently to the form of ${\overline{\alpha }}_1\le {\Xi }_5$ with ${\Xi }_5=\frac{P_u{\left|h_g\right|}^2-{\gamma }_{\mathrm{th}}{\sigma }_g^2}{(1+{\gamma }_{\mathrm{th}}){\left|h_g\right|}^2{P}_u}$. Then, problem (P8) can be simplified as:

$$\begin{array}{}\mathrm{(36a)}& \begin{array}{ccc}\hfill \left(\mathrm{P}9\right):& \underset{{\overline{\alpha }}_1,{\overline{\alpha }}_2}{\max }{R}_{b,s_b\left(k\right)}^{{}^{\mathrm{NOMA}}}\hfill & \hfill \end{array}\mathrm{(36b)}& \begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& {\overline{\alpha }}_1\le \min \left\{\frac{1}{2},{\Xi }_1,{\Xi }_5\right\},\hfill & \hfill \end{array}\end{array}$$

Since $R_{g,s_b\left(k\right)}^{{}^{\mathrm{NOMA}}}$ is a monotonically increasing function with respect to ${\overline{\alpha }}_1$, the optimal transmit power allocation in the UAV-NOMA system is given by ${\overline{\alpha }}_1^{*}=\min \left\{\frac{1}{2},{\Xi }_1,{\Xi }_5\right\}$ and ${\overline{\alpha }}_2^{*}=1-{\overline{\alpha }}_1^{*}$.

For any given ${\overline{\alpha }}_1$ and ${\overline{\alpha }}_2$, problem (P7) reduces to only optimize the UAV hovering height. Similarly to problem (P6), we transform the covert rate maximization problem to a minimization problem as follows:

$$\begin{array}{}\mathrm{(37a)}& \begin{array}{ccc}\hfill \left(\mathrm{P}10\right):& \underset{H}{\min }\frac{1}{{\gamma }_{b,s_b\left(k\right)}^{{}^{\mathrm{NOMA}}}}\hfill & \hfill \end{array}\mathrm{(37b)}& \begin{array}{}\begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& R_g\ge R_g^{\min },\hfill & \hfill \end{array}\end{array}\mathrm{(37c)}& \begin{array}{}\begin{array}{ccc}\hfill & H_{min}\le H\le H_{max},\hfill & \hfill \end{array}\end{array}\mathrm{(37d)}& \begin{array}{}\begin{array}{ccc}\hfill & \frac{P_u{\mathcal{L}}_0}{\widehat{\theta }}\le \frac{(\kappa -1){\sigma }_w^2}{1-\kappa {\overline{\alpha }}_2}.\hfill & \hfill \end{array}\end{array}\end{array}$$

Now, problem (P10) can be solved by using a convex optimization tool such as CVX. Based on the decoupled problems (P9) and (P10), the optimized system parameters can be obtained in an AO way, and the corresponding algorithm is omitted here to save space.

Compared to the UAV-RSMA scheme, the UAV-NOMA scheme only has two transmit power allocation coefficients ${\overline{\alpha }}_1$ and ${\overline{\alpha }}_2$, which results in reduced degrees of freedom (DoF). Consequently, the transmit power can be allocated only between the signals of Bob and Grace, while the SIC processing is conducted only at Bob. On the contrary, the UAV-RSMA scheme achieves a higher DoF than the UAV-NOMA scheme by utilizing rate splitting. The transmit power can be allocated not only between the private messages of Bob and Grace but also between the common and private messages. Accordingly, common rate allocation is conducted between Bob and Grace. With the increased DoF, the covert rate of the UAV-RSMA scheme can be effectively increased compared to the UAV-NOMA scheme, as will be verified by the simulation results in the next section.

4. Simulation Results

In this section, the simulation results are presented to establish the covert performance achieved by the proposed UAV-RSMA scheme. Unless otherwise stated, the following system parameter setup is assumed in the simulations: ${\mathcal{L}}_0=-30$ dB, ${\sigma }_b^2={\sigma }_g^2={\sigma }_w^2=-80$ dBm, $H_{min}=40$ m, and $H_{max}=100$ m. The horizontal locations of Bob and Grace are set as ${\mathbf{q}}_b=[0,0]$ and ${\mathbf{q}}_b=[200,0]$, respectively. The horizontal location of the UAV is positioned at ${\mathbf{q}}_u=[55,0]$ such that both Bob has a better LoS channel connecting to the UAV than that of Grace. Since it is hard for the UAV to locate Willie, we consider the worst scenario in which Willie is positioned right below the UAV.

In Figure 3, the trends of covert rate varying with the UAV’s transmit power are depicted. In the simulations, we set $R_g^{min}=3$ bps/Hz and $\epsilon =0.01$. For comparison purposes, the results achieved by the proposed AO scheme with fixed H and random H are presented. In addition, the results achieved by the benchmark scheme are also presented. From Figure 3, it can be seen that the proposed UAV-RSMA with the AO algorithm achieves the highest covert rate among all the schemes, which verifies the effectiveness of the proposed AO algorithm. Since the UAV-RSMA scheme obtains a higher DoF than the UAV-NOMA scheme by optimizing transmit power and common rate allocations, the covert rate can be significantly improved by applying the UAV-RSMA scheme. At the covert rate level of $R_b=10$ bps/Hz, about 5 dBm transmit power gain can be achieved by the proposed scheme. With the increase in $P_u$, the covert rates achieved by all the schemes also increase. Moreover, the gap between the covert rates of the UAV-RSMA and UAV-NOMA schemes increases with increasing $P_u$, which further verifies the superior covert rate performance of the UAV-RSMA scheme over that of the UAV-NOMA scheme. By adopting the fixed H and random H in the proposed AO algorithm, in which only the transmit power and common rate allocations are optimized, the obtained covert rate is smaller than the highest curve of $R_b$. For example, at the covert rate level of $R_b=10$ bps/Hz, about 7 dBm transmit power gain can be achieved by the AO algorithm with optimizing H compared to the AO algorithm with the fixed $H=55$ m. Moreover, the AO algorithm with the random H achieves a smaller $R_b$ than the AO algorithm with the fixed $H=55$ m in the low and middle transmit power regions, whereas a higher $R_b$ is achieved by the AO algorithm with the random H than the AO algorithm with fixed $H=55$ m in the high transmit power region. However, the AO algorithm with optimizing H obviously achieves a higher $R_b$ than the AO algorithms with the fixed H and random H. Thus, the importance of optimizing the UAV hovering height is verified by the results in Figure 3. For the considered benchmark scheme, the achieved covert rate is much lower than that of the UAV-RSMA scheme, which establishes the superior covert performance of the proposed UAV-RSMA scheme.

In Figure 4, the covert rate versus the covertness parameter $\epsilon$ is investigated. For the simulations corresponding to Figure 4, we set $P_u=20$ dBm and $R_g^{min}=2$ bps/Hz unless otherwise stated. The results in Figure 4 show that the covert rate achieved by all the schemes increase monotonically with the increasing $\epsilon$. Since a larger $\epsilon$ represents a smaller detection error probability, a looser covert constraint results in a larger $R_b$. Moreover, the proposed UAV-RSMA with the AO algorithm achieves the largest $R_b$ among all the schemes. The reason for this phenomenon is that the UAV-RSMA scheme obtains the additional DoF by optimizing not only the transmit power allocation among common and private messages but also the common rate allocation between Bob and Grace compared to that of the UAV-NOMA scheme. Although the covert rates achieved by the AO algorithms with the fixed H and random H increase with the increasing $\epsilon$, the corresponding $R_b$ is smaller than that of the AO algorithm with optimizing H. For example, at the covertness level of $\epsilon =0.1$, which is a typical covertness requirement in practice, the covert rate achieved by the AO algorithm with the optimizing H is 1 bps/Hz larger than that of the random H and 0.5 bps/Hz larger than that of the fixed $H=60$ m. Thus, the results in Figure 4 verify that the covert rate performance in the case of not optimizing the UAV hovering height suffers a decrease compared to the optimized H in the considered $\epsilon$ region. Furthermore, the superiority of the proposed scheme over that of the benchmark scheme is also verified by the curves in Figure 4. As can be seen from Figure 4, although the covert rates achieved by the UAV-NOMA scheme increase with the increasing $\epsilon$, the values of $R_b$ are much smaller than those of the UAV-RSMA schemes. For example, at the covertness level of $\epsilon =0.1$, the covert rate achieved by the UAV-RSMA scheme with the AO algorithm is 3.5 bps/Hz larger than that of the UAV-NOMA scheme with the AO algorithm. In the cases of fixed $H=60$ m and random H, the covert rates achieved by the UAV-RSMA scheme are 3.5 bps/Hz and 3 bps/Hz larger than those achieved by the UAV-NOMA scheme, respectively. In addition, the results in Figure 4 show that with the fixed H and random H, the covert rates achieved by the UAV-NOMA scheme is decreased compared to that achieved by the UAV-NOMA with optimizing H, which also verifies that the optimization of the UAV hovering height is important to enhance the covert rate performance.

To further reveal the impacts of the hovering height of the UAV on the covert rate, we plot the covert rate versus hovering height in Figure 5. In the simulations, we set $P_u=30$ dBm and $R_g^{min}=3$ bps/Hz. The curves in Figure 5 show that the covert rates achieved by all the schemes first increase with the increase in the hovering height. Then, the covert rates decrease with the increase in the hovering height after H exceeds a certain value. In the low H region, the covert communication performance is greatly limited by the detection of Willie, such that a lower H results in a smaller covert rate and a higher H corresponds to a larger covert rate. On the other hand, with the increase in H, more transmit power is needed to guarantee the Qos requirements of the public user, which results in a reduced transmit power for the covert user such that the covert rate is decreased. In the high H region, we can see that the covert rates achieved with different $\epsilon$ tend to be the same. The reason for this phenomenon is that an extremely high H results in a very small detection probability at Willie, so that the covert constraint (33) does not affect the achievable covert rate. Moreover, the curves in Figure 5 also show that a smaller $\epsilon$ results in a smaller covert rate and vice verse. Also, the results in Figure 5 verify that the UAV-RSMA scheme achieves a larger covert rate than the UAV-NOAM scheme.

The impacts of Grace’s QoS requirements on the covert rate are depicted in Figure 6, where we set $P_u=20$ dBm, $R_g^{min}=2.5$ bps/Hz, and $\epsilon =0.1$. It is shown in Figure 6 that the covert rates achieved by all the schemes decrease monotonically with increasing $R_g^{min}$. Moreover, the covert rate achieved by the proposed UAV-RSMA scheme with the AO algorithm is the highest among all the schemes, which is also no less than those of the benchmark schemes even in the high $R_g^{min}$ region. At the level of $R_g^{min}=1.5$ bps/Hz, the covert rate achieved by the UAV-RSMA scheme with optimizing H is 2.2 bps/Hz higher than the UAV-RSMA scheme with the fixed H and 3 bps/Hz higher than the UAV-RSMA scheme with the random H. These results verify that jointly optimizing the transmit power allocation, common rate allocation, and hovering height is necessary to improve the covert rate. Also, the effectiveness of the proposed AO algorithm is verified. Compared to the covert rate achieved by the UAV-NOMA scheme with the AO algorithm, the covert rate achieved by the UAV-RSMA scheme with the AO algrithm is 3.6 bps/Hz higher at the level of $R_g^{min}=1.5$ bps/Hz, which also verifies that the proposed UAV-RSMA scheme achieves a better covert rate performance than the UAV-NOMA scheme.

5. Conclusions

In this paper, the covert communications in air-to-ground UAV-RSMA have been investigated with the aim of maximizing the covert rate. To achieve the maximum covert rate, the system parameters, including the transmit power allocation, common rate allocation, and UAV hovering height, have been jointly considered in the optimization. Due to the non-convexity of the formulated covert rate maximization problem and the highly coupled system parameters, an AO algorithm has been designed to obtain the optimized solution with the aid of decoupling the original problem into three subproblems of the transmit power allocation, common rate allocation, and UAV hovering height. We derive the optimal solution of the transmit power allocation and common rate allocation for the corresponding subproblems and propose an AO approach to obtain the optimized UAV hovering height. The simulation results verify the superior covert performance achieved by the proposed AO algorithm. The superiority of the proposed scheme over the benchmark schemes has also been established by the simulation results.