© The Author(s) 2024. Published by EDP Sciences and China Science Publishing & Media Ltd.

1. Introduction

In the B5G/6G wireless communication network, unmanned aerial vehicles (UAVs) are prominent owing to their advanced agility and potential capability in disasters, transportation, or complex environments[1–3]. Unlike terrestrial communication facilities, UAVs can improve the quality of service, due to the high controllability and dominance of line-of-sight (LoS) links at moderate heights [4–6]. Thus, UAVs not only act as aerial base stations to send information to ground users [7–9] but also are utilized to collect data from terrestrial nodes [10–12].

Recently, many studies have been devoted to the optimization of UAV-based uplink (UL) and downlink (DL) communication [13–15]. Based on half-duplex communication, resource allocation has been optimized to improve the network throughput [16]. For simultaneous transmission and reception (STR), frequency band-division-duplex (FDD) with fixed bandwidth allocation was considered [17]. To perform STR in the whole bandwidth, two UAVs’ cooperative communication network was considered in [18], where one UAV was dispatched to send information and the other UAV was used to receive information from ground users. By applying the full-duplex (FD) technology, the optimization of resource allocation between DL and UL communication is important [19–24]. In [19], one UAV was deployed to simultaneously serve multiple downlink users (DLUs) and uplink users (ULUs), where four different STR schemes were investigated to maximize the throughput of the system. A novel joint uplink-downlink resource allocation algorithm that optimizes the power allocation for uplink and downlink to achieve balanced average rates for both UL and DL communications was proposed [20]. A cellular network in [21] was investigated comprising a UAV-mounted aerial base station and multiple terrestrial base stations, each serving multiple users. The balance between DL throughput and UL throughput in the UAV-based communication networks was investigated [22]. The UL and DL resource allocation were jointly optimized to improve spectrum efficiency [23]. Similarly, [24] improved the overall capacity by considering scheduling, UAV trajectory, and ULU power allocation. Full-duplex communication faces challenges due to self-interference, where the transmit signal can interfere significantly with the received signal. This strong interference hinders the performance of FD systems [25]. It is a remarkable fact that the quality of service in FD-based communication depends on self-interference, which cannot be neglected in the network. Instead of utilizing the FD communication, FDD and time-fraction (TF) are investigated to implement the simultaneous transmission and reception within a time slot in this paper. Besides, resource allocation such as bandwidth allocation and time slot scheduling are considered in the optimization [26, 27].

In addition, the security of UAV-based communications is more susceptible to interference from malicious terrestrial nodes due to the physical properties of the air-ground broadcast channel [28]. In [29], the secrecy rate of DL and UL communication was optimized considering the security of UAV-based communication in the existence of eavesdroppers. Additionally, considering both unfriendly jammers and eavesdroppers in [30], the UAV trajectory and power allocation were optimized to improve the secrecy rate of DL and UL communications. However, [29] and [30] considered scenarios where users can only act as uplink or downlink nodes during a one-time slot (TS). The scenario of the UAV simultaneously serving ULUs and DLUs in the presence of malicious jammers has not been investigated.

Motivated by the aforementioned issues, a UAV-assisted STR communication system in the existence of multiple jammers is considered. We aim to maximize the worst communication throughput among DLUs and ULUs. The main contributions of this investigation can be captured as follows:

• To perform the STR, we first investigate the FDD-based scheme for UAV-assisted downlink-and-uplink communication. The available bandwidth is divided for DL and UL communication within a single time slot. The presence of multiple jammers adds complexity to the system, as they send malicious interference signals to both the UAV and users. To address these challenges, we propose a joint optimization approach. Specifically, we optimize the UAV’s three-dimensional trajectory, downlink scheduling, uplink scheduling, bandwidth allocation, and UAV’s transmission power. Our objective is to maximize the worst throughput among DLUs and ULUs, considering the impact of interference from the jammers.

• Furthermore, instead of the bandwidth allocation for DL and UL communication, a time fraction (TF) based transmission scheme to maximize the worst throughput by utilizing the fraction of a time slot for the DL communication and the remaining fraction for the UL communication [31]. In the TF-based scheme, we optimize the time fraction to balance DL and UL communication. To address each non-convex optimization problem, we develop the mathematically soluble optimization algorithm for each scheme based on the block coordinate decomposition (BCD) method and successive convex approximation (SCA) technique.

• Finally, we analyze the convergence of the proposed approaches and examine the system performance by adopting different schemes through numerical simulations.

The organization of the paper is as follows. The channel model and FDD-based UAV-assisted system model are presented in Section 2, whereas TF-based UAV-assisted STR is constructed in Section 3 to optimize downlink scheduling, uplink scheduling, 3D trajectory, UAV transmission power, and time fraction. Simulation results are presented in Section 4 to evaluate the performance of the proposed algorithms, and finally, the paper is concluded in Section 5.

2. Frequency band-division-duplex based scheme

We establish a UAV-based joint DL and UL communication network with K_D terrestrial downlink users and K_U uplink users in the presence of K_M jammers as shown in Figure 1.

Figure 1. FDD-based UAV-assisted communication

The frequency band-division-duplex is proposed for simultaneous transmission and reception, where the downlink channel is orthogonal to the uplink channel [32]. The planned flight time T is evenly separated into N time slots δ and FDD-based UAV can communicate with a DLU and a ULU during one TS. Therefore, the constraints for DLU’s scheduling and ULU’s scheduling can be given by

$$\begin{array}{c}\hfill \sum _{j=1}^{K_D}{x}_{j,n}^{\mathrm{DL}}\le 1,\forall n,j\in K_D,\end{array}$$(1)

$$\begin{array}{c}\hfill \sum _{i=1}^{K_U}{x}_{i,n}^{\mathrm{UL}}\le 1,\forall n,i\in K_U,\end{array}$$(2)

$$\begin{array}{c}\hfill x_{j,n}^{\mathrm{DL}}\in \{0,1\},\forall n,j\in K_D,\end{array}$$(3)

$$\begin{array}{c}\hfill x_{i,n}^{\mathrm{UL}}\in \{0,1\},\forall n,i\in K_U,\end{array}$$(4)

where $x_{j,n}^{\mathrm{DL}}$ and $x_{i,n}^{\mathrm{UL}}$ represent user scheduling of j-th DLU j ∈ {1, …, K_D} and i-th ULU i ∈ {1, …, K_U}, respectively.

Different from the fixed bandwidth segmentation in [17], we define τ_n as the portion of the normalized bandwidth, which is allocated to the downlink communication during n-th TS. Meanwhile, ζ denotes the portion for guarding, while (η − τ_n) with η = 1 − ζ is denoted by us as the rest of the bandwidth for uplink communication. Define τ  = {τ_n, ∀n}, ${\mathcal{X}}_{\mathit{D}}=\{x_{j,n}^{\mathrm{DL}},\forall n,j\}$, ${\mathcal{X}}_{\mathit{U}}=\{x_{i,n}^{\mathrm{UL}},\forall n,i\}$, 𝒫 = {p_n^b, ∀n} and 𝒬 = {[Q_n H_n],∀n}. 𝒫 is the transmit power of UAV. 𝒬 is the 3D UAV-assisted coordinate, where Q_n and H_n are denoted as the horizontal coordinate and the flight height, respectively. Let [Q_ini H_ini] denote the initial point, while [Q_end H_end] denotes the end location. The UAV’s mobility is limited by

$$\begin{array}{c}\hfill \Vert {\mathit{Q}}_n-{\mathit{Q}}_{n-1}\Vert \le V_{\max }^{\mathit{xy}}\delta ,\forall n,\end{array}$$(5)

$$\begin{array}{c}\hfill {\mathit{Q}}_{\mathrm{ini}}={\mathit{Q}}_0,{\mathit{Q}}_{\mathrm{end}}={\mathit{Q}}_N,\end{array}$$(6)

$$\begin{array}{c}\hfill |H_n-H_{n-1}|\le V_{\max }^z\delta ,\forall n,\end{array}$$(7)

$$\begin{array}{c}\hfill H_{\mathrm{ini}}=H_0,H_{\mathrm{end}}=H_N,\end{array}$$(8)

where $V_{\max }^{\mathit{xy}}$ is the maximum horizontal velocity and $V_{\max }^z$ is the maximum vertical velocity. The UAV-to-ground (U2G) channel, and the ground-to-UAV (G2U) channel are assumed to be dominated by LoS [4]. Due to the characteristics of wireless channels, the channel from the UAV to the j-th DLU, the channel from the i-th ULU to the UAV and the channel from the m-th jammer to the UAV during the n-th TS can be regarded as Rician fading [18]:

$$\begin{array}{c}\hfill {\widehat{g}}_{n,j}^{\mathrm{DL}}(\mathcal{Q})={\rho }_n\sqrt{g_{n,j}^{\mathrm{DL}}(\mathcal{Q})},\forall n,j\in K_D,\end{array}$$(9)

$$\begin{array}{c}\hfill {\widehat{g}}_{i,n}^{\mathrm{UL}}(\mathcal{Q})={\rho }_n\sqrt{g_{i,n}^{\mathrm{UL}}(\mathcal{Q})},\forall n,i\in K_U,\end{array}$$(10)

$$\begin{array}{c}\hfill {\widehat{g}}_{m,n}^{\mathrm{UL}}(\mathcal{Q})={\rho }_n\sqrt{g_{m,n}^{\mathrm{UL}}(\mathcal{Q})},\forall n,m\in K_M,\end{array}$$(11)

where the large-scale attenuation can be expressed as

$$\begin{array}{c}\hfill g_{n,j}^{\mathrm{DL}}(\mathcal{Q})=\frac{{\beta }_0}{{\Vert {\mathit{Q}}_n-{\mathit{w}}_j\Vert }^2+{H_n}^2},\end{array}$$

$$\begin{array}{c}\hfill g_{i,n}^{\mathrm{UL}}(\mathcal{Q})=\frac{{\beta }_0}{{\Vert {\mathit{Q}}_n-{\mathit{w}}_i\Vert }^2+{H_n}^2},\end{array}$$

$$\begin{array}{c}\hfill g_{m,n}^{\mathrm{UL}}(\mathcal{Q})=\frac{{\beta }_0}{{\Vert {\mathit{Q}}_n-{\mathit{w}}_m\Vert }^2+{H_n}^2}.\end{array}$$

Let K_r be the Rician factor, while ${\widehat{\rho }}_n$ denotes the deterministic LoS component with $|{\widehat{\rho }}_n|=1$, ${\overline{\rho }}_n\sim \mathcal{C}\mathcal{N}(0,1)$ is the small-fading fraction and β₀ indicates the channel gain at the reference distance. The small-scale fading can be expressed as ${\rho }_n=\sqrt{\frac{K_r}{K_r+1}}{\widehat{\rho }}_n+\sqrt{\frac{1}{K_r+1}}{\overline{\rho }}_n$ [18].

For the ground-to-ground (G2G) channels between m-th jammer and j-th DLU within one TS, the channels are largely subject to Rayleigh fading, which can be defined as $h_{m,j}^{\mathrm{DL}}={\beta }_0{d}_{m,j}^{-\alpha }\varrho$ [24]. Additionally, 𝜚 is a random value that follows an exponential distribution with a unit mean.

The FDD-based network downlink throughput within the n-th TS can be formulated as [33]

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\dot{R}}_{j,n}^{\mathrm{FDD}}(\mathcal{Q},\mathcal{P})={\tau }_n{log}_2(1+\frac{p_n^b{|{\widehat{g}}_{n,j}^{\mathrm{DL}}|}^2}{{\sum }_{m=1}^{K_M}{P}_m{\beta }_0{d}_{m,j}^{-\alpha }+{\tau }_n{\sigma }^2}),\forall n,j\in K_D,\end{array}\end{array}$$(12)

where $p_n^b$ is denoted as the transmission power of the UAV, while P_m is the interference power of m-th jammer. σ² is the white Gaussian noise power. Also, the FDD-based network uplink throughput is given by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\dot{R}}_{i,n}^{\mathrm{FDD}}(\mathcal{Q})=(\eta -{\tau }_n){log}_2(1+\frac{P_i{|{\widehat{g}}_{i,n}^{\mathrm{UL}}|}^2}{{\sum }_{m=1}^{K_M}{P}_m{|{\widehat{g}}_{m,n}^{\mathrm{UL}}|}^2+(\eta -{\tau }_n){\sigma }^2}),\forall n,i\in K_U,\end{array}\end{array}$$(13)

where P_i represents the ULU’s transmission power.

However, because of the randomness of the channel gain, the achievable throughput is random. We aim to investigate the average throughput of the network and the approximated throughput expression is given by [18], the FDD throughput of the downlink during n-th TS is formulated as

$$\begin{array}{c}\hfill \begin{array}{cc}& \mathbb{E}[{\dot{R}}_{j,n}^{\mathrm{FDD}}(\mathcal{Q},\mathcal{P})]={\tau }_n{log}_2(1+\frac{p_n^b{g}_{n,j}^{\mathrm{DL}}}{{\sum }_{m=1}^{K_M}{P}_m{\beta }_0{d}_{m,j}^{-\alpha }+{\tau }_n{\sigma }^2})\triangleq R_{j,n}^{\mathrm{FDD}}(\mathcal{Q},\mathcal{P},\mathit{\tau }),\hfill \end{array}\end{array}$$(14)

where σ² denotes the white Gaussian noise power. Besides, the FDD throughput of the uplink during n-th TS is given by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathbb{E}[{\dot{R}}_{i,n}^{\mathrm{FDD}}(\mathcal{Q},,\mathit{\tau })]=(\eta -{\tau }_n){log}_2(1+\frac{p_i{g}_{i,n}^{\mathrm{UL}}}{{\sum }_{m=1}^{K_M}{P}_m{g}_{m,n}^{\mathrm{UL}}+(\eta -{\tau }_n){\sigma }^2})\triangleq R_{i,n}^{\mathrm{FDD}}(\mathcal{Q},\mathit{\tau }).\end{array}\end{array}$$(15)

The binary constraints (3)–(4) make it inefficient to obtain a global feasible solution. According to the approach in the study [24], the binary constraints can be rewritten as

$$\begin{array}{c}\hfill 0\le x_{j,n}^{\mathrm{DL}}\le 1,\forall n,j\in K_D,\end{array}$$(16)

$$\begin{array}{c}\hfill 0\le x_{i,n}^{\mathrm{UL}}\le 1,\forall n,i\in K_U.\end{array}$$(17)

Considering the 3D trajectory design, the ULU/DLU scheduling, and the UAV transmission power, to find the balance between DL and UL communication, we aim to maximize the worst average throughput of the network. The optimization problem of FDD-based communication can be formulated as

$$\begin{array}{ccc}& \underset{{\mathcal{X}}_{\mathit{D}},{\mathcal{X}}_{\mathit{U}},\mathcal{Q},\mathcal{P},\mathit{\tau }}{max}\hfill & \hfill min\{\underset{j}{min}{R}_j^{\mathrm{FDD},DL}({\mathcal{X}}_{\mathit{D}},\mathcal{Q}\mathbf{,}\mathcal{P},\mathit{\tau }),\underset{i}{min}{R}_i^{\mathrm{FDD},UL}(({\mathcal{X}}_{\mathit{U}},\mathit{Q},\mathit{\tau })\}\end{array}$$(18a) $$\begin{array}{ccc}\hfill & \text{s.t.}\hfill & \hfill 0\le p_n^b\le P_{max},\forall n,\end{array}$$(18b) $$\begin{array}{cc}\hfill & 0\le {\tau }_n\le 1,\forall n,\hfill \end{array}$$(18c) $$\begin{array}{cc}\hfill & (1)-(2),(5)-(8),(16)-(17),\hfill \end{array}$$(18d)

where

$$\begin{array}{c}\hfill R_j^{\mathrm{FDD},DL}({\mathcal{X}}_{\mathit{D}},\mathcal{Q}\mathbf{,}\mathcal{P},\mathit{\tau })=\frac{1}{N}\sum _{n=1}^N{x}_{j,n}^{\mathrm{DL}}{R}_{j,n}^{\mathrm{FDD}}(\mathcal{Q}\mathbf{,}\mathcal{P},\mathit{\tau })\end{array}$$

and

$$\begin{array}{c}\hfill R_i^{\mathrm{FDD},UL}({\mathcal{X}}_{\mathit{U}},\mathit{Q},\mathit{\tau })=\frac{1}{N}\sum _{n=1}^N{x}_{i,n}^{\mathrm{UL}}{R}_{i,n}^{\mathrm{FDD}}(\mathit{Q},\mathit{\tau }).\end{array}$$

denote the average throughput of j-th downlink and i-th uplink communication, respectively. P_max denotes the transmission power allocation budget.

Since the optimization problem (18) cannot be solved efficiently, we introduce the slack variable ϑ and the problem (18) is reformulated as follows

$$\begin{array}{ccc}& \underset{{\mathcal{X}}_{\mathit{D}},{\mathcal{X}}_{\mathit{U}},\mathcal{Q},\mathcal{P},\mathit{\tau },\vartheta }{max}\hfill & \hfill \vartheta \end{array}$$(19a) $$\begin{array}{ccc}\hfill & \text{s.t.}\hfill & \hfill 0\le p_n^b\le P_{max},\forall n,\end{array}$$(19b) $$\begin{array}{cc}\hfill & 0\le {\tau }_n\le 1,\forall n,\hfill \end{array}$$(19c) $$\begin{array}{cc}\hfill & (1)-(2),(5)-(8),(16)-(17),\hfill \end{array}$$(19d) $$\begin{array}{cc}\hfill & R_j^{\mathrm{FDD},\mathrm{DL}}({\mathcal{X}}_{\mathit{D}},\mathcal{Q}\mathbf{,}\mathcal{P},\mathit{\tau })\ge \vartheta ,j\in K_D,\hfill \end{array}$$(19e) $$\begin{array}{cc}\hfill & R_i^{\mathrm{FDD},\mathrm{UL}}({\mathcal{X}}_{\mathit{U}},\mathit{Q},\mathit{\tau })\ge \vartheta ,i\in K_U.\hfill \end{array}$$(19f)

We decompose the optimization problem (19) into four subproblems and alternately optimize these subproblems by using the BCD technique.

2.1. Subproblem for the user scheduling

We fixed the feasible point the transmission power of the UAV 𝒫, the 3D trajectory 𝒬, and the allocation portion τ to obtain the optimal DLU’s scheduling 𝒳_D and ULU’s scheduling 𝒳_U. The optimization subproblem can be expressed as

$$\begin{array}{ccc}& \underset{{\mathcal{X}}_{\mathit{D}},{\mathcal{X}}_{\mathit{U}},\vartheta }{max}\hfill & \hfill \vartheta \end{array}$$(20a) $$\begin{array}{ccc}\hfill & \text{s.t.}\hfill & \hfill (16)-(17),(1)-(2),\end{array}$$(20b) $$\begin{array}{cc}\hfill & R_j^{\mathrm{FDD},DL}({\mathcal{X}}_{\mathit{D}})\ge \vartheta ,j\in K_D,\hfill \end{array}$$(20c) $$\begin{array}{cc}\hfill & R_i^{\mathrm{FDD},UL}({\mathcal{X}}_{\mathit{U}})\ge \vartheta .i\in K_U.\hfill \end{array}$$(20d)

Obviously, the subproblem for user scheduling is a standard linear problem, which can be solved efficiently by using an optimization package [24].

2.2. Subproblem for the portion of bandwidth

With the achievable DLU’s scheduling 𝒳_D, ULU’s scheduling 𝒳_U, the transmission power of the UAV 𝒫 and the 3D trajectory 𝒬, the objective function (19) can be reformulated as follows

$$\begin{array}{ccc}& \underset{\mathit{\tau },\vartheta }{max}\hfill & \hfill \vartheta \end{array}$$(21a) $$\begin{array}{ccc}\hfill & \text{s.t.}\hfill & \hfill 0\le {\tau }_n\le 1,\forall n,\end{array}$$(21b) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^N{x}_{j,n}^{\mathrm{DL},r}{R}_{j,n}^{\mathrm{FDD}}(\mathit{\tau })\ge \vartheta ,\forall n,j\in K_D,\hfill \end{array}$$(21c) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^N{x}_{i,n}^{\mathrm{UL},r}{R}_{i,n}^{\mathrm{FDD}}(\mathit{\tau })\ge \vartheta ,\forall n,i\in K_U.\hfill \end{array}$$(21d)

To deal with constraint (21c) and constraint (21d) more efficiently, we apply the inequality (66) with [τ_n^r, ∀n] for them respectively. Next, the lower bound ${\tilde{R}}_{j,n}^{\mathrm{FDD}}(\mathit{\tau })$ and the lower bound ${\tilde{R}}_{i,n}^{\mathrm{FDD}}(\mathit{\tau })$ yield as follows

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill R_{j,n}^{\mathrm{FDD}}(\mathit{\tau })=& {\tau }_n{log}_2(1+A_{j,n}^{\mathrm{DL}}(\mathit{\tau }))\ge \hfill \\ \hfill & {log}_{2}e\ast (2ln(1+A_{j,n}^{\mathrm{DL},r}){\tau }_n^r\hfill \\ \hfill & +\frac{A_{j,n}^{\mathrm{DL},r}}{A_{j,n}^{\mathrm{DL},r}+1}{\tau }_n^r(1-\frac{A_{j,n}^{\mathrm{DL},r}}{A_{j,n}^{\mathrm{DL}}(\mathit{\tau })})\hfill \\ \hfill & -\frac{ln(1+A_{j,n}^{\mathrm{DL},r})}{{\tau }_n}{({\tau }_n^r)}^2)\triangleq {\tilde{R}}_j^{\mathrm{FDD}}(\mathit{\tau }),\hfill \end{array}\end{array}$$(22)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill R_{i,n}^{\mathrm{FDD}}(\mathit{\tau })=& (\eta -{\tau }_n){log}_2(1+B_{i,n}^{\mathrm{UL}}(\mathit{\tau }))\ge \hfill \\ \hfill & {log}_{2}e\ast (2ln(1+B_{i,n}^{\mathrm{UL},r})(\eta -{\tau }_n^r)\hfill \\ \hfill & +\frac{B_{i,n}^{\mathrm{UL},r}}{B_{i,n}^{\mathrm{UL},r}+1}(\eta -{\tau }_n^r)(1-\frac{B_{i,n}^{\mathrm{UL},r}}{B_{i,n}^{\mathrm{UL}}(\mathit{\tau })})\hfill \\ \hfill & -\frac{ln(1+B_{i,n}^{\mathrm{UL},r})}{\eta -{\tau }_n}{(\eta -{\tau }_n^r)}^2)\triangleq {\tilde{R}}_i^{\mathrm{FDD}}(\mathit{\tau }),\hfill \end{array}\end{array}$$(23)

where

$$\begin{array}{c}\hfill \begin{array}{c}\hfill A_{j,n}^{\mathrm{DL}}(\mathit{\tau })=\frac{p_n^{b,r}{g}_{n,j}^{\mathrm{DL},r}}{{\sum }_{m=1}^{K_M}{P}_m{\beta }_0{d}_{m,j}^{-\alpha }+{\tau }_n{\sigma }^2},\\ \hfill A_{j,n}^{\mathrm{DL},r}=\frac{p_n^{b,r}{g}_{n,j}^{\mathrm{DL},r}}{{\sum }_{m=1}^{K_M}{P}_m{\beta }_0{d}_{m,j}^{-\alpha }+{\tau }_n^r{\sigma }^2},\end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill B_{i,n}^{\mathrm{UL}}(\mathit{\tau })=\frac{p_i{g}_{i,n}^{\mathrm{UL},r}}{{\sum }_{m=1}^{K_M}{P}_m{g}_{m,n}^{\mathrm{UL},r}+(\eta -{\tau }_n){\sigma }^2},\\ \hfill B_{i,n}^{\mathrm{UL},r}=\frac{p_i{g}_{i,n}^{\mathrm{UL},r}}{{\sum }_{m=1}^{K_M}{P}_m{g}_{m,n}^{\mathrm{UL},r}+(\eta -{\tau }_n^r){\sigma }^2}.\end{array}\end{array}$$

Analogously, the optimal subproblem (21) can be reformulated as

$$\begin{array}{ccc}& \underset{\mathit{\tau },\vartheta }{max}\hfill & \hfill \vartheta \end{array}$$(24a) $$\begin{array}{ccc}\hfill & \text{s.t.}\hfill & \hfill 0\le {\tau }_n\le 1,\forall n,\end{array}$$(24b) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^N{x}_{j,n}^{\mathrm{DL},r}{\tilde{R}}_{j,n}^{\mathrm{FDD}}(\mathit{\tau })\ge \vartheta ,\forall n,j\in K_D,\hfill \end{array}$$(24c) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^N{x}_{i,n}^{\mathrm{UL},r}{\tilde{R}}_{i,n}^{\mathrm{FDD}}(\mathit{\tau })\ge \vartheta ,\forall n,i\in K_U.\hfill \end{array}$$(24d)

The above subproblem (24) can be solved steadily and the local solution can be obtained efficiently.

2.3. Subproblem for the 3D trajectory design

In this part, we aim to optimize the 3D trajectory 𝒬, while DLU’s scheduling 𝒳_D, ULU’s scheduling 𝒳_U, the transmission power of the UAV 𝒫, and the portion of bandwidth τ are fixed. The objective function (19) can be rewritten as

$$\begin{array}{ccc}& \underset{\mathcal{Q},\vartheta }{max}\hfill & \hfill \vartheta \end{array}$$(25a) $$\begin{array}{ccc}\hfill & \text{s.t.}\hfill & \hfill (5)-(8),\end{array}$$(25b) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^N{x}_{j,n}^{\mathrm{DL},r}{R}_{j,n}^{\mathrm{FDD}}(\mathcal{Q})\ge \vartheta ,\forall n,j\in K_D,\hfill \end{array}$$(25c) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^N{x}_{i,n}^{\mathrm{UL},r}{R}_{i,n}^{\mathrm{FDD}}(\mathcal{Q})\ge \vartheta .\forall n,j\in K_D.\hfill \end{array}$$(25d)

The constraint (25c) and the constraint (25d) are non-convex w.r.t the 3D trajectory 𝒬. Firstly, we define the lower bound for the term $R_{j,n}^{\mathrm{FDD}}$ in the constraint (25c) with [Q_n^r H_n^r, ∀n] as follows

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill R_{j,n}^{\mathrm{FDD}}(\mathcal{Q})& \ge {\overline{R}}_{j,n}^{\mathrm{FDD}}(\mathcal{Q})\hfill \\ \hfill & \triangleq {\tau }_n^r({log}_2(1+\frac{D_{j,n}^{\mathrm{FDD}}}{||{\mathit{Q}}_n^r-{\mathit{w}}_j{||}^2+{H_n^r}^2})\hfill \\ \hfill & -{\mathrm{\Gamma }}_{j,n}^{\mathrm{FDD}}\times (||{\mathit{Q}}_n-{\mathit{w}}_j{||}^2-||{\mathit{Q}}_n^r-{\mathit{w}}_j{||}^2\hfill \\ \hfill & +H_n^2-{H_n^r}^2)),\hfill \end{array}\end{array}$$(26)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Gamma }}_{j,n}^{\mathrm{FDD}}=& \frac{D_{j,n}^{\mathrm{FDD}}{log}_{2}e}{(D_{j,n}^{\mathrm{FDD}}+||{\mathit{Q}}_n^r-{\mathit{w}}_j{||}^2+{H_n^r}^2)(||{\mathit{Q}}_n^r-{\mathit{w}}_j{||}^2+{H_n^r}^2)}\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill D_{j,n}^{\mathrm{FDD}}=\frac{p_n^{b,r}{\beta }_0}{{\sum }_{m=1}^M{P}_m{\beta }_0{d}_{m,j}^{-\alpha }+{\tau }_n^r{\sigma }^2}.\end{array}$$

With reference to the term $R_{i,n}^{\mathrm{FDD}}(\mathcal{Q})$ in the constraint (25d), by introducing slack variables $\mathit{L}=\{L_{i,n}^{\mathrm{FDD}},\forall n,i\}$ and $\mathit{I}=\{I_n^{\mathrm{FDD}},\forall n\}$, $R_{i,n}^{\mathrm{FDD}}(\mathcal{Q})$ can be substituted as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\overline{R}}_{i,n}^{\mathrm{FDD}}(\mathit{L},\mathit{I})=(\eta -{\tau }_n^r)({log}_2(1+\frac{1}{L_{i,n}^{\mathrm{FDD}}{I}_n^{\mathrm{FDD}}})),\forall n,i\in K_U,\end{array}\end{array}$$(27)

with extra constraints

$$\begin{array}{c}\hfill p_i{g}_{i,n}^{\mathrm{UL}}\ge {L_{i,n}^{\mathrm{FDD}}}^{-1},\forall n,i\in K_U,\end{array}$$(28)

and

$$\begin{array}{c}\hfill \sum _{m=1}^{K_M}{P}_m{g}_{m,n}^{\mathrm{UL}}+(\eta -{\tau }_n^r){\sigma }^2\le I_n^{\mathrm{FDD}},\forall n,m\in K_M.\end{array}$$(29)

Next, we apply inequality (64) for (27), and the lower bound ${\overline{R}}_{i,n}^{\mathrm{FDD},lb}(\mathit{L},\mathit{I})$ of (27) can be expressed as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\overline{R}}_{i,n}^{\mathrm{FDD},lb}(\mathit{L},\mathit{I})=& (\eta -{\tau }_n^r)({log}_2(\frac{1}{L_{i,n}^{\mathrm{FDD},r}{I}_n^{\mathrm{FDD},r}})\hfill \\ \hfill & +{\varphi }_{i,n}^{\mathrm{FDD}}(L_{i,n}^{\mathrm{FDD}}-L_{i,n}^{\mathrm{FDD},r})\hfill \\ \hfill & +{\psi }_{i,n}^{\mathrm{FDD}}(I_n^{\mathrm{FDD}}-I_n^{\mathrm{FDD},r})),\hfill \end{array}\end{array}$$(30)

where

$$\begin{array}{c}\hfill {\varphi }_{i,n}^{\mathrm{FDD}}=-\frac{{log}_{2}e}{(L_{i,n}^{\mathrm{FDD},r}+{(L_{i,n}^{\mathrm{FDD},r})}^2{I}_n^{\mathrm{FDD},r})}\end{array}$$

and

$$\begin{array}{c}\hfill {\psi }_{i,n}^{\mathrm{FDD}}=-\frac{{log}_{2}e}{(I_n^{\mathrm{FDD},r}+{(I_n^{\mathrm{FDD},r})}^2{L}_{i,n}^{\mathrm{FDD},r})}.\end{array}$$

To handle the extra non-convex constraint (29), we introduce the slack variable $\mathit{D}=\{d_{m,n}^{\mathrm{FDD}},\forall n,m\}$ to transform the constraint (29) into the new constraints as follows

$$\begin{array}{c}\hfill \sum _{m=1}^{K_M}{P}_m{\beta }_0{d_{m,n}^{\mathrm{FDD}}}^{-1}+(\eta -{\tau }_n^r){\sigma }^2\le I_n^{\mathrm{FDD}},\forall n,m\in K_M,\end{array}$$(31)

$$\begin{array}{c}\hfill d_{m,n}^{\mathrm{FDD}}\le ||{\mathit{Q}}_n-{\mathit{w}}_m{||}^2+H_n^2,\forall n,m\in K_M,\end{array}$$(32)

$$\begin{array}{c}\hfill d_{m,n}^{\mathrm{FDD}}\ge 0,\forall n,m\in K_M.\end{array}$$(33)

The right-hand side (RHS) of the constraint (32) is non-concave, by using the first-order Taylor expansion, the lower bound is derived as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_{m,n}^{\mathrm{FDD},l}(\mathcal{Q})& =||{\mathit{Q}}_n^r-{\mathit{w}}_m{||}^2\hfill \\ \hfill & +2{({\mathit{Q}}_n^r-{\mathit{w}}_m)}^T({\mathit{Q}}_n-{\mathit{Q}}_n^r)\hfill \\ \hfill & +{H_n^r}^2+2H_n^r(H_n-H_n^r).\hfill \end{array}\end{array}$$(34)

The constraint (32) can be rewritten as

$$\begin{array}{c}\hfill d_{m,n}^{\mathrm{FDD}}\le C_{m,n}^{\mathrm{FDD},l}(\mathcal{Q}),\forall n,m\in K_M.\end{array}$$(35)

The optimization problem (25) can be revised as

$$\begin{array}{ccc}& \underset{\mathcal{Q},\mathit{D},\mathit{L},\mathit{I},\vartheta }{max}\hfill & \hfill \vartheta \end{array}$$(36a) $$\begin{array}{ccc}\hfill & \text{s.t.}\hfill & \hfill (5)-(8),\end{array}$$(36b) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^N{x}_{j,n}^{\mathrm{DL},r}{\overline{R}}_{j,n}^{\mathrm{FDD}}(\mathit{Q})\ge \vartheta ,\forall n,j\in K_D,\hfill \end{array}$$(36c) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^N{x}_{i,n}^{\mathrm{UL},r}{\overline{R}}_{i,n}^{\mathrm{FDD},lb}(\mathit{L},\mathit{I})\ge \vartheta ,\forall n,i\in K_U,\hfill \end{array}$$(36d) $$\begin{array}{cc}\hfill & p_i{g}_{i,n}^{\mathrm{UL}}\ge {L_{i,n}^{\mathrm{FDD}}}^{-1},\forall n,i\in K_U,\hfill \end{array}$$(36e) $$\begin{array}{cc}\hfill & (31),(33),(35).\hfill \end{array}$$(36f)

2.4. Subproblem for UAV transmission power allocation

With the given DLU’s scheduling 𝒳_D, ULU’s scheduling 𝒳_U and the 3D trajectory of the UAV 𝒬, the optimization problem (19) can be reformulated as follows

$$\begin{array}{ccc}& \underset{\mathcal{P},\vartheta }{max}\hfill & \hfill \vartheta \end{array}$$(37a) $$\begin{array}{ccc}\hfill & \text{s.t.}\hfill & \hfill 0\le p_n^b\le P_{max},\forall n,\end{array}$$(37b) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^N{x}_{j,n}^{\mathrm{DL},r}{R}_{j,n}^{\mathrm{FDD}}(\mathcal{P})\ge \vartheta ,\forall n,j\in K_D,\hfill \end{array}$$(37c) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^N{x}_{i,n}^{\mathrm{UL},r}{R}_{i,n}^{\mathrm{FDD}}\ge \vartheta ,\forall n,i\in K_U.\hfill \end{array}$$(37d)

The left-hand side (LHS) of the constraint (37c) is concave w.r.t the UAV transmission power p_n^b. We can obtain the local solution by using CVX [34] expeditiously.

3. Time-fraction-based scheme

However, the FDD-based scheme cannot satisfy the STR in the whole bandwidth within one TS. Considering the similar treatment in [31], we now consider the TF-based UAV-assisted system serving both the downlinks and the uplinks, as shown in Figure 2.

Figure 2. TF-based UAV-assisted communication

The time fraction 0 <  μ_n <  1 is used for the downlink communication, while the remaining time fraction of (1 − μ_n) is used for the uplink communication. Define μ = [μ_n, ∀n]. Different from the FDD-based UAV-assisted system, the TF-based UAV-assisted system can transmit the signal to the DLU and receive the signal from the ULU within a single TS by utilizing the complete bandwidth. We assume that 1/p_n^b is the UAV power allocation within one TS for sending information to DLUs. The constraint of 1/p_n^b is given by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mu }_n/p_n^b\le P_{\max },\forall n.\end{array}\end{array}$$(38)

By using the channel models (9)–(11), the throughput expressions of the DL and UL communication can be derived [19], respectively,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill R_{j,n}^{\mathrm{TF}}(\mathcal{Q},\mathcal{P},\mathit{\mu })={\mu }_n{log}_2(1+\frac{g_{n,j}^{\mathrm{DL}}/p_n^b}{{\sum }_{m=1}^{K_M}{P}_m{\beta }_0{d}_{m,j}^{-\alpha }+{\sigma }^2})\end{array}\end{array}$$(39)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill R_{i,n}^{\mathrm{TF}}(\mathcal{Q},\mathit{\mu })=(1-{\mu }_n){log}_2(1+\frac{p_i{g}_{i,n}^{\mathrm{UL}}}{{\sum }_{m=1}^{K_M}{P}_m{g}_{m,n}^{\mathrm{UL}}+{\sigma }^2})\end{array}\end{array}$$(40)

Then, the optimization problem is given by

$$\begin{array}{ccc}& \underset{{\mathcal{X}}_{\mathit{D}},{\mathcal{X}}_{\mathit{U}},\mathcal{Q},\mathcal{P},\mathit{\mu }}{max}\hfill & \hfill min\{\underset{j}{min}{R}_j^{TF,DL}({\mathcal{X}}_{\mathit{D}},\mathcal{Q},\mathcal{P},\mathit{\mu }),\\ \hfill & \underset{i}{min}{R}_i^{TF,UL}({\mathcal{X}}_{\mathit{U}},\mathcal{Q},\mathit{\mu })\}\hfill \end{array}$$(41a) $$\begin{array}{ccc}\hfill & \text{s.t.}\hfill & \hfill {\mu }_n/p_n^b\le P_{max},\forall n,\end{array}$$(41b) $$\begin{array}{cc}\hfill & 0\le {\mu }_n\le 1,\forall n,\hfill \end{array}$$(41c) $$\begin{array}{cc}\hfill & (1)-(2),(5)-(8),(16)-(17)\hfill \end{array}$$(41d)

where

$$\begin{array}{c}\hfill R_j^{\mathrm{TF},\mathrm{DL}}({\mathcal{X}}_{\mathit{D}},\mathcal{Q},\mathcal{P},\mathit{\mu })=\frac{1}{N}\sum _{n=1}^N{\mu }_n{x}_{j,n}^{\mathrm{DL}}{R}_{j,n}^{\mathrm{TF}}(\mathcal{Q},\mathcal{P}),\end{array}$$

and

$$\begin{array}{c}\hfill R_i^{\mathrm{TF},\mathrm{UL}}({\mathcal{X}}_{\mathit{U}},\mathcal{Q},\mathit{\mu })=\frac{1}{N}\sum _{n=1}^N(1-{\mu }_n)x_{i,n}^{\mathrm{UL}}{R}_{i,n}^{\mathrm{TF}}(\mathcal{Q}).\end{array}$$

Similarly, by introducing the slack variable ϑ, the objective function (41) can be rewritten as follows

$$\begin{array}{ccc}& \underset{{\mathcal{X}}_{\mathit{D}},{\mathcal{X}}_{\mathit{U}},\mathcal{Q},\mathcal{P},\mathit{\mu },\vartheta }{max}\hfill & \hfill \vartheta \end{array}$$(42a) $$\begin{array}{ccc}\hfill & \text{s.t.}\hfill & \hfill 0\le {\mu }_n/p_n^b\le P_{max},\forall n,\end{array}$$(42b) $$\begin{array}{cc}\hfill & 0\le {\mu }_n\le 1,\forall n,\hfill \end{array}$$(42c) $$\begin{array}{cc}\hfill & (1)-(2),(5)-(8),(16)-(17),\hfill \end{array}$$(42d) $$\begin{array}{cc}\hfill & R_j^{TF,DL}({\mathcal{X}}_{\mathit{D}},\mathcal{Q},\mathcal{P},\mathit{\mu })\ge \vartheta ,j\in K_D,\hfill \end{array}$$(42e) $$\begin{array}{cc}\hfill & R_i^{TF,UL}({\mathcal{X}}_{\mathit{U}},\mathcal{Q},\mathit{\mu })\ge \vartheta ,i\in K_U.\hfill \end{array}$$(42f)

Again, by using the BCD technique, we alternately optimize the user scheduling 𝒳_D, 𝒳_U, the time fraction μ, the 3D trajectory of the UAV 𝒬 and the UAV transmission power 𝒫 to obtain the optimal solution.

3.1. Subproblem for User Scheduling

Define the time fraction μ, 3D trajectory 𝒬 and the transmission power of the UAV 𝒫 be fixed, the objective function (42) can be revised as

$$\begin{array}{ccc}& \underset{{\mathcal{X}}_{\mathit{D}},{\mathcal{X}}_{\mathit{U}},\vartheta }{max}\hfill & \hfill \vartheta \end{array}$$(43a) $$\begin{array}{ccc}\hfill & \text{s.t.}\hfill & \hfill (1)-(2),(16),(17),\end{array}$$(43b) $$\begin{array}{cc}\hfill & R_j^{\mathrm{TF},\mathrm{DL}}({\mathcal{X}}_{\mathit{D}})\ge \vartheta ,j\in K_D,\hfill \end{array}$$(43c) $$\begin{array}{cc}\hfill & R_i^{\mathrm{TF},\mathrm{UL}}({\mathcal{X}}_{\mathit{U}})\ge \vartheta ,i\in K_U.\hfill \end{array}$$(43d)

The linear subproblem (19) for DLU’s scheduling 𝒳_D and ULU’s scheduling 𝒳_D can be solved efficiently by using CVX [34].

3.2. Subproblem for time fraction

With the provided user scheduling 𝒳_D, 𝒳_U, 𝒫 and the 3D trajectory of the UAV 𝒬, the optimization problem (42) can be rewritten as follows

$$\begin{array}{ccc}& \underset{\mathit{\mu },\vartheta }{max}\hfill & \hfill \vartheta \end{array}$$(44a) $$\begin{array}{ccc}\hfill & \text{s.t.}\hfill & \hfill {\mu }_n/p_n^{b,r}\le P_{max},\forall n,\end{array}$$(44b) $$\begin{array}{cc}\hfill & 0\le {\mu }_n\le 1,\forall n,\hfill \end{array}$$(44c) $$\begin{array}{cc}\hfill & R_j^{TF,DL}(\mathit{\mu })\ge \vartheta ,j\in K_D,\hfill \end{array}$$(44d) $$\begin{array}{cc}\hfill & R_i^{TF,UL}(\mathit{\mu })\ge \vartheta ,i\in K_U.\hfill \end{array}$$(44e)

Obviously, the subproblem for the time fraction μ is a linear programming problem and can be solved directly.

3.3. Subproblem for 3D Trajectory Design

With feasible DLU’s scheduling 𝒳_D, ULU’s scheduling 𝒳_U, time fraction μ and the transmission power of the UAV 𝒫, the objective function (42) can be expressed as

$$\begin{array}{ccc}& \underset{\mathcal{Q},\vartheta }{max}\hfill & \hfill \vartheta \end{array}$$(45a) $$\begin{array}{ccc}\hfill & \text{s.t.}\hfill & \hfill (5)-(8),\end{array}$$(45b) $$\begin{array}{cc}\hfill & R_j^{\mathrm{TF},\mathrm{DL}}(\mathcal{Q})\ge \vartheta ,\forall n,j\in K_D,\hfill \end{array}$$(45c) $$\begin{array}{cc}\hfill & R_i^{\mathrm{TF},\mathrm{UL}}(\mathcal{Q})\ge \vartheta ,\forall n,i\in K_U.\hfill \end{array}$$(45d)

Since the constraints (45c) and (45d) are non-convex, this subproblem seems non-trivial and nonlinear. About the constraint (45c), by employing the first-order Taylor expansion, the lower bound ${\overline{R}}_{j,n}^{\mathrm{TF},lb}(\mathcal{Q})$ with feasible point [Q_n^r H_n^r, ∀n] can be derived by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill R_{j,n}^{\mathrm{TF}}(\mathcal{Q})& \ge {\overline{R}}_{j,n}^{\mathrm{TF},lb}(\mathcal{Q})\hfill \\ \hfill & \triangleq {log}_2(1+\frac{D_{j,n}^{\mathrm{TF}}}{\Vert {\mathit{Q}}_n^r-{\mathit{w}}_j{\Vert }^2+{H_n^r}^2})-{\mathrm{\Gamma }}_{j,n}^{\mathrm{TF}}\hfill \\ \hfill & \times (\Vert {\mathit{Q}}_n-w_j{\Vert }^2-\Vert {\mathit{Q}}_n^r-w_j{\Vert }^2+H_n^2-{H_n^r}^2),\hfill \end{array}\end{array}$$(46)

where

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathrm{\Gamma }}_{j,n}^{\mathrm{TF}}=\frac{D_{j,n}^{\mathrm{TF}}{log}_{2}e}{(D_{j,n}^{\mathrm{TF}}+\Vert {\mathit{Q}}_n^r-w_j{\Vert }^2+{H_n^r}^2)(\Vert {\mathit{Q}}_n^r-{\mathit{w}}_j{\Vert }^2+{H_n^r}^2)}\end{array}\end{array}$$

and

$$\begin{array}{c}\hfill D_{j,n}^{\mathrm{TF}}=\frac{{\beta }_0/p_n^{b,r}}{{\sum }_{m=1}^{K_M}{P}_m{\beta }_0{d}_{m,j}^{-\alpha }+{\sigma }^2}.\end{array}$$

To handle the non-convex constraint (45d), introducing slack variables $\mathit{L}=\{L_{i,n}^{\mathrm{TF}},\forall n,i\}$ and I = {I_n^TF, ∀n}, $R_{i,n}^{\mathrm{TF}}(\mathcal{Q})$ of constraint (45d) can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\overline{R}}_{i,n}^{\mathrm{TF}}(\mathit{L},\mathit{I})={log}_2(1+\frac{1}{L_{i,n}^{\mathrm{TF}}{I}_n^{\mathrm{TF}}}),\forall n,i\in K_U,\end{array}\end{array}$$(47)

with extra constraints

$$\begin{array}{c}\hfill p_i{g}_{i,n}^{\mathrm{UL}}\ge {L_{i,n}^{\mathrm{TF}}}^{-1},\forall n,i\in K_U,\end{array}$$(48)

and

$$\begin{array}{c}\hfill \sum _{m=1}^{K_M}{P}_m{g}_{m,n}^{\mathrm{UL}}+{\sigma }^2\le I_n^{\mathrm{TF}},\forall n,m\in K_M.\end{array}$$(49)

To solve the problem (45), utilize the inequality (64) for (47) with $[L_{i,n}^{\mathrm{TF},r}{I}_n^{\mathrm{TF},r},\forall n,i\in K_U]$ to get the lower bound of ${\overline{R}}_{i,n}^{\mathrm{TF}}$ as follows

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\overline{R}}_{i,n}^{\mathrm{TF},lb}(\mathit{L},\mathit{I})={log}_2(\frac{1}{L_{i,n}^{\mathrm{TF},r}{I}_n^{\mathrm{TF},r}})\\ \hfill +{\varphi }_{i,n}^{\mathrm{TF}}(L_{i,n}^{\mathrm{TF}}-L_{i,n}^{\mathrm{TF},r})\\ \hfill +{\psi }_{i,n}^{\mathrm{TF}}(I_n^{\mathrm{TF}}-I_n^{\mathrm{TF},r}),\end{array}\end{array}$$(50)