Issue 
Security and Safety
Volume 3, 2024
Security and Safety in Physical Layer Systems



Article Number  2023031  
Number of page(s)  20  
Section  Information Network  
DOI  https://doi.org/10.1051/sands/2023031  
Published online  24 January 2024 
Research Article
Optimization for UAVassisted simultaneous transmission and reception communications in the existence of malicious jammers
^{1}
Key Laboratory of Specialty Fiber Optics and Optical Access Networks, Shanghai University, Shanghai, 200444, China
^{2}
School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, BT7 1NN, UK
^{*} Corresponding author (email: zcsheng@shu.edu.cn)
Received:
18
May
2023
Revised:
29
August
2023
Accepted:
21
September
2023
In this paper, we study an unmanned aerial vehicle (UAV)assisted communication system, where the UAV is dispatched to implement simultaneous transmission and reception (STR) in the existence of multiple malicious jammers. Two schemes are investigated, namely frequency banddivisionduplex (FDD) and timefraction (TF). Based on the FDD scheme, the UAV can transmit information by using the portion of the bandwidth and receive information within the remaining portion of the bandwidth simultaneously. To perform the STR within the whole bandwidth, the TFbased scheme is considered by using a fraction of a time slot for the downlink, while the remaining fraction of the time slot is allocated for the uplink. We aim to maximize the worstcase throughput by optimizing the UAV threedimensional (3D) trajectory and resource allocation for each scheme. The optimization problem is nonconvex and thus computationally intractable. To handle the nonlinear problem, we use the block coordinate decomposition method to disaggregate the optimization problem into four subproblems and adopt the successive convex approximation technique to tackle nonconvex problems. The simulation results demonstrate the performance of the TFbased scheme over the benchmark schemes.
Key words: Unmanned aerial vehicle (UAV)assisted communication / simultaneous transmission and reception (STR) / throughput optimization / nonconvex optimization
Citation: Huang Z, Liu S, Sheng Z, Yu H and Masaracchia A. Optimization for UAVassisted simultaneous transmission and reception communications in the existence of malicious jammers. Security and Safety 2024; 3: 2023031. https://doi.org/10.1051/sands/2023031
© The Author(s) 2024. Published by EDP Sciences and China Science Publishing & Media Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 lineofsight (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 UAVbased uplink (UL) and downlink (DL) communication [13–15]. Based on halfduplex communication, resource allocation has been optimized to improve the network throughput [16]. For simultaneous transmission and reception (STR), frequency banddivisionduplex (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 fullduplex (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 uplinkdownlink 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 UAVmounted aerial base station and multiple terrestrial base stations, each serving multiple users. The balance between DL throughput and UL throughput in the UAVbased 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. Fullduplex communication faces challenges due to selfinterference, 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 FDbased communication depends on selfinterference, which cannot be neglected in the network. Instead of utilizing the FD communication, FDD and timefraction (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 UAVbased communications is more susceptible to interference from malicious terrestrial nodes due to the physical properties of the airground broadcast channel [28]. In [29], the secrecy rate of DL and UL communication was optimized considering the security of UAVbased 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 onetime 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 UAVassisted 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 FDDbased scheme for UAVassisted downlinkanduplink 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 threedimensional 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 TFbased scheme, we optimize the time fraction to balance DL and UL communication. To address each nonconvex 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 FDDbased UAVassisted system model are presented in Section 2, whereas TFbased UAVassisted 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 banddivisionduplex based scheme
We establish a UAVbased 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. FDDbased UAVassisted communication 
The frequency banddivisionduplex 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 FDDbased 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 jth DLU j ∈ {1, …, K_{D}} and ith 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 nth 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 UAVassisted 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
$$\begin{array}{c}\hfill \Vert {\mathit{Q}}_{n}{\mathit{Q}}_{n1}\Vert \le {V}_{\mathrm{max}}^{\mathit{xy}}\delta ,\forall n,\end{array}$$(5)
$$\begin{array}{c}\hfill {\mathit{Q}}_{\mathrm{ini}}={\mathit{Q}}_{0},{\mathit{Q}}_{\mathrm{end}}={\mathit{Q}}_{N},\end{array}$$(6)
$$\begin{array}{c}\hfill {H}_{n}{H}_{n1}\le {V}_{\mathrm{max}}^{z}\delta ,\forall n,\end{array}$$(7)
$$\begin{array}{c}\hfill {H}_{\mathrm{ini}}={H}_{0},{H}_{\mathrm{end}}={H}_{N},\end{array}$$(8)
where ${V}_{\mathrm{max}}^{\mathit{xy}}$ is the maximum horizontal velocity and ${V}_{\mathrm{max}}^{z}$ is the maximum vertical velocity. The UAVtoground (U2G) channel, and the groundtoUAV (G2U) channel are assumed to be dominated by LoS [4]. Due to the characteristics of wireless channels, the channel from the UAV to the jth DLU, the channel from the ith ULU to the UAV and the channel from the mth jammer to the UAV during the nth 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 largescale 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 smallfading fraction and β_{0} indicates the channel gain at the reference distance. The smallscale 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 groundtoground (G2G) channels between mth jammer and jth 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 FDDbased network downlink throughput within the nth 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 mth jammer. σ^{2} is the white Gaussian noise power. Also, the FDDbased 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 nth 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 σ^{2} denotes the white Gaussian noise power. Besides, the FDD throughput of the uplink during nth 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 FDDbased 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) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill 0\le {p}_{n}^{b}\le {P}_{max},\forall n,\end{array}$$(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 jth downlink and ith 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
$$\begin{array}{ccc}& \underset{{\mathcal{X}}_{\mathit{D}},{\mathcal{X}}_{\mathit{U}},\mathcal{Q},\mathcal{P},\mathit{\tau},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(19a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill 0\le {p}_{n}^{b}\le {P}_{max},\forall n,\end{array}$$(19b) $$\begin{array}{cc}\hfill & 0\le {\tau}_{n}\le 1,\forall n,\hfill \end{array}$$(19c) $$\begin{array}{cc}\hfill & (1)(2),(5)(8),(16)(17),\phantom{\rule{1em}{0ex}}\hfill \end{array}$$(19d) $$\begin{array}{cc}\hfill & {R}_{j}^{\mathrm{FDD},\mathrm{DL}}({\mathcal{X}}_{\mathit{D}},\mathcal{Q}\mathbf{,}\mathcal{P},\mathit{\tau})\ge \vartheta ,j\in {K}_{D},\phantom{\rule{2em}{0ex}}\hfill \end{array}$$(19e) $$\begin{array}{cc}\hfill & {R}_{i}^{\mathrm{FDD},\mathrm{UL}}({\mathcal{X}}_{\mathit{U}},\mathit{Q},\mathit{\tau})\ge \vartheta ,i\in {K}_{U}.\phantom{\rule{2em}{0ex}}\hfill \end{array}$$(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
$$\begin{array}{ccc}& \underset{{\mathcal{X}}_{\mathit{D}},{\mathcal{X}}_{\mathit{U}},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(20a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill (16)(17),(1)(2),\end{array}$$(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
$$\begin{array}{ccc}& \underset{\mathit{\tau},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(21a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill 0\le {\tau}_{n}\le 1,\forall n,\end{array}$$(21b) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}{x}_{j,n}^{\mathrm{DL},r}{R}_{j,n}^{\mathrm{FDD}}(\mathit{\tau})\ge \vartheta ,\forall n,j\in {K}_{D},\phantom{\rule{1em}{0ex}}\hfill \end{array}$$(21c) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}{x}_{i,n}^{\mathrm{UL},r}{R}_{i,n}^{\mathrm{FDD}}(\mathit{\tau})\ge \vartheta ,\forall n,i\in {K}_{U}.\phantom{\rule{1em}{0ex}}\hfill \end{array}$$(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 ${\stackrel{~}{R}}_{j,n}^{\mathrm{FDD}}(\mathit{\tau})$ and the lower bound ${\stackrel{~}{R}}_{i,n}^{\mathrm{FDD}}(\mathit{\tau})$ yield as follows
$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {R}_{j,n}^{\mathrm{FDD}}(\mathit{\tau})=& {\tau}_{n}{log}_{2}(1+{A}_{j,n}^{\mathrm{DL}}(\mathit{\tau}))\ge \hfill \\ \hfill & {log}_{2}e\ast (2ln(1+{A}_{j,n}^{\mathrm{DL},r}){\tau}_{n}^{r}\hfill \\ \hfill & +\frac{{A}_{j,n}^{\mathrm{DL},r}}{{A}_{j,n}^{\mathrm{DL},r}+1}{\tau}_{n}^{r}(1\frac{{A}_{j,n}^{\mathrm{DL},r}}{{A}_{j,n}^{\mathrm{DL}}(\mathit{\tau})})\hfill \\ \hfill & \frac{ln(1+{A}_{j,n}^{\mathrm{DL},r})}{{\tau}_{n}}{({\tau}_{n}^{r})}^{2})\triangleq {\stackrel{~}{R}}_{j}^{\mathrm{FDD}}(\mathit{\tau}),\hfill \end{array}\end{array}$$(22)
$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {R}_{i,n}^{\mathrm{FDD}}(\mathit{\tau})=& (\eta {\tau}_{n}){log}_{2}(1+{B}_{i,n}^{\mathrm{UL}}(\mathit{\tau}))\ge \hfill \\ \hfill & {log}_{2}e\ast (2ln(1+{B}_{i,n}^{\mathrm{UL},r})(\eta {\tau}_{n}^{r})\hfill \\ \hfill & +\frac{{B}_{i,n}^{\mathrm{UL},r}}{{B}_{i,n}^{\mathrm{UL},r}+1}(\eta {\tau}_{n}^{r})(1\frac{{B}_{i,n}^{\mathrm{UL},r}}{{B}_{i,n}^{\mathrm{UL}}(\mathit{\tau})})\hfill \\ \hfill & \frac{ln(1+{B}_{i,n}^{\mathrm{UL},r})}{\eta {\tau}_{n}}{(\eta {\tau}_{n}^{r})}^{2})\triangleq {\stackrel{~}{R}}_{i}^{\mathrm{FDD}}(\mathit{\tau}),\hfill \end{array}\end{array}$$(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
$$\begin{array}{ccc}& \underset{\mathit{\tau},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(24a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill 0\le {\tau}_{n}\le 1,\forall n,\end{array}$$(24b) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}{x}_{j,n}^{\mathrm{DL},r}{\stackrel{~}{R}}_{j,n}^{\mathrm{FDD}}(\mathit{\tau})\ge \vartheta ,\forall n,j\in {K}_{D},\phantom{\rule{1em}{0ex}}\hfill \end{array}$$(24c) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}{x}_{i,n}^{\mathrm{UL},r}{\stackrel{~}{R}}_{i,n}^{\mathrm{FDD}}(\mathit{\tau})\ge \vartheta ,\forall n,i\in {K}_{U}.\phantom{\rule{1em}{0ex}}\hfill \end{array}$$(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
$$\begin{array}{ccc}& \underset{\mathcal{Q},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(25a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill (5)(8),\end{array}$$(25b) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}{x}_{j,n}^{\mathrm{DL},r}{R}_{j,n}^{\mathrm{FDD}}(\mathcal{Q})\ge \vartheta ,\forall n,j\in {K}_{D},\phantom{\rule{1em}{0ex}}\hfill \end{array}$$(25c) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}{x}_{i,n}^{\mathrm{UL},r}{R}_{i,n}^{\mathrm{FDD}}(\mathcal{Q})\ge \vartheta .\forall n,j\in {K}_{D}.\phantom{\rule{1em}{0ex}}\hfill \end{array}$$(25d)
The constraint (25c) and the constraint (25d) are nonconvex 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
$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {R}_{j,n}^{\mathrm{FDD}}(\mathcal{Q})& \ge {\overline{R}}_{j,n}^{\mathrm{FDD}}(\mathcal{Q})\hfill \\ \hfill & \triangleq {\tau}_{n}^{r}({log}_{2}(1+\frac{{D}_{j,n}^{\mathrm{FDD}}}{{\mathit{Q}}_{n}^{r}{\mathit{w}}_{j}{}^{2}+{{H}_{n}^{r}}^{2}})\hfill \\ \hfill & \phantom{\rule{1em}{0ex}}{\mathrm{\Gamma}}_{j,n}^{\mathrm{FDD}}\times ({\mathit{Q}}_{n}{\mathit{w}}_{j}{}^{2}{\mathit{Q}}_{n}^{r}{\mathit{w}}_{j}{}^{2}\hfill \\ \hfill & \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}+{H}_{n}^{2}{{H}_{n}^{r}}^{2})),\hfill \end{array}\end{array}$$(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 nonconvex 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 righthand side (RHS) of the constraint (32) is nonconcave, by using the firstorder Taylor expansion, the lower bound is derived as
$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {C}_{m,n}^{\mathrm{FDD},l}(\mathcal{Q})& ={\mathit{Q}}_{n}^{r}{\mathit{w}}_{m}{}^{2}\hfill \\ \hfill & +2{({\mathit{Q}}_{n}^{r}{\mathit{w}}_{m})}^{T}({\mathit{Q}}_{n}{\mathit{Q}}_{n}^{r})\hfill \\ \hfill & +{{H}_{n}^{r}}^{2}+2{H}_{n}^{r}({H}_{n}{H}_{n}^{r}).\hfill \end{array}\end{array}$$(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
$$\begin{array}{ccc}& \underset{\mathcal{Q},\mathit{D},\mathit{L},\mathit{I},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(36a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill (5)(8),\end{array}$$(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) $$\begin{array}{cc}\hfill & {p}_{i}{g}_{i,n}^{\mathrm{UL}}\ge {{L}_{i,n}^{\mathrm{FDD}}}^{1},\forall n,i\in {K}_{U},\phantom{\rule{2em}{0ex}}\hfill \end{array}$$(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
$$\begin{array}{ccc}& \underset{\mathcal{P},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(37a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill 0\le {p}_{n}^{b}\le {P}_{max},\forall n,\end{array}$$(37b) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}{x}_{j,n}^{\mathrm{DL},r}{R}_{j,n}^{\mathrm{FDD}}(\mathcal{P})\ge \vartheta ,\forall n,j\in {K}_{D},\phantom{\rule{2em}{0ex}}\hfill \end{array}$$(37c) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}{x}_{i,n}^{\mathrm{UL},r}{R}_{i,n}^{\mathrm{FDD}}\ge \vartheta ,\forall n,i\in {K}_{U}.\phantom{\rule{2em}{0ex}}\hfill \end{array}$$(37d)
The lefthand 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.
1: Initialization: Set r = 0. Find initial feasible points {Q^{r}_{n} H^{r}[n]}, {x^{UL,r}_{i,n}}, {x^{DL,r}_{j,n}}, {p^{b,r}_{n}}, {τ^{r}_{n}} for (19), set ∈ > 0.
2: repeat.
3: Directly Solve the subproblem (20) for χ_{D} and χ_{U} to update the optimal {x^{DL,r+1}_{j,n}} and {x^{UL,r+1}_{i,n}} with {Q^{r}_{n} H^{r}_{n}}, {p^{b,r}_{n}}, {T^{r}_{n}}.
4: Solve the problem (24) to update the optimal {τ^{r+1}_{n}} with {x^{UL,r+1}_{i,n}} {Q^{r}[n] H^{r}[n]}, {x^{DL,r+1}_{j,n}}, {p^{b,r}_{n}}.
5: Solve the problem (36) to update the optimal {Q^{r+1}_{n} H^{r+1}_{n}} with {x^{UL,r+1}_{i,n}}, {x^{DL,r+1}_{j,n}}, {p^{b,r}_{n}}, {τ^{r+1}_{n}}.
6: Directly Solve the problem (37) to update the optimum point as {p^{b,r+1}_{n}} with {x^{UL,r+1}_{i,n}}, {x^{DL,r+1}_{j,n}}, {Q^{r+1}_{n} H^{r+1}_{n}}, {τ^{r+1}_{n}}.
7: Set r := r + 1
8: Until the fractional growth of the objective value of (19) is within the tolerance ∈
3. Timefractionbased scheme
However, the FDDbased scheme cannot satisfy the STR in the whole bandwidth within one TS. Considering the similar treatment in [31], we now consider the TFbased UAVassisted system serving both the downlinks and the uplinks, as shown in Figure 2.
Figure 2. TFbased UAVassisted 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 FDDbased UAVassisted system, the TFbased UAVassisted 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
$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mu}_{n}/{p}_{n}^{b}\le {P}_{\mathrm{max}},\forall n.\end{array}\end{array}$$(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
$$\begin{array}{ccc}& \underset{{\mathcal{X}}_{\mathit{D}},{\mathcal{X}}_{\mathit{U}},\mathcal{Q},\mathcal{P},\mathit{\mu}}{max}\hfill & \hfill min\{\underset{j}{min}{R}_{j}^{TF,DL}({\mathcal{X}}_{\mathit{D}},\mathcal{Q},\mathcal{P},\mathit{\mu}),\phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}\underset{i}{min}{R}_{i}^{TF,UL}({\mathcal{X}}_{\mathit{U}},\mathcal{Q},\mathit{\mu})\}\phantom{\rule{2em}{0ex}}\hfill \end{array}$$(41a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill {\mu}_{n}/{p}_{n}^{b}\le {P}_{max},\forall n,\end{array}$$(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
$$\begin{array}{ccc}& \underset{{\mathcal{X}}_{\mathit{D}},{\mathcal{X}}_{\mathit{U}},\mathcal{Q},\mathcal{P},\mathit{\mu},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(42a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill 0\le {\mu}_{n}/{p}_{n}^{b}\le {P}_{max},\forall n,\end{array}$$(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
$$\begin{array}{ccc}& \underset{{\mathcal{X}}_{\mathit{D}},{\mathcal{X}}_{\mathit{U}},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(43a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill (1)(2),(16),(17),\end{array}$$(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
$$\begin{array}{ccc}& \underset{\mathit{\mu},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(44a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill {\mu}_{n}/{p}_{n}^{b,r}\le {P}_{max},\forall n,\end{array}$$(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
$$\begin{array}{ccc}& \underset{\mathcal{Q},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(45a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill (5)(8),\end{array}$$(45b) $$\begin{array}{cc}\hfill & {R}_{j}^{\mathrm{TF},\mathrm{DL}}(\mathcal{Q})\ge \vartheta ,\forall n,j\in {K}_{D},\phantom{\rule{2em}{0ex}}\hfill \end{array}$$(45c) $$\begin{array}{cc}\hfill & {R}_{i}^{\mathrm{TF},\mathrm{UL}}(\mathcal{Q})\ge \vartheta ,\forall n,i\in {K}_{U}.\phantom{\rule{2em}{0ex}}\hfill \end{array}$$(45d)
Since the constraints (45c) and (45d) are nonconvex, this subproblem seems nontrivial and nonlinear. About the constraint (45c), by employing the firstorder 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 nonconvex 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 $[{L}_{i,n}^{\mathrm{TF},r}\phantom{\rule{1em}{0ex}}{I}_{n}^{\mathrm{TF},r},\forall n,i\in {K}_{U}]$ 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)
where
$$\begin{array}{c}\hfill {\varphi}_{i,n}^{\mathrm{TF}}=\frac{{log}_{2}e}{({L}_{i,n}^{\mathrm{TF},r}+{({L}_{i,n}^{\mathrm{TF},r})}^{2}{I}_{n}^{\mathrm{TF},r})}\end{array}$$
and
$$\begin{array}{c}\hfill {\psi}_{i,n}^{\mathrm{TF}}=\frac{{log}_{2}e}{({I}_{n}^{\mathrm{TF},r}+{({I}_{n}^{TF,r})}^{2}{L}_{i,n}^{\mathrm{TF},r})}.\end{array}$$
Given that the additional constraint (49) is nonconvex, the slack variable $\mathit{D}=\{{d}_{m,n}^{\mathrm{TF}},\forall n,m\}$ is introduced to replace the constraint (49) and new constraints can be given by
$$\begin{array}{c}\hfill \sum _{m=1}^{{K}_{M}}{P}_{m}{\beta}_{0}{{d}_{m,n}^{\mathrm{TF}}}^{1}+{\sigma}^{2}\le {I}_{n}^{\mathrm{TF}},\forall n,m\in {K}_{M},\end{array}$$(51)
$$\begin{array}{c}\hfill {d}_{m,n}^{\mathrm{TF}}\le {\mathit{Q}}_{n}{\mathit{w}}_{m}{}^{2}+{H}_{n}^{2},\forall n,m\in {K}_{M},\end{array}$$(52)
$$\begin{array}{c}\hfill {d}_{m,n}^{\mathrm{TF}}\ge 0,\forall n,m\in {K}_{M}.\end{array}$$(53)
The RHS of the constraint (52) is approximated by the firstorder Taylor expansion
$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {C}_{m,n}^{\mathrm{TF},l}(\mathcal{Q})& ={\mathit{Q}}_{n}^{r}{\mathit{w}}_{m}{}^{2}\hfill \\ \hfill & +2{({\mathit{Q}}_{n}^{r}{\mathit{w}}_{m})}^{T}({\mathit{Q}}_{n}{\mathit{Q}}_{n}^{r})\hfill \\ \hfill & +{{H}_{n}^{r}}^{2}+2{H}_{n}^{r}({H}_{n}{H}_{n}^{r}).\hfill \end{array}\end{array}$$(54)
The constraint (52) is revised as
$$\begin{array}{c}\hfill {d}_{m,n}^{\mathrm{TF}}\le {C}_{m,uav}^{\mathrm{TF},l}(\mathcal{Q}),\forall n,m\in {K}_{M}.\end{array}$$(55)
Finally, the approximation problem (45) can be reformulated as follows
$$\begin{array}{ccc}& \underset{\mathcal{Q},\mathit{D},\mathit{L},\mathit{I},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(56a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill (5)(8),\end{array}$$(56b) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}{x}_{j,n}^{\mathrm{DL},r}{\overline{R}}_{j,n}^{\mathrm{TF},lb}(\mathit{Q})\ge \vartheta ,\forall n,j\in {K}_{D},\hfill \end{array}$$(56c) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}{x}_{i,n}^{\mathrm{UL},r}{\overline{R}}_{i,n}^{\mathrm{TF},lb}(\mathit{L},\mathit{I})\ge \vartheta ,\forall n,i\in {K}_{U},\hfill \end{array}$$(56d) $$\begin{array}{cc}\hfill & {p}_{i}{g}_{i,n}^{\mathrm{UL}}\ge {{L}_{i,n}^{\mathrm{TF}}}^{1},\forall n,m\in {K}_{M},\phantom{\rule{2em}{0ex}}\hfill \end{array}$$(56e) $$\begin{array}{cc}\hfill & (51),(53),(55).\hfill \end{array}$$(56f)
3.4. Subproblem for UAV transmission power allocation
With the given 𝒳_{D}, 𝒳_{U}, 𝒬 and μ, the optimization problem (42) can be rewritten as
$$\begin{array}{ccc}& \underset{\mathcal{P},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(57a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill {\mu}_{n}^{r}/{p}_{n}^{b}\le {P}_{max},\forall n,\end{array}$$(57b) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}{\mu}_{n}^{r}{x}_{j,n}^{\mathrm{DL},r}{R}_{j,n}^{\mathrm{TF}}(\mathcal{P})\ge \vartheta ,\forall n,j\in {K}_{D},\hfill \end{array}$$(57c) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}(1{\mu}_{n}^{r}){x}_{i,n}^{\mathrm{UL},r}{R}_{i,n}^{\mathrm{TF}}\ge \vartheta ,\forall n,i\in {K}_{U}.\hfill \end{array}$$(57d)
The constraint (57c) is nonconvex. We also apply the firstorder Taylor expansion at the feasible point $\{{p}_{n}^{b,r}\}$ to be approximated as:
$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}{R}_{j,n}^{\mathrm{TF}}(\mathcal{P})\ge & {log}_{2}(1+\frac{{Z}_{j,n}^{r}}{{p}_{n}^{b,r}})\frac{{Z}_{j,n}^{r}{log}_{2}e}{{p}_{n}^{b,r}({Z}_{j,n}^{r}+{p}_{n}^{b,r})}({p}_{n}^{b}{p}_{n}^{b,r})\triangleq {\widehat{R}}_{j,n}^{\mathrm{TF}}(\mathcal{P}),\hfill \end{array}\end{array}$$(58)
where ${Z}_{j,n}^{r}=\frac{{g}_{n,j}^{\mathrm{DL}}}{{\sum}_{m=1}^{{K}_{M}}{P}_{m}{\beta}_{0}{d}_{m,j}^{\alpha}+{\sigma}^{2}}$.
The subproblem (57) can be rewritten as:
$$\begin{array}{ccc}& \underset{\mathcal{P},\vartheta}{max}\hfill & \hfill \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\vartheta \end{array}$$(59a) $$\begin{array}{ccc}\hfill & \phantom{\rule{0.333333em}{0ex}}\text{s.t.}\hfill & \hfill {\mu}_{n}^{r}/{p}_{n}^{b}\le {P}_{max},\forall n,\end{array}$$(59b) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}{x}_{j,n}^{\mathrm{DL},r}{\widehat{R}}_{j,n}^{\mathrm{TF}}(\mathcal{P})\ge \vartheta ,\forall n,j\in {K}_{D},\hfill \end{array}$$(59c) $$\begin{array}{cc}\hfill & \frac{1}{N}\sum _{n=1}^{N}{x}_{i,n}^{\mathrm{UL},r}{R}_{i,n}^{\mathrm{TF}}\ge \vartheta ,\forall n,i\in {K}_{U}.\hfill \end{array}$$(59d)
Then, the local optimal solution can be obtained by solving the approximate problem (59).
1: Initialization: Set r = 0. Find initial feasible points {Q^{r}_{n} H^{r}_{n}} {x^{UL,r}_{i,n}}, {x^{DL,r}_{j,n}}, {μ^{r}_{n}}, {p^{b,r}_{n}} for (42), set ∈ > 0.
2: repeat.
3: Directly solve the subproblem (43) to update the optimum point {x^{DL,r+1}_{j,n}} and {x^{UL,r+1}_{i,n}} with {Q^{r}_{n} H^{r}_{n}}, {μ^{r}_{n}}, {p^{b,r}_{n}}.
4: Directly solve the subproblem (44) to update the optimum point {μ^{r+1}_{n}} with {Q^{r}_{n} H^{r}_{n}}, {x^{DL,r+1}_{j,n}}, {x^{UL,r+1}_{i,n}} and {p^{b,r}_{n}}.
5: Solve the problem (56) to update the optimum point as {Q^{r+1}_{n} H^{r+1}_{n}} with {x^{UL,r+1}_{i,n}}, {x^{DL,r+1}_{j,n}}, {p^{b,r}_{n}} and {μ^{r+1}_{n}}.
6: Solve the problem (59) to update the optimum point as {p^{b,r+1}_{n}} with {x^{UL,r+1}_{i,n}}, {x^{DL,r+1}_{j,n}}, {μ^{r+1}_{n}} and {Q^{r+1}_{n} H^{r+1}_{n}}.
7: Set r := r + 1
8: Until the fractional growth of the objective value of (42) is within the tolerance ∈
4. Numerical result
This section demonstrates our numerical outcomes to characterize the performance of our proposed algorithms. The locations of DLUs are set as D_{1} = (200, 700, 0), D_{2} = (800, 100, 0), D_{3} = (400, 700), D_{4} = (600, 100), while the coordinates of ULUs are set as U_{1} = (200, 800, 0), U_{2} = (800, 0, 0), U_{3} = (400, 800), U_{4} = (600, 0), respectively. Besides, the coordinates of jammers are set as: J_{1} = (300, 300, 0), J_{1} = (600, 800, 0). The maximal horizontal speed and vertical speed are ${V}_{\mathrm{max}}^{\mathit{xy}}=40\phantom{\rule{0.166667em}{0ex}}$m/s and ${V}_{\mathrm{max}}^{z}=30$m/s, respectively. The location of the UAV initial point is fixed at (0, 500, 100) and the final coordinate of the UAV is set to be (1000, 500, 100). The altitude constraints of the UAV are H_{min} = 100 m and H_{max} = 300 m [36]. We assume that P_{max} = P_{m} = P_{i} = 20dBm. In addition, the bandwidth of the communication is B = 1 MHz with σ^{2} = −110dBm and the channel gain at the reference is β_{0} = −60dB [5, 24, 37]. The G2G path loss coefficients α are determined as α = 3 and the Rician factor is K_{a} = 3dB [18]. The maximum tolerable value is ϵ = 10^{−3}. Additionally, the time slot is δ = 0.5 and the allocation ζ = 0.1 [38]. In the simulation results, “TF” refers to the proposed algorithm for timefractionbased UAVassisted communication, while “FDD” refers to the proposed algorithm for frequency banddivisionduplex UAVassisted communication.
In Figure 3, we plot the UAV 2D trajectory and 3D trajectory for different schemes within “T = 30s” and “T = 90s", respectively. During a long flight time such as “T = 90s", compared with the FDDbased system, the UAV would like to adjust the flight altitude to reduce the interference from jammers in the TFbased system. It can be supposed that the UAV prefers moving closer to the ULUs during the flight. This is because the channels between the UAV and malicious jammers are also mainly dominated by LoS which enhances the negative interference in the uplink communication. And, the UAV has to guarantee the fairness between the DL and UL communication. In order to investigate the fairness among users, we investigate the average throughput of each user.
Figure 3. Optimized UAV trajectory of different algorithms: (a) Horizontal plane; (b) 3D plane 
In Figure 4, the benchmark “FDD Fixed Band” refers to the traditional algorithm in [17] without optimal bandwidth allocation, in which bandwidth for DL and UL communication is divided into two fixed parts, while “TF Fixed Time Fraction” refers to the algorithm in [19] with equal time fraction. Obviously, the average throughput of each ULU is less than or equal to each DLU’s throughput. Therefore, the UAV flies closer to the ULUs to improve the throughput of the uplink. Besides, in Figure 4, it can be seen that each user is fair and enjoys the same throughput in the “TF” scheme.
Figure 4. Average throughput of users with T = 90s 
Meanwhile, we can observe that the “FDD” scheme has better performance than the “FDD Fixed Band” scheme in Figure 4. As demonstrated in Figure 5, the “FDD” scheme is able to adjust its bandwidth allocation to improve the throughput of the system to guarantee fairness between the DL and UL communication. Especially, with the jammer transmission power increasing, the adjustment of the bandwidth allocation is more evident. Also, as can be seen in Figure 6, TFbased UAV communication can improve the throughput by changing the time fraction in different slots.
Figure 5. Bandwidth allocation versus time slots with T = 90s 
Figure 6. Optimal Time Fraction versus time slots with T = 90s 
Figure 7 shows the average throughput achieved by different schemes versus the flight time T. One can observe that the performance of the TFbased scheme outperforms other schemes. As expected, the “FDD Fixed Band” scheme is the worst performer. Besides, we compare the TFbased average throughput achieved by the polyblock outer approximation method (POA) in [18] and our proposed scheme. Considering the same trajectory, our proposed scheme demonstrates a performance that is closely aligned with that of the POAbased method.
Figure 7. Average throughput versus Flight time T(s) 
Furthermore, to examine the performance of different schemes in the existence of malicious jammers, we simulate average throughput versus the transmission power of jammers in Figure 8. As can be seen, the average throughput of each proposed algorithm decreases when the jammer transmission power gradually rises. Nevertheless, the performance of the TFbased scheme is superior to other schemes.
Figure 8. Average throughput versus Jammer transmission power P_{m}(dBm) with T = 90s 
Finally, Figure 9 characterizes the convergence of the proposed algorithms. The “FDD Fixed Band” algorithm and the “TF Fixed Fraction” algorithm achieve the same convergence throughput. The TFbased algorithm and FDDbased algorithm require 40 and 19 iterations, respectively.
Figure 9. Convergence of the proposed algorithms 
5. Conclusions
In this paper, a joint downlink and uplink communication system in the presence of multiple malicious jammers has been considered, where a UAV is designed to transmit the signal to DLUs and receive the signal from ULUs simultaneously. The possible schemes for jointly optimizing the DL and UL communication to maximize the worst throughput among users are proposed, namely the FDDbased scheme and TFbased scheme. Our numerical results demonstrate the advantage of the TFbased scheme over other schemes.
6. Appendix: Rate function approximation
Let g_{1}(x)=log_{2}(1 + x) and ${g}_{2}(x)={log}_{2}(1+\frac{1}{x})$, x > 0. It can be verified that g_{1}(x) is concave with respect to x and g_{2}(x) is convex with respect to x. Based on Jensen’s inequality, the inequalities are given by [18]
$$\begin{array}{c}\hfill {log}_{2}(1+\frac{1}{\mathbb{E}\{\frac{1}{x}\}})\le \mathbb{E}\{{log}_{2}(1+x)\}\le {log}_{2}(1+\mathbb{E}\{x\}).\end{array}$$(60)
Define $x=\frac{X}{Y}(X>0,Y>0)$, and X and Y are independent, we have
$$\begin{array}{c}\hfill {log}_{2}(1+\frac{1}{\mathbb{E}\{\frac{Y}{X}\}})\le \mathbb{E}\{{log}_{2}(1+\frac{X}{Y})\}\le {log}_{2}(1+\mathbb{E}\{\frac{X}{Y}\}).\end{array}$$(61)
$$\begin{array}{c}\hfill \mathbb{E}\{\frac{X}{Y}\}\ge \frac{\mathbb{E}\{X\}}{\mathbb{E}\{Y\}},\end{array}$$(62)
where the (62) follows the convexity of function $\frac{1}{Y}$. Hence we can derive the following approximation result
$$\begin{array}{c}\hfill {log}_{2}(1+\frac{1}{\mathbb{E}\{\frac{Y}{X}\}})\le {log}_{2}(1+\frac{\mathbb{E}\{X\}}{\mathbb{E}\{Y\}})\le {log}_{2}(1+\mathbb{E}\{\frac{X}{Y}\}).\end{array}$$(63)
From (61) and (63), it can be seen that $\mathbb{E}\{{log}_{2}(1+\frac{X}{Y})\}$ and ${log}_{2}(1+\frac{\mathbb{E}\{X\}}{\mathbb{E}\{Y\}})$ have the same lower bound and upper bound. Therefore, $\mathbb{E}\{{log}_{2}(1+\frac{X}{Y})\}\approx {log}_{2}(1+\frac{\mathbb{E}\{X\}}{\mathbb{E}\{Y\}})$ can be derived.
Lemma. With any given achievable point (L^{r}[n],I^{r}[n]), $\overline{R}$ should be lower bounded by [39]
$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \overline{R}[n]={log}_{2}(1+\frac{1}{I[n]L[n]})\ge & {log}_{2}(\frac{1}{{L}^{r}[n]{I}^{r}[n]})\hfill \\ \hfill & +\varphi (L[n]{L}^{r}[n])\hfill \\ \hfill & +\psi ({I}_{[}n]{I}^{r}[n])\triangleq {\overline{R}}^{\mathit{lb}}[n]\hfill \end{array}\end{array}$$(64)
where $\varphi =\frac{{log}_{2}e}{({L}^{r}[n]+{({L}^{r}[n])}^{2}{I}^{r}[n])}$ and $\psi =\frac{{log}_{2}e}{({I}^{r}[n]+{({I}^{r}[n])}^{2}{L}^{r}[n])}$
It is true that
$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{ln(1+\mathit{x})}{\mathit{t}}\ge & 2\frac{ln(1+\overline{x})}{\overline{t}}+\frac{\overline{x}}{(\overline{x}+1)\overline{t}}(1\frac{\overline{x}}{\mathit{x}})\frac{ln(1+\overline{x})}{{\overline{t}}^{2}}\mathit{t},\forall (\mathit{x},\mathit{t})\in {\mathbb{R}}_{+}^{2},(\overline{x},\overline{t})\in {\mathbb{R}}_{+}^{2}\hfill \end{array}\end{array}$$(65)
which can be proved in [38],
Substituting $\mathit{t}\to \frac{1}{\mathit{t}}$ and $\overline{t}\to \frac{1}{\overline{t}}$ in (65) leads to
$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathit{t}ln(1+\mathit{x})\ge & 2\overline{t}ln(1+\overline{x})+\frac{\overline{t}\overline{x}}{(\overline{x}+1)}(1\frac{\overline{x}}{\mathit{x}})\frac{{\overline{t}}^{2}ln(1+\overline{x})}{\mathit{t}},\forall (\mathit{x},\mathit{t})\in {\mathbb{R}}_{+}^{2},(\overline{x},\overline{t})\in {\mathbb{R}}_{+}^{2}\hfill \end{array}\end{array}$$(66)
for all x ≥ 0, y > 0, and $\overline{x}\ge 0$, $\overline{y}>0$.
Conflict of Interest
The authors declare no conflict of interest.
Data Availability
No data are associated with this article.
Authors’ Contributions
Zhiyu Huang and Zhichao Sheng designed the system model and wrote this paper. Hongwen Yu and Antonino Masaracchia discussed the recent developments. Shuzhen Liu checked and corrected the mistakes in the paper.
Acknowledgments
Thanks to anonymous reviewers for their helpful comments and suggestions.
Funding
This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 61901254 and in part by the Aeronautical Science Foundation of China under Grant 2020Z0660S6001.
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Zhiyu Huang received the B.S. degrees from Shanghai University, Shanghai, China, in 2021. He is currently pursuing the M.S. degree with the Shanghai University. His current research interests include optimization methods for UAV communication and fullduplex communication.
Shuzhen Liu received the B.S. degrees from Anhui Jianzhu University, Hefei, China, in 2019. She is currently pursuing the M.S. degree with the Shanghai University. Her current research interests include UAV communications, physical layer security and intelligent reflecting surface.
Zhichao Sheng received the Ph.D. degree in electrical engineering from the University of Technology Sydney, Sydney, NSW, Australia, in 2018. From 2018 to 2019, he was a Research Fellow with the School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, U.K. He is currently a Lecturer with Shanghai University, Shanghai, China. His research interests include optimization methods for wireless communication and signal processing.
Hongwen Yu received his B.Eng., M.Eng. and Ph.D. degree in communication and information engineering from Shanghai University, Shanghai, China, in 2011, 2014 and 2020, respectively, and his Ph.D. degree in electronic engineering from the University of Technology Sydney, NSW, Australia in 2022. Currently, he is an Associate Professor in the Department of Communication and Information Engineering, Shanghai University. His research interests include reconfigurable intelligent surface, Wave communications and B5G/6G wireless communications.
Antonino Masaracchia received the Ph.D. degree in electronics and telecommunications engineering from the University of Palermo, Italy, in 2016. Since 2018, he has been a Research Fellow with the Centre for Wireless Innovation, Queen’s University Belfast, U.K. His research interests include heterogeneous networks, convex optimization and machine learning techniques, wireless communications, and green communication networking.
All Figures
Figure 1. FDDbased UAVassisted communication 

In the text 
Figure 2. TFbased UAVassisted communication 

In the text 
Figure 3. Optimized UAV trajectory of different algorithms: (a) Horizontal plane; (b) 3D plane 

In the text 
Figure 4. Average throughput of users with T = 90s 

In the text 
Figure 5. Bandwidth allocation versus time slots with T = 90s 

In the text 
Figure 6. Optimal Time Fraction versus time slots with T = 90s 

In the text 
Figure 7. Average throughput versus Flight time T(s) 

In the text 
Figure 8. Average throughput versus Jammer transmission power P_{m}(dBm) with T = 90s 

In the text 
Figure 9. Convergence of the proposed algorithms 

In the text 
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