# Joint Position and Time Allocation Optimization of UAV-Aided Wireless Powered Relay Communication Systems

Authors: Xiaofei Di; Yang Chen
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

The Internet of things (IoT) has emerged as a platform for connecting massive physical devices to collect and analyze data for decision-making. Wireless devices in IoT are usually energy-constrained and thus need to be powered by a stable and reliable energy source in order to maintain a long network lifetime. An unmanned aerial vehicle (UAV) as an energy source is a proper and applicable way to supply energy to wireless devices in IoT, due to its flexibility and potential of providing line-of-sight (LOS) links for wireless air-to-ground channels. In this paper, a UAV-aided wireless powered relay communication system is presented, where a UAV firstly emits energy to a source and a relay, and then, the source and relay cooperatively transmit information to their destination. To explore the performance limit of the system, a problem is formulated by jointly optimizing the position of the UAV and time allocation to maximize the achievable information rate of the system. By deriving the explicit expressions of the optimal position of UAV and optimal time fraction, the nonconvex optimization problem is efficiently solved. Simulation results show that our proposed method significantly outperforms the benchmark methods.


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## Body

## 1. Introduction

With the development of the fifth generation (5G) mobile networks, including machine-type communications (MTC), the Internet of things (IoT) has gradually become reality [1, 2]. In typical applications of IoT, a large number of sensors collect and transmit information to the sink node, and then, the sink node processes the received information to assist analysis and decision. Sensors in IoT generally need to work for 10 to 15 years [3]. However, they are usually energy-constrained nodes, so it is very important to supply sensors with stable and reliable energy in order to prolong the lifetime of IoT. Conventionally replacing batteries of the sensors is inconvenient for the massive sensors usually sparsely distributed in a wide area.The wireless power transfer (WPT) technology provides a feasible way to satisfy the requirement of energy supply of sensors in IoT, where sensors are powered by radio frequency (RF) energy transfer [4]. Recently, a wireless powered communication network (WPCN) has emerged as a promising technique by combining WPT and communication technologies. In WPCN, energy-constrained sensors have to complete information transmissions by using the energy harvested from a dedicated power beacon (PB) [5] or a hybrid access point (H-AP) [6]. In [5], a dedicated PB was deployed to supply energy to two communication nodes, and then, the two communication nodes exchanged information by using their harvested energies. In [6], a H-AP firstly transferred energy to multiple users and then received the information which the users transmitted by using the harvested energy. In [7], WPCN was combined with a relay technique, where a source and a relay harvested energy from a signal emitted by a PB, then cooperatively transmitted information to a destination. However, in the existing works, PB or H-AP is fixed at a position; it is difficult to satisfy energy supply requirements of a massive number of sensors in IoT. On the one hand, due to widely and sparsely distributed sensors, a lot of fixed PBs or HAPs have to be applied, which significantly increases the deployment cost. On the other hand, fixed PBs or HAPs are hard to apply in some special environments such as toxic or remote forest regions.Recently, unmanned aerial vehicle- (UAV-) aided WPCNs have attracted considerable attention, where UAV acts as an aerial PB or HAP to help energy and information transmission. Due to the inherent flexibility and mobility, UAV-aided WPCNs can efficiently solve the problems of fixed PBs or HAPs mentioned above. Moreover, UAV has potential to provide line-of-sight (LOS) links for wireless air-to-ground channels, so it can significantly enhance the energy transfer or communication performance [8]. According to [9], UAVs can be classified into two types, that is, mobile UAVs and static UAVs. The two types of UAVs have different advantages and applications, so both of them have attracted much interest.Mobile UAV-aided WPCNs were studied in [10–13]. In [10, 11], a mobile UAV-aided multiuser WPCN was studied, where a mobile UAV first wirelessly charged multiple users on the ground and then received the information from the users, and the packet loss rate and energy consumption of the UAV performance were minimized, respectively. In [12], two UAVs wirelessly charged two users on the ground and collected information from them, respectively. For the two-UAV two-user interference channel, the common throughput of two uses was maximized. [13] further investigated multiple UAV multiple user scenarios and minimized the communication delay. Mobile UAVs are not capable of hovering, which limits the power transfer efficiency. Moreover, mobile UAV-aided WPCNs involve the optimization of continuous functions on trajectory, and its complexity is high [11–13]. So, mobile UAVs must be equipped with large computation capacity to satisfy the requirement of optimization algorithms.In this paper, we consider static UAV communication systems, where static UAVs can hover over a fixed position by equipping propellers on UAVs [14–16]. Recently, static UAVs have been considered to apply to WPCNs. Compared with mobile UAVs, static UAVs can supply more stable and reliable services due to their hovering. Moreover, static UAV-aided WPCNs only need to design the hovering positions of UAVs, which simplifies the system design, and therefore can deploy UAVs without large computation capacity in networks. In addition, static UAVs can also be realized by the tethered UAVs, which are connected by a cable/wire with a ground control platform to satisfy their requirement of energy supply, such as AT&T’s flying cell-on-wings (COWs) and Facebook’s Tether-Tenna [17].Due to the reasons above, static UAV-aided WPCNs have also attracted research interest [18–21]. In [18], a static UAV acting as H-AP firstly emitted energy to multiple users and then received the information from the users, and the performance of throughput was investigated and optimized by jointly optimizing UAV position and time allocation. In [19], a UAV-aided WPCN was investigated, where a UAV as relay firstly simultaneously received energy and information from the signal transmitted by a source and then helped to forward received information to a destination by using the harvested energy. The outage performance of the system was analyzed. This way is referred to as simultaneous wireless information and power transfer (SWIPT) [22]. The authors in [20] considered the identical system model with [19] and further minimized the outage probability by jointly optimizing UAV position and system parameters. [21] extended [19] to a multiple destination scenario and reduced UAV’s energy consumption by optimizing the UAV position.Different from the existing works mentioned above, in this paper, we consider a static UAV-aided wireless powered relay communication system, where a static UAV serves as a PB to provide energy to a source and a relay, and then, the source and relay cooperatively transmit information to their destination, as shown in Figure1. As a practical application of the presented system in IoT, the source and relay are energy-constrained sensors, and the destination is a sink node with a fixed energy source. They cannot directly communicate due to long distances between sensors and sink, so sensors have to cooperatively transmit information by the relaying way. Moreover, the sensors may be located in a toxic environment, and therefore, UAV is a better way to transfer power to the energy-constrained sensors. To explore the performance limit of the system, a problem is formulated by jointly optimizing the position of the UAV and time allocation to maximize the achievable information rate of the system.Figure 1
UAV-aided wireless powered relay communication network.Our contributions are summarized as follows. Firstly, we propose a UAV-aided wireless powered relay communication system, which reduces the deployment cost thanks to the flexibility of UAV, compared with fixed PB. Secondly, due to the coupling of optimized variables, the optimization problem is nonconvex. To this end, we decompose the problem into two subproblems to, respectively, derive the explicit expressions of the optimal position of UAV and optimal time allocation. Thirdly, simulation results show that our proposed method is significantly superior to the benchmark methods. Finally, the effect of relay position on the system is demonstrated, and it provides some insights for future application of the UAV-aided wireless powered relay communication system.The rest of this paper is organized as follows. In Section2, the system model is given and an optimization problem is formulated. Section 3 proposes an efficient method to solve the optimization problem. Section 4 gives simulation results to illustrate the system performance. Finally, Section 5 concludes this paper.


## 2. System Model

We consider a UAV-aided wireless powered relay communication network consisting of a source (S), a relay (R), and a destination (D) on the ground, as well as a UAV (U) in the air, as shown in Figure 1. The source wishes to transmit the information to the destination with the assistance of the half-duplex decode-and-forward (DF) relay. It is assumed that the direct link from the source to the destination does not exist. The source and relay are energy-constrained nodes, and thus, they have to harvest energy from the signal emitted by the UAV. All the nodes of the network are deployed with a single antenna.A three-dimensional (3D) Cartesian coordinate system is considered, since there simultaneously exist air-to-ground (ATG) channels, i.e., the channels from the UAV to the nodes on the ground (S,R), and ground-to-ground (GTG) channels, i.e., the channel from S to R and that from R to D. The UAV is assumed to hover at a fixed altitude H, and its position is denoted as dU=xU,yU,H. The positions of the nodes on the ground are denoted as dv=xv,yv,0,v∈S,R,D.According to the preliminary channel measurement results, the ATG channels typically consist of strong LoS links [23]. Further, according to 3GPP, when the altitude of UAV exceeds 40 m [24], the LOS probability is 100% in a rural environment. Therefore, in this paper, the channel power gain from the UAV to S and R is, respectively, modeled as
(1)gv=βA∥dU−dv∥αA,v∈S,R,where αA and βA represent the path loss factor and the channel power gain at unit reference distance for the ATG channel, respectively. For the GTG channel, the channel power gain between two nodes is modeled as
(2)huv=βGρ2∥du−dv∥αG,u∈S,R,v∈R,D,where ρ2 is a random variable following exponential distribution with unit mean. αG and βG represent the path loss factor and the channel power gain at unit reference distance for the GTG channel, respectively.The transmission procedure is divided into three phases, that is, the power transfer (PT) phase, source transmission (ST) phase, and relay transmission (RT) phase, as shown in Figure2. The duration of each transmission procedure is denoted as T. In the first phase of duration τ0T0<τ0<1, i.e., the PT phase, the UAV emits the energy to the source and the relay. The signal received by the source and the relay can be, respectively, given as
(3)ev=gvPx0+nv,v∈S,R,where P is the transmit power of UAV, x0 is the signal of unit energy transmitted by UAV, and nv is the additive white Gaussian noise (AWGN) with power σ2.Figure 2
The transmission procedure.Thus, in the PT phase, the energy harvested byS and R can be expressed as
(4)Ev=ητ0TgvP,v∈S,R,whereη is the energy harvesting efficiency and satisfies 0≤η≤1.In the second phase of duration1−τ0T/2, i.e., ST phase, S transmits information to R, and the signal received by R can be expressed as
(5)yR=hSRPSxS+nR,where nR is AWGN with power σ2 and PS=ES/1−τ0T/2 is the transmit power of S. By using (1) and (4), PS can be further calculated as
(6)PS=2ητ0βAP1−τ0dU−dSαA.From (5), the received signal-to-noise ratio (SNR) at R can be calculated as
(7)γR=hSRPSσ2.By using (6), (7) can be transformed into
(8)γR=2τ01−τ0L1dU−dSαA,where L1=ηβAhSRP/σ2.Therefore, the achievable information rate fromS to R can be expressed as
(9)RSR=1−τ02log21+γR.In the third phase of duration1−τ0T/2, i.e., RT phase, R first decodes the information and then forwards decoded information to D, and the signal received by D can be expressed as
(10)yD=hRDPRxR+nD,where nD is AWGN with power σ2 and PR=ER/1−τ0T/2 is the transmit power of R. By using (1) and (4), PR can be expressed as
(11)PR=2ητ0βAP1−τ0dU−dRαA.From (10) and (11), the received SNR at D can be calculated as
(12)γD=hRDPRσ2=2τ01−τ0L2dU−dRαA,where L2=ηβAhRDP/σ2, and therefore, the achievable information rate from R to D can be expressed as
(13)RRD=1−τ02log21+γD.The end-to-end achievable information rate of the system is the minimum ofRSR and RRD, that is,
(14)R=1−τ02log21+2τ01−τ0minL1dU−dSαA,L2dU−dRαA.Our objective is to maximize the end-to-end achievable information rateR by jointly optimizing the position of UAV and time allocation, and therefore, the corresponding optimal problem is formulated as
(15)P1:maxτ0,xU,yUR,s.t.0<τ0<1.


## 3. Problem Solution

The optimization problemP1 is a nonconvex problem, due to the coupling of optimized variables. In this section, problem P1 is first transformed into an equivalent problem, and then, it is efficiently solved by our proposed algorithm.


### 3.1. Equivalent Problem of ProblemP1

In this subsection, the original problemP1 is transformed into an equivalent problem, and the result is given in Proposition 2 as follows.Algorithm 1.
Algorithm of solving problemP1.
(1)
Obtain the optimal UAV position (xU∗,yU∗,H) and the optimal value d∗ by solving problem P3(2)
Obtain the optimalτ0∗ by solving problem P2 with the obtained d∗(3)
Calculate the optimal value of problemP1 by using the obtained xU∗,yU∗ and τ0∗Proposition 2.
ProblemP1 is equivalent to the following problem P2:
(16)P2:maxτ01−τ02log21+2τ01−τ0d∗,(17)s.t.0<τ0<1,where d∗ is the optimal value of the following problem P3:
(18)P3:d∗=maxxU,yUminL1∥dU−dS∥αA,L2∥dU−dR∥αA.Proof.
Proposition2 can be proved by the similar method adopted in [7], and thus, the proof is omitted here.By using Proposition2, the original problem P1 can be decomposed into two successive subproblems P3 and P2 to solve. Firstly, by solving problem P3, the optimal UAV position is obtained. Then, by substituting the obtained optimal value d∗ of P3 into (16) and solving problem P2, the optimal time allocation is obtained. Finally, by using obtained optimal UAV position and time allocation, the maximal achievable information rate can be calculated. Algorithm 1 gives the detailed procedure.In the rest of this section, problemP3 and P2 are dealt with sequentially to obtain their explicit optimal solution.


### 3.2. Solution of ProblemP3

From problemP3, it can be observed that the solution of the problem depends on the distances from UAV to S (i.e., U‐S) and from UAV to R (i.e., U‐R). Without loss of generality, the considered coordinate system is set as shown in Figure 3. S is placed at the origin with coordinate (0, 0, 0). R is located at the positive axis of the X-axis, and its coordinate is (xR, 0, 0), where xR>0. D is located on the X‐Y plane, and its coordinate is (xD,yD,0). It is worth noting if system nodes are in a practical geographical coordinate system; it can be changed to our consider coordinate system by translation and rotation transformations [25]. Figure 3 plots two possible positions of UAV, i.e., a general position U1xU,yU,H and a special position U2xU,0,H. We can prove the following lemma, which shows that the position U1 is superior to U2.Lemma 3.
There exists a positionxU,yU,H of UAV satisfying the following condition, which can achieve the optimum of problem P3:
(19)0≤xU≤xR,yU=0.Proof.
Firstly, we prove that at the optimum of problemP3, yU=0. That is, in Figure 3, the position of U2 is superior to U1. Observing problem P3, it can be found that the solution of the problem depends on the distances of U‐S and U‐R, and less distances lead to a greater objective function. According to the geometrical relations, one can easily see that the distance from U2 to S is less than that from U1 to S. Similarly, the distance from U2 to R is less than that from U1 to R. So, at the optimum, yU=0.Figure 3
The considered 3D coordinate system.Secondly, we prove that there exists a positionxU,0,H of UAV satisfying (19), which can achieve the optimum of problem P3. Figure 4 plots two possible positions of UAV, i.e., U1xU1,0,H satisfying (19) and U2xU2,0,H not satisfying (19). It is noting that for the cases not satisfying (19), we only consider the case xU2<0, and the other case xU2>xR is neglected, due to the symmetry.Figure 4
TheX‐Z plane of the considered 3D coordinate system.Further, we dividexU2<0 into the following two intervals to discuss. On the one hand, when xU2<−xU1, it can be easily seen that the distance of U2‐S is greater than that of U1‐S and the distance of U2‐R is greater than that of U1‐R. This means that for problem P3, the position of U1 is superior to U2. On the other hand, when −xU1≤xU2≤0, we can adjust the position of U1 in order so that xU1=−xU2. At this moment, the distance of U2‐S is equal to that of U1‐S, and the distance of U2‐R is greater than or equal to that of U1‐R. Therefore, the first term of the objective function of problem P3 with position U1 is equal to that with U2 while the second term of the objective function of problem P3 with U1 is larger than that with U2. This means that for problem P3, when UAV is located at U1, it makes the objective function of problem P3 either become larger than U2 or be equal to U2. So, this lemma is proved.Lemma3 shows that the UAV should be deployed at some position with height H above the line between S and R. In the following, we derive the optimal X-axis cordinate xU of the UAV.Using Lemma3, problem P3 can be transformed into the following problem P4:
(20)P4:d∗=maxxUminL1xU2+H2αA/2,L2xU−xR2+H2αA/2,s.t.0≤xU≤xR.For problemP4, the following proposition can be derived.Proposition 4.
The optimalxU∗ of problem P4 is given by
(21)xU∗=0,ifL>1andxR<HL−1,xR,ifL<1andxR<H1L−1,xR2,ifL=1,κ,else,where κ=LxR2−L−12H2−xR/L−1 and L=L2/L12/αA.Proof.
LetT1xU=L1/xU2+H2αA/2 and T2xU=L2/xU−xR2+H2αA/2. It can be easily found that when 0≤xU≤xR, T1 is a decreasing function of xU while T2 is an increasing function of xU. Figure 5 plots all the possible cases of two functions T1xU and T2xU.Figure 5
The four cases for functionT1xU and T2xU. The red (∗) represents the optimal point of problem P4.
(a)(b)(c)(d)For Figure5(a), this case happens when T10<T20, i.e., L>1 and xR<HL−1. For this case, we can have the optimal xU∗=0 and the corresponding d∗=L1/HαA.For Figure5(b), this case happens when T1xR>T2xR, i.e., L<1 and xR<H1/L−1. For this case, we can have the optimal xU∗=xR and the corresponding d∗=L2/HαA.For the remaining Figures5(c) and 5(d), problem P4 achieves the optimum, when
(22)T1xU=T2xU.After some simple algebraic manipulations, (22) is transformed into
(23)L−1xU2+2xRxU+L−1H2−xR∗=0.IfL=1, corresponding to Figure 5(c), (23) becomes 2xRxU−xR∗=0, and thus, xU∗=xR/2.Otherwise, corresponding to Figure5(d), (23) is a quadratic equation of xU, and we can obtain the root of this equation as
(24)xU∗=LxR∗−L−12H2−xRL−1.Substituting the obtainedxU∗ into T1xU or T2xU, the corresponding d∗ can be calculated. Thus, this proposition is proved.Lemma3 and Proposition 4 show that to achieve the best system performance, the UAV should be placed at the position above the line between S and R, where its height is H and horizontal distance away from S is x∗U.


### 3.3. Solution of ProblemP2

For problemP2, we can derive the following proposition to obtain the optimal solution.Proposition 5.
The optimalτ0∗ of problem P2 is given by
(25)τ0∗=eW02d∗−1/e+1−1eW02d∗−1/e+1+2d∗−1,where W0x is the branch satisfying Wx≥−1 of Lambert W function Wx [26].Proof.
The objective function of problemP2 is concave due to its second derivative −2d∗2/1−τ0+2τ0d∗21−τ0ln2≤0. To obtain the maximal value of the objective function, let its first derivative be equal to zero, so we have
(26)1+2d∗τ01−τ0ln1+2d∗τ01−τ0=2d∗+2d∗τ01−τ0.
Let(27)1+2d∗τ01−τ0=y,and we can easily obtain that y>1, due to d∗>0 and 0<τ0<1. After some simple algebraic manipulations for (26), we can have
(28)ylny=y+2d∗−1.
(28) can be further transformed into
(29)lnyeelny/e=2d∗−1e.
According to the definition of the LambertW function, we can have
(30)lnye=W2d∗−1e.
Then, we can have(31)y=eW2d∗−1/e+1.
According to the obtained conditiony>1, we can have W2d∗−1/e>−1. This means that the W0x branch is adopted in (31). Then, after some simple algebraic manipulations for (27), we can obtain
(32)τ0∗=eW02d∗−1/e+1−1eW02d∗−1/e+1+2d∗−1.
This proposition is thus proved.Finally, we analyze the computation complexity of Algorithm1. Algorithm 1 is composed of three successive steps. In the first step and the second step, the explicit expressions of the optimal UAV position and optimal time fraction are, respectively, used, and thus, the complexity is O1. By using the obtained optimal UAV position and time fraction, the optimal value can be calculated by using (14), and thus, its complexity is also O1. So, the total complexity of Algorithm 1 is O1.


## 3.1. Equivalent Problem of ProblemP1

In this subsection, the original problemP1 is transformed into an equivalent problem, and the result is given in Proposition 2 as follows.Algorithm 1.
Algorithm of solving problemP1.
(1)
Obtain the optimal UAV position (xU∗,yU∗,H) and the optimal value d∗ by solving problem P3(2)
Obtain the optimalτ0∗ by solving problem P2 with the obtained d∗(3)
Calculate the optimal value of problemP1 by using the obtained xU∗,yU∗ and τ0∗Proposition 2.
ProblemP1 is equivalent to the following problem P2:
(16)P2:maxτ01−τ02log21+2τ01−τ0d∗,(17)s.t.0<τ0<1,where d∗ is the optimal value of the following problem P3:
(18)P3:d∗=maxxU,yUminL1∥dU−dS∥αA,L2∥dU−dR∥αA.Proof.
Proposition2 can be proved by the similar method adopted in [7], and thus, the proof is omitted here.By using Proposition2, the original problem P1 can be decomposed into two successive subproblems P3 and P2 to solve. Firstly, by solving problem P3, the optimal UAV position is obtained. Then, by substituting the obtained optimal value d∗ of P3 into (16) and solving problem P2, the optimal time allocation is obtained. Finally, by using obtained optimal UAV position and time allocation, the maximal achievable information rate can be calculated. Algorithm 1 gives the detailed procedure.In the rest of this section, problemP3 and P2 are dealt with sequentially to obtain their explicit optimal solution.


## 3.2. Solution of ProblemP3

From problemP3, it can be observed that the solution of the problem depends on the distances from UAV to S (i.e., U‐S) and from UAV to R (i.e., U‐R). Without loss of generality, the considered coordinate system is set as shown in Figure 3. S is placed at the origin with coordinate (0, 0, 0). R is located at the positive axis of the X-axis, and its coordinate is (xR, 0, 0), where xR>0. D is located on the X‐Y plane, and its coordinate is (xD,yD,0). It is worth noting if system nodes are in a practical geographical coordinate system; it can be changed to our consider coordinate system by translation and rotation transformations [25]. Figure 3 plots two possible positions of UAV, i.e., a general position U1xU,yU,H and a special position U2xU,0,H. We can prove the following lemma, which shows that the position U1 is superior to U2.Lemma 3.
There exists a positionxU,yU,H of UAV satisfying the following condition, which can achieve the optimum of problem P3:
(19)0≤xU≤xR,yU=0.Proof.
Firstly, we prove that at the optimum of problemP3, yU=0. That is, in Figure 3, the position of U2 is superior to U1. Observing problem P3, it can be found that the solution of the problem depends on the distances of U‐S and U‐R, and less distances lead to a greater objective function. According to the geometrical relations, one can easily see that the distance from U2 to S is less than that from U1 to S. Similarly, the distance from U2 to R is less than that from U1 to R. So, at the optimum, yU=0.Figure 3
The considered 3D coordinate system.Secondly, we prove that there exists a positionxU,0,H of UAV satisfying (19), which can achieve the optimum of problem P3. Figure 4 plots two possible positions of UAV, i.e., U1xU1,0,H satisfying (19) and U2xU2,0,H not satisfying (19). It is noting that for the cases not satisfying (19), we only consider the case xU2<0, and the other case xU2>xR is neglected, due to the symmetry.Figure 4
TheX‐Z plane of the considered 3D coordinate system.Further, we dividexU2<0 into the following two intervals to discuss. On the one hand, when xU2<−xU1, it can be easily seen that the distance of U2‐S is greater than that of U1‐S and the distance of U2‐R is greater than that of U1‐R. This means that for problem P3, the position of U1 is superior to U2. On the other hand, when −xU1≤xU2≤0, we can adjust the position of U1 in order so that xU1=−xU2. At this moment, the distance of U2‐S is equal to that of U1‐S, and the distance of U2‐R is greater than or equal to that of U1‐R. Therefore, the first term of the objective function of problem P3 with position U1 is equal to that with U2 while the second term of the objective function of problem P3 with U1 is larger than that with U2. This means that for problem P3, when UAV is located at U1, it makes the objective function of problem P3 either become larger than U2 or be equal to U2. So, this lemma is proved.Lemma3 shows that the UAV should be deployed at some position with height H above the line between S and R. In the following, we derive the optimal X-axis cordinate xU of the UAV.Using Lemma3, problem P3 can be transformed into the following problem P4:
(20)P4:d∗=maxxUminL1xU2+H2αA/2,L2xU−xR2+H2αA/2,s.t.0≤xU≤xR.For problemP4, the following proposition can be derived.Proposition 4.
The optimalxU∗ of problem P4 is given by
(21)xU∗=0,ifL>1andxR<HL−1,xR,ifL<1andxR<H1L−1,xR2,ifL=1,κ,else,where κ=LxR2−L−12H2−xR/L−1 and L=L2/L12/αA.Proof.
LetT1xU=L1/xU2+H2αA/2 and T2xU=L2/xU−xR2+H2αA/2. It can be easily found that when 0≤xU≤xR, T1 is a decreasing function of xU while T2 is an increasing function of xU. Figure 5 plots all the possible cases of two functions T1xU and T2xU.Figure 5
The four cases for functionT1xU and T2xU. The red (∗) represents the optimal point of problem P4.
(a)(b)(c)(d)For Figure5(a), this case happens when T10<T20, i.e., L>1 and xR<HL−1. For this case, we can have the optimal xU∗=0 and the corresponding d∗=L1/HαA.For Figure5(b), this case happens when T1xR>T2xR, i.e., L<1 and xR<H1/L−1. For this case, we can have the optimal xU∗=xR and the corresponding d∗=L2/HαA.For the remaining Figures5(c) and 5(d), problem P4 achieves the optimum, when
(22)T1xU=T2xU.After some simple algebraic manipulations, (22) is transformed into
(23)L−1xU2+2xRxU+L−1H2−xR∗=0.IfL=1, corresponding to Figure 5(c), (23) becomes 2xRxU−xR∗=0, and thus, xU∗=xR/2.Otherwise, corresponding to Figure5(d), (23) is a quadratic equation of xU, and we can obtain the root of this equation as
(24)xU∗=LxR∗−L−12H2−xRL−1.Substituting the obtainedxU∗ into T1xU or T2xU, the corresponding d∗ can be calculated. Thus, this proposition is proved.Lemma3 and Proposition 4 show that to achieve the best system performance, the UAV should be placed at the position above the line between S and R, where its height is H and horizontal distance away from S is x∗U.


## 3.3. Solution of ProblemP2

For problemP2, we can derive the following proposition to obtain the optimal solution.Proposition 5.
The optimalτ0∗ of problem P2 is given by
(25)τ0∗=eW02d∗−1/e+1−1eW02d∗−1/e+1+2d∗−1,where W0x is the branch satisfying Wx≥−1 of Lambert W function Wx [26].Proof.
The objective function of problemP2 is concave due to its second derivative −2d∗2/1−τ0+2τ0d∗21−τ0ln2≤0. To obtain the maximal value of the objective function, let its first derivative be equal to zero, so we have
(26)1+2d∗τ01−τ0ln1+2d∗τ01−τ0=2d∗+2d∗τ01−τ0.
Let(27)1+2d∗τ01−τ0=y,and we can easily obtain that y>1, due to d∗>0 and 0<τ0<1. After some simple algebraic manipulations for (26), we can have
(28)ylny=y+2d∗−1.
(28) can be further transformed into
(29)lnyeelny/e=2d∗−1e.
According to the definition of the LambertW function, we can have
(30)lnye=W2d∗−1e.
Then, we can have(31)y=eW2d∗−1/e+1.
According to the obtained conditiony>1, we can have W2d∗−1/e>−1. This means that the W0x branch is adopted in (31). Then, after some simple algebraic manipulations for (27), we can obtain
(32)τ0∗=eW02d∗−1/e+1−1eW02d∗−1/e+1+2d∗−1.
This proposition is thus proved.Finally, we analyze the computation complexity of Algorithm1. Algorithm 1 is composed of three successive steps. In the first step and the second step, the explicit expressions of the optimal UAV position and optimal time fraction are, respectively, used, and thus, the complexity is O1. By using the obtained optimal UAV position and time fraction, the optimal value can be calculated by using (14), and thus, its complexity is also O1. So, the total complexity of Algorithm 1 is O1.


## 4. Simulation Results

This section gives simulation results to illustrate the performance of the presented system and proposed algorithm. In the simulations, the considered 3D coordinate system is shown in Figure3, where S is located at (0, 0, 0), R is at (0, 100, 0), and D is at (200, 200, 0). The UAV is set to hover at the altitude of H=50 m. The noise power is σ2=−130 dBm. According to [5], the path loss factor αA and the channel power gain βA at unit reference distance for the ATG channel are set to be αA=2 and βA=30 dB, respectively. The path loss factor αG and the channel power gain βG at unit reference distance for the GTG channel are set to be αG=3 and βG=−30 dB, respectively. The energy harvesting efficiency η is set to 1 in order to explore the performance limit of the system.


### 4.1. Performance Comparisons

In this subsection, the performance of the proposed method is compared with three other methods, i.e., (1) The benchmark method: fixed time fractionτ0=1/3 and fixed position of UAV dU=50,0,50; that is, UAV hovers above the middle point between S and R; (2) The optimal time fraction and fixed position (OTFP) method: fixed position of UAV dU=50,0,50 and optimal time fraction τ which is set according to (25); (3) The optimal position and fixed time fraction (OPFT) method: fixed time fraction τ0=1/3 and the optimal position of UAV which is set according to (21). Figure 6 shows the achievable information rates of the proposed method and three other methods with respect to the power of UAV P. It can be found that the achievable information of the proposed method is significantly superior to the three other methods.Figure 6
Performance comparisons of the proposed method and three other methods.


### 4.2. The Effect of UAV Altitude on System Performance

In this subsection, the effect of UAV altitude on the achievable information rate of the system is demonstrated in Figure7, where the power of UAV is set to be P=40 dBm. One can observe that with the increase of the altitude of UAV, the achievable information rates of the four methods decrease. Moreover, our proposed method outperforms the three other methods, which agrees with Figure 6. It can also be found that the achievable information rate of the proposed method is reduced faster than the OTFP one as the UAV altitude increases. The reason may be as follows. From (14), it can be seen that the optimal UAV position helps to reduce the distances from UAV to S and from UAV to R, and thus, the transmit power of S and R can be significantly increased. This further leads to very higher SNR, which is a more key factor for the achievable information rate of the system compared with time allocation. As the UAV altitude increases, the distance from UAV to S and that from UAV to R will be dominated by the altitude of UAV, and therefore, the gap of the two methods will gradually decrease.Figure 7
The effect of UAV altitude on achievable information rate performance.


### 4.3. The Effect of Relay Position on System Performance

In this subsection, we discuss the effect of relay position on the achievable information rate of the system. The UAV is set to hover at the altitude ofH=50 m. S is set to be located at (0, 0, 0), and D is at (200, 0, 0), which means that both S and D are located at the X-axis. The relay is also set to be located at the X-axis, and its position is denoted by the distance from S to R (i.e., S‐R). The power of UAV is set to be P=40 dBm. Figure 8 shows the achievable information rates of the four methods with respect to the distance of S−R. It can be seen that with the increase of S‐R distance, the achievable information rate of the system firstly increases and then gradually decreases. Moreover, when R is at the position of about 90 m away from S, the system performance achieves the maximum.Figure 8
The effect of relay position on achievable information rate performance.


## 4.1. Performance Comparisons

In this subsection, the performance of the proposed method is compared with three other methods, i.e., (1) The benchmark method: fixed time fractionτ0=1/3 and fixed position of UAV dU=50,0,50; that is, UAV hovers above the middle point between S and R; (2) The optimal time fraction and fixed position (OTFP) method: fixed position of UAV dU=50,0,50 and optimal time fraction τ which is set according to (25); (3) The optimal position and fixed time fraction (OPFT) method: fixed time fraction τ0=1/3 and the optimal position of UAV which is set according to (21). Figure 6 shows the achievable information rates of the proposed method and three other methods with respect to the power of UAV P. It can be found that the achievable information of the proposed method is significantly superior to the three other methods.Figure 6
Performance comparisons of the proposed method and three other methods.


## 4.2. The Effect of UAV Altitude on System Performance

In this subsection, the effect of UAV altitude on the achievable information rate of the system is demonstrated in Figure7, where the power of UAV is set to be P=40 dBm. One can observe that with the increase of the altitude of UAV, the achievable information rates of the four methods decrease. Moreover, our proposed method outperforms the three other methods, which agrees with Figure 6. It can also be found that the achievable information rate of the proposed method is reduced faster than the OTFP one as the UAV altitude increases. The reason may be as follows. From (14), it can be seen that the optimal UAV position helps to reduce the distances from UAV to S and from UAV to R, and thus, the transmit power of S and R can be significantly increased. This further leads to very higher SNR, which is a more key factor for the achievable information rate of the system compared with time allocation. As the UAV altitude increases, the distance from UAV to S and that from UAV to R will be dominated by the altitude of UAV, and therefore, the gap of the two methods will gradually decrease.Figure 7
The effect of UAV altitude on achievable information rate performance.


## 4.3. The Effect of Relay Position on System Performance

In this subsection, we discuss the effect of relay position on the achievable information rate of the system. The UAV is set to hover at the altitude ofH=50 m. S is set to be located at (0, 0, 0), and D is at (200, 0, 0), which means that both S and D are located at the X-axis. The relay is also set to be located at the X-axis, and its position is denoted by the distance from S to R (i.e., S‐R). The power of UAV is set to be P=40 dBm. Figure 8 shows the achievable information rates of the four methods with respect to the distance of S−R. It can be seen that with the increase of S‐R distance, the achievable information rate of the system firstly increases and then gradually decreases. Moreover, when R is at the position of about 90 m away from S, the system performance achieves the maximum.Figure 8
The effect of relay position on achievable information rate performance.


## 5. Conclusions

In this paper, a UAV-aided wireless powered relay communication system has been presented, where a UAV firstly supplies energy to a source and a relay, and then, the source and relay cooperatively transmit information to their destination. An efficient algorithm has been proposed to maximize the achievable information rate of the system by deriving the explicit expressions of the optimal position of UAV and optimal time fraction. Simulation results have shown that our proposed algorithm significantly outperforms the benchmark methods. The effect of relay position on the system has also been demonstrated, which provides some insights for the deployment of relay in UAV-aided wireless powered relay communication systems in IoT.



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*Source: 5537517-2021-04-19.xml*