I Introduction
Recently, deploying unmanned aerial vehicles (UAVs) in wireless communication networks for coverage and throughput enhancement has attracted significant attention from both the industry and academia [1, 2, 3]
. The swift mobility of UAV enables fast deployment and establishment of communications in emergency situations such as for rescue after hurricane or earthquake. The lower cost of UAV than the traditional communication infrastructure also makes UAV a costeffective option for the network coverage and throughput enhancement in coveragelimited zones like the rural or mountainous areas. Besides, UAVs in general have better airtoground (A2G) channels due to a high probability of line of sight (LOS) link with ground users
[4]. Therefore, the UAV has been considered for being an aerial base station (BS) [5, 6, 7, 8, 9, 10], wireless relay [11, 12, 13, 14, 15, 16], and for networking [17, 18] as well as for data collection and dissemination in wireless sensor networks [19, 20, 21, 22, 23]. Several industrial projects that leverage the UAV for enhanced wireless communications, like the Facebook’s laser drone test [24] and Qualcomm’s drone communication plan [3], are also proposed.Ia Related Works
There are still many technical challenges to overcome in order to harvest the benefits of UAVenabled wireless communications [2]. Specifically, the airtoground (A2G) channel is different from the existing groundtoground channel, and is highly dependent on the position of UAV. In addition, due to limited battery energy, joint positioning/flying trajectory design and transmission power control are critical to achieve high spectral efficiency and energy efficiency in UAVenabled communication systems. For example, reference [5] derived a fixwing UAV propulsion energy consumption model and studied the joint UAV trajectory and transmission power control problem for maximizing the system energy efficiency. By deploying the UAV as an aerial BS, reference [6] studied the trajectory and power control problem for maximizing the minimum downlink rate of ground users over orthogonal channels. By assuming that the aerial BS has multiple antennas, reference [7] considered joint optimization of the UAV flying altitude and beamwidth for throughput maximization in multicast, broadcast and uplink scenarios, respectively. Reference [8] considered the placement of a minimum number UAVmounted BSs for providing required quality of service for the ground users, while [9] studied the joint scheduling, flying trajectory and power control of multiple UAVmounted BSs for maximizing the minimum rate of served ground users. Unlike [8, 9], by modeling the positions of the UAVs as a 3dimensional Poisson point process, the work of [10] considered the spectrum sharing problem between the cellular network and drone small cells, and investigated the deployment density of UAVs to maximize the outageconstrained throughput. While most of the aforementioned works have assumed deterministic LOS links, the work [25] has studied the optimal flying altitude of a UAV for coverage maximization under a probabilistic LOS channel model [26].
When the UAV is deployed as a wireless relay, the position and flying trajectory design are also of great importance [11]. The work [12] considered an uplink relaying system and optimized the flying heading of the UAV for maximizing an ergodic transmission rate. In [13], a decodeandforward relay system is considered, and the UAV flying trajectory and transmission power are jointly optimized for maximizing the throughput between the ground BS and user equipment (UE). In [15], the authors considered the UAV positioning problem in a relay system by incorporating the local topological information, where the UAV is aimed to be deployed in a position that can enjoy LOS links. The work [14] considered an uplink multiUAV relaying system under the LOS channels with random phase. The UAV positions and UE transmission powers are jointly optimized to maximize the minimum ergodic throughput of ground UEs. Reference [16] considered the use of a relay UAV for communicating with another observation UAV and studied the optimal positioning of the relay UAV for throughput maximization. The works [17] and [18] considered the deployment of multiple relay UAVs to form an adhoc network and achieve long distance communications, respectively.
IB Contributions
In this paper, we consider a wireless relay network where the UAV is used to extend the service of a BS for a set of distant ground UEs, as shown in Fig. 1. Different from the aforementioned works where either uplink or downlink transmission is considered, we consider the twoway communications between the BS and ground UEs. Besides, unlike [12, 6] which consider only onehop communication between the UEs and UAV, we consider the twohop communications where the relay UAV amplifies and forwards (AF) the signals from one side to the other.
We assume the LOS channels and aim to optimize the UAV position and transmission powers of the BS, UEs and the UAV jointly, for maximizing the sum rate of the twoway communication links. Except for the maximum transmission power constraints, we also consider the quality of service (QoS) constraint on the control link between the BS and the relay UAV. In practice, the control link is used for control and command signaling between the relay UAV and the BS, and is essential to the UAV motion control. The formulated joint UAV positioning and power control (JPPC) problem has a complicated nonconcave sum rate function and is difficult to solve in general. The main contributions are summarized as below.

We first consider a simple scenario with only one UE [11], and present a semianalytical solution to the JPPC problem. It is shown that the optimal position of the relay UAV, when projected onto the xy plane, must lie on the line segment between the BS and UE.

For the general case with multiple UEs, we employ the successive convex approximation (SCA) technique [27]. In SCA, one solves a surrogate convex optimization problem iteratively by replacing the nonconcave objective by a concave surrogate function. Interestingly, according to [28], the curvature of the surrogate function has a direct impact on the convergence behavior of the SCA iterations. By carefully exploiting the function structure, we propose a concave surrogate function for the SCA algorithm. Moreover, we show that the proposed surrogate function has a smaller curvature than the one that is obtained by following the recent work [23], and can lead to a fast convergence of the SCA iterations.

The SCA algorithm requires one to globally solve the convex surrogate problem in every iteration. Since the convex surrogate problem does not admit closedfrom solutions, it requires one to employ another powerful optimization method in order to solve the surrogate problem, which may not be efficient especially when the number of UEs is large. To improve the computation efficiency, we adopt a recently proposed algorithm by [29] which combines the SCA iteration with the FISTAtype accelerated gradient projection (AGP) algorithm [30]. By applying the algorithm in [29] to our JPPC problem, one of the step involves projection onto a set of quadratic constraints. We further employ the AGP method to solve the Lagrange dual problem of the projection step. Thus, the proposed algorithm for the JPPC problem involves double loops of AGP iterations.

Simulation results are presented to show that the SCA algorithm using the proposed surrogate function exhibits a significantly faster convergence behavior than that using [23]. Besides, the doubleloop AGP algorithm can further reduce the computation time by more than an order of magnitude. Simulation results also reveal that fact that the optimal positioning of the relay UAV is not trivial since the optimized solution can greatly outperform simple strategies that deploy the relay UAV on top of the BS or in a geometric center of the network.
The remainder of the paper is organized as follows. Section II presents the twoway relay system model and formulates the JPPC problem. The scenario with only one UE is studied in Section III. In Section IV, the proposed SCA algorithm and doubleloop AGP algorithm are presented. The simulation results are given in Section V. Finally, conclusions are drawn in Section VI.
Ii System Model and Problem Formulation
Iia System Model
As illustrated in Fig. 1, we consider a UAVenabled wireless twoway relaying communication network constituted by UEs, one UAV and one BS. All the nodes are equipped with single antenna. It is assumed that there is no direct communication link between the UEs and the BS, and the UAV plays a role relaying the uplink signals from the UEs to the BS as well as relaying the downlink signals from the BS to the UEs. So the UAV extends the service coverage of the BS, and its flying and communication are controlled by the BS. Without lose of generality, we assume that all the UEs and the BS are located at the same ground plane. Denote by a three dimension (3D) location of the th UE and by the 3D location of the BS. The UAV flies in the sky with a fixed altitude (meters) and its 3D location is denoted by .
We assume that the frequency division duplex (FDD) is used for uplink and downlink communications. The UAV works as a twoway relay which amplifies and forwards the uplink and downlink signals to the BS and UEs, respectively. Besides, the frequency division multiplxing (FDM) is used so that the communication links of different UEs are orthogonal to each other and have no crosslink interference. For the uplink transmission, i.e., the UEUAVBS link, we denote as the transmission power of each UE , where . The transmission power allocated by the UAV for relaying the uplink signals from UE to the BS is denoted by . For the downlink transmission, i.e., the BSUAVUE link, the transmission power of the BS for UE is . The downlink relaying power of the UAV for UE is denoted as . Since the airtoground (A2G) channel between the UAV and BS and that between the UAV and UEs usually consist of a strong lineofsight (LOS) link [14], we adopt this model throughout this paper.
Uplink signal model: Denote as the Gaussian information signal sent by UE . In the first time slot of the AF transmission, the signals received by the UAV are given by
(1) 
where is the reference channel gain at the distance meter from the UE, is the Euclidean distance between UE and the UAV, and
is the additive noise with zero mean and variance
. In the second time slot, the UAV amplifies the received signal and transmits it to the BS. In particular, by assuming that the channel state information (CSI) is available at the UAV, the UAV can amplify the signal with a gain where(2) 
is the inverse of the signal power of , and is the uplink transmission power of the UAV. Thus the received signal at the BS for UE is given by
(3) 
where is the distance between the UAV and the BS and is the additive noise at the BS. By (3), the uplink signaltonoise ratio (SNR) for the th UE can be expressed as
(4) 
where (2) is applied to obtain the second equality and is defined in the last equality.
Downlink signal model: In the downlink transmission, given the information signal for UE sent from the BS in the first time slot, the received signal at the UAV is given by
(5) 
In the second time slot, the UAV amplifies by the gain , where
(6) 
and forwards it to UE . The received signal at UE is given by
(7) 
where is the additive noise at UE . By (7), the downlink SNR for the th UE is thus given by
(8) 
Denote by , , and
the vectors that collect the transmission powers of the UAV, BS and the UEs, respectively. Based on the uplink and downlink SNR expressions in (
4) and (8), the sum rate of the network is(9) 
where is the frequency bandwidth allocated for each UE, and and are respectively the uplink and downlink transmission rates of each UE . As the AF relay transmission requires two time slots, the rate is divided by 2 in (9).
Control link: Besides the data transmission, signaling on the control link between the UAV and the BS requires stringent communication quality. Let us denote as the trasmisssion power for the control signaling between the BS and the UAV. Then, the received SNR for the control link is
(10) 
Note that the control link is symmetric between the BS and the UAV under the LOS channel model. Thus, both the UAV and BS use the same power for control signaling.
IiB Problem Formulation
Denote by and the maximum transmission powers of the UAV, the BS and each UE , respectively. By (4), (8), (9) and (10), we consider the following joint UAV positioning and power control (JPPC) problem
(11a)  
s.t.  (11b)  
(11c)  
(11d)  
(11e) 
where is the allone vector, and is the SNR requirement of the downlink and uplink control signaling. The constraints (11b) and (11c) are the total transmission power constraints at the UAV and the BS, respectively; (11d) constrains the maximum transmission power of each UE .
Proof: It is easy to verify that in (4) is an increasing function of and , respectively; similary, in (8) is an increasing function of and , respectively. Thus, constraints (11b) to (11d) must hold with equality at the optimum. If (11e) holds with strict inequality at the optimum, then one can reduce and it makes (11b) to (11d) inactive. Then, either , or can be further increased to improve the sum rate. As a result, (11e) must also hold with equality at the optimum.
Iii UAV Positioning and Power Control: Single UE Case
To gain more insights, let us first study a special instance of problem (12) with only one UE (). For the signle UE case, problem (12) reduces to
(16a)  
s.t.  (16b)  
(16c) 
where
(17) 
Here, the subscript of all variables is removed for notation simplicity; besides, each is replaced by in which is the 3D location of the UE.
It is not surprising to see that the following statement is true.
Property 2
When projected onto the xy plane, the optimal UAV position is on the line segment connecting the BS and the UE.
Proof: The proof is presented in Appendix A.
By Property 2, we can write , where , and . By this expression and Property 1, we have
(18)  
(19)  
(20) 
Thus, problem (16) is equivalent to the following problem
(21)  
(22) 
where is obtained by substituting (18), (19) and (20) into (III), and denotes the optimal sum rate of the inner problem in (21) with a given value of . It is worth noting that, while problem (21) is not a convex problem, the inner problem with a given value of is a convex problem (since is a concave function for ), which can be efficiently solved. Therefore, one can globally solve problem (16) by searching the optimal value of in (22).
Iv UAV Positioning and Power Control: Multiple User Case
In this section, we study efficient algorithms to solve the UAV positioning and power control problem (12) with multiple UEs. Unlike the single user case, (12) is much more challenging to solve due to the nonconcave objective function. Our aim is to develop computationally efficient algorithms for (12). Specifically, the proposed approach is based on the successive convex approximation (SCA) method [27, 28, 31], where one obtains a suboptimal solution by solving a sequence of convex surrogate problems. For our problem (12), since the constraints (12b) and (12c) are both convex, we need to find a proper concave surrogate function for the nonconcave sum rate function . Next, we propose such a concave surrogate function that is amenable for fast SCA convergence.
Iva Proposed SCA Algorithm
Let us present a concave surrogate function for in (13). Let be a feasible point to problem (12). Define
(23a)  
(23b) 
and
(24)  
(25) 
for all .
Proposition 1
Proof: The derivations of (28) and (29) are technical. They are obtained by carefully examining the function structure and applying the firstorder Taylor lower bound of convex components in the rate functions. The details are given in Appendix B.
(28)  
(29) 
By replacing the objective function of (12) by (26), we obtain the following convex optimization problem
(30a)  
s.t.  (30b)  
(30c) 
The proposed SCA algorithm for solving problem (12) then iteraively solves (30) with a given feasible point obtained in the previous iteration, as shown in Algorithm 1. Since the constraint set of problem (12) is compact and convex, according to [32, Corollary 1], it can be shown that the variables yielded by Algorithm 1 converges to a stationary point of problem (12) as the iteration number goes to infinity.
Remark 1
It is worthwhile to mention that, except for using the offtheshelf convex solvers such as CVX [33] to solve (30), it would be more efficient to develop a customized algorithm. For example, because the Slater’s condition holds for (30), one may consider the Lagrange dual problem of (30), i.e.,
(31) 
where
(32) 
is the Lagrangian function, and and are the dual variables associated with (30b) and (30c), respectively. The dual subgradient ascent (DSA) method [34] can be applied to (31) while the inner minimization problem can be solved by applying the gradient projection (GP) method [35]. Since has a separable strucutre (it is a summation and each of the terms involves variables of either one UE or the UAV only), the GP method for the inner minimization problem can inherently be implemented in a fully parallel manner. The resultant algorithm is therefore more time efficient than the generalpurpose solvers.
IvB Comparison with the Surrogate Function in [23]
It is important to notice that the locally tight surrogate functions presented in Proposition 1 are simply one of the choices for SCA optimization, and a different surrgorate function may be obtained by another approach. From a theoretical point of view, as long as the surrogate function is a locally tight lower bound, i.e., satisfies (27), convergence of the SCA algorithm is guaranteed. Nevertheless, different surrograte functions may result in quite different convergence behavior. In accordance with [28, Theorem 3], the iteration complexity of the SCA algorithm is in the order of , where is a solution accuracy and is the gradient Lipschitz constant of the employed surrogate function. The constant represents the curvature and is also the spectral radius of the Hessian matrix of the surrogate function provided that it is twice differentiable.
In this subsection, we aim to demonstrate that the proposed surrogate functions in Proposition 1 is good in the sense that it has a faster convergence behavior than a surrogate function that is deduced following the idea in a recent work [23]. In particular, as we show in Appendix C, by following a similar method as in [23, Eqn. (20)], one can obtain
(33) 
as another concave and locally tight lower bound for in (13). Here
(34)  
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