I Introduction
Unmanned aerial vehicles (UAVs) or drones are expected to have a lot of applications in beyondfifthgeneration (B5G) and sixthgeneration (6G) wireless networks as dedicatedly deployed aerial wireless platforms (such as aerial base stations (BSs) [2, 3, 4, 5, 6], cellularconnected users [7], energy transmitters (ETs) [8, 9, 10], relays [11, 12], and mobile edge computing (MEC) servers [13, 14]). Among others, there has been an upsurge of interest in using UAVs as aerial data collectors (or fusion centers) to collect data from largescale wireless sensor networks (WSNs). In the upcoming Internet of Things (IoT) era, WSNs have been widely deployed for applications such as surveillance and environmental, agricultural, and traffic monitoring [15, 16, 17, 18], by collecting, e.g., geographical and environmental information, as well as images and videos. How to collect the data in a fast and reliable manner is one of key challenges faced in the design of WSNs. Different from the conventional designs using onground fusion centers for data collection, the UAVs in the sky can exploit the fullycontrollable mobility in the threedimensional (3D) space to fly close to sensors for collecting data more efficiently. UAVs can also leverage the strong lineofsight (LoS) groundtoair (G2A) channels for increasing the communication quality.
In the literature, there have been a handful of prior works studying the UAVenabled data collection [19, 20, 21, 22, 23, 24, 25], in which the UAV’s trajectory is designed for enhancing the system performance. For example, the authors in [19] and [20] jointly designed the UAV’s flight trajectory and wireless resource allocation/scheduling to minimize the mission completion time, in the scenarios when the sensors are deployed in the onedimensional (1D) and twodimensional (2D) spaces, respectively. The authors in [21] and [22] optimized the UAV’s trajectory and the sensors’ transmission/wakeup scheduling, in order to maximize the energy efficiency of the WSNs while ensuring the collected data amounts from sensors. The authors in [23] jointly designed the sensors’ transmission scheduling, power allocations, and UAV’s trajectory to maximize the minimum data collection rate from the ground sensors to a multiantenna UAV. Furthermore, [24] exploited the UAV’s 3D trajectory optimization for maximizing the minimum average rate for data collection, by considering angledependent Rician fading channels. In addition, [25] characterized the fundamental rate limits of UAVenabled multiple access channels (MAC) for data collection in a simplified scenario with linearly deployed sensors on the ground. In these prior works, the authors considered the adaptiverate transmission at the sensors, such that the sensors on the ground can adaptively adjust their transmission rate based on the wireless channel fluctuations due to the mobility of the UAVs. Furthermore, these prior works assumed that the onground devices (or sensors) send independent messages to the UAV under different multiple access techniques.
In contrast to the communicating independently, distributed beamforming has been recognized as a promising technique to enhance the data rate and energy efficiency in WSNs (see, e.g., [26, 27, 28, 29, 30] and the references therein), in which a large number of sensors are enabled to coordinate in transmitting common or shared messages to a fusion center (the UAV of our interest). By properly controlling the phases, the signals transmitted from different sensors can be coherently combined at the fusion center, thus increasing the communication range and enhancing the energy efficiency via exploiting the distributed beamforming gain [26]. For example, the authors in [27] and [28] investigated the distributed carrier synchronization, in which the fusion center broadcasts reference signals periodically, such that the sensors can synchronize their signal phases to facilitate the distributed beamforming. The authors in [29] considered a wireless powered communication networks system, in which the sensors first harvest energy from dedicated ETs and then transmit information to a fixed access point (AP), to enhance the transmission performance via designing the distributed beamforming. The authors in [30] designed the distributed beamforming in order to maximize the network lifetime under the requirement of a prespecified quality of service. In these prior works, the authors assumed that the fusion centers are fixed on the ground. By contrast, under the mobile fusion center deployed at a UAV of our interest, how to jointly design the UAV’s trajectory and the sensors’ wireless resource allocation for improving the data collection performance is a new problem that has not been investigated yet.
Motivated by this, this paper focuses on a new UAVenabled data collection system with distributed beamforming, in which the UAV collects data from multiple singleantenna sensors via the distributed beamforming. Different from the existing works focusing on the adaptiverate transmissions at the sensors, we consider two scenarios with the adaptiverate and fixedrate transmissions, respectively. These two scenarios may correspond to the delaytolerant applications (e.g., for delayinsensitive measurement information delivery) and the delaysensitive applications (e.g., for realtime video delivery), respectively. For the two scenarios, our objectives are to maximize the average datarate throughput and minimize the transmission outage probability, respectively, by jointly optimizing the UAV’s trajectory design and the sensors’ transmit power allocation over time, subject to the UAV’s flight speed constraints and the sensors’ individual average power constraints. However, due to the infinite number of optimization variables for the sensors’ power allocation and UAV’s trajectory over continuous time, how to jointly optimize them is a difficult problem.
To deal with this issue, we first consider the relaxed problems in the ideal case without considering the UAV’s flight speed constraints, for which the wellstructured optimal solutions are obtained via the Lagrange duality method to reveal the fundamental performance upper bounds. It is observed that for the two scenarios, the optimal trajectory solutions follow the same multilocationhovering structure, but the optimal power allocation solutions are distinct. In particular, in the first scenario for rate maximization, the sensors transmit their messages based on the waterfillinglike power allocation over time; while in the second scenario for outage probability minimization, the sensors adopt an onoff power allocation over time, where the sensors may remain silent in the outage status when the wireless channels become bad, such that the transmit power can be reserved for nonoutage transmission at other time instants. Next, motivated by the obtained optimal trajectories for the above special problems, we propose efficient approaches to obtain highquality solutions to the general problems with the UAV’s flight speed constraints considered, by using techniques from convex optimization and approximation. In the proposed approaches, we solve a series of approximated convex optimization problems to update the UAV’s flight trajectories and the sensors’ power allocations towards efficient solutions. Finally, we provide numerical simulations to validate the effectiveness of our proposed schemes. It is shown that our proposed designs significantly outperform the benchmark schemes in terms of the achieved datarate throughput and outage probability under the two scenarios. It is also shown that when the communication duration becomes sufficiently long, the proposed designs approach the performance upper bounds achieved when the UAV’s flight speed constraints are ignored.
The remainder of this paper is organized as follows. Section II introduces the system model of our considered UAVenabled data collection system with distributed beamforming. Section III solves the average datarate throughput maximization problem in the delaytolerant application scenario. Section IV solves the outage probability minimization problem in the delaysensitive application scenario. Section V presents numerical results. Finally, Section VI concludes this paper.
Notations:
The vectors (lower case) or matrices (upper case) are denoted by the letters in bold. For a square matrix
, refers to its trace. For a nonsingular square matrix , denotes the inverse matrix of . For a vector , , , , and denote its transpose, norm, norm, and Euclideannorm, respectively. denotes the componentwise no smaller than zero. denotes the space of realvalued matrices. denotes the statistical expectation.denotes the distribution of a circularly symmetric complex Gaussian (CSCG) random variable with mean
and variance
. stands for “distributed as”. denotes an diagonal matrix with being the diagonal elements. and return the maximum and minimum elements in a set , respectively, and .Ii System Model and Problem Formulation
As shown in Fig. 1, we consider a UAVenabled data collection system, in which one singleantenna UAV acts as an aerial mobile date collector to periodically collect data from a set of singleantenna sensors on the ground. We assume that all the sensors collaborate as a cluster to transmit common/shared sensing messages towards the UAV with distributed beamforming employed. It is assumed that each sensor is deployed at a fixed location
on the ground in the 3D Cartesian coordinate system. For notational convenience, let
denote the horizontal location of sensor , which is assumed to be known by the UAV apriori to facilitate the trajectory design.We focus on one particular mission period of the UAV with finite duration in second (s), denoted by . The UAV is assumed to fly at a fixed altitude , with the timevarying horizontal location for any time instant . Suppose that and denote the UAV’s initial and final locations, respectively. Let denote the UAV’s maximum flight speed. Thus, we have
(1)  
(2) 
where and denote the firstderivatives of and with respect to , respectively. We denote the region as the UAV’s desirable flight region in the horizontal plane, where , , , and . We also assume that the UAV’s mission duration satisfies , in order for the trajectory from the initial to final locations to be feasible. Accordingly, the distance between the UAV and sensor at any time instant is given by
As the G2A channels from sensors to UAV are LoS dominated, we consider a channel model with LoS path loss together with random phases. Consequently, the channel coefficient between the UAV and sensor at any time instant is given by
where denotes the channel power gain at the reference distance of m, denotes the imaginary unit, denotes the channel phase shift at any time instant [26], and denotes the path loss exponent.
In particular, we consider that all the sensors collaborate as a cluster to transmit a common message , which is a CSCG random variable with zero mean and unit variance (i.e., ). Such common information can be obtained at different sensors either by their independent sensing (e.g., the common temperature information) or via sharing with each other. At any time instant , the transmit signal of sensor is , where and denote sensor ’s transmit power and signal phase, respectively. Suppose that each sensor is subject to a maximum average power budget . Therefore, the average transmit power constraint for each sensor is given by
(3) 
Then, the received signal at the UAV at any time instant is given by
Here, denotes the additive white Gaussian noise (AWGN) at the UAV’s information receiver, which is a CSCG random variable with zero mean and variance (i.e., ). In order to achieve the maximum received signal power at the UAV, we design the signal phase as
. Thus, the received signaltonoise ratio (SNR) by the UAV at any time instant
is given by(4) 
Consequently, the datarate throughput from the sensors to the UAV in bits/second/Hertz (bps/Hz) at time instant is given by
(5) 
In the following, we will formulate the optimization problems for rate maximization in the delaytolerant application scenario and outage probability minimization in the delaysensitive application scenario, respectively.
Iia Rate Maximization in DelayTolerant Application Scenario
In the delaytolerant application scenario, we assume that the sensors can adaptively adjust the communication rate based on channel variations due to the timevarying locations of the UAV. In this case, the average or ergodic datarate throughput is used as the performance metric. According to (5), the average datarate throughput from sensors to the UAV over the whole duration in bps/Hz is given by
(6) 
Our objective is to maximize the average datarate throughput , by jointly optimizing the UAV’s trajectory and sensors’ power allocation over time, subject to the UAV’s flight speed constraints in (1), the UAV’s initial and final locations constraints in (2), and the sensors’ average transmit power constraints in (3). Consequently, the average datarate throughput maximization problem is formulated as
It is worth noting that the objective function of problem is nonconcave, due to the complicated datarate throughput expression with respect to coupled variables ’s and ’s. Moreover, problem contains an infinite number of optimization variables over continuous time. As a result, problem is difficult to be solved optimally. We will deal with this issue in Section III.
IiB Outage Probability Minimization in DelaySensitive Application Scenario
In the delaysensitive application scenario, we assume that the sensors use a fixed transmission rate for delivering the delaysensitive information. In order for the UAV to successfully decode the message (with fixed rate) at any given time instant, the received SNR must be no smaller than a certain threshold . In this case, the transmission outage occurs if the received SNR at the UAV falls below . Therefore, we use the following indicator function to indicate the transmission outage at any time instant .
Accordingly, we define the outage probability as the probability that the transmission is in outage over the whole duration , which is expressed as
Our objective is to minimize the outage probability , by jointly optimizing the UAV’s trajectory and sensors’ power allocation over time, subject to the UAV’s flight speed constraints in (1), the UAV’s initial and final locations constraints in (2), and the sensors’ average transmit power constraints in (3). Consequently, the outage probability minimization problem is formulated as
It is worth noting that the objective function of problem is nonconvex and even nonsmooth due to the indicator function with coupled variables ’s and ’s. In addition, problem contains an infinite number of optimization variables over continuous time. As a result, problem is even more challenging to be solved optimally than problem . We will deal with this issue in Section IV.
Iii Proposed Solution to Problem
In this section, we solve the datarate throughput maximization problem in the delaytolerant scenario. We first obtain the optimal solution to a relaxed problem of in the special case with to gain key engineering insights. Then, based on the optimal solution under the special case, we propose an alternatingoptimizationbased algorithm to obtain an efficient solution to the original problem under any finite .
Iiia Optimal Solution to Relaxed Problem of with
In this subsection, we consider the special case when the UAV’s flight duration is sufficiently large (i.e., ), such that we can ignore the finite flight time of the UAV from one location to another. As a result, the UAV’s flight speed constraints in (1) as well as the initial and final locations constraints in (2) can be neglected. Therefore, problem can be relaxed as
Though problem is still nonconvex, it satisfies the socalled timesharing condition [31]. Therefore, the strong duality holds between problem and its Lagrange dual problem. As a result, we can optimally solve problem by using the Lagrange duality method [32]. Let denote the dual variable associated with the th constraint in (3). For notational convenience, we define . The partial Lagrangian of problem is given as
The dual function is
(7) 
The dual problem of problem is given by
In the following, we solve problem by first obtaining the dual function via solving problem (7) and then solving the dual problem ().
First, we solve problem (7) for finding under given . For notational convenience, let and denote the sensors’ distributed beamforming vector and the combined channel vector at any time instant , respectively. To obtain , we decompose problem (7) into a set of subproblems, each for one time instant, which are presented in the following with the index dropped for facilitating the analysis.
(8) 
where denotes a vector with only the th element being and the others being . Under any given , problem (8) is simplified as
(9) 
where . In general, we must have , since otherwise, is not upper bounded. Let and . Then, problem (9) is recast into
(10) 
Notice that the maximum value of problem (10) is attained at with . Therefore, problem (10) can be reexpressed as
(11) 
Problem (11) is convex. Hence, by checking the firstderivative of the objective function, we obtain the optimal solution to problem (11) as
Accordingly, the optimal solution to problem (9) is given as
Thus, each sensor’s optimal power allocation is
After substituting into problem (8), we can obtain the optimal location for problem (8) by using the 2D exhaustive search over the region , given as
(12) 
Without loss of generality, suppose that the set of the optimal locations in (12) are given as , with denoting the number of optimal locations for problem (12). Note that when the optimal solution to problem (12) is nonunique, we can arbitrarily choose any one of ’s for obtaining .
Next, we solve the dual problem by minimizing the dual function . This is implemented via using subgradientbased methods, such as the ellipsoid method [33], with the subgradient being .
After solving the dual problem , it remains to construct the optimal primal solution to , denoted by . In this case, since the optimal solution to problem (7) is nonunique in general, we need to time share among these hovering locations to construct the optimal primal solution to . Let denote the hovering duration at the optimal location . In the following, we solve the following problem to obtain the optimal hovering durations for time sharing.
(13)  
As problem (13
) is a linear program, the optimal hovering durations
can be obtained by CVX [32]. As a result, is optimally solved.It is observed that the optimal UAV trajectory solution to problem has a multilocation hovering structure, while the sensors’ optimal power allocation follows a waterfillinglike pattern, dependent on the value of .
Example 1: For obtaining more insights, we consider the special case with two sensors. Without loss of generality, we suppose that the two sensors are deployed at and , where denotes the distance between the two sensors.
Fig. 2 shows the UAV’s optimal hovering locations with different sensors’ distances m in subfigure (a) and m in subfigure (b), where m and dBm. It is observed that if the two sensors are far away (i.e., m), then the UAV should hover at two symmetric locations with the same hovering time; while if the two sensors are close (i.e., m), the UAV should hover at the middle point of them. Table I shows the optimal power and hovering time allocations in Fig. 2(a) with s. It is observed that the UAV’s optimal hovering durations at the two hovering locations are equal due to the symmetric nature of the considered setup; while the sensors’ transmit power allocations are different, which have a symmetric structure. Moreover, the optimal power allocation of each sensor in Fig. 2(b) is obtained at the power .
Hovering duration  s  s 

Sensor 1’s transmit power  dBm  dBm 
Sensor 2’s transmit power  dBm  dBm 
IiiB Proposed Solution to Problem with Finite
In this subsection, we consider problem in the general case with finite . Motivated by the optimal solution to the relaxed problem in the previous subsection, we propose an efficient solution based on the techniques of convex optimization and successive convex approximation (SCA). Towards this end, we first discretize the whole duration into a finite number of time slots denoted by the set , each with equal duration . Let and denote the UAV’s horizontal location and sensor ’s transmit power at time slot , , . Accordingly, problem can be approximated as
(14a)  
(14b)  
(14c) 
Problem is nonconvex due to the nonconcave objective function. To tackle this issue, we introduce two sets of auxiliary variables and , . Problem is reexpressed as
(15a)  
(15b)  
Problem is still nonconvex due to the nonconvex constraints in (15a) and (15b). Next, we solve problem by alternatively optimizing the UAV’s trajectory and the sensors’ power allocation.
IiiB1 Trajectory Optimization
Under given sensors’ power allocation , we optimize the UAV’s trajectory with variables , , and for problem by adopting the SCA technique. To deal with the nonconvex constraints in (15a) and (15b), we update the UAV’s trajectory and in an iterative manner by approximating the nonconvex problem into a convex problem. Let and denote the local points at the th iteration. Under given UAV’s trajectory and , since any convex function is globally lowerbounded by it firstorder Taylor expansion at any point, we have the lower bounds for and as follows.
(16) 
(17) 
In each iteration with given local points and , we replace and as their lower bounds and , respectively. As a result, the trajectory optimization problem is changed to a convex optimization problem, which can be optimally solved by CVX [32].
IiiB2 Power Allocation
Under any given UAV trajectory , we optimize the sensors’ power allocation together with and for problem by using the SCA technique as well. In this case, only the constraints in (15a) are nonconvex. Similarly as for optimizing the UAV trajectory, we replace in (15a) as its lower bound in (17) to approximate the nonconvex terms into convex forms, so as to optimize the UAV trajectory iteratively, for which the details are omitted for brevity.
By alternately optimizing the UAV trajectory and sensors’ power allocation, we can obtain a converged solution to problem , thus efficiently solving problem .
Remark iii.1
It is worth nothing that the performance of the alternating optimizationbased approach critically depends on the initial point chosen for iteration. In this paper, we consider the following three trajectory designs as the potential initial point.

Flyhoverfly trajectory with power design: The UAV first flies straightly from the initial location to one optimized fixed location , and hovers at this location as long as possible, and finally flies to the final location at the maximum flight speed. The fixed location is obtained by using a 2D exhaustive search over the region in during the mission time to maximize the received SNR at the UAV, during which each sensor employs the fixed power . Thus, the flying time is and the hovering duration at the optimized location is given as . Under such a trajectory, the sensors’ power allocation can be obtained by solving the power allocation problem in .

Successive hoverandfly trajectory with power design: The UAV flies from the initial location to successively visit the optimized hovering locations to problem , then hovers at these locations, and finally flies to the final location at the maximum flight speed. During the flight, we choose the minimum flying path by solving the traveling salesman problem (TSP) [8]. Then, we have the minimum flying time and the hovering duration at each optimized location can be obtained similarly by solving problem , with total hovering time given by . Under such a trajectory, the sensors’ power allocation can be obtained by solving the power allocation problem in problem .

Power design only: The UAV flies from the initial location to the final location directly with a constant flight speed . Under such a trajectory, the sensors’ power allocation can be obtained by solving the power allocation problem in .
Note that the minimum flying time in each trajectory design should be no larger than the UAV flight duration to guarantee a feasible trajectory. In this case, under any given , we choose the one which has the best performance as the initial point of our proposed SCAbased algorithm.
Iv Proposed Solution to Problem
In this section, we address the outage probability minimization problem in the delaysensitive application scenario. We first obtain the optimal solution to a relaxed problem of in the special case with to gain key engineering insights. Then, based on the optimal solution under the special case, we propose an alternatingoptimizationbased algorithm to obtain an efficient solution to the original problem under any finite .
Iva Optimal Solution to Relaxed Problem of with
In this subsection, we consider the special case that the UAV’s flight duration is sufficiently large (i.e., ). Similarly as for problem , problem can be relaxed as
Though problem is nonconvex, it satisfies the socalled timesharing condition [31]. Therefore, the strong duality holds between problem and its Lagrange dual problem. As a result, we can optimally solve problem by using the Lagrange duality method [32] as follows. Let denote the dual variable associated with the th constraint in (3). For notational convenience, we define . The partial Lagrangian of problem is given as
The dual function is
(18) 
The dual problem of problem is given by
In the following, we solve problem by first obtaining the dual function and then solving the dual problem (). First, to obtain , we solve problem (18) by solving a set of subproblems, each for a time instant in the following, in which the index is dropped for facilitating the analysis.
(19) 
To solve problem (19), we consider the following two cases when equals one and zero, respectively.
IvA1 Outage case
First, consider that . In this case, the outage occurs, and thus we have , and can be any arbitrary value. Accordingly, the optimal value for problem (19) is .
IvA2 Nonoutage case
Next, consider that . In this case, we solve problem (19) by first deriving the sensors’ power allocation under any given UAV’ location and then searching over via a 2D exhaustive search over . Under given and defining , problem (19) is reduced as
(20)  
If , then problem (20) is a convex problem. By checking the KarushKuhnTucker (KKT) conditions, we have the optimal solution as
(21) 
If there exists any such that , then the optimal value of problem (20) is zero, which is attained by setting to be sufficiently large and . Therefore, we obtain . By substituting into problem (19), we obtain the optimal UAV location by using the 2D exhaustive search over , given as
(22) 
Accordingly, the obtained power allocation is given by and the optimal value for problem (19) is . Without loss of generality, suppose that the set of the optimal locations is , with denoting the number of optimal locations for problem (22). Note that when the optimal location for problem (22) is nonunique, we can arbitrarily choose any one of ’s for obtaining .
By comparing the corresponding optimal values under and , we can obtain the optimal solution to problem (19) as the one achieving the smaller optimal value. Therefore, the dual function is obtained.
Next, we solve the dual problem by maximizing the dual function . This is implemented via using subgradientbased methods, such as the ellipsoid method [33], with the subgradient being . We denote the optimal dual solution to as
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