I Introduction
On November 8, 2017, “Drone Integration Pilot Program” [1] was launched under a presidential memorandum from the White House, which aimed at further exploring expanded use of unmanned aerial vehicles (UAVs) including beyondvisuallineofsight flights, nighttime operations, flights over people, etc. [2]. In fact, the past several years have witnessed an unprecedented growth on the use of UAVs in a wide range of civilian and defense applications such as search and rescue, aerial filming and inspection, cargo/packet delivery, precise agriculture, etc. [3]. In particular, there has been a fastgrowing interest in utilizing UAVs as aerial communication platforms to help enhance the performance and/or extend the coverage of existing wireless networks on the ground [4, 5]. For example, UAVs such as drones, helikites, and balloons, could be deployed as aerial base stations (BSs) and/or relays to enable/assist the terrestrial communications. UAVenabled/aided wireless communications possess many appealing advantages such as swift and costeffective deployment, lineofsight (LoS) aerialtoground link, and controllable mobility in threedimensional (3D) space, thus highly promising for numerous use cases in wireless communications including ground BS traffic offloading, mobile relaying and edge computing, information/energy broadcasting and data collection for InternetofThings (IoT) devices, fast network recovery after natural disasters, etc. [6, 7, 8, 9, 10, 11, 12]. For example, Facebook has even ambitiously claimed that “Building drones is more feasible than covering the world with ground signal towers” [13]. By leveraging the aerial BSs along with terrestrial and satellite communications, Europe has established an industrydriven project called “ABSOLUTE” with the ultimate goal of enhancing the ground network capacity to many folds, especially for public safety in emergency situations [14]. At present, there are two major ways to practically implement aerial BSs/relays by using tethered and untethered UAVs, respectively, which are further explained as follows.
A tethered UAV literally means that the UAV is connected by a cable/wire with a ground control platform (e.g., a custombuilt trailer). Although it may sound ironic for a UAV to be on a tethering cable, this practice is very common due to many advantages including stable power supply and hence unlimited endurance, more affordable payload (e.g., more antennas), ultrahigh speed backhaul with secured data transmission (e.g., realtime highdefinition video), robustness to wind, etc. All these evident benefits have triggered a great interest in testing tethered UAV BSs, such as Facebook’s “TetherTenna”, AT&T’s “flying cellonwings (COWs)”, and EverythingEverywhere’s (UK’s largest mobile network operator, EE) “Air Masts”. However, such a tethering feature also limits the operations of UAVs to taking off, hovering, and landing only, thus rendering the wireless networks employing tethered UAV BSs like “hovering cells” over the air. As a result, the research efforts in this paradigm mainly focus on the UAV deployment/placement optimization in a given target area to meet the ground data traffic demand [15, 16, 17, 18, 10, 19]. In particular, [15] provides an analytical approach to optimize the altitude of a UAV for providing the maximum coverage for ground users (GUs). Alternatively, by fixing the altitude, the horizontal positions of UAVs are optimized in [16] to minimize the number of required UAVs to cover a given set of GUs.
In contrast to tethered UAVs, the untethered UAVs generally rely on onboard battery and/or solar energy harvesting for power supply [20]
and wireless links for data backhaul. Although untethered UAVs in general have smaller payload and limited endurance/backhaul rate as compared to their tethered counterparts, they have fully controllable mobility in 3D space which can be exploited for communication performance enhancement. First, an untethered UAV not only can hover above a fixed ground location like tethered UAVs, but also can fly freely over a wide ground area to significantly extend the communication coverage. Furthermore, the freeflying feature enables the UAV BS to timely adjust its position according to the dynamic distributions of the GUs, and even follow closely some specific GUs, to achieve a new “usercentric and cellfree” communication service. In fact, the reduced UAVGU distance not only decreases the signal attenuation but also increases the probability of the LoS link between them, which is particularly crucial for highrate communication. As a result, untethered UAV BSs/relays have been envisioned to be a revolutionizing technology for future wireless communication systems
[21] and preliminary industry prototypes have been built and tested including e.g. Facebook’s Aquila and Nokia’s flyingcell (FCell). This has also inspired a proliferation of studies recently on the new research paradigm of jointly optimizing the UAV trajectory design and communication resource allocation, for e.g. mobile relaying channel [22, 23], multiple access channel (MAC) and broadcast channel (BC) [24, 25, 26], interference channel (IFC)[27], and wiretap channel [28]. In particular, as shown in [25] and [26], significant communication throughput gains can be achieved by mobile UAVs over static UAVs/fixed terrestrial BSs by exploiting the new design degree of freedom of UAV trajectory optimization, especially for delaytolerant applications. In
[27], a joint UAV trajectory, user association, and power control scheme is proposed for cooperative multiUAV enabled wireless networks. A multiobjective path planning (MOPP) framework is proposed in [29] to explore a suitable path for a UAV operating in a dynamic urban environment. In [30], an efficient mobile architecture is proposed for uplink data collection application in IoT networks.To optimize the wireless system performance by exploiting UAVenabled BSs, assorted UAV trajectory designs have been proposed in the literature [23, 24, 27, 25, 26, 31, 22, 32, 30, 33], based on optimization techniques such as successive convex optimization (SCA) and solutions for Travelling Salesman Problem (TSP) as well as its various extensions. However, all these works either assume time division multiple access (TDMA) [23, 24, 27, 25, 32] or frequency division multiple access (FDMA) [26, 31, 22, 30, 33] to simplify the multiuser communication design, which, however, are in general suboptimal from an informationtheoretic perspective. As a result, the fundamental capacity limits of UAVenabled multiuser communication systems still remain largely unknown, which thus motivates this work.
In this paper, we aim to characterize the capacity region of a UAVenabled BC and reveal the capacityoptimal joint UAV trajectory and communication design. As an initial study, we consider the simplified setup with two GUs, as shown in Fig. 1. Specifically, it is assumed that a UAV with the maximum speed meter/second (m/s) flies at a constant altitude of m to serve two GUs at fixed locations with a distance of m. We consider the communication within a given UAV flight duration of s. Note that if , the considered system simplifies to e.g. a tethered UAV BS, which can be placed above a fixed ground location. On one hand, we assume that is sufficiently large such that the channels from the UAV to both GUs are dominated by the LoS link, according to the recently conducted experimental results by Qualcomm [7]. On the other hand, we assume that is sufficiently large and comparable to so that the UAV’s horizontal position can have a nonnegligible impact on the UAVGU channel strength.^{1}^{1}1Otherwise, if , the UAV trajectory design becomes trivial and it should simply stay above any point along the line between the two GUs and their channels can be regarded as constant irrespective of . As a result, an effective timevarying BC can be generally established between the UAV and the two GUs as the UAV moves horizontally above them. Given the UAV trajectory, the UAVGU BC resembles the conventional fading BC with a terrestrial BS [34]. However, their fundamental difference lies in that the UAV trajectory and hence its induced timevarying channel are controllable and thus can be proactively designed to maximize the capacity of the BC, while this is impossible for conventional fading channels due to the randomness in the propagation environment. As such, the joint UAV trajectory and communication design can exploit this additional degree of freedom to enlarge the capacity region compared to the conventional BC with a static BS on the ground.
To this end, we characterize the capacity region of this new UAVenabled BC over a given UAV flight duration , by jointly optimizing the UAV’s trajectory and transmit power/rate allocations over time, subject to the practical UAV’s maximum speed and transmit power constraints. Specifically, we adopt the rateprofile approach as in [35] to maximize the sum rate of the two GUs under their different rate ratios, which leads to a complete characterization of all the achievable ratepairs for the two GUs on the socalled Pareto boundary of the capacity region. However, such a joint optimization problem is shown to be nonconvex and difficult to solve in general. Nevertheless, we obtain the optimal solution of the considered problem by exploiting its particular structure and applying tools from convex optimization. The main results of this paper are summarized as follows:

First, to draw essential insights, we consider two special cases with asymptotically large UAV flight duration, i.e., (or equivalently ) and asymptotically low UAV speed, i.e., (or equivalently ), respectively. We introduce a simple and practical UAV trajectory called hoverflyhover (HFH), where the UAV successively hovers at a pair of initial and final locations above the line segment connecting the two GUs each with a certain amount of time, and flies unidirectionally between them at its maximum speed. Then, for the case of , we show that the HFH trajectory with hovering locations above the two GUs together with the TDMA based orthogonal multiuser transmission is capacityachieving. In contrast, for the case of , it is shown that the UAV should hover at a fixed location that is nearer to the GU with larger achievable rate and in general superposition coding (SC) based nonorthogonal transmission with interference cancellation at the receiver of the nearer GU is required. Furthermore, it is shown that in general there exists a significant capacity gap between the above two cases, which demonstrates the potential of exploiting the UAV’s trajectory design and motivates the study for the general case with finite UAV maximum speed and flight duration.

Next, for the case of finite UAV speed and flight duration, we prove that the proposed HFH trajectory is also optimal while SC is generally required to achieve the capacity. In addition, the initial and final hovering locations need to be properly selected from the points above the line segment between the two GUs to achieve the capacity region Pareto boundary. It is also observed that by increasing the UAV maximum speed and/or flight duration, the capacity region is effectively enlarged, especially for the low signaltonoise ratio (SNR) case. To gain more insights, we further analyze the high SNR case and it is shown that the HFH trajectory reduces to a static point above one of the two GUs. This result implies that dynamic UAV movement is less effective for capacity enhancement as SNR increases.

Last, for the sake of comparison, we further characterize the achievable rate region of the UAVenabled twouser BC with the TDMAbased (instead of the optimal SCbased) transmission with finite UAV speed and flight duration. It is shown that the optimal UAV trajectory still follows the HFH structure as in the capacityachieving case with SCbased transmission, while the difference lies in that the hovering locations can only be those above the two GUs in the TDMA case. It is also revealed that the capacity gain of the optimal SCbased transmission over the suboptimal TDMAbased transmission decreases as the UAV maximum speed and/or flight duration increases.
The rest of this paper is organized as follows. Section II introduces the system model and presents the problem formulation for capacity region characterization. In Sections IIIV, we study the capacity region for two special cases and the general case, respectively. Section VI addresses the case with TDMAbased transmission. Finally, we conclude the paper in Section VI.
Notations:
In this paper, scalars are denoted by italic letters, vectors and matrices are denoted by boldface lowercase and uppercase letters, respectively.
denotes the space of dimensional realvalued vectors. For a vector , represents its Euclidean norm and denotes its transpose. For a timedependent function , denotes the derivative with respect to time . For a set , denotes its cardinality, and represent the interior and boundary of a set , represents the convex hull of a set , which is the set of all the convex combinations of the points in , i.e., . For two sets and , is the set of all elements in excluding those in . Notation indicates that vector is elementwisely less than or equal to vector .Ii System Model and Problem Formulation
As shown in Fig. 1, we consider a UAVenabled BC with one UAV transmitting independent information to two GUs at fixed locations. Without loss of generality, we consider a twodimensional (2D) Cartesian coordinate system. Let the location of each GU be denoted by , where m and m with denoting their distance. The UAV is assumed to fly at a constant altitude of m. In practice, the value of is set based on regulations on the minimum UAV height as well as the communication system requirement. We focus on a particular UAV flight duration of s and denote the UAV’s timevarying location at time instant by . The system bandwidth is denoted by in Hertz (Hz) and hence the symbol period is s. We assume that is sufficiently large such that the UAV can adopt the Gaussian signaling with a sufficiently long symbol block length to achieve the channel capacity. It is also assumed that the UAV’s location change within a symbol period is negligible compared to the altitude , i.e., , where denotes the UAV maximum speed in m/s. Thus, the UAVGU channel is assumed to be constant within each symbol interval. Mathematically, we express the UAV speed constraint as [27, 26],
(1) 
For the purpose of exposition, we consider the freespace path loss model for the airtoground wireless communication channels from the UAV to the two GUs, as justified in Section I. It is assumed that the Doppler effect induced by the UAV mobility can be perfectly compensated at the user receivers [4, 36]. As a result, the channel power gain from the UAV to each GU at time instant is modeled as
(2) 
where denotes the channel power gain at the reference distance m.
At time instant , let and denote the UAV’s transmitted informationbearing symbols for GUs 1 and 2, respectively. Accordingly, the received signal at GU is expressed as
(3) 
where denotes the additive white Gaussian noise (AWGN) at the receiver of GU . For simplicity, the noise power is assumed to be equal for the two GUs, denoted by . With given , the signal model in (3) resembles a conventional fading BC consisting of one transmitter (the UAV) and two receivers (GUs) [34]. In order to achieve the capacity region of this channel, the UAV transmitter should employ Gaussian signaling by setting
’s as independent circularly symmetric complex Gaussian (CSCG) random variables with zero mean and variances
. Suppose that at each time instant , the UAV is subject to a maximum transmit power constraint , similarly as assumed in [37, 38], i.e.,(4) 
In this paper, we are interested in characterizing the capacity region of the UAVenabled twouser BC, which consists of all the achievable average ratepairs for the two GUs over the duration , subject to the UAV’s maximum speed constraint in (1) and maximum power constraint in (4). For given UAV trajectory and power allocation , let denote the set of all achievable average ratepairs ( in bits per second per Hertz (bps/Hz) for the two GUs, respectively, which need to satisfy the following inequalities [37, 39]:
(5)  
(6)  
(7) 
where . Denote by and the feasible sets of and specified by the UAV’s speed constraint (1) and maximum power constraint (4), respectively. Then, the capacity region of the UAVenabled twouser BC is defined as
(8) 
Our objective is to characterize the Pareto boundary (or the upperright boundary) of the capacity region by jointly optimizing the UAV trajectory and power allocation . The Pareto boundary consists of all the achievable average ratepairs at each of which it is impossible to improve the average rate of one GU without simultaneously decreasing that of the other GU. Since it remains unknown yet whether the capacity region is a convex set or not, we apply the rateprofile technique in [35] that ensures a complete characterization of the capacity region, even if it is nonconvex.^{2}^{2}2Another commonly adopted approach for characterizing the Pareto boundary is to maximize the weighted sum of the average rates of the two GUs. Although this approach is effective in characterizing Pareto boundary points for convex capacity regions, it may fail to obtain all the boundary points when the capacity region is a nonconvex set [40]. Specifically, let denote a rateprofile vector which specifies the rate allocation between the two GUs with , and . Here, a larger value of indicates that GU has a higher priority in information transmission to achieve a larger average rate. Then, the characterization of each Paretoboundary point corresponds to solving the following problem,
(9)  
(10)  
(11)  
(12)  
(13)  
(14) 
where denotes the achievable sum average rate of the two GUs. Problem (P1) is challenging to be solved optimally due to the following two reasons. First, the constraints in (11) are nonconvex, as the rate functions in (i.e., the righthandsides (RHSs) of (5), (6), and (7)) are nonconcave with respect to . Second, problem (P1) involves an infinite number of optimization variables (e.g., ’s over continuous time ). As a result, (P1) is a highly nonconvex optimization problem and in general, there is no standard method to solve such problem efficiently. For notational convenience, the optimal UAV trajectory for problem (P1) under any given is denoted by , and the optimal power allocation is denoted by . Furthermore, the corresponding ratepair achieved by the above optimal UAV trajectory and power allocation is denoted by , which is on the (Pareto) boundary of the capacity region .
Iia Capacity Region Properties and HFH Trajectory
Before explicitly characterizing the capacity region , we first provide some interesting properties of this region, which can be used to simplify the optimization of the UAV trajectory in (P1) later.
Lemma 1.
The capacity region is symmetric with respect to the line .
Proof.
Suppose that a ratepair is achieved by and . Then we can construct another solution and with , , and , , which can be easily shown to achieve the symmetric ratepair . As the newly constructed solution is also feasible to (P1), this lemma is thus proved. ∎
Based on Lemma 1, it is evident that the boundary of is also symmetric with respect to the line .
Lemma 2.
For problem (P1), the optimal UAV trajectory satisfies , i.e., the UAV should stay above the line segment between the two GUs.
Proof.
Supposing that the optimal UAV trajectory does not lie within the interval , we can always construct a new trajectory with , which simultaneously decreases the distances from the UAV to both GUs, thus resulting in a strictly componentwise larger ratepair based on (2) and (5)(7). This thus completes the proof. ∎
Lemma 3.
For problem (P1), there always exists an optimal UAV trajectory that is unidirectional, i.e., if , .
Proof.
Suppose that is a nonunidirectional optimal UAV trajectory to problem (P1), which implies that the UAV visits some locations more than one times. We denote the total time that the UAV stays at such a location () by . Then, we show that there always exists an alternative unidirectional UAV trajectory that achieves the same objective value of (P1). Specifically, we construct a unidirectional UAV trajectory with and as its initial and final locations, respectively, where the UAV stays at location with a duration , i.e., , , with being the time instant once the UAV reaches location . It is easy to show that is feasible to (P1) and always achieves the same objective value as . This thus completes the proof. ∎
With Lemmas 2 and 3, we only need to consider the unidirectional UAV trajectory between in the rest of the paper. In addition, as implied by Lemma 1, we only need to obtain the boundary point with at one side of the line . This corresponds to the optimal solution to problem (P1) in the case with . When or , it is easy to show that the optimal ratepair is or where .
Next, we introduce a simple and yet practical HFH UAV trajectory, which will be shown optimal for (P1) in the sequel. Specifically, with the HFH trajectory, the UAV successively hovers at a pair of initial and final locations, denoted by and , respectively, with , each for a certain amount of time, denoted by and with , and flies at the maximum speed between them. Mathematically, the HFH trajectory is generally given by
(15) 
where , , , and . It follows from (15) that under the HFH trajectory, the UAV hovers at two different locations at most; while if , then the UAV trajectory reduces to hovering at a fixed location during the entire flight duration . In the following two sections, we first investigate the solutions to (P1) for the two special cases with and , respectively.
Iii Capacity Characterization with Large Flight Duration
In this section, we study the special case when the UAV flight duration is asymptotically large, i.e., , where the corresponding capacity region is denoted by . To this end, we first ignore the UAV maximum speed constraint (13) in (P1) and derive its optimal solution for any . Then, we show that the resulting capacity region is equal to as .
By dropping constraint (13) under finite , problem (P1) is reduced to
(16)  
(17) 
whose optimal objective value serves as an upper bound of that of problem (P1). Although problem (P2) is a nonconvex optimization problem, we obtain its optimal solution as in the following lemma.
Lemma 4.
Under given with , the optimal trajectory and power allocation solution to (P2) is given as , , , , and , where , and . Accordingly, the optimal ratepair is obtained as and .
Proof.
Please refer to Appendix A. ∎
Based on Lemma 4 and by changing the values of and for (P2), the capacity region without considering the UAV maximum speed constraint (13), denoted by , can be easily obtained in the following proposition.
Proposition 1.
In the absence of constraint (13), the capacity region of the UAVenabled twouser BC is given by
(18) 
which is an equilateral triangle.
From Lemma 4 and Proposition 1, the optimal UAV trajectory for achieving the boundary points of is to let the UAV successively hover above each of the two GUs for communication in a TDMA manner. It is worth pointing out that the UAV maximum speed is finite in practice and thus constraint (13) cannot be ignored in general. As a result, the capacity region in (18) generally serves as an “upper bound” of the capacity region with finite . However, as shown in the following theorem, can be asymptotically achieved when is sufficiently large for any .
Theorem 1.
As , we have , , where the optimal UAV trajectory follows the HFH structure in (15) with and , and the TDMAbased transmission is capacityachieving.
Proof.
First, it is evident that for any and . Next, we show that the HFH trajectory in (15) with and together with TDMAbased transmission achieves the boundary of as , as follows. For any boundary point in (18) satisfying , , , we can construct a feasible solution for (P1) where the UAV flies at the maximum speed between the two GUs and hovers above GUs 1 and 2 for and proportion of the remaining time. In addition, the UAV only transmits information to GU 1 or 2 when hovering above that GU via TDMA. Thus, the corresponding achievable ratepair of the two GUs, denoted by , are given by and , where corresponds to the proportion of time for the UAV’s maximumspeed flying from to . As , for any and since . Based on the facts that and (due to ), we have as . Thus, the unidirectional HFH trajectory with and with TDMA is asymptotically optimal and . ∎
For the purpose of illustration, Fig. 2 shows the capacity region with dBm, dB, m, m, and dBm. For comparison, we also show the capacity region achieved when the UAV is fixed at a given location , where , , or . In this case, the system becomes a conventional twouser AWGN BC with constant channel gains and its capacity region is denoted by . Based on the uplinkdownlink duality [39], we have , where denotes the capacity region of the dual twouser MAC specified by the following inequalities [37] (or equivalently with ):
(19)  
(20)  
(21) 
where . It is known that the capacity region is convex and its boundary is generally achieved by SCbased nonorthogonal transmission with interference cancellation at the receiver of the GU with higher channel gain (or nearer to the UAV in our context) [34].^{3}^{3}3For degraded BC, dirty paper coding (DPC) also achieves the same capacity boundary as SC [34]. Without loss of optimality, we consider SC in this paper. For example, in Fig. 2, when or , the corresponding boundary points of can only be achieved by applying SC while TDMA is strictly suboptimal (except for the two extreme points)[34]. By contrast, when , the two GUs have the same channel gain and thus both SC and TDMA are optimal. Interestingly, based on Theorem 1, when the UAV mobility can be fully exploited (say, with untethered UAVs) with (or equivalently ), TDMAbased orthogonal transmission along with the simple HFH UAV trajectory is capacityachieving whereas SC is not required. Nevertheless, it is also worth pointing out that the significant capacity gain by exploiting the high mobility UAV over the static UAV comes at the cost of transmission delay at one of the two GUs (e.g., GU 2 needs to wait for about to be scheduled for transmission, which can be substantial when becomes large). Therefore, there is a fundamental throughputdelay tradeoff in wireless communications enabled by highmobility UAVs [25, 26].
Remark 1.
Based on Proposition 1, we next provide a property of capacity region for finite flight duration, which helps reveal the fundamental reason why the UAV mobility can potentially enlarge the capacity region: is nonconvex for . The nonconvexity of the capacity region essentially suggests that it can be enlarged if the convex combinations of the ratepairs can be achieved (generally achieved at different locations). This is the reason why the UAV mobility/movement can be helpful, leading to a timesharing of multiple locations.
Iv Capacity Characterization with Limited UAV Mobility
In this section, we study the other special case with limited UAV mobility, i.e., (or equivalently ). In this case, the UAV’s horizontal movement has negligible impact on the UAVGU channels with (since is comparable with ). As a result, the UAV should hover at a fixed location during the entire once it is deployed (e.g., a tethered UAV), i.e., , which becomes a special case of the proposed HFH trajectory in (15) with . In this case, solving (P1) is equivalent to finding the optimal hovering location of the UAV, , given the rateprofile parameters , as well as the corresponding transmission power, and , , and rates and for GUs 1 and 2, respectively. As such, holds and problem (P1) is reformulated as
(22)  
(23)  
(24)  
(25)  
(26) 
Proposition 2.
For problem (P3) with , there always exists an optimal UAV hovering location , such that .
Proof.
Please refer to Appendix B. ∎
Proposition 2 suggests that when , the UAV should be placed closer to GU 2 such that it has a larger channel gain than GU 1. As a result, GU 2 needs to decode GU 1’s signal first and then decodes its own signal after canceling the interference from GU 1’s signal. Therefore, we have and , where and . As the inequalities in (23) must be tight at the optimality, (P3) can be transformed into the following problem,
(27)  
(28)  
(29) 
Note that for , , the LHS (RHS) of (28) increases (decreases) with . It thus follows that under any given , the optimal solution of is unique and can be directly obtained by solving the equality in (28) with a bisection search, and thus the objective value of (P4) can be obtained accordingly. Therefore, to solve problem (P4), we only need to apply the onedimensional search over , together with a bisection search for under each given .
Fig. 3 shows the capacity region of the UAVenabled twouser BC as . The parameters are same as those for Fig. 2. It is observed that is a nonconvex set that is larger than the twouser AWGN BC capacity region at any fixed location , thanks to the location optimization for the UAV based on the GU rate requirements (or rateprofile vector). In addition, it is interesting to observe that at some locations, e.g., and , the fixedlocation capacity region touches the boundary of , while at other locations, e.g., , lies strictly inside . The latter observation suggests that some locations are inferior to the others in the sense that they achieve componentwise smaller ratepairs. This further implies that when , the UAV should hover at such superior locations rather than the inferior locations in order to maximize the GU average rates. This is illustrated by observing in Fig. 3. However, since the UAV’s speed is finite in practice, it may need to fly over some inferior locations (e.g., ), in order to travel between and hover over different superior locations that are far apart (e.g., and ) in a timesharing manner. This intuitively explains why the UAV should fly at the maximum speed in the optimal HFH trajectory for the general case with finite UAV maximum speed and flight duration , as will be rigorously proved in the next section. Finally, by comparing with , it is observed that significant capacity improvement can be achieved by increasing the UAV maximum speed and/or flight duration.
V Capacity Characterization for Finite UAV Speed and Flight Duration
In this section, we characterize the capacity region by solving problem (P1) for the general case with finite UAV maximum speed and flight duration .
Va Capacity Region Characterization
First, we reveal an important property of the optimal UAV trajectory solution to problem (P1) with any given and , based on which, we show that the HFH UAV trajectory is capacityachieving. According to Lemmas 2 and 3, we consider a unidirectional UAV trajectory without loss of generality, in which the initial and final locations are denoted by and , respectively, with .
Intuitively, the fixedlocation capacity regions of and , i.e., and , should have rate superiority over those of the other locations on the line between them, which is affirmed by the following proposition.
Proposition 3.
At the optimal UAV trajectory solution to problem (P1), it must hold that
(30) 
for any location between the initial and final locations and , i.e., .
Proof.
Please refer to Appendix C. ∎
Proposition 3 essentially implies there always exists a ratepair in the boundary of the convex hull, , which is componentwise no smaller than any given ratepair in the fixedlocation capacity region at location (i.e., ), as illustrated by Fig. 4. Based on Proposition 3, the optimal UAV trajectory to problem (P1) is obtained as follows.
Theorem 2.
For problem (P1) with given and , the HFH trajectory in (15) is optimal.
Proof.
Please refer to Appendix D. ∎
Based on Theorem 2, the optimal UAV trajectory is determined only by the initial and final hovering locations and as well as the hovering time at location . Accordingly, the optimal hovering time can be obtained as . Therefore, we can solve problem (P1) by first optimizing the power allocation under any given UAV trajectory, and then searching over the three variables , , and to obtain the optimal UAV trajectory for any given . Specifically, based on a fixed HFH UAV trajectory , (P1) is reduced to the following problem,
(31)  
(32)  
(33)  
(34)  
(35) 
Note that (P5) is a convex optimization problem and thus can be solved efficiently by applying the wellestablished polymatroid structure and the Lagrange duality method [37, 40]. Therefore, the optimal ratepair corresponding to rateprofile can be found by applying a threedimensional search on , , and , and selecting their values to maximize in (P5). The details are omitted here due to the space limitation. Notice that in this case, SCbased transmission is generally required.
VB Numerical Results
In Fig. 6, the capacity region for finite UAV maximum speed and flight duration is shown under different setups. The parameters are same as those for Figs. 2 and 3. It is interesting to observe that although is generally a nonconvex set, its boundary has a general concaveconvexconcave shape when and are finite. This observation helps explain whether the UAV movement is able to enlarge the capacity region or not. Specifically, when , the boundary is convex for bps/Hz, which implies that the convex combination of any two boundary points (ratepairs) in this regime always achieves a componentwise larger ratepair than any boundary points between them. As such, increasing the UAV maximum speed and/or flight duration enables the UAV to fly closer to each GU and thus achieves higher ratepairs, leading to an enlarged capacity region. By contrast, the boundary is concave for bps/Hz, which means that the convex combination of any two boundary points within this regime will achieve a componentwise smaller ratepair than any boundary points between them. This suggests that if and are small such that the UAV can only fly locally among these locations, it is not desirable for the UAV to move in terms of achieving a componentwise larger ratepair. This is in fact the reason why in Fig. 6, when bps/Hz, the boundary for m/s and s remains the same as that for . However, when the duration is further increased from s to s, it is observed that the boundary for bps/Hz shifts towards the upperright direction, which means that the UAV movement becomes helpful. This is because with sufficiently large , the UAV is able to fly over its nearby locations to reach some superior locations and hence can achieve a componentwise larger ratepair. In fact, with any given , as long as is sufficiently large, the UAV movement is always beneficial to enlarge the capacity region.
VC High SNR Case
Lastly, we consider the asymptotically high SNR case with such that can be assumed, to provide more insights. This assumption means that if the UAV is placed above one (near) GU, the SNR of the other (far) GU (with maximum UAVGU distance ) is still sufficiently large when is used.
Theorem 3.
Under the assumption of , the optimal HFH UAV trajectory to (P1) is simplified to if ; and if . Accordingly, the capacity region is given by
(36) 
Proof.
Please refer to Appendix E. ∎
Similar to Proposition 1, Theorem 3 shows the superiority of the UAV hovering locations right above the GUs. However, unlike , the capacity region is independent of both UAV flight duration and maximum speed , which suggests that the UAV movement is less effective to enlarge the capacity region as the SNR becomes large.
In Fig. 6, we plot the capacity region for the same setup of Fig. 6 except that is increased from dBm to dBm. In this case, we have , i.e., the high SNR assumption for Theorem 3 approximately holds. It is observed that when , the UAV always hovers above the GU that requires larger rate, e.g., , for all ratepairs satisfying , and SC is needed. Furthermore, it is observed that hovering at the middle location, i.e., , where the two GUs have equal channel gains, suffers a significant capacity loss in the high SNR regime even for maximizing the equal rate with . Finally, it is observed that the capacity region improvement is very limited by increasing the UAV maximum speed and/or flight duration in this case since the gap between and is already very small, i.e., the gain achieved by exploiting the UAV movement is not appealing.
Vi Achievable Rate Region with TDMA
In this section, we consider the UAVenabled twouser BC with TDMAbased communication.
Via Achievable Rate Region Characterization
For TDMA, the UAV can communicate with at most one GU at any time instant. Denote by
the binary variable which indicates that GU
is scheduled for communication at time instant if ; otherwise, . Accordingly, the achievable average rate region with TDMA is given by(37)  
(38)  
(39)  
(40) 
We denote the achievable rate region characterized by (37) and (38) subject to (39) and (40) as . Let . Similarly as for problem (P1), we can apply the rateprofile approach to characterize with rateprofile parameters and the optimization problem is formulated as
(41)  
(42)  
(43)  
(44)  
(45) 
Note that problem (P6) is a nonconvex optimization problem since it involves binary variables in and the LHSs of (42) and (43) are not concave with respect to even for given . Nevertheless, we show in the following proposition that the optimal UAV trajectory still follows the HFH structure as for (P1), except that the UAV only hovers at and/or .
Proposition 4.
The optimal UAV trajectory to problem (P6) satisfies the HFH trajectory in (15). Furthermore, the following properties hold: 1) if , then and ; 2) if , then
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