I Introduction
Unmanned aerial vehicles (UAVs) assisted wireless communications have attracted considerable attention recently due to its maneuverability and increasing affordability [1]. Compared to conventional wireless communications, UAVenabled wireless communications can provide higher wireless connectivity in areas without infrastructure coverage and achieve higher capacity for lineofsight (LoS) communication links with the ground terminals (GTs).
To fully exploit the design degrees of freedom for UAVenabled communications, it is crucial to investigate the UAV mobility in the threedimensional space.
In [2], the altitude of UAV was optimized to provide maximum radio coverage on the ground. For an underlaid devicetodevice (D2D) communication network with one UAV, the optimal values for the UAV altitude were analyzed in [3]for the maximum system sum rate and coverage probability. Considering the adjustable UAVs’ locations over time, the UAV number and trajectory optimization problems were respectively considered in
[4] and [5]. Further optimizing userUAV association, [6] investigated the sum power minimization problem of the UAV. Different from [2, 3, 4, 5, 6] with fixedbeamwidth antenna, the beamwidth of the directional antenna and the altitude of the UAV were jointly optimized in [7] to improve the system throughput. However, the optimal beamwidth was only examined numerically and simple equal bandwidth allocation was assumed in [7], even though proper bandwidth allocation can further enhance the system performance.In this letter, we aim to minimize the sum power for an uplink UAVenabled wireless communication. There are two main contributions. One contribution is that we consider joint altitude, beamwidth, location and bandwidth allocation, and an algorithm is proposed by solving three subproblems iteratively, where each subproblem can be solved optimally. We also provide the complexity analysis of the proposed algorithm. Numerical results verify that the proposed algorithm outperforms the existing algorithms with fixed beamwidth or bandwidth allocation in terms of sum power, especially when the minimal rate demand is high. The other contribution is to effectively obtain the optimal beamwidth with the bisection method when the pathloss exponent is two, and to obtain the optimal solution in closed form for bandwidth allocation subproblem.
Ii System Model and Problem Formulation
Consider an uplink UAVenabled wireless communication system with one flying UAV and GTs. The UAV is deployed as a flying BS with horizontal and vertical location at hight . The horizontal and vertical location of GT is denoted by , and the hight of each GT is assumed to be zero compared with the hight of the UAV.
Assume that the UAV is equipped with a directional antenna with adjustable beamwidth, while each GT is equipped with an omnidirectional antenna with unit gain. The azimuth and elevation halfpower beamwidths of the UAV antenna are equal, which are both denoted by . According to [8, Eq. (251)], the antenna gain in the direction with azimuth angle and elevation angle can be modeled as
(1) 
where , and means the channel gain outside the beamwidth of the antenna. For simplify, we set . We consider the case that the GTs are located outdoors, and the channel between each GT and the UAV is mainly a LoS path. The uplink channel gain between GT and the UAV is
(2) 
where denotes the Euclidian norm, is the channel power gain at the reference distance 1 m, is the hight of the UAV, is the distance between GT and the UAV, and is the pathloss exponent. Based on (1) and (2), the uplink achievable rate of GT in the coverage area of the UAV is
(3) 
where is the allocated bandwidth for GT , is the transmit power of GT , is the noise power density and is the noise power for decoding the information of GT at the UAV side. For GT , the minimal rate constraint should be satisfied. Since we aim at minimizing uplink sum power of all GTs, it is always energy saving to transmit with minimal rate. Setting in (3) , we have
(4) 
where .
We aim at minimizing the uplink sum power of all GTs whilst satisfying the minimal rate constraints. Mathematically, the sum power minimization problem is formulated as
(5a)  
s.t.  
(5b)  
(5c)  
(5d)  
(5e)  
(5f) 
where , is the maximal bandwidth of the system, is the maximum transmit power of GT , is the feasible region of height determined by obstacle heights and authority regulations, and is the feasible region of halfbeamwidth determined by practical antenna beamwidth tuning technique. Constraints in (5c) ensure that all GTs are in the coverage area of the UAV.
Iii Proposed Algorithm
Due to nonconvex objective function (5a) and constraints (5b)(5c), Problem (5) is a nonconvex problem. It is generally hard to obtain the globally optimal solution to Problem (5). To solve this problem, we propose an iterative algorithm with low complexity through sequently optimizing , and . It is fortunate that we can globally optimize each variable with other variables fixed in each step.
Iiia Optimal Altitude and Beamwidth
With fixed and , Problem (5) is formulated as
(6a)  
s.t.  (6b)  
(6c)  
(6d) 
where , , and . Denoting as optimal value of and observing that the objective function (6a) is an increasing function in with given , we can claim that
(7) 
for the optimal solution. This claim can be proved by the contradiction method. If is the optimal solution of Problem (6) with , we find that solution is also a feasible solution of Problem (6) with , which contradicts the hypothesis that is the optimal solution. Based on (7), we consider the value of in the following two cases.
1) Case 1: With , Problem (6) is equivalent to
(8a)  
s.t.  (8b)  
(8c)  
(8d) 
Since Problem (8) is a minimization of , the optimal solution is thus
(9) 
which is the minimal value of satisfying (8b), (8c) and (8d). Note that Problem (8) is feasible if and only if
(10) 
2) Case 2: With , Problem (6) becomes
(11a)  
s.t.  (11b)  
(11c)  
(11d) 
Due to the complicated objective function (11a), it is generally difficult to obtain the optimal of Problem (11) in closed form. can be obtained via a onedimension exhaustive search over .
Specifically, for the special case where pathloss exponent , we can fortunately obtain the optimal through a simple bisection method. When the GTs are located outdoors in rural areas, and the communication channel between the UAV and each GT is dominated by the LoS path, i.e., [7, Eq. (2)]. For , we define function
(12) 
(13) 
(14) 
and then we have
(15) 
(16) 
(17) 
for . Since for , we have , i.e., . As a result, , , i.e., is an increasing function, and . Due to that , is equivalent to . To show the monotonicity of , we consider the following two situations:

If , then for all , i.e., is monotonically increasing.

If , there must exist one solution such that due to the fact that and is a continuous function. In this situation, first decreases for and then increases with .
According to the above analysis, is equivalent to , where and can be obtained by using the bisection method. As a result, constraints (11b)(11d) can be equivalently transformed to
(18) 
where , .
Based on the definition of , the objective function (11a) can be expressed as . We have . To show the monotonicity of in , we also consider the following three situations:

If , then for , i.e., is monotonically increasing. The optimal beamwidth is .

If , is monotonically decreasing and .

If and , there must exist one solution such that . In this situation, first decreases for and then increases with , i.e., .
IiiB Optimal Location Planning
For Problem (5) with fixed and , the location planning problem can be formulated as
(19a)  
s.t.  (19b)  
(19c) 
where , and . Since is a convex function and is convex and nondecreasing, is convex based on the scalar composition property of convex functions [9]. As a result, Problem (19) is a convex problem, which can be effectively solved via the standard interior point method.
IiiC Optimal Bandwidth Allocation
It remains to investigate the bandwidth allocation with fixed location, altitude and beamwidth. Define function for , and we have
(20) 
From (20), we observe that function is a convex function, which indicates that Problem (5) is a convex problem with fixed and . Based on (20),we have , i.e., is a monotonically decreasing function, which is helpful in transforming constraints (5b). With optimized and , Problem (5) is equivalent to
(21a)  
s.t.  (21b)  
(21c) 
where , , and is the inverse function of . The lagrangian of convex Problem (21) is
(22) 
where is the nonnegative Lagrange multiplier associated with constraint (21b). According to [9] and [10, Appendix A], the KKT conditions of (21) are
(23) 
From (23), we have
(24) 
Define function , . We have , . Thus, function is strictly increasing and , . Based on (24) and (21c), we have
(25) 
where is the inverse function of . According to (24), , which implies that (21b) holds with equality. Plugging (25) into (21b) yields
(26) 
Equation (26) has a unique solution . Since is strictly increasing, inverse function is also strictly increasing in . Thus, is a strictly decreasing function in . Owing to the fact that and , there exists one unique satisfying , and the solution can be obtained by using the bisection method. Having obtained the value of , the optimal can be obtained from (25).
IiiD Iterative Algorithm and Complexity Analysis
The iterative procedure for solving Problem (5) is given in Algorithm 1. The main complexity of Algorithm 1 lies in Problem (6) and Problem (21). For Problem (6), the major computation comes from case 2, which needs to solve Problem (11) via a onedimension exhaustive search method with complexity for minimal step . To solve Problem (21), the major complexity lies in solving (26) with complexity for the bisection method with accuracy . As a result, the total complexity of Algorithm 1 is , where is the number of iterations of the iterative Algorithm 1.
Iv Numerical Results
We consider that there are
GTs uniformly distributed in a circular area with radius
m. We set , MHz, dBm, dBm/Hz, m, m, , and rad. We consider equal minimal rate demand, i.e., .In Fig. 1, we consider the sum power (11a), which equals to (5a) with fixed and , versus for various minimal rate demands with equal bandwidth allocation and the UAV located at the center of the circle. With given minimal rate demand, it is observed that the sum power first decreases and then increases with the increase of , which verifies the theoretical analysis in Section III.A.
We compare the proposed algorithm with the following four methods: fixed location method with optimized altitude, beamwidth and bandwidth (labeled as ‘FL’), fixed altitude and beamwidth method with optimized location and bandwidth (labeled as ‘FAB’), fixed bandwidth method with optimized location altitude and beamwidth (labeled as ‘FB’), and exhaustive method via running Algorithm 1 with 1000 initial points (labeled as ‘Exhaustive’). We investigate the sum power versus the minimal rate demand in Fig. 2. It can be seen that the proposed algorithm outperforms FL, FAB and FB, especially when the minimal rate demand is large. This is because the proposed algorithm jointly optimizes altitude, beamwidth, location and bandwidth. It can be seen that the sum power of the exhaustive method is slightly lower than that of the proposed algorithm, which indicates that the proposed algorithm approaches the globally optimal solution.
V Conclusion
In this letter, we investigated the sum power minimization problem in uplink UAVenabled communications. We showed that the sum power first decreases and then decreases with the beamwidth. Numerical results showed that the uplink sum power performance can be improved by the proposed algorithm.
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