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1 King s Research Portal DOI: /LCOMM Document Version Peer reviewed version Lin to publication record in King's Research Portal Citation for published version APA: Yang, Z., Pan, C., Shih-Bahaei, M. R., Xu, W., Chen, M., Elashlan, M., & Nallanathan, A Joint Altitude, Beamwidth, Location and Bandwidth Optimization for UAV-Enabled Communications. IEEE COMMUNICATIONS LETTERS. Citing this paper Please note that where the full-tet provided on King's Research Portal is the Author Accepted Manuscript or Post-Print version this may differ from the final Published version. If citing, it is advised that you chec and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to chec the publisher's website for any subsequent corrections. General rights Copyright and moral rights for the publications made accessible in the Research Portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognize and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the Research Portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-maing activity or commercial gain You may freely distribute the URL identifying the publication in the Research Portal Tae down policy If you believe that this document breaches copyright please contact librarypure@cl.ac.u providing details, and we will remove access to the wor immediately and investigate your claim. Download date: 17. Mar. 019

2 1 Joint Altitude, Beamwidth, Location and Bandwidth Optimization for UAV-Enabled Communications Zhaohui Yang, Cunhua Pan, Mohammad Shih-Bahaei, Wei Xu, Ming Chen, Maged Elashlan, and Arumugam Nallanathan, Fellow, IEEE Abstract This letter investigates an uplin power control problem for unmanned aerial vehicles UAVs assisted wireless communications. We jointly optimize the UAV s flying altitude, antenna beamwidth, UAV s location and ground terals allocated bandwidth and transmit power to imize the sum uplin power subject to the imal rate demand. An iterative algorithm is proposed with low compleity to obtain a suboptimal solution. Numerical results show that the proposed algorithm can achieve good performance in terms of uplin sum power saving. Inde Terms UAV communications, altitude optimization, beamwidth optimization, location placement, bandwidth allocation. 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, UAV-enabled wireless communications can provide higher wireless connectivity in areas without infrastructure coverage and achieve higher capacity for line-of-sight LoS communication lins with the ground terals GTs. To fully eploit the design degrees of freedom for UAVenabled communications, it is crucial to investigate the UAV mobility in the three-dimensional space. In [], the altitude of UAV was optimized to provide maimum radio coverage on the ground. For an underlaid device-to-device communication networ with one UAV, the optimal values for the UAV altitude were analyzed in [3] for the maimum 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 user-uav association, [6] investigated the sum power imization problem of the UAV. Different from [] [6] with fied-beamwidth 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 eaed numerically and simple equal bandwidth allocation was assumed in [7], even This wor was supported in part by the Engineering and Physical Science Research Council under grant EP/P003486/1, grant EP/N09666/1 and grant EP/N0970/1, in part by the National Nature Science Foundation of China under Grant , Grant and Grant , and in part by the Si Talent Peas project in Jiangsu Province under GDZB-005 Corresponding author: Cunhua Pan. Z. Yang and M. Shih-Bahaei are with Centre for Telecommunications Research, King s College London, London, U.K. s: {yang.zhaohui, m.sbahaei}@cl.ac.u. W. Xu and M. Chen are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 10096, China, s: {wu, cheng}@seu.edu.cn. C. Pan, M. Elashlan, and A. Nallanathan are with the School of Electronic Engineering and Computer Science, Queen Mary, University of London, London E1 4NS, U.K. s: {c.pan, maged.elashlan, a.nallanathan}@qmul.ac.u. though proper bandwidth allocation can further enhance the system performance. In this letter, we aim to imize the sum power for an uplin UAV-enabled 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. The other contribution is to effectively obtain the optimal beamwidth with the bisection method when the pathloss eponent is two, and to obtain the optimal solution in closed form for bandwidth allocation subproblem. II. SYSTEM MODEL AND PROBLEM FORMULATION Consider an uplin UAV-enabled wireless communication system with one flying UAV and K GTs. The UAV is deployed as a flying BS with horizontal and vertical location y = y1, y at hight H. The horizontal and vertical location of GT is denoted by = 1,, 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 half-power beamwidths of the UAV antenna are equal, which are both denoted by 0, π. According to [8, Eq. -51], the antenna gain in the direction with azimuth angle θ and elevation angle φ can be modeled as G = { G0 if 0 θ and 0 φ g 0 otherwise, where G 0.846, and g means the channel gain outside the beamwidth of the antenna. For simplify, we set g = 0. 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 uplin channel gain between GT and the UAV is g 0 1 g =, y + H α where denotes the Euclidian norm, g 0 is the channel power gain at the reference distance 1 m, H is the hight of the UAV, y + H 1 is the distance between GT and the UAV, and α is the pathloss eponent. Based on 1 and, the uplin achievable rate of GT in the coverage area of the UAV is p g 0 G 0 r = w log 1 +, 3 w σ y + H α where w is the allocated bandwidth for GT, p is the transmit power of GT, σ is the noise power density and

3 w σ is the noise power for decoding the information of GT at the UAV side. For GT, the imal rate constraint r R should be satisfied. Since we aim at imizing uplin sum power of all GTs, it is always energy saving to transmit with imal rate. Setting r = R in 3, we have p = aw R y + H α, 4 where a = σ g 0G 0. We aim at imizing the uplin sum power of all GTs whilst satisfying the imal rate constraints. Mathematically, the sum power imization problem is aw R y + H α 5a H,,y,w s.t. aw R y + H α P, = 1,, K 5b y H tan, = 1,, K 5c w B 5d H H H ma, ma 5e w 0, = 1,, K. 5f where w = w 1,, w K, B is the maimal bandwidth of the system, P is the maimum transmit power of GT, [H, H ma ] is the feasible region of height H detered by obstacle heights and authority regulations, and [, ma ] is the feasible region of half-beamwidth detered 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 nonconve objective function 5a and constraints 5b-5c, Problem 5 is a nonconve 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 compleity through sequently optimizing H,, y and w. It is fortunate that we can globally optimize each variable with other variables fied in each step. A. Optimal Altitude and Beamwidth With fied y and w, Problem 5 is formulated as H, A D + H α 6a s.t. A D + H α P, = 1,, K 6b H tan D ma 6c H H H ma, ma, 6d where A = aw R, D = y, and D ma = ma,,k D. Denoting H as optimal value of H and observing that the objective function 6a is an increasing function in H with given, we can claim that { } H Dma = ma tan, H, 7 for the optimal solution. This claim can be proved by the contradiction method. If H, is the optimal solution of Problem 6 with H > H, we find that solution H, is also a feasible solution of Problem 6 with K A D + H α < K A D + H α, which contradicts the hypothesis that H, is the optimal solution. Based on 7, we consider the value of H in the following two cases. 1 Case 1: With H = H, Problem 6 is equivalent to 8a s.t. A D + H α P, = 1,, K 8b H tan D ma ma. 8c 8d Since Problem 8 is a imization of, the optimal solution is thus { } Dma = ma arctan,, 9 H which is the imal value of satisfying 8b, 8c and 8d. Note that Problem 8 is feasible if and only if { } P,,K A D + H, α ma. 10 Case : With H = D ma tan, Problem 6 becomes A D + D α ma tan s.t. A D + D ma tan H tan D ma ma. 11a α P, = 1,, K 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 one-dimension ehaustive search over [, ma ]. Specifically, for the special case where pathloss eponent α =, 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 doated by the LoS path, i.e., α = [7, Eq. ]. For α =, we define function f = D + D ma tan, [0, π/, 1 and then we have h 1 = cot cot cot, 13 h = + cos 3/ sin, 14 f = D + D ma h 1, 15 h 1 = csc 4 h, 16 h = 4 cos sin sin, 17

4 3 for [0, π/. Since tan for [0, π/, we have cos sin 0, i.e., h 0. As a result, h h 0 = 0, h 1 = csc 4 h 0, i.e., h 1 is an increasing function, and /3 = lim 0+ h 1 h 1 lim π/ h 1 = 0. Due to that 0, f 0 is equivalent to D + D ma h 1 0. To show the monotonicity of f, we consider the following two situations: If D 3 D ma 0, then D + D ma h 1 0 for all [0, π/, i.e., f is monotonically increasing. If D 3 D ma < 0, there must eist one solution such that D + D ma h 1 = 0 due to the fact that D 0 and h 1 is a continuous function. In this situation, f first decreases for [0, ] and then increases with, π/. According to the above analysis, f P is equivalent to ma, where and ma can be obtained by using the bisection method. As a result, constraints 11b- 11d can be equivalently transformed to ma, 18 where = ma{ma,,k, }, ma = {,,K ma, ma, arctan D ma /H }. Based on the definition of f, the objective function 11a can be epressed as f = K f. We have f = K f = K D + KD ma h 1. To show the monotonicity of in [, ma ], we also consider the following three situations: D + KD ma h 1 0, then K D + KD ma h 1 0 for [, ma ], i.e., f is monotonically increasing. The optimal beamwidth is =. D + KD ma h 1 ma < 0, f is monotonically decreasing and = ma. D + KD ma h 1 < 0 and K D + KD ma h 1 ma > 0, there must eist one solution such that K D + KD ma h 1 = 0. In this situation, f first decreases for [, ] and then increases with, ma ], i.e., =. Note that Problem 11 is feasible if and only if ma. By comparing the objective values of the solutions obtained in the above two cases, the one with lower objective value is chosen as the optimal solution to Problem 6. B. Optimal Location Planning For Problem 5 with fied H, and w, the location planning problem can be formulated as y C y + H α 19a s.t. y + H Ē, = 1,, K 19b y H tan, = 1,, K, 19c where C = aw R, and Ē = P α. Since C y is a conve function and α is conve and nondecreasing, y + H α is conve based on the scalar composition property of conve functions [9]. As a result, Problem 19 is a conve problem, which can be effectively solved via the standard interior point method. C. Optimal Bandwidth Allocation It remains to investigate the bandwidth allocation with fied location, altitude and beamwidth. Define function u = for > 0, and we have R u = R ln R R 1, u = ln R R 3 > 0. 0 From 0, we observe that function u is a conve function, which indicates that Problem 5 is a conve problem with fied H, and y. Based on 0,we have u < lim + u = 0, i.e., u is a monotonically decreasing function, which is helpful in transforg constraints 5b. With optimized H, and y, Problem 5 is equivalent to w s.t. F w R w B where F = a y + H α, W = u 1 1a 1b w W = 1,, K, 1c P b, and u 1 is the inverse function of u. The lagrangian of conve Problem 1 is Lw, λ = F w R K + λ w B, where λ is the non-negative Lagrange multiplier associated with constraint 1b. According to [9] and [10, Appendi A], the KKT conditions of 1 are L w = F R w From 3, we have λ = F e ln R w ln R R w + λ = ln R e ln R w w Define function u = e e +1, 0. We have u = e > 0, > 0. Thus, function u is strictly increasing and u > u0 = 0, > 0. Based on 4 and 1c, we have ln R w = ma, W, = 1,, K, 5 u 1 λ F where u 1 is the inverse function of u. According to 4, λ = F u ln R w > 0, which implies that 1b holds with equality. Plugging 5 into 1b yields ln R B = ma, W ûλ. 6 u 1 λ F Equation 6 has a unique solution λ > 0. Since u is strictly increasing, inverse function u 1 is also strictly

5 4 increasing in 0, +. Thus, ûλ i is a strictly decreasing function in 0, +. Owing to the fact that lim λ 0+ ûλ = + and lim λ + ûλ = 0, there eists one unique λ satisfying ûλ = B, and the solution can be obtained by using the bisection method. Having obtained the value of λ, the optimal w can be obtained from 5. D. Iterative Algorithm and Compleity Analysis Algorithm 1: Iterative Algorithm 1: Set the initial solution H 0, 0, y 0, w 0, and iteration number n = 1. : repeat 3: With fied y n 1 and w n 1, obtain the optimal H n, n of problem 6. 4: With fied H n, n and w n 1, obtain the optimal y n of problem 19. 5: With fied H n, n and y n, obtain the optimal w n of problem 1. 6: Set n = n : until the objective function 5a converges. The iterative procedure for solving Problem 5 is given in Algorithm 1. The main compleity of Algorithm 1 lies in Problem 6 and Problem 1. For Problem 6, the major computation comes from case, which needs to solve Problem 11 via a one-dimension ehaustive search method with compleity O ma δ for imal step δ. To solve Problem 1, the major compleity lies in solving 6 with compleity Olog 1/ɛ for the bisection method with accuracy ɛ. As a result, the total compleity of Algorithm 1 is OL ma it δ + L it log 1/ɛ, where L it is the number of iterations of the iterative Algorithm 1. IV. NUMERICAL RESULTS We consider that there are K = 0 GTs uniformly distributed in a circular area with radius 300 m. We set g 0 = , B = 10 MHz, P 1 = = P K = 0 dbm, σ = 169 dbm/hz, H = 50 m, H ma = 500 m, = 0, and ma = π/ rad. We consider equal imal rate demand, i.e., R 1 = = R K = R. In Fig. 1, we consider the sum power 11a, which equals to 5a with fied y and w, versus for various imal rate demands with equal bandwidth allocation and the UAV located at the center of the circle. With given imal 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: fied location method with optimized altitude, beamwidth and bandwidth labeled as FL, fied altitude and beamwidth method with optimized location and bandwidth labeled as FAB, fied bandwidth method with optimized location altitude and beamwidth labeled as FB, and ehaustive method via running Algorithm 1 with 1000 initial points labeled as Ehaustive. We investigate the sum power versus the imal rate demand in Fig.. It can be seen that the proposed algorithm outperforms FL, FAB and FB, especially when the imal rate demand is large. This is because the proposed algorithm jointly optimizes altitude, The objective value 6a Fig. 1. Sum power versus. R=0.6 Mbps, α= R=0.8 Mbos, α= R=0.6 Mbps, α=.5 R=0.8 Mbos, α= rad Sum power W Proposed FL FAB FB Ehaustive Minimal rate demand Mbps Fig.. Sum rate versus the imal rate demand with α =. beamwidth, location and bandwidth. It can be seen that the sum power of the ehaustive 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 imization problem in uplin UAV-enabled communications. We showed that the sum power first decreases and then decreases with the beamwidth. Numerical results showed that the uplin sum power performance can be improved by the proposed algorithm. REFERENCES [1] Y. Zeng, R. Zhang, and T. J. Lim, Wireless communications with unmanned aerial vehicles: Opportunities and challenges, IEEE Commun. Mag., vol. 54, no. 5, pp. 36 4, May 016. [] A. Al-Hourani, S. Kandeepan, and S. 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