Joint Trajectory and Hybrid Beamforming Design for Multi Antenna UAV Enabled Network

In this paper, we investigate the transmission design in a multi antenna unmanned aerial vehicle (UAV)-enabled network, where the flight trajectory and hybrid digital and analog beamforming (BF) of the UAV are jointly designed, for both the fully-connected structure and sub-connected structure. Our goal is to maximize the weighted sum rate for multiply ground users, subject to the transmit power and trajectory constraints. Due to the non-convex property of the formulated problem, we propose to linearize the objective by using a newly Lagrangian dual transform. Then, an alternating optimization (AO) method is proposed, where the digital BF can be obtained by utilizing the bisection search method, while the analog BF and the flight trajectory are handled by the alternating direction of multipliers method (ADMM) algorithm. Finally, simulation results verify the performance of the proposed algorithm.

trajectory, and power control for UAV assisted wireless networks. In addition, in [9], the authors studied the optimal three dimensional trajectory design and resource allocation for solar-powered UAV communication systems, where a mixed-integer non-convex optimization problem was proposed, taking into account the aerodynamic power consumption, the solar energy harvesting, the finite energy storage capacity, and the quality-of-service (QoS) requirements of the users. Also, energy efficient UAV communication have been investigated in [10] and [11], where the Dinkebach method was commonly used to transform the fractional programming to a linear optimization, but the computational complexity is high. Recently, in [12], the authors investigated the trajectory and transmit power optimization in secrecy UAV communication network, and was extended in [13] with considering robust design to handle the uncertainty ground user locations.
However, these works focus on the single antenna UAV case. In fact, large scale multiple antenna technique has attracted considerable interest for the development of the next generation wireless networks, since it can dramatically increase the system capacity and mitigate the spectrum shortage [14]. Recently, the multiple antenna technique enabled UAV communication has been investigated in several works VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ such as [15]- [17]. For instance, in [15], the authors studied the robust trajectory and resource allocation design in multiple antenna UAV networks. While in [16], the authors investigated the joint trajectory and resource allocation design for energy-efficient secure multiple antenna UAV communication system. Also in [17], the authors studied the energyefficient and secure multiple antenna UAV communication with jittering. Nevertheless, the conventional fully digital beamforming (BF) structure imposes an excessive fabrication cost and energy consumption owing to using numerous costly radio frequency (RF) chains and analog-digital (A/D) converters.
To overcome this obstacle, the newly hybrid A/D BF technique has been proposed as a promising way, which relies on a lower number of RF chains than the fully digital BF method [18]. However, the main challenges in the design of hybrid BF arise from the transmit power constraint with coupling components, imposed by the product of digital and analog BF matrices, and by the unit modulus constraint (UMC) on the elements of the analog BF matrix [19].
To solve this problem, several methods were proposed in related literatures. Specifically, in [20], the authors proposed a matrix factorization method to obtain the hybrid BF matrices based on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. While in [21], the digital BF was derived according to the original objective, while the analog BF was handled by several iterative algorithms. Also in [22], the authors investigated the hybrid BF design based on a joint iterative algorithm. In [23], secure communication for spatially sparse multiple-input multiple-output (MIMO) channel with hybrid precoding was proposed. What's more, in [24], the authors investigated the hybrid precoding using deep learning method for MIMO network. Recently in [25], the authors proposed a hybrid BF method in multipleinput single-output (MISO) wiretap channel to improve the security.
Due to the limited power, volume and cost, it may be impractical to equipment many RF links in a UAV [26]. Thus, the hybrid BF technique is a promising way to improve the spectrum efficiency in a multi antenna UAV-enabled network. However, such work has not been studied yet. Besides, the trajectory optimization method in related work mainly based on the successive convex approximation (SCA) and inner point method, which may lead to high computational complexity, thus is hard to conduct on the UAV. Motivated by this, this paper investigates a new paradise of the transmission in a multi antenna UAV network, where the UAV provide service for multiple ground users. Specifically, we aim to maximize the weighted sum rate (WSR) by jointly designing the hybrid BF and flight trajectory of the UAV, where both the fully-connected structure and subconnected structure are considered. To solve the formulated non-convex problem, we propose to linearize the objective by a newly Lagrangian dual transform. Then, an alternating optimization (AO) method is proposed to obtain the solution, where the analog BF matrix and the trajectory are handled by the alternating direction of multipliers method (ADMM) technique. Finally, simulation results are provided to verify the proposed algorithm.
We summarize our main contributions as follows: • To the best of our knowledge, similar work about exploiting the hybrid BF in UAV communication network to improve the spectrum efficiency has not been investigated yet. The formulated WSR maximization design is non-convex, due to the complex objective and multiple coupled variables. Since it is difficult to obtain the globally optimal solution, we propose an AO algorithm to obtain a suboptimal solution, where each subproblem can be solved efficiently and the variables are alternately optimized in an iterative manner until convergence.
• By a newly Lagrangian dual transform, we transform the original problem into a quadratic form, which is beneficial for optimization. Specifically, the bisection search method is proposed to design the digital BF matrix. Then, an ADMM method is proposed to handle the UMC of the analog BF matrix, which is suitable for both the fully connected case and the sub connected case. While for the trajectory design, by decoupling the maximum speed constraints into a linear constraint, a low complexity ADMM method is proposed.
• The computational complexity and the convergence of the AO method are analyzed. In fact, the proposed method has lower complexity than the interior-point method, which is beneficial for implementation online and onboard on the UAV. Besides, simulation results suggest that the proposed algorithm can apparently improve the spectrum efficiency, when compared to other benchmark schemes such as the no trajectory design or the random BF method. The rest of this paper is organized as follows. A system model and problem statement is given in Section II. Section III investigates the joint design problem, wherein a Lagrangian dual transform based method is proposed. Simulation results are illustrated in Section IV. Section V concludes this paper.
Notations: Throughout this paper, the upper case boldface letters and lower case boldface letters are used to denote matrices and vectors, respectively. A H , A T , A † , Tr (A) and vec (A) denote the Hermitian transpose, the transpose, the conjugate, the trace and the vectorization of matrix A, respectively. · denotes the Euclidean norm of a vector, or the Frobenius norm of a matrix. {a} and |a| denote the real part and the modulus of a complex a, respectively.
x ∼ CN (u, ) denotes that x is a circularly symmetric complex Gaussian (CSCG) random vector with mean u and covariance . denotes the optimal value of a variable.

A. SYSTEM MODEL
Let us consider a UAV-enabled communication system as shown in Fig. 1, which consists of a UAV and multiply single antenna ground users. The UAV is equipped with N t antennas and N RF RF links. Without loss of generality, we consider a three-dimensional (3D) Cartesian coordinate system. The coordinate of the k-th user is denoted as where S max = V max δ T is the maximum distance that the UAV can fly within one slot on the assumption of maximum speed V max in meter/second (m/s). The analog precoding matrices map the signals from the RF chains to the transmit antennas using the phase shifting networks. Therefore, every phase shifter in the network should satisfy the UMC, which is translated as a constraint on each entry of the analog precoding matrices, that is, F u,v = 1, ∀u ∈ N t , ∀v ∈ N RF . We consider the two most common hybrid architectures namely, e.g., the fully-connected and sub-connected ones, which are shown in Fig. 2 and Fig. 3, respectively. In the former case, the signal from each RF chain is connected to all the antennas via the phase shifting network, whereas in latter, the signal from each RF chain is connected to a subset of antennas. The total number of phase shifters required to implement a fullyconnected architecture on the transmitter sides is N t × N RF . On the other hand, for the sub-connected architecture, each RF chain is connected to N t N RF antennas on the transmitter,  as shown below, where each f i is N t N RF × 1 dimensional vector consisting unit modulus entries [29]. Thus, this architecture requires fewer phase shifters compared to the fully-connected architecture, but at the expense of loss in spectrum efficiency. The transmit signal by the UAV in the n-th slot can be expressed as [18] x where denote the digital BF vector and the intended signal for the i-th user at the n-th slot, respectively. In this paper, we assume that the air-to-ground communication links between the UAV and the ground users are line-of-sight (LoS) channels, as [12] and [15]. In fact, it is shown that for the rural macro with UAV and urban macro with UAV scenarios, if the UAV height is larger than 40 m and 100 m, respectively, the LoS probability is almost 100% [27]. In particular, the channel vector between the UAV and the k-th user in time slot n is given by [13] where ρ = λ c 4π 2 is a constant with λ c being the wavelength of the center frequency of the carrier [28]. Moreover, a k [n] is VOLUME 9, 2021 the antenna array response (AAR) between the UAV and user k in time slot n. For the AAR, when the uniform linear arrays (ULA) is utilized by the UAV, then, a k [n] is given by where ϕ k and θ k (in rads) are the azimuth and elevation angles of departure (AoD) of the path between the ULA and the k-th user in the n-th slot, respectively. In addition, b is the antenna spacing of the ULA [28]. It should be noted that in this work, the jittering of the UAV is neglected since we assume that the AoD is mainly determined by the locations of the UAV and the users. Besides, for some other UAV-ground channel models such as the altitude dependent model [17], the proposed hybrid BF method can still work. Thus, the received signal by the k-th user in the n-th slot can be expressed as [15] The signal-to-interference-and-noise-ratio (SINR) for the k-th user at the n-th slot is [15] γ k [n]

B. PROBLEM STATEMENT
In this work, we investigate the average WSR maximization problem by jointly designing the hybrid BF and flight trajectory. Mathematically, the problem is given by where = 1 denotes the weighted factor for the k-th user at the n-th slot,P andP denote the average power and the peak power, respectively. In addition, (8c) is the UMC for the analog BF matrix, which is the main difference for the hybrid BF design with the fully digital BF design. In order to simplify the symbolic expression, in the following, we will neglect the slot index n in the associated variables.

III. THE JOINT BF AND TRAJECTORY DESIGN
A. THE LAGRANGIAN DUAL TRANSFORMATION (8) is hard to solve due to the non-convex objective and constraints [30]. In this section, we will propose an effective way to handle (8).
Then, for fixed α, (9) is reduced to whereα k = ω k (1 + α k ). More details can refer [31]. (10) is the sum of multiple-ratio fractional programming problems, while the non-convexity introduced by the ratio operation can be solved by the recently proposed technique in [32]. Then, (10) can be solved in an AO method. In particular, in each iteration, we first update the SINR α, and then update {W, F, q}, respectively. The process is repeated until no further improvement is obtained.
Specifically, using γ k in (7), the objective function of (10) is rewritten as Thus, with given α, (9) becomes max W,F,q It is known that (12) is a multiple-ratio fractional programming (FP) problem. Using the quadratic transform proposed in [32], f 1 (W, F, q) can be equivalently reformulated as where β = [β 1 , . . . , β K ] T , with β k ∈ C is the auxiliary variable. Then, based on [32], solving (13) (14) is a biconvex optimization problem, and a common method for solving it is alternatively updating {W, F, q, β} by optimizing one of them while fixing the other variables [33]. First, we introduce the following Theorem.
Theorem 1: With given {W, F, q}, the optimal β k is Proof: β k can be obtained by setting ∂f 2 ∂β k to zero.

B. THE HYBRID BF DESIGN
In this subsection, we will investigate the hybrid BF design.

1) THE DIGITAL BF DESIGN
Firstly, we solve the digital BF design. By fixing {F, q, β}, the subproblem w.r.t. w k is given by For (16), the optimal w k is where λ is the dual variable introduced for the power constraint, which is determined by and can be obtained by the bisection search method.

2) THE ANALOG BF DESIGN
In this subsection, we propose the ADMM method to optimize the analog BF F with fixed {W, q, β}. Specifically, the subproblem w.r.t. F is given by In fact, the only non-convexity in (19) is the UMC (8c). To handle this, we introduce the auxiliary variables Q and convert (19) into the following problem The augmented Lagrangian function of (20) is given by where ρ 0 ≥ 0 is the scalar penalty factor, and P ∈ C N t ×N RF is the Lagrangian multiplier matrix associated with (20b). In addition, A, B = u,v A † u,v B u,v for matrices A and B. Then, the following iterative steps derives from (21) are applied to solve (20) for m = 1, . . . until certain stop criterion is meet. The main merit of the ADMM method is that each subproblem has closed-form solution. Firstly, via the following relationship (22a) can be equivalently rewritten as Via the first order optimization condition, the optimal f m+1 is given by On the other hand, (22b) is equivalent to and the optimal Q m+1 is To this end, we have obtained the optimal hybrid BF for the fully-connected structure. In the following, we will handle the sub-connected design. In fact, the proposed ADMM method can be used to the sub-connected case with some improvement. Specifically, the augmented Lagrangian function is L (F, Q, P) where the indicator function of an arbitrary set I F {Q} is defined as and F denotes the sub-connected matrix constraint as shown in (2). We now proceed to derive the solution of for the subconnected architecture. Firstly, the subproblem w.r.t. Q is written in the following compact form By exploiting the block diagonal structure of Q, (30) can be further rewritten as and A (u, ;) is the u-th row of a matrix A. (31) is actually a vector approximation under phase rotation problem [34], which solution is given by (32) where F is the projection onto the set of F for an arbitrary matrix A, which is given by The remain step to update F and P are the same with the fully-connected case, thus we omit the detail for brevity.
To this end, the ADMM algorithm to obtain F is summarized in Algorithm 1. In addition, when the penalty parameter ρ 0 satisfies: ρ 0 I 2 − 0, the convergence of the ADMM method can be guaranteed by the analysis in [25], we omit the details for brevity.

C. THE TRAJECTORY DESIGN
With given {W, F, β}, the subproblem w.r.t. the trajectory q is formulated as: It is known that (34) is a convex problem w.r.t. q, which can be solved by the standard convex solver such as CVX [35], as the commonly used method in several literatures about trajectory design. However, in the work, we will propose a low complexity method to design the trajectory based on the ADMM method.
Specifically, the Lagrangian of (34) is given by where x is the copy of the original trajectory, {z, p, D} are the slack variables to decouple (1), one can refer the trajectory optimization in page 6 of [36] for more details about {z, p, D}.
The main advantage of the ADMM algorithm is that all the subproblems have closed-form solution. Specifically, the subproblem w.r.t. x is via the first order condition, the optimal solution is Then, the subproblem w.r.t. q is and the optimal solution is and the optimal solution is where P Z (x) = min S max |x| , 1 x denotes the projector associated with the linear space Z.
To this end, the ADMM algorithm to obtain q is summarized in Algorithm 2.
Finally, the entail procedure for (8) is summarized in Algorithm 3, where w l k , F l , q l and WSR l are the optimal solution and optimal value of (8) in the l-th iteration, respectively. In addition, denotes the stopping criterion, namely the tolerance. (8) 1: Initialization: Set P s , h k , the initial w 1 k , F 1 , q 1 , and l = 1. 2: repeat 3: Update w l k by solving (16).

4:
Update F l by solving (19) with the ADMM method.

D. OPTIMALITY AND COMPUTATIONAL COMPLEXITY ANALYSIS
In this subsection, we analyze the optimality and computational complexity of the proposed AO method. Firstly, the optimality and convergence of each subproblem can be guaranteed by the conclusion in related works. Specifically, the optimality and convergence of the Lagrangian dual transform is analyzed in [31]. Also, the optimality and convergence of the digital BF method is analyzed in [32]. While the optimality and convergence of the ADMM method to design the analog BF is given in [25]. Besides, the optimality and convergence of the ADMM method to design the trajectory is given in [36]. Then, based on the convergence of each subproblem and the feature of AO, the convergence of the AO method can be guaranteed [34]. However, since the original problem (8) is not convex w.r.t. all these variables, thus only suboptimal solution can be obtained by the AO method [30].
Then, we focus on the computational complexity of the proposed AO method, which is mainly determined by the complexity of each subproblem. Specifically, for the optimization of w k , in each step of the bisection search method, the main complexity is to calculate w k in (17), which is given by O NK N 3 t N 3 RF [31]. The required number of iterations for Algorithm 1 to converge is log 2 where ε denotes the accuracy tolerance, λ u and λ l are the bounds of λ. Hence, the total complexity of Algorithm 1 is O log 2 [32]. Then, for the ADMM algorithm to update F, according to [25], the complexity is O T F N N 3 t N 3 RF , where T F denotes the number of iterations required for convergence. As for the optimization of trajectory q, according to [36], the complexity is given by O T q N 2 , where T q denotes the number of iterations required for convergence. However, when CVX is utilized to obtain q, according to [35] and [36], the complexity is given by O N 7 / 2 , which will be higher than the ADMM method, especially for large N .
From the above analysis, the overall complexity of the proposed AO method can be estimated by [30] From this analysis, we can see that the proposed algorithm obtain polynomial time complexity, which is beneficial for practical implementation.

IV. SIMULATION RESULTS
In this section, we provide some Monte-Carlo (MC) simulations to testify the performance of the proposed scheme. Unless specified, the simulation setting are set as: T = 80s, δ T = 0.5s,P = 4P withP = 20dBm, while the noise power is

A. CONVERGENCE BEHAVIOUR
Firstly, we investigate the convergence behaviour of the proposed AO algorithm. Fig. 4 shows the convergence behaviour of the inner ADMM algorithm with different N t and N RF . From Fig. 4, we can see that the WSR increases with the number of iterations, and finally converges. Moreover, higher WSR can be achieved by using larger N t or N RF . However, larger N t or N RF leads to slower convergence, since more variables need to be optimized. Now, we investigate the convergence behaviour of the outer AO algorithm. Fig. 5 shows the WSR versus the number of iterations with different N t and N RF . It can be seen from Fig. 5 that, for different values of N t and N RF , the AO algorithm  always converge within 20 iterations, which confirms the practicality of the proposed design.

B. PERFORMANCE EVALUATION
In this subsection, we investigate the WSR performance versus the main system parameters. In addition, we compare the proposed method with the following benchmark schemes: 1) the fully digital BF method, i.e., the BF is obtained by the fully digital method, which can be seen as the upper bound of the hybrid BF method; 2) the fully-connected BF with the CVX based trajectory design, which is a comparison to the ADMM based trajectory design; 3) the no trajectory design method, i.e., the UAV fly directly from the initial location to the final location. This comparison can show the effect of the trajectory design in UAV communication system; 4) the random hybrid BF method, i.e., choose the BF vector randomly. This comparison can show the effect of the proposed BF method. These methods are labelled as ''Fullyconnected case'', ''Sub-connected case'', ''Fully digital BF method'', ''CVX based trajectory design'', ''No trajectory design method'', and ''Random BF method'', respectively.
Firstly, we show the WSRs of these schemes versus the transmit power budget of the UAVP. It can be seen from    6 that, with the increase ofP, the WSRs increase for all these methods, while the proposed method outperforms the random BF method and the no trajectory design method, which shows the importance of joint BF and trajectory design. In addition, the ADMM-based trajectory design obtains a nearly performance with the CVX based trajectory design, which suggests the effectiveness of the ADMM method, while the fully-connected structure outperforms the subconnected structure, and the performance gap between the proposed method and the fully digital BF method is small, which will be further confirmed by the following simulations.
Then, in Fig. 7, we show the WSRs of these schemes versus the RF chain number N RF . From this figure, we can see that more N RF leads to higher WSR. In addition, with the increase of N RF , the performance gap between the proposed hybrid BF and the fully digital BF method tends to small, when N RF = N t , the proposed method can obtain similar performance with the fully digital BF method, which shows the superiority of the proposed design. Besides, with the increase of N RF , the sub-connected structure can achieve nearly performance to the fully-connected structure, which is consistent with the results in related literatures such as [21]. This result suggests the effectiveness for the ADMM method in the sub-connected structure.
Lastly, in Fig. 8, we show the WSRs of these schemes versus the UAV antenna number N r . From this figure, we can see that more N r leads to higher WSR, since higher spatial degree of freedom can be achieved. In addition, with the increase of N r , the performance gap between the fully digital BF method and the hybrid BF method tends to larger, since with the increase of N r , the dimensions of the variables in the fully digital BF method will be more greater than the hybrid BF method, at the expenses of higher hardware cost.

V. CONCLUSION
This work has studied the transmission design in multi antenna UAV-enabled downlink multiuser network. Specifically, we aimed to maximize the WSR by jointly designing the hybrid BF and flight trajectory of the UAV. To handle the formulated non-convex problem, we proposed an AO method which linearize the objective by a newly Lagrangian dual transform, where each subproblem can be solved with low complexity. Simulation results verified the performance the proposed method. In our future work, to further reduce the cost and energy consumption of the UAV, another alternative technique such as the intelligent reflecting surface to achieve high beamforming gain will be investigated.