Backhaul-Aware Resource Allocation and Optimum Placement for UAV-Assisted Wireless Communication Network

: Driven by its agile maneuverability and deployment, the unmanned aerial vehicle (UAV) becomes a potential enabler of the terrestrial networks. In this paper, we consider downlink communications in a UAV-assisted wireless communication network, where a multi-antenna UAV assists the ground base station (GBS) to forward signals to multiple user equipments (UEs). The UAV is associated with the GBS through in-band wireless backhaul, which shares the spectrum resource with the access links between UEs and the UAV. The optimization problem is formulated to maximize the downlink ergodic sum-rate by jointly optimizing UAV placement, spectrum resource allocation and transmit power matrix of the UAV. The deterministic equivalents of UE’s achievable rate and backhaul capacity are ﬁrst derived by utilizing large-dimensional random matrix theory, in which, only the slowly varying large-scale channel state information is required. An approximation problem of the joint optimization problem is then introduced based on the deterministic equivalents. Finally, an algorithm is proposed to obtain the optimal solution of the approximate problem. Simulation results are provided to validate the accuracy of the deterministic equivalents, and the effectiveness of the proposed method.


Introduction
Recently, unmanned aerial vehicles (UAVs) has been widely investigated and applied to provide seamless coverage and capacity enhancement in wireless communication systems [1][2][3]. Deployed as aerial base stations or mounted with access points, UAVs can provide flexible and on-demand services to ground users dynamically by leveraging its agile mobility and maneuverability. As typical applications, UAVs can be exploited to load traffic in temporary or unexpected circumstances when the ground base stations (GBSs) are congested or broken. UAVs can also assist the GBSs to relay signals to remote users who are out of the coverage provided by terrestrial infrastructure [4].
In the UAV-assisted networks, wireless backhaul and cellular-connected UAVs have been proposed as potential solutions to connect UAV networks with terrestrial networks [5]. However, differing from the GBSs, the lack of fixed backhaul link for UAVs has emerged as a challenge in UAV-assisted wireless communication network, which needs to be further investigated [6]. In addition, an efficient resource allocation strategy is essential to improve the resource utilization and is connected to the GBS through in-band wireless backhaul link, which shares the spectrum resource with the access links of UEs as the spectrum resource is limited. • Next, we consider a composite channel model in which both large-scale and small-scale channel fading are considered. The RZF precoding is performed across the transmitted signals at the UAV to mitigate inter-user interference. In addition, to mitigate the computational complexity, we introduce the large-dimensional random matrix theory and derive the deterministic equivalents of UE's achievable rate and UAV's backhaul capacity which depends on only slowly-varying statistical channel information. The accuracy of the deterministic equivalents is verified. • Last, we formulate an optimization problem to maximize the downlink network sum-rate of UEs by jointly optimizing UAV placement, spectrum resource allocation and the transmit power matrix of the UAV. Based on the deterministic equivalents, an approximation problem of the joint optimization problem is proposed, from which the optimal solution of the approximation problem can be obtained. The effectiveness of the proposed method is also validated by simulations.
The remainder of this paper is organized as follows. In Section 2, the system model is described and the optimization problem is formulated. In Section 3, the deterministic equivalents of the ergodic rates is derived. In Section 4, we propose the algorithms to solve the approximate problem based on the deterministic results. In Section 5, simulation results for the proposed methods are provided, and the conclusion is drawn in Section 6.
Notation: In this paper, vectors, and matrices are denoted by lower-case, and upper-case bold-face letters, respectively. The N × N identity matrix is denoted by I N . The superscripts (·) H , (·) T , and (·) * represent the conjugate-transpose, transpose, and conjugate operations, respectively. E{·} is used to denote expectation with respect to all random variables within the brackets. The matrix principal square root, inverse, trace, and determinant are represented by (·) 1 2 , (·) −1 , tr(·), and det(·), respectively.

System Model and Problem Formulation
As shown in Figure 1, a downlink UAV-assisted wireless communication network is considered, consisting of an N-antenna GBS and a rotary-wing UAV equipped with M antennas. The UAV is connected to the GBS through in-band wireless backhaul link. The UAV acting as a remote relay, assists the GBS to forward signals for K UEs in remote areas outside the coverage of the GBS. The UEs have a single antenna each and they simultaneously receive messages from the UAV.
We assume the messages sent to the UEs cannot be transmitted by the GBS directly due to blockage and far distance. The messages for UEs are received by the UAV first and then relayed to UEs. We consider the wireless backhaul link between the UAV and the GBS is in-band along with the UEs' access links. In the frequency domain, the dynamic allocation of bandwidth resource for wireless backhaul and user communication is adopted. We assume that the total available bandwidth for the whole network is F Hz, from which is divided into two orthogonal parts F 1 = ηF and F 2 = F − F 1 , where η ∈ [0, 1] is a scale factor. The former is designated to the UAV to serve the UEs and the latter is designed to the backhaul link.

Transmission Model under Polar Coordinate
According to a three-dimensional Cartesian coordinate system, we consider the GBS is located at the origin (0, 0, 0). The UAV flies at a fixed altitude of l m. The distance between the projection of the UAV on the ground and the GBS is denoted as r. The UAV flies to a dispatched position and then hovers to transmit data for a group of K UEs. The polar coordinate of UE k with respect to the origin is denoted as (z k , θ k , 0), thus, the corresponding cartesian coordinate is (z k cos θ k , z k sin θ k , 0). The azimuth angle of the UAV's projection with respect to the horizontal axis is set as [15] Thus, the cartesian coordinate of the position at which the UAV hovers is expressed as (r cos φ, r sin φ, l). As the downlink transmission from the UAV to the served UEs is considered, the distance between the UAV and UE k is denoted as Considering low-altitude UAV scenario [18], the UAV-ground channels contain some multi-path components due to reflection and scattering. We take into account both the large-scale and the small-scale channel fading. The channel between the UAV and UE k is defined as h k = βd −α kh k , where β denotes the channel power gain at the reference distance 1m, α denotes the path loss exponent between the UAV and UE k, andh k ∈ C M accounts for the small-scale fading which has independent and identically distributed (i.i.d.) zero-mean entries of unit variance.
We assume the channel state information is available in the considered system. The received signal y k of UE k from the UAV is given by where w k ∈ C M is the precoding matrix for UE k, s k ∼ CN (0, 1) denotes the transmitted symbols for UE k, and n k ∼ CN (0, σ 2 ) is the additive white Gaussian noise (AWGN) at the receiver. We define W = [w 1 , w 2 , ..., w K ] ∈ C M×K as the precoding matrix for all UEs, and H = [h 1 , h 2 , ..., h K ] ∈ C M×K as the channel matrix which collects the channels between all the UEs and the UAV. H is modeled as whereH = h 1 ,h 2 , ...,h K ∈ C M×K accounts for small-scale fading, and D is a K × K diagonal matrix whose diagonal elements βd −α k 's denote large-scale channel fading. Here, the RZF precoder is considered [22], and given by where P (U) ∈ R K×K is a diagonal matrix of transmit power at the UAV whose k-th element is p k , ω is the regularization scalar, and ξ is a normalization scalar satisfying The normalized ergodic sum-rate of UEs is given as where γ k is the received signal to interference plus noise ratio (SINR) of UE k, which is expressed as The location of the GBS is considered at the origin of the coordinate system, thus, the distance between the UAV and the GBS is denoted as d u = √ r 2 + l 2 . The UAV receives the UEs' messages from the GBS through wireless backhaul link. The large-scale and the small-scale channel fading elements between the GBS and the UAV are both considered. We denote the channel from the GBS to the UAV through wireless backhaul link as G ∈ C M×N , which is given by whereG ∈ C M×N accounts for the small-scale fading channel in which the elements are i.i.d. complex random variables with zero mean and unit variance, and α denotes the path loss exponent from the GBS to the UAV. The received signal y u ∈ C M at the UAV from the GBS through wireless backhaul link can be expressed as where s u ∈ C N ∼ CN (0, I N ) is the signal vector transmitted from the GBS, the diagonal matrix P (B) denotes the transmit power allocation at the GBS, and n u ∼ CN 0, σ 2 I M is the independent AWGN. The normalized ergodic capacity rate of the backhaul link from the GBS to the UAV can be expressed as R bh

Problem Formulation
In the considered UAV-assisted network, by jointly designing the spectrum resource allocation factor η, the UAV projection distance r, and the transmit power matrix at the UAV P (U) , the achievable ergodic sum-rate maximization problem can be formulated as where the constraint (12b) indicates the sum-rate of UEs is restricted by the wireless backhaul capacity since the signals transmitted to all UEs are first received by the UAV through wireless backhaul. The constraint (12e) describes the transmit power constraint for the UAV. Noting that the optimization problem is based on the ergodic sum-rate, Monte Carlo averaging over channels is time-consuming and compute-complicated. Moreover, the small-scale channel fading is difficult to acquire during the UAV's flight. Therefore, the deterministic equivalents of the ergodic sum-rate and wireless backhaul capacity in the large-system regime are introduced, and the optimization problem can be solved based on the approximations in the following sections.

Deterministic Equivalent
In this section, we derive the deterministic equivalents of R sum and R bh in the large-system regime where N, M, and K are assumed to approach infinity with the ratio M/N and K/M fixed.

Deterministic Equivalent of R sum
The following lemma is provided to give the deterministic equivalentγ k of the SINR γ k by mean of large dimensional random matrix theory tools. Lemma 1. We assume that 1 M HH H has uniformly bounded spectral norm with respect to M. Therefore, we have γ k −γ k → 0 almost surely as M → ∞, wherē e k forms the unique solution of the following equations Define e = [e 1 , ..., e k ] T where the term e k is the derivative of e k , J, and v as Therefore, e is given explicitly as Proof. Refer to Appendix A.
Utilizing the continuous mapping theorem and the deterministic equivalent of γ k , we obtain Therefore, the deterministic equivalent of the normalized ergodic sum-rate R sum is given byR sum = ηR 0,sum withR 0,sum = K ∑ k=1 log 2 (1 +γ k ).

Deterministic Equivalent of R bh
According to Reference [23], the following lemma is provided to give the deterministic equivalent of the normalized backhaul ergodic rate R bh .

Lemma 2.
Consider that the elements of G are i.i.d. complex Gaussian variables with independent real and imaginary parts. For brevity, the transmit power budget at the GBS is assumed to be p b . Therefore, we have R bh −R bh → 0 almost surely as M → ∞.R bh = (1 − η)R 0,bh , and where with Proof. Refer to Appendix B.
The deterministic equivalentsR sum andR bh are determined by statistical channel knowledge which varies much slower than small-scale channel fading.

Optimization Design
Based on the deterministic equivalents derived in Section 3, an alternative sum-rate maximization problem is given as (12c), (12d), (12e).
Note that the objective function and backhaul constraint are related to the statistical channel knowledge, thus, the redundant Monte Carlo averaging computation can be avoided. However, the optimal solution of the maximization problem (23) is still hard to find as the non-convexity of the objective function.
In the following, we first reformulate problem (23) as a minimization problem of two fractions added. The transmit power matrix of UAV P (U) is then analyzed in an independent sum-rate maximization problem. Finally, the solution to the placement problem is obtained using the first order Taylor approximation.

Optimization of Bandwidth Allocation
According to the backhaul constraint, the optimal value of spectrum allocation η can be obtained when ηR 0,sum = (1 − η)R 0,bh , thus we have By substituting the optimal value of η into (23), we reformulate an equivalent problem as min r,P (U) We observe that, in the equivalent minimization problem,R 0,sum andR 0,bh are separated in two fractions. Therefore, the transmit power matrix of the UAV can be analyzed in an independent sum-rate maximization problem.

Optimization of Transmit Power Matrix
Based on the discussion in previous subsection, the transmit power matrix of the UAV P (U) can be solved by the following independent sum-rate maximization problem as This sum-rate maximization problem can be solved by the WMMSE algorithm described in References [24,25], where we define the equivalent channelĥ kj = h H kŵ j . Based on the deterministic equivalents proposed in Section 3, we have the corresponding deterministic equivalents ofĥ kk and ĥ kj 2 , which are respectively given byĥ as M → ∞.
Note that with the consideration of RZF precoding regime, the equivalent channels of different UEs are almost equal, thus the near-optimal power allocation scheme is to allocate the total power to each UE equally.

Optimization of UAV Placement
Given the equal power allocation is adopted at the UAV, the problem (25) can be reformulated as the optimum placement problem given by Note that R 0,sum is concave with respect to r [26,27], as the consistency ofR 0,sum and R 0,sum is verified in Section 5, thus,R 0,sum is concave with respect to r. However,R 0,bh is neither concave nor convex with respect to r. Thus, an approximation ofR 0,bh is introduced based on its first-order Taylor expansion. We achieve the approximation as followed.
Give a point r(n) in the n-th iteration. According to the first-order Taylor expansion at the given local point r(n), we obtain the approximationR appro 0,bh as R appro 0,bh = ∂R 0,bh ∂r (r(n)) (r − r(n)) +R 0,bh (r(n)), where ∂R 0,bh ∂r is the first-order derivative, which is given by with ∂ϕ ∂r Then, with the given local point r(n) and the approximationR appro 0,bh , problem (28) can be approximated as Note that the problem (32) is a convex problem with the linear constraint, thus, it can be solved by some optimization tools, such as CVX [28]. Therefore, an iterative algorithm is proposed to find the solution of problem (32), which is shown in Algorithm 1.

Numerical Results and Discussion
In this section, the accuracy of the deterministic equivalents of the ergodic sum-rate and wireless backhaul capacity are first verified, and then we use the deterministic equivalents to investigate the system performance and the effectiveness of the proposed algorithm. We consider a downlink UAV-assisted wireless communication network where a GBS is located at the origin with N = 12 antennas. The UEs are randomly distributed in the region of z k ∈ [2000, 2200] m, and θ k ∈ [0, π/2] with a uniform distribution under the polar coordinate. The height of the UAV l = 100 m, the noise power is σ 2 = −110 dBm, the path loss exponent α = 2, and the channel power gains at the reference distance β = 0.1 W [15,26].
In Figures 2-4, we compare the ergodic sum-rate of UEs R 0,sum and the ergodic capacity rate of the backhaul R 0,bh with their deterministic equivalentsR 0,sum ,R 0,bh under various system settings, respectively. Monte-Carlo simulation results is obtained by averaging over a large number of i.i.d. small-scale fading channels. The accuracy of the deterministic equivalents is validated even for practical system dimensions.  Figure 2 compares R 0,sum over the numbers of UEs with different values of the UAV's projection distance with {M = 10, p k = 0.2 W, r = (500, 1000, 1500)m}. Owing to the RZF precoding, inter-user interference can be mitigated, thus, higher sum-rate can be achieved as the numbers of UEs grows. Moreover, the larger UAV's projection distance indicates the UAV is closer to the UEs and better channel condition can be obtained. Figure 3 shows the backhaul rate versus different numbers of UAV antennas with the fixed number {N = 12} of the GBS antennas. It states that the UAV with more number of antennas provides better performance due to higher multi-antenna gain, meanwhile, the larger UAV's projection distance results in less backhaul capacity.  Figure 4 illustrates the impact of UAV's projection distance on R 0,sum with {M = 10, p k = 0.2 W, K = (4, 5)}. The optimal sum-rate can be found as the UAV hovers approaching to the UEs. However, when the backhaul capacity constraint and the dynamic spectrum allocation are considered, the achievable rate of UEs' access links and UAV's backhaul link can be both affected by the placement of the UAV. The approximation problem (23) based on the deterministic equivalents is effective as the accuracy of the deterministic equivalents is verified. It can avoid complicated computation resulting from Monte-Carlo averaging over small-scale fading channels, and the optimization problem becomes a slow time-scale issue.
In Figure 5, we compare the optimal normalized sum-rate of UEs R sum and the optimal UAV's projection distance r with respect to the transmit power of the GBS under the scenario of {N = 8, M = 6, K = 4, p k = 0.2W}. The optimal UAV's projection distance is found by solving problem (32), and the optimal normalized sum-rate can be obtained, accordingly. The line corresponds to its coordinate with the same color. Figure 5 indicates that with the transmit power of the GBS growing, the optimal UAV's projection distance increases fast especially in the low GBS transmit power region, since the system performance is restricted by the backhaul capacity. The backhaul capacity becomes abundant when the transmit power of the GBS is large. Therefore, the UAV trends to approach to UEs where the higher rates of access links can be achieved. Note that the optimal UAV's projection distance is determined by both the rates of access links and backhaul. The optimal normalized sum-rate also increases fast with the transmit power of the GBS grows, and then it will be degraded by the rates of access links. In Figure 6, it represents that the normalized sum-rate and the optimal spectrum allocation η are affected by the UAV's projection distance with a fixed GBS transmit power p b = 0.2 W. The line corresponds to the coordinate with the same color. From Figure 6, as the UAV is near to the GBS at the beginning, the backhaul capacity becomes dominant compared with the rates of access links. Therefore, less spectrum resource is allocated to the backhaul. With the UAV's projection distance becomes large, the UAV is far away from the GBS, the limited backhaul capacity will significantly restrict the system performance. In order to satisfy the backhaul constraint, more spectrum resource will be designated to the backhaul. Optimal spectrum allocation 9 Figure 6. Normalized sum-rate and optimal spectrum allocation versus UAV's projection distance. Figure 7 illustrates that using the optimal dynamic spectrum resource allocation can achieve higher normalized sum-rate, as the optimal dynamic spectrum allocation factor always achieves the balance between the rates of backhaul and access links. When η = 0.5, the normalized sum-rate is always less than that of backhaul. As η = 0.6, more spectrum resource is allocated to the access links, and the normalized sum-rate is larger than that of backhaul when the UAV' projection distance is large where the UAV is near the UEs. Therefore, the normalized sum-rate is then degraded by the normalized backhaul rate in this case.  Figure 8 presents the convergence performance of proposed algorithm with respect to the residual norm which is defined as the norm of the difference between the n-th iterative and the prior iterative value of the objective function. The tolerance is set as 10 −3 , and we consider the scenarios under different number of users and UAV's antennas which are labeled in the figure. From the figure, it indicates that the residual norm will convergence to zero after a few number of iterations for different numbers of users and UAV's antennas, as a result, the proposed algorithm ensures a fast convergence.

Conclusions
In this paper, we considered a UAV-assisted wireless communication network where a multi-antenna UAV utilized as a remote relay, assisted the GBS to forward signals for UEs in remote areas outside the coverage of the GBS. The UAV was connected to the GBS through in-band wireless backhaul, which shared the spectrum resource with the access links of UEs due to the limited spectrum resource. To mitigate the inter-user interference, the RZF precoding was adopt. A sum-rate maximization problem was formulated by jointly designing the spectrum resource allocation factor, the UAV's projection distance, and the transmit power matrix at the UAV. The deterministic equivalents of the ergodic sum-rate and wireless backhaul capacity were derived using large dimensional random matrix theory. Based on the deterministic equivalents, an approximation problem of the original optimization problem was formulated and the suboptimal solutions were obtained. The simulation results validated the accuracy of the deterministic equivalents and showed the system performance with respect to different variables, that is, the user number, the number of the UAV antennas, the transmit power of the GBS, and UAV projection distance.
In this work, the elastic UAV was deployed to assist the static terrestrial network. How to utilize the UAV to serve dynamic network is still an open issue. In addition, due to the on-board battery of the UAV is limited, the energy efficiency problem of the UAV-assisted network is still a challenge which needs more efforts in future research.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
Substituting the deterministic equivalents of desired signal, interference, and noise part into (8), then the deterministic equivalent of the SINR γ k is obtained in Lemma 1, hence the proof is completed.

Appendix B. Proof of Lemma 2
We consider that R 0,bh (σ 2 ) is a function of σ 2 . The derivative of R 0,bh (σ 2 ) with respect to σ 2 is expressed as The mutual information can be equivalently written as [23] R 0,bh (σ 2 ) Utilizing the large dimensional random matrix theory, we obtain as M → ∞, where Υ is given by (22). We define Ω = p b β d α u I M and R(σ 2 , x) given by The explicit expression can be proved according to References [23], as where ϕ is given by (22), hence the proof is completed.