1. 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 to enhance the system capacity among UEs’ access links and UAV’s backhaul links in the UAV-assisted networks [
7].
The main characteristic of the UAV applications in wireless communication system is agile deployment. Therefore, the placement and trajectory optimization of the UAVs have been researched in consideration of the quality of access links of user equipments (UEs). In Reference [
8], the minimum throughput of all UEs was maximized by jointly optimizing UEs’ scheduling and association along with UAVs’ power control and trajectory. In Reference [
9], the authors minimized the total UAV energy consumption with both propulsion and communication related energy by jointly optimizing the UAV’s trajectory, communication time allocation, and the total mission completion time. The optimum placement of a relaying UAV was studied in Reference [
10] to maximum reliability for both static and mobile UAVs. In Reference [
11], the optimum placement was considered to minimize the number of UAVs and the system cost while guaranteeing the wireless coverage performance.
However, in the UAV-assisted wireless communication network, the signals of UEs are forwarded through the wireless backhaul links between the UAV and the GBS, thus, the limited backhaul capacity would degrade the overall system performance [
7]. Meanwhile, the spectrum efficiency of UEs’ access links and UAV’s backhaul links can be both affected by the placement of the UAVs. To this end, the optimal placement of UAV in the UAV-assisted network requires to consider both access and backhaul links. In some existing works, the out-band wireless backhaul is considered. In Reference [
12], the millimeter wave frequency band is adopted for UAVs, and the microwave frequency band is adopted for GBSs. However, the dedicated spectrum band for backhaul link may be unavailable when the spectrum resources are limited. Therefore, the in-band backhaul link can be adopted where the spectrum resources are shared among UEs’ access links and UAV’s backhaul links. However, sharing of spectrum resources among access and backhaul links may induce additional interference. The orthogonal time-frequency resource allocation and user scheduling are effective methods to mitigate the interference [
8,
13,
14]. Different from a number of previous studies where the UAV was equipped with single antenna, the multi-antenna UAV was considered in References [
15,
16,
17] to form an MIMO communication link.
The multi-antenna UAV has rarely been analyzed in the UAV-assisted wireless communication network with the consideration of the in-band backhaul link. With the precoding method performed, the UAV can serve multiple UEs simultaneously, meanwhile, the inter-user interference can be well mitigated. Benefiting from the large-dimensional random matrix theory, the deterministic equivalents can be obtained without requiring the instantaneous channel state information. Therefore, the system performance in regard to different variables will be numerically analyzed avoiding Monte Carlo averaging which leads to high computational complexity. In addition, once the locations of UEs are determined, the optimal UAV placement as well as the spectrum resource allocation between the backhaul and access links can be pre-designated which can maximize the network throughput.
In this paper, we aim to maximize the downlink ergodic sum-rate by jointly optimizing UAV placement, spectrum resource allocation and transmit power matrix of the UAV. The composite channel model proposed in Reference [
15] is adopted which considerers both large-scale and small-scale fading. The Rayleigh small-scale fading is taken into account in accord with low-altitude UAV scenario [
18]. Note that computing the ergodic sum-rate relies on the instantaneous channel state information, which may lead to prohibitively computational complexity. Motivated by this challenge, the large-dimensional random matrix theory [
19,
20,
21] is utilized to derive the deterministic equivalents of UE’s achievable rate and UAV’s backhaul capacity, which relies on only statistical channel information. In this paper, we adopt regularized zero-forcing (RZF) precoding to mitigate inter-user interference [
22]. The main contributions of this paper are summarized as follows.
First, we consider the downlink transmission in a UAV-assisted wireless communication network. Differing from previous works, a multi-antenna UAV is considered to assist a multi-antenna GBS to forward signals for remote UEs which are out of the coverage provided from the GBS. The UAV 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 identity matrix is denoted by . The superscripts , , and represent the conjugate-transpose, transpose, and conjugate operations, respectively. 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 , , , and , respectively.
4. Optimization Design
Based on the deterministic equivalents derived in
Section 3, an alternative sum-rate maximization problem is given as
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 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.
4.1. Optimization of Bandwidth Allocation
According to the backhaul constraint, the optimal value of spectrum allocation
can be obtained when
, thus we have
By substituting the optimal value of
into (23), we reformulate an equivalent problem as
We observe that, in the equivalent minimization problem, and are separated in two fractions. Therefore, the transmit power matrix of the UAV can be analyzed in an independent sum-rate maximization problem.
4.2. Optimization of Transmit Power Matrix
Based on the discussion in previous subsection, the transmit power matrix of the UAV
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 channel
. Based on the deterministic equivalents proposed in
Section 3, we have the corresponding deterministic equivalents of
and
, which are respectively given by
as
.
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.
4.3. 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
is concave with respect to
r [
26,
27], as the consistency of
and
is verified in
Section 5, thus,
is concave with respect to
r. However,
is neither concave nor convex with respect to
r. Thus, an approximation of
is introduced based on its first-order Taylor expansion. We achieve the approximation as followed.
Give a point
in the
n-th iteration. According to the first-order Taylor expansion at the given local point
, we obtain the approximation
as
where
is the first-order derivative, which is given by
with
Then, with the given local point
and the approximation
, 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.
Algorithm 1 Iterative Algorithm for Problem (32) |
- Input:
Initial solution and tolerance factor ; - Output:
optimal . - 1:
repeat - 2:
Obtain ,,, by solving (16) and (22) with ; - 3:
Obtain the approximation with ; - 4:
Find the solution of problem (32); - 5:
Update ; - 6:
until the residual of objective function is less than .
|
5. 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
antennas. The UEs are randomly distributed in the region of
m, and
with a uniform distribution under the polar coordinate. The height of the UAV
m, the noise power is
dBm, the path loss exponent
, and the channel power gains at the reference distance
W [
15,
26].
In
Figure 2,
Figure 3 and
Figure 4, we compare the ergodic sum-rate of UEs
and the ergodic capacity rate of the backhaul
with their deterministic equivalents
,
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
over the numbers of UEs with different values of the UAV’s projection distance with
. 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
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
with
. 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
and the optimal UAV’s projection distance
r with respect to the transmit power of the GBS under the scenario of
. 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
are affected by the UAV’s projection distance with a fixed GBS transmit power
. 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.
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
, the normalized sum-rate is always less than that of backhaul. As
, 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
, 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.
6. 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.