# An Energy-efficient Aerial Backhaul System with Reconfigurable Intelligent
Surface

Hong-Bae Jeon, , Sung-Ho Park, , Jaedon Park,
Kaibin Huang, , and Chan-Byoung Chae This research was supported by the Agency
for Defense Development, Rep. of Korea. This article was presented in part at
the IEEE Globecom 2021, Madrid, Spain, December 7–11, 2021 [1]. (Corresponding
Author: Chan-Byoung Chae.)H.-B. Jeon and C.-B. Chae are with the School of
Integrated Technology, Yonsei University, Seoul 03722, Rep. of Korea (e-mail:
{hongbae08, cbchae}@yonsei.ac.kr).S.-H. Park is with the Datasolution, Inc.,
Seoul 06101, Rep. of Korea (e-mail: shpark@datasolution.kr).J. Park is with
the Agency for Defense Development, Daejeon 34186, Rep. of Korea (e-mail:
jaedon2@add.re.kr).K. Huang is with the Department of Electrical and
Electronic Engineering, The University of Hong Kong, Pok Fu Lam, Hong Kong
(e-mail: huangkb@eee.hku.hk).

###### Abstract

In this paper, we propose a novel wireless architecture, mounted on a high-
altitude aerial platform, which is enabled by reconfigurable intelligent
surface (RIS). By installing RIS on the aerial platform, rich line-of-sight
and full-area coverage can be achieved, thereby, overcoming the limitations of
the conventional terrestrial RIS. We consider a scenario where a sudden
increase in traffic in an urban area triggers authorities to rapidly deploy
unmanned-aerial vehicle base stations (UAV-BSs) to serve the ground users. In
this scenario, since the direct backhaul link from the ground source can be
blocked due to several obstacles from the urban area, we propose reflecting
the backhaul signal using aerial-RIS so that it successfully reaches the UAV-
BSs. We jointly optimize the placement and array-partition strategies of
aerial-RIS and the phases of RIS elements, which leads to an increase in
energy-efficiency of every UAV-BS. We show that the complexity of our
algorithm can be bounded by the quadratic order, thus implying high
computational efficiency. We verify the performance of the proposed algorithm
via extensive numerical evaluations and show that our method achieves an
outstanding performance in terms of energy-efficiency compared to benchmark
schemes.

###### Index Terms:

Reconfigurable intelligent surface, unmanned aerial vehicle, wireless
backhaul, half-power beamwidth, non-convex optimization, multi-objective
optimization, energy-efficient communication.

##  I Introduction

WHEN there are issues concerning the wireless network in terms of ground base
stations (BSs) malfunctioning or a massive increase in network traffic in a
wireless network, a promising solution is to use unmanned-aerial-vehicles
(UAVs) as temporary BSs [2, 3, 4]. The mobility and high-altitudes of UAV-BSs
allow flexible deployment deployment and provide rich line-of-sight (LoS)
links [5]. To efficiently deploy UAV-BSs, researchers have focused on
designing the deployment strategies [6, 7], trajectory optimization [8, 9, 10]
and power control [11, 12, 13].

One of the implicit conditions for the UAV-BS deployment is that the backhaul
rate should be high enough to support the traffic of the UAV-BSs [14]. To
satisfy this requirement, the energy-efficiency of the backhaul source should
also be considered [15, 16]. However, providing backhaul links directly from
the ground source is inefficient owing to potential blockages, which generate
non-LoS (NLoS) links and significantly degrade the system’s energy-efficiency.
This gives rise to a critical issue for future wireless networks with higher
frequencies and wider bandwidths [17, 18, 19, 20, 21].

Figure 1: Proposed UAV-BS access network with aerial-RIS backhaul link. The
source is equipped with $M$ directional antennas and sends the backhaul signal
to $M_{0}$ UAV-BSs. The RIS is assumed to be implemented on a voluminous
aerial platform placed at an altitude of $H$, and reflects the backhaul signal
from the source with high-LoS probability.

Addressing the issue motivates researchers to design reconfigurable
intelligent surfaces (RISs) to create LoS links by smart reflection and to
direct them in desired directions [22, 23]. RIS is an artificial metasurface
composed of passive reflecting elements that can adjust the amplitude and
phase of a reflected signal [24, 25]. Moreover, since the array architecture
of the RIS is passive, it features low power consumption [26]. Therefore, RISs
are expected to be deployed in future wireless networks to enhance their
performance. In [27, 28], the authors improved the energy-efficiency by
jointly allocating transmit power and adjusting phase shifts of RIS. In [29,
30, 31], the researchers proposed various RIS transmission strategies,
including a combined design of continuous transmit precoder and discrete phase
shifts [29], single-to-multi-beam training methods by partitioning RIS array
and designing their beam directions [30], and constructing multiple-input-
multiple-output quadrature amplitude modulation (MIMO-QAM) under the RIS
hardware constraints such as discrete phase shift and phase-dependent
amplitude [31]. Moreover, maximizing the spectral efficiency is studied in
[32, 33, 34]. The prior works, however, mainly focuses on deploying RIS in a
terrestrial environment that includes buildings and walls. Such an environment
poses several limitations on communication performance. First of all, in urban
areas with many buildings, reliable communication requires multiple
reflections, which requires the installing of many RISs to avoid severe signal
attenuation [35]. Moreover, the terrestrial RIS can only reflect the signal
from the source to the destination on the same side of the RIS, which cannot
achieve isotropic reflection with $0^{\circ}\sim 360^{\circ}$ arrival-angles.
Although some attempts have been made to implement the RIS on the aerial
platforms [36, 37, 38], only the 2D coverage is considered or the fixed
transmit power assumed, without considering the energy-efficiency problem.

To address these issues, we, in this paper, propose an energy-efficient aerial
backhaul structure [39] by mounting RIS on a sizable aerial platform, as
motivated by the results in [37] and illustrated in Fig. 1. By combining the
properties of the RIS with the rich-LoS capability of the aerial platform, the
aerial-RIS gives rise to a favorable channel with a reduced number of
reflections. This reduces transmit power for a given data rate and thereby
increases the energy-efficiency. Moreover, since the aerial-RIS can reflect
the signal in an arbitrary direction [37], it supports the full-3D coverage,
including the receivers (UAV-BSs) in the air. Exploiting these benefits, we
further propose an energy-efficiency enhancement strategy. The contributions
of this paper are summarized as:

  1. 1.

We propose to deploy RIS on the aerial platform to establish backhaul links
with rich-LoS component to the UAV-BSs, which leverages the advantages of both
a high-altitude platform and the RIS. We first adapt the maximum-ratio
transmission (MRT) strategy to maximize the received signal-to-noise ratio
(SNR) and thereby lower the transmit power.

  2. 2.

To achieve a high energy-efficiency, we derive a non-zero lower-bound on the
source transmit power for every UAV-BS and minimize the bounds by minimizing
and maximizing their numerator and denominator, respectively. It consists of
achieving the Pareto-optimum using the method of global criterion, array-
partition-mode selection with our proposed “reverse-waterfilling” partitioning
strategy, and the exploitation of the relationship between the passive
beamforming gain and the sin-angle-deviations.

  3. 3.

Through our novel framework of aerial-RIS deployments, we optimize the
placement, array-partition, and phase alignment of aerial-RIS to minimize the
source transmit power under the fronthaul-rate-ensuring constraints.
Furthermore, we show that the computational complexity of the proposed
algorithm is upper-bounded by the quadratic order, which shows a high
computational efficiency. We numerically estimate the performance of the
proposed algorithm in a realistic urban outdoor scenario with randomly
distributed ground users and corresponding UAV-BSs. By extensive simulations,
we demonstrate substantial improvements of energy-efficiency and considerable
diminishment of the optimality gap by the proposed algorithm.

The rest of the paper is organized as follows: In Section II, the system model
is described, including the target power minimization problem with the
fronthaul rate constraints. In Section III, we present our proposed algorithm
based on maximizing the lower-bound of the transmit power in the sense of the
placement of aerial-RIS, phase alignment, and the RIS array-partition. The
computational complexity of the proposed algorithm is also analyzed. We
present the numerical results of the proposed algorithm in Section IV,
followed by the conclusion and future works in Section V.

Notations: In this paper, $a$ is a scalar, $\mathbf{a}$ is a column vector,
$\mathbf{A}$ is a matrix, and $\mathbb{A}$ is a set.
$\mathbf{A}^{\mathrm{T}},\mathbf{A}^{*}$ denote the transpose and Hermitian of
$\mathbf{A}$, respectively. $\mathrm{diag}\left(\cdot\right)$ denotes a
diagonal matrix with given diagonal elements, and $\mathbf{0}$ denotes an all-
zero matrix. $\left|\cdot\right|$ denotes the modulus of a complex number or
the cardinality of a set and $\left|\left|\cdot\right|\right|_{2}$ denotes the
$\ell_{2}$-norm of the vector. $\left(a,b\right)$ and $\left[a,b\right]$ are
the open and closed interval on the real line, respectively. $\mathbb{R}$ and
$\mathbb{C}$ denote the real and complex number sets, respectively, and
$\mathbb{C}^{M\times N}$ denotes the space of $M\times N$ complex-valued
matrices for natural number $M$ and $N$. $j\triangleq\sqrt{-1}$ is the
imaginary unit, and $\mathcal{O}\left(\cdot\right)$ denotes the big-O
notation. $\mathcal{E}\left(\cdot\right)$ and $\mathrm{tr}\left(\cdot\right)$
denotes the expectation and trace operator, respectively, and
$\mathcal{U}\left(\cdot\right)$ denotes the uniform probability distribution
with given interval.

TABLE I: Summary of System Parameters Parameter |  Description
---|---
$\mathbb{M}=\left\\{1,\cdots,M_{0}\right\\}$ |  Index set of UAV-BSs
$\mathbb{U}_{m}$ |  Set of users serviced by UAV-BS $m$
$\bm{\rho}_{m}\triangleq\left[\mathbf{w}_{m}^{\mathrm{T}}~{}h_{m}\right]^{\mathrm{T}}$ |  3D coordinate of UAV-BS $m$
$\left(M,N\right)$ |  Number of antenna/RIS elements
$\lambda$ |  Link wavelength
$G_{\mathrm{s}}$ |  Antenna gain
$\left(d_{\mathrm{s}},d_{\mathrm{RIS}}\right)$ |  Antenna and RIS-element separation
$\left(\bar{d_{\mathrm{s}}},\bar{d}\right)$ |  $\lambda$-normalized
antenna/RIS-element separation |
$\bm{\rho}_{\mathrm{RIS}}\triangleq\left[\mathbf{q}^{\mathrm{T}}~{}H\right]^{\mathrm{T}}$ |  3D coordinate of aerial-RIS
$\beta_{0}$ |  Reference path loss
at a distance of 1 m |
$\left(\beta_{\mathrm{s}}\left(\mathbf{q}\right),\beta\left(\mathbf{q},\bm{\rho}_{m}\right)\right)$ |  Path loss
(source-RIS, RIS-UAV-BS $m$) |
$R_{m}$ |  Backhaul rate of UAV-BS $m$
$C_{m}$ |  Throughput of UAV-BS $m$
$\sigma^{2}$ |  Noise power
$N_{\mathrm{psd}}$ |  Noise PSD ($=-174~{}\mathrm{dBm}$)
$\bm{\Theta}$ |  RIS phase shift matrix
$\theta_{n}$ |  Phase shift of the $n$th element of RIS
$\mathbf{v}_{m}$ |  Unit-magnitude precoding vector intended to UAV-BS $m$
$s_{m}$ |  Unit-power signal
intended to UAV-BS $m$ |
$P_{m}$ |  Source transmit power
intended to UAV-BS $m$ |
$P_{\max}$ |  Feasible threshold of the
source transmit power |
$\mathbf{x}$ |  Total transmit signal of the source
$\bar{N}\left(\leq N\right)$ |  Number of activated RIS elements
$\left(\mathbf{H}\left(\mathbf{q}\right),\mathbf{h}^{*}\left(\mathbf{q},\cdot\right)\right)$ |  Channel models
(source-RIS, RIS-destination) |
$\left(\mathbf{a}_{\mathrm{s}}\left(\cdot\right),\mathbf{a}_{\mathrm{RIS}}\left(\cdot\right)\right)$ |  Array response (source, RIS)
$\left(\phi_{\mathrm{t,s}}\left(\mathbf{q}\right),\phi_{\mathrm{t,RIS}}\left(\mathbf{q},\cdot\right)\right)$ |  AoD of the source-RIS and RIS-destination link
$\phi_{\mathrm{r,RIS}}\left(\mathbf{q}\right)$ |  AoA of the source-RIS link
$\gamma_{m}$ |  Received SNR of UAV-BS $m$
$g\left(\cdot\right)$ |  Passive beamforming gain by aerial-RIS
$\Delta\phi_{\mathrm{HPBW}}\left(\cdot\right)$ |  Half-power beamwidth of $g$
$\Delta\phi_{m}\left(\bar{\bm{\rho}}\right)$ |  sin-AoD deviation between $\bm{\bar{\rho}}$ and $\bm{\rho}_{m}$

##  II System Model

###  II-A Aerial Backhaul Model

The important notations used in this paper are summarized in Table I. As
illustrated in Fig. 1, we consider the urban area $\mathcal{G}$ on the
horizontal plane with the origin $\bm{\rho}_{\mathcal{G}}$ and containing
$N_{0}$ users equipped with an omnidirectional antenna and serviced by
stationary UAV-BSs $\mathbb{M}=\left\\{1,\cdots,M_{0}\right\\}$. Additionally,
we assume that the UAV-BSs are equipped with a directional antenna where its
azimuth and elevation half-power beamwidth (HPBW) are not equal [40]. The 3D
coordinate of UAV-BS $m$ is given by
$\bm{\rho}_{m}\triangleq\left[\mathbf{w}_{m}^{\mathrm{T}}~{}h_{m}\right]^{\mathrm{T}}$,
which consists of the 2D location $\mathbf{w}_{m}$ and its height $h_{m}$.

To avoid the inter-cell interference, we assume that each UAV-BS services a
different non-overlapping and non-empty user subset. Here, the Ellipse
Clustering [11] is applied for servicing the ground users, which determines
$M_{0}$ and extensively lowers the total transmit power of UAV-BSs by
determining their 3D locations. In this case, the throughput of UAV-BS $m$ is
given by

$C_{m}=\sum_{n\in\mathbb{U}_{m}}\frac{B_{\mathrm{f}}}{\left|\mathbb{U}_{m}\right|}\log_{2}\left(1+\frac{P_{\mathrm{f},n}}{\sigma_{\mathrm{f}}^{2}}\right),$
(1)

where $\mathbb{U}_{m}$ is the set of users serviced by UAV-BS $m$,
$B_{\mathrm{f}}$ is the transmission bandwidth of the fronthaul link equally
divided into $\left|\mathbb{U}_{m}\right|$ bands for each user in
$\mathbb{U}_{m}$, $P_{\mathrm{f},n}$ is the received power of user
$n\in\mathbb{U}_{m}$, and
$\sigma_{\mathrm{f},m}^{2}=\frac{B_{\mathrm{f}}}{\left|\mathbb{U}_{m}\right|}N_{\mathrm{psd}}$
is the noise power of the fronthaul link for UAV-BS $m$ with the power
spectral density (PSD) of noise $N_{\mathrm{psd}}=-174~{}\mathrm{dBm}$.

For a backhaul link via an aerial-RIS, we assume that the source is located at
the origin and equipped with uniform-linear-array (ULA) with $M$ antennas.
Each antenna has a gain of $G_{\mathrm{s}}$ and is separated by
$d_{\mathrm{s}}$. We assume that the source-to-center distance
$d_{\mathcal{G}}\triangleq\left|\left|\bm{\rho}_{\mathcal{G}}\right|\right|_{2}$
is sufficiently large and that the direct link from the source to the UAV-BS
is blocked. The aerial-RIS also consists of ULA with $N$ reflecting elements
without power amplification, separated by
$d_{\mathrm{RIS}}\left(\in\left[\frac{\lambda}{10},\frac{\lambda}{5}\right]\right)$
[37, 34], where $\lambda$ is the carrier wavelength. Note that by apposing the
parallel ULA-structured RIS by $N^{\prime}$ times, it is feasible to extend
our scenario to aerial-RIS equipped with an $N\times N^{\prime}$ uniform-
planar array (UPA) with expected performance gain of $N^{\prime}$ times, which
can be regarded as the subset optimization of UPA-RIS. With its fixed altitude
$H$ and by letting the first reflection element be the reference point, the 3D
coordinate of the aerial-RIS is given by
$\bm{\rho}_{\mathrm{RIS}}\triangleq\left[\mathbf{q}^{\mathrm{T}}~{}H\right]^{\mathrm{T}}$.
Without loss of generality, we set the RIS array parallel to the x-axis.
Moreover, due to the altitude of aerial-RIS, we assume that the backhaul link
is dominated by an LoS component. Owing to the sufficiently small
$d_{\mathrm{RIS}}$ compared to $H$ and $d_{\mathcal{G}}$, which determine the
link distance, the backhaul link can be well approximated as uniform plane
waves and we assume that the attenuation of the path can be considered as
identical for all RIS-element pairs. Hence, the path loss of source-to-RIS
$\beta_{\mathrm{s}}\left(\mathbf{q}\right)$ link and RIS-to-UAV-BS $m$ link
$\beta\left(\mathbf{q},\bm{\rho}_{m}\right)$ are given by [41]

$\beta_{\mathrm{s}}\left(\mathbf{q}\right)=\frac{\beta_{0}}{\left|\left|\bm{\rho}_{\mathrm{RIS}}\right|\right|_{2}^{2}},~{}\beta\left(\mathbf{q},\bm{\rho}_{m}\right)=\frac{\beta_{0}}{\left|\left|\bm{\rho}_{\mathrm{RIS}}-\bm{\rho}_{m}\right|\right|_{2}^{2}},$
(2)

where $\beta_{0}$ is the reference path loss at a link distance of 1 m given
by $\beta_{0}=-20\log_{10}f-32.45$ [dB] for the link frequency $f$ measured in
GHz. As we assumed ULA for both the source and the RIS, the channels between
source-RIS $\mathbf{H}\left(\mathbf{q}\right)\in\mathbb{C}^{N\times M}$ and
RIS-destination
$\mathbf{h}^{*}\left(\mathbf{q},\cdot\right)\in\mathbb{C}^{1\times N}$ can be
modeled by the angle-of-departure and arrival (AoD/AoA) of the communication
links, thus containing every information about the azimuth and zenith angles
of departure and arrival. Since the links are dominated by LoS path only, they
are given as

$\begin{split}\begin{cases}\mathbf{H}\left(\mathbf{q}\right)\\\
=\sqrt{\beta_{\mathrm{s}}\left(\mathbf{q}\right)}e^{j\Phi_{\mathbf{H}}}e^{-j\frac{2\pi\left|\left|\bm{\rho}_{\mathrm{RIS}}\right|\right|_{2}}{\lambda}}\mathbf{a}_{\mathrm{RIS}}\left(\phi_{\mathrm{r,RIS}}\left(\mathbf{q}\right)\right)\mathbf{a}_{\mathrm{s}}^{*}\left(\phi_{\mathrm{t,s}}\left(\mathbf{q}\right)\right),\\\
\mathbf{h}^{*}\left(\mathbf{q},\bm{\rho}_{m}\right)\\\
=\sqrt{\beta\left(\mathbf{q},\bm{\rho}_{m}\right)}e^{j\Phi_{\mathbf{h}}}e^{-j\frac{2\pi\left|\left|\bm{\rho}_{\mathrm{RIS}}-\bm{\rho}_{m}\right|\right|_{2}}{\lambda}}\mathbf{a}_{\mathrm{RIS}}^{*}\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q},\bm{\rho}_{m}\right)\right),\end{cases}\end{split}$
(3)

where $\Phi_{\mathbf{H}}$ and $\Phi_{\mathbf{h}}$ are the random phase which
are independent and generated with $\mathcal{U}\left[0,2\pi\right)$, and
$\mathbf{a}_{\mathrm{s}}\left(\cdot\right)\in\mathbb{C}^{M}$,
$\mathbf{a}_{\mathrm{RIS}}\left(\cdot\right)\in\mathbb{C}^{N}$ are the array
response of the source and the aerial-RIS, respectively. They are given by

$\begin{split}\begin{cases}\mathbf{a}_{\mathrm{s}}\left(\cdot\right)=\left[1~{}e^{-j2\pi\bar{d}_{\mathrm{s}}\left(\sin\left(\cdot\right)\right)}~{}\cdots~{}e^{-j2\pi\left(M-1\right)\bar{d}_{\mathrm{s}}\left(\sin\left(\cdot\right)\right)}\right]^{\mathrm{T}},\\\
\mathbf{a}_{\mathrm{RIS}}\left(\cdot\right)=\left[1~{}e^{-j2\pi\bar{d}\left(\sin\left(\cdot\right)\right)}~{}\cdots~{}e^{-j2\pi\left(N-1\right)\bar{d}\left(\sin\left(\cdot\right)\right)}\right]^{\mathrm{T}},\end{cases}\end{split}$
(4)

where $\bar{d}_{\mathrm{s}}\triangleq\frac{d_{\mathrm{s}}}{\lambda}$ and
$\bar{d}\triangleq\frac{d_{\mathrm{RIS}}}{\lambda}$. Finally,
$\phi_{\mathrm{t,s}}\left(\mathbf{q}\right)$,
$\phi_{\mathrm{t,RIS}}\left(\mathbf{q},\cdot\right)$ and
$\phi_{\mathrm{r,RIS}}\left(\mathbf{q}\right)$ are the AoD of the source-RIS
and RIS-destination link and the AoA of the source-RIS link, respectively. To
make the targeted problem more tractable, we assume that the ground source
perfectly knows all the RIS channels
$\left\\{\mathbf{h}^{*}\left(\mathbf{q},\bm{\rho}_{m}\right)\right\\}_{m=1}^{M_{0}}$
and $\mathbf{H}\left(\mathbf{q}\right)$, which can be acquired by the methods
described in e.g. [42, 43].

By concatenating the channels in (3) and the phase shift of the RIS, the
backhaul rate of UAV-BS $m$ is given by

$R_{m}=\frac{B_{\mathrm{b}}}{M_{0}}\log_{2}\left(1+\underbrace{\frac{P_{m}G_{\mathrm{s}}\left|\mathbf{h}^{*}\left(\mathbf{q},\bm{\rho}_{m}\right)\bm{\Theta}\mathbf{H}\left(\mathbf{q}\right)\mathbf{v}_{m}\right|^{2}}{\sigma^{2}}}_{\triangleq\gamma_{m}}\right),$
(5)

where $B_{\mathrm{b}}$ is the transmission bandwidth of the backhaul link
equally divided into $M_{0}$ bands for each UAV-BS,
$\sigma^{2}=\frac{B_{\mathrm{b}}}{M_{0}}N_{\mathrm{psd}}$ is the noise power
of the backhaul link, $\gamma_{m}$ is the received SNR of UAV-BS $m$, and
$\mathbf{v}_{m}\in\mathbb{C}^{M}$ is a unit-magnitude precoding vector
corresponding to unit-power signal $s_{m}$ intended to UAV-BS $m$ with source
transmit power $P_{m}$. The total transmit signal $\mathbf{x}$ of the source
is

$\mathbf{x}=\sum_{m\in\mathbb{M}}\mathbf{v}_{m}\sqrt{P_{m}G_{\mathrm{s}}}s_{m}.$
(6)

Finally,
$\bm{\Theta}\triangleq\mathrm{diag}\left(\left\\{\alpha_{n}e^{j\theta_{n}}\right\\}_{n=1}^{N}\right)\in\mathbb{C}^{N\times
N}$ is a diagonal phase shift matrix with the amplitude reflection coefficient
$\alpha_{n}\in\left[0,1\right]$ and phase shift
$\theta_{n}\in\left[0,2\pi\right)$ of the $n$th element. Since we assume no
power amplification of RIS, we can set $\alpha_{n}=1\left(\forall
n=1,\cdots,N\right)$. Note that the equivalent channel matrix $\mathbf{X}$ in
(5) that concatenates the the channels in (3) and the phase shift information
is given by
$\mathbf{X}\triangleq\mathbf{h}^{*}\left(\mathbf{q},\bm{\rho}_{m}\right)\bm{\Theta}\mathbf{H}\left(\mathbf{q}\right)$.

Figure 2: Passive beamforming gain $g$ and the region of full/sub-array
structure. If the sin-AoD deviation exceeds the HPBW of the beamforming gain
with full-array structure, we apply the sub-array structure to include the
deviated point.

Here, to minimize the source transmit power $\sum_{m\in\mathbb{M}}P_{m}$ for
high energy-efficiency [11, 12], we first determine the precoding vector
$\mathbf{v}_{m}$ of UAV-BS $m$ that maximizes the received SNR
$\gamma_{m}\triangleq\frac{P_{m}G_{\mathrm{s}}\left|\mathbf{h}^{*}\left(\mathbf{q},{\bm{\rho}}_{m}\right)\bm{\Theta}\mathbf{H}\left(\mathbf{q}\right)\mathbf{v}_{m}\right|^{2}}{\sigma^{2}}$
for fixed $P_{m}$, which leads to a smaller transmit power while achieving the
same rate. Lemma 1 presents the result.

######  Lemma 1.

To maximize $\gamma_{m}$, the source should apply an MRT strategy. In other
words, $\mathbf{v}_{m}$ is given by

$\mathbf{v}_{m}=\mathbf{v}\triangleq\frac{\mathbf{a}_{\mathrm{s}}\left(\phi_{\mathrm{t,s}}\left(\mathbf{q}\right)\right)}{\left|\left|\mathbf{a}_{\mathrm{s}}\left(\phi_{\mathrm{t,s}}\left(\mathbf{q}\right)\right)\right|\right|_{2}}~{}\left(\forall
m\in\mathbb{M}\right).$ (7)

###### Proof.

By definition of $\mathbf{X}$, it is evident that

$\gamma_{m}=\frac{P_{m}G_{\mathrm{s}}}{\sigma^{2}}\mathbf{v}_{m}^{*}\mathbf{X}^{*}\mathbf{X}\mathbf{v}_{m}.$
(8)

By (3), $\mathbf{X}^{*}\mathbf{X}$ has a form of

$\mathbf{X}^{*}\mathbf{X}=C_{0}\mathbf{a}_{\mathrm{s}}\left(\phi_{\mathrm{t,s}}\left(\mathbf{q}\right)\right)\mathbf{a}_{\mathrm{s}}^{*}\left(\phi_{\mathrm{t,s}}\left(\mathbf{q}\right)\right),$
(9)

where $C_{0}$ is a positive constant. Equation (9) implies that
$\mathbf{X}^{*}\mathbf{X}$ is a rank-1 matrix with corresponding non-zero
eigenpair:

$\left(C_{0}\left|\left|\mathbf{a}_{\mathrm{s}}\left(\phi_{\mathrm{t,s}}\left(\mathbf{q}\right)\right)\right|\right|_{2}^{2},\frac{\mathbf{a}_{\mathrm{s}}\left(\phi_{\mathrm{t,s}}\left(\mathbf{q}\right)\right)}{\left|\left|\mathbf{a}_{\mathrm{s}}\left(\phi_{\mathrm{t,s}}\left(\mathbf{q}\right)\right)\right|\right|_{2}}\right).$
(10)

Hence, by using $\left|\left|\mathbf{v}_{m}\right|\right|_{2}=1$ and the
Rayleigh-Ritz theorem, $\mathbf{v}_{m}$ is given by

$\mathbf{v}_{m}=\frac{\mathbf{a}_{\mathrm{s}}\left(\phi_{\mathrm{t,s}}\left(\mathbf{q}\right)\right)}{\left|\left|\mathbf{a}_{\mathrm{s}}\left(\phi_{\mathrm{t,s}}\left(\mathbf{q}\right)\right)\right|\right|_{2}}$
(11)

to maximize $\mathbf{v}_{m}^{*}\mathbf{X}^{*}\mathbf{X}\mathbf{v}_{m}$ and the
theorem follows. ∎

Note that the optimal transmission strategy is given by MRT since we have
eliminated the interference by considering frequency division multiple access
(FDMA) in (5). By Lemma 1, MRT is adopted toward the aerial-RIS and
$\gamma_{m}$ can be transformed into

$\begin{split}\gamma_{m}=&\frac{P_{m}G_{\mathrm{s}}}{\sigma^{2}}\left|\mathbf{h}^{*}\left(\mathbf{q},\bm{\rho}_{m}\right)\bm{\Theta}\mathbf{H}\left(\mathbf{q}\right)\frac{\mathbf{a}_{\mathrm{s}}\left(\phi_{\mathrm{t,s}}\left(\mathbf{q}\right)\right)}{\left|\left|\mathbf{a}_{\mathrm{s}}\left(\phi_{\mathrm{t,s}}\left(\mathbf{q}\right)\right)\right|\right|_{2}}\right|^{2}\\\
=&\bar{\gamma}\frac{\biggl{|}\sum_{n=1}^{N}e^{j\left(\theta_{n}+2\pi\left(n-1\right)\bar{d}\left(\sin\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q},\bm{\rho}_{m}\right)\right)-\sin\left(\phi_{\mathrm{r,RIS}}\left(\mathbf{q}\right)\right)\right)\right)}\biggr{|}^{2}}{\left|\left|\bm{\rho}_{\mathrm{RIS}}\right|\right|_{2}^{2}\left|\left|\bm{\rho}_{\mathrm{RIS}}-\bm{\rho}_{m}\right|\right|_{2}^{2}},\end{split}$
(12)

where
$\bar{\gamma}\triangleq\frac{P_{m}G_{\mathrm{s}}\beta_{0}^{2}M}{\sigma^{2}}$.
From (12), we can notice that $\gamma_{m}$ is invariant of the antenna spacing
$d_{\mathrm{s}}$.

Figure 3: Block diagram of the proposed algorithm.

###  II-B Aerial-RIS Beamforming Model

To maximize $\gamma_{m}$ for a given $P_{m}$, we have to align the phase
$\left\\{\theta_{n}^{*}\right\\}_{n=1}^{N}$ such that the reflected signals
are coherently added to point $\bm{\rho}_{m}$:

$\begin{split}&\theta_{n}^{*}\left(\mathbf{q},\bm{\rho}_{m}\right)\\\
&=\bar{\theta}-2\pi\left(n-1\right)\bar{d}\left(\sin\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q},\bm{\rho}_{m}\right)\right)-\sin\left(\phi_{\mathrm{r,RIS}}\left(\mathbf{q}\right)\right)\right),\end{split}$
(13)

where $\bar{\theta}\in\left[0,2\pi\right)$ is an arbitrary phase shift.
However, as there are multiple $M_{0}$ UAV-BSs to serve,
$\left\\{\theta_{n}^{*}\right\\}_{n=1}^{N}$ differs for every
$m\in\mathbb{M}$. Hence, we should determine the phase align point
$\bm{\bar{\rho}}$ that leads to a Pareto-optimum for
$\left\\{\gamma_{m}\right\\}_{m\in\mathbb{M}}$. In other words, for a given
$\mathbf{q}$ and $\bm{\bar{\rho}}$, we set
$\left\\{\theta_{n}^{*}\right\\}_{n=1}^{N}$ as

$\begin{split}&\theta_{n}^{*}\left(\mathbf{q},\bm{\bar{\rho}}\right)\\\
&=\bar{\theta}-2\pi\left(n-1\right)\bar{d}\left(\sin\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q},\bm{\bar{\rho}}\right)\right)-\sin\left(\phi_{\mathrm{r,RIS}}\left(\mathbf{q}\right)\right)\right),\end{split}$
(14)

which coherently adds the reflected signal to $\bm{\bar{\rho}}$. By
substituting (LABEL:phase) into (12), $\gamma_{m}$ becomes

$\gamma_{m}=\bar{\gamma}\frac{g\left(\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)\right)}{\left|\left|\bm{\rho}_{\mathrm{RIS}}-\bm{\rho}_{m}\right|\right|_{2}^{2}\left|\left|\bm{\rho}_{\mathrm{RIS}}\right|\right|_{2}^{2}},$
(15)

where $g\left(\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)\right)$ is the
passive beamforming gain by aerial-RIS to $\bm{\rho}_{m}$ assuming the phases
are aligned to $\bm{\bar{\rho}}$, which is obtained by manipulating
$\biggl{|}\sum_{n=1}^{N}e^{j\left(\theta_{n}^{*}\left(\mathbf{q},\bm{\bar{\rho}}\right)+2\pi\left(n-1\right)\bar{d}\left(\sin\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q},\bm{\rho}_{m}\right)\right)-\sin\left(\phi_{\mathrm{r,RIS}}\left(\mathbf{q}\right)\right)\right)\right)}\biggr{|}^{2}$
in (12), with considering $\bar{N}\left(\leq N\right)$ activated RIS elements,
and given by

$g\left(\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)\right)\triangleq\left|\frac{\sin\left(\pi\bar{N}\bar{d}\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)\right)}{\sin\left(\pi\bar{d}\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)\right)}\right|^{2},$
(16)

where $\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)$ is the sin-AoD deviation
between $\bm{\bar{\rho}}$ and $\bm{\rho}_{m}$, that is,

$\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)\triangleq\sin\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q},\bm{\rho}_{m}\right)\right)-\sin\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q},\bm{\bar{\rho}}\right)\right).$
(17)

As illustrated in Fig. 2, $g$ diminishes to 0 out of its HPBW, which is given
by [44]

$\begin{cases}\Delta\phi_{\mathrm{HPBW}}\left(\bar{N}\right)\approx\frac{0.8858}{\bar{N}\bar{d}},\\\
\Delta\phi_{\mathrm{HPBW}}\left(N\right)\triangleq\Delta\phi_{\mathrm{HPBW}},\end{cases}$
(18)

and the peak gain at $\Delta\phi_{m}(\cdot)=0$ is ${\bar{N}}^{2}$. Therefore,
we have to fine tune $\bm{\bar{\rho}}$ and determine maximum $\bar{N}$ such
that every UAV-BS locates in the HPBW of $g$, which leads to the maximization
of
$\left\\{g\left(\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)\right)\right\\}_{m\in\mathbb{M}}$
in Section III.B.

###  II-C Problem Formulation

To reliably support the UAV-BSs, the backhaul rate should not be less than the
throughput of the fronthaul link $\left\\{C_{m}\right\\}_{m\in\mathbb{M}}$:
$R_{m}\geq C_{m}~{}\left(\forall m\in\mathbb{M}\right)$. By manipulating the
equation using
$R_{m}=\frac{B_{\mathrm{b}}}{M_{0}}\log_{2}\left(1+\gamma_{m}\right)$, the
transmit power $P_{m}$ must satisfy the following constraint:

$\displaystyle
P_{m}\geq\left(2^{\frac{M_{0}}{B_{\mathrm{b}}}C_{m}}-1\right)\frac{\sigma^{2}\left|\left|\bm{\rho}_{\mathrm{RIS}}-\bm{\rho}_{m}\right|\right|_{2}^{2}\left|\left|\bm{\rho}_{\mathrm{RIS}}\right|\right|_{2}^{2}}{G_{\mathrm{s}}\beta_{0}^{2}Mg\left(\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)\right)}~{}\left(\forall
m\in\mathbb{M}\right).$ (19)

Moreover, by (6) and Lemma 1, the maximum transmit constraint of the ground
source is given by [27]

$\mathcal{E}\left[\left|\bm{\mathrm{x}}\right|^{2}\right]=\mathrm{tr}\left(G_{\mathrm{s}}\bm{\mathrm{PV^{\mathrm{H}}V}}\right)=G_{\mathrm{s}}\sum_{m\in\mathbb{M}}P_{m}\leq
P_{\mathrm{max}},$ (20)

wherein
$\bm{\mathrm{V}}\triangleq\left[\bm{\mathrm{v}}_{1}\cdots\bm{\mathrm{v}}_{M_{0}}\right]\in\mathbb{C}^{M\times
M_{0}}$ derived by Lemma 1,
$\bm{\mathrm{P}}\triangleq\mathrm{diag}\left(\left\\{P_{m}\right\\}_{m\in\mathbb{M}}\right)\in\mathbb{R}^{M_{0}\times
M_{0}}$ and $P_{\mathrm{max}}$ is the feasible threshold of the source
transmit power. Hence, the source power minimization problem can be formulated
of

$\displaystyle\underset{\mathbf{q},\bm{\bar{\rho}},\left\\{\bar{N}\right\\},\left\\{P_{m}\right\\}_{m\in\mathbb{M}}}{\texttt{min}}~{}\sum\limits_{m\in\mathbb{M}}P_{m}$
(21)
$\displaystyle\text{~{}~{}s.t.}~{}P_{m}\geq\left(2^{\frac{M_{0}}{B_{\mathrm{b}}}C_{m}}-1\right)\frac{\sigma^{2}\left|\left|\bm{\rho}_{\mathrm{RIS}}-\bm{\rho}_{m}\right|\right|_{2}^{2}\left|\left|\bm{\rho}_{\mathrm{RIS}}\right|\right|_{2}^{2}}{G_{\mathrm{s}}\beta_{0}^{2}Mg\left(\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)\right)}$
$\displaystyle~{}~{}~{}~{}~{}\left(\forall
m\in\mathbb{M}\right),\sum_{m\in\mathbb{M}}P_{m}\leq
G_{\mathrm{s}}^{-1}P_{\mathrm{max}}.$

Problem (21) is non-convex due to the product
$\left|\left|\bm{\rho}_{\mathrm{RIS}}-\bm{\rho}_{m}\right|\right|_{2}^{2}\left|\left|\bm{\rho}_{\mathrm{RIS}}\right|\right|_{2}^{2}$
and the beamforming function $g$ in (16). We propose to solve it by minimizing
the numerator and maximizing the denominator of the lower-bound of $P_{m}$ in
(21) to minimize the sum. Fig. 3 shows a block diagram that illustrates the
main steps for solving (21). In the following section, we discuss, in detail,
each block of the proposed algorithm in Fig. 3.

######  Remark 1.

Since $C_{m}>0$, $\left|\left|\bm{\rho}_{\mathrm{RIS}}\right|\right|_{2}\geq
H>0$ and
$\left|\left|\bm{\rho}_{\mathrm{RIS}}-\bm{\rho}_{m}\right|\right|_{2}>0$
($\because\bm{\rho}_{\mathrm{RIS}}$ is extremely close to the origin. See
Theorem 2 and Fig. 7), we can conclude that the right-hand side of (19) is not
zero, which therefore guarantees non-zero transmit power for every UAV-BS
through aerial-RIS.

##  III Proposed Aerial-RIS Setup Algorithm

###  III-A Minimizing the Numerator: Determining $\mathbf{q}$

By letting $\mathbf{q}_{m}$ the 2D location of the aerial-RIS considering UAV-
BS $m$
$\left(\bm{\rho}_{\mathrm{RIS}}=\left[\mathbf{q}_{m}^{\mathrm{T}}~{}H\right]^{\mathrm{T}}\right)$,
we can express the numerator minimization problem in regard to
$\mathbf{q}_{m}$, which is provided by

$\displaystyle\underset{\mathbf{q}_{m}}{\texttt{min}}~{}\left|\left|\bm{\rho}_{\mathrm{RIS}}-\bm{\rho}_{m}\right|\right|_{2}^{2}\left|\left|\bm{\rho}_{\mathrm{RIS}}\right|\right|_{2}^{2}$
(22)
$\displaystyle~{}~{}~{}~{}~{}~{}=\left(H^{2}+\left|\left|\mathbf{q}_{m}\right|\right|_{2}^{2}\right)\left(\left(H-h_{m}\right)^{2}+\left|\left|\mathbf{q}_{m}-\mathbf{w}_{m}\right|\right|_{2}^{2}\right)$
$\displaystyle\text{~{}~{}s.t.}~{}\left|\left|\mathbf{q}_{m}\right|\right|_{2}\ll\delta\left|\left|\mathbf{w}_{m}\right|\right|_{2},$

where $\delta$ is a sufficiently small positive constant. The constraint
“$\left|\left|\mathbf{q}_{m}\right|\right|_{2}\ll\delta\left|\left|\mathbf{w}_{m}\right|\right|_{2}$”
is added to restrict $\mathbf{q}_{m}$ to be around the origin (source), as
placing the RIS close to the source leads to the almost sure use of full-array
RIS architecture $\left(\bar{N}=N\right)$, which maximizes the minimum SNR
[37] and therefore leads to smaller $\sum_{m\in\mathbb{M}}P_{m}$. Fortunately,
we can find a practical solution for the problem, as stated in Theorem 2.

######  Theorem 2.

The solution of Problem (22) is given by

$\mathbf{q}_{m}^{*}=\xi_{m}\mathbf{w}_{m},$ (23)

where

$\begin{split}&\xi_{m}=\frac{1}{2}+2\sqrt{-\frac{a}{3}}\cos\left(\frac{1}{3}\cos^{-1}\left(\frac{3b}{2a}\sqrt{-\frac{3}{a}}\right)-\frac{4}{3}\pi\right),\end{split}$
(24)

$a$ and $b$ are given by

$a\triangleq\frac{1}{2}\left(\zeta_{1}^{2}+\zeta_{2}^{2}\right)-\frac{1}{4},~{}b\triangleq\frac{1}{4}\left(\zeta_{2}^{2}-\zeta_{1}^{2}\right),$
(25)

and

$\zeta_{1}\triangleq\frac{H}{\left|\left|\mathbf{w}_{m}\right|\right|_{2}},~{}\zeta_{2}\triangleq\frac{\left|H-h_{m}\right|}{\left|\left|\mathbf{w}_{m}\right|\right|_{2}}.$
(26)

###### Proof.

See Appendix A. ∎

In Section IV.B, we numerically verify that $\xi_{m}$ is greater than 0 but
extremely close to 0 under the assumptions, which ascertains that
$\mathbf{q}_{m}=\xi_{m}\mathbf{w}_{m}$ is a proper solution for our power-
minimization procedure.

######  Remark 2.

For $h_{m}=0$, the following holds:

$\zeta_{1}=\zeta_{2}\triangleq\zeta\left(=\frac{H}{\left|\left|\mathbf{w}_{m}\right|\right|_{2}}\right)\rightarrow
a=\frac{1}{2}\zeta^{2}-\frac{1}{4},~{}b=0.$ (27)

Therefore, $\xi_{m}$ becomes

$\xi_{m}=\frac{1}{2}-\sqrt{\frac{1}{4}-\zeta^{2}}.$ (28)

Equation (28) implies that $\xi_{m}$ goes to 0 for sufficiently large
$\left|\left|\mathbf{w}_{m}\right|\right|_{2}$ $\left(\because\zeta\rightarrow
0\right)$, which also corresponds to the derived result.

After obtaining $\left\\{\mathbf{q}_{m}^{*}\right\\}_{m\in\mathbb{M}}$, we
have to determine the location $\mathbf{q}^{*}$ from the result, which
achieves a Pareto-optimum of (22) for every $m\in\mathbb{M}$. Under the prior
knowledge of $\left\\{\mathbf{q}_{m}^{*}\right\\}_{m\in\mathbb{M}}$, we
consider the following method of the global criterion that minimizes the sum
of $\ell_{2}$-deviation and leads to the Pareto front [45]:

$\displaystyle\underset{\mathbf{q}}{\texttt{min}}~{}\sum_{m\in\mathbb{M}}\left|\left|\mathbf{q}_{m}^{*}-\mathbf{q}\right|\right|_{2}.$
(29)

Problem (29) is called the Fermat-Torricelli problem, which is convex and can
therefore be solved efficiently using the Weiszfeld’s algorithm [46], which is
proven to strictly converge to the optimal point of the problem [47]. Hence,
we can determine $\mathbf{q}^{*}$ by (29), which leads to a suboptimal
solution of the numerator minimization.

######  Remark 3.

To maintain small $\left|\left|\mathbf{q}^{*}\right|\right|_{2}$, although an
outlier in $\left\\{\mathbf{q}_{m}^{*}\right\\}_{m\in\mathbb{M}}$ exists, we
minimize the sum of the norm instead of the squared norm in (29), which
additionally guarantees the robustness of the solution [45].

(a) Full-array structure.

(b) Sub-array structure with the proposed array-partition strategy.

Figure 4:  Array-structure selection of aerial-RIS with respect to HPBW of $g$
and maximum sin-AoD deviation. Due to the peak gain ${\bar{N}}^{2}$ placed in
the denominator in (19), the full-array structure leads to a further reduction
in the transmit power.

###  III-B Maximizing the Denominator: Determining $\\{\bar{N}\\}$,
$\boldsymbol{\bar{\rho}}$, and $\bm{\Theta}$

For a given $m\in\mathbb{M}$, it is known that
$\Delta\phi_{m}\left(\cdot\right)=0$ leads to a maximum denominator, which is
equivalent to $\bm{\bar{\rho}}=\bm{\rho}_{m}$. By the same logic, we have to
determine $\bm{\bar{\rho}}$ that maximizes
$\left\\{g\left(\Delta\phi_{m}(\bm{\bar{\rho}})\right)\right\\}_{m\in\mathbb{M}}$.
As $\bm{\rho}_{m}$ and $\bm{\bar{\rho}}$ are sufficiently close, $\xi_{m}\ll
1$ and $d_{\mathcal{G}}$ is sufficiently large, we can approximate the
absolute value of the sin-AoD deviation by the first-order Taylor
approximation with respect to $\bm{\rho}_{m}$:

$\begin{split}&\left|\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)\right|\\\
&=\left|\sin\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q}^{*},\bm{\rho}_{m}\right)\right)-\sin\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q}^{*},\bm{\bar{\rho}}\right)\right)\right|\\\
&\approx\left|\cos\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q}^{*},\bm{\rho}_{m}\right)\right)\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q}^{*},\bm{\rho}_{m}\right)-\phi_{\mathrm{t,RIS}}\left(\mathbf{q}^{*},\bm{\bar{\rho}}\right)\right)\right|\\\
&\approx\left|\cos\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q}^{*},\bm{\rho}_{m}\right)\right)\right|\cdot\left|\left|\bm{\rho}_{m}-\bm{\bar{\rho}}\right|\right|_{2}~{}\left(\forall
m\in\mathbb{M}\right).\end{split}$ (30)

By (LABEL:approxAoD), we aim to minimize the weighted $\ell_{2}$-deviation

$\left|\cos\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q}^{*},\bm{\rho}_{m}\right)\right)\right|\cdot\left|\left|\bm{\rho}_{m}-\bm{\bar{\rho}}\right|\right|_{2}$
(31)

for every $m\in\mathbb{M}$. Similar to (29), to find the Pareto-optimum of
$\left\\{g\left(\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)\right)\right\\}_{m\in\mathbb{M}}$,
we first find the phase align point $\bm{\bar{\rho}}^{*}$ assuming that every
RIS array is activated for backhauling every UAV-BS in $\mathbb{M}$ (i.e., a
full-array scenario), as illustrated in Fig. 4a. This is performed by solving
the following weighted Fermat-Torricelli problem:

$\displaystyle\underset{\bm{\bar{\rho}}}{\texttt{min}}~{}\sum_{m\in\mathbb{M}}w_{m}\left|\left|\bm{\rho}_{m}-\bm{\bar{\rho}}\right|\right|_{2}\left(\approx\left|\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)\right|\right),$
(32)

where
$w_{m}\triangleq\left|\cos\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q}^{*},\bm{\rho}_{m}\right)\right)\right|~{}\left(\forall
m\in\mathbb{M}\right)$. We can apply the weighted version of the Weiszfeld’s
algorithm given in Algorithm 1, which also converges to the solution of (32)
at the Pareto front [47].

######  Remark 4.

Note that the original Fermat-Torricelli problem can also be solved by letting
$w_{i}=1~{}\left(\forall i=1,\cdots,m\right)$ in Algorithm 1.

Algorithm 1 Weighted Version of the Weiszfeld’s Algorithm

1:procedure Minimizing the sum of $\ell_{2}$-deviation
Input: $\left\\{\mathbf{z}_{i}\right\\}_{i=1}^{m}$ (no three points are
colinear), $\epsilon_{0}\ll 1$,
$\left\\{w_{i}\right\\}_{i=1}^{m}\left(>0\right)$,
$\mathbf{x}_{0}\notin\left\\{\mathbf{z}_{i}\right\\}_{i=1}^{m}$

2: Define $\mathbb{I}=\left\\{1,\cdots,m\right\\}$ and
$W:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ by
$W\left(\mathbf{x}\right)=\frac{\sum_{i=1}^{m}w_{i}\frac{\mathbf{z}_{i}}{\left|\left|\mathbf{x}-\mathbf{z}_{i}\right|\right|_{2}}}{\sum_{i=1}^{m}w_{i}\frac{1}{\left|\left|\mathbf{x}-\mathbf{z}_{i}\right|\right|_{2}}}$
(33) For continuity, define
$W\left(\mathbf{z}_{i}\right)=\mathbf{z}_{i}\left(\forall
i\in\mathbb{I}\right)$
---

3: if $\left|\left|\sum_{j=1,j\neq
i}^{m}w_{j}\frac{\mathbf{z}_{j}-\mathbf{z}_{i}}{\left|\left|\mathbf{z}_{j}-\mathbf{z}_{i}\right|\right|_{2}}\right|\right|_{2}>w_{i}\left(\forall
i\in\mathbb{I}\right)$ then

4: Choose an initial point
$\mathbf{x}_{0}\in\mathbb{R}^{3}\backslash\left\\{\mathbf{z}_{i}\right\\}_{i=1}^{m}$

5: Set $k=0$

6: while
$\left|\left|\mathbf{x}_{k+1}-\mathbf{x}_{k}\right|\right|_{2}>\epsilon_{0}$
do

7: $\mathbf{x}_{k+1}=W\left(\mathbf{x}_{k}\right)$

8: $k=k+1$

9: end while

10: $\mathbf{x}^{*}=\mathbf{x}_{k}$

11: else $\left(\exists i~{}\mathrm{s.t.}\left|\left|\sum_{j=1,j\neq
i}^{m}w_{j}\frac{\mathbf{z}_{j}-\mathbf{z}_{i}}{\left|\left|\mathbf{z}_{j}-\mathbf{z}_{i}\right|\right|_{2}}\right|\right|_{2}\leq
w_{i}\right)$

12: $\mathbf{x}^{*}=\mathbf{z}_{i}$

13: end if

14:end procedure

15:Output: $\mathbf{x}^{*}$

After deriving $\bm{\bar{\rho}}^{*}$, if there is a UAV-BS that locates out of
the HPBW of $g$:

$\Delta\phi_{\max}\triangleq\max_{m\in\mathbb{M}}\left|\Delta\phi_{m}\left(\bm{\bar{\rho}}^{*}\right)\right|>\frac{\Delta\phi_{\mathrm{HPBW}}}{2},$
(34)

we adopt the sub-array scenario $\left(\bar{N}<N\right)$, where the RIS array
is divided into several sub-arrays and serve the UAV-BSs in each subset of
$\mathbb{M}$, instead of full-array and vice versa. It is because the deviated
UAV-BS has almost zero passive beamforming gain and significantly increases
the corresponding transmit power (see (19) and Fig. 2).

####  III-B1 Full-array scenario $\left(\bar{N}=N\right)$

For the full-array case, we use the phase align point $\bm{\bar{\rho}}^{*}$
derived by (32) and the RIS phase
$\left\\{\theta_{n}^{*}\left(\mathbf{q}^{*},\bm{\bar{\rho}}^{*}\right)\right\\}_{n=1}^{N}$
is derived by (LABEL:phase).

####  III-B2 Sub-array scenario $\left(\bar{N}<N\right)$

For the sub-array case, as illustrated in Fig. 4b, we divide the RIS array
into $L$ partitions with $\left\\{N_{i}\right\\}_{i=1}^{L}$ elements for given
$L$ and $\mathbb{M}$ into $\left\\{\mathbb{M}_{i}\right\\}_{i=1}^{L}$
corresponding to each $N_{i}$, where $\mathbb{M}_{i}$ is the set of UAV-BS
$m_{0}$ that corresponds to one of the $L$ equal partitions of the feasible
sin-AoD deviation range of $\mathbb{M}$:

$\Delta\phi_{m_{0}}\left(\bm{\bar{\rho}}^{*}\right)\in\frac{\Delta\phi_{\max}}{L}\left(2\left(i-1\right)-L,2i-L\right]\triangleq\mathbb{I}_{i}.$
(35)

Next, we have to determine $\left\\{N_{i}\right\\}_{i=1}^{L}$ that leads to
minimum transmit power. Assuming that $\bm{\rho}_{\mathrm{RIS}}$ is determined
in Section III.A, and every UAV-BS locates in the HPBW so that
$g\approx\bar{N}^{2}$ holds for every sub-array, the lower-bound of $P_{m}$ in
(21) can be approximated by

$\left(2^{\frac{M_{0}}{B_{\mathrm{b}}}C_{m}}-1\right)\frac{\sigma^{2}\left|\left|\bm{\rho}_{\mathrm{RIS}}-\bm{\rho}_{m}\right|\right|_{2}^{2}\left|\left|\bm{\rho}_{\mathrm{RIS}}\right|\right|_{2}^{2}}{G_{\mathrm{s}}\beta_{0}^{2}Mg\left(\Delta\phi_{m}\left(\bm{\bar{\rho}}\right)\right)}\approx\frac{A_{m}}{{\bar{N}}^{2}},$
(36)

where $A_{m}$ is a positive constant. Moreover, we should prevent the “HPBW-
outlier” in every sub-array that satisfies the following:

$\left|\Delta\phi_{m}\left(\bm{\bar{\rho}}_{i}^{*}\right)\right|>\frac{\Delta\phi_{\mathrm{HPBW}}\left(N_{i}\right)}{2}~{}\left(m\in\mathbb{M}_{i}\right),$
(37)

where $\bm{\bar{\rho}}_{i}^{*}$ is the phase align point of $i$th sub-array
which will be determined by maximizing
$\left\\{g\left(\Delta\phi_{m}\left(\bm{\bar{\rho}}_{i}^{*}\right)\right)\right\\}_{m\in\mathbb{M}_{i}}$
and described at the end of the section. By (35), the maximum sin-AoD
deviation
$\max_{m\in\mathbb{M}_{i}}\left|\Delta\phi_{m}\left(\bm{\bar{\rho}}_{i}^{*}\right)\right|$
of $\mathbb{M}_{i}$ is reduced by $L$ times [37], which is
$\left|\mathbb{I}_{i}\right|=\frac{\Delta\phi_{\max}}{L}$. Therefore, to
prevent the outlier, $N_{i}$ should satisfy the following:

$\begin{split}&\frac{\Delta\phi_{\max}}{L}\leq\frac{\Delta\phi_{\mathrm{HPBW}}\left(N_{i}\right)}{2}\\\
&~{}\rightarrow N_{i}\leq
L\frac{\Delta\phi_{\mathrm{HPBW}}/2}{\Delta\phi_{\max}}N\left(\triangleq
kN\right).\end{split}$ (38)

Hence, to minimize the sum of the lower-bound of $P_{m}$ by determining
$\left\\{N_{i}\right\\}_{i=1}^{L}$, we consider the following problem.

$\displaystyle\underset{\left\\{N_{i}\right\\}_{i=1}^{L}}{\texttt{min}}~{}\sum_{i=1}^{L}\sum_{m\in\mathbb{M}_{i}}\frac{A_{m}}{N_{i}^{2}}$
(39) $\displaystyle\text{~{}~{}s.t.}~{}\sum_{\ell=1}^{L}N_{\ell}=N,~{}0\leq
N_{i}\leq kN~{}\left(i=1,\cdots,L\right).$

For fixed $L$, Problem (39) is convex and therefore can be directly solved
[48]. The result is given by Theorem 3.

Algorithm 2 Proposed Aerial-RIS Setup Algorithm

1:procedure Source power minimization

2: Find $\mathbf{q}_{m}^{*}=\xi_{m}\mathbf{w}_{m}$ by (52) and determine
$\mathbf{q}^{*}$ by (29)

3: Determine $\bm{\bar{\rho}}^{*}$ by (32)

4: Compare $\frac{\Delta\phi_{\mathrm{HPBW}}}{2}$ and
$\Delta\phi_{\max}\triangleq\max_{m\in\mathbb{M}}|\Delta\phi_{m}(\boldsymbol{\bar{\rho}}^{*})|$

5: if $\frac{\Delta\phi_{\mathrm{HPBW}}}{2}>\Delta\phi_{\max}$ then

6: Apply full-array structure $\left(\bar{N}=N\right)$

7: Find the RIS phase
$\left\\{\theta_{n}^{*}\left(\mathbf{q}^{*},\bm{\bar{\rho}}^{*}\right)\right\\}_{n=1}^{N}$
by (LABEL:phase)

8: else (sub-array structure $\left(\bar{N}<N\right)$)

9: for $L=2$ to $L_{\max}$ do

10: Divide $\mathbb{M}$ into $\left\\{\mathbb{M}_{i}\right\\}_{i=1}^{L}$ based
on (35)
---

11: Divide the RIS array into $L$ sub-arrays by (40)

12: end for

13: Determine $L^{*}\in\left\\{2,\cdots,L_{\max}\right\\}$ by (41)
---

14: Round the solution
$\left\\{N_{i}^{*}\right\\}_{i=1}^{L^{*}}\rightarrow\left\\{\bar{N}_{i}^{*}\right\\}_{i=1}^{L^{*}}$

15: Evaluate $\left\\{\bm{\bar{\rho}}_{i}^{*}\right\\}_{i=1}^{L^{*}}$ by (42)

16: Find
$\left\\{\left\\{\theta_{n}^{*}\left(\mathbf{q}^{*},\bm{\bar{\rho}}_{i}^{*}\right)\right\\}_{n\in
i\textrm{th~{}sub-array}}\right\\}_{i=1}^{L^{*}}$ by (LABEL:phase)
---

17: end if

18: Determine $\left\\{P_{m}^{*}\right\\}_{m\in\mathbb{M}}$ by (43)

19: if $\sum_{m\in\mathbb{M}}P_{m}^{*}>G_{\mathrm{s}}^{-1}P_{\mathrm{max}}$
then

20: Break and declare infeasibility

21: end if

22:end procedure

######  Theorem 3.

The solution of Problem (39) is given by

$N_{i}^{*}=\min\left(kN,\sqrt[3]{\frac{\sum_{m\in\mathbb{M}_{i}}2A_{m}}{\mu}}\right)\left(i=1,\cdots,L\right),$
(40)

where $\mu$ is chosen such that $\sum_{\ell=1}^{L}N_{\ell}^{*}=N$ is met.

###### Proof.

See Appendix B. ∎

Intuitively, we set initial $N_{i}^{*}$ by
$\sqrt[3]{\frac{\sum_{m\in\mathbb{M}_{i}}2A_{m}}{\mu}}$, but if the value
exceeds the threshold $kN$, we automatically restrict $N_{i}^{*}$ by
$N_{i}^{*}=kN$. We repeat the procedure for
$L\in\left\\{2,\cdots,L_{\max}\right\\}$ for positive integer
$L_{\max}\left(\geq 2\right)$, and find $L^{*}$ that leads to minimum
objective function in (39) via one-dimensional search over
$\left\\{2,\cdots,L_{\max}\right\\}$, that is,

$L^{*}\triangleq\operatorname*{\arg\min}_{L\in\left\\{2,\cdots,L_{\max}\right\\}}\sum_{i=1}^{L}\sum_{m\in\mathbb{M}_{i}}\frac{A_{m}}{N_{i}^{*2}}.$
(41)

Finally, since the number of RIS elements is an integer, we can round the
result to give the integer solution
$\left\\{\bar{N}_{i}^{*}\right\\}_{i=1}^{L^{*}}$. The proposed array-partition
is given in Fig. 5. The proposed partition can be considered as the “reversed”
version of the waterfilling algorithm [49, 50], where the role between the
vessel and the water is changed.

For phase align point $\left\\{\bm{\bar{\rho}}_{i}^{*}\right\\}_{i=1}^{L^{*}}$
of sub-arrays corresponding to
$\left\\{\mathbb{M}_{i}\right\\}_{i=1}^{L^{*}}$, by following
(LABEL:approxAoD) and (32), we have to solve the following weighted Fermat-
Torricelli problem for every $i\in\left\\{1,\cdots,L^{*}\right\\}$.

$\displaystyle\underset{\bm{\bar{\rho}}_{i}}{\texttt{min}}~{}\sum_{m\in\mathbb{M}_{i}}w_{m}\left|\left|\bm{\rho}_{m}-\bm{\bar{\rho}}_{i}\right|\right|_{2}~{}\left(i=1,\cdots,L^{*}\right).$
(42)

Note that
$w_{m}=\left|\cos\left(\phi_{\mathrm{t,RIS}}\left(\mathbf{q}^{*},\bm{\rho}_{m}\right)\right)\right|$
is independent of $\bm{\bar{\rho}}$. By (42), we can find the suboptimal
solution $\left\\{\bm{\bar{\rho}}_{i}^{*}\right\\}_{i=1}^{L^{*}}$ for each
sub-array and their corresponding RIS phases
$\left\\{\left\\{\theta_{n}^{*}\left(\mathbf{q}^{*},\bm{\bar{\rho}}_{i}^{*}\right)\right\\}_{n\in
i\textrm{th~{}sub-array}}\right\\}_{i=1}^{L^{*}}$ by (LABEL:phase).

Figure 5: RIS array-partition strategy for sub-array scenario. We restrict the length of the sub-array by $kN$, and hence the lighter region of the graph is ignored in the final result $\left\\{N_{i}^{*}\right\\}_{i=1}^{L}$. TABLE II: Complexity of the Proposed Algorithm Section III.A | Section III.B | Section III.C
---|---|---
Parameter | Complexity | Parameter | Complexity | Parameter | Complexity
$\left\\{\xi_{m}\right\\}_{m\in\mathbb{M}}$ | $\mathcal{O}(M_{0})$ | $\bm{\bar{\rho}}^{*}$ | $\leq\mathcal{O}\left(I_{\mathbb{M}}M_{0}+M_{0}^{2}\right)$ | $\left\\{P_{m}^{*}\right\\}_{m\in\mathbb{M}}$ | $\mathcal{O}\left(M_{0}\right)$
$\left\\{N_{i}^{*}\right\\}_{i=1}^{L},~{}\sum_{i=1}^{L}\sum_{m\in\mathbb{M}}\frac{A_{m}}{N_{i}^{*2}}$ | $\mathcal{O}\left(M_{0}\right)$
$L^{*}$ | $\mathcal{O}\left(L_{\max}M_{0}\right)$
$\mathbf{q}^{*}$ | $\leq\mathcal{O}\left(I_{\mathbb{M}}M_{0}\right)$ | $\left\\{N_{i}^{*}\right\\}_{i=1}^{L^{*}}\rightarrow\left\\{\bar{N}_{i}^{*}\right\\}_{i=1}^{L^{*}}$ | $\mathcal{O}\left(L^{*}\right)$
$\left\\{\bm{\bar{\rho}}_{i}^{*}\right\\}_{i=1}^{L^{*}}$ | $\leq\mathcal{O}\left(I_{L^{*}}M_{0}+M_{0}^{2}\right)$
$\bm{\Theta}\triangleq\mathrm{diag}\left(\left\\{e^{j\theta_{n}}\right\\}_{n=1}^{N}\right)\left(\because\forall\alpha_{n}=1\right)$ | $\mathcal{O}\left(N\right)$
TABLE III: Simulation Parameters

Parameter |  Description |  Value
---|---|---
$B_{\mathrm{b}}$ |  Bandwidth of the backhaul link |
(unless referred) |  50 (MHz) |
$M_{0}$ |  Number of UAV-BSs |  $\gtrapprox 4$ [11]
$\beta_{0}$ |  Reference path loss |
for sub-6 GHz backhaul |  -43.32 (dB) |
$\mathcal{G}$ |  Targeted urban region |  $500\times 500$ (m)
$\bm{\rho}_{\mathcal{G}}$ |  Center of $\mathcal{G}$ (unless referred) |  $[1000~{}0]^{\mathrm{T}}$ (m)
$\delta$ |  Upper-bound of restricting $\mathbf{q}_{m}^{*}$ |  $10^{-1}$
$P_{\max}$ |  Feasible threshold of |
source transmit power |  30 (dBW) |
$G_{\max}$ |  Maximum directional gain |  8 (dB)
$\left(\mathrm{SLA_{v}},A_{\max}\right)$ |  Vertical side-lobe attenuation and maximum attenuation |  30 (dB) [51]
$H$ |  Height of aerial-RIS |
(unless referred) |  150 (m) |
$N$ |  Number of RIS elements |
(unless referred) |  300 |
$M$ |  Number of source antennas |  16
$L_{\max}$ |  Upper-bound of |
one-dimensional search |  5 |
$\left(d_{\mathrm{s}},d_{\mathrm{RIS}}\right)$ |  Source antenna and |
RIS element separations |  $\left(\frac{\lambda}{2},\frac{\lambda}{10}\right)$ [24] |

###  III-C Determining $\left\\{P_{m}\right\\}_{m\in\mathbb{M}}$

Since we determine the placement of aerial-RIS and the phase align
point/points in Sections III.A and B, we apply the result to the constraints
in (21). In turn, Problem (21) becomes linear programming with respect to
$\left\\{P_{m}\right\\}_{m\in\mathbb{M}}$. The feasible solution satisfies the
equality of the lower-bound of the problem for every $m\in\mathbb{M}$. It is
given by

$P_{m}^{*}=\left(2^{\frac{M_{0}}{B_{\mathrm{b}}}C_{m}}-1\right)\frac{\sigma^{2}\left|\left|\bm{\rho}_{\mathrm{RIS}}^{*}-\bm{\rho}_{m}\right|\right|_{2}^{2}\left|\left|\bm{\rho}_{\mathrm{RIS}}^{*}\right|\right|_{2}^{2}}{G_{\mathrm{s}}\beta_{0}^{2}Mg\left(\Delta\phi_{m}^{*}\right)}~{}\left(\forall
m\in\mathbb{M}\right),$ (43)

where
$\bm{\rho}_{\mathrm{RIS}}^{*}=\left[\mathbf{q}^{*\mathrm{T}}~{}H\right]^{\mathrm{T}}$
and $\Delta\phi_{m}^{*}$ is defined by

$\Delta\phi_{m}^{*}\triangleq\begin{cases}\Delta\phi_{m}\left(\bm{\bar{\rho}}^{*}\right)&\left(\text{full-
array scenario}\right)\\\
\Delta\phi_{m}\left(\bm{\bar{\rho}}_{i}^{*}\right)&\left(\text{sub-array
scenario},m\in\mathbb{M}_{i}\right).\end{cases}$ (44)

The complete algorithm is summarized in Algorithm 2.

######  Remark 5.

Although the power is guaranteed to be positive in Remark 1, no strict
feasibility claim can be made since we cannot always guarantee
$\sum_{m\in\mathbb{M}}P_{m}^{*}\leq G_{\mathrm{s}}^{-1}P_{\mathrm{max}}$ by
(43). In Section IV, the practical transmit power given by the proposed
algorithm is way less (below 40$\sim$45 dBm) than the feasible threshold (the
feasibility rate is 100$\%$ for our algorithm), and this ensures the almost-
sure feasibility. Therefore, we can conclude that the consideration of the
infeasibility scenario is negligible.

###  III-D Complexity Analysis

The computational complexity of the proposed algorithm consists of three
parts. In Section III.A, we first choose
$\left\\{\xi_{m}\right\\}_{m\in\mathbb{M}}$ by (52), which requires a
complexity of $\mathcal{O}\left(M_{0}\right)$. Next, we determine
$\mathbf{q}^{*}$ by (29) and Algorithm 1 with complexity upper-bounded by
$\mathcal{O}\left(I_{\mathbb{M}}M_{0}\right)$, where $I_{\left(\cdot\right)}$
is the number of iterations from procedure 4$\sim$7 in the Weiszfeld’s
algorithm for given set. In Section III.B, we first find $\bm{\bar{\rho}}^{*}$
by (32) with complexity upper-bounded by
$\mathcal{O}\left(I_{\mathbb{M}}M_{0}+M_{0}^{2}\right)$. For full-array
scenarios, we align the RIS phases by (LABEL:phase), which requires a
complexity of $\mathcal{O}\left(N\right)$. For sub-array scenarios, we find
$\left\\{N_{i}^{*}\right\\}_{i=1}^{L}$ for given $L$ by (40) with a complexity
of
$\sum_{i=1}^{L}\mathcal{O}\left(\left|\mathbb{M}_{i}\right|\right)=\mathcal{O}\left(M_{0}\right)$,
and compute the objective function
$\sum_{i=1}^{L}\sum_{m\in\mathbb{M}_{i}}\frac{A_{m}}{N_{i}^{*2}}$, which also
has a complexity of
$\sum_{i=1}^{L}\mathcal{O}\left(\left|\mathbb{M}_{i}\right|\right)=\mathcal{O}\left(M_{0}\right)$.
We repeat the procedure for $L=2,\cdots,L_{\max}$ and search $L^{*}$ by (41),
which has a complexity of $\mathcal{O}\left(L_{\max}M_{0}\right)$. We finally
round the solution $\left\\{N_{i}^{*}\right\\}_{i=1}^{L^{*}}$ with a
complexity of $\mathcal{O}\left(L^{*}\right)$. Next, we find the phase align
point $\left\\{\bm{\bar{\rho}}_{i}^{*}\right\\}_{i=1}^{L^{*}}$ by (42) with
complexity upper-bounded by
$\sum_{i=1}^{L^{*}}\mathcal{O}\left(I_{\mathbb{M}_{i}}\left|\mathbb{M}_{i}\right|+\left|\mathbb{M}_{i}\right|^{2}\right)\leq\mathcal{O}\left(I_{L^{*}}M_{0}+M_{0}^{2}\right)$,
where
$I_{L^{*}}\triangleq\max_{i\in\left\\{1,\cdots,L^{*}\right\\}}I_{\mathbb{M}_{i}}$.
Then, we align the RIS phases by (LABEL:phase), which has a complexity of
$\mathcal{O}\left(N\right)$. Finally, in Section III.C, we determine
$\left\\{P_{m}^{*}\right\\}_{m\in\mathbb{M}}$ by (43), which requires a
complexity of $\mathcal{O}\left(M_{0}\right)$.

Hence, the total complexity is upper-bounded by

$\begin{split}&\mathcal{O}\left(\left(I_{\mathbb{M}}+L_{\max}+I_{L^{*}}+M_{0}\right)M_{0}+L^{*}+N\right)\\\
&\approx\mathcal{O}\left(\left(I_{\mathbb{M}}+I_{L^{*}}+M_{0}\right)M_{0}+N\right)~{}\left(\because
I_{\left(\cdot\right)}>L_{\max}\geq L^{*}\right),\end{split}$ (45)

which does not exceed the quadratic order. We can therefore conclude that our
proposed algorithm is not only energy-efficient but also complexity-efficient.
The complexity of each step of the proposed algorithm is summarized in Table
II.

Figure 6: Simulated aerial-RIS setup comprising of a $N$-element RIS and
$M_{0}$ UAV-BSs.

##  IV Simulation Results

###  IV-A Simulation Setup

For numerical analysis, we considered $10^{3}$ independent random user
distributions and corresponding UAV-BSs [11] in $\mathcal{G}$ and assumed 2
GHz and sub-6 GHz channels for the fronthaul and the backhaul link,
respectively [51]. We assumed that the directional antennas installed on
source are equipped with the pattern in [51]. That is, the vertical and
horizontal attenuations of the directional antenna are given by

$\begin{split}\begin{cases}A_{\mathrm{v}}\left(\theta\right)=\min\left(12\left(\frac{\theta-90^{\circ}}{\theta_{\mathrm{H}}}\right)^{2},~{}\mathrm{SLA}_{\mathrm{v}}\right),\\\
A_{\mathrm{h}}\left(\phi\right)=\min\left(12\left(\frac{\phi}{\phi_{\mathrm{H}}}\right)^{2},~{}A_{\max}\right),\end{cases}\end{split}$
(46)

where $\theta\in\left[0^{\circ},180^{\circ}\right]$ and
$\phi\in\left[-180^{\circ},180^{\circ}\right)$ are the ranges of vertical and
horizontal angle, respectively, $\theta_{\mathrm{H}}$ and $\phi_{\mathrm{H}}$
are vertical and horizontal HPBWs of the antenna pattern, respectively,
$\mathrm{SLA}_{\mathrm{v}}$ is vertical side-lobe attenuation, and $A_{\max}$
is maximum attenuation. Hence, the antenna gain
$G_{\mathrm{s}}\left(\theta,\phi\right)$ is obtained as

$G_{\mathrm{s}}\left(\theta,\phi\right)=G_{\max}-\min\left(A_{\mathrm{v}}\left(\theta\right)+A_{\mathrm{h}}\left(\phi\right),A_{\max}\right),$
(47)

where $G_{\max}$ is a maximum directional gain.

For fair comparison, we considered the benchmark schemes with
$(\mathbf{q},\bar{\bm{\rho}})$ given by
$\left(\frac{1}{2}\bm{\rho}_{\mathcal{G}},\bar{\bm{\rho}}^{*}~{}\textrm{(full-
array)}~{}\textrm{or}~{}\left\\{\bar{\bm{\rho}}_{i}^{*}\right\\}_{i=1}^{L}~{}\textrm{(sub-
array)}\right)$ and $\left(\mathbf{0},{\bm{\rho}}_{\mathcal{G}}\right)$,
respectively. The former benchmark assumes the same RIS-partition structure as
our algorithm. We also considered the conventional terrestrial-based backhaul
link where the ground source directly transmits the backhaul signal to the
UAV-BSs based on the LoS probability and channel parameters given in [52].

Moreover, we analyzed the optimality of our algorithm based on exhaustive
search. Since it is infeasible to search every possible
$\left\\{\bar{N}\right\\}$, $\mathbf{q}$, and $\bm{\rho}$ that reach the
global optimum of (21), we conducted exhaustive subspace search with respect
to: 1) full-array structure $\left(\bar{N}=N\right)$, 2)
$\mathbf{q}\in\left\\{\mathbf{q}\in\mathbb{R}^{2}:\left|\left|\mathbf{q}\right|\right|_{2}\leq\delta\cdot\max_{m\in\mathbb{M}}\left|\left|\mathbf{w}_{m}\right|\right|_{2}\right\\}\cap\left\\{\left(x,y\right):x\geq
0\right\\}$ (based on (22)), 3)
$\bm{\rho}\in\mathcal{C}\cap\left\\{\left(x,y,z\right):z\geq 0\right\\}$,
where $\mathcal{C}$ is the smallest 3D cube for which every edge is parallel
to x, y, or z-axis and contains every
$\left\\{\bm{\rho}_{m}\right\\}_{m\in\mathbb{M}}$. Since adopting a full-array
structure, placing aerial-RIS near the backhaul source, and setting the phase
align point close to every UAV-BS lead to better performance, we believe that
our subspace search indeed leads to a reliable suboptimal solution. The
simulation environment based on the parameters is illustrated in Fig. 6, and
the detailed parameters are given in Table III-B2.

Figure 7: Behavior of $\xi_{m}$ with $h_{m}=45~{}\textrm{m}$ with respect to the height of aerial-RIS and the source-UAV 2D distance. Figure 8: CDF of the absolute value of the sin-AoD deviations with the phase align point $\bm{\bar{\rho}}^{*}$ $\left(N=400\right)$. TABLE IV: Empirical Probability of $\left|\Delta\phi_{m}(\bm{\bar{\rho}}^{*})\right|>\frac{\Delta\phi_{\mathrm{HPBW}}}{2}$ in Fig. 8 Scenario |  Probability
---|---
Proposed aerial-RIS setup |  $\mathbf{7.50\times 10^{-3}}$
RIS at $\frac{1}{2}\rho_{\mathcal{G}},\bm{\bar{\rho}}=\bm{\bar{\rho}}^{*}~{}\textrm{or}~{}\left\\{\bm{\bar{\rho}}_{i}^{*}\right\\}_{i=1}^{L}$ |  $7.22\times 10^{-1}$
RIS at the origin, $\bm{\bar{\rho}}=\bm{\bar{\rho}}_{\mathcal{G}}$ |  $2.24\times 10^{-2}$
Figure 9: Array-partition result with $N=400$ and the corresponding transmit
power for each number of partitions. Figure 10: Transmit power with respect to
the bandwidth of the backhaul link.

###  IV-B Reliability of Aerial-RIS Placement and Phase Alignment

Fig. 7 shows the behavior of $\xi_{m}$ according to the height of aerial-RIS
$\left(H\right)$ and the 2D distance of the UAV-BS from the source
$\left(\left|\left|\mathbf{w}_{m}\right|\right|_{2}\right)$. It is shown that
$\xi_{m}$ has an order of $10^{-2}$ for sufficiently large
$\left|\left|\mathbf{w}_{m}\right|\right|_{2}$, which implies that
$\mathbf{q}_{m}^{*}$ is extremely close to the origin compared to
$\mathbf{w}_{m}$. Besides, $\xi_{m}$ increases according to $H$. It is because
the increase of $H$ leads to larger $a$ and smaller
$\sqrt{-\frac{a}{3}}~{}\left(a<0\right)$. Furthermore, because the posterior
term of the $\sqrt{-\frac{a}{3}}$ in (LABEL:xixixixi), which is also
approximated by
$\left(-1+\frac{\epsilon}{3\sqrt{3}}\right)~{}\left(|\epsilon|\ll 1\right)$ in
(53), is negative, we can interpret that the decrease in the term
$\sqrt{-\frac{a}{3}}$ significantly affects the increase of $\xi_{m}$. It also
coincides with Remark 2, where $\xi_{m}$ increases and converges to
$\frac{1}{2}$ when
$H\rightarrow\frac{1}{2}\left|\left|\mathbf{w}_{m}\right|\right|_{2}$.
Moreover, as $\xi_{m}\ll 1$, $\Delta\phi_{m}\left(\cdot\right)$ stays nearly
the same as the $\mathbf{q}_{m}^{*}=\mathbf{0}$ case, where the numerically
evaluated difference is given by less than $10^{-4}$, and thus negligible for
the same environment. Nevertheless, the proposed method leads to energy-
efficiency enhancement compared to the case of simply letting
$\mathbf{q}_{m}^{*}=\mathbf{0}$.

Fig. 8 illustrates the cumulative distribution function (CDF) of the absolute
value of the sin-AoD deviations between the UAV-BS and the phase align point
$\bm{\bar{\rho}}^{*}$ with $N=400$. We can apply the full-array RIS structure,
which leads to the maximum energy-efficiency, when every absolute value of
sin-AoD deviation is less than
$\frac{\Delta\phi_{\mathrm{HPBW}}}{2}\approx\frac{1}{2}\frac{0.8858}{400*0.1}\approx
0.011$. As shown from the CDF plot, the proposed algorithm outperforms every
benchmark distributed in “$\leq\frac{\Delta\phi_{\mathrm{HPBW}}}{2}$” region
colored in blue for almost 100% of probability, where for the worst case of
the benchmarks, only about 30% of them are contained in the region. For more
accurate analysis, we have also computed the empirical probability that the
absolute value exceeds $\frac{\Delta\phi_{\mathrm{HPBW}}}{2}$ based on the CDF
and organized in Table IV. As we have conducted the numerical simulation for
$10^{3}$ independent user distributions with more than 8,000 UAV-BSs in total,
from the result, we can conclude that the excess of the absolute value occurs
for about 60 UAV-BSs by the proposed algorithm, which implies that the
proposed algorithm guarantees the full-array scenario almost completely. As
for the benchmarks, the value is larger than the proposed algorithm in every
case. For the worst scenario, the probability of excess exceeds 0.5.
Therefore, it is forced to apply the sub-array architecture and leads to the
loss of energy-efficiency.

###  IV-C Transmit Power Comparisons under Several Effects

Fig. 9 illustrates the array-partition result
$\left(\left\\{N_{i}^{*}\right\\}_{i=1}^{L}\right)$ with $N=400$ and the
transmit power of the source corresponding to each number of partitions. We
considered the number of partition $L$ by $L=2,\cdots,L_{\max}\left(=5\right)$
and compute the optimal array-partition for each $L$ by (40). We can find that
the upper-bound of $\left\\{N_{i}^{*}\right\\}_{i=1}^{L}\left(kN\right)$ is
meaningful at $L=2$ case, because the increase of $L$ leads to the rise of
$k=L\frac{\Delta\phi_{\mathrm{HPBW}}/2}{\Delta\phi_{\max}}$ in (LABEL:constN),
reducing the significance of the upper-bound $kN$. Additionally, It is
observed that the transmit power increases according to the number of array-
partitions. The result agrees with our derived result in that, the array-
partition leads to the decrease of the peak gain of $g$ and the increase of
the transmit power. Moreover, by searching $L^{*}$, the proposed algorithm can
achieve even less transmit power for approximately 0.5 dB compared to the
benchmark with the same $N=400$ case, which is given by 48.15 dBm.

Fig. 10 shows the transmit power according to the bandwidth of the backhaul
link $\left(B_{\mathrm{b}}\right)$. It is clear that when the bandwidth
increases, by (5), we can achieve the same backhaul rate
$\left\\{R_{m}\right\\}_{m\in\mathbb{M}}$ with less transmit power, thereby
increasing energy-efficiency. In fact, due to the efficient setup of the
aerial-RIS by the proposed algorithm, the transmit power decreases further
than the benchmarks, where the gain is up to 12 dB for every scenario with
RIS. Moreover, the gain is up to 44 dB for terrestrial backhaul scenario
without RIS, which suffers from extreme path loss due to high NLoS probability
[5, 52] and leads to infeasible transmit power.

Fig. 11 compares the transmit power according to the number of RIS elements
$\left(N\right)$. Here, we considered the upper-bound of $N$ due to the area
limitation of the aerial platform and the high possibility of stability loss
of the aerial platform that carries a vast number of RIS elements [8]. As
shown in the figure, the proposed method significantly reduces the transmit
power compared to the benchmarks, where the gain is up to 10$\sim$13 dB for
RIS-equipped cases and up to 32 dB for the terrestrial backhaul scenario,
which suffers from high NLoS component. When $N$ goes significantly higher, in
fact, the HPBW is reduced by the order of $N^{-1}$, which leads to the
decrease of the probability of using the full-array structure and consequently
causes the increase of a transmit power. Nevertheless, even for large $N$, our
algorithm guarantees reliable transmit power, whereas for the benchmarks the
power continues to increase. We can therefore conclude that for the feasible
range of $N$ that we have considered, the performance of our algorithm
monotonically increases with the number of RIS elements.

Fig. 12 shows the transmit power according to the distance of
$\bm{\rho}_{\mathcal{G}}$ from the source $\left(d_{\mathcal{G}}\right)$. It
is clear that by applying the proposed algorithm, we can achieve an additional
reduction of transmit power compared to the benchmarks. Even for low
$d_{\mathcal{G}}$, our algorithm guarantees reliable transmit power, where for
RIS-equipped benchmarks the power increases due to the larger possibility of
using sub-array structure or the “HPBW-outlier” by simply letting
$\bm{\bar{\rho}}=\bm{\rho}_{\mathcal{G}}$.

Figure 11: Transmit power with respect to the number of RIS elements.

Fig. 13 illustrates the transmit power according to feasible $H$ with the
guarantee of high LoS probability [5, 11]. Clearly, by applying our algorithm,
we can reduce the transmit power by 6$\sim$7 dB compared to the RIS-equipped
benchmark algorithms, and by about 23 dB compared to the terrestrial backhaul
solution. Although the power increases for high $H$ owing to the increase of
path loss, our algorithm can maintain a reliable level of transmit power
comparable with the low-$H$ scenario. The power growth becomes larger for the
benchmarks, resulting in lower energy-efficiencies. Hence, we can conclude
that we should choose sufficiently low $H$ with the consideration of the
height of skyscrapers in the urban area and the maintenance of the high
probability of LoS links.

Fig. 14 shows a comparison between the solution obtained by the proposed
algorithm and the suboptimal solution obtained from the exhaustive subspace
search according to $d_{\mathcal{G}}$ and two different values of $H$.
Evidently, owing to the alternative optimization and considered
approximations, the proposed solution has a slight error with the suboptimal
solution. However, the maximal optimality gap of the simulation is about
1$\%$, which is negligible. Therefore, we can deduce that there is no
significant impact on the performance of energy-efficiency.

Figure 12: Transmit power with respect to the distance of
$\bm{\rho}_{\mathcal{G}}$ from the source.

##  V Conclusion

In this paper, we have proposed an aerial-RIS backhaul architecture that can
support UAV-BSs with a high energy-efficiency. We have derived the lower-bound
on source transmit power for backhauling UAV-BSs for a given data rate and
minimized the lower-bound. Through balancing the beamforming gain of aerial-
RIS and the product of squared distances between multiple UAV-BSs by
minimizing the sum of $\ell_{2}$-deviations and considering the proposed
“reverse” waterfilling array-partition, we determined the placement of aerial-
RIS and the suboptimal phase align point/points that significantly enhance
energy-efficiency. Additionally, the total computational complexity does not
exceed the quadratic order, which also guarantees the efficiency of
computation. Simulation results confirm that, compared to the benchmark
schemes, the proposed method can achieve additional energy-efficiency by
ensuring a source-nearby aerial-RIS placement and consequently a high
probability of selecting the full-array RIS structure. We anticipate our
analysis providing a deep insight for successfully integrating RIS with high-
altitude platform and the performance analysis concerning several parameters,
including the 3D placement and the phase align point/points of aerial-RIS.

For future work, our research can be broadened by extending the RIS structure,
i.e., the UPA structure [38, 53], and considering the physical and hardware
characteristics of the RIS dependent on the size and number of the RIS
elements [24, 54, 26, 55]. Furthermore, one can design the aerial-RIS
trajectory to adapt to moving UAV-BSs under the mobility scenario [36, 56, 57,
58]. It is also essential to study other factors that affect the energy-
efficiency, such as the battery consumption of the aerial platforms [8, 59],
assistance of the multiple RIS deployment including the terrestrial RIS [60,
61], and the power dissipation in the RIS hardware [27, 62]. Moreover, the
work on the methodology of RIS installation on the aerial platform can be
performed [63], which is still in its infancy. Comparative research of our
proposed aerial-RIS setup algorithm and other relaying methods, including
amplify-and-forward (AF) [27] and decode-and-forward (DF) architectures [64,
62, 27] can be taken into account. Furthermore, to confront the higher-data-
rate fronthaul/backhaul links for the future wireless network [65, 66, 67, 39,
68, 69, 70, 71, 72], the application of signals with a higher spectrum can be
contemplated for our proposed aerial-RIS backhaul system [56, 73, 74],
followed by the software simulation and prototype measurement of its
performance and reliability [75, 76, 77].

Figure 13: Transmit power with respect to the height of aerial-RIS. Figure 14:
The optimality validation of the proposed solution with respect to
$d_{\mathcal{G}}$ for $H=150,200~{}\textrm{m}$.

## Acknowledgement

The authors would like to thank Prof. Robert W. Heath Jr., Dr. Min Soo Sim and
Mr. Yonghwi Kim for their helpful discussions.

Figure 15: The orthogonal projection of $\mathbf{q}_{m}$ and illustration of
(49) in Theorem 2.

## Appendix A

## Proof of Theorem 2

Let the orthogonal projection of $\mathbf{q}_{m}$ onto the line segment that
links the source in the origin and $\mathbf{w}_{m}$ be $\mathbf{\bar{q}}_{m}$.
Then, by the definition of the orthogonal projection, the following holds:

$\begin{split}\begin{cases}\left|\left|\mathbf{q}_{m}-\mathbf{w}_{m}\right|\right|_{2}^{2}=\left|\left|\mathbf{q}_{m}-\mathbf{\bar{q}}_{m}\right|\right|_{2}^{2}+\left|\left|\mathbf{\bar{q}}_{m}-\mathbf{w}_{m}\right|\right|_{2}^{2},\\\
\left|\left|\mathbf{q}_{m}\right|\right|_{2}^{2}=\left|\left|\mathbf{q}_{m}-\mathbf{\bar{q}}_{m}\right|\right|_{2}^{2}+\left|\left|\mathbf{\bar{q}}_{m}\right|\right|_{2}^{2}.\end{cases}\end{split}$
(48)

Here, to minimize the objective function of (22), we need to minimize the
left-hand side of both equations in (48). Hence, we have to choose
$\mathbf{q}_{m}$ that satisfies the following:

$\left|\left|\mathbf{q}_{m}-\mathbf{\bar{q}}_{m}\right|\right|_{2}^{2}=0~{}\leftrightarrow~{}\mathbf{q}_{m}=\xi_{m}\mathbf{w}_{m}~{}\left(\xi_{m}>0\right),$
(49)

which is clarified in Fig. 15. Hence, the solution $\mathbf{q}_{m}^{*}$ of the
problem is given by $\mathbf{q}_{m}^{*}=\xi_{m}\mathbf{w}_{m}$, and by
substituting we can denote the objective function by $f\left(\xi_{m}\right)$,
that is,

$f\left(\xi_{m}\right)\triangleq\left|\left|\mathbf{w}_{m}\right|\right|_{2}^{4}\left(\xi_{m}^{2}+\zeta_{1}^{2}\right)\left(\left(1-\xi_{m}\right)^{2}+\zeta_{2}^{2}\right),$
(50)

where $\zeta_{1}=\frac{H}{\left|\left|\mathbf{w}_{m}\right|\right|_{2}}$ and
$\zeta_{2}=\frac{\left|H-h_{m}\right|}{\left|\left|\mathbf{w}_{m}\right|\right|_{2}}$.
To find the minimum of $f\left(\xi_{m}\right)$, we should find the root of
$f^{\prime}\left(\xi_{m}\right)$. The discriminant $\Delta$ of the cubic
equation $f^{\prime}\left(\xi_{m}\right)=0$ is given by [78]

$\Delta=\left(\frac{a}{3}\right)^{3}+\left(\frac{b}{2}\right)^{2},$ (51)

where $a=\frac{1}{2}\left(\zeta_{1}^{2}+\zeta_{2}^{2}\right)-\frac{1}{4}$ and
$b=\frac{1}{4}\left(\zeta_{2}^{2}-\zeta_{1}^{2}\right)$. Because we assume
that $d_{\mathcal{G}}$ is sufficiently large, it is reasonable to assume that
$\left|\left|\mathbf{w}_{m}\right|\right|_{2}$ follows the same scale, which
leads to $a<0\left(\approx-\frac{1}{4}\right)$ and $|b|\ll 1$. In this case,
$\Delta<0$ holds and the three real solutions
$\left\\{\xi_{m,k}\right\\}_{k=0}^{2}$ of $f^{\prime}\left(\xi_{m}\right)=0$
are given by [78]

$\begin{split}\xi_{m,k}=\frac{1}{2}+2\sqrt{-\frac{a}{3}}\cos\left(\frac{1}{3}\cos^{-1}\left(\frac{3b}{2a}\sqrt{-\frac{3}{a}}\right)-\frac{2}{3}\pi
k\right)&\\\ \left(k=0,1,2\right)&.\end{split}$ (52)

As $a\approx-\frac{1}{4}$ and $|b|\ll 1$, by letting
$\frac{3b}{2a}\sqrt{-\frac{3}{a}}=\epsilon~{}\left(|\epsilon|\ll 1\right)$ and
successively applying the first-order Taylor approximation to (52), it becomes

$\begin{split}\begin{cases}\xi_{m,0}\approx\frac{1}{2}+\sqrt{-a}\left(1+\frac{\epsilon}{3\sqrt{3}}\right),\\\
\xi_{m,1}\approx\frac{1}{2}+\sqrt{-a}\left(-\frac{\epsilon}{3\sqrt{3}}\right),\\\
\xi_{m,2}\approx\frac{1}{2}+\sqrt{-a}\left(-1+\frac{\epsilon}{3\sqrt{3}}\right).\end{cases}\end{split}$
(53)

Moreover, by substituting
$a=\frac{1}{2}\left(\zeta_{1}^{2}+\zeta_{2}^{2}\right)-\frac{1}{4}$ into (53)
we obtain

$\begin{split}\begin{cases}\xi_{m,0}\approx\frac{1}{2}-\sqrt{\frac{1}{4}-\frac{1}{2}\left(\zeta_{1}^{2}+\zeta_{2}^{2}\right)}\left(-1-\frac{\epsilon}{3\sqrt{3}}\right),\\\
\xi_{m,1}\approx\frac{1}{2}-\sqrt{\frac{1}{4}-\frac{1}{2}\left(\zeta_{1}^{2}+\zeta_{2}^{2}\right)}\left(\frac{\epsilon}{3\sqrt{3}}\right),\\\
\xi_{m,2}\approx\frac{1}{2}-\sqrt{\frac{1}{4}-\frac{1}{2}\left(\zeta_{1}^{2}+\zeta_{2}^{2}\right)}\left(1-\frac{\epsilon}{3\sqrt{3}}\right).\\\
\end{cases}\end{split}$ (54)

From (54), we can obtain the following outcomes:

  1. 1.

It is clear that $\xi_{m,0}>\xi_{m,1}>\xi_{m,2}$ for small $|\epsilon|$. Owing
to the characteristics of the quartic equation [78], $f$ achieves its local
minimum at $\xi_{m,0}$ and $\xi_{m,2}$, and either of two achieves the global
minimum.

  2. 2.

Since $|\epsilon|\ll 1$, the following holds for $\xi_{m,2}$:

$\xi_{m,2}\approx\frac{1}{2}-\sqrt{\frac{1}{4}-\frac{1}{2}\left(\zeta_{1}^{2}+\zeta_{2}^{2}\right)}\left(1\right)=\frac{\frac{1}{2}\left(\zeta_{1}^{2}+\zeta_{2}^{2}\right)}{\sqrt{\frac{1}{4}+\frac{1}{2}\left(\zeta_{1}^{2}+\zeta_{2}^{2}\right)}},$
(55)

which implies that $\xi_{m,2}$ is greater than 0 for small $|\epsilon|$.
Moreover, since $\zeta_{1},\zeta_{2}$ is sufficiently small
($\because\left|\left|\mathbf{w}_{m}\right|\right|_{2}$ is sufficiently
large), by (55) it can be deduced that $\xi_{m,2}$ is close to the origin.

From above results, we should choose $\xi_{m}$ as

$\xi_{m}\triangleq\xi_{m,2}=\frac{1}{2}+2\sqrt{-\frac{a}{3}}\cos\left(\frac{1}{3}\cos^{-1}\left(\frac{3b}{2a}\sqrt{-\frac{3}{a}}\right)-\frac{4}{3}\pi\right),$
(56)

and determine $\mathbf{q}_{m}^{*}=\xi_{m}\mathbf{w}_{m}$, which is the result
of the theorem. $\blacksquare$

## Appendix B

## Proof of Theorem 3

The Lagrangian $\mathcal{L}$ of Problem (39) with dual variable
$\left[\left\\{\lambda_{1i}\right\\}_{i=1}^{L},\left\\{\lambda_{2i}\right\\}_{i=1}^{L},\mu\right]$
is given by [48]

$\begin{split}&\mathcal{L}\left(\left\\{\lambda_{1i}\right\\}_{i=1}^{L},\left\\{\lambda_{2i}\right\\}_{i=1}^{L},\mu,\left\\{N_{i}\right\\}_{i=1}^{L}\right)\\\
&=\sum_{i=1}^{L}\sum_{m\in\mathbb{M}_{i}}\frac{A_{m}}{N_{i}^{2}}+\sum_{i=1}^{L}\lambda_{1i}\left(N_{i}-kN\right)\\\
&~{}~{}~{}+\sum_{i=1}^{L}\lambda_{2i}\left(-N_{i}\right)+\mu\left(\sum_{i=1}^{L}N_{i}-N\right).\end{split}$
(57)

Hence, by differentiating $\mathcal{L}$ into $N_{i}$ and applying the Karush-
Kuhn-Tucker (KKT) conditions [48] we obtain

$\begin{split}\begin{cases}\frac{\partial\mathcal{L}}{\partial
N_{i}}=\sum_{m\in\mathbb{M}_{i}}\frac{-2A_{m}}{N_{i}^{3}}+\left(\lambda_{1i}-\lambda_{2i}\right)+\mu=0~{}\left(\star\right),\\\
\lambda_{1i}\geq 0,\lambda_{2i}\geq
0,\lambda_{1i}\left(N_{i}-kN\right)=0,\lambda_{2i}N_{i}=0.\end{cases}\end{split}$
(58)

However, since every RIS partition, in fact, should have a positive number of
elements $\left(N_{i}\neq 0\right)$, $\lambda_{2i}=0$ holds. For the case of
$\lambda_{1i}=0$, $\left(\star\right)$ in (58) is equivalent to

$\begin{split}&\frac{\partial\mathcal{L}}{\partial
N_{i}}=\sum_{m\in\mathbb{M}_{i}}\frac{-2A_{m}}{N_{i}^{3}}+\mu=0\\\
&~{}\rightarrow
N_{i}=\sqrt[3]{\sum_{m\in\mathbb{M}_{i}}{\frac{2A_{m}}{\mu}}}\leq
kN.\end{split}$ (59)

However, for the case of $N_{i}-kN=0$, by substituting $N_{i}=kN$ into
$\left(\star\right)$ we obtain

$\begin{split}&\mu+\sum_{m\in\mathbb{M}_{i}}\frac{-2A_{m}}{\left(kN\right)^{3}}=-\lambda_{1i}\leq
0\\\ &~{}\rightarrow
N_{i}=kN\leq\sqrt[3]{\sum_{m\in\mathbb{M}_{i}}{\frac{2A_{m}}{\mu}}}.\end{split}$
(60)

Hence, from (LABEL:muu) and (LABEL:bigsmall), the optimal
$\left\\{N_{i}^{*}\right\\}_{i=1}^{L}$ is given by

$N_{i}^{*}=\min\left(kN,\sqrt[3]{\frac{\sum_{m\in\mathbb{M}_{i}}2A_{m}}{\mu}}\right)\left(i=1,\cdots,L\right),$
(61)

where $\mu$ is determined so that $\sum_{\ell=1}^{L}N_{\ell}^{*}=N$ holds and
the theorem follows. $\blacksquare$

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|  Hong-Bae Jeon (S’19) received the B.S degree in electrical and electronic
engineering and mathematics from Yonsei University, South Korea, in 2017. He
is currently pursuing the Ph.D degree in the School of Integrated Technology,
Yonsei University, South Korea. His research interests include applied
mathematics and emerging technologies for 5G/B5G communications, such as
reconfigurable intelligent surface, UAV network, radio resource management,
and free-space optical communications.
---|---
|  Sung-Ho Park (M’87) received the B.S. degree in mathematics from Korea
University, Seoul, South Korea, in 1984. From 1985 to 2005, he was working in
a software company, dealing with word-processor, middle-ware, and software for
financial industry. Since 2005, he has been developing spectrum management and
analysis software for government and military. He is currently developing
spectrum analysis software for a military operation.
---|---
|  Jaedon Park received the B.S. degree in electronics engineering from
Hanyang University, Seoul, South Korea, in 2000, and the M.S. and Ph.D.
degrees from the School of Electrical Engineering, Korea Advanced Institute of
Science and Technology (KAIST), Daejeon, South Korea, in 2002 and 2016,
respectively. He is currently a Senior Researcher with the Agency for Defense
Development (ADD), Daejeon. His research interests include MIMO systems, relay
systems, and FSO systems.
---|---
|  Kaibin Huang (M’08-SM’13-F’21) received the B.Eng. and M.Eng. degrees from
the National University of Singapore and the Ph.D. degree from The University
of Texas at Austin, all in electrical engineering. He is currently an
Associate Professor with the Department of Electrical and Electronic
Engineering, The University of Hong Kong, Hong Kong. He received the IEEE
Communication Society’s 2021 Best Survey Paper, 2019 Best Tutorial Paper, 2019
Asia–Pacific Outstanding Paper, 2015 Asia–Pacific Best Paper Award, and the
best paper awards at IEEE GLOBECOM 2006 and IEEE/CIC ICCC 2018. He received
the Outstanding Teaching Award from Yonsei University, South Korea, in 2011.
He has been named as a Highly Cited Researcher by the Clarivate Analytics in
2019 and 2020. He served as the Lead Chair for the Wireless Communications
Symposium of IEEE GLOBECOM 2017 and the Communication Theory Symposium of IEEE
GLOBECOM 2014, and the TPC Co-chair for IEEE PIMRC 2017 and IEEE CTW 2013. He
is also an Executive Editor of IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS,
an Associate Editor of IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, and
an Area Editor for IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING.
Previously, he served on the Editorial Board for IEEE WIRELESS COMMUNICATIONS
LETTERS. He has guest edited special issues of IEEE JOURNAL ON SELECTED AREAS
IN COMMUNICATIONS, IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, and
IEEE COMMUNICATIONS MAGAZINE. He is also a Distinguished Lecturer of the IEEE
Communications Society and the IEEE Vehicular Technology Society, and a
Research Fellow of Hong Kong Research Grants Council.
---|---
|  Chan-Byoung Chae (S’06-M’09-SM’12-F’21) received the Ph.D. degree in
electrical and computer engineering from The University of Texas (UT) at
Austin, in 2008. From 2001 to 2005, he was a Research Engineer with the
Telecommunications Research and Development Center, Samsung Electronics,
Suwon, South Korea. From 2008 to 2009, he was a Postdoctoral Research Fellow
with Harvard University, Cambridge, MA, USA. From 2009 to 2011, he was MTS
with Bell Labs, Alcatel-Lucent, Murray Hill, NJ, USA. He was a member of the
Wireless Networking and Communications Group (WNCG) at UT. He is currently an
Underwood Distinguished Professor with the School of Integrated Technology,
Yonsei University, South Korea. Dr. Chae was a recipient/co-recipient of the
IEEE WCNC Best Demo Award in 2020, the Young Engineer Award from the National
Academy of Engineering of Korea (NAEK) in 2019, the IEEE DySPAN Best Demo
Award, in 2018, the IEEE/KICS JOURNAL OF COMMUNICATIONS AND NETWORKS Best
Paper Award in 2018, the Award of Excellence in Leadership of 100 Leading Core
Technologies for Korea 2025 from the NAEK in 2017, the Yonam Research Award
from the LG Yonam Foundation, in 2016, the IEEE INFOCOM Best Demo Award in
2015, the IEIE/IEEE Joint Award for Young IT Engineer of the Year in 2014, the
KICS Haedong Young Scholar Award in 2013, the IEEE Signal Processing Magazine
Best Paper Award in 2013, the IEEE ComSoc AP Outstanding Young Researcher
Award, in 2012, the IEEE VTS Dan. E. Noble Fellowship Award, in 2008. He is
currently the Editor-in-Chief of the IEEE TRANSACTIONS ON MOLECULAR,
BIOLOGICAL, AND MULTI-SCALE COMMUNICATIONS and a Senior Editor of the IEEE
WIRELESS COMMUNICATIONS LETTERS. He has been serving as an Editor for the IEEE
COMMUNICATIONS MAGAZINE, since 2016 and has worked with the IEEE TRANSACTIONS
ON WIRELESS COMMUNICATIONS from 2012 to 2017. He is an IEEE ComSoc
Distinguished Lecturer for the term 2020–2021.
---|---