# UAV Trajectory and Multi-User Beamforming Optimization for Clustered Users
Against Passive Eavesdropping Attacks With Unknown CSI

Aly Sabri Abdalla, Ali Behfarnia, and Vuk Marojevic Manuscript received 8
September 2022; revised 3 November 2022; accepted 4 June 2023. Date of
publication - - 2023; date of current version 10 June 2023. This work was
supported in part by the NSF PAWR program, under grant number CNS-1939334. The
work of A. Behfarnia is supported in part by the Faculty Research Program
under the Office of Sponsored Programs and Research at the University of
Tennessee at Martin. Aly Sabri Abdalla and Vuk Marojevic are with the
Department of Electrical and Computer Engineering, Mississippi State
University, MS, USA e-mail<EMAIL_ADDRESS>$vm602$@msstate.edu).  Ali
Behfarnia is with the Department of Engineering, University of Tennessee at
Martin, TN, USA e-mail: (a.behfarnia@tennessee.edu).

###### Abstract

This paper tackles the fundamental passive eavesdropping problem in modern
wireless communications in which the location and the channel state
information (CSI) of the attackers are unknown. In this regard, we propose
deploying an unmanned aerial vehicle (UAV) that serves as a mobile aerial
relay (AR) to help ground base station (GBS) support a subset of vulnerable
users. More precisely, our solution (1) clusters the single-antenna users in
two groups to be either served by the GBS directly or via the AR, (2) employs
optimal multi-user beamforming to the directly served users, and (3) optimizes
the AR’s 3D position, its multi-user beamforming matrix and transmit powers by
combining closed-form solutions with machine learning techniques.
Specifically, we design a plain beamforming and power optimization combined
with a deep reinforcement learning (DRL) algorithm for an AR to optimize its
trajectory for the security maximization of the served users. Numerical
results show that the multi-user multiple input, single output (MU-MISO)
system split between a GBS and an AR with optimized transmission parameters
without knowledge of the eavesdropping channels achieves high secrecy
capacities that scale well with increasing the number of users.

Index Terms: UAV-assisted, beamforming, DRL, eavesdropping, MU-MISO, physical
layer security, power control, trajectory optimization.

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##  I Introduction

Unmanned aerial vehicles (UAVs) are envisioned to improve the next generation
of wireless communication systems, 6G and beyond, by providing flexible,
intelligent, secure, and limitless connectivity [1, 2, 3]. Steps to identify
the challenges and solutions of emerging cellular networks to serve UAVs are
being undertaken by the 3rd Generation Partnership Project (3GPP) [4]. A
prominent use case for a UAV is the aerial relay (AR) which supports extended
coverage or higher system capacity at low-cost. However, the largely line of
sight (LoS) air-to-ground (A2G) communications between UAVs and user equipment
(UEs) make the system vulnerable to a variety of attacks [5]. Eavesdropping is
a major passive attack that can compromise communications channels and gain
access to private and sensitive user information.

Physical layer security has been introduced as a powerful tool to secure
communication links by using the physical characteristics of wireless
communication channels [6, 7, 8]. UAVs can benefit from physical layer
security by applying the latest technologies such as artificial intelligent
(AI) methods as well as various communication techniques to mitigate the
compromise of malicious behavior in the network. However, applying such
techniques is complicated by three factors: (i) requiring to coordinate
between a ground base station (GBS) and the AR to determine which users should
be served by which stations, (ii) handling resource limitations and trajectory
of ARs to serve specific UE(s), and (iii) choosing the well-suited learning
and communication techniques for the GBS and the AR to dynamically maximize
the security metrics.

###  I-A Related Work

The recent related works can be classified into three groups: i) beamforming-
aided secure communications; ii) UAV-aided secure communications; iii)
beamforming and UAV-aided secure communications. Table I provides a summery of
the prior art and proposed research related to the work presented in this
paper.

Table I: Prior Art and Proposed Research. Category |  Ref. |  Objective Metric |  Attack type |  Strategy |  Attackers’ CSI
---|---|---|---|---|---
Beamforming |  [9] |  Secrecy sum rate |  Passive eavesdroppers |  RCI is adopted to drive the power allocation between AN and information signal. |  Perfect CSI with channel errors
|  [10] |  Secrecy rate |  Active eavesdropper |  Analytical framework to find the best combination of AN and beamforming. |  Perfect CSI
|  [11] |  Secrecy rate |  Active and passive eavesdroppers |  Analytical framework to design the beamforming. |  Statistical CSI
UAV |  [12] |  Average secrecy rate |  Passive eavesdropper |  Optimizing the UAV’s trajectory and AN allocation via iterative algorithm. |  Perfect CSI
|  [13] |  Secrecy rate |  Passive eavesdropper |  Jointly optimizing the source/ UAV relay transmit power and the UAV trajectory through an iterative algorithm. |  Perfect CSI
|  [14] |  Secrecy rate |  Passive eavesdropper |  The UAV’s trajectory and transmit power allocation are jointly optimized by applying DQL algorithm. |  Unknown CSI
Beamforming and UAV |  [15] |  Minimum secrecy rate |  Passive eavesdropper |  Jointly optimizing the UAV beamforming and position to enhance UE’s secrecy rate through applying multi-objective dragonfly algorithm. |  Perfect CSI
|  [16] |  Secrecy capacity |  Passive eavesdroppers |  A DRL is proposed to optimize the UAV trajectory and transmitter and jammer UAVs beamforming. |  Perfect CSI
|  [17] |  Secrecy rate |  Passive eavesdropper |  Jointly optimize the beamforming of multi-beam satellite and the power allocation of UAV through an iterative alternating optimization approach. |  Perfect CSI
This work |  |  Secrecy sum capacity |  Passive eavesdroppers |  User clustering for association with the GBS and AR, where a DQL is designed to optimize the UAV trajectory, beamforming, and power control without knowledge of the wiretap CSI. |  Unknown CSI

Beamforming-aided Secure Communications: Transmit beamforming limit the radio
frequency (RF) propagation footprint and thus implicitly enable a secure the
propagation channel without high computational requirements a the receiver as
compared to cryptographic security schemes. Carefully designing the beam
patterns of antennas at the transmitter, receiver, or both can enhancing
system performance and security parameter, such as signal-to-interference plus
noise ratio (SINR) and secrecy rate, respectively. Researchers have studied
how to leverage and optimize beamforming for improving the PLS of current and
future wireless communication networks. The work presented in [9] investigates
the achieved secrecy sum rate for a multi-cell multiple-input multiple-output
(MIMO) system which is under a passive eavesdropper attack. The power
allocation between artificial noise (AN) and information signal is managed to
maximize the sum secrecy rate with imperfect channel state information (CSI),
which is derived using regularized channel inversion (RCI) precoding.
Reference [10] proposes strategies of combining AN and beamforming to achieve
high secrecy performance for massive MIMO systems in spite of single-antenna
active eavesdropping attacks that attempt to spoil the channel estimation
acquisition at the BS. Reference [11] derives the multiple-input, single-
output (MISO) beamforming design for random wireless networks with statistical
CSI in an environment with eavesdroppers and interferers.

UAV-aided Secure Communications: Reference [12] demonstrates the applicability
of maximizing the achievable average secrecy rate by optimizing the AN
transmission and UAV trajectory. Reference [13] proposes using the UAV as a
relay to improve the secrecy rate by jointly optimizing the source/relay
transmit power and the UAV trajectory. In our previous work [14], we have
proposed a deep Q-learning (DQL) algorithm to optimize the secrecy rate by
optimizing the trajectory of the UAV relay and the transmit power without the
availability of the CSI of the wiretap channel.

Beamforming Plus UAV-Aided Secure Communications: Combining both beamforming
and UAV has been considered as an enhanced PLS technique in advanced wireless
communications. For example, [15] introduces the multi-objective dragonfly
algorithm (MODA) to solve the multi-objective optimization problem for
enhancing the minimum secrecy rate between the UAV node and a single UE for
different clusters. The work presented in that paper assumes perfect CSI
conditionsfor the BS and focuses on optimizing the UAV performance. Reference
[16] proposes a multi-agent deep reinforcement learning (DRL) algorithm to
maximize the secrecy capacity of a multi user system by optimizing the
trajectory of the aerial BS and the beamforming matrix of the jammer UAV
interfering with the eavesdroppers. The authors of [17] propose an iterative
optimization approach that alternately optimizes the beamforming of satellite
transmitters and the power allocation of the UAV acting as an aerial relay and
friendly jammer supporting multi-beam satellite-enabled vehicle communication
in the presence of eavesdropping.

###  I-B Contribution

In this paper, we aim to mitigate passive eavesdropping attacks, where an
eavesdropper illegitimately wiretaps the legitimate wireless communications
links. To this end, we propose a combination of machine learning, deep
reinforcement learning, and multi-antennas techniques at the BS and the AR to
maximize the security of UEs in a wireless communication network. The
contributions of this paper are:

  * •

We define a practical optimization problem to maximize the channel secrecy
capacity without CSI knowledge of the wiretap channel.

  * •

We introduce a framework for effectively solving this problem by means of user
clustering, beamforming and power control, and AR trajectory optimization. We
design a DRL solution for the trajectory optimization and leverage the closed
form solutions for the beamforming and transmit and relay power allocation.

  * •

We provide a comprehensive numerical analysis that demonstrates the
effectiveness of the proposed tools.

The rest of paper is organized as follows. Section II presents the system
model. Section III formulates the problem and defines the relevant metrics.
Section IV derives the solution. Numerical results and analyses are presented
in Section V. Section VI provides the concluding remarks.

##  II System Model

We consider a ground base station (GBS) serving ground UEs where the
communication links are subject to passive eavesdropping attacks. The
eavesdroppers have a radio receiver and can wiretap the downlink transmission.
A UAV acting as an AR is dispatched to support secure communications. This
scenario is illustrated in Fig. 1.

Figure 1: System model.

We use the following notation: lower-case letters represent scalars and bold
lower-case letters denote vectors. Bold upper-case letters are used for
matrices. Tr($\mathbf{S}$) and $\mathbf{S}^{-1}$ represent the trace and the
inverse of a square matrix $\mathbf{S}$, respectively. The operator (.)T
denotes transpose, and the operator (.)† denotes conjugate transpose.
$\mathbf{S}(i,j)$ shows the $(i,j)$th element of matrix $\mathbf{S}$ and
Rank($\mathbf{S}$) shows the rank of the matrix. $||\mathbf{v}||$ represents
the Euclidean norm of a complex vector $\mathbf{v}$. Also, $|v|$ denotes the
norm of a complex number $v$. $\mathbb{C}^{a\times b}$ denotes the dimension
of $a\times b$ for a complex vector or matrix. Complex normal distribution
vector with the mean vector $\mathbf{m}$ and the covariance matrix
$\mathbf{\Sigma}$ is denoted by $\mathcal{CN}(\textbf{m},\mathbf{\Sigma})$,
and $\sim$ implies ”distributed as”.

###  II-A Channel model

####  II-A1 Air-to-ground

In terms of modelling the A2G communication channel between the UAV and ground
receivers, we consider small-scale Rician fading where the line of sight (LoS)
component coexist with non-LoS (NLoS) components [18]. The GBS and AR have
both a uniform linear array (ULA) of $M$ and $N$ antennas, respectively. The
A2G channel model,

$\vspace{-4 pt}{\bf\it
G}_{TR}=\frac{\sqrt{\lambda_{0}}}{d^{\alpha}_{TR}}\bigg{(}\sqrt{\frac{\beta}{1+\beta}}\
{\bf\it G^{LoS}_{TR}}+\sqrt{\frac{1}{\beta+1}}\ {\bf\it
G^{NLoS}_{TR}}\bigg{)},$ (1)

is obtained as the superposition of the LoS and NLoS channel components, where
$\lambda_{0}$ is the path loss at the reference distance of $1\ m$, $d_{TR}$
is the 3D distance between the GBS and AR, $\alpha$ is the path loss exponent,
and $\beta$ is the Rician factor. Without loss of generality, the entries of
${\bf\it G^{NLoS}_{TR}}$ are assumed to be independent and identically
distributed (i.i.d.) zero-mean and unit variance circularly symmetric complex
Gaussian (CSCG), i.e., $\sim\mathcal{CN}(0,1)$. The LoS component,

$\vspace{-6 pt}{\bf\it G}^{LoS}_{TR}={\bf\it g^{(A)}_{TR}}\ {\bf\it
g^{(D)}_{TR}},$ (2)

where

$\vspace{-4 pt}{\bf\it
g^{(A)}_{TR}}=\Big{[}1,e^{-j\frac{2\pi}{\lambda}\Upsilon\Lambda^{TR}},\cdots,e^{-j\frac{2\pi}{\lambda}(N-1)\Upsilon\Lambda^{TR}}\Big{]}$
(3)

and

$\vspace{-4 pt}{\bf\it
g^{(D)}_{TR}}=\Big{[}1,e^{-j\frac{2\pi}{\lambda}\Upsilon\Gamma^{TR}},\cdots,e^{-j\frac{2\pi}{\lambda}(M-1)\Upsilon\Gamma^{TR}}\Big{]}$
(4)

correspond to channel contributions from the angel-of-arrival (AoA) and angel-
of-departure (AoD) between the GBS and the AR. Parameter $\lambda$ is the
carrier wavelength, $\Upsilon$ is the antenna separation, $\Lambda^{TR}=cos\
\Theta\ sin\ \varphi$ is the AoA component ($\Theta$–azimuth and
$\varphi$–elevation AoA), and $\Gamma^{TR}=sin\ \vartheta\ cos\ \psi$ is the
AoD component ($\vartheta$–elevation and $\psi$–azimuth AoD) of the
transmitted signal from the GBS to the AR.

The A2G channel between the AR and the ground users,

$\vspace{-4 pt}{\bf\it G}_{RK}=\frac{\sqrt{\lambda_{0}}}{{\bf\it
d^{\alpha}_{RK}}}\bigg{(}\sqrt{\frac{\beta}{1+\beta}}\ {\bf\it
g^{LoS}_{RK}}+\sqrt{\frac{1}{\beta+1}}\ {\bf\it G^{NLoS}_{RK}}\bigg{)},$ (5)

has an LoS and an NLoS term, where ${\bf\it d_{RK}}$ is the 3D distance
between the AR and the ground user cluster. The ${\bf\it G^{NLoS}_{RK}}$
entries follow the same CSCG distribution as ${\bf\it G^{NLoS}_{TR}}$. The LoS
term,

$\vspace{-4 pt}{\bf\it
g^{LoS}_{RK}}=\Big{[}1,e^{-j\frac{2\pi}{\lambda}\Upsilon\chi^{RK}},\cdots,e^{-j\frac{2\pi}{\lambda}(N-1)\Upsilon\chi^{RK}}\Big{]},$
(6)

defines the AoD components $\chi^{RK}=cos\ \Phi\ sin\ \Omega$ ($\Phi$–azimuth
and $\Omega$–elevation AoD) of the transmitted signal from the ULA of the AR
to the single-antenna users.

####  II-A2  Ground-to-ground

the Alpha-beta-gamma (ABG) [19] channel model is adopted for the ground-to-
ground (G2G) communication channels between the GBS and the eavesdropper and
between the UEs and the eavesdropper. It is the closest path-loss model
approximation to the actual 5G ground communications measurement results and
it is employed by standard organizations such as ITU-R, 3GPP, mmMAGIC, and
QuaDRiGa [20]. It is defined as

$\displaystyle h_{G2G}(f,d)$ $\displaystyle=10\hskip 3.0pt\rho_{G}\times
log\Big{(}\frac{d_{gg}}{1\hskip 1.0ptm}\Big{)}+\jmath$ (7)
$\displaystyle+10\hskip 3.0pt\gamma_{G}\times log\Big{(}\frac{f_{c}}{1\hskip
1.0ptGHz}\Big{)}+\chi_{\sigma}^{G2G},$

where $d_{gg}$ is the 2D distance between the transmitter and receiver nodes,
$\jmath$ is the intercept, and $\rho_{G}$ and $\gamma_{G}$ correspond to the
distance and the frequency-dependent exponents. Shadow fading,
$\chi_{\sigma}^{G2G}$, is modeled as a Gaussian random variable of zero-mean
and standard deviation $\sigma_{sh}$.

###  II-B Communication Model for Legitimate Users

The $M$-antenna GBS can communicate with the $K$ single-antenna UEs either
directly or using the $N$-antenna AR at the same frequency, employing space-
division multiple access (SDMA) and time-division multiple access (TDMA) [21].
The GBS serves $K_{b}$ users directly and $K_{r}$ users via the AR, where
$K=K_{b}+K_{r}$. In what follows, we provide the corresponding communication
models and channel capacities.

####  II-B1  Direct communication from GBS

For the direct communication, the GBS forms multiple simultaneous beams to
spatially separated users employing SDMA. The transmit beamforming assigns one
beam vector for each user. However, transmit power leakage can occur between
beams causing multi-user interference.

We consider the downlink transmission, where the GBS transfers $K_{b}$ data
streams to $K_{b}$ users. The transmitted signal model is

$\vspace{-4 pt}{\bf\it x}_{b}=\sum\limits_{k=1}^{K_{b}}{\bf\it
w}_{b,k}\,s_{k},$ (8)

where ${\bf\it x}_{b}\in\mathbb{C}^{M\times 1}$, ${\bf\it
w}_{b,k}\in\mathbb{C}^{M\times 1}$ is the beamforming vector and $s_{k}$ the
transmitted information symbol for the $k$th user. The beamforming, or
precoding, matrix of the GBS contains $K_{b}$ beamforming vectors, ${\bf\it
W}_{bk}\in\mathbb{C}^{M\times K_{b}}$, where ${\bf\it W}_{bk}=[{\bf\it
w}_{b,1},\cdots,{\bf\it w}_{b,K_{b}}]$. The allocated transmit power for the
$k$th user can then be calculated by the squared norm of the beamforming
vector $\parallel{\bf\it w}_{b,k}\parallel^{2}$. The received signal at the
$K_{b}$ users can is expressed as

$\displaystyle\vspace{-6 pt}{\bf\it y}_{0}={\bf\it H}_{0}\,{\bf\it
x}_{b}\,+\,{\bf\it n}_{0},$ (9)

where ${\bf\it y}_{0}\in\mathbb{C}^{K_{b}\times 1}$, ${\bf\it
H}_{0}\in\mathbb{C}^{K_{b}\times M}$ represents the channel between the $M$
antennas of the GBS and the $K_{b}$ single-antenna users, and ${\bf\it
n}_{0}\in\mathbb{C}^{K_{b}\times 1}$ represents noise. It is assumed that the
distribution of noise at each user is complex normal with zero-mean and unit
variance, i.e., $n_{k}\sim\mathcal{CN}(0,1)$. The received signal at user $k$,

$\displaystyle\vspace{-6 pt}y_{0,k}$ $\displaystyle={\bf\it h}_{0,k}\,{\bf\it
x}_{b}\,+\,n_{k},$ $\displaystyle={\bf\it
h}_{0,k}\Big{(}\sum\limits_{k=1}^{K_{b}}{\bf\it w}_{b,k}\
s_{k}\Big{)}+\,n_{k},$ $\displaystyle={\bf\it h}_{0,k}{\bf\it
w}_{b,k}\,s_{k}\,+{\bf\it
h}_{0,k}\Big{(}\sum\limits_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}^{K_{b}}{\bf\it w}_{b,i}\,s_{i}\Big{)}\,+\,n_{k}$ (10)

has the signal-to-interference-plus-noise-ratio (SINR)

$\displaystyle\gamma_{b,k}$ $\displaystyle=\frac{\mid{\bf\it h}_{0,k}{\bf\it
w}_{b,k}\mid^{2}}{\sum\limits_{{i\neq k}}\mid{\bf\it h}_{0,k}{\bf\it
w}_{b,i}\mid^{2}+1}\,,$ (11)

where ${\bf\it h}_{0,k}\in\mathbb{C}^{1\times M}$ denotes the MISO channel
from the GBS to the $k$th user. The channel capacity of the direct link is
obtained from

$\displaystyle C_{b,k}=\log_{2}\big{(}1+\gamma_{b,k}\big{)}.$ (12)

####  II-B2  Indirect communication via AR

We assume a time-slot based synchronization between the GBS transmission and
the AR transmission [22, 23]. In odd time-slots (phase), the BS transmits
$K_{r}$ data streams to the AR, each of which is destined to one UE. The
transmission between the GBS and AR can be modeled as a standard point-to-to-
point MIMO channel. The received signal at the AR can be written as

$\displaystyle{\bf\it y}_{1}$ $\displaystyle={\bf\it H}_{1}{\bf\it
x}_{b}+{\bf\it n}_{1},$ $\displaystyle={\bf\it
H}_{1}\Big{(}\sum\limits_{k=1}^{K_{r}}{\bf\it w}_{b,k}\ s_{k}\Big{)}+\,{\bf\it
n}_{1},$ (13)

where ${\bf\it y}_{1}\in\mathbb{C}^{N\times 1}$, ${\bf\it
H}_{1}\in\mathbb{C}^{N\times M}$ is the MIMO communication channel between the
BS and the AR, and ${\bf\it n}_{1}\sim\mathcal{CN}(0,{\bf\it
I})\in\mathbb{C}^{N\times 1}$ is the noise vector.

In the even time-slots, the AR transmits

$\displaystyle{\bf\it x}_{r}={\bf\it W}_{r}\ {\bf\it y}_{1},$ (14)

where ${\bf\it x}_{r}\in\mathbf{C}^{N\times 1}$ and ${\bf\it
W}_{r}\in\mathbb{C}^{N\times N}$ is the beamforming matrix. The received
signals at the $K_{r}$ UEs are modeled as

$\displaystyle{\bf\it y}_{2}$ $\displaystyle={\bf\it H}_{2}\ {\bf\it
x}_{r}+{\bf\it n}_{2},$ $\displaystyle={\bf\it H}_{2}\,\Bigg{(}{\bf\it
W}_{r}\,\bigg{(}{\bf\it H}_{1}\Big{(}\sum\limits_{k=1}^{K_{r}}{\bf\it
w}_{b,k}\ s_{k}\Big{)}+\,{\bf\it n}_{1}\bigg{)}\Bigg{)}+{\bf\it n}_{2},$ (15)

where ${\bf\it y}_{2}\in\mathbb{C}^{K_{r}\times 1}$, ${\bf\it
H}_{2}\in\mathbb{C}^{K_{r}\times N}$ is the A2G communication channel between
the AR and the $K_{r}$ UEs, and ${\bf\it n}_{2}\sim\mathcal{CN}(0,{\bf\it
I})\in\mathbb{C}^{K_{r}\times 1}$ is the noise vector. The $k$th user receives

$\displaystyle y_{2,k}$ $\displaystyle={\bf\it h}_{2,k}{\bf\it W}_{r}\ {\bf\it
H}_{1}\ {\bf\it w}_{b,k}\ s_{k}$ $\displaystyle+{\bf\it h}_{2,k}{\bf\it
W}_{r}\ {\bf\it H}_{1}\Big{(}\sum\limits_{i\neq k}^{K_{r}}{\bf\it w}_{b,i}\
s_{i}\Big{)}$ $\displaystyle+{\bf\it h}_{2,k}{\bf\it W}_{r}{\bf\it
n}_{1}+n_{2,k},$ (16)

where ${\bf\it h}_{2,k}\in\mathbb{C}^{1\times N}$ denotes the MISO channel
from the AR to the $k$th UE and $n_{2,k}\sim\mathcal{CN}(0,1)$ is the additive
noise. The SINR of this relayed communication link from the BS to the $k$th UE
via the AR can then be calculated as

$\displaystyle\gamma_{r,k}$ $\displaystyle=\frac{\mid{\bf\it h}_{2,k}{\bf\it
W}_{r}{\bf\it H}_{1}{\bf\it w}_{b,k}\mid^{2}}{\sum\limits_{i\neq k}\mid{\bf\it
h}_{2,k}{\bf\it W}_{r}{\bf\it H}_{1}{\bf\it w}_{b,i}\mid^{2}+\|{\bf\it
h}_{2,k}{\bf\it W}_{r}\|^{2}+1}\,.$ (17)

The channel capacity $C_{r,k}$ of the indirect link is obtained from (12)
using $\gamma_{r,k}$ instead of $\gamma_{b,k}$. Note that ${\bf\it H}_{1}$ and
${\bf\it H}_{2}$ are directly influenced by UAV mobility due to changes in
distance, altitude, and orientation relative to ground receivers.

###  II-C Communication Model for Eavesdroppers

####  II-C1 Eavesdropping on the direct communication link

The eavesdropper listens on the the direct link between the GBS and the
associated UEs and receives

$\displaystyle y_{0,e}$ $\displaystyle={\bf\it h}_{0,e}\ {\bf\it
x}_{b}+n_{e},$ $\displaystyle={\bf\it h}_{0,e}\
\big{(}\sum\limits_{k=1}^{K}{\bf\it w}_{b,k}\,s_{k}\big{)}+n_{e},$
$\displaystyle={\bf\it h}_{0,e}{\bf\it w}_{b,k}\,s_{k}\,+{\bf\it
h}_{0,e}\Big{(}\sum\limits_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}^{K}{\bf\it w}_{b,i}\,s_{i}\Big{)}+n_{e},$ (18)

where ${\bf\it h}_{0,e}\in\mathbb{C}^{1\times M}$ is the G2G communication
channel between the BS and the eavesdropper and $n_{e}$ is the noise at the
eavesdropper such that $n_{e}\sim\mathcal{CN}(0,1)$. The SINR associated with
the direct link between the GBS and the eavesdropper—for the beam formed to
user $k$—can be calculated as

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma_{b,e,k}}$
$\displaystyle=\frac{\mid{\bf\it h}_{0,e}{\bf\it
w}_{b,k}\mid^{2}}{\sum\limits_{{i\neq k}}\mid{\bf\it h}_{0,e}{\bf\it
w}_{b,i}\mid^{2}+1}\,.$ (19)

Consequently, the capacity of the eavesdropper associated with the direct link
from the BS to the $k^{th}$ user can be derived as

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C_{b,e,k}}$
$\displaystyle=\log_{2}\big{(}1+\gamma_{b,e,k}\big{)},$
$\displaystyle=\log_{2}\bigg{(}1+\frac{\mid{\bf\it h}_{0,e}{\bf\it
w}_{b,k}\mid^{2}}{\sum\limits_{{i\neq k}}\mid{\bf\it h}_{0,e}{\bf\it
w}_{b,i}\mid^{2}+1}\bigg{)}.$ (20)

####  II-C2 Eavesdropping from relay communication link

The eavesdropper can wiretap the A2G relay communication link between the UAV
and the UEs. Similar to the section II-B2, the capacity of the eavesdropper
associated with the relay link can be derived as

$\displaystyle C_{r,e}=\log_{2}(1+\gamma_{r,e})$
$\displaystyle=\log_{2}\bigg{(}1+\frac{\mid{\bf\it h}_{2,e}{\bf\it
W}_{r}{\bf\it H}_{1}{\bf\it w}_{b,k}\mid^{2}}{\sum\limits_{i\neq k}\mid{\bf\it
h}_{2,e}{\bf\it W}_{r}{\bf\it H}_{1}{\bf\it w}_{b,i}\mid^{2}+\|{\bf\it
h}_{2,e}{\bf\it W}_{r}\|^{2}+1}\bigg{)},$ (21)

where $\gamma_{r,e}$ is the SINR, and ${\bf\it h}_{2,e}\in\mathbb{C}^{1\times
N}$ denotes the A2G channel between the UAV and the eavesdropper, and
$n_{2,e}$ is the noise at the eavesdropper such that
$n_{2,e}\sim\mathcal{CN}(0,1)$.

###  II-D Secrecy Capacity

The term secrecy capacity is a measure of the information rate that can be
transmitted securely without being intercepted. It is obtained as the
difference between the achievable data rate of a legitimate receiver and the
achievable data rate of an eavesdropper, taking into account the channel
conditions and the employed security measures. It corresponds to the rate at
which no data will be decoded by the eavesdropper [24].

For the system model of Section II, the average sum-secrecy capacity of the
$K_{b}$ UEs that are directly served by the GBS over $T$ time slots is

$\displaystyle{C_{sec,b}}$
$\displaystyle=\frac{1}{T}\sum\limits_{t=1}^{T}\sum\limits_{k=1}^{K_{b}}\Big{(}C_{b,k}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-C_{b,e}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{)}^{+}$
$\displaystyle=\frac{1}{T}\sum\limits_{t=1}^{T}\sum\limits_{k=1}^{K_{b}}\Bigg{[}\log_{2}\bigg{(}1+\frac{\mid{\bf\it
h}_{0,k}{\bf\it w}_{b,k}\mid^{2}}{\sum\limits_{{i\neq k}}\mid{\bf\it
h}_{0,k}{\bf\it w}_{b,i}\mid^{2}+1}\bigg{)}-$
$\displaystyle\qquad\qquad\quad\log_{2}\bigg{(}1+\frac{\mid{\bf\it
h}_{0,e}{\bf\it w}_{b,k}\mid^{2}}{\sum\limits_{{i\neq k}}\mid{\bf\it
h}_{0,e}{\bf\it w}_{b,i}\mid^{2}+1}\bigg{)}\Bigg{]}^{+}.$ (22)

Likewise, the average sum-secrecy capacity of the $K_{r}$ UEs served via the
AR over $T$ time slots is obtained as

$\displaystyle{C_{sec,r}}$
$\displaystyle=\frac{1}{T}\sum\limits_{t=1}^{T}\sum\limits_{k=1}^{K_{r}}\Big{(}C_{r,k}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-C_{r,e}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{)}^{+}=\frac{1}{T}\sum\limits_{t=1}^{T}$
$\displaystyle\sum\limits_{k=1}^{K_{r}}\Bigg{[}\log_{2}\bigg{(}1+\frac{\mid{\bf\it
h}_{2,k}{\bf\it W}_{r}{\bf\it H}_{1}{\bf\it
w}_{b,k}\mid^{2}}{\sum\limits_{i\neq k}\mid{\bf\it h}_{2,k}{\bf\it
W}_{r}{\bf\it H}_{1}{\bf\it w}_{b,i}\mid^{2}+\|{\bf\it h}_{2,k}{\bf\it
W}_{r}\|^{2}+1}\bigg{)}$ $\displaystyle-\log_{2}\bigg{(}1+\frac{\mid{\bf\it
h}_{2,e}{\bf\it W}_{r}{\bf\it H}_{1}{\bf\it
w}_{b,k}\mid^{2}}{\sum\limits_{i\neq k}\mid{\bf\it h}_{2,e}{\bf\it
W}_{r}{\bf\it H}_{1}{\bf\it w}_{b,i}\mid^{2}+\|{\bf\it h}_{2,e}{\bf\it
W}_{r}\|^{2}+1}\bigg{)}\Bigg{]}^{+}.$ (23)

where $[\omega]^{+}\triangleq max(\omega,0)$. The total secrecy capacity is

$C_{T}=C_{sec,b}+C_{sec,r}.\vspace{-6pt}$ (24)

Formulas (22)-(24) are derived from information theory and provide
quantitative measures of the level of secrecy achieved in the communications
channels according to the system model of Fig. 1. The secrecy capacity is
maximized when maximizing the SINRs at the legitimate receivers and minimizing
the SINRs at the eavesdroppers.

##  III Problem Formulation

This paper aims to maximize the total secrecy capacity of the UEs whether they
are directly served by the GBS or through the AR. Considering the degrees of
freedom for serving UEs directly or via the AR, we formulate two optimization
problems.

####  III-1 Direct communication

For the directly served UEs, the optimization problem is defined as

$\displaystyle\hskip 40.0pt\underset{{\bf\it w}_{b,k}}{\text{max}}$
$\displaystyle{C_{sec,b}}$ (25) subject to (s.t.) $\displaystyle P_{b}\leq
P_{b,max},$

where $C_{sec,b}$ is the secrecy capacity defined in (II-D), $P_{b,max}$ is
the maximum transmit power of the GBS, and $P_{b}$ is the transmit power of
the GBS, that is, $P_{b}=$ Tr (${\bf\it x}_{b}{\bf\it x}_{b}^{\dagger}$).

Problem (25) requires the knowledge of the eavesdropping channel. We assume
the location of the eavesdropper and thus its CSI to be unknown, which is the
scenario of interest in practice where it is difficult to detect or estimate
the presence, location, or channel of eavesdroppers because of their passive
nature. Therefore, we can only consider the capacity of the legitimate user
and reformulate the optimization problem:

$\displaystyle\underset{{\bf\it w}_{b,k}}{\text{max}}$
$\displaystyle\frac{1}{T}\sum\limits_{t=1}^{T}\sum\limits_{k=1}^{K}\Bigg{[}\log_{2}\bigg{(}1+\frac{\mid{\bf\it
h}_{0,k}{\bf\it w}_{b,k}\mid^{2}}{\sum\limits_{{i\neq k}}\mid{\bf\it
h}_{0,k}{\bf\it w}_{b,i}\mid^{2}+1}\bigg{)}\Bigg{]}$ (26) s.t.
$\displaystyle\text{Tr}\big{(}{\bf\it x}_{b}{\bf\it
x}_{b}^{\dagger}\big{)}\leq P_{b,max}.$

The eavesdropper location and channel are used only for calculating the
resulting secrecy capacity for performance evaluation.

####  III-2 Relay communication

For the UEs that are served via the AR, the optimization problem is defined as

$\displaystyle\underset{\\{{\bf\it w}_{b,k},x_{r},y_{r},z_{r}\\}}{\text{max}}$
$\displaystyle{C_{sec,r}}$ (27) s.t. $\displaystyle P_{r}\leq P_{r,max}$
$\displaystyle(x_{r},y_{r},z_{r})\leq(L_{x},L_{y},L_{z}),$

where $C_{sec,r}$ is the secrecy capacity defined in (II-D), $P_{r,max}$ is
the maximum transmit power of the AR, $P_{r}=$ Tr (${\bf\it x}_{r}{\bf\it
x}_{r}^{\dagger}$) is the transmit power of the AR and $P_{r,max}$ the maximum
transmit power, and $(x_{r},y_{r},z_{r})$ are the 3D coordinates of the UAV
bound to $(L_{x},L_{y},L_{z})$. Because of the unknown eavesdropper location
and CSI, the optimization problem is rewritten as

$\displaystyle\underset{\\{{\bf\it W}_{r},x_{r},y_{r},z_{r}\\}}{\text{max}}$
$\displaystyle\frac{1}{T}\sum\limits_{t=1}^{T}\sum\limits_{k=1}^{K}\Bigg{[}\log_{2}\bigg{(}1+$
(28) $\displaystyle\frac{\mid{\bf\it h}_{2,k}{\bf\it W}_{r}{\bf\it
H}_{1}{\bf\it w}_{b,k}\mid^{2}}{\sum\limits_{i\neq k}\mid{\bf\it
h}_{2,k}{\bf\it W}_{r}{\bf\it H}_{1}{\bf\it w}_{b,i}\mid^{2}+\|{\bf\it
h}_{2,k}{\bf\it W}_{r}\|^{2}+1}\bigg{)}\Bigg{]}$ s.t.
$\displaystyle\text{Tr}\big{(}{\bf\it x}_{r}{\bf\it
x}_{r}^{\dagger}\big{)}\leq P_{r,max}$
$\displaystyle(x_{r},y_{r},z_{r})\leq(L_{x},L_{y},L_{z}).$

Although we have incorporated practical system constraints in our model, we
acknowledge that there are additional operational aspects, such as UAV energy
consumption, flight time, and speed [25], which are not optimized in this
paper.

##  IV  Proposed Solution

Given the available resources, which are one multi-antenna GBS and one multi-
antenna AR, the secrecy capacity optimization problem becomes a user
association and transmission parameter optimization problem.

Figure 2: Proposed solution flowchart.

We perform UE clustering for user association, followed by GBS and AR
beamforming and transmit power control, and UAV trajectory optimization.
Figure 2 illustrates this. It is important to mention that the
beamforming/power control and the UAV trajectory optimization are done through
an iterative process. That is, the algorithm obtains the optimal power
coefficients for every 3D location of the UAV. Hence, the beamforming and
transmit power control of the UAV affects its trajectory adjustment. The
details are discussed in Sections IV.B and IV.C

###  IV-A User Clustering

The goal of user clustering is to divide $K$ users into two clusters, one
cluster is to be served by the GBS and the other cluster is served the UAV.

For solving the user clustering problem, we can employ an exhaustive search,
but it entails a high computational complexity, which increases exponentially
with the number of users. We instead apply K-means clustering, a unsupervised
machine learning algorithm that is used for grouping a set of objects so that
the similarity criterion of members in a group and the dissimilarity with
members of other groups is maximized. K-mean works with any single or multi-
dimensional metric that the data captures and user-defined target number of
clusters [26]. It is a computationally efficient method compared to other
techniques such as graph theory, fuzzy c-means clustering, and hierarchical
clustering [27].

We consider the characteristics of wireless communication systems to determine
the similarities of the data points. Because the objective it to associate
users to base stations, one fixed (GBS) and one mobile (UAV), we take the
normalized channel coefficients between UEs and GBS as the data points of the
K-means clustering algorithm. This captures the variations of channel gains
resulting from different RF propagation effects such as small-scale fading and
shadow fading. Hence, we can define

$h^{n}_{b,k}=\frac{h_{b,k}}{\parallel h_{b,k}\parallel_{2}},\vspace{-4pt}$
(29)

where $h^{n}_{b,k}$ is the normalized channel gain, $h_{b,k}$ is the channel
gain between the GBS and $k$-th UE, and $\parallel.\parallel_{2}$ is the
$L_{2}$ vector norm. Having these channel gains as data points, we apply
K-mean clustering algorithm to determine the cluster centers, or centroids,
and consequently the UEs associated with each centroid. The goal is to
leverage the similarity of channels between the GBS and the UEs to create two
UE clusters, where the UEs of one cluster are to be served by the GBS and the
UEs of the other cluster by the UAV. This approach is applying the same
clustering principle as other studies in the literature [28, 29, 30, 31].

The K-mean clustering algorithm can be done as follows [32, 33]: (i) the
initial centroids $C=\\{c_{1},c_{2},\dots c_{n}\\}$ are randomly selected as
the $n$ cluster centers of the $K$ available data points:
$U=\\{u_{1},u_{2},\dots,u_{k}\dots,u_{K}\\}$. Here, we consider two clusters
$c_{1}$ and $c_{2}$ for the GBS and the UAV, and $K$ users where
$u_{k}=h^{n}_{b,k}$ can be considered as a data point of the $k$-th user. (ii)
Distance between each data point, e.g., channel status, and the cluster
centers is calculated to assign the data point to the nearest center.
Different metrics can be used to measure the distance between data points such
as Euclidean distance, Manhattan distance, etc. In this paper, we use
$L^{2}_{2}$ norm or Euclidean distance. (iii) the centroids are updated to
minimize the sum of squared distances between a user and its centroid,

$\min\limits_{C}\sum\limits_{k}\min\limits_{r\in R}\ d_{r,k}\vspace{-7pt}$
(30)

where $d_{r,k}=\ \parallel u_{k}-c_{r}\parallel^{2}_{2}$ is the Euclidean
distance between $C\triangleq\\{c_{r}\mid r\in R\\}$ and $R=2$ represents the
number of clusters. For example, the distance between the normalized channel
gains and the centroids is $d_{r,k}=\ \parallel
h^{n}_{b,k}-c_{r}\parallel^{2}_{2}$. Algorithm 1 represents a pseudocode for
the K-mean clustering algorithm. The implementation of algorithm 1 for one
scenario is shown in Figure 3 wherein data points are $h^{n}_{b,k}$

In addition to UEs’ channel status, other data points can also be considered
in the algorithm to study the problem. For example, distance based clustering
or rate based clustering, where UEs with the nearest distance to the GBS and
the highest downlink rates, respectively, would be grouped. Each scenario can
have an effect on the system performance and should be chosen based on the
objective and the ability or simplicity to obtain the necessary information to
calculate the value for each UE. In Section V.C, we discuss about the results
of different scenarios.

Figure 3: Channel-based clustering.

###  IV-B Optimal Beamforming and Power Control

Problem (26) is a traditional beamforming power control problem between a GBS
and a user. This problem has been extensively studied in the literature [34,
35]. The beamforming vectors of the GBS are obtained by applying the weighted
minimum mean square error (WMMSE) algorithm [36]. The WMMSE is an iterative
closed form solution that optimizes the transmitter and receiver precoding
vectors to maximize the sum rate of all UEs for a GBS power constraint. The
precoding solution for the direct communication links is then [37]

${\bf\it W_{bk}}=\Big{(}{\bf\it H_{0}^{H}}{\bf\it Q^{H}}{\bf\it F}{\bf\it
Q}{\bf\it H_{0}}+\frac{\text{Tr}({\bf\it F}{\bf\it Q}{\bf\it
Q^{H}})}{P_{b,max}}{\bf\it I_{M}}\Big{)}^{-1}{\bf\it H_{0}^{H}}{\bf\it
Q^{H}}{\bf\it F}\vspace{-2pt}$ (31)

where $Q=diag\\{q_{1},\cdots,q_{k}\\}$ is the receiver precoding,
$F=diag\\{f_{1},\cdots,f_{k}\\}$ is the weight matrix, and ${\bf\it I_{M}}$ is
the covariance matrix.

The beamforming and power control for the relay communications problem is
solved in the remainder of this section. Inspired by the zero-forcing (ZF)
criterion and the channel singular value decomposition (SVD) based structure
introduced in [21], we first propose a beamforming matrix structure for the
UAV (i.e., ${\bf\it W}_{r}$) to eliminate interference among users. This
converts the optimization problem (28) into a simplified convex optimization
problem. Then, we solve the modified optimization problem using the Lagrangian
function and Karush-Kuhn-Tucker (KKT) conditions to obtain the UAV’s optimal
beamforming matrix.

####  IV-B1 Beamforming Matrices

Beamforming is done at the GBS and the AR, each serving a distinct set of
users. By using the concepts of channel inversion, ZF, and linear algebra, the
multi-user interference can be minimized. From (II-B2), the received signal at
$K$ UEs transmitted from the UAV can be written as

$\displaystyle{\bf\it y}_{2}$ $\displaystyle={\bf\it H}_{2}{\bf\it
W}_{r}{\bf\it H}_{1}{\bf\it W}_{br}\,{\bf\it s}_{K}\,+\,{\bf\it H}_{2}{\bf\it
W}_{r}\,{\bf\it n}_{1}\,+\,{\bf\it n}_{2},$ (32)

where ${\bf\it s}_{K}\in\mathbb{C}^{K\times 1}$ corresponds to the $K$
transmit signals to the $K$ UEs. The ZF criterion requires that ${\bf\it
H}_{2}{\bf\it W}_{r}{\bf\it H}_{1}{\bf\it W}_{br}$ is to be a diagonal matrix
with rank $K$, which implies that Rank$({\bf\it H}_{1})\geq K$ and
Rank$({\bf\it H}_{2})\geq K$ [21]. By applying the SVD, ${\bf\it H}_{2}$ and
${\bf\it H}_{1}$ can be expressed as

$\displaystyle{\bf\it H}_{1}$ $\displaystyle={\bf\it
U}_{1}\,{\bf\it\Sigma}_{1}\,{\bf\it V}_{1}^{†},$ (33) $\displaystyle{\bf\it
H}_{2}$ $\displaystyle={\bf\it U}_{2}\,{\bf\it\Sigma}_{2}\,{\bf\it
V}_{2}^{†},$ (34)

where ${\bf\it U}_{i}$ and ${\bf\it V}_{i}$, for $i=1,2$, are unitary matrices
and ${\bf\it\Sigma}_{i}\in\mathbb{C}^{K\times K}$ is a diagonal matrix with
positive diagonal elements. Knowing the channel coefficients at the BS and at
the UAV, to satisfy the ZF criterion, we propose the following beamforming
matrices for the GBS and the UAV

$\displaystyle{\bf\it W}_{br}$ $\displaystyle={\bf\it
V}_{1}\,{\bf\it\Lambda}_{b}\,{\bf\it U}_{2}^{†},$ (35) $\displaystyle{\bf\it
W}_{r}$ $\displaystyle={\bf\it V}_{2}\,\hat{{\bf\it\Lambda}}_{r}\,{\bf\it
U}_{1}^{†}\,{\bf\it\Lambda}_{r},$ (36)

where ${\bf\it\Lambda}_{b},\hat{{\bf\it\Lambda}}_{r}$, and
${\bf\it\Lambda}_{r}$ are all $K\times K$ diagonal matrices. Without loss of
generality, it can be assumed that the elements of these two diagonal matrices
are non-negative, representing the allocated beamforming power at the BS and
the UAV, respectively.

####  IV-B2 Optimal AR Transmit Power Allocation

######  Lemma IV.1.

The objective function defined in (28) can be written as

$\displaystyle\frac{1}{T}\sum\limits_{t=1}^{T}\sum\limits_{k=1}^{K}log_{2}\Big{(}1+\frac{\lambda_{r,k}^{2}}{\lambda_{r,k}^{2}+1}\Big{)},$
(37)

where $\lambda_{r,k}$ is the ${\bf\it\Lambda}_{r}(k,k)$.

######  Lemma IV.2.

The beamforming power constraint defined in (28) can be expressed as

$\displaystyle 2\,\sum\limits_{m=1}^{K}\sum\limits_{n=1}^{K}|{\bf\it
U}_{2}(m,n)|^{2}\,\sigma_{2,n}^{-2}\,\lambda_{r,m}^{2}\leq P_{r,max},$ (38)

where $\sigma_{2,n}$ is the ${\bf\it\Sigma}_{2}(n,n)$.

The lemmas are proved in the Appendix (Section VI). Leveraging (37) and (38),
the beamforming power optimization problem for the UAV at any location can be
written as

$\displaystyle\underset{\\{\lambda_{r,m}\\}}{\text{max}}$
$\displaystyle\frac{1}{T}\sum\limits_{t=1}^{T}\sum\limits_{m=1}^{K}log_{2}\Big{(}1+\frac{\lambda_{r,m}^{2}}{\lambda_{r,m}^{2}+1}\Big{)}$
(39) s.t. $\displaystyle 2\,\sum\limits_{m=1}^{K}\sum\limits_{n=1}^{K}|{\bf\it
U}_{2}(m,n)|^{2}\,\sigma_{2,n}^{-2}\,\lambda_{r,m}^{2}\leq P_{r,max},$ (40)
$\displaystyle 0\leq\lambda_{r,m}\leq\lambda_{r,max},\quad m\in\\{1,...,K\\},$
(41)

where $P_{r,max}$ is the maximum available transmit power at the UAV, and
$\lambda_{r,max}$ is the maximum allocated power for each antenna.

The Lagrangian function of the optimization problem can be expressed as

$\displaystyle\mathcal{L}(\lambda_{r,l},\alpha_{1},\alpha_{2,l},\alpha_{3,l})=+\sum\limits_{l=1}^{K}log_{2}\Big{(}1+\frac{\lambda_{r,l}^{2}}{\lambda_{r,l}^{2}+1}\Big{)}$
$\displaystyle-\alpha_{1}\bigg{(}2\,\sum\limits_{l=1}^{K}\sum\limits_{n=1}^{K}|{\bf\it
U}_{2}(l,n)|^{2}\,\sigma_{2,n}^{-2}\,\lambda_{r,l}^{2}-P_{r,max}\bigg{)}$
$\displaystyle-\bigg{(}\alpha_{2,l}\Big{(}\sum\limits_{l=1}^{K}\lambda_{r,l}-\lambda_{r,max}\Big{)}\bigg{)}-\bigg{(}\alpha_{3,l}\sum\limits_{l=1}^{K}-\lambda_{r,l}\bigg{)},$
(42)

where $\alpha_{1}$, $\alpha_{2,l}$, and $\alpha_{3,l}$ are the non-negative
Lagrangian multipliers corresponding to the first and the second constraints,
respectively.

######  Theorem IV.3.

The optimal beamforming power for the $l$th antenna of the UAV can be obtained
as

$\lambda_{r,l}^{*}=\begin{cases}\small
0\qquad&\quad\lambda_{r,l}^{\dagger}\leq
0,(\alpha_{1}^{*}\,F\,\text{ln}\,2)>0.25\\\
\lambda_{r,l}^{\dagger}\qquad&\qquad
0<\lambda_{r,l}^{\dagger}<\lambda_{r,max}\\\
\lambda_{r,max}\qquad&\qquad\quad\ \
\lambda_{r,l}^{\dagger}\geq\lambda_{r,max}\end{cases}$

in which

$\displaystyle\lambda_{r,l}^{\dagger}=\sqrt{\frac{1}{4}\Big{(}\sqrt{1+\frac{2}{\alpha_{1}^{*}\,F\,\text{ln}\,2}}-3\Big{)}},$
(43)

where $F$ is a constant and equals to $\sum\limits_{n=1}^{K}|{\bf\it
U}_{2}(l,n)|^{2}\,\sigma_{2,n}^{-2}$, and the Lagrangian multiplier
$\alpha_{1}^{*}$ can be obtained by replacing (43) into the first constraint
of (40) when the equality holds.

The optimal beamforming matrix and transmit power formulations for the AR
defined above are used for the UAV trajectory optimization.

###  IV-C UAV Trajectory Optimization

The objective function of (28) is non-convex with respect to parameters
$x_{r}$, $y_{r}$, $z_{r}$, $P$, and the constraints, and the problem is NP-
hard [38, 39, 40, 21]. We, therefore, propose a machine learning solution
where the UAV trajectory is updated through a transition process based on the
current system state. Since the next system state is independent from the
previous state and action, the process can be modeled as a Markov decision
process (MDP). In order to avoid intractably high dimensionality for the high
state-action space, we propose a DQN. It is noteworthy that the following
proposed DQN is based on basic reinforcement learning algorithms such as
Q-learning and deep reinforcement learning. The aim is to use DQN as an
alternative tool for solving this NP-hard optimization problem while consuming
less power and computational resources [41, 42, 43]. Depending on an
application, one can extend the following framework to more advanced learning
models that suit particular use cases.

####  IV-C1 MDP Settings

The MDP for the UAV agent is composed of the state space $\mathcal{S}$, the
action space $\mathcal{A}$, the reward space $\mathcal{R}$, and the transition
probability space $\mathcal{T}$. At time slot $t$, the agent observes the
state $s_{t}\in\mathcal{S}$, and takes action $a_{t}\in\mathcal{A}$ based on
its policy. Depending on the distribution of the transition probability
$\mathcal{T}(s_{t+1}|s_{t},a_{t})$, the agent is then transferred to the new
state $s_{t+1}$. Since the transition probability is specific to the
operational environment, we choose the Q-learning method as a model-free
algorithm to find the best policy for each action in each state. This means
that we do not need to know $\mathcal{T}$, but we need to carefully define the
states, the actions, and the reward.

State: The set of states is defined as
$\mathcal{S}=\\{s_{1},s_{2},...,s_{t},..,s_{T}\\},$ where $t$ is the time slot
index. Each state $s_{t}$ corresponds to the 3D coordinates of the UAV and the
users served by the AR.

Action: The states are transitioned according to the defined set of actions
defined as $\mathcal{A}=\\{a_{1},a_{2},...,a_{t},..,a_{T}\\}$, where each
action consists of three parts related to the UAV movement,
$a_{t}=\\{\delta_{x},\delta_{y},\delta_{z}\\}$, where $\delta_{x}$ ,
$\delta_{y}$, and $\delta_{z}$ represent the movement in the $x$, $y$ and $z$
directions. The movement along each axis is assumed to change positively or
negatively, or remain in the original position. Hence, here we consider $3$
possible directional movements for the $3$ axes of the AR trajectory,
resulting in $27$ possible actions for the AR.

Reward: After taking action $a_{t}$ in state $s_{t}$, the UAV agent will
receive a reward $R_{t}(s_{t},a_{t})$. The UAV gets more rewards for actions
that lead to higher legitimate user rates. We define the reward function
accordingly:

${R_{t}(s_{t},a_{t})}=\sum\limits_{k=1}^{K_{r}}C_{r,k}$ (44)

####  IV-C2 Deep Q-Network Method

The DQN, initially proposed by Google Deep Mind [44], integrates the RL and
deep learning methods. This technique uses the power of nonlinear functions,
specifically DNNs, in order to approximate the Q-values and handle highly
dimensional state-action problems.

There are two DNNs of the same structure: a training network and a target
network. The training network outputs the Q-values associated with the actions
of the UAV in each state. The target network supervises the training network
by providing the target Q-values obtained from the Bellman equation [45],

$\displaystyle
Q^{*}(s,a)=E_{s^{\prime}}\bigg{[}R(s,a)+\gamma\times\max_{a\in\mathcal{A}}\
Q(s^{\prime},a^{\prime})\bigg{]},$ (45)

which provides the optimal state-action pairs, where $s^{\prime}$ and
$a^{\prime}$ symbolize the next state and action. Parameter $\gamma\in(0,1)$
denotes the discount factor that affects the importance of the future reward.

The target values are compared with the outputs of the training network to
minimize the loss function,

$\displaystyle
L(\theta)=\mathbb{E}\Bigg{[}\bigg{(}\Big{[}r_{t}+\gamma\times\max_{a\in\mathcal{A}}\
Q(s_{t+1},a_{t+1};\theta^{\dagger})\Big{]}-$
$\displaystyle\Big{[}Q(s_{t},a_{t};\theta)\Big{]}\bigg{)}^{2}\Bigg{]},$ (46)

where the Q-value of the first term is obtained from the target network and
the Q-value of the second term is obtained from the training network.
Parameters $\theta^{\dagger}$ and $\theta$ denote the weights of the target
network and training network, respectively. The $\theta^{\dagger}$
coefficients are updated every few time slots in order to ensure the stability
of the target values and, hence, facilitate stable learning.

As the UAV takes an action, the system generates a record of experience. At
time step $t$, the experience contains the current state $s_{t}$, the action
$a_{t}$, the reward $r_{t}$, and the next state $s_{t+1}$, formed as a tuple
$e_{t}=(s_{t},a_{t},r_{t},s_{t+1})$. Each such experience is stored in a
replay memory with the capacity of $N$, such that
$\mathcal{M}=\\{e_{1},...,e_{t},...,e_{N}\\}$. The memory is a queue-like
buffer that stores the latest N experience vectors. We use a mini-batch sample
from the replay memory to feed the input of the training network. The main
reason for using the mini-batch samples from the reply memory is to break
possible correlations between sequential states of the environment, and
thereby facilitate generalization.

The UAV applies a gradient descent algorithm,

$\displaystyle\nabla_{\theta}\,L(\theta)$ $\displaystyle=-\mathbb{E}\Bigg{[}2\
\nabla_{\theta}Q(s_{t},a_{t};\theta)\bigg{(}\,r_{t}+\,\gamma\ \times$
$\displaystyle\qquad\max_{a\in\mathcal{A}}\
Q(s_{t+1},a_{t+1};\theta^{\dagger})-Q(s_{t},a_{t};\theta)\bigg{)}\Bigg{]},$
(47)

to update $\theta$ an $\theta^{\dagger}$ as the weights of the DNNs with the
aim of minimizing the prediction error.

Finally, we apply the $\epsilon-$greedy algorithm to select an action while
balancing the exploration and the exploitation of the UAV in the environment.
In this algorithm, the UAV explores the environment with the probability of
$\epsilon$ by choosing a random action. More precisely, the UAV exploits the
environment with the probability of $1-\epsilon$ by choosing the actions that
maximize the Q-value function, i.e., $a^{*}=\text{argmax}_{a\in\mathcal{A}}\
Q(s,a;\theta)$. A high value of $\epsilon$ is initially set in the model for
the UAV to spend more time for the exploration. As the agent obtains more
knowledge about the environment, the $\epsilon$ value is gradually decreased
to leverage the experience and choose the best actions for the UAV, rather
than continuing with the exploration.

Algorithm 2 details the DQN-based algorithm used by the UAV agent for
optimizing the sum-rate of the UEs that are served via the AR.

In summary, the proposed techniques accomplish the following: i) User-BS
association employing K-means clustering (e.g., channel, rate, or distance
based), ii) multi-user beamforming and power management for the GBS, iii) UAV
trajectory optimization in conjunction with multi-user beamforming and power
management. The objective is to maximize the secrecy rate which can be used to
evaluate the effectiveness of the proposed security measurein protecting
against eavesdropping attacks [46, 47], especially in the context of wireless
communication [48, 49, 50, 51]. Since the CSI of the eavesdropping channel
cannot be obtained for passive, receive-only eavesdroppers, our solution
maximizes the user rate for it to be generally applicable without requiring
collaboration with eavesdroppers or wasting power for generating artificial
noise in random directions, because of the unknown eavesdropper locations, as
opposed to using this power to increase the user rate. The secrecy capacity
also provides a unified measurement framework for the numerical analyses
presented in the following section.

##  V Numerical Analysis and Discussion

In this section, we present simulation results to evaluate the secrecy
performance of the UAV-assisted communications system, where users are
clustered and served by a fixed and a mobile access point. In the presence of
an eavesdropping attack, our solution jointly optimizes of the UAV trajectory,
GBS beamforming, and AR beamforming coefficients. The numerical analysis
quantifies the impact of different user clustering technique, discount factor
(Gamma) values, learning rates, and number of users on the achievable secrecy
capacity of the system.

The simulation scenario is illustrated in Fig. 1 and consist of multiple
single antenna ground UEs, an AR, and a group of malicious nodes that is
performing a passive eavesdropping attack on the downlink transmission. The
terrestrial users and the eavesdroppers are randomly distributed in a 2D area.
The AR is launched at a random location and height and is equipped with an
antenna array to enable communications with the GBS and the UEs. Table II
captures the simulation parameters. The simulations are performed with Python
3.6 and PyTorch 1.7.

###  V-A Hyper-parameters

The hyper-parameters of the learning algorithm need to be optimized for our
specific problem and environment. Therefore, Fig. 4 and Fig. 5 numerically
evaluate the secrecy capacity of the UEs served through the AR for different
discount factors Gamma and learning rates (LRs). Additionally, Fig. 4 and Fig.
5 verify the convergence of the proposed solution across these different
settings. The results presented in both figures are for the case of 16 ground
users where there are 8 users in each clusters as shown earlier in Fig. 3.
These figures plot the total achieved secrecy capacity of the user cluster
served by the AR over the training time for different hyper-parameter values.

Figure 4: DQL performance for different discount factors Gamma for a learning
rate of $10^{-4}$.  Figure 5: DQL performance for different learning rates
(LRs) for Gamma = 0.9.  Figure 6: Comparison of the learning performance of
the DQL and Q-learning for the UAV trajectory optimization.

When the discount factor is very high, the agent equally considers the future
and current rewards. Fig. 4 shows that this leads to low performance. The best
result for our scenario is achieved by slightly discounting the future reward,
corresponding to a Gamma of 0.9.

By configuring higher LRs, the agent becomes increasingly biased to take the
same action that will enforce the learning policy to be particular to a
deterministic environment. On the other hand, for very low LRs the DQL agent
keeps exploring the environment in a complete random behavior without
learning. A moderate LR provides the equilibrium between a deterministic and
stochastic environment. Fig. 5 compares the learning outcome for three LRs,
where a LR of $10^{-4}$ provides the best result.

Note that one reason of DQN failure is related to the choice of the
hyperparameters. There are a number of search techniques that can be used to
adjust hyperparameters. In this analysis, we have employed the grid search
technique that involves specifying a range of values for each hyperparameter
and then training the DQN with all possible combinations of these values. The
combination of hyperparameters that produces the best performance is then
selected.

###  V-B Learning Performance Evaluation

Figure 6 compares of learning and convergence performance of the proposed DQL
scheme with the Q-learning as a benchmark learning algorithm. It plots the
total secrecy capacity of the user cluster served by the AR over the number of
learning episodes for the DQL and Q-learning trajectory and power
optimization. The curves show that the total secrecy capacity of user served
by the UAV tends to increase over the episodes until convergence. This
validates our approach to define the reward so as to maximize the user rates
with unknown CSI of the eavesdropping channel. Initially, the total secrecy
capacities match for the two algorithms. This is so because of lacking
interaction with the environment to provide enough data for training the
learning agents. As the learning evolves, favorable actions become more easily
discriminated from the unfavorable ones by exploring the environment. It is
noticeable that the DQL performance substantially exceeds the Q-learning
performance due to its ability to approximate the Q-values instead of an
inefficient Q-table of the QL, which allows learning gigantic state and action
in fewer episodes.

(a)

(b)

Figure 7: Comparison of the MSE (a) and MAE (b) losses of the DQL and
Q-learning over the number of episodes.

In order to further analyze the effectiveness and the convergence performance
of the learning performance of the DQL design for optimizing the UAV
trajectory, Fig. 7 plots the mean square error (MSE) and mean absolute error
(MAE) of the DQL and Q-learning solutions over the number of episodes. The
results show how the loss error is minimized by adjusting the weights of the
neural network used to approximate the Q-function. This indicates how well the
proposed algorithm is performing and illustrates its convergence toward an
optimal policy. The MSE and MAE are calculated by comparing the estimated
Q-values using the learned DQL and Q-learning models versus their actually
computed values. Those figures reveal that the quality of the UAV actions are
rather poor during the early training phase. As the learning continues, more
measurements are accumulated that yield to improved actions taken by the UAV
agent for both algorithms. The DQL method meets a MSE target of 50 an order of
magnitude faster than Q-learning (Fig. 7a). It converges faster and achieves a
35% higher secrecy capacity after 20000 episodes (Fig. 6).

Figure 8: Comparison of different clustering techniques.

###  V-C Clustering Performance Evaluation

Here we evaluate the performance of the proposed user clustering scheme and
the employed metric on the total secrecy rate performance of the system. We
consider three metrics for the clustering algorithm presented in Algorithm 1:
distance clustering, where UEs are grouped based on their distances to the
GBS, rate clustering, where UEs are grouped based on their downlink rates
while being served by the GBS, and channel clustering, where UEs are grouped
based on the normalized channel coefficients.

Figure 8, shows the total secrecy capacity of the system after clustering,
beamforming, and UAV trajectory optimization for the three clustering metrics.
We observe that the proposed channel clustering metric outperforms the rate
and distance clustering metrics in terms of total secrecy capacity.

###  V-D Overall Performance Evaluation

We explain the overall performance evaluation of the proposed method in two
parts. In the first part, the impact of the optimal beamforming and power
control performance evaluation is studied. In particular, the proposed method
is compared with three scenarios where our optimal beamforming and power
control are only partially implemented. In the second part, the impact of the
UAV trajectory on the secrecy capacity is studied. Specifically, the UAV’s 3D
movement is shown in a scenario in which two clusters of UEs are
simultaneously served, one by the UAV, which relocates to best serve the UEs
in the cluster, and the other by the GBS.

The context that we study is unique compared to other studies as captured in
Table I. We develop a framework that involves user clustering, multi-user
beamforming, power control, and reinforcement learning for solving the problem
and there are no existing studies that propose comparable solution. Therefore,
we define our own benchmarks to evaluate the proposed framework and the
importance of each component comprising it. The baseline techniques are: AR
deployment without optimal GBS beamforming (UAV+NoBF), no AR deployment with
optimal GBS beamforming (NoUAV+BF), and no AR deployment and without optimal
GBS beamforming (NoUAV+NoBF). In all cases where the AR is deployed the
optimal beamforming and power control of the AR is activated.

Figure 9 shows the achieved total secrecy capacity over the number of users.
The secrecy capacity improves with the number of users for all schemes. The
proposed solution clearly outperforms the other techniques. The UAV+NoBF
scheme achieves a better secrecy capacity than the NoUAV+BF scheme. That is,
deploying an AR is more useful for improving the secrecy capacity than
employing optimal multi-user beamforming at the GBS. Nevertheless, beamforming
and power control schemes have a notable contribution to the secrecy
performance of the system as can be observed when comparing the performance of
the NoUAV+NoBF scheme with the proposed solution and other benchmark
techniques. The optimal beamforming and power control increases the SNR and
reduces the multi-user interference while minimizing the likelihood of
eavesdropping and improving the overall secrecy capacity. The addition of the
UAV as an AR allows to serve those users effectively that have a worse channel
to the GBS. This is accomplished by the proposed clustering method and the UAV
trajectory optimization along with beamforming and power control.

Figure 9: Comparison of different techniques Vs number of users for total
secrecy capacity of the system.  Figure 10: Illustration of channel based user
clustering of Algorithm 1 and UAV trajectory optimization of Algorithm 2.
Figure 11: Secrecy capacity of the proposed optimization framework for the
cases of known and unknown CSI of eavesdroppers, employing the secrecy
capacity and user capacity as the reward function, respectively.

Figure 10 illustrates the dynamic 3D trajectory optimization process for the
case of 16 users where there are 8 users in each cluster as a result of
Algorithm 1 that employs channel based clustering. We observe that the UAV
moves toward the center of the area where the users are located, and its final
position is near the minimum height to be as close as possible to the UEs
served by the AR for ensuring good channels to be able to lower the
transmission power and thus increase the secrecy capacity, while also ensuring
that the UAV stays within its operational limits and avoids ground obstacles.
Overall, the dynamic 3D trajectory optimization process, combined with optimal
multi-user beamforming and power control, helps achieve high secrecy
capacities (Fig. 9) in the presence of eavesdroppers.

###  V-E Known vs. Unknown Information of Eavesdroppers

In order to put our contribution in context and provide further justification
for our optimization framework, we consider the case where the location and
the channel states of eavesdroppers are known to the GBS and the UAV in the
proposed communication context. Knowning the CSI of eavesdropping channels
allows employing the secrecy capacity (24) as the reward function.

We simulate 16 users that are clustered in two groups to be served by the GBS
or AR resulting from the channel based clustering with known eavesdropper
locations and CSI. Figure 11 shows the resulting average secrecy capacity per
user with and without eavesdropping information available to the network. For
the case of unknown eavesdropping channels, we employ the proposed
optimization solution and reward function based on the legitimate user rate.
The UAV adjusts its power and trajectory according to the available
information. As expected, having the information of malicious actors
eavesdropping on the wireless links allows the network to adjust its
parameters better and increase the secrecy capacity.

Figure 11 also indicates that the secrecy capacity performance gap of not
knowing the channel characteristics of eavesdroppers is not significant. In
other words, blindly optimizing the secrecy capacity by focusing on the
legitimate user rates produces an outcome that is very close to an
optimization framework that has and leverages the full information about
eavesdroppers. The reason for this is that the proposed practical solution
with unknown CSI imperceptibly considers the possible CSI between the base
stations and the eavesdroppers.

We conclude that despite the practical assumption of not knowing the CSI of
eavesdropping channels and not exploring methods other than optimizing the
user rates, the proposition of this paper, the proposed communications and
optimization framework can accomplish a performance that is very close to a
network that has access to the full information about eavesdroppers. This
encourages doing further research on improving the proposed technique, for
example, by considering partially known information about eavesdroppers or
other reasonable assumptions, or even exploring new physical layer security
metrics.

###  V-F User Mobility

In this subsection, we examine the secrecy capacity of mobile users with a
static eavesdropper. Without loss of generality, the mobility of the ground
users will be over the x-axis with a fixed y-position. We define the distance
step parameter ($dx$), which corresponds to the granularity of movement. Then,
$UE_{X_{C}^{t+1}}=UE_{X_{C}^{t}}+dx$, can be defined to model the mobility of
the ground users,

Figure 12: Comparing the performance of the proposed solution under user
mobility scenario with other benchmark techniques.

where $UE_{X_{C}^{t+1}}$ is the next center x-positions of all the ground
users in the next time step and for each new center the users are
redistributed randomly around the center. This process enables the users to
simulate a realistic movement pattern that reflects their actual movement.

We consider 16 ground users to be served either directly by the GBS or through
the AR. Additionally, with each movement of step $dx$ , the proposed solution
of Fig. 2 re-clusters the users, re-performs beamforming and power control for
the GBS and UAV transmissions, and re-optimizes the trajectory of the UAV
given the new positions of users. Figure 12 presents the obtained total
secrecy capacity over the center position of the moving user cluster for the
proposed solution and the benchmarks introduced in Fig. 9.

The results of Fig. 12 show that the proposed solution achieves a higher total
secrecy capacity compared to the other schemes. By optimizing the trajectory
of the UAV, the system ensures that it flies as close as possible to the users
being served by the AR enabling good channels and lower transmit powers. We
observe that the secrecy capacity of the proposed solution drops to zero only
for the case where the center of the user cluster matches the eavesdropper
position. After the users pass the eavesdropper, the secrecy capacity rapidly
recovers.

Notice the secrecy capacity after passing the eavesdroppers is lower than
before reaching it. This is because of the lower data rates achieved by the
direct GBS links experiencing a higher path loss with increasing distance. On
the other hand, the NoUAV+NoBF scheme has reached the zero secrecy capacity
much earlier and remains at this state even after leaving the eavesdropper
behind. When comparing the performance of the UAV+NoBF and the NoUAV+BF
schemes, we again realize the effectiveness of deploying the AR for achieving
a higher secrecy capacity.

##  VI Conclusions

This paper addressed the major eavesdropping problem in present-day wireless
communications. We developed a practical framework against passive
eavesdroppers in multi-user cellular networks without knowledge of the
eavesdroppers’ locations and CSI channels. Considering the unknowns, we
optimized the user rates employing advanced wireless techniques at the
physical layer to improve the sum-secrecy capacity among all users in a cell.
Specifically, we suggested employing multi-user beamforming and deploying a
UAV that serves as an AR. We clustered the users into two groups wherein users
are either served by the GBS or by the AR, whose 3D position, multiuser
beamforming matrix, and transmit powers are optimized combining closed-form
expressions with machine learning techniques. Specifically, we designed and
analyzed a DQN for the UAV trajectory optimization subproblem. Numerical
results showed that the proposed system achieves highest secrecy capacities
and scales well over the number of users to be served.

Lessons learned from this work can lead to a number of research directions for
solving open research challenges. We will examine additional UAV specific
operational constraints, including energy consumption, flight time, and speed,
in future work. These are especially important for implementation and
deployments of ARs with today’s small UAVs. One can prototype and validate the
presented techniques on the Aerial Experimentation and Research Platform for
Advanced Wireless (AERPAW) [53], which facilitates implementing the proposed
communications system with software radios and conducting different types of
mobility experiments by leveraging AERPAW’s unmanned ground vehicles.

##  Appendix

###  VI-A Proof of Lemma IV.1

###### Proof.

The substitution of ${\bf\it H}_{1}$, ${\bf\it H}_{2}$, ${\bf\it W}_{br}$, and
${\bf\it W}_{r}$, which are defined in (33), (34), (35), and (36),
respectively, in ${\bf\it y}_{2}$ defined in (32) yields

$\displaystyle{\bf\it y}_{2}$ $\displaystyle={\bf\it
U}_{2}\,{\bf\it\Sigma}_{2}\,\underbrace{{\bf\it V}_{2}^{†}\,{\bf\it
V}_{2}}_{{\bf\it I}}\,\hat{{\bf\it\Lambda}}_{r}\,{\bf\it
U}_{1}^{†}\,{\bf\it\Lambda}_{r}\,{\bf\it
U}_{1}\,{\bf\it\Sigma}_{1}\,\underbrace{{\bf\it V}_{1}^{†}\,{\bf\it
V}_{1}}_{{\bf\it I}}\,{\bf\it\Lambda}_{b}\,{\bf\it U}_{2}^{†}\,{\bf\it
s}_{K}\,$ $\displaystyle\ +\,{\bf\it
U}_{2}\,{\bf\it\Sigma}_{2}\,\underbrace{{\bf\it V}_{2}^{†}\,{\bf\it
V}_{2}}_{{\bf\it I}}\,\hat{{\bf\it\Lambda}}_{r}\,{\bf\it
U}_{1}^{†}\,{\bf\it\Lambda}_{r}\,{\bf\it n}_{1}\,+\,{\bf\it n}_{2}.$ (48)

Therefore we have

$\displaystyle{\bf\it y}_{2}$ $\displaystyle=\underbrace{{\bf\it
U}_{2}\,{\bf\it\Sigma}_{2}\,\hat{{\bf\it\Lambda}}_{r}\,{\bf\it
U}_{1}^{†}\,}_{{\bf\it I}}{\bf\it\Lambda}_{r}\,\underbrace{{\bf\it
U}_{1}\,{\bf\it\Sigma}_{1}\,{\bf\it\Lambda}_{b}\,{\bf\it U}_{2}^{†}}_{{\bf\it
I}}\,{\bf\it s}_{K}\,$ $\displaystyle\quad+\,{\bf\it
U}_{2}\,{\bf\it\Sigma}_{2}\,\hat{{\bf\it\Lambda}}_{r}\,{\bf\it
U}_{1}^{†}\,{\bf\it\Lambda}_{r}\,{\bf\it n}_{1}\,+\,{\bf\it n}_{2},$ (49)

where the matrices ${\bf\it I}$ are obtained using the ZF criterion, i.e.,
${\bf\it U}_{2}\,{\bf\it\Sigma}_{2}\,\hat{{\bf\it\Lambda}}_{r}\,{\bf\it
U}_{1}^{†}={\bf\it I}$ and ${\bf\it
U}_{1}\,{\bf\it\Sigma}_{1}\,{\bf\it\Lambda}_{b}\,{\bf\it U}_{2}^{†}={\bf\it
I}$. As a result, the simplified equation is

$\displaystyle{\bf\it y}_{2}$ $\displaystyle={\bf\it\Lambda}_{r}\,{\bf\it
s}_{K}\,+\,{\bf\it\Lambda}_{r}\,{\bf\it n}_{1}\,+\,{\bf\it n}_{2}$ (50)

in which

${\bf\it\Lambda}_{r}=\begin{pmatrix}\lambda_{r,1}&0&\ldots&0&0\\\
\vdots&\vdots&\ldots&\vdots&0\\\ 0&\ldots&\lambda_{r,k}&0&0\\\
0&\vdots&\ldots&\vdots&0\\\ 0&0&\ldots&0&\lambda_{r,K}\end{pmatrix}_{K\times
K},{\bf\it s}_{K}=\begin{pmatrix}s_{1,1}\\\ \vdots\\\ s_{k,1}\\\ \vdots\\\
s_{K,1}\end{pmatrix}_{K\times 1}.$

${\bf\it\Lambda}_{r}$ is a diagonal matrix and ${\bf\it n}_{1},{\bf\it
n}_{2}\in\mathbb{C}^{K\times 1}$ are the noise vectors. The simplified SINR
can then be written as

$\displaystyle SINR=\frac{\lambda_{r,k}^{2}}{\lambda_{r,k}^{2}+1},$ (51)

which proves Lemma IV.1.

∎

###  VI-B Proof of Lemma IV.2

###### Proof.

The beamforming power at the relay can be simplified as follows

$\displaystyle P_{r}$ $\displaystyle=Tr\big{(}{\bf\it x}_{r}{\bf\it
x}_{r}^{\dagger}\big{)}$ (52) $\displaystyle=Tr\bigg{(}{\bf\it
W}_{r}\Big{(}\underbrace{{\bf\it H}_{1}{\bf\it W}_{b}{\bf\it
W}_{b}^{\dagger}{\bf\it H}_{1}^{\dagger}}_{term\ i}+{\bf\it I}\Big{)}{\bf\it
W}_{r}^{\dagger}\bigg{)},$ (53) $\displaystyle=Tr\bigg{(}{\bf\it
W}_{r}\Big{(}\underbrace{{\bf\it
U}_{1}{\bf\it\Sigma}_{1}{\bf\it\Lambda}_{b}{\bf\it\Lambda}_{b}^{\dagger}{\bf\it\Sigma}_{1}^{\dagger}{\bf\it
U}_{1}^{\dagger}}_{{\bf\it I}}+{\bf\it I}\Big{)}{\bf\it
W}_{r}^{\dagger}\bigg{)},$ (54) $\displaystyle=2\times Tr\Big{(}{\bf\it
W}_{r}\,{\bf\it W}_{r}^{\dagger}\Big{)}$ (55)

where (53) is obtained by replacing (II-B2) and (14) in (52), (54) is derived
by substituting (33) and (35) in term i of (53), and (54) is obtained from
${\bf\it U}_{1}\,{\bf\it\Sigma}_{1}\,{\bf\it\Lambda}_{b}\,{\bf\it
U}_{2}^{†}={\bf\it I}$. Subsequently, by replacing (36) into (55), we have

$\displaystyle P_{r}$ $\displaystyle=2\times Tr\Big{(}{\bf\it W}_{r}\,{\bf\it
W}_{r}^{\dagger}\Big{)}$ $\displaystyle=2\times Tr\Big{(}{\bf\it
V}_{2}\,\underbrace{\hat{{\bf\it\Lambda}}_{r}\,{\bf\it U}_{1}^{†}}_{term\
ii}\,{\bf\it\Lambda}_{r}^{2}\,\underbrace{{\bf\it
U}_{1}\,\hat{{\bf\it\Lambda}}_{r}}_{term\ iii}\,{\bf\it V}_{2}^{†}\Big{)}$
(56) $\displaystyle=2\times Tr\Big{(}{\bf\it
V}_{2}{\bf\it\Sigma}_{2}^{-1}{\bf\it U}_{2}^{†}{\bf\it\Lambda}_{r}^{2}{\bf\it
U}_{2}{\bf\it\Sigma}_{2}^{-1}{\bf\it V}_{2}^{†}\Big{)}$ (57)
$\displaystyle=2\,\sum\limits_{m=1}^{K}\sum\limits_{n=1}^{K}|{\bf\it
U}_{2}(m,n)|^{2}\,\sigma_{2,n}^{-2}\,\lambda_{r,m}^{2},$ (58)

where considering ${\bf\it
U}_{2}\,{\bf\it\Sigma}_{2}\,\hat{{\bf\it\Lambda}}_{r}\,{\bf\it
U}_{1}^{†}={\bf\it I}$ forterm ii and term iii of (56) yields (57). ∎

###  VI-C Proof of Theorem IV.3

###### Proof.

We need to obtain the optimum beamforming power elements and Lagrange
multipliers, i.e., $\lambda_{r,l}^{*}$, $\alpha_{1}^{*}$, $\alpha_{2,l}^{*}$,
and $\alpha_{3,l}^{*}$, where $l=\\{1,\cdots,K\\}$. To this end, we apply the
Karush-Kuhn-Tucker (KKT) conditions to this problem as has been applied to
similar ones [54] [55] [56]. From the gradient condition and the complementary
slackness condition, we have

$\displaystyle\nabla_{\lambda_{r,l}}\mathcal{L}(\lambda_{r,l}^{*},\alpha_{1}^{*},\alpha_{2,l}^{*},\alpha_{3,l}^{*})=0,$
(59)
$\displaystyle-\alpha_{1}^{*}\bigg{(}2\,\sum\limits_{l=1}^{K}\sum\limits_{n=1}^{K}|{\bf\it
U}_{2}(l,n)|^{2}\,\sigma_{2,n}^{-2}\,\lambda_{r,l}^{*\
2}-P_{r,max}\bigg{)}=0,$ (60)
$\displaystyle-\alpha_{2,l}^{*}\Big{(}\lambda_{r,l}^{*}-\lambda_{r,max}\Big{)}=0,$
(61) $\displaystyle-\alpha_{3,l}^{*}\Big{(}-\lambda_{r,l}^{*}\Big{)}=0.$ (62)

By simplifying (59), we obtain

$\displaystyle\frac{2\,\lambda_{r,l}^{*}}{\big{(}2\lambda_{r,l}^{*\,2}+1\big{)}\,\big{(}\lambda_{r,l}^{*\,2}+1\big{)}\,\text{ln}\,2}\
-4\,\alpha_{1}^{*}\,\lambda_{r,l}^{*}$
$\displaystyle\times\,\sum\limits_{n=1}^{K}|{\bf\it
U}_{2}(l,n)|^{2}\,\sigma_{2,n}^{-2}-\,\alpha_{2,l}^{*}\,+\,\alpha_{3,l}^{*}=0.$
(63)

Applying the KKT conditions yields the optimal beamforming power as follow

$\lambda_{r,l}^{*}=\begin{cases}\small
0\qquad&\quad\lambda_{r,l}^{\dagger}\leq
0,(\alpha_{1}^{*}\,F\,\text{ln}\,2)>0.25\\\
\lambda_{r,l}^{\dagger}\qquad&\qquad
0<\lambda_{r,l}^{\dagger}<\lambda_{r,max}\\\
\lambda_{r,max}\qquad&\qquad\quad\ \
\lambda_{r,l}^{\dagger}\geq\lambda_{r,max}\end{cases}\vspace{-7pt}$

in which

$\displaystyle\vspace{-5pt}\lambda_{r,l}^{\dagger}=\sqrt{\frac{1}{4}\Big{(}\sqrt{1+\frac{2}{\alpha_{1}^{*}\,F\,\text{ln}\,2}}-3\Big{)}},$
(64)

where $F$ is a constant that equals to $\sum\limits_{n=1}^{K}|{\bf\it
U}_{2}(l,n)|^{2}\,\sigma_{2,n}^{-2}$, and the Lagrangian multiplier
$\alpha_{1}^{*}$ can be obtained by replacing (43) into the first constraint
of (40) when the equality holds: $\alpha_{1}^{*}=f_{1}(P_{r,max},{\bf\it
U}_{2},\Sigma_{2})$. Note that if the last constraint defined in the condition
(62) is binding, i.e., if $\lambda_{r,l}^{*}=0$, then
$\alpha_{1}^{*}=\alpha_{2,l}^{*}=0$ due to the complementary slackness
conditions. Substituting these multipliers in (59) results in
$\alpha_{3,l}^{*}=0$. Also, replacing $\lambda_{r,l}^{*}=0$ in the objective
function of (39) results in a zero capacity rate, which is not desired. In the
same way as in [55, 56], it can be considered that
$\lambda_{r,l}^{*}=\lambda_{r,max}$ for the values of beamforming powers above
than the maximum. In addition, the value of $\lambda_{r,l}^{\dagger}$ in (64)
can be numerically obtained for different values of channel coefficients and
the UAV’s power limitation, i.e., ${\bf\it U}_{2}$, $\sigma_{2,n}^{-2}$, and
$P_{r,max}$. However, using the Taylor series in (64) at
$x=\frac{2}{\alpha_{1}^{*}\,F\,\text{ln}\,2}$ with negligible $O(x^{2})$, one
can further simplify the obtained $\lambda_{r,l}^{\dagger}$ as

$\displaystyle\vspace{-5pt}\lambda_{r,l}^{\dagger
2}=\frac{1}{4\,f_{1}\,\sum\limits_{n=1}^{K}|{\bf\it
U}_{2}(l,n)|^{2}\,\sigma_{2,n}^{-2}\ \text{ln}\,2}-0.5,$ (65)

where $\lambda_{r,l}^{\dagger}$ is a positive number and $f_{1}$ is the above
defined function of the UAV power and channel coefficients. Finally, it is
worth pointing out that if the UAV uses only one antenna for communicating
with its users, then the beamforming power for the antenna is set to be
$\lambda_{r,max}$. ∎

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|  Aly Sabri Abdalla received the Ph.D. degree in Electrical and Computer
Engineering at Mississippi State University, MS, USA in 2023. He received the
B.S. and M.S. degrees in Electronics and Communications Engineering from the
Arab Academy for Science Technology and Maritime Transport, Egypt, in 2014 and
2019, respectively. Since 2019 he has been a Research Assistant in the
Department of Electrical and Computer Engineering at Mississippi State
University. His research interests include wireless communication and
networking, software radio, spectrum sharing, wireless testbeds and testing,
and wireless security with application to mission-critical communications,
open radio access network (O-RAN), unmanned aerial vehicles (UAVs), and
reconfigurable intelligent surfaces (RISs). He is serving as a student member
of the IEEE Vehicular Technology Society’s Ad Hoc Committee on Drones and IEEE
1920.1: Aerial Communications and Networking.
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|  Ali Behfarnia received the Ph.D. degree in electrical engineering and
computer science from Wichita State University, KS, USA in 2020. He is
currently serving as an Assistant Professor in the Department of Engineering
with the University of Tennessee at Martin. His research interests include
wireless communication systems, game theory, error correction coding, cyber-
physical systems, and the applications of machine and deep learning in
communications over wireless networks.
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|  Vuk Marojevic (Senior Member, IEEE) received the M.S. degree from the
University of Hanover, Germany, in 2003, and the Ph.D. degree from Barcelona
Tech-UPC, Spain, in 2009, all in electrical engineering. He is currently an
Associate Professor with the Department of Electrical and Computer
Engineering, Mississippi State University. His research interests include
4G/5G security, spectrum sharing, software radios, testbeds, resource
management, and vehicular and aerial communications technologies and systems.
Prof. Marojevic is an Editor of the IEEE Transactions on Vehicular Technology,
an Associate Editor of IEEE Vehicular Technology Magazine, and an Officer of
the IEEE ComSoc Aerial Communications Emerging Technology Initiative.
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