# Reflections in the Sky: Joint Trajectory and Passive Beamforming Design for
Secure UAV Networks with Reconfigurable Intelligent Surface

Hui Long1, Ming Chen1, Zhaohui Yang2, Bao Wang1,Zhiyang Li1, Xu Yun1, and
Mohammad Shikh-Bahaei2
1National Mobile Communications Research Laboratory, Southeast University,
Nanjing 210096, China.
2Centre for Telecommunications Research, Department of Informatics, King’s
College London, London WC2B 4BG, UK
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>220180884<EMAIL_ADDRESS><EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
###### Abstract

This paper investigates a problem of secure energy efficiency maximization for
a reconfigurable intelligent surface (RIS) assisted uplink wireless
communication system, where an unmanned aerial vehicle (UAV) equipped with an
RIS works as a mobile relay between the base station (BS) and a group of
users. We focus on maximizing the secure energy efficiency of the system via
jointly optimizing the UAV’s trajectory, the RIS’s phase shift, user
association and transmit power. To tackle this problem, we divide the original
problem into three sub-problems, and propose an efficient iterative algorithm.
In particular, the successive convex approximation (SCA) method is applied to
solve the nonconvex UAV trajectory, the RIS’s phase shift, and transmit power
optimization sub-problems. We further provide two schemes to simplify the
solution of phase and trajectory sub-problem. Simulation results demonstrate
that the proposed algorithm converges fast, and the proposed design can
enhance the secure energy efficiency by up to 38% gains, as compared to the
traditional schemes without any RIS.

###### Index Terms:

UAV communications, reconfigurable intelligent surface, secure communication.

##  I Introduction

Unmanned aerial vehicle (UAV) is playing an increasingly important role in
line of sight (LoS) instant communication[1]. Typical applications of UAV-
assisted communication have established the safe LoS communication links with
ground nodes utilizing its flexible networking structure and feasible low
deployment cost [2, 3, 4, 5, 6, 7, 8, 9]. Achieving secure transmission of
confidential information and avoiding eavesdropping have always been an
important problem in the design of wireless communication systems. In the
existing research on physical layer security, the communication nodes are
static which means the channel quality of eavesdropper-base station (BS) or
legal user-BS mainly depends on the location of eavesdropper and legal users.
If the distance between the BS and the legitimate receiver/edvesdropper is
fixed, the achievable secrecy rate will be limited even if techniques such as
artificial noise (AN) and power control are applied. However, UAV not only
establishes stronger legitimate links with the ground nodes by designing its
trajectory, but also can detect any potential eavesdroppers by equipping with
optical cameras. As a result, in this paper, we focus on the application of
UAV on the physical layer security.

A number of existing works such as [10, 11, 12] have studied on security for
various UAV communication systems. Considering a typical three-node
eavesdropping scenario, the work in [10] improved the average secrecy rate
through optimizing the UAV trajectory under total transmit power constraint of
the BS. Considering the general multi-user scenario, the authors in [11]
maximized the minimum secrecy rate by controlling the user association as well
as considering the trajectory and transmit power. Different from the works on
the ground to air communication system in 2-dimensional (2D) space [10][11],
the work in [12] investigated the more complicated air-to-air (A2A) systems in
3-dimensional (3D) space. By characterizing the statistical characteristics of
the signal-to-noise-ratio (SNR) over the A2A links, the authors in [12]
obtained the closed-form expressions for secrecy outage probability.

Recently, reconfigurable intelligent surface (RIS) has been widely used in
improving the energy efficiency and communication equality in wireless
networks [13, 14, 15, 16, 17, 18, 19, 20]. An RIS is made up of a number of
configurable elements that can reflect the incident signal by controlling its
phase shifts appropriately. Different from the traditional amplify and forward
(AF) relays, RIS is almost passive, and does not incur energy cost[21].
Besides, an RIS will not induce or amplify noise while reflecting signals. It
is worth noticing that there have been several works on RIS taking the secrecy
into consideration including multiple-input single-output (MISO) systems[22,
23, 24, 25] and multiple-input multiple-output (MIMO) systems [26]. The
authors in [22] maximized the power of received signal under the transmit
power and the unit modulus constraints, and further improved secrecy rate for
a MISO system. For the MIMO case, the authors in [26] proposed an iterative
optimization method to maximize the secrecy rate with respect to the RIS’s
phase shift coefficient and the transmit covariance. However, the
aforementioned works in [22, 23, 24, 25, 26] do not exploit the RIS’s
characteristic in the mobile UAV-enabled communication system even though UAV
can further improve the secrecy rate performance. Meanwhile, energy efficiency
has emerged as an important performance index for deploying green and
sustainable wireless networks [27],[28]. In consequence, it is of importance
to investigate the application of RIS in improving secrecy energy efficiency,
which is defined as the ratio of the minimum secrecy rate to the total power
consumption.

Motivated by the previous works, we investigate the secrecy energy efficiency
of an UAV-enabled system by taking the advantages of RIS in this paper.
Considering the complex outdoor environment, tall building may block the LoS
communication links between ground users and BS which seriously affects the
channel quality. To improve the channel quality between users and the BS, an
UAV relay with one RIS is considered in this paper. Our goal is to maximize
the secrecy energy efficiency. The main contributions of this paper are
summarized as follows:

  * •

To solve this nonconvex secrecy energy efficiency maximization problem, an
alternating method is proposed with solving three sub-problems iteratively.
For the integer user association sub-problem, it is relaxed to a linear
problem. For the power control sub-problem, the successive convex
approximation (SCA) method is adopted. For the phase and trajectory
optimization sub-problem, the optimal phase to maximize the user rate is
derived first, and then an alternating method is proposed by analyzing the
convexity of the secrecy rate expression with respect to the trajectory
variable.

  * •

To compare the method for the phase and trajectory optimization sub-problem,
we also provide the detailed procedures to jointly optimize phase and
trajectory via SCA.

  * •

Simulation results show that our proposed approach can enhance the secrecy
energy efficiency by up to 38% gains compared to the conventional AF relay
scheme.

The rest of this paper is organized as follows. System model and problem
formulation are described in Section II. Section III provides the algorithm
design and analysis. Section IV presents the simulation results to demonstrate
the performance of the proposed algorithm. Conclusions are drawn in Section V.

##  II System Model and Problem Formulation

###  II-A System Model

Figure 1: An uplink RIS-assisted wireless communication system.

Consider an uplink wireless communication system with one UAV, one
eavesdropper (Eve), $K$ users, and one BS, as shown in Fig. 1. The set of $K$
users is denoted by $\mathcal{K}=\\{1,\cdots,K\\}$. Due to the obstacle of
high walls, there is non LoS channel between the BS and each user. The UAV
equipped with one RIS serves as a passive relay to assist the communication
between the users and the BS. The RIS is equipped with a uniform linear array
(ULA) of $M$ reflecting elements and the phase of each element can be
controlled by the UAV.

Eve, all users, and the BS are located at the ground. The horizontal
coordinates of user $k$ where $k\in\mathcal{K}$ , Eve, and BS are denoted by
$\bm{w}_{k}=[x_{k},y_{k}]^{T}$, $\bm{w}_{e}=[x_{e},y_{e}]^{T}$, and
$\bm{w}_{b}=[x_{b},y_{b}]^{T}$, respectively. In this system, multiple users
are served in different time intervals. The UAV flies at a fixed altitude $H$
with flight period $T$. To facilitate the analysis, the UAV flight period $T$
is divided into $N$ equally-spaced time slots with step size $\delta$, i.e.,
$T=N\delta$. Denote $\mathcal{N}=\\{1,\cdots,N\\}$ as the set of all discrete
time slots. The time-variant horizontal coordinate of the UAV in time slot $n$
is denoted by $\bm{q}[n]=[x[n],y[n]]^{T}$, $n\in\mathcal{N}$.

To serve the users periodically, the UAV needs to return back to the initial
position by the end of period $T$, i.e.,

$\bm{q}[N]=\bm{q}[0],$ (1)

where $\bm{q}[0]=[x[0],y[0]]^{T}$ is the predetermined initial horizontal
coordinate of the UAV. With given maximal UAV speed $V_{\max}$, the number of
time slots $N$ can be chosen properly such that the time for UAV location
changing within time $\delta$ can be negligible. As a result, we have

$\|\bm{q}[n]-\bm{q}[{n-1}]\|\leq S_{\max},$ (2)

where $S_{\max}=V_{\max}\delta$ is the maximal horizontal distance that the
UAV can travel within one time slot.

The channel gain between user $k$ and the UAV in time slot $n$ can be
expressed as [13]

$\bm{g}_{k}[n]=\sqrt{h_{0}d_{k}^{-\alpha}[n]}[1,e^{-j\frac{2\pi
d}{\lambda}\phi_{k}[n]},\cdots,e^{-j\frac{2(M-1)\pi
d}{\lambda}\phi_{k}[n]}]^{T},$ (3)

where $h_{0}$ is the channel gain at a reference distance $d_{0}=1$ m,
$d_{k}[n]=\sqrt{\|\bm{q}[n]-\bm{w}_{k}\|^{2}+H^{2}}$ is the distance between
user $k$ and the UAV in time slot $n$, $\alpha\geq 2$ is the pathloss
exponent, $\phi_{k}[n]=\frac{x[n]-x_{k}}{d_{k}[n]}$ represents the cosine of
the angle of arrival (AoA) of the signal from user $k$ to the ULA at the RIS
in time slot $n$, $d$ is the antenna separation, and $\lambda$ is the carrier
wavelength.

Similarly, the channel gain between the UAV and the BS in time slot $n$ can is
given by

$\bm{g}_{b}[n]=\sqrt{h_{0}d_{b}^{-\alpha}[n]}[1,e^{-j\frac{2\pi
d}{\lambda}\phi_{b}[n]},\cdots,e^{-j\frac{2(M-1)\pi
d}{\lambda}\phi_{b}[n]}]^{T},$ (4)

where $d_{b}[n]=\sqrt{\|\bm{q}[n]-\bm{w}_{b}\|^{2}+H^{2}}$ and
$\phi_{b}[n]=\frac{x[n]-x_{b}}{d_{b}[n]}$. The channel gain between the UAV
and Eve in time slot $n$ can be expressed as

$\bm{g}_{e}[n]=\sqrt{h_{0}d_{e}^{-\alpha}[n]}[1,e^{-j\frac{2\pi
d}{\lambda}\phi_{e}[n]},\cdots,e^{-j\frac{2(M-1)\pi
d}{\lambda}\phi_{e}[n]}]^{T},$ (5)

where $d_{e}[n]=\sqrt{\|\bm{q}[n]-\bm{w}_{e}\|^{2}+H^{2}}$ and
$\phi_{e}[n]=\frac{x[n]-x_{e}}{d_{e}[n]}$.

Let the binary variable $a_{k}[n]$ denote the association of user $k$ in time
slot $n$, i.e., $a_{k}[n]=1$ represents that user $k$ is associated with the
UAV; otherwise $a_{k}[n]=0$. Assume that at most one user is served in each
time slot, i.e.,

$\sum_{k=1}^{K}a_{k}[n]\leq 1,\quad\forall n\in\mathcal{N}$ (6)

Based on (3) and (4), the achievable rate from user $k$ to the BS via RIS in
time slot $n$ can be given by

$r_{k}[n]=\log_{2}\left(1+\frac{p_{k}[n]|\bm{g}_{b}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]|^{2}}{\sigma^{2}}\right),$
(7)

where $\bm{\Theta}[n]$ is the phase shift matrix of the RIS in time slot $n$,
$\sigma^{2}$ is the noise power, and $p_{k}[n]$ is the transmit power of user
$k$ in time slot $n$. Matrix
$\bm{\Theta}[n]=\text{diag}(\text{e}^{j\theta_{1}[n]},\cdots,\text{e}^{j\theta_{M}[n]})\in\mathbb{C}^{M\times
M}$ with $\theta_{m}[n]\in[0,2\pi]$, which captures the effective phase shifts
applied by all reflecting elements of the RIS. If user $k$ transmits data to
the BS in time slot $n$, the achievable rate at Eve is

$c_{k}[n]=\log_{2}\left(1+\frac{p_{k}[n]|\bm{g}_{e}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]|^{2}}{\sigma^{2}}\right).$
(8)

According to [11], the secrecy rate of user $k$ in time slot $n$ can be
expressed as

$R_{k}[n]=a_{k}[n]\max\\{r_{k}[n]-c_{k}[n],0\\}.$ (9)

Considering the fairness among all users, we provide the minimum secrecy rate
of all users as follow

$\zeta=\min_{k\in\mathcal{K}}\frac{1}{N}\sum_{n=1}^{N}R_{k}[n].$ (10)

###  II-B Problem Formulation

Our aim is to maximize the fair secrecy energy efficiency, via joint
trajectory, transmit power, and passive beamforming optimization.
Mathematically, the optimization problem can be formulated as

$\displaystyle\\!\\!\mathop{\max}_{\zeta,\bm{Q},\bm{A},\bm{P},\bm{\Theta}}\quad$
$\displaystyle\frac{\zeta}{\sum_{k=1}^{K}\sum_{n=1}^{N}p_{k}[n]+P_{0}}$ (11a)
s.t. $\displaystyle\zeta\leq\frac{1}{N}\sum_{n=1}^{N}R_{k}[n],\quad\forall
k\in\mathcal{K}$ (11b) $\displaystyle\|\bm{q}[n]-\bm{q}[{n-1}]\|\leq
S_{\max},\quad\forall n\in\mathcal{N}$ (11c)
$\displaystyle\sum_{k=1}^{K}a_{k}[n]\leq 1,\quad\forall n\in\mathcal{N}$ (11d)
$\displaystyle a_{k}[n]\in\\{0,1\\},\quad\forall
n\in\mathcal{N},k\in\mathcal{K}$ (11e) $\displaystyle p_{k}[n]\leq
P_{k},\quad\forall n\in\mathcal{N},k\in\mathcal{K}$ (11f)
$\displaystyle\bm{q}[N]=\bm{q}[0],$ (11g)
$\displaystyle\theta_{m}[n]\in[0,2\pi],\quad\forall
m\in\mathcal{M},n\in\mathcal{N},$ (11h)

where $\bm{Q}=\\{\bm{q}[n]\\}_{\forall n}$, $\bm{A}=\\{a_{k}[n]\\}_{\forall
k,n}$, $\bm{P}=\\{p_{k}[n]\\}_{\forall k,n}$,
$\bm{\Theta}=\\{\theta_{m}[n]\\}_{\forall m,n}$, $P_{0}$ is the circuit power
consumption of the system, $P_{k}$ is the maximum transmit power of user $k$,
and $\mathcal{M}=\\{1,\cdots,M\\}_{\forall m\in\mathcal{M}}$.

##  III Proposed Algorithm

Problem (11) is a nonconvex problem due to the nonconvex and discrete
constraints. In this section, a sub-optimal solution that contains the SCA and
alternating methods is provided to tackle with problem (11).

###  III-A User Association Optimization

To make problem (11) tractable, we relax the binary variables in (11e) into
continuous variables. Thus, with given transmit power $\bm{P}$, RIS’s phase
shift matrix $\bm{\Theta}$, and UAV trajectory $\bm{Q}$, the user association
problem can be optimized by solving the following problem

$\displaystyle\\!\\!\mathop{\max}_{\zeta,\bm{A}}\quad$ $\displaystyle\zeta$
(12a) s.t.
$\displaystyle\zeta\leq\frac{1}{N}\sum_{n=1}^{N}R_{k}[n],\quad\forall
k\in\mathcal{K}$ (12b) $\displaystyle\sum_{k=1}^{K}a_{k}[n]\leq 1,\quad\forall
n\in\mathcal{N}$ (12c) $\displaystyle 0\leq a_{k}[n]\leq 1,\quad\forall
n\in\mathcal{N},k\in\mathcal{K}.$ (12d)

Problem (12) is a standard linear programming and can be solved efficiently by
existing optimization tools such as CVX. We can obtain the optimal solution of
problem (12), and then using the rounding method to further get the integer
solution.

###  III-B Power Optimization

With fixed user association $\bm{A}$, UAV trajectory $\bm{Q}$, and RIS’s phase
shift matrix $\bm{\Theta}$, the transmit power optimization problem reduces to

$\displaystyle\\!\\!\mathop{\max}_{\zeta,\bm{P}}\quad$
$\displaystyle\frac{\zeta}{\sum_{k=1}^{K}\sum_{n=1}^{N}p_{k}[n]+P_{0}}$ (13a)
s.t. $\displaystyle\zeta\leq\frac{1}{N}\sum_{n=1}^{N}R_{k}[n],\quad\forall
k\in\mathcal{K}$ (13b) $\displaystyle p_{k}[n]\leq P_{k},\quad\forall
n\in\mathcal{N},k\in\mathcal{K}.$ (13c)

Problem (13) is nonconvex as $R_{k}[n]$ in the right hand side of constraint
(13b) is a difference of two concave functions with respect to the power
control variables $p_{k}[n]$. We can use the SCA method to solve problem (13)
by sequentially solving a series of convex approximation problems. Let
$p_{k}^{(r)}[n]$ denote the transmit power of user $k$ in the $r$-th iteration
of SCA method. As the $c_{k}[n]$ is the concave function of $p_{k}[n]$, we
have

$\displaystyle c_{k}[n]$
$\displaystyle=\log_{2}\left(1+\frac{p_{k}[n]|\bm{g}_{e}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]|^{2}}{\sigma^{2}}\right)$
$\displaystyle\leq\log_{2}\left(1+\frac{p_{k}^{(r)}[n]|\bm{g}_{e}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]|^{2}}{\sigma^{2}}\right)$
$\displaystyle+\frac{|\bm{g}_{e}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]|^{2}\left(p_{k}[n]-p_{k}^{(r)}[n]\right)}{\ln
2\left(\sigma^{2}+p_{k}^{(r)}[n]|\bm{g}_{e}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]|^{2}\right)}$
$\displaystyle\triangleq\hat{c}_{k}[n].$ (14)

Replacing $c_{k}[n]$ with $\hat{c}_{k}[n]$, problem (13) is equivalent to

$\displaystyle\\!\\!\mathop{\max}_{\zeta,\bm{P}}\quad$
$\displaystyle\frac{\zeta}{\sum_{k=1}^{K}\sum_{n=1}^{N}p_{k}[n]+P_{0}}$ (15a)
s.t.
$\displaystyle\zeta\leq\frac{1}{N}\sum_{n=1}^{N}a_{k}[n]\max\\{r_{k}[n]-\hat{c}_{k}[n],0\\},\quad\forall
k\in\mathcal{K}$ (15b) $\displaystyle p_{k}[n]\leq P_{k},\quad\forall
n\in\mathcal{N},k\in\mathcal{K}.$ (15c)

Problem (15) is a nonlinear fractional programming[29]. We solve this problem
by adding a multiplication factor $\eta$, and problem (15) is approximated as
the following problem

$\displaystyle\\!\\!\mathop{\max}_{\zeta,\bm{P}}\quad$
$\displaystyle\zeta-\eta\left(\sum_{k=1}^{K}\sum_{n=1}^{N}p_{k}[n]+P_{0}\right)$
(16a) s.t.
$\displaystyle\zeta\leq\frac{1}{N}\sum_{n=1}^{N}a_{k}[n]\max\\{r_{k}[n]-\hat{c}_{k}[n],0\\},\forall
k\in\mathcal{K}$ (16b) $\displaystyle p_{k}[n]\leq P_{k},\quad\forall
n\in\mathcal{N},k\in\mathcal{K},$ (16c)

where $\eta$ can be obtained by Algorithm 1. It is proved that the solution of
problem (15) can be obtained by solving problem (16) according to reference
[29].

Algorithm 1 Iterative Algorithm for Problem (13)

1:Initialize $\zeta^{(0)}$, $\bm{P}^{(0)}$, the tolerance $\varepsilon$, and
the iteration number $t=1$.

2:repeat

3:Calculate
$\vspace{-.5em}\eta^{(t)}=\frac{\zeta^{(t-1)}}{\sum_{k=1}^{K}\sum_{n=1}^{N}p_{k}^{(t-1)}[n]+P_{0}}.$
(17)

4:Solve the following optimization problem
$\displaystyle\left({{\zeta^{(t)}},{\bm{P}^{(t)}}}\right)=\mathop{\arg\max}_{\zeta,\bm{P}}\quad\zeta-\eta^{(t)}\left(\sum_{k=1}^{K}\sum_{n=1}^{N}p_{k}[n]+P_{0}\right)$
s.t. $\displaystyle\zeta\leq\frac{1}{N}\sum_{n=1}^{N}\max
a_{k}[n]\\{r_{k}[n]-\hat{c}_{k}[n],0\\},\quad\forall k\in\mathcal{K}$
$\displaystyle p_{k}[n]\leq P_{k},\quad\forall
n\in\mathcal{N},k\in\mathcal{K}.$ (18)

5:Update $t=t+1$.

6:until $\left|\zeta^{(t)}-\zeta^{(t-1)}\right|\leq\varepsilon$.

The suboptimal solution of problem (13) can be obtained through the iterative
algorithm in Algorithm 1. Note that problem (4) in Algorithm 1 is a convex
problem, which can be easily solved by the standard toolbox, such as CVX.

###  III-C Joint Phase and UAV Trajectory Optimization (Scheme I)

For any given user association $\bm{A}$ and transmit power $\bm{P}$, the
optimization problem of UAV trajectory $\bm{Q}$ and RIS’s phase shift matrix
$\bm{\Theta}$ can be reformulated as

$\displaystyle\\!\\!\mathop{\max}_{\zeta,\bm{Q},\bm{\Theta}}\quad$
$\displaystyle\zeta$ (19a) s.t.
$\displaystyle\zeta\leq\frac{1}{N}\sum_{n=1}^{N}R_{k}[n],\quad\forall
k\in\mathcal{K}$ (19b) $\displaystyle\|\bm{q}[n]-\bm{q}[{n-1}]\|\leq
S_{\max},\quad\forall n\in\mathcal{N}$ (19c)
$\displaystyle\bm{q}[N]=\bm{q}[0]$ (19d)
$\displaystyle\theta_{m}[n]\in[0,2\pi],\quad\forall
m\in\mathcal{M},n\in\mathcal{N}.$ (19e)

Problem (19) can be solved in two steps: passive beamforming optimization and
UAV trajectory optimization.

####  III-C1 Passive Beamforming Optimization

For the optimization of $\bm{\Theta}$, we can align the phases of the received
signal at BS to maximize the received signal energy.

Firstly, considering the optimization of $\bm{\Theta}\ $ with any given
$\bm{Q}$, expression $\bm{g}_{b}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]$ can be
written as

$\bm{g}_{b}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]=\frac{h_{0}\sum\limits_{m=1}^{M}e^{j\left(\theta_{m}[n]+\frac{2(m-1)\pi
d}{\lambda}\left(\phi_{b}[n]-\phi_{k}[n]\right)\right)}}{\sqrt{d^{\alpha}_{b}[n]d^{\alpha}_{k}[n]}}.$
(20)

To maximize the received signal energy, we combine the signals from different
paths coherently at BS. Thus, we set $\theta_{1}[n]=\theta_{2}[n]+\frac{2\pi
d}{\lambda}\left(\phi_{b}[n]-\phi_{k}[n]\right)=...=\theta_{M}[n]+\frac{2\pi(M-1)d}{\lambda}\left(\phi_{b}[n]-\phi_{k}[n]\right)=\omega$,
or re-expressed as

$\theta_{m}[n]=\frac{2\pi(m-1)d}{\lambda}\left(\phi_{k}[n]-\phi_{b}[n]\right)+\omega,\quad\forall
m,n,k$ (21)

where $\omega\in[0,2\pi]$. In that way, we achieve the phase alignment of the
received signal and further maximize the received signal energy. Thus,
$\bm{g}_{b}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]$ can be rewritten as

$\bm{g}_{b}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]=\frac{h_{0}Me^{j\omega}}{\sqrt{d^{\alpha}_{b}[n]d^{\alpha}_{k}[n]}}.$
(22)

Based on (22), the achievable rate of user $k$ in time slot $n$ can be
rewritten as

$r_{k}[n]=\log_{2}\left(1+\frac{B}{d^{\alpha}_{b}[n]d^{\alpha}_{k}[n]}\right),$
(23)

where $B=\frac{p_{k}[n]\left|h_{0}\right|^{2}M^{2}}{\sigma^{2}}$.

Given the phase shift results in (21),
$\left|\bm{g}_{e}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]\right|^{2}$ can be written
as

$\left|\bm{g}_{e}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]\right|^{2}=\frac{\left|h_{0}\right|^{2}C^{2}}{d^{\alpha}_{k}[n]d^{\alpha}_{e}[n]},$
(24)

where $C=\left|\sum\limits_{m=1}^{M}e^{j\left(\theta_{m}[n]+\frac{2(m-1)\pi
d}{\lambda}\left(\phi_{e}[n]-\phi_{k}[n]\right)\right)}\right|$.

The effective channel gain between the user and Eve in (24) is a complicated
function of the UAV trajectory. To make the problem tractable, we provide the
following upper bound, i.e.,

$\left|\bm{g}_{e}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]\right|^{2}\leq\frac{\left|h_{0}\right|^{2}M^{2}}{d^{\alpha}_{k}[n]d^{\alpha}_{e}[n]}.$
(25)

####  III-C2 UAV Trajectory Optimization

Based on (23) and (25), we first introduce slack variables
$\bm{Z}=\left\\{z_{k}[n]\right\\}_{\forall k,n}$ and
$\bm{V}=\left\\{v_{k}[n]\right\\}_{\forall k,n}$. Problem (19) can be
reformulated as

$\displaystyle\\!\\!\mathop{\max}_{\zeta,\bm{Q},\bm{Z},\bm{V}}\quad$
$\displaystyle\zeta$ (26a) s.t.
$\displaystyle\zeta\leq\frac{1}{N}\sum_{n=1}^{N}a_{k}[n]\max\left\\{r_{k}[n]-c_{k}[n],0\right\\},\forall
k\in\mathcal{K}$ (26b) $\displaystyle\|\bm{q}[n]-\bm{q}[{n-1}]\|\leq
S_{\max},\quad\forall n\in\mathcal{N}$ (26c)
$\displaystyle\bm{q}[N]=\bm{q}[0]$ (26d) $\displaystyle
z^{\frac{2}{\alpha}}_{k}[n]\geq d_{k}^{2}[n]d_{b}^{2}[n],\quad\forall
n\in\mathcal{N},k\in\mathcal{K}$ (26e) $\displaystyle
v^{\frac{2}{\alpha}}_{k}[n]\leq d_{k}^{2}[n]d_{e}^{2}[n],\quad\forall
n\in\mathcal{N},k\in\mathcal{K},$ (26f)

where $r_{k}[n]\\!=\\!\log_{2}\left(1+\frac{B}{z_{k}[n]}\right)$ and
$c_{k}[n]\\!=\\!\log_{2}\left(1+\frac{B}{v_{k}[n]}\right)$. To maximize the
objective value $\zeta$, constraints (26e) and (26f) must hold with equality
at the optimal solution of problem (26). To solve problem (26), we introduce
an important lemma as follows.

###### Lemma 1

Given $B>0$, function $r_{k}[n]$ is convex with respect to $z_{k}[n]>0$.

###### Proof III.1

See Appendix A.

Using the same method in Appendix A, $c_{k}[n]$ can be proved to be convex
with respect to $v_{k}[n]$.

With Lemma 1, constraint (26b) is the difference of two convex functions. Set
$z^{(l)}_{k}[n]$ as the given points
$\bm{Z}=\left\\{z_{k}[n]\right\\}_{\forall k,n}$ in the $l$-th iteration, we
obtain the following lower bound for $r_{k}[n]$ in (26b), i.e.,

$\displaystyle r_{k}[n]$
$\displaystyle=\log_{2}\left(1+\frac{B}{z_{k}[n]}\right)$
$\displaystyle\geq\log_{2}\left(1+\frac{B}{z^{(l)}_{k}[n]}\right)$
$\displaystyle+\frac{-B\left(z_{k}[n]-z^{(l)}_{k}[n]\right)}{z^{(l)}_{k}[n](z^{(l)}_{k}[n]+A)\ln
2}\triangleq\hat{r}_{k}[n].$ (27)

Therefore, the nonconvexity of constraint (26b) can be handled based on
(III-C2). Before tackling constraint (26e) and (26f), we show another
important lemma as follows.

###### Lemma 2

$d_{k}^{2}[n]$, $d_{k}^{4}[n]$ are convex functions of $\bm{q}[n]$.

###### Proof III.2

See Appendix B.

Lemma 2 is proved by checking the Hessian matrix of $d_{k}^{2}[n]$,
$d_{k}^{4}[n]$. Similarly, $d_{b}^{2}[n]$, $d_{e}^{2}[n]$, $d_{b}^{4}[n]$,
$d_{e}^{4}[n]$ are all convex functions of $\bm{q}[n]$.

Problem (26) is still nonconvex due to the nonconvex terms
$d_{k}^{2}[n]d_{b}^{2}[n]$ and $d_{e}^{2}[n]d_{b}^{2}[n]$. We can obtain the
upper bound function of $d_{k}^{2}[n]d_{b}^{2}[n]$ via its first-order Taylor
expansion at any given point $d^{(l)}_{b}[n]$, $d^{(l)}_{k}[n]$,
$d^{(l)}_{e}[n]$ and $\bm{q}^{(l)}[n]$, i.e.,

$\displaystyle d_{k}^{2}[n]d_{b}^{2}[n]$
$\displaystyle=\frac{1}{2}\left[\left(d_{k}^{2}[n]+d_{b}^{2}[n]\right)^{2}-\left(d_{k}^{4}[n]+d_{b}^{4}[n]\right)\right]$
$\displaystyle\leq\frac{1}{2}\left[\left(d_{k}^{2}[n]+d_{b}^{2}[n]\right)^{2}-\left(\left(d_{k}^{(l)}[n]\right)^{4}+\left(d_{b}^{(l)}[n]\right)^{4}\right)\right]$
$\displaystyle-2\left(d_{k}^{(l)}[n]\right)^{2}\left(\bm{q}^{(l)}[n]-\bm{w}_{k}\right)^{T}\cdot\left(\bm{q}[n]-\bm{q}^{(l)}[n]\right)$
$\displaystyle-2\left(d_{b}^{(l)}[n]\right)^{2}\left(\bm{q}^{(l)}[n]-\bm{w}_{b}\right)^{T}\cdot\left(\bm{q}[n]-\bm{q}^{(l)}[n]\right)$
$\displaystyle\triangleq f\left(\bm{q}[n]\right).$ (28)

Similarly, the lower bound function of $d_{k}^{2}[n]d_{e}^{2}[n]$ can be
written as

$\displaystyle d_{k}^{2}[n]d_{e}^{2}[n]$
$\displaystyle=\frac{1}{2}\left[\left(d_{k}^{2}[n]+d_{e}^{2}[n]\right)^{2}-\left(d_{k}^{4}[n]+d_{e}^{4}[n]\right)\right]$
$\displaystyle\geq\frac{1}{2}\left[\left(\left(d_{k}^{(l)}[n]\right)^{2}+\left(d_{b}^{(l)}[n]\right)^{2}\right)^{2}-\left(d_{k}^{4}[n]+d_{e}^{4}[n]\right)\right]$
$\displaystyle+2\left(d_{k}^{(l)}[n]\right)^{2}\left(2\bm{q}^{(l)}[n]-\bm{w}_{k}-\bm{w}_{e}\right)^{T}\cdot\left(\bm{q}[n]-\bm{q}^{(l)}[n]\right)$
$\displaystyle+2\left(d_{e}^{(l)}[n]\right)^{2}\left(2\bm{q}^{(l)}[n]-\bm{w}_{k}-\bm{w}_{e}\right)^{T}\cdot\left(\bm{q}[n]-\bm{q}^{(l)}[n]\right)$
$\displaystyle\triangleq g\left(\bm{q}[n]\right).$ (29)

Note that $v^{\frac{2}{\alpha}}_{k}[n]$ is concave, the first-order Taylor
expansions of $v^{\frac{2}{\alpha}}_{k}[n]$ at the given point
$v^{(l)}_{k}[n]$ is applied to make constraint (26f) feasible for convex
optimization, thus

$\displaystyle v^{\frac{2}{\alpha}}_{k}[n]\leq$
$\displaystyle\left(v^{(l)}_{k}[n]\right)^{\frac{2}{\alpha}}+\frac{2}{\alpha}\left(v^{(l)}_{k}[n]\right)^{\frac{2}{\alpha}-1}\left(v_{k}[n]-v^{(l)}_{k}[n]\right)$
$\displaystyle\triangleq$ $\displaystyle h\left(v_{k}[n]\right).$ (30)

With (III-C2)-(III-C2), the optimization of UAV trajectory can be formulated
as

$\displaystyle\\!\\!\mathop{\max}_{\zeta,\bm{Q},\bm{Z},\bm{V}}\quad$
$\displaystyle\zeta$ (31a) s.t.
$\displaystyle\zeta\leq\frac{1}{N}\sum_{n=1}^{N}a_{k}[n]\left[\hat{r}_{k}[n]-\log_{2}\left(1+\frac{B}{v_{k}[n]}\right)\right]^{+}$
(31b) $\displaystyle\|\bm{q}[n]-\bm{q}[{n-1}]\|\leq S_{\max},\quad\forall
n\in\mathcal{N}$ (31c) $\displaystyle\bm{q}[N]=\bm{q}[0]$ (31d) $\displaystyle
z^{\frac{2}{\alpha}}_{k}[n]\geq f\left(\bm{q}[n]\right)\quad\forall
n\in\mathcal{N},k\in\mathcal{K}$ (31e) $\displaystyle
h\left(v_{k}[n]\right)\leq g\left(\bm{q}[n]\right)\quad\forall
n\in\mathcal{N},k\in\mathcal{K}.$ (31f)

Since all the constraints in (31) are convex and the objective function is
linear, problem (31) is convex, which can be effectively solved via the
conventional methods such as CVX. Note that problem (19) also can be solved by
joint optimization with SCA method (scheme II) as shown in Appendix C.

###  III-D Overall Algorithm and Analysis

In summary, the overall algorithm for solving problem (11) is given in
Algorithm 2. The convergence as well as complexity of the proposed Algorithm 2
are given as follows.

###### Lemma 3

Algorithm 2 is guaranteed to converge.

###### Proof III.3

See Appendix D.

From Algorithm 2, the complexity of solving problem (11) is dominated by the
complexity of solving three sub-problems : user association sub-problem (12),
power control sub-problem (13), the phase and trajectory optimization sub-
problem (19).

The complexity of user association sub-problem is the total number of
variables $KN$, as problem (12) is a standard linear programing problem. The
power control sub-problem is solved by SCA method and its complexity depends
on its variables and constraints. Since there are $KN+1$ constraints in
problem (13), the number of iteration required for SCA method is
$\mathcal{O}\left(\sqrt{KN+1}\log_{2}{(1/\epsilon_{1}})\right)$, where
$\epsilon_{1}$ is the accuracy of SCA method for solving problem (13). Note
that $S_{1}=KN+1$ is the total number of variables and $S_{2}=K(N+1)$ is the
total number of constraints. Thus, the complexity of solving problem (16) at
each iteration is $\mathcal{O}\left(S_{1}^{2}S_{2}\right)$. In consequence,
the complexity of solving problem (13) is
$\mathcal{O}\left((KN+1)^{2.5}(KN+K)\log_{2}{(1/\epsilon_{1}})\right)$ ( For
simplicity, it can be equivalent to
$\mathcal{O}\left((KN)^{3.5}\log_{2}{(1/\epsilon_{1}})\right)$. We only
provide the simplidied forms in the following analysis.). In scheme I, the
complexity for solving problem (19) is dominated by the complexity of solving
problem (31) $\mathcal{O}\left((2KN)^{3.5}\log_{2}{(1/\epsilon_{2}})\right)$,
where $\epsilon_{2}$ is the accuracy of SCA method for solving problem (31).
Similarly, in scheme II the complexity of solving problem (57) is
$\mathcal{O}\left(((6K+M)N)^{3.5}\log_{2}{(1/\epsilon_{3}}\right)$, where
$\epsilon_{3}$ is the accuracy of SCA method for solving problem (57).

In conclusion, the total complexity of Algorithm 2 with scheme I for solving
problem (11) is
$\mathcal{O}\left(S_{3}\left(KN+(KN)^{3.5}\log_{2}{(1/\epsilon_{1}})+(2KN)^{3.5}\log_{2}{(1/\epsilon_{2}})\right)\right)$,
where $S_{3}$ is the number of iteration for Algorithm 2. It is obvious that
the complexity of the proposed scheme I is lower than that of the proposed
scheme II.

Algorithm 2 Proposed Algorithm for Problem (11)

1:Initialize $\bm{P}^{(0)}$, $\bm{Q}^{(0)}$, $\bm{\Theta}^{(0)}$,
$\Gamma^{(0)}$, the tolerance $\varepsilon$, the iteration number $t=0$, the
convergence accuracy $\epsilon$.

2:repeat

3: With given $\left(\bm{P}^{(t)},\bm{Q}^{(t)},\bm{\Theta}^{(t)}\right)$,
obtain $\bm{A}^{(t+1)}$ by solving problem (12).

4: With given
$\left(\bm{A}^{(t+1)},\bm{Q}^{(t)},\bm{\Theta}^{(t)}\right)$,obtain
$\bm{P}^{(t+1)}$ by Algorithm 1.

5: With given $\left(\bm{A}^{(t+1)},\bm{P}^{(t+1)}\right)$, obtain
$\bm{Q}^{(t+1)}$ by solving problem (31).

6: With $\bm{Q}^{(t+1)}$, update $\bm{\Theta}^{(t+1)}$ by using (21).

7: Calculate
$\Gamma^{(t+1)}=\frac{\zeta^{(t+1)}}{\sum_{k=1}^{K}\sum_{n=1}^{N}p^{(t+1)}_{k}[n]+P_{0}}$.

8: If $\Gamma^{(t+1)}<\Gamma^{(t)}$, we set
$Q^{(t+1)}=Q^{(t)},\Theta^{(t+1)}=\Theta^{(t)}$.

9: Set $t\leftarrow t+1$.

10:Until $\left|\Gamma^{(t+1)}-\Gamma^{(t)}\right|\leq\epsilon$.

##  IV Numerical Results

Figure 2: Convergence behavior with different optimization schemes.

In this section, we analyze the performance of the proposed algorithm through
numerical results. Considering a square area of 300 m $\times$ 300 m with the
BS located at the center. There are 4 users located at the four vertices of
the square area with the coordinates $(\pm 300,\pm 300)$ m, and the location
of Eve is (0, 200) m. The flight period $T$ is fixed to 80 s, and the time
slot is taken as $N$ = 12. We set the pathloss exponent $\alpha$ = 2.2, the
RIS’s elements $M=10$, $d/\lambda=0.5$, the channel gain $h_{0}=-80$ dB, noise
power $\sigma^{2}$ = -120 dBm, and the system circuit power $P_{0}=$ 1 W. Each
user’s maximum transmit power $P_{k}$ is set to 1 W. The initial trajectory
points of UAV flight trajectory are equally spaced on a polygon.

Fig. 2 compare the convergence performance between the proposed scheme I and
scheme II in different heights. As shown in Fig. 2, the performance gap
between Scheme I and Scheme II shows the secrecy energy efficiency gain
brought by the proposed scheme I. This is because that too many approximations
are applied (e.g. the cosine of the AoA $\phi_{k}[n]$, $\phi_{b}[n]$,
$\phi_{e}[n]$ in (51)-(53), and the first order Taylor expansion in (C) and
(C) in the proposed optimization scheme II. Hence, we only consider the scheme
I in the following simulation.

Fig. 3 shows the UAV optimal trajectory solved by the proposed algorithm. In
this simulation, the maximal UAV speed $V_{\max}$ is 50 m/s, which means that
the maximal horizontal distance $S_{\max}$ in Fig. 3 is 333.3 m. The UAV
flight height $H$ is fixed at 100 m. The UAV flight trajectory can be
optimized by Algorithm 2 to achieve the secrecy rate within the $S_{\max}$. It
is observed that UAV visits all user sequentially and its trajectory is a
closed loop. On one hand, it is worth noting that UAV travels less distance
when approaching user 4 and user 1, comparing with the initial trajectory.
This shows that the UAV can establish communication with the users away from
eavesdroppers Eve and achieve high secrecy rates. On the other hand, some
points are marked with black in the optimal trajectory, which indicates that
in these time slots there is no communication between any user and the BS.
This is because UAV at these black points is close to the eavesdropper Eve
which can not establish secure communication with users.

Fig. 4 presents the trajectory when the maximal UAV speed $v_{\max}$ is fixed
at 30 m/s, and the maximal horizontal distance $S_{\max}$ = 200 m. It can be
found that the trajectory is smaller than the trajectory at $v_{\max}=50$ m/s,
which means with the decrease of the maximal UAV speed $v_{max}$, UAV will
adjust it trajectory and fly closer to BS to achieve secure transmission of
confidential information. Similar to the situation $v_{\max}=50$ m/s, when UAV
is close to the eavesdropper Eve which is marked with black points in Fig 4,
UAV keeps silence to avoid information leakage.

Figure 3: Optimized UAV trajectory by Algorithm 2 ($v_{\max}=50$ m/s). Figure
4: Optimized UAV trajectory by Algorithm 2 ($v_{\max}=30$ m/s). Figure 5:
Convergence behavior versus the UAV’s flying height $H$. Figure 6: Secrecy
energy efficiency behavior versus the pathloss exponent $\alpha$. Figure 7:
Secrecy energy efficiency behavior versus the maximum transmit power. Figure
8: Minimum secrecy rate behavior versus the maximum transmit power.

Fig. 5 illustrates the convergence of the Algorithm 2 in different heights
under 2 two case: with RIS (labeled ’w RIS’) and the conventional AF relay
scheme without RIS (labeled ’w/o RIS’) mentioned in [11], respectively. In w/o
RIS case, the UAV works as an AF relay, and its wireless transmit power
$P_{UAV}$ is set to 0.2 W. It can be seen that Algorithm 2 converges fast
which reveals the effectiveness of the proposed algorithm. Besides, Fig. 5
also shows the relationship between the secrecy energy efficiency and the UAV
flight height: the secrecy energy efficiency decreases with the increase of
height. This is because with the increase of the distance of UAV-BS and UAV-
users, the channel gain decreases subsequently. Thus, the achievable secrecy
rates reduce correspondingly. In addition, by comparing the RIS’s performance
with no-RIS in Fig. 5, the advantage of RIS in improving energy efficiency is
obvious. Obviously, the UAV equipped with RIS achieves higher secrecy energy
efficiency compared to the case without RIS.

The secrecy energy efficiency versus the pathloss exponent $\alpha$ is shown
in Fig. 6. In this figure, we can see that the energy efficiency of different
UAV flying height decreases as the pathloss exponent $\alpha$ increases. This
is because that both the channel gain between the UAV and BS/Eve is a
decreasing function of the pathloss exponent $\alpha$. It is also demostrated
that the lower flying height achieves better performance than the higher
flying height. The reason is that the channel gain is a decreasing function of
the distance of UAV-BS and UAV-users. When the flying height get larger, the
achievable rate decreases correspondingly, as well the secrecy energy
efficiency.

In Fig. 7, we study the variation of the secure energy efficiency performance
in different heights with the maximum transmit power of users (we assume the
maximum transmit power $P_{k}$ for user $k$ are all the same) varies. It can
be seen that secure energy efficiency first increases fast and then the
increasing speed is slower. This is because secure energy efficiency is an
non-decreasing function of the maximum transmit power $P_{k}$. As shown in
Fig. 7, when $P_{k}>$ 0.9 W ($H=50$ m) or $P_{k}>$ 0.8 W ($H=100$ m and
$H=150$ m) the secrecy energy efficiency keeps stable at a certain value. The
result can be explained as follow. When maximum transmit power $P_{k}$ gets
larger, the exceed transmit power is not used and it will not increase the
energy efficiency. The result indicates that providing more transmit power for
the system does not always obtain additional secure energy efficiency gains,
and reasonable designing maximum transmit power will save energy. Fig. 8
presents how the minimum secrecy rate changes with the increase of transmit
power. It is obvious that when the transmit power becomes larger, e.g.
$P_{k}>$ 0.7 W, the minimum secrecy rate increases slower than before. That
means we do not need as much transmit power as possible to achieve the minimum
secrecy rate.

In the final set of experiments shown in Fig. 9, We compare the secrecy energy
efficiency performance of the proposed scheme with RIS (labeled ’w RIS’) and
the traditional AF scheme without RIS (labeled ’w/o RIS’). First, Fig. 9 shows
secrecy energy efficiency in w RIS case increases as the number of reflecting
elements $M$ becomes large. Second, we can see that the w RIS case performs
better with the increase of reflecting elements compared to w/o RIS case. This
is reasonable as RIS is an passive reflecting structure and does not cost any
specific energy. In contrast, in w/o RIS case UAV needs extra energy to relay
the received signal to BS which increase the energy consumption. From Fig. 9,
the proposed scheme can increase up to 49.5% ($H=50$ m) and 56.7% ($H=100$ m)
compared to the traditional AF scheme, which demonstrates the RIS’s
characteristic in improving energy efficiency.

Figure 9: Secrecy energy efficiency behavior versus the number of reflecting
elements $M$.

##  V Conclusion

In this paper, we have investigated an UAV equipped with a RIS wiretap
wireless network. The RIS’s phase shifts, UAV association, trajectory and the
user’s transmit power have been jointly optimized to maximize the system
secrecy energy efficiency. To solve this problem, we have proposed an
efficient iterative algorithm by applying the SCA and alternating methods. In
particular, we proposed two optimization schemes to solve the joint phase and
UAV trajectory optimization problem. Numerical results have shown the fast
convergence of the proposed algorithm, and our proposed method provides an
excellent plan for the outdoor communication scene with obstacles.

##  Appendix A Proof of Lemma 1

Lemma 1 is proved by the definition of convex function. First, the first-order
partial derivatives of $r_{k}[n]$ with respect to $z_{k}[n]$ is given as

$\displaystyle\frac{\partial}{\partial
z_{k}[n]}\left\\{\log_{2}\left(1+\frac{A}{z_{k}[n]}\right)\right\\}=\frac{-1}{\ln
2}\frac{A}{\left(z_{k}[n]+A\right)z_{k}[n]}$ (32)

Then, the second-order partial derivative of of $r_{k}[n]$ with respect to
$z_{k}[n]$ is given as

$\displaystyle\frac{\partial}{\partial^{2}z_{k}[n]}\left\\{\log_{2}\left(1+\frac{A}{z_{k}[n]}\right)\right\\}^{2}=\frac{1}{\ln
2}\frac{A^{2}+2Az_{k}[n]}{z^{2}_{k}[n]\left(z_{k}[n]+A\right)^{2}}$ (33)

Since $z_{k}[n]>0$ and $A>0$, the second-order partial derivative
$\frac{\partial r_{k}[n]^{2}}{\partial^{2}z_{k}[n]}>0$. Thus, $r_{k}[n]$ is a
convex function of $z_{k}[n]$.

##  Appendix B Proof of Lemma 2

Lemma 2 is proved by the definition of convex function. As
$d_{k}[n]=\sqrt{\|\bm{q}[n]-\bm{w}_{k}\|^{2}+H^{2}}$, we can get

$\displaystyle d_{k}^{2}[n]$
$\displaystyle=\|\bm{q}[n]-\bm{w}_{k}\|^{2}+H^{2}$
$\displaystyle=\left(x[n]-x_{k}\right)^{2}+\left(y[n]-y_{k}\right)^{2}+H^{2}.$
(34)

The first-order partial derivatives of $d_{k}^{2}[n]$ with respect to $x[n]$
and $y[n]$ are given by

$\frac{\partial d_{k}^{2}[n]}{\partial x[n]}=2\left(x[n]-x_{k}\right),$ (35)
$\frac{\partial d_{k}^{2}[n]}{\partial y[n]}=2\left(y[n]-y_{k}\right).$ (36)

So we can get the the first-order derivatives of $d_{k}^{2}[n]$ with respect
to $\bm{q}[n]$ is given as

$\displaystyle\frac{\partial
d_{k}^{2}[n]}{\partial\bm{q}[n]}=\left(\begin{array}[]{c}\frac{\partial
d_{k}^{2}[n]}{\partial x[n]}\\\ \\\ \frac{\partial d_{k}^{2}[n]}{\partial
y[n]}\end{array}\right)=\left(\begin{array}[]{c}2\left(x[n]-x_{k}\right)\\\
\\\ 2\left(y[n]-y_{k}\right)\end{array}\right)=2(\bm{q}[n]-\bm{w}_{k}).$ (43)

It can be easily know the the Hessian of $d_{k}^{2}[n]$ is

$\bigtriangledown d_{k}^{2}[n]=\left[\begin{array}[]{ccc}2&0\\\
0&2\end{array}\right].$ (44)

Obviously, it is positive definite. Consequently, $d_{k}^{2}[n]$ is a convex
function.

The first-order partial derivatives of $d_{k}^{4}[n]$ with respect to $x[n]$
and $y[n]$ are given by

$\frac{\partial d_{k}^{4}[n]}{\partial x[n]}=2d_{k}^{2}[n]\frac{\partial
d_{k}^{2}[n]}{\partial x[n]}=4d_{k}^{2}[n]\left(x[n]-x_{k}\right),$ (45)
$\frac{\partial d_{k}^{4}[n]}{\partial y[n]}=2d_{k}^{2}[n]\frac{\partial
d_{k}^{2}[n]}{\partial y[n]}=4d_{k}^{2}[n]\left(y[n]-y_{k}\right).$ (46)

The second-order partial derivatives of $d_{k}^{4}[n]$ with respect to $x[n]$
and $y[n]$ are given by

$\frac{\partial\left(d_{k}^{4}[n]\right)^{2}}{\partial^{2}x[n]}=4\left(d_{k}^{2}[n]+2\left(x[n]-x_{k}\right)^{2}\right),$
(47)
$\frac{\partial\left(d_{k}^{4}[n]\right)^{2}}{\partial^{2}y[n]}=4\left(d_{k}^{2}[n]+2\left(y[n]-y_{k}\right)^{2}\right),$
(48) $\frac{\partial\left(d_{k}^{4}[n]\right)^{2}}{\partial x[n]\partial
y[n]}=8\left(x[n]-x_{k}\right)\left(y[n]-y_{k}\right)=\frac{\partial\left(d_{k}^{4}[n]\right)^{2}}{\partial
y[n]\partial x[n]},$ (49)
$\displaystyle\frac{\partial\left(d_{k}^{4}[n]\right)^{2}}{\partial^{2}x[n]}\frac{\partial\left(d_{k}^{4}[n]\right)^{2}}{\partial^{2}y[n]}-\frac{\partial\left(d_{k}^{4}[n]\right)^{2}}{\partial
x[n]\partial y[n]}\frac{\partial\left(d_{k}^{4}[n]\right)^{2}}{\partial
y[n]\partial x[n]}$ $\displaystyle=$ $\displaystyle
16d_{k}^{4}[n]+32\left(x[n]-x_{k}\right)^{2}\left(y[n]-y_{k}\right)^{2}>0.$
(50)

Since $\frac{\partial\left(d_{k}^{4}[n]\right)^{2}}{\partial^{2}x[n]}>0$ and
(B), $d_{k}^{4}[n]$ is a convex function.

##  Appendix C Joint Phase and UAV Trajectory Optimization (scheme II)

To reveal the hidden convexity in
$\bm{g}_{b}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]$ and
$\bm{g}_{b}^{H}[n]\bm{\Theta}[n]\bm{g}_{k}[n]$, we first introduce two slack
variables $u_{k}[n]=\phi_{e}[n]-\phi_{k}[n]$,
$l_{k}[n]=\phi_{b}[n]-\phi_{k}[n]$. To make problem more tackleable, we
introduce an approximation as follow,

$\phi_{k}[n]=\frac{x[n]-x_{k}}{d_{k}[n]}\approx\frac{x[n]-x_{k}}{d_{k}^{(r-1)}[n]},$
(51)
$\phi_{b}[n]=\frac{x[n]-x_{b}}{d_{b}[n]}\approx\frac{x[n]-x_{b}}{d_{b}^{(r-1)}[n]},$
(52)

and

$\phi_{e}[n]=\frac{x[n]-x_{e}}{d_{e}[n]}\approx\frac{x[n]-x_{e}}{d_{e}^{(r-1)}[n]},$
(53)

where $d_{k}^{(r-1)}[n],d_{b}^{(r-1)}[n],d_{e}^{(r-1)}[n]$ is the solution in
the $(r-1)$-th iteration. It can be found that both $u_{k}[n]$ and $l_{k}[n]$
are the linear functions of $x[n]$ as well as $q[n]$ based on (51)-(53). Note
that the approximations (51)-(53) are reasonable as the distance between the
users and the UAV is large and it will not change a lot in the $(r-1)$-th
iteration. To further solve the problem, we set
$t_{mk}[n]=\theta_{m}[n]+\frac{2(m-1)\pi d}{\lambda}u_{k}[n]$,
$s_{mk}[n]=\theta_{m}[n]+\frac{2(m-1)\pi d}{\lambda}l_{k}[n]$. It is obvious
that $t_{mk}[n]$, and $s_{mk}[n]$ are linear functions of variables
$\theta_{m}[n]$, $u_{k}[n]$ and $l_{k}[n$], Consequently, we can obtain

$\displaystyle\left|\sum\limits_{m=1}^{M}e^{jt_{mk}[n]}\right|^{2}$
$\displaystyle=\left[\sum\limits_{m=1}^{M}cos(t_{mk}[n])\right]^{2}+\left[\sum\limits_{m=1}^{M}sin(t_{mk}[n])\right]^{2}$
$\displaystyle\approx\left[\sum\limits_{m=1}^{M}cos(t_{mk}^{(r)}[n])\right]^{2}+\left[\sum\limits_{m=1}^{M}sin(t_{mk}^{(r)}[n])\right]^{2}$
$\displaystyle-2\sum\limits_{m=1}^{M}\left(\sum\limits_{i=1}^{M}cos(t_{ik}^{(r)}[n])\right)sin(t_{mk}^{(r)}[n])\left(t_{mk}[n]-t_{mk}^{(r)}[n]\right)$
$\displaystyle+2\sum\limits_{m=1}^{M}\left(\sum\limits_{i=1}^{M}sin(t_{ik}^{(r)}[n])\right)cos(t_{mk}^{(r)}[n])\left(t_{mk}[n]-t_{mk}^{(r)}[n]\right)$
$\displaystyle\triangleq g_{ek}[n]$ (54)

and

$\displaystyle\left|\sum\limits_{m=1}^{M}e^{js_{mk}[n]}\right|^{2}$
$\displaystyle=\left[\sum\limits_{m=1}^{M}cos(s_{mk}[n])\right]^{2}+\left[\sum\limits_{m=1}^{M}sin(s_{mk}[n])\right]^{2}$
$\displaystyle\approx\left[\sum\limits_{m=1}^{M}cos(s_{mk}^{(r)}[n])\right]^{2}+\left[\sum\limits_{m=1}^{M}sin(s_{mk}^{(r)}[n])\right]^{2}$
$\displaystyle-2\sum\limits_{m=1}^{M}\left(\sum\limits_{i=1}^{M}cos(s_{ik}^{(r)}[n])\right)sin(s_{mk}^{(r)}[n])\left(s_{mk}[n]-s_{mk}^{(r)}[n]\right)$
$\displaystyle+2\sum\limits_{m=1}^{M}\left(\sum\limits_{i=1}^{M}sin(s_{ik}^{(r)}[n])\right)cos(s_{mk}^{(r)}[n])\left(s_{mk}[n]-s_{mk}^{(r)}[n]\right)$
$\displaystyle\triangleq g_{bk}[n].$ (55)

Note that the approximations in (C) and (C) are derived based on the first-
order Taylor expansions, and both $g_{bk}[n]$ and $g_{ek}[n]$ are linear
functions with respect to $t_{mk}[n]$ and $s_{mk}[n]$.

Two slack variables $\bm{G_{e}}=\left\\{g_{ek}[n]\right\\}_{\forall
k,n}$,$\bm{G_{b}}=\left\\{g_{bk}[n]\right\\}_{\forall k,n}$ are introduced to
assist the problem solution. Thus, we get

$\displaystyle c_{k}[n]$
$\displaystyle=\log_{2}\left(1+\frac{h_{0}^{2}g_{ek}[n]}{d_{k}^{a}[n]d_{e}^{a}[n]}\right)$
(52)
$\displaystyle=\log_{2}\left(d_{k}^{a}[n]d_{e}^{a}[n]+h_{0}^{2}g_{ek}[n]\right)-\log_{2}\left(d_{k}^{a}[n]d_{e}^{a}[n]\right)$

and

$\displaystyle r_{k}[n]$
$\displaystyle=\log_{2}\left(1+\frac{h_{0}^{2}g_{bk}[n]}{d_{k}^{a}[n]d_{b}^{a}[n]}\right)$
(53)
$\displaystyle=\log_{2}\left(d_{k}^{a}[n]d_{b}^{a}[n]+h_{0}^{2}g_{eb}[n]\right)-\log_{2}\left(d_{k}^{a}[n]d_{b}^{a}[n]\right),$

where $g_{ek}[n]$ and $g_{eb}[n]$ both contain the variable $q[n]$. We
introduce another two slack variables
$\bm{Z}=\left\\{z_{k}[n]\right\\}_{\forall k,n}$ and
$\bm{V}=\left\\{v_{k}[n]\right\\}_{\forall k,n}$. We reformulate the problem
as

$\displaystyle\mathop{\max}_{\zeta,\bm{Q},\bm{\Theta},\bm{Z},\bm{V},\bm{U},\bm{L},\bm{G_{e}},\bm{G_{b}}}\quad\zeta$
(54a) s.t.
$\displaystyle\zeta\leq\frac{1}{N}\sum_{n=1}^{N}a_{k}[n]\left[r_{k}[n]-c_{k}[n]\right]^{+},\forall
k\in\mathcal{K}$ (54b) $\displaystyle\|\bm{q}[n]-\bm{q}[{n-1}]\|\leq
S_{\max},\quad\forall n\in\mathcal{N}$ (54c)
$\displaystyle\bm{q}[N]=\bm{q}[0]$ (54d) $\displaystyle
z^{\frac{2}{\alpha}}_{k}[n]\geq d_{k}^{2}[n]d_{b}^{2}[n],\quad\forall
n\in\mathcal{N},k\in\mathcal{K}$ (54e) $\displaystyle
v^{\frac{2}{\alpha}}_{k}[n]\leq d_{k}^{2}[n]d_{e}^{2}[n],\quad\forall
n\in\mathcal{N},k\in\mathcal{K}$ (54f) $\displaystyle
u_{k}[n]=\phi_{e}[n]-\phi_{k}[n],\quad\forall n\in\mathcal{N},k\in\mathcal{K}$
(54g) $\displaystyle l_{k}[n]=\phi_{b}[n]-\phi_{k}[n],\quad\forall
n\in\mathcal{N},k\in\mathcal{K}$ (54h)
$\displaystyle\theta_{m}[n]\in[0,2\pi],\quad\forall
m\in\mathcal{M},n\in\mathcal{N}$ (54i)
$\displaystyle\eqref{eq48},\eqref{eq49},\quad\forall
n\in\mathcal{N},k\in\mathcal{K}.$ (54j)

To handle the nonconvexity of (54b), we have

$\displaystyle R_{k}[n]$ $\displaystyle=r_{k}[n]-c_{k}[n]$ (55)
$\displaystyle=\log_{2}\left(z_{k}[n]+h_{0}^{2}g_{bk}[n]\right)-\log_{2}\left(z_{k}[n]\right)$
$\displaystyle-\log_{2}\left(v_{k}[n]+h_{0}^{2}g_{ek}[n]\right)+\log_{2}\left(v_{k}[n]\right)$
$\displaystyle\geq\log_{2}\left(z_{k}[n]+h_{0}^{2}g_{bk}[n]\right)+\log_{2}\left(v_{k}[n]\right)$
$\displaystyle-\log_{2}\left(z_{k}^{(l)}[n]\right)-\frac{1}{\ln
2z_{k}^{(l)}[n]}\left(z_{k}[n]-z_{k}^{(l)}[n]\right)$
$\displaystyle-\log_{2}\left(v_{k}^{(l)}[n]+h_{0}^{2}g_{ek}^{(l)}[n]\right)$
$\displaystyle-\frac{1}{(\ln
2)(v_{k}^{(l)}[n]+h_{0}^{2}g_{ek}^{(l)}[n])}\left(v_{k}[n]-v_{k}^{(l)}[n]\right)$
$\displaystyle-\frac{h_{0}^{2}}{(\ln
2)(v_{k}^{(l)}[n]+h_{0}^{2}g_{ek}^{(l)}[n])}\left(g_{ek}[n]-g_{ek}^{(l)}[n]\right)$
$\displaystyle\triangleq\hat{R}_{k}[n].$ (56)

To handle the nonconvexity of (54e) and (54f), the technique applied is the
same as the (III-C2)-(III-C2). Hence, we can solve the following problem

$\displaystyle\mathop{\max}_{\zeta,\bm{Q},\bm{\Theta},\bm{Z},\bm{V},\bm{U},\bm{L},\bm{G_{e}},\bm{G_{b}}}\quad\zeta$
(57a) s.t.
$\displaystyle\zeta\leq\frac{1}{N}\sum_{n=1}^{N}a_{k}[n]\max\left\\{\hat{R}_{k}[n],0\right\\},\forall
k\in\mathcal{K}$ (57b) $\displaystyle\|\bm{q}[n]-\bm{q}[{n-1}]\|\leq
S_{\max},\quad\forall n\in\mathcal{N}$ (57c)
$\displaystyle\bm{q}[N]=\bm{q}[0]$ (57d) $\displaystyle
z^{\frac{2}{\alpha}}_{k}[n]\geq f\left(\bm{q}[n]\right)\quad\forall
n\in\mathcal{N},k\in\mathcal{K}$ (57e) $\displaystyle
h\left(v_{k}[n]\right)\leq g\left(\bm{q}[n]\right)\quad\forall
n\in\mathcal{N},k\in\mathcal{K}$ (57f) $\displaystyle
u_{k}[n]=\phi_{e}[n]-\phi_{k}[n],\forall n\in\mathcal{N},k\in\mathcal{K}$
(57g) $\displaystyle l_{k}[n]=\phi_{b}[n]-\phi_{k}[n],\forall
n\in\mathcal{N},k\in\mathcal{K}$ (57h)
$\displaystyle\theta_{m}[n]\in[0,2\pi],\quad\forall
m\in\mathcal{M},n\in\mathcal{N}$ (57i)
$\displaystyle\eqref{eq48},\eqref{eq49},\quad\forall
n\in\mathcal{N},k\in\mathcal{K}.$ (57j)

Since (57g), (57h) and (57i) are all linear constraints and other constrains
are convex, problem (57) is a convex optimization problem which can be
efficiency solved by standard convex optimization solver. As too many
approximation are applied in the proposed optimization scheme II, the optimal
objective value obtained from problem (57) can serve as a lower bound of
problem (19).

##  Appendix D Proof of Lemma 3

The proof is eastablished by showing that the secrecy energy efficiency is
non-decreasing after each iteration. Considering in the $t$-th iteration, we
proved following update rules based on Algorithm 2

$\displaystyle\Gamma^{(t)}$
$\displaystyle=f(\bm{A}^{(t)},\bm{P}^{(t)},\bm{Q}^{(t)},\bm{\Theta}^{(t)})$
$\displaystyle\overset{(a)}{\leq}f(\bm{A}^{(t+1)},\bm{P}^{(t)},\bm{Q}^{(t)},\bm{\Theta}^{(t)})$
$\displaystyle\overset{(b)}{\leq}f(\bm{A}^{(t+1)},\bm{P}^{(t+1)},\bm{Q}^{(t)},\bm{\Theta}^{(t)})$
$\displaystyle\overset{(c)}{\leq}f(\bm{A}^{(t+1)},\bm{P}^{(t+1)},\bm{Q}^{(t+1)},\bm{\Theta}^{(t+1)})$
$\displaystyle=\Gamma^{(t+1)}$ (58)

where
$\Gamma=f(\bm{A},\bm{P},\bm{Q},\bm{\Theta})=\frac{\zeta}{\sum_{k=1}^{K}\sum_{n=1}^{N}p_{k}[n]+P_{0}}$.
Inequality $(a)$ follows that in step 3 in Algorithm 2 problem (12) is solved
optimally with solution $\bm{A}^{(t+1)}$. In the proposed Algorithm 2, for the
power control optimization problem (13) and trajectory and phift optimization
(19), we only solved the their approximate problem (16) and problem (31).
Define $\Gamma_{pow}=f_{pow}(\bm{A},\bm{P},\bm{Q},\bm{\Theta})$ , where
$\Gamma_{pow}$ is respectively the objective values of problem (16). The
inequality $(b)$ can be explained as follow

$\displaystyle f(\bm{A}^{(t+1)},\bm{P}^{(t)},\bm{Q}^{(t)},\bm{\Theta}^{(t)})$
$\displaystyle\overset{(d)}{=}f_{pow}(\bm{A}^{(t+1)},\bm{P}^{(t)},\bm{Q}^{(t)},\bm{\Theta}^{(t)})$
$\displaystyle\overset{(e)}{\leq}f_{pow}^{lb}(\bm{A}^{(t+1)},\bm{P}^{(t+1)},\bm{Q}^{(t)},\bm{\Theta}^{(t)})$
$\displaystyle\overset{(f)}{\leq}f(\bm{A}^{(t+1)},\bm{P}^{(t+1)},\bm{Q}^{(t)},\bm{\Theta}^{(t)})$
(59)

where inequality $(d)$ holds since the SCA method in (III-B), which means
problem (16) at the $p_{k}^{(r)}$ has the same objective value as problem
(13); inequality $(e)$ follows that the step 4 of Algorithm 2 with given
$\bm{A}^{(t+1)}$, $\bm{Q}^{(t)}$ and $\bm{\Theta}^{(t)}$, $\bm{P}^{(t+1)}$ is
the optimal solution of problem (16); inequality (f) is due to the fact that
the objective vaule of problem (31) is the lower bound of the original problem
(19) at $\bm{P}^{(t+1)}$. The inequality $(c)$ can hold due to the fact that
we set $Q^{(t+1)}=Q^{(t)},\Theta^{(t+1)}=\Theta^{(t)}$ if
$\Gamma^{(t+1)}<\Gamma^{(t)}$ in Algorithm 2. Thus the object vaule is non-
decreasing after each iteration of Algorithm 2. Furthermore, the objective
vaule of problem (11) is upper bounded by a finite value, the proposed
Algorithm is guaranteed to converge.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
(NSFC) under grant 61871128 and the Fundamental Research on Foreword Leading
Technology of Jiangsu Province under grant BK20192002.

## References

  * [1] X. Sun, D. W. K. Ng, Z. Ding, Y. Xu, and Z. Zhong, “Physical layer security in UAV systems: Challenges and opportunities,” _IEEE Wireless Communications_ , vol. 26, no. 5, pp. 40–47, Oct. 2019.
  * [2] Y. Zeng, R. Zhang, and T. J. Lim, “Throughput maximization for UAV-enabled mobile relaying systems,” _IEEE Transactions on Communications_ , vol. 64, no. 12, pp. 4983–4996, Dec. 2016.
  * [3] T. A. Johansen, A. Zolich, T. Hansen, and A. J. Sørensen, “Unmanned aerial vehicle as communication relay for autonomous underwater vehicle — field tests,” in _2014 IEEE Globecom Workshops (GC Wkshps)_ , Dec. 2014, pp. 1469–1474.
  * [4] J. Lyu, Y. Zeng, R. Zhang, and T. J. Lim, “Placement optimization of UAV-mounted mobile base stations,” _IEEE Communications Letters_ , vol. 21, no. 3, pp. 604–607, Mar. 2017.
  * [5] M. Chen, M. Mozaffari, W. Saad, C. Yin, M. Debbah, and C. S. Hong, “Caching in the sky: Proactive deployment of cache-enabled unmanned aerial vehicles for optimized quality-of-experience,” _IEEE Journal on Selected Areas in Communications_ , vol. 35, no. 5, pp. 1046–1061, 2017.
  * [6] Z. Yang, C. Pan, M. Shikh-Bahaei, W. Xu, M. Chen, M. Elkashlan, and A. Nallanathan, “Joint altitude, beamwidth, location, and bandwidth optimization for uav-enabled communications,” _IEEE Communications Letters_ , vol. 22, no. 8, pp. 1716–1719, 2018.
  * [7] F. Zhou, Y. Wu, R. Q. Hu, and Y. Qian, “Computation rate maximization in UAV-enabled wireless-powered mobile-edge computing systems,” _IEEE Journal on Selected Areas in Communications_ , vol. 36, no. 9, pp. 1927–1941, 2018\.
  * [8] Y. Wang, M. Hua, Z. Liu, D. Zhang, B. Ji, and H. Dai, “UAV-based mobile wireless power transfer systems with joint optimization of user scheduling and trajectory,” _Mobile Networks and Applications_ , pp. 1–15, 2019.
  * [9] Z. Yang, C. Pan, K. Wang, and M. Shikh-Bahaei, “Energy efficient resource allocation in UAV-enabled mobile edge computing networks,” _IEEE Transactions on Wireless Communications_ , vol. 18, no. 9, pp. 4576–4589, 2019.
  * [10] G. Zhang, Q. Wu, M. Cui, and R. Zhang, “Securing UAV communications via trajectory optimization,” in _GLOBECOM 2017 - 2017 IEEE Global Communications Conference_ , Dec. 2017, pp. 1–6.
  * [11] Z. Li, M. Chen, C. Pan, N. Huang, Z. Yang, and A. Nallanathan, “Joint trajectory and communication design for secure UAV networks,” _IEEE Commun. Lett._ , vol. 23, no. 4, pp. 636–639, Feb. 2019.
  * [12] J. Ye, C. Zhang, H. Lei, G. Pan, and Z. Ding, “Secure UAV-to-UAV systems with spatially random UAVs,” _IEEE Wireless Communications Letters_ , vol. 8, no. 2, pp. 564–567, Apr. 2019.
  * [13] S. Li, B. Duo, X. Yuan, Y.-C. Liang, and M. Di Renzo, “Reconfigurable intelligent surface assisted UAV communication: Joint trajectory design and passive beamforming,” _IEEE Wireless Commun. Lett._ , Jan. 2020.
  * [14] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and C. Yuen, “Reconfigurable intelligent surfaces for energy efficiency in wireless communication,” _IEEE Transactions on Wireless Communications_ , vol. 18, no. 8, pp. 4157–4170, 2019.
  * [15] D. Xu, X. Yu, Y. Sun, D. W. K. Ng, and R. Schober, “Resource allocation for secure irs-assisted multiuser miso systems,” in _2019 IEEE Globecom Workshops (GC Wkshps)_ , 2019, pp. 1–6.
  * [16] Z. Yang, M. Chen, W. Saad, W. Xu, M. Shikh-Bahaei, H. V. Poor, and S. Cui, “Energy-efficient wireless communications with distributed reconfigurable intelligent surfaces,” 2020.
  * [17] Z. Yang, J. Shi, Z. Li, M. Chen, W. Xu, and M. Shikh-Bahaei, “Energy efficient rate splitting multiple access (RSMA) with reconfigurable intelligent surface,” in _Proc. IEEE Int. Conf. Commun. Workshop_ , 2020 (To appear), pp. 1–6.
  * [18] C. Huang, S. Hu, G. C. Alexandropoulos, A. Zappone, C. Yuen, R. Zhang, M. D. Renzo, and M. Debbah, “Holographic mimo surfaces for 6g wireless networks: Opportunities, challenges, and trends,” 2019.
  * [19] C. Pan, H. Ren, K. Wang, W. Xu, M. Elkashlan, A. Nallanathan, and L. Hanzo, “Multicell mimo communications relying on intelligent reflecting surfaces,” _IEEE Transactions on Wireless Communications_ , pp. 1–1, 2020\.
  * [20] C. Pan, H. Ren, K. Wang, M. Elkashlan, A. Nallanathan, J. Wang, and L. Hanzo, “Intelligent reflecting surface aided mimo broadcasting for simultaneous wireless information and power transfer,” 2019.
  * [21] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M. Alouini, and R. Zhang, “Wireless communications through reconfigurable intelligent surfaces,” _IEEE Access_ , vol. 7, pp. 116 753–116 773, 2019.
  * [22] H. Shen, W. Xu, S. Gong, Z. He, and C. Zhao, “Secrecy rate maximization for intelligent reflecting surface assisted multi-antenna communications,” _IEEE Communications Letters_ , vol. 23, no. 9, pp. 1488–1492, Sep. 2019.
  * [23] Z. Chu, W. Hao, P. Xiao, and J. Shi, “Intelligent reflecting surface aided multi-antenna secure transmission,” _IEEE Wireless Communications Letters_ , vol. 9, no. 1, pp. 108–112, 2020.
  * [24] X. Yu, D. Xu, and R. Schober, “Enabling secure wireless communications via intelligent reflecting surfaces,” in _2019 IEEE Global Communications Conference (GLOBECOM)_ , 2019, pp. 1–6.
  * [25] M. Cui, G. Zhang, and R. Zhang, “Secure wireless communication via intelligent reflecting surface,” _IEEE Wireless Communications Letters_ , vol. 8, no. 5, pp. 1410–1414, 2019.
  * [26] L. Dong and H. Wang, “Secure mimo transmission via intelligent reflecting surface,” _IEEE Wireless Communications Letters_ , pp. 1–1, 2020.
  * [27] M. Chen, Z. Yang, W. Saad, C. Yin, H. V. Poor, and S. Cui, “A joint learning and communications framework for federated learning over wireless networks,” 2019\.
  * [28] Y. Mao, B. Clerckx, and V. O. K. Li, “Rate-splitting for multi-antenna non-orthogonal unicast and multicast transmission: Spectral and energy efficiency analysis,” _IEEE Transactions on Communications_ , vol. 67, no. 12, pp. 8754–8770, 2019.
  * [29] W. Dinkelbach, “On nonlinear fractional programming,” _Management Science_ , vol. 13, no. 7, pp. 492–498, 1967.