# Energy Efficient Design in IRS-Assisted UAV Data Collection System under
Malicious Jamming

Zhi Ji, Jia Tu, Xinrong Guan, Wendong Yang, Weiwei Yang, and Qingqing Wu  Zhi
Ji, Xinrong Guan, Wendong Yang, and Weiwei Yang are with the College of
Communications Engineering, Army Engineering University of PLA, Nanjing,
210007, China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>wwyang1981@163.com). (Corresponding author: Xinrong Guan.) J. Tu is with
International Studies College, National University of Defense Technology,
Nanjing, 210007, China (Email: tujia$\\_$666@163.com). Q. Wu is with the State
Key Laboratory of Internet of Things for Smart City, University of Macau,
Macau, 999078, China (email: qingqingwu@um.edu.mo).

###### Abstract

In this paper, we study an unmanned aerial vehicle (UAV) enabled data
collection system, where an intelligent reflecting surface (IRS) is deployed
to assist in the communication from a cluster of Internet-of-Things (IoT)
devices to a UAV in the presence of a jammer. We aim to improve the energy
efficiency (EE) via the joint design of UAV trajectory, IRS passive
beamforming, device power allocation and communication scheduling. However,
the formulated non-linear fractional programming problem is challenging to
solve due to its non-convexity and coupled variables. To overcome the
difficulty, we propose an alternating optimization based algorithm to solve it
sub-optimally by leveraging the Dinkelbach’s algorithm, successive convex
approximation (SCA) technique, and block coordinate descent (BCD) method.
Extensive simulation results show that the proposed design can significantly
improve the anti-jamming performance. In particular, for the remote jammer
case, the proposed design can largely shorten the flight path and thus
decrease the energy consumption via the signal enhancement; while for the
local jammer case, which is deemed highly challenging in conventional systems
without IRS since the retreating away strategy becomes ineffective, our
proposed design even achieves a higher performance gain owing to the efficient
jamming signal mitigation.

###### Index Terms:

anti-jamming, trajectory design, IRS, UAV communication.

##  I Introduction

Benefiting from the flexible mobility and deployment, unmanned aerial vehicles
(UAVs) have been employed in numerous applications in recent years, such as
remote surveillance, photography, agricultural irrigation, traffic control,
and cargo transportation, etc [1]. By the properly designing its trajectory,
the UAV tends to obtain more dominant line-of-sight (LoS) channels and thus
greatly improves the communication performance [5, 2, 3, 4]. In particular,
exploiting the UAV as an aerial data collector for Internet-of-Things (IoT)
devices is expected to bring significant benefits for transmission rate,
energy consumption, and service coverage [6, 7, 8].

However, due to the broadcast nature of wireless communications, the strong
LoS channels also make UAVs more vulnerable to attacks from ground, e.g.,
jamming, surveillance, and eavesdropping [9, 10, 13, 14, 15, 16, 11, 12]. In
particular, the malicious jamming would severely distort the signal received
at the UAV and even thus disable the UAV’s data collection. To tackle this
challenge, in [13], a multi-UAV data collection system was studied where the
jammers are assumed to be located randomly. Via the joint design of UAVs’
trajectories, ground nodes’ (GN) scheduling and power allocation, the minimum
throughput, the average throughput, and the delay-constrained minimum
throughput of all GNs are improved, respectively. In [14], turning and
climbing angle-constrained are considered to improve the uplink throughput by
the joint design of UAV and transmit power under malicious jamming. In
particular, authors in [15] studied an anti-jamming trajectory of UAV to
prevent jammers from attacking the legitimate transmission in UAV-enabled
communication systems under probabilistic LoS channel. Furthermore, [16]
proposed an iterative algorithm which strikes a better balance between the
throughput and energy consumption under malicious jamming by the optimized UAV
trajectory and thus improves the energy efficiency (EE) significantly.
However, in all of the above works, the UAV’s anti-jamming strategies are
mostly based on retreating away from the jammer for suppressing the undesired
signal, which renders not only complex trajectories but also high flight
energy consumption. Even worse, considering a challenging scenario when the
jammer is located nearby the legitimate terrestrial user, the above retreating
away strategy becomes ineffective because when the UAV retreats away from the
jammer, it also keeps away from the user, which thus degrades the reception of
the desired signal as well.

Recently, intelligent reflecting surface (IRS), due to its capability of
smartly configuring the wireless propagation environment, has been
investigated extensively [17, 18, 19]. Specifically, IRS comprises a large
number of passive units, which can reflect the wireless signal with adjustable
amplitude and/or phase shift, thus achieving the so-called passive/reflective
beamforming gain. The main advantage of IRS-assisted wireless communications
lies in that it considerably increases the channel capacity with low power
consumption and can be flexibly deployed in the radio environment. Benefiting
from the above superiority, IRS has been studied under various wireless system
setups [20, 21, 22, 23, 24, 25, 26, 27]. In particular, extensive research
efforts have been devoted to employing IRS for improving the transmission rate
of UAV communications and/or safeguarding them from security threats. It
showed that by joint design of the UAV trajectory and IRS passive beamforming,
the average achievable rate of the system can be significantly improved [28,
29]. Facing with the threat of eavesdropping, the average secrecy rate was
maximized by jointly optimizing the UAV trajectory, the transmit beamforming
and the phase shift of IRS in [30, 31]. Furthermore, for the case suffered
from jamming, it was shown that IRS can help to improve the transmission rate
for mitigating the jamming signal from the malicious jammer [32]. However, it
should be noted that all these works only focus on the IRS passive beamforming
gain in terms of rate but ignore the benifits of IRS on UAV’s energy
consumption and EE, which is more practically important in UAV communication
systems and thus need further study.

Motivated by the above, in this paper, we focus on how to exploit the IRS for
improving the EE of UAV in an UAV data collection system, wherein a cluster of
IoT devices (IDs) tend to transmit data to the UAV in the presence of
malicious jamming. The main contributions are summarized as follows.

Figure 1: An IRS-assisted UAV data collection system in the presence of a
jammer.

  * •

We propose an IRS-enabled novel anti-jamming method for the UAV data
collection system instead of solely relying the conventional retreating away
strategy. Specifically, by deploying an IRS nearby the IDs, the signal
reception at the UAV can be significantly improved via the enhancement of the
desired signal and/or the mitigation of the jamming signal, via exploiting the
channel configuring capacity of the IRS.

  * •

We propose an alternating optimization (AO) based algorithm to solve the
formulated challenging problem with coupled optimization variables and non-
convex objective function. Specifically, by leveraging the block coordinate
descend (BCD) method, we divide the original problem into four sub-problems
that can be optimized in an iterative manner. Then, successive convex
optimization (SCA) and the Dinkelbach’s algorithm are applied to settle the
non-convexity of sub-problems. In particular, a low-complexity algorithm is
introduced to handle the phase shifts of the IRS.

  * •

Numerical results show that the proposed design can greatly improve the anti-
jamming performance as compared to the benchmark ones. Specifically, two
different setups corresponding to a remote jammer (i.e., the jammer is far
away from the IDs) and a local jammer (i.e., the jammer is nearby the IDs) are
respectively considered for drawing some useful insights. It is found that for
the remote jammer case, the proposed design can largely shorten the flight
path and thus decrease the energy consumption via the signal enhancement;
while for the local jammer case when the conventional retreating away strategy
becomes ineffective, our proposed design even achieves a higher performance
gain owing to the efficient jamming signal mitigation.

Notations: Boldface lower-case letters and boldface upper-case letters are
used to denote vectors and matrices. ${\mathbb{C}^{L\times 1}}$ denotes the
$L$-dimensional complex-valued vector. ${\left(\cdot\right)^{T}}$,
${\left(\cdot\right)^{H}}$ and ${\left(\cdot\right)^{*}}$ represent the
transopse, the Hermitian operations and the conjugate, respectively.
$\left|\cdot\right|$ and $\left\|\cdot\right\|$ stand for the absolute value
of a scalar and the Euclidean norm of a vector, respectively.
${\text{diag}}\left\\{\cdot\right\\}$ returns the diagonal matrix whose
diagonals are the elements of input vector. $\Re\left(\cdot\right)$ and
$\arg\left(\cdot\right)$ represent the real part and the phase of the input
complex value. ${\lambda_{\max}}(\Phi)$ returns the maximum eigenvalue of
$\Phi$.

##  II System Model and Problem Formulation

###  II-A System Model

In this paper, an IRS-aided UAV data collection system is considered as shown
in Fig. 1, where an IRS is deployed to assist in the data transmission from a
cluster of IDs to a rotary-wing UAV in the presence of a jammer. The set of
the IDs is denoted by $\cal K$. Without loss of generality, we consider that
the UAV shares the same frequency band for its communications and the time-
division multiple access (TDMA) protocol is applied to serve the $K$ IDs. It
is assumed that both the IDs and the UAV are equipped with single omni-
directional antenna, and the IRS composes of a uniform planar array (UPA) with
$M=M_{x}\times M_{z}$ elements, where $M_{x}$ and $M_{z}$ denote the number of
elements along the x-axis and z-axis, respectively. Then we denote the
diagonal phase-shift matrix for the IRS as
${\mathbf{\Gamma}}\triangleq\left\\{{{\mathbf{\Gamma}}[n]={\text{diag}}\left({{e^{j{\theta_{1}}[n]}},{e^{j{\theta_{2}}[n]}},...,{e^{j{\theta_{M}}[n]}}}\right),\forall
n}\right\\}$, where ${\theta_{i}}[n]\in[0,2\pi),i\in\\{1,...,M\\}$ is the
phase shift of the $i$-th reflecting element in time slot $n$. All
communication nodes are placed in the three dimensional (3D) Cartesian
coordinates. The location of ID $k$ is represented by
${\bf{q_{k}}}=[{x_{k}},{y_{k}},0]$, where $k\in{\cal{K}}=\\{1,2,...,K\\}.$ The
location of the jammer and IRS are denoted by
${\bf{q_{j}}}=[{x_{j}},{y_{j}},{z_{j}}]$ and
${\bf{q_{r}}}=[{x_{r}},{y_{r}},{z_{r}}]$, respectively. We assume that the
location of the jammer is known by leveraging existing jammer localization
techniques, such as the jamming signal strength (JSS)-based localization
methods [33]. Moreover, considering the high-altitude UAV and flexibly
deployed IRS, LoS channels are assumed for all the considered links.

The UAV is assumed to fly at a fixed height $H_{u}$. The flying time of the
UAV is $T$. For ease of handling, $T$ is divided into $N$ time slots, i.e.,
$\Delta t=T/N$, where ${\Delta t}$ is the length of one time slot. The UAV has
the fixed starting point and destination, which are denoted by
${{\mathbf{q}}_{S}}$ and ${{\mathbf{q}}_{F}}$, respectively. The trajectory of
the UAV can be expressed by ${\bf
q_{u}}[n]={[x_{u}[n],y_{u}[n],z_{u}[n]]^{T}},n\in{\cal{N}}=\\{1,2,...,N\\}$,
${\bf{Q}}\buildrel\Delta\over{=}\\{{\bf q_{u}}[n],\forall n\\}$, which meets
the mobility constraints as

$\displaystyle{\bf{q}}_{u}\left[0\right]={{\bf{q}}_{S}},{\bf{q}}_{u}\left[N\right]={{\bf{q}}_{F}},$
(1a)
$\displaystyle\left\|{{\mathbf{q}}_{u}[n]-{\mathbf{q}}_{u}[n-1]}\right\|\leqslant{D_{\max}},n=1,...,N,$
(1b)

 where ${{D_{\max}}}$ is the maximum flying length in each time slot.
Assuming that $p_{k}[n]$ is the transmit power of $k$-th device in time slot
$n$ and ${\mathbf{P}}=\left\\{{p_{k}\left[n\right],\forall n,\forall
k}\right\\}$, we have the following power constraints as

$\frac{1}{N}\sum\limits_{n=1}^{N}{p_{k}[n]}\leqslant\bar{p},\>\>p_{k}[n]\leqslant{p_{{\text{max}}}},\forall
n,$ (2)

where $\bar{p}$ and ${p_{max}}$ are the average transmit power and the maximum
transmit power of the IDs, respectively. Considering the high-altitude UAV and
flexibly deployed IRS, LoS channels are supposed for the ID-UAV/IRS links, the
jammer-UAV/IRS links, and the IRS-UAV links. Specifically, the ID-UAV channel
in time slot $n$ is expressed as

${{\text{h}}_{ku}}[n]=\sqrt{\rho
d_{ku}^{-2}\left[n\right]}{e^{{{-j2\pi{d_{ku}}\left[n\right]}\mathord{\left/{\vphantom{{-j2\pi{d_{ku}}\left[n\right]}\lambda}}\right.\kern-1.2pt}\lambda}}},$
(3)

where
${d_{ku}}\left[n\right]=\left\|{{\bf{q}}_{u}\left[n\right]-{{\bf{q}}_{k}}}\right\|$
is the distance between the ID $k$ and the UAV. $\lambda$ is the carrier
wavelength and $\rho$ is the path loss at the reference distance
${D_{0}}=1{\rm{m}}$. The same channel model is adopted for the channel from
the jammer to the UAV, i.e., ${h_{ju}[n]}$. Furthermore, the reflecting
channel, such as ID-IRS-UAV channel, is composed of two parts, namely, the ID-
IRS channel and the IRS-UAV channel. Specifically, the IRS-UAV channel denoted
by ${{\bf{h}}_{ru}}\left[n\right]\in{\mathbb{C}}^{M\times 1}$, can be given by

${{\mathbf{h}}_{ru}}[n]=\sqrt{\rho
d_{ru}^{-2}\left[n\right]}{{\mathbf{\tilde{h}}}_{ru}}\left[n\right],$ (4)

where ${{\mathbf{\tilde{h}}}_{ru}}\left[n\right]$ denotes the phase of the
channel, which is expressed by

${{\mathbf{\tilde{h}}}_{ru}}\left[n\right]={{{e}}^{{{-j2\pi{d_{ru}}\left[n\right]}\mathord{\left/{\vphantom{{-j2\pi{d_{ru}}\left[n\right]}\lambda}}\right.\kern-1.2pt}\lambda}}}{u_{x}}\left[n\right]\otimes{u_{z}}\left[n\right],$
(5)

where

${u_{x}}\left[n\right]\\!=\\!\left[{1,{e^{-j{{{2\pi
d{\phi_{ru,x}}\left[n\right]}\mathord{\left/{\vphantom{{2\pi
d{\phi_{ru,x}}\left[n\right]}\lambda}}\right.\kern-1.2pt}\lambda}}}},...,{e^{-j\left({{M_{x}}-1}\right){{{2\pi
d{\phi_{ru,x}}\left[n\right]}\mathord{\left/{\vphantom{{2\pi
d{\phi_{ru,x}}\left[n\right]}\lambda}}\right.\kern-1.2pt}\lambda}}}}}\right]^{T}\\!,\\!$
${u_{z}}\left[n\right]\\!=\\!\left[{1,{e^{-j{{{2\pi
d{\phi_{ru,z}}\left[n\right]}\mathord{\left/{\vphantom{{2\pi
d{\phi_{ru,z}}\left[n\right]}\lambda}}\right.\kern-1.2pt}\lambda}}}},...,{e^{-j\left({{M_{z}}-1}\right){{{2\pi
d{\phi_{ru,z}}\left[n\right]}\mathord{\left/{\vphantom{{2\pi
d{\phi_{ru,z}}\left[n\right]}\lambda}}\right.\kern-1.2pt}\lambda}}}}}\right]^{T}\\!,\\!$
${{\phi_{ru,x}}}\left[n\right]=\sin\varphi_{ru}^{\left(v\right)}\left[n\right]\cos\varphi_{ru}^{\left(h\right)}\left[n\right]=\frac{{{x\left[n\right]}-{x_{r}}}}{{{d_{ru}\left[n\right]}}},$
${\phi_{ru,z}}\left[n\right]=\sin\varphi_{ru}^{\left(v\right)}\left[n\right]\sin\varphi_{ru}^{\left(h\right)}\left[n\right]=\frac{{{H_{u}}-{z_{r}}}}{{{d_{ru}\left[n\right]}}}.$

$\varphi_{ru}^{\left(v\right)}$ and $\varphi_{ru}^{\left(h\right)}$ represent
the vertical and horizontal angle of departure (AoD) at the IRS, respectively.
${d_{ru}}\left[n\right]=\left\|{{\mathbf{q}}_{u}\left[n\right]-{{\mathbf{q}}_{r}}}\right\|$
is the distance between the IRS and the UAV. $d$ is the antenna distance. The
same model is adopted for ID-IRS channel and jammer-IRS channel. Then the
channel can be expressed by

$\displaystyle{g_{k}}\left[n\right]{\text{ =
}}{\left|{{h_{ku}}\left[n\right]+{\mathbf{h}}_{kr}^{H}\left[n\right]{\mathbf{\Gamma}}\left[n\right]{{\mathbf{h}}_{ru}}\left[n\right]}\right|^{2}},$
(6a) $\displaystyle{g_{j}}\left[n\right]{\text{ =
}}{\left|{{h_{ju}}\left[n\right]+{\mathbf{h}}_{jr}^{H}\left[n\right]{\mathbf{\Gamma}}\left[n\right]{{\mathbf{h}}_{ru}}\left[n\right]}\right|^{2}}.$
(6b)

Then, the throughput during each time slot $\Delta t$ is given by

${R_{k}}\left[n\right]=\Delta
tB{\log_{2}}\left({1+\frac{{{p_{k}}[n]{g_{k}}\left[n\right]}}{{{p_{j}}{g_{j}}\left[n\right]+{\sigma^{2}}}}}\right),$
(7)

where ${p_{j}}$ denotes the transmit power of the jammer, ${\sigma^{2}}$
denotes the power of additive white Gaussian noise (AWGN) at the UAV, and $B$
denotes the system bandwidth. Accordingly, we define a binary variable
${\mathbf{U}}{\text{ =
}}\left\\{{{u_{k}}\left[n\right]\in\left\\{{0,1}\right\\},\forall
n,k}\right\\}$, which indicates whether ID $k$ is scheduled for communicating
with the UAV in time slot $n$ or not, i.e., ID $k$ transmits data to the UAV
if ${u_{k}}\left[n\right]=1$ and keeps silent otherwise. Moreover, we assume
that only one ID is allowed to transmit data in one time slot so that the
constraints can be expressed by

$\displaystyle\sum\limits_{k=1}^{K}{{u_{k}}\left[n\right]}\leqslant 1,\forall
n,$ (8a) $\displaystyle{u_{k}}\left[n\right]\in\left\\{{0,1}\right\\},\forall
k,n.$ (8b)

In general, the UAV energy consumption is composed of two main parts. One is
the communication-related energy and the other is the propulsion energy. Note
that compared with the propulsion energy of UAVs, the communication-related
energy is much smaller (e.g., by three orders of magnitude) in practice. Thus,
we only consider the propulsion energy which is required for supporting UAV’s
mobility and making it remain aloft during the flight period [34]. The
propulsion energy consumption $E_{p}[n]$ in Joule (J) for the rotary-wing UAV
with speed $v_{u}\left[n\right]$ in time slot $n$ can be modeled as

${E_{u}}\left[n\right]=\Delta
t\left({{P_{0}}{\vartheta_{0}}\left[n\right]+{P_{1}}{\vartheta_{1}^{\frac{1}{2}}}\left[n\right]+\frac{1}{2}d^{\prime}\rho^{\prime}s^{\prime}Av_{u}^{3}\left[n\right]}\right),$
(9)

where $d^{\prime}$ and $s^{\prime}$ represent fuselage drag ratio and the
rotor solidity, respectively. $\rho^{\prime}$ and $A$ denote the air density
and the rotor disc area, respectively. $P_{0}$ and $P_{1}$ are the blade
profile power and induced power which are constants related to the UAV itself.
$v_{u}\left[n\right]$ denotes the UAV horizontal flying speed, which is given
by

${v_{u}}\left[n\right]=\frac{{\left\|{{\mathbf{q}}_{u}\left[{n+1}\right]-{\mathbf{q}}_{u}\left[n\right]}\right\|}}{{\Delta
t}}.$ (10)

$\vartheta_{0}\left[n\right]$ and $\vartheta_{1}\left[n\right]$ are given by

$\displaystyle{\vartheta_{0}}\left[n\right]=1+\frac{{3v_{u}^{2}\left[n\right]}}{{U_{tip}^{2}}},$
(11a)
$\displaystyle{\vartheta_{1}}\left[n\right]=\sqrt{1+\frac{{v_{u}^{4}\left[n\right]}}{{v_{0}^{4}}}}-\frac{{v_{u}^{2}\left[n\right]}}{{2v_{0}^{4}}},$
(11b)

 where $U_{tip}$ represents the tip speed of the UAV’s rotor blade.
$v_{0}$ is the mean rotor induced speed when the UAV is hovering.

To ensure the efficient data collection, our goal is to maximize the EE of the
UAV via the joint design of the UAV trajectory $\bf{Q}$, ID’s transmit power
$\bf{P}$, communication scheduling $\bf{U}$, and the IRS phase shift matrix
${\mathbf{\Gamma}}$. Thus, the optimization problem is formulated as

$\displaystyle\mathop{\max}\limits_{{\mathbf{Q}},{\mathbf{P}},{\mathbf{\Gamma}}}$
$\displaystyle\;\frac{{\sum\limits_{n=1}^{N}{\sum\limits_{k=1}^{K}{{u_{k}}\left[n\right]{R_{k}}\left[n\right]}}}}{{\sum\limits_{n=1}^{N}{{E_{u}}\left[n\right]}}}$
(12a) $\displaystyle{\rm s.t}.\>0\leqslant{\theta_{i}}\left[n\right]\leqslant
2\pi,\forall i,\forall n,$ (12b)
$\displaystyle\left({\rm{1}}\right),\left({\rm{2}}\right),\left({\rm{8}}\right).$

Problem (12) is challenging to solve optimally since the optimization
variables are coupled in the objective function with a fractional programming.
However, it can be effectively solved by dividing the problem into four sub-
problems by applying the BCD. Thus, this motivates us to propose an algorithm
based on AO to solve problem (12) sub-optimally, i.e., by iteratively
optimizing one of sub-problems and with the other three being fixed at each
iteration until convergence is achieved, as detailed in the next section.

##  III The proposed alternating algorithm

####  III-1 Transmit Power Optimization

For given the UAV trajectory ${\bf{Q}}$, communication scheduling ${\bf{U}}$,
and IRS phase shift matrix $\bf{\Gamma}$, the problem (12) can be reduced to

$\displaystyle\mathop{\max}\limits_{\mathbf{P}}\sum\limits_{n=1}^{N}\sum\limits_{k=1}^{K}{{u_{k}}\left[n\right]{{\log}_{2}}\left({1+\frac{{{p_{k}}[n]{g_{k}}\left[n\right]}}{{{{p_{j}}{g_{j}}\left[n\right]}+{\sigma^{2}}}}}\right)}$
(13a)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!{\text{s}}{\text{.t}}{\text{.}\>}\frac{1}{N}\sum\limits_{n=1}^{N}{p[n]}\leqslant\bar{p},$
(13b) $\displaystyle p[n]\leqslant{p_{{\text{max}}}},\forall n.$ (13c)

 This is a standard convex optimization problem that can be
efficiently solved by CVX.

####  III-2 Communication Scheduling Optimization

For any given UAV trajectory ${\bf{Q}}$, IRS passive beamforming
$\bf{\Gamma}$, and device transmit power ${\bf{P}}$, we relax the binary
variables in (12) into continuous variables, which yields the following
problem

$\displaystyle\mathop{\max}\limits_{\mathbf{U}}\sum\limits_{n=1}^{N}\sum\limits_{k=1}^{K}{{u_{k}}\left[n\right]{{\log}_{2}}\left({1+\frac{{{p_{k}}[n]{g_{k}}\left[n\right]}}{{{p_{j}}{g_{j}}\left[n\right]+{\sigma^{2}}}}}\right)}$
(14a)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!{\text{s}}{\text{.t}}{\text{.}\>}\sum\limits_{k=1}^{K}{{u_{k}}\left[n\right]}\leqslant
1,\forall n,$ (14b) $\displaystyle 0\leqslant{u_{k}}\left[n\right]\leqslant
1,\forall n,k.$ (14c)

 Note that (14) is a standard linear program (LP), it can be
efficiently solved by CVX. Moreover, it is easy to observed that the
constraints (14b) and (14c) are met with equalities when the optimal solution
$\bf U$ is attained.

####  III-3 IRS Passive Beamforming Optimization

To solve the subproblem with given any trajectory Q, communication scheduling
U, and transmit power P, the optimal phase shifts of IRS can be obtained by
solving

$\displaystyle\mathop{\max}\limits_{\mathbf{\Gamma}}\frac{{{\text{
}}{{\left|{{h_{ku}}\left[n\right]+{\mathbf{h}}_{kr}^{H}\left[n\right]{\mathbf{\Gamma}}\left[n\right]{{\mathbf{h}}_{ru}}\left[n\right]}\right|}^{2}}}}{{{{\left|{{h_{ju}}\left[n\right]+{\mathbf{h}}_{jr}^{H}\left[n\right]{\mathbf{\Gamma}}\left[n\right]{{\mathbf{h}}_{ru}}\left[n\right]}\right|}^{2}}}}$
(15a)
$\displaystyle{\text{s}}{\text{.t}}{\text{.}}\;{\theta_{i}}\left[n\right]\in\left[{0,2\pi}\right),\forall
i,\forall n.$ (15b)

 It is quite hard to obtain the optimal solution due to the unit
modulus constraints. By invoking the equality
${{\mathbf{a}}^{H}}{\mathbf{\Gamma
b}}={{\mathbf{v}}^{H}}{\text{diag}}\left\\{{{{\mathbf{a}}^{H}}}\right\\}{\mathbf{b}}$,
where
$v[n]=[{e^{j{\theta_{1}}[n]}},{e^{j{\theta_{2}}[n]}},...,{e^{j{\theta_{M}}[n]}}]$,
(6a), (6b) can be rewritten as

${g_{k}}\left[n\right]{\text{ = }}{\left|{{\mathbf{v}}{{[n]}^{H}}{{\bf
a}_{k}}[n]+{{\tilde{\bf a}}_{k}}[n]}\right|^{2}},$ (16)
${g_{j}}\left[n\right]{\text{ = }}{\left|{{\mathbf{v}}{{[n]}^{H}}{{\bf
a}_{j}}[n]+{{\tilde{\bf a}}_{j}}[n]}\right|^{2}},$ (17)

where ${{{\bf
a}}_{k}}[n]={\text{diag}}\left\\{{{\mathbf{h}}_{kr}^{H}[n]}\right\\}{{\mathbf{h}}_{ru}}[n]\sqrt{p_{k}[n]}$,
${{{\bf
a}}_{j}}[n]={\text{diag}}\left\\{{{\mathbf{h}}_{jr}^{H}[n]}\right\\}{{\mathbf{h}}_{ru}}[n]\sqrt{{p_{j}}}$,
${\tilde{\bf a}_{k}}[n]=h_{ku}^{H}[n]\sqrt{p_{k}[n]}$, ${\tilde{\bf
a}_{j}}[n]=h_{ju}^{H}[n]\sqrt{{p_{j}}}$. The problem is non-convex and is not
easy to solve directly. As such, problem (15) in each time slot can be
equivalently written as

$\displaystyle\mathop{\max}\limits_{\mathbf{v}}$
$\displaystyle\frac{{{{{\left|{{\mathbf{v}}{{[n]}^{H}}{{\bf
a}_{k}}[n]+{{\tilde{\bf
a}}_{k}}[n]}\right|}^{2}}}}}{{{{{\left|{{\mathbf{v}}{{[n]}^{H}}{{\bf
a}_{j}}[n]+{{\tilde{\bf a}}_{j}}[n]}\right|}^{2}}}+{\sigma^{2}}}},\forall n$
(18a)
$\displaystyle{\text{s}}{\text{.t}}{\text{.}}\left|{{v_{i}}[n]}\right|=1,\forall
i,\forall n.$ (18b)

 By assuming the transmit power of ID $k$ in time slot $n$ is not
zero, problem (18) can be equivalently transformed into

$\displaystyle\mathop{\min}\limits_{\mathbf{v}}$
$\displaystyle\frac{{{{{\left|{{\mathbf{v}}{{[n]}^{H}}{{\bf
a}_{j}}[n]+{{\tilde{\bf
a}}_{j}}[n]}\right|}^{2}}}+{\sigma^{2}}}}{{{{{\left|{{\mathbf{v}}{{[n]}^{H}}{{\bf
a}_{k}}[n]+{{\tilde{\bf a}}_{k}}[n]}\right|}^{2}}}}},\forall n$ (19a)
$\displaystyle{\text{s}}{\text{.t}}{\text{.}}\left|{{v_{i}}[n]}\right|=1,\forall
i,\forall n.$ (19b)

 The problem above belongs to fractional programming and can be
converted into a series of parametric sub-problems. In the $i$-th iteration,
we consider the following parametric problem.

$\displaystyle\mathop{\min}\limits_{\mathbf{v}}{\left|{{\mathbf{v}}{{[n]}^{H}}{{\bf
a}_{j}}[n]\\!+\\!{{\tilde{\bf
a}}_{j}}[n]}\right|^{2}}\\!+\\!{\sigma^{2}}\\!-\\!{\eta^{(i-1)}}{\left|{{\mathbf{v}}{{[n]}^{H}}{{\bf
a}_{k}}[n]\\!+\\!{{\tilde{\bf a}}_{k}}[n]}\right|^{2}}$ (20a)
$\displaystyle{\text{s}}{\text{.t}}{\text{.}}\left|{{v_{i}}[n]}\right|=1,\forall
i,\forall n.$ (20b)

 where $\eta^{(i-1)}$ is an introduced parameter with an initial value
$\eta^{(0)}=0$. $\eta^{(i)}$ can be updated by

${\eta^{\left(i\right)}}=\frac{{{{{\left|{{{\mathbf{v}}^{\left(i\right)H}}[n]{{\bf
a}_{j}}[n]+{{\tilde{\bf
a}}_{j}}[n]}\right|}^{2}}}+{\sigma^{2}}}}{{{{{\left|{{{\mathbf{v}}^{\left(i\right)H}}{{[n]}^{H}}{{\bf
a}_{k}}[n]+{{\tilde{\bf a}}_{k}}[n]}\right|}^{2}}}}},$ (21)

where ${{{\mathbf{v}}^{\left(i\right)H}}[n]}$ can be obtained by solving
problem (20). To obtain the optimal value of $\bf{v}$, we transform the
objective function and minimize an upper bound as follows [35]

$\displaystyle{\left|{{{\mathbf{v}}^{H}}[n]{{\bf a}_{j}}[n]+{{\tilde{\bf
a}}_{j}}[n]}\right|^{2}}+{\sigma^{2}}-\eta{\left|{{{\mathbf{v}}^{H}}{{[n]}^{H}}{{\bf
a}_{k}}[n]+{{\tilde{\bf a}}_{k}}[n]}\right|^{2}}$
$\displaystyle={{\mathbf{v}}^{H}}\left({{{\mathbf{{\bf a}}}_{j}}{\mathbf{{\bf
a}}}_{j}^{H}-\eta{{\mathbf{{\bf a}}}_{k}}{\mathbf{{\bf
a}}}_{k}^{H}}\right){\mathbf{v}}+{\left|{{{\tilde{\bf
a}}_{j}}}\right|^{2}}+{\sigma^{2}}$ $\displaystyle-\eta{\left|{{{\tilde{\bf
a}}_{k}}}\right|^{2}}-2\Re\left\\{{{{\mathbf{v}}^{H}}\left({\eta\tilde{\bf
a}_{k}^{*}{{{\bf a}}_{k}}-\tilde{\alpha}_{j}^{*}{{{\bf
a}}_{j}}}\right)}\right\\}$
$\displaystyle\leqslant{\lambda_{\max}}({\mathbf{\Phi}}){\left\|{\mathbf{v}}\right\|^{2}}-2\Re\left\\{{{{\mathbf{v}}^{H}}{\mathbf{\omega}}}\right\\}+c,$
(22a)

where

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\bf
a}_{j}=[{\bf a}_{j}[1],...,{\bf a}_{j}[N]],$ (23a)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\bf
a}_{k}=[{\bf a}_{k}[1],...,{\bf a}_{k}[N]],\forall k,$ (23b)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\mathbf{\Phi}}={{\bf
a}_{j}}{{\bf a}}_{j}^{H}-\eta{{\bf a}_{k}}{{\bf a}}_{k}^{H},$ (23c)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\mathbf{\omega}}=\left({{\lambda_{\max}}\left({\mathbf{\Phi}}\right){\mathbf{I}}-\Phi}\right){\mathbf{\tilde{v}}}+\eta\tilde{\bf
a}_{k}^{*}{{{\bf a}}_{k}}-\tilde{\bf a}_{j}^{*}{{{\bf a}}_{j}},$ (23d)
$\displaystyle
c={{\mathbf{\tilde{v}}}^{H}}\left({{\lambda_{\max}}\left({\mathbf{\Phi}}\right){\mathbf{I}}-\Phi}\right){\mathbf{\tilde{v}}}+{\left|{{{\tilde{\bf
a}}_{j}}}\right|^{2}}+{\sigma^{2}}-\eta{\left|{{{\tilde{\bf
a}}_{k}}}\right|^{2}},$ (23e)

where $\tilde{v}$ is the previous iteration of the solution to $v$. Therefore,
the final optimization problem is given by

$\displaystyle\min\;\;{\lambda_{\max}}({\mathbf{\Phi}}){\left\|{\mathbf{v}}\right\|^{2}}-2\Re\left\\{{{{\mathbf{v}}^{H}}{\mathbf{\omega}}}\right\\}$
(24a)
$\displaystyle{\text{s}}{\text{.t}}{\text{.}}\left|{{v_{i}}\left[n\right]}\right|=1,\forall
i,\forall n.$ (24b)

 It is obvious that the optimal value is obtained when
$\Re\left\\{{{{\mathbf{v}}^{H}}{\mathbf{\omega}}}\right\\}$ is maximized,
i.e., the phase of ${v_{i}}$ and ${\omega_{i}}$ are equal, where
${\omega_{i}}$ is the $i$-th element of ${\mathbf{\omega}}$. Accordingly, we
obtain the final solution in the time slot $n$ as

$\displaystyle{\mathbf{v}}\left[n\right]={\left[{{e^{-j\arg\left({{\omega_{1}}}\right)}},...,{e^{-j\arg\left({{\omega_{M}}}\right)}}}\right]^{T}}.$
(25)

Finally, we obtain a closed-form solution in each iteration and the
computational complexity is significantly reduced.

Algorithm 1 An alternating algorithm for solving (15)

1: Input: devicess transmit power ${\bf P}$, communication sheduling ${\bf
U}$, and UAV trajectory ${\bf Q}$.

2: Initialization: set initial $\mu_{1}$, $i_{max}$, and $\eta^{(0)}$. Set
$i=0$, as iteration index, ${\bf{\tilde{\Gamma}}}$ as initial IRS phase shift.

3: repeat.

4: Fix $\bf\Gamma$ and calculate $\eta^{(i)}$ using (19) and ${\bf\omega}$
using (21).

5: Set ${\mathbf{\tilde{\Gamma}}}{\text{ =
diag}}\left\\{{{e^{-j{\mathbf{\omega}}}}}\right\\}$.

6: Update $i\leftarrow i+1\ $.

7: until: convergence or $i={i_{\max}}$.

8: Output:${\bf\Gamma}$.

####  III-4 UAV Trajectory Optimization

For given transmit power ${\bf{P}}$, communication scheduling ${\bf{U}}$, and
phase shift $\bf{\Gamma}$, (12) can be rewritten by

$\displaystyle\mathop{\max}\limits_{\mathbf{Q}}\frac{{B\sum\limits_{n\in\cal
N}{\sum\limits_{k\in\cal K}{\Delta
t{{\log}_{2}}\left({1+\frac{{{p_{k}}[n]{g_{k}}\left[n\right]}}{{{{p_{j}}{g_{j}}\left[n\right]}+{\sigma^{2}}}}}\right)}}}}{{\sum\limits_{n\in\cal
N}{\Delta
t\left({{P_{0}}{\vartheta_{0}}\left[n\right]+{P_{1}}\vartheta_{1}^{\frac{1}{2}}\left[n\right]+\frac{1}{2}d^{\prime}\rho^{\prime}s^{\prime}Av_{u}^{3}\left[n\right]}\right)}}}$
(26a) s.t.
$\displaystyle{\bf{q}}_{u}\left[0\right]={{\bf{q}}_{S}},{\bf{q}}_{u}\left[N\right]={{\bf{q}}_{F}},$
(26b)
$\displaystyle\left\|{{\mathbf{q}}_{u}[n]-{\mathbf{q}}_{u}[n-1]}\right\|\leqslant{D_{\max}},n=1,...,N.$
(26c)

 Problem (26) is difficult to solve due to its non-convex objective
function with a fractional objective function. Note that
${{{\mathbf{\tilde{h}}}}_{ru}}\left[n\right]$,
${{{\mathbf{\tilde{h}}}}_{ku}}\left[n\right]$ and
${{{\mathbf{\tilde{h}}}}_{ju}}\left[n\right]$ are relative to the trajectory
variables. However, it is observed that those are more complex to handle due
to their non-linear formation. To overcome such a problem, we use ($i-1$)th
iteration to obtain the approximate
${{{\mathbf{\tilde{h}}}}_{ru}}\left[n\right]$,${{{\mathbf{\tilde{h}}}}_{ku}}\left[n\right]$
and ${{{\mathbf{\tilde{h}}}}_{ju}}\left[n\right]$ [31]. Based on the
consideration above, we consider rewriting the objective function. By denoting

$\begin{gathered}{{\bf{H}}_{k}}\left[n\right]=\left[{\sqrt{\rho}\tilde{h}_{ku}^{\left(i-1\right)}\left[n\right],\sqrt{\rho}{\mathbf{h}}_{kr}^{H}\left[n\right]{\mathbf{\Gamma}}\left[n\right]{{{\mathbf{\tilde{h}}}}_{ru}^{\left(i-1\right)}}\left[n\right]}\right],\hfill\\\
{{\bf{H}}_{j}}\left[n\right]=\left[{\sqrt{\rho}\tilde{h}_{ju}^{\left(i-1\right)}\left[n\right],\sqrt{\rho}{\mathbf{h}}_{jr}^{H}\left[n\right]{\mathbf{\Gamma}}\left[n\right]{{{\mathbf{\tilde{h}}}}_{ru}^{\left(i-1\right)}}\left[n\right]}\right],\hfill\\\
\end{gathered}$

where
${\tilde{h}_{ku}}[n]={e^{-j\frac{{2\pi{d_{ku}}\left[n\right]}}{\lambda}}}$,
(6) can be transformed as

$\displaystyle{g_{k}}\left[n\right]={\mathbf{d}}_{k}^{T}[n]{\mathbf{H}}_{k}^{H}[n]{{\mathbf{H}}_{k}}[n]{{\mathbf{d}}_{k}}[n],$
(27a)
$\displaystyle{g_{j}}\left[n\right]={\mathbf{d}}_{j}^{T}[n]{\mathbf{H}}_{j}^{H}[n]{{\mathbf{H}}_{j}}[n]{{\mathbf{d}}_{j}}[n],$
(27b)

where
${{\mathbf{d}}_{k}}[n]\\!=\\!{\left[{{{{d_{ku}^{-1}}[n]}},{{{d_{ru}^{-1}}[n]}}}\right]^{T}}$,
${{\mathbf{d}}_{j}}[n]={\text{
}}{\left[{{{{d_{ju}^{-1}}[n]}},{{{d_{ru}^{-1}}[n]}}}\right]^{T}}$.

Moreover, the other problem is the non-convex objective function which is
composed of the coupled variables. To transform the numerator of the objective
function of (26) into concave, we consider introducing the slack variables and
leveraging the SCA method. By introducing the slack variables
${\mathbf{S}}=\\{S_{k}[n],\forall n,k\\}$ and ${\mathbf{G}}=\\{G[n],\forall
n\\}$, the throughput of ID $k$ in time slot $n$, i.e.,
${R_{k}}\left[n\right]$, can be rewritten as follows

${R_{k}}\left[n\right]=\Delta
tB{\log_{2}}\left({1+\frac{{{S^{-1}_{k}}\left[n\right]}}{{G\left[n\right]}}}\right),\forall
n,k,$ (28)

with two additional constraints

$p_{k}[n]{\mathbf{d}}_{k}^{T}[n]{\mathbf{H}}_{k}^{H}[n]{{\mathbf{H}}_{k}}[n]{{\mathbf{d}}_{k}}[n]\geqslant{S^{-1}}[n],\forall
n,k,$ (29)

and

${p_{j}}{\mathbf{d}}_{j}^{T}[n]{\mathbf{H}}_{j}^{H}[n]{{\mathbf{H}}_{j}}[n]{{\mathbf{d}}_{j}}[n]+{\sigma^{2}}\leqslant
G[n],\forall n.$ (30)

The equivalence has been proved by contradiction in [13]. Specifically, if the
constraints hold with inequalities, we can improve the value of the objective
function by decreasing variable ${\mathbf{S}}$ and ${\mathbf{G}}$.

In the next step, we focus on dealing with problem (28). Note that constraints
(29) and (30) are still non-convex, thus, with the property of the convex
function’s first-order Taylor expansion and successive convex optimization
method, we first use the following Lemma to obtain a lower bound of the
objective function (26).

###### Lemma 1

The numerator of the objective function (26) is lower bounded at the feasible
point $(S_{k,0}[n],{G_{0}}[n])\ $ by

$\begin{array}[]{l}\tilde{R}_{k}\left[n\right]={\log_{2}}\left({1+\frac{1}{{{S_{k,0}}[n]{G_{0}}[n]}}}\right)+{\zeta_{k,1}}\left[n\right]\left({S_{k}\left[n\right]-{S_{k,0}}\left[n\right]}\right)\\\
+{\zeta_{k,2}}\left[n\right]\left({G\left[n\right]-{G_{0}}\left[n\right]}\right),\forall
n,k,\end{array}\ $ (31)

where
${\zeta_{k,1}}[n]=-{\log_{2}}\left({\frac{e}{{{S_{k,0}}[n]+{{({S_{k,0}}[n])}^{2}}{G_{0}}[n]}}}\right)$,
${\zeta_{k,2}}[n]=-{\log_{2}}\left({\frac{e}{{{G_{0}}[n]+{{({G_{0}}[n])}^{2}}{S_{k,0}}[n]}}}\right)$.

_Proof:_ Since $f(x,y)={\log_{2}}(1+1/xy)$ is a convex function [36], its
first-order Taylor expansion provides a global under-estimator at a feasible
point $({x_{0}},{y_{0}})$, i.e.,

$\begin{array}[]{l}{\log_{2}}(1+1/xy)\geq{\log_{2}}(1+1/{x_{0}}{y_{0}})\\\
-(x-{x_{0}}){\log_{2}}e/({x_{0}}+{({x_{0}})^{2}}{y_{0}})\\\
-(y-{y_{0}}){\log_{2}}e/({y_{0}}+{({y_{0}})^{2}}{x_{0}}).\end{array}\ $ (32)

Thus, by applying $x={S_{k}}[n]\ $, $y=G[n]\ $,  _Lemma_ _1_ is proved.

Therefore, we can obtain a lower bound of objective function. However, the
constraints (29) and (30) are non-convex because of the coupled variables. To
deal with the non-convexity, we introduce the slack variables
${{\bf{\xi}}_{1}}=\left\\{{{\xi_{1}}\left[n\right],\forall n}\right\\}$,
${{\mathbf{\xi}}_{2}}=\left\\{{{\xi_{2}}\left[n\right],\forall n}\right\\}$,
${{\mathbf{\xi}}_{3}}=\left\\{{{\xi_{3}}\left[n\right],\forall n}\right\\}$,
${{\mathbf{\xi}}_{4}}=\left\\{{{\xi_{4}}\left[n\right],\forall n}\right\\}$,
and by denoting
${{\mathbf{\tilde{d}}}_{k}}={\left[{{\xi_{1}}\left[n\right],{\xi_{2}}\left[n\right]}\right]^{T}}$,
${{\mathbf{\tilde{d}}}_{j}}={\left[{{\xi_{3}}\left[n\right],{\xi_{4}}\left[n\right]}\right]^{T}}$,
the constraints (29) and (30) are rewritten as

$p_{k}[n]\tilde{\bf{d}}_{k}^{T}[n]{\bf{H}}_{k}^{H}[n]{{\bf{H}}_{k}}[n]{{\tilde{\bf{d}}}_{k}}[n]\geq{S^{-1}_{k}}[n],\forall
n,k\\\ $ (33)

and

${p_{j}}\tilde{\bf{d}}_{j}^{T}[n]{\bf{H}}_{j}^{H}[n]{{\bf{H}}_{j}}[n]{{\tilde{\bf{d}}}_{j}}[n]+{\sigma^{2}}\leq
G[n],\forall n\\\ $ (34)

with for additional constraints

$\displaystyle{d_{ku}^{-1}}[n]\geq{\xi_{1}}[n],\forall n,\forall k,$ (35a)
$\displaystyle{d_{ru}^{-1}}[n]\geq{\xi_{2}}[n],\forall n,$ (35b)
$\displaystyle{d_{ju}^{-1}[n]}\leq{\xi_{3}}[n],\forall n,$ (35c)
$\displaystyle{d_{ru}^{-1}}[n]\leq{\xi_{4}}[n],\forall n.$ (35d)

In order to conveniently tackle the non-convex variables, we unfold as follows

$\displaystyle{F_{k}}\left[n\right]-{{{\xi_{1}}^{-2}\left[n\right]}}\leqslant
0,\forall n,k,$ (36a)
$\displaystyle{F_{r}}\left[n\right]-{{{\xi_{2}}^{-2}\left[n\right]}}\leqslant
0,\forall n,$ (36b)
$\displaystyle{{\xi_{3}}^{-2}\left[n\right]}-{F_{j}}\left[n\right]\leqslant
0,\forall n,$ (36c)
$\displaystyle{{\xi_{4}}^{-2}\left[n\right]}-{F_{r}}\left[n\right]\leqslant
0,\forall n,$ (36d)

 where
${F_{k}}\left[n\right]={\left({x\left[n\right]-{x_{k}}}\right)^{2}}+{\left({y\left[n\right]-{y_{k}}}\right)^{2}}+{H_{u}^{2}}$,
${F_{r}}\left[n\right]={\left({x\left[n\right]-{x_{r}}}\right)^{2}}+{\left({y\left[n\right]-{y_{r}}}\right)^{2}}+{\left({H_{u}-{z_{r}}}\right)^{2}}$,
${F_{j}}\left[n\right]={\left({x\left[n\right]-{x_{j}}}\right)^{2}}+{\left({y\left[n\right]-{y_{j}}}\right)^{2}}+{\left({H_{u}-{z_{j}}}\right)^{2}}$.
However, there still exists several non-convex feasible regions in constraint
(36). We use SCA technique to address the non-convex regions. The first-order
Taylor expansions of ${{{\xi_{1}}^{-2}\left[n\right]}}$,
${{{\xi_{2}}^{-2}\left[n\right]}}$,
${\mathbf{d}}_{k}^{T}[n]{\mathbf{H}}_{k}^{H}[n]{{\mathbf{H}}_{k}}[n]{{\mathbf{d}}_{k}}[n]$
at the feasible points
${{\mathbf{\xi}}_{{\mathbf{1}},{\mathbf{0}}}}=\left\\{{{\xi_{1,0}}\left[n\right],\forall
n}\right\\}$,
${{\mathbf{\xi}}_{{\mathbf{2}},{\mathbf{0}}}}=\left\\{{{\xi_{2,0}}\left[n\right],\forall
n}\right\\}$ and
${{\mathbf{\tilde{d}}}_{k,0}}=\left\\{{{{\tilde{d}}_{k,0}}[n],\forall
n}\right\\}$, which is given by

$\displaystyle\xi_{1}^{-2}\left[n\right]\geqslant\xi_{1,0}^{-2}\left[n\right]-2\xi_{1,0}^{-2-1}\left[n\right]\left({{\xi_{1}}\left[n\right]-{\xi_{1,0}}\left[n\right]}\right),$
(37a)
$\displaystyle\xi_{2}^{-2}\left[n\right]\geqslant\xi_{2,0}^{-2}\left[n\right]-2\xi_{2,0}^{-2-1}\left[n\right]\left({{\xi_{2}}\left[n\right]-{\xi_{2,0}}\left[n\right]}\right),$
(37b)
$\displaystyle{\mathbf{\tilde{d}}}_{k}^{T}[n]{\mathbf{H}}_{k}^{H}[n]{{\mathbf{H}}_{k}}[n]{{{\mathbf{\tilde{d}}}}_{k}}[n]\geqslant-{\mathbf{\tilde{d}}}_{k,0}^{T}[n]{\mathbf{H}}_{k}^{H}[n]{{\mathbf{H}}_{k}}[n]{{{\mathbf{\tilde{d}}}}_{k,0}}[n]$
$\displaystyle+2\Re\left[{{\mathbf{\tilde{d}}}_{k,0}^{T}[n]{\mathbf{H}}_{k}^{H}[n]{{\mathbf{H}}_{\text{k}}}[n]{{{\mathbf{\tilde{d}}}}_{k}}[n]}\right].$
(37c)

To make the problem solvable, we use the first-order Taylor expansion of the
convex function to obtain the lower bound. At the given feasible points
${{\mathbf{x}}_{0}}=\left\\{{{x_{0}}\left[n\right],\forall n}\right\\}$ and
${{\mathbf{y}}_{0}}=\left\\{{{y_{0}}\left[n\right],\forall n}\right\\}$, the
first-order Taylor expansions are given by

$\displaystyle{x^{2}}\left[n\right]\geqslant-
x_{0}^{2}\left[n\right]+{x_{0}}\left[n\right]x\left[n\right],$ (38a)
$\displaystyle{y^{2}}\left[n\right]\geqslant-
y_{0}^{2}\left[n\right]+{y_{0}}\left[n\right]y\left[n\right].$ (38b)

As such, ${F_{r}}\left[n\right]$ and ${F_{j}}\left[n\right]$ can be
transformed as

$\begin{gathered}{{\tilde{F}}_{r}}\left[n\right]=-x_{0}^{2}\left[n\right]+2{x_{0}}\left[n\right]x\left[n\right]+x_{r}^{2}-y_{0}^{2}\left[n\right]+\\\
2{y_{0}}\left[n\right]y\left[n\right]-y_{r}^{2}\\!+\\!H_{u}^{2}-2{x_{r}}x\left[n\right]-2{y_{r}}y\left[n\right]-2{z_{r}}H_{u}.\hfill\\\
\end{gathered}$ (39)
$\begin{gathered}{{\tilde{F}}_{j}}\left[n\right]=-x_{0}^{2}\left[n\right]+2{x_{0}}\left[n\right]x\left[n\right]+x_{j}^{2}-y_{0}^{2}\left[n\right]+\\\
2{y_{0}}\left[n\right]y\left[n\right]-y_{j}^{2}\\!+\\!H_{u}^{2}-2{x_{j}}x\left[n\right]-2{y_{j}}y\left[n\right]-2{z_{j}}H_{u}.\hfill\\\
\end{gathered}$ (40)

Thus (29) can be writen as

$\displaystyle p_{k}[n]$
$\displaystyle(2\Re\left[{{\mathbf{\tilde{d}}}_{g,0}^{T}[n]{\mathbf{H}}_{g}^{H}[n]{{\mathbf{H}}_{\text{g}}}[n]{{{\mathbf{\tilde{d}}}}_{g}}[n]}\right]-$
(41)
$\displaystyle{{\tilde{\bf{d}}}_{g,0}}^{T}[n]{\bf{H}}_{g}^{H}[n]{{\bf{H}}_{g}}[n]{{\tilde{\bf{d}}}_{g,0}}[n])\geq{S^{-1}}[n],\forall
n,k.$

Further more, by denoting

$\displaystyle{{{{\tilde{\xi}}_{1}}\left[n\right]}}={\xi_{1,0}^{-2}\left[n\right]}-2{{\xi_{1,0}^{-3}\left[n\right]}}\left({{\xi_{1}}\left[n\right]-{\xi_{1,0}}\left[n\right]}\right),$
(42a)
$\displaystyle{{{{\tilde{\xi}}_{2}}\left[n\right]}}={{\xi_{2,0}^{-2}\left[n\right]}}-2{{{\xi_{2,0}^{-3}}\left[n\right]}}\left({{\xi_{2}}\left[n\right]-{\xi_{2,0}}\left[n\right]}\right),$
(42b)

 we can substitute all concave points to solvable ones. Then, to deal
with the non-convexity of $E_{u}[n]$ in the objective function of problem
(26), we introduce the other slack variable
${\mathbf{e}}=\left\\{{e\left[n\right],\forall n}\right\\}$ such that

$e\left[n\right]\geqslant{\left({\sqrt{1+\frac{{v_{u}^{4}\left[n\right]}}{{v_{0}^{4}}}}-\frac{{v_{u}^{2}\left[n\right]}}{{2v_{0}^{4}}}}\right)^{\frac{1}{2}}}.$
(43)

Next, we can rewrite (43) as

$\frac{1}{{{e^{2}}\left[n\right]}}\leqslant{e^{2}}\left[n\right]+\frac{{v_{u}^{2}}}{{v_{0}^{2}}}={e^{2}}\left[n\right]+\frac{{{{\left\|{{\mathbf{q}}_{u}\left[{n+1}\right]-{\mathbf{q}}_{u}\left[n\right]}\right\|}^{2}}}}{{v_{0}^{2}\Delta{t^{2}}}}.$
(44)

Note that the constraint (44) should hold with equality when the optimal
solution is obtained. If the constraint (44) holds with inequality, we can
improve the value of the objective function (26) by decreasing variable
$e[n]$. Since that both sides of constraint (44) are convex, we should
transform the right-hand side into concave formations. By applying the first-
order Taylor expansion at given points
${{\mathbf{s}}_{0}}=\left\\{{{s_{0}}\left[n\right],\forall n}\right\\}$ and
${{\mathbf{Q}}_{0}}=\left\\{{{{\mathbf{q}}_{0}}\left[n\right],\forall
n}\right\\}$,

$\begin{gathered}{e^{2}}\left[n\right]+\frac{{{{\left\|{{\mathbf{q}}_{u}\left[{n+1}\right]-{\mathbf{q}}_{u}\left[n\right]}\right\|}^{2}}}}{{v_{0}^{2}\Delta{t^{2}}}}\hfill\\\
\geqslant
e_{0}^{2}\left[n\right]+2{e_{0}}\left[n\right]\left({e\left[n\right]-{e_{0}}\left[n\right]}\right)-\frac{{{{\left\|{{{\mathbf{q}}_{0}}\left[{n+1}\right]-{{\mathbf{q}}_{0}}\left[n\right]}\right\|}^{2}}}}{{v_{0}^{2}\Delta{t^{2}}}}\hfill\\\
+\frac{2}{{v_{0}^{2}\Delta{t^{2}}}}{\left({{{\mathbf{q}}_{0}}\left[{n+1}\right]\\!-\\!{{\mathbf{q}}_{0}}\left[n\right]}\right)^{T}}\left({{\mathbf{q}}_{u}\left[{n+1}\right]\\!-\\!{\mathbf{q}}_{u}\left[n\right]}\right)\triangleq{{\cal
L}_{0}}\left[n\right].\hfill\\\ \end{gathered}$ (45)

As such, all the constraints are convex. Problem (26) is transformed into

$\displaystyle\mathop{\max}\limits_{\begin{subarray}{\mathbf{Q}}{\mathbf{Q}},{\mathbf{S}},{\mathbf{G}},{\mathbf{e}},\\\
{\mathbf{\xi_{1}}},{\mathbf{\xi_{2}}},{\mathbf{\xi_{3}}},{\mathbf{\xi_{4}}}\end{subarray}}\>\frac{{\Delta
tB\sum{{}_{n=1}^{N}}\tilde{R}_{u}[n]}}{{\sum{{}_{n=1}^{N}}\Delta
t\left({{P_{0}}{\vartheta_{0}}\left[n\right]+{P_{1}}e\left[n\right]+\frac{1}{2}d^{\prime}\rho^{\prime}s^{\prime}Av_{u}^{3}\left[n\right]}\right)}}$
(46a) s.t. $\displaystyle\frac{1}{{{e^{2}}\left[n\right]}}\leqslant{{\cal
L}_{0}}\left[n\right],\forall n,$ (46b) $\displaystyle
e\left[n\right]>0,\forall n,$ (46c)
$\displaystyle{F_{k}}\left[n\right]-\tilde{\xi}_{1}^{-2}\left[n\right]\leqslant
0,\forall n,k,$ (46d)
$\displaystyle{F_{r}}\left[n\right]-\tilde{\xi}_{2}^{-2}\left[n\right]\leqslant
0,\forall n,$ (46e)
$\displaystyle\xi_{3}^{-2}\left[n\right]-{\tilde{F}_{j}}\left[n\right]\leqslant
0,\forall n,$ (46f)
$\displaystyle\xi_{4}^{-2}\left[n\right]-{\tilde{F}_{r}}\left[n\right]\leqslant
0,\forall n,$ (46g) $\displaystyle(26),(\ref{jammer_power}),(\ref{GN_power}).$

Until now, all constraints are convex, the objective function composed of a
linear numerator and a convex denominator. The problem (46) is a quasi-convex
optimization problem so that it can thus be solved by employing fractional
programming methods, e.g., the Dinkelbach’s algorithm.

Algorithm 2 An alternating algorithm for solving (12)

1: Input: ${\mu_{2}}$, ${i_{\max}}$

2: Initialization: Set $i=0$, as iteration index, ${\mu_{2}}$ as the threshold
and original points
${\Upsilon_{0}}=\left\\{{{{\mathbf{Q}}^{\left(0\right)}},{{\mathbf{\Gamma}}^{\left(0\right)}},{{\mathbf{P}}^{\left(0\right)}},{{\mathbf{U}}^{\left(0\right)}},}\right\\}$,
thus obtaining the EE by using (12) and generating a series of initial points
${\mathbf{S}}_{k,0}^{\left(0\right)},{\mathbf{G}}_{0}^{\left(0\right)},{\mathbf{\xi}}_{1,0}^{\left(0\right)},{\mathbf{\xi}}_{2,0}^{\left(0\right)},{\mathbf{h}}_{ru}^{\left(0\right)},{\mathbf{d}}_{k,0}^{\left(0\right)},{\mathbf{d}}_{j,0}^{\left(0\right)}$.

3: repeat.

4: With given ${{\mathbf{Q}}^{\left(i\right)}}$,
${{\mathbf{\Gamma}}^{\left(i\right)}}$, and ${{\mathbf{U}}^{\left(i\right)}}$,
update ${{\mathbf{P}}^{\left(i\right)}}$ to
${{\mathbf{P}}^{\left(i+1\right)}}$ by solving sub-problem (13).

5: With given ${{\mathbf{Q}}^{\left(i\right)}}$,
${{\mathbf{\Gamma}}^{\left(i\right)}}$, update
${{\mathbf{U}}^{\left(i\right)}}$ to ${{\mathbf{U}}^{\left(i+1\right)}}$ by
solving sub-problem (14).

6: With given ${{\mathbf{Q}}^{\left(i\right)}}$,
${{\mathbf{p}}^{\left({i+1}\right)}}$, and
${{\mathbf{U}}^{\left({i+1}\right)}}$, update
${{\mathbf{\Gamma}}^{\left(i\right)}}$ to
${{\mathbf{\Gamma}}^{\left(i+1\right)}}$ by solving sub-problem (15).

7: With given ${{\mathbf{\Gamma}}^{\left({i+1}\right)}}$,
${{\mathbf{P}}^{\left({i+1}\right)}}$, and
${{\mathbf{U}}^{\left({i+1}\right)}}$, update
${{\mathbf{Q}}^{\left(i\right)}}$ to ${{\mathbf{Q}}^{\left(i+1\right)}}$ by
solving sub-problem (46).

8: With given ${{\mathbf{Q}}^{\left(i+1\right)}}$,
${{\mathbf{P}}^{\left(i+1\right)}}$, ${{\mathbf{\Gamma}}^{\left(i+1\right)}}$,
${{\mathbf{U}}^{\left({i+1}\right)}}$, compute EE and
${\mathbf{S}}_{0}^{\left({i+1}\right)}$,
${\mathbf{T}}_{0}^{\left({i+1}\right)}$,
${\mathbf{\xi}}_{10}^{\left({i+1}\right)}$,
${\mathbf{\xi}}_{20}^{\left({i+1}\right)}$,
${\mathbf{h}}_{ru}^{\left({i+1}\right)}$,
${\mathbf{d}}_{k,0}^{\left({i+1}\right)}$,
${\mathbf{d}}_{j,0}^{\left({i+1}\right)}$.

9: Update $i\leftarrow i+1\ $.

10: until: Convergence or $i=i_{\max}$.

11:
Output:$\left\\{{{{\mathbf{Q}}},{{\mathbf{\Gamma}}},{{\mathbf{P}}},{{\mathbf{U}}},}\right\\}$.

####  III-5 Overall Algorithm

In summary, by exploiting the BCD method, we divide the original problem into
four blocks. Furthermore, with the aid of Dinkelbach’s algorithm and SCA, four
sub-problems are solved as shown. The overall algorithm for solving (12) is
summarized in Algorithm 2.

Note that there are two layers in Algorithm 2, including an inner iteration as
shown in Algorithm 1 and an outer iteration. In fact, solving sub-problem 3
and sub-problem 4 dominates the complexity of Algorithm 2. In the inner layer,
the fraction programing in sub-problem 3 and 4 can always converge to the
stationary and sub-optimal solution [30]. In the outer layer, BCD technique is
adopted, and we update
$\left\\{{{\bf{P}},{\bf{U}},{\bf{\Gamma}},{\bf{Q}}}\right\\}$ in an
alternating manner. Consequently, the computational complexities of Algorithm
2 is approximately ${{\cal
O}_{sub2}}\left({N{{{L}}^{3}}{I_{1}}{I_{2}}}\right)+{{\cal
O}_{sub3}}\left({8{N^{3.5}}{I_{2}}{I_{3}}}\right)$, where $I_{1}$ is the
number of iterations required for solving (23d), $I_{2}$ is for (46), and
$I_{3}$ is for the Dinkelbach’s algorithm.

##  IV Numerical Results

TABLE I: SIMULATIONS PARAMETERS UAV flight altitulde |  $H_{u}=100$ m
---|---
Maximum UAV speed |  $V_{max}=60$ m/s
Number of IoT devices |  $k=5$
Element spacing of IRS |  $d=0.1$ m
Number of IRS elements |  $M=150$
Noise power |  $\sigma^{2}=-80$ dBm
System bandwidth |  $B=1$ MHz
Channel power gain at the reference distance |  $\rho=-30$ dB
Average transmit power |  $\bar{p}=20$ dBm
Maximum transmit power |  $p_{max}=26$ dBm
Jamming power |  $\bar{p}_{j}=30$ dBm
Flight time of UAV |  $T=20$ s
Time slot |  ${\Delta t}=0.5$ s

In this part, we give the simulation results to verify the effectiveness of
the proposed design. “Proposed” refers to the case we proposed for maximizing
the EE by the jointly optimizing communication scheduling, ID transmit power
allocation, IRS passive beamforming, and UAV trajectory. On the other hand, we
compared our proposed algorithm with the case without IRS proposed in [15],
denoted by “w/o IRS”. Moreover, we consider two setups for the location of the
jammer as shown in Fig. 2. For Setup (a), the location of the jammer is
${{\rm{{\bf{q}}}}_{j}}=\left({250,50,0}\right)$ m, i.e., it is relatively far
away from the cluster of the IDs. While in Setup (b), the location of the
jammer is ${{\rm{{\bf{q}}}}_{j}}=\left({210,100,0}\right)$ m, i.e., the jammer
is located nearby the cluster of the IDs. The initial and final horizontal
coordinates of UAV are set as ${{\rm{{\bf{q}}}}_{S}}=\left({0,0,100}\right)$ m
and ${{\rm{{\bf{q}}}}_{F}}=\left({400,200,100}\right)$ m, respectively. We
assume that line segment from the start point to the end point is used here to
initialize the feasible point of the variables. The location of IRS is set as
${{\rm{{\bf{q}}}}_{r}}=\left({205,100,3}\right)$ m. All the values of required
parameters in (9) are set in [34].

Figure 2: UAV trajectory for different cases.

Figure 3: UAV trajectories for different cases.

Figure 4: Performance as (a) the throughput, (b) the energy assumption of UAV,
and (c) the EE of the UAV, versus the transmit power of the jammer, i.e.,
$p_{j}$, when $M=150$ and $\bar{p}=20$ dBm.

Fig. 3 shows the UAV’s trajectories for different cases. It can be observed
that by deploying the IRS to assist in the UAV data collection, our proposed
design achieves much shorter flight path than the cases without IRS [15],
regardless the location of the jammer. Specifically, in Setup (a), for the
case without IRS, the UAV should retreat far away from the jammer for reducing
the jamming signal; while for the case with IRS, the received signal at the
UAV can be significantly enhanced by the IRS passive beamforming, which can
effectively compensate for the jamming and thus enables the UAV to fly much
closer to the cluster of IDs. Moreover, one can observe that in Setup (b), the
UAV in our proposed design flies straightly from the start point to the end
point. The reason is that when the jammer is also located in the coverage of
the IRS, the jamming signal can be effectively mitigated at the UAV by IRS
passive beamforming, thus the UAV can fly over the IDs for data collection as
if there is no jammer. However, for the conventional design without IRS, the
retreating away strategy becomes much less effective in Setup (b) (since such
behavior would also degrade the signal from the IDs), thus the UAV no more
retreats as far as in Setup (a) for saving flight energy.

Fig. 4 shows the throughput, and energy consumption, and EE versus $p_{j}$ for
different cases. In Fig. 4(a), for the benchmark without IRS, Setup (a) facing
a remote jammer is more favorable for the UAV data collection than Setup (b)
facing a local jammer; however, for the case with IRS, the observation is
opposite. The reason is that in our proposed design, when the jammer is
located in the coverage of IRS as in Setup (b), the additional gain from
reducing the jamming signal via passive beamforming is achieved except that
from enhancing the desired signal as in Setup (a), which thus renders a higher
SINR at the UAV. While for the benchmark, the retreating away strategy is
effective only in the remote jammer case for reducing the jamming signal; when
the jammer is co-located with the IDs, retreating away becomes much less
effective because such behavior also degrades the reception of the desired
signal. In Fig. 4(b), one can observe that our proposed design achieves lower
energy consumption than the benchmark scheme in both setups. Moreover, in
Setup (b), the energy consumption for both our proposed and the benchmark
designs almost keeps unchanged as $p_{j}$ increases. This is because in our
proposed design, the UAV flies straightly from the start point to the end
point regardless of $p_{j}$ (thanks to the effectively jamming mitigation);
while in the benchmark design, the UAV also would not retreating further as
$p_{j}$ increases since the retreating away strategy becomes less effective.
On contrast, in Setup (a) facing remote jammer, retreating away is beneficial
for increasing the SINR for both designs. Owing to the above, we can conclude
that by deploying the IRS nearby the IDs for assisting in the UAV data
collection, the EE of the UAV can be significantly improved, especially for
the case when the jammer is co-located with the IDs (deemed challenging in the
conventional systems without IRS), which is also validated by the observation
in Fig. 4(c).

Figure 5: The EE of UAV versus the number of IRS elements, i.e., $M$.

In Fig. 5, the EE of the UAV for different cases versus the number of IRS
elements is illustrated. It is observed that with the increasing of IRS
element $M$, the EE for the cases with IRS are both improved, which validates
the performance gain by enlarging the IRS size. Moreover, one can observe that
in our proposed design, Setup (b) achieves higher EE of UAV than Seutp (a)
(the reason is as similar as that in Fig. 4 and thus omitted here) and the
performance gap enlarges as $M$ increases. This is because a larger number of
IRS elements enables a larger jamming signal mitigation gain in Setup (b),
which is however unavailable in Setup (a).

Figure 6: The EE of UAV versus the average transmit power of IDs, i.e.,
$\bar{p}$ .

Fig. 6 shows the EE of the UAV versus the average transmit power of IDs
$\bar{p}$. It can be observed that as $\bar{p}$ increases, the EE of UAV for
all cases are improved since the received signal at the UAV are enhanced by
increasing the uplink transmit power. More importantly, one can observe that
in Setup (a), for achieving the same EE of UAV as 4 kbits/J, a transmit power
of $26$ dBm is required at the IDs for the benchmark while this value is
reduced to $19$ dBm for our proposed design, which suggest a 7 dB gain by
deploying the IRS nearby the IDs. Note that such performance improvement even
reaches up to almost 10 dB in Setup (b) due to the additional jamming
mitigation gain. The above observation reveals that the proposed design can
not only help to improve the EE of UAV, but also is beneficial for saving the
transmit power of the IDs and thus prolong their life time.

##  V Conclusions

In this paper, we study an IRS-aided UAV data collection system in the
presence of a jammer. By considering the communication scheduling, IDs’ power
allocation, IRS passive beamforming, and UAV trajectory, an alternating
optimization based algorithm is proposed to solve the problem by exploiting
the SCA and BCD techniques. Simulation results showed that by deploying the
IRS in our proposed design, the EE of the UAV has improved significantly,
especially for the case facing a local jammer, which is deemed highly
challenging in the conventional systems without IRS.

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