# Active Intelligent Reflecting Surface Assisted Secure Air-to-Ground
Communication with UAV Jittering

Yimeng Ge and Jiancun Fan, _Senior Member, IEEE_ This work is partially
supported by the National Natural Science Foundation of China under Grants No.
61671367, the Key Research and Development Plan of Shaanxi Province under
Grant No. 2018GY-003, the Research Foundation of Science and Technology on
Communication Networks Laboratory, and the Fundamental Research Funds for the
Central Universities. (Corresponding author: Jiancun Fan.)Y. Ge and J. Fan are
with the School of Information and Communication Engineering, Xi’an Jiaotong
University, Xi’an, Shaanxi 710049, P. R. China (e-mail:
gym0415@stu.xjtu.edu.cn;fanjc0114@gmail.com)Copyright (c) 2015 IEEE. Personal
use of this material is permitted. However, permission to use this material
for any other purposes must be obtained from the IEEE by sending a request to
<EMAIL_ADDRESS>
###### Abstract

Unmanned Aerial Vehicles (UAV)-enabled communication is a promising solution
for secure air-to-ground (A2G) networks due to the additional secure degrees
of freedom afforded by mobility. However, the jittering characteristics caused
by the random airflow and the body vibration of the UAV itself have a non-
negligible impact on the performance of UAV communication. Considering the
impact of UAV jittering, this paper propose a robust and secure transmission
design assisted by an novel active intelligent reflecting surface (IRS), where
the reflecting elements in IRS not only adjust the phase shift but also
amplify the amplitude of signals. Specifically, under the worst-case secrecy
rate constraints, we aim to minimize the transmission power by the robust
joint design of active IRS’s reflecting coefficient and beamforming at the
UAV-borne base station (UBS). However, it is challenging to solve the joint
optimization problem due to its non-convexity. To tackle this problem, the
non-convex problem is reformulated with linear approximation for the channel
variations and linear matrix inequality transformed by S-procedure and Schur’s
complement. Then, we decouple this problem into two sub-problems, namely,
passive beamforming and active IRS’s reflecting coefficient optimization, and
solve them through alternate optimization (AO). Finally, the numerical results
demonstrate the potential of active IRS on power saving under secure
transmission constraints and the impact of UAV jittering.

###### Index Terms:

active intelligent reflecting surface, secure UAV communications, robust
beamforming, worst-case model.

##  I Introduction

Unmanned aerial vehicle (UAV) communication has received widespread attention
due to its low cost, high flexibility and mobility, which will play an
important role in the fifth-generation (5G) networks [1]. Compared with
traditional terrestrial communications, air-to-ground (A2G) channels are
usually dominated by line of sight (LOS) links, which helps to establish long-
range coverage and reliable transmission [2]. In addition, the UAV can fly
close to its intended ground users to obtain better channel conditions.
Despite the above benefits, A2G channels are vulnerable to various attacks,
especially eavesdropping attacks from illegal eavesdroppers [3]. In this
regard, the physical layer security (PLS) technology is considered as a
promising method to ensure secure UAV communication. Most of the related
researches jointly optimize the UAV trajectory and resource allocation to
improve the secure communication quality. Among them, one type of effective
cooperative jamming method is to prevent eavesdropping by using the
interference of other nearby UAV to ensure the safety of unmanned aerial
vehicle (UAV) communications [4, 5, 6, 7, 8], where one UAV transmitter
delivers the confidential information to a ground node, and the other UAV
jammer cooperatively sends the artificial noise to confuse the ground
eavesdropper. In particular, the secure UAV communication in different
scenarios was studied in [9, 10, 11, 12]. A secure UAV-enabled mobile edge
computing (MEC) system has been proposed in the presence of multiple
eavesdropping UAVs with imperfect locations [9]. Then, in [10] the authors
investigates the secrecy performance of a UAV-to-vehicle communication system.
Furthermore, a secure downlink multi-user transmission scheme enabled by a
flexible UAV borne base station (BS) and non-orthogonal multiple access (NOMA)
was rigorously studied in [11]. In [12], a secure UAV-enabled cognitive radio
network is studied by robustly optimizing the UAV’s trajectory and transmit
power. However, the secrecy performance in these UAV communications is still
unsatisfactory because all the above methods are essentially unable to improve
the secure degrees of freedom [13].

In this situation, intelligent reflecting surface (IRS) as a promising
technology has attracted great attention. By independently controlling passive
reflective elements on the plane, it can produce a desired phase shift on the
impinging electromagnetic waves [14], thereby providing more secure degrees of
freedom for the wireless link. Furthermore, each reflected signals can be
added constructively at the legitimate receiver to enhance its reception
quality, while being received destructively at eavesdropper to degrade its
signal reception [15]. This indicates IRS is an excellent solution to
significantly improve the PLS performance [16]. Thus, the design of IRS-
assisted secure UAV communication systems has attracted increasing attention.
In [17], a theoretical framework was presented to analyze the performance of
an integrated UAV-IRS relaying system. Furthermore, in [18], the secure
transmission problem was considered in an UAV and IRS assisted millimeter wave
(mmWave) networks in the presence of an eavesdropper. In addition, [19] showed
that the use of IRSs can effectively improve the coverage and reliability of
mmWave UAV communication systems. In fact, all these results indicate that IRS
greatly boosts the UAV transmission performance compared with no IRS case.

###  I-A Backround and Motivation

Although the integration of IRS technology with UAV secure communication is a
great step toward the realization of efficient and flexible aerial networks.
It is observed that among the current related works, there exists many
bottlenecks to be overcome on the design of secure UAV communication. How to
fully exploit the potential of the IRS and make it applicable to practical
scenarios to further improve the secrecy performance of UAVs is an important
issue. Especially in multi-UAV networks, the communication nodes are usually
scattered in a relatively wide geographic range and long-distance transmission
will bring more serious challenges.

In fact, all these aforementioned existing works have ignored an unavoidable
problem in IRS-assisted UAV communication: Although IRS brings new reliable
reflection link for signal transmission in addition to the direct link, a
“double fading” effect always exists in this reflection link, i.e., the
signals received via this link suffer from large-scale fading twice [20]. And
if the fading coefficient is large, the signals from this longer reflection
link lose more power than that from the short direct link, resulting in a
limited secrecy performance gain compared with the one without IRS. Recently,
a new concept of active IRS has been proposed [20, 21, 22, 23, 24] which can
combat this “double fading” effect. Compared with passive IRS, the active IRS
directly reflects the incident signal with the power amplification in the
electromagnetic (EM) level, and it still takes the advantage of IRS that no
complex and power-hungry radio frequency (RF) chain components are needed.
Specifically, in [20, 21, 22], an active IRS assisted single-input multiple-
output uplink system and downlink multiple-input single-output multi-user
system were studied respectively, and numerical algorithms were developed to
optimize the phase shift and amplification factor of active IRS. Based on the
numerical examples, this active IRS design can realize a significantly higher
transmission rate compared with those via passive IRS, thus overcoming the
fundamental limit of “double fading” effect.

Essentially, the key of active IRS is that after the incident signal that pass
through the BS-IRS subchannel, it typically has a weaker signal power and can
be easily amplified with a considerable gain at the expense of low power
consumption. Therefore, the physical size of the active IRS can be reduced,
making it more suitable for UAV communication limited by the payload.
Furthermore, it offers more freedom to design secure beamforming by optimizing
the amplitudes of the reflection coefficients instead of just phases,
resulting in more flexibility to reconfigure the wireless propagation
environment [24], [25]. As a result, the secure UAV communication enhanced by
active IRSs is a promising solution for providing high-efficient communication
services. However, all of active IRS study [20, 21, 22, 23, 24] did not
consider secure UAV communications. Although active IRS greatly helps
enhancing the quality of communications of legitimate user, it also reduces
“double fading” effect in the reflection link of UAV-IRS-eavesdropper due to
the broadcast nature of wireless channels, resulting larger information
leakage to eavesdropper. Hence, the current proposed algorithms is not
applicable in secure UAV communication case, and the transmit beamformer at
UAV-borne base station (UBS) as well as active IRS’s reflecting coefficient
should be carefully re-designed via new efficient algorithms.

On the other hand, the jittering characteristics caused by the random airflow
and the body vibration of the UAV itself have a non-negligible impact on the
establishment of a robust and secure communication link. Unlike terrestrial
cellular networks with fixed and stable infrastructure, UAVs are susceptible
to airflow and body vibration, resulting in random vibrations such as yaw
jittering in the horizontal direction or pitch jittering in the vertical
direction [26]. It should also be pointed out that the structure of most BS is
not specifically designed and optimized for UAV. The weight of the UBS is
large and its center of gravity is often not in the middle of the equipment,
which makes it very easy to shake, especially when the UAV is vibrating. As a
result, the vibration introduces a non-negligible channel estimation error,
which results in the deviation of the directional signal beam, and further
causes severe performance loss [27]. It must be pointed out that the
equivalent noise due to the channel estimation error will also be amplified by
active IRS, which means the robust design considering the impact of UAV
jittering is essential in active IRS-assisted UAV secure communication.

However, most existing UAV communication research assumes the perfect channel
state information (CSI) for A2G links. Only a few papers have studied UAV
communication under imperfect CSI caused by channel estimation error. For
example, [28] maximized the average worst-case secrecy rate by the robust
joint design of the UAV’s trajectory, IRS’s passive beamforming, and
transmission power of the legitimate transmitters under the premise of
imperfect CSI. In [29], a precisely designed neural network is adopted to
optimize the beamformer for confidential signal and artificial noise, with
partial CSI of legitimate UAV and eavesdropping UAV. In adittion, the authors
of [30] studied the robust and secure transmission in the mmWave UAV
communication assisted by IRS under imperfect CSI. In fact, it can be found
from the above research that the secrecy performance of IRS-assisted
communication is largely limited by the CSI error, and robust design needs to
be considered in all scenarios. Therefore, we also need to consider the impact
of CSI error caused by UAV jittering in follow-up research.

###  I-B Contribution

Motivated by the aforementioned facts, this paper develops a comprehensive
mathematical framework to characterize the robust secrecy performance of an
active IRS-assisted UAV communication, where the impact of UAV jittering on
secure beamforming is also considered. To the best of our knowledge, in most
current research on secure UAV communication, maximizing achievable secure
rate (ASR) is usually the optimization objective. However, maximizing ASR will
only guarantee the legitimate user’s information security while ignoring their
data rates restriction and the UAV’s energy efficiency. In fact, an UAV-
enabled system should be designed to satisfy the data-link security
requirements of its legitimate users as well as to improve the energy
efficiency or reduce transmit power, which drives us to investigate a more
general scenario and adapt to some complex scenarios, such as power-
constrained UAV and distributed UAV system where the communication nodes are
usually scattered in a relatively wide geographic range so that long-distance
transmission will bring high power consumption. Therefore, considering the
legitimate user data rate restriction and the total power constraints of UBS,
we study the transmission power minimization problem for active IRS-assisted
UAV communication while ensuring ASR. In the proposed solution, power-
constrained UAV can use less transmission power to achieve the same ASR, which
greatly improves the power-efficiency. Specifically, we formulated the problem
of minimizing UBS transmission power by jointly designing the active IRS’s
reflecting coefficient and beamforming at the UBS, revealing the positive
effect of active IRS on the secure UAV communications and the impact of UAV
jittering on beamforming. The main contributions of this paper are summarized
as follows:

(1) To the best of our knowledge, this is the first work to consider active
IRS-assisted UAV secure communication with consideration of UAV jittering,
which is more practical and power-efficient than the previous related works.
Note that all the existing numerical solutions cannot be directly applied to
this setting. Due to the uncertainty factors of UAV jittering including yaw
vibration and pitch vibration, accurate estimates of the CSI are usually not
available. Specifically, imperfect estimation of antenna array response is
characterized with variations on the elevation angle-of-departure (AOD) and
azimuth AOD at UAV. Hence, we assume imperfect CSI acquisition of the channels
and use a bounded model [31] to describe the CSI uncertainty. To make the
subsequent optimization problems easier to solve, linear approximation
applying by Taylor expansion is used to characterize the impact of the
variations of elevation AOD and azimuth AOD on the channel response. On this
basis, we formulate a non-convex robust secure beamforming optimization
problem to jointly design the transmit precoding and active IRS’s reflecting
coefficient matrix.

(2) Because the problem involves optimization of non-concave objective
functions subject to non-convex constraints, for which the optimal solution is
computationally difficult to find. Thus, S-procedure, Schur’s complement and
slack variables are firstly applied to transform the restrictions into linear
matrix inequalities (LMI). Then, the optimization problem is decoupled into
two sub-problems: 1) Optimize the active IRS’s reflecting coefficient matrix
for given beamforming matrix; 2) Optimize the beamforming matrix for given
IRS’s reflecting coefficient matrix. After that, we solved them effectively
through the alternating optimization (AO) techniques, in which each set of
variables is alternately optimized. Specifically, S-procedure is adopted to
approximate the semi-infinite inequality constraints for the first sub-
problem. Subsequently, the second sub-problem of transmit precoding is solved
by Schur’s complement. Thus, it is a convex optimization problem with a linear
objective function and LMI constraints, which can be easily solved by CVX.
(3) Finally, numerical results are provided to show the efficiency of the
proposed optimization scheme and to demonstrate the impact of active IRS on
the power saving. Specifically, compared with passive IRS, the proposed active
IRS-assisted scheme can significantly reduce transmitting power at UBS while
ensuring ASR, even in the complex situation of UAV’s power limitation, severe
jittering, and high flight altitude. Besides, the secure UAV communication can
get a significant performance improvement by the active IRS with a small
number of elements, which is more suitable for drones. It is worth pointing
out that unlike passive IRS, the performance gain from active IRS will not
continue to improve with the increase of the number of elements and the
maximum amplification gain, that is, the best performance requires a
reasonable design. Our solution also provides a design basis for the active
IRS-assisted UAV communication.
The rest part of our paper is organized as follows. Section II introduces the
system model including A2G wireless channel Model, jittering UAV model, and
signal model. Sections III formulates the worst-case robust beamforming
problem to minimize the transmit power, and presents the AO algorithm to solve
the problems in the cases of active IRS and passive IRS, respectively. Section
VI presents numerical results to evaluate the secrecy performance of the
proposed algorithms and finally Section V concludes the paper.
_Notations_ : Boldface lowercase and uppercase letters denote vectors and
matrices, respectively. For a vector $\mathbf{x}=[x_{1},\cdots,x_{n}]$,
diag$(\mathbf{x})$ denotes a diagonal matrix whose entries are
$x_{1},\cdot\cdot\cdot,x_{n}$. The symbols $\|\mathbf{X}\|_{F}$ denote
Frobenius norm of matrix $\mathbf{X}$ and the symbol $\|\mathbf{x}\|_{2}$
denotes 2-norm of vector $\mathbf{x}$. Moreover, $(\cdot)^{H}$ is the
conjugate transpose, $(\cdot)^{T}$ is the transpose, $(\cdot)^{*}$ is the
conjugate matrix, and $|\cdot|$ is the absolute value. $\mathbf{I}$ and
$(\cdot)^{-1}$ denote the identity matrix and the inverse of a matrix,
respectively. The Kronecker product and Hadamard product are denoted by
$\otimes$ and $\odot$, respectively. By $\mathbf{X}\succeq 0$ or
$\mathbf{X}\succ\mathbf{0}$, we mean that $\mathbf{X}$ is positive semi-
definite or positive definite, respectively. $\mathbb{C}^{x\times y}$ denotes
the $x\times y$ domain of complex matrices, $\operatorname{Re}\\{\cdot\\}$
denotes the real part. $\mathcal{C}\mathcal{N}\left(0,\delta^{2}\right)$
represent the real-valued and complex-valued Gaussian distribution with mean 0
and covariance $\delta^{2}$.

##  II System Model

In this section, we establish the A2G wireless channel model, CSI error models
considering UAV jittering, and signal model in an active IRS-assisted UAV
secure communication system as follows.

###  II-A A2G Wireless Channel Model

Consider an active IRS-assisted UAV communication system, as in Fig. 1,
including a single-antenna legitimate users Alice and a single-antenna
eavesdropper Eve.

Figure 1: A2G wiretap model.

Both the UBS and the active IRS are equipped with uniform rectangular arrays
(URAs) with the size of $N=N_{x}\times N_{y}$ and $M=M_{x}\times M_{y}$
respectively, where $N_{x}\left(N_{y}\right)$ and $M_{x}\left(M_{y}\right)$
denote the numbers of UBS antennas and REs along the $x(y)$ axis,
respectively. All communication nodes are placed in the three-dimensional (3D)
Cartesian coordinate system. The coordinations of UAV, Alice and Eve are
$\left(x_{\mathrm{U}},y_{\mathrm{U}},z_{\mathrm{U}}\right)$,
$\left(x_{\mathrm{A}},y_{\mathrm{A}},0\right)$ and
$\left(x_{\mathrm{E}},y_{\mathrm{E}},0\right)$, respectively. The reflecting
coefficient matrix for the active IRS is denoted by
$\mathbf{\Theta}=\operatorname{diag}\left\\{\tau_{1}e^{j\theta_{1}},\tau_{2}e^{j\theta_{2}},\ldots,\tau_{M}e^{j\theta_{M}}\right\\}\in\mathbb{C}^{M\times
M}$, where $\tau_{m}$ and $\theta_{m}$ represents the amplitude and the phase,
respectively, and $\tau_{m}$ can be greater than 1 with active load. By
contrast, the passive IRS cannot amplify the incident signal, so the amplitude
of each RE is limited with $\tau_{m}\leqslant 1$. Furthermore, the first
element of the IRS is regarded as the reference point whose coordinates are
denoted by $\left(x_{\mathrm{I}},y_{\mathrm{I}},z_{\mathrm{I}}\right)$.
Therefore, the distance between the IRS and a certain communication node can
be approximated by that between the reference point and the corresponding
node. We assume that both UBS-Alice/Eve links and UBS-IRS link contain LOS and
non-LOS (NLOS) components, which follow the free-space path loss model [19].
Following [32], we consider that all the channels are formulated as the
product of large scale fading and small scale fading, and the small scale
fading in the A2G links are assumed to be Rician fading. In particular,
$\mathbf{h}_{\mathrm{U}}$, $\mathbf{h}_{\mathrm{E}}$ and
$\mathbf{H}_{\mathrm{I}}$ can be modeled as

$\displaystyle\mathbf{h}_{\mathrm{U}}=\sqrt{\frac{A_{\mathrm{L}}d_{\mathrm{U}}^{-\alpha_{\mathrm{L}}}K_{\mathrm{U}}}{1+K_{\mathrm{U}}}}\mathbf{h}_{\mathrm{U},\mathrm{L}}+\sqrt{\frac{A_{\mathrm{N}}d_{\mathrm{U}}^{-\alpha_{\mathrm{N}}}}{1+K_{\mathrm{U}}}}\mathbf{h}_{\mathrm{U},\mathrm{N}},$
(1)
$\displaystyle\mathbf{h}_{\mathrm{E}}=\sqrt{\frac{A_{\mathrm{L}}d_{\mathrm{E}}^{-\alpha_{\mathrm{L}}}K_{\mathrm{E}}}{1+K_{\mathrm{E}}}}\mathbf{h}_{\mathrm{E},\mathrm{L}}+\sqrt{\frac{A_{\mathrm{N}}d_{\mathrm{E}}^{-\alpha_{\mathrm{N}}}}{1+K_{\mathrm{E}}}}\mathbf{h}_{\mathrm{E},\mathrm{N}},$
$\displaystyle\mathbf{H}_{\mathrm{I}}=\sqrt{\frac{A_{\mathrm{L}}d_{\mathrm{I}}^{-\alpha_{\mathrm{L}}}K_{\mathrm{I}}}{1+K_{\mathrm{I}}}}\mathbf{H}_{\mathrm{I},\mathrm{L}}+\sqrt{\frac{A_{\mathrm{N}}d_{\mathrm{I}}^{-\alpha_{\mathrm{N}}}}{1+K_{\mathrm{I}}}}\mathbf{H}_{\mathrm{I},\mathrm{N}},$

where $K_{\text{U}}$, $K_{\text{E}}$ and $K_{\text{I}}$ are the Rician
K-factors of the UBS-Alice channel, UBS-Eve channel and UBS-IRS channel,
respectively. $A_{\text{L}}$ and $A_{\text{N}}$ are the path loss factors of
the LOS and NLOS, respectively, $\alpha_{\mathrm{L}}$ and
$\alpha_{\mathrm{N}}$ are the path loss exponents of the LOS and NLOS,
respectively. The distance between the UBS-Alice, UBS-Eve and UBS-IRS, e.i.,
$d_{\mathrm{U}}$, $d_{\mathrm{E}}$ and $d_{\mathrm{I}}$ are expressed as

$\displaystyle
d_{\text{U}}=\sqrt{\left(z_{\text{U}}\right)^{2}+\left(y_{\text{U}}-y_{\text{A}}\right)^{2}+\left(x_{\text{U}}-x_{\text{A}}\right)^{2}},$
(2) $\displaystyle
d_{\text{E}}=\sqrt{\left(z_{\text{U}}\right)^{2}+\left(y_{\text{U}}-y_{\text{E}}\right)^{2}+\left(x_{\text{U}}-x_{\text{E}}\right)^{2}},$
$\displaystyle
d_{\text{I}}=\sqrt{\left(z_{\text{U}}-z_{\text{I}}\right)^{2}+\left(y_{\text{U}}-y_{\text{I}}\right)^{2}+\left(x_{\text{U}}-x_{\text{I}}\right)^{2}}.$

In addition, the channel fading between the IRS and Alice/Eve also follows
Rician distribution which can be generated with a similar procedure, and
denoted as $\mathbf{h}_{\mathrm{IU}}$ and $\mathbf{h}_{\mathrm{IE}}$,
respectively. Furthermore, in (1), the LOS and NLOS components of UBS-Alice
channel $\mathbf{h}_{\mathrm{U}}$ are denoted as
$\mathbf{h}_{\mathrm{U},\mathrm{L}}$ and $\mathbf{h}_{\mathrm{U},\mathrm{N}}$,
respectively, and the UBS-Eve and UBS-IRS channels are also represented in a
similar way. The NLOS component $\mathbf{h}_{\mathrm{U},\mathrm{N}}$ with the
variables independently is drawn from the circularly symmetric complex
Gaussian distribution with zero mean and unit variance, i.e.,
$\mathcal{C}\mathcal{N}(0,1)$. The deterministic LOS component
$\mathbf{h}_{\mathrm{U},\mathrm{L}}$ and $\mathbf{h}_{\mathrm{U},\mathrm{L}}$
are expressed as

$\displaystyle\mathbf{h}_{\mathrm{U},\mathrm{L}}\\!=$
$\displaystyle\left[1,e^{-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(\cos\omega_{\mathrm{U}}+\sin\phi_{\mathrm{U}}\sin\omega_{\mathrm{U}}\right)},\cdots,\right.$
(3) $\displaystyle\left.e^{-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(N_{x}-1\right)\cos\omega_{\mathrm{U}}+\left(N_{y}-1\right)\sin\phi_{\mathrm{U}}\sin\omega_{\mathrm{U}}}\right]^{T},$
$\displaystyle\mathbf{h}_{\mathrm{E},\mathrm{L}}\\!=$
$\displaystyle\left[1,e^{-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(\cos\omega_{\mathrm{E}}+\sin\phi_{\mathrm{E}}\sin\omega_{\mathrm{E}}\right)},\cdots,\right.$
$\displaystyle\left.e^{-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(N_{x}-1\right)\cos\omega_{\mathrm{E}}+\left(N_{y}-1\right)\sin\phi_{\mathrm{E}}\sin\omega_{\mathrm{E}}}\right]^{T},$

where $\omega_{\mathrm{U}}(\omega_{\mathrm{E}})$ and
$\phi_{\mathrm{U}}(\phi_{\mathrm{E}})$ are the azimuth and elevation AOD of
the path between the URA at UBS and Alice (Eve), respectively,
$b_{\mathrm{B}}$ is the distance between two adjacent UBS antennas, and
$\lambda$ is the wavelength of the center frequency of the carrier. Similarly,
$\mathbf{H}_{\mathrm{I},\mathrm{L}}$ is given by
$\mathbf{H}_{\mathrm{I},\mathrm{L}}=\mathbf{h}_{\mathrm{I},\mathrm{L}}^{(\text{A})}\mathbf{h}_{\mathrm{I},\mathrm{L}}^{(\text{D})}$,
where

$\displaystyle\mathbf{h}_{\mathrm{I},\mathrm{L}}^{(\text{A})}\\!=\\!\left[1,e^{-2\pi
j\frac{b_{\mathrm{IRS}}}{\lambda}\left(\cos\varphi_{\mathrm{I}}\sin\vartheta_{\mathrm{I}}\\!+\sin\varphi_{\mathrm{I}}\sin\vartheta_{\mathrm{I}}\right)},\right.$
(4) $\displaystyle\left.\\!\cdots,\\!e^{-2\pi
j\frac{b_{\mathrm{IRS}}}{\lambda}\left(M_{x}-1\right)\cos\varphi_{\mathrm{I}}\sin\vartheta_{\mathrm{I}}\\!+\left(M_{y}-1\right)\sin\varphi_{\mathrm{I}}\sin\vartheta_{\mathrm{I}}}\right]^{T}$

is the array response at IRS with $\varphi_{\mathrm{I}}$ and
$\vartheta_{\mathrm{I}}$ defined as the azimuth and elevation angle-of-arrival
elevation (AOA) of the path between the URA at UBS and IRS. $b_{\mathrm{IRS}}$
is the antenna spacing of the URA at IRS.

$\displaystyle\mathbf{h}_{\mathrm{I},\mathrm{L}}^{(\text{D})}\\!=$
$\displaystyle\\!\left[1,\\!\cdots,\\!e^{-2\pi\frac{b_{\mathrm{B}}}{\lambda}\left(\cos\omega_{\mathrm{I}}\\!+\sin\phi_{\mathrm{I}}\sin\omega_{\mathrm{I}}\right)},\\!\cdots,\\!\right.$
(5) $\displaystyle\left.e^{-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(N_{x}-1\right)\cos\omega_{\mathrm{I}}+\left(N_{y}-1\right)\sin\phi_{\mathrm{I}}\sin\omega_{\mathrm{I}}}\right]^{T}$

is the array response at UBS with $\omega_{\mathrm{I}}$ and
$\phi_{\mathrm{I}}$ defined as the azimuth and elevation AOD of the path
between the URA at UBS and IRS, respectively.

###  II-B CSI Error Models Considering UAV Jittering

There is no doubt that the vibration characteristics of UAV are an important
factor affecting the stability of its secrecy communication performance. In
order to fully evaluate the impact of UAV jittering, we will establish a
bounded CSI error model considering the UAV jittering in this section.

The vibration characteristics of UAVs are caused by multiple factors. In
addition to the torsional vibration excitation generated during the rotation
of the engine-driven transmission system, the rotor-winged is also one of the
vibration excitations. That means the UAV itself is a vibration system with
free vibration and forced vibration [3]. Moreover, due to the randomness of
wind gusts, rotation, yaw, and pitch motions, the stability of the UAV
platform cannot be guaranteed. These random vibration characteristics will
lead to imperfect CSI estimation and unstable wireless transmission,
especially when equipped with a large antenna array. As shown in the Fig. 2(b)
and Fig. 2(c), for a URA, varying elevation angles capture the UAV jittering
in pitch angle and roll angle, while varying azimuth angles capture the UAV
jittering in yaw angle.

Figure 2: (a) Jittering of elevation angle. (b) Jittering of azimuth angle.

To capture the jittering effect, the actual elevation AOD $\phi_{\mathrm{U}}$,
$\phi_{\mathrm{E}}$ and $\phi_{\mathrm{I}}$, and the azimuth angle AOD
$\omega_{\mathrm{U}}$ , $\omega_{\mathrm{E}}$ and $\omega_{\mathrm{I}}$ are
modeled as

$\displaystyle\omega_{\mathrm{U}}=\bar{\omega}_{\mathrm{U}}+\Delta\omega_{\mathrm{U}},\phi_{\mathrm{U}}=\bar{\phi}_{\mathrm{U}}+\Delta\phi_{\mathrm{U}}.$
(6)
$\displaystyle\Omega_{\mathrm{U}}=\left\\{\Delta\omega_{\mathrm{U}},\Delta\phi_{\mathrm{U}}\in\mathbb{R}\mid\left(\Delta\omega_{\mathrm{U}}\right)^{2}\leqslant\beta_{\mathrm{U}1}^{2},\left(\Delta\phi_{\mathrm{U}}\right)^{2}\leqslant\beta_{\mathrm{U}2}^{2}\right\\}.$
$\displaystyle\omega_{\mathrm{E}}=\bar{\omega}_{\mathrm{E}}+\Delta\omega_{\mathrm{E}},\phi_{\mathrm{E}}=\bar{\phi}_{\mathrm{E}}+\Delta\phi_{\mathrm{E}}.$
$\displaystyle\Omega_{\mathrm{E}}=\left\\{\Delta\omega_{\mathrm{E}},\Delta\phi_{\mathrm{E}}\in\mathbb{R}\mid\left(\Delta\omega_{\mathrm{E}}\right)^{2}\leqslant\beta_{\mathrm{E}1}^{2},\left(\Delta\phi_{\mathrm{E}}\right)^{2}\leqslant\beta_{\mathrm{E}2}^{2}\right\\}.$
$\displaystyle\omega_{\mathrm{I}}=\bar{\omega}_{\mathrm{I}}+\Delta\omega_{\mathrm{I}},\phi_{\mathrm{I}}=\bar{\phi}_{\mathrm{I}}+\Delta\phi_{\mathrm{I}}.$
$\displaystyle\Omega_{\mathrm{I}}=\left\\{\Delta\omega_{\mathrm{I}},\Delta\phi_{\mathrm{I}}\in\mathbb{R}\mid\left(\Delta\omega_{\mathrm{I}}\right)^{2}\leqslant\beta_{\mathrm{I}1}^{2},\left(\Delta\phi_{\mathrm{I}}\right)^{2}\leqslant\beta_{\mathrm{I}2}^{2}\right\\}.$

where
$\bar{\omega}_{\mathrm{U}}(\bar{\omega}_{\mathrm{E}},\bar{\omega}_{\mathrm{I}})$
and
$\Delta\omega_{\mathrm{U}}(\Delta\omega_{\mathrm{E}},\Delta\omega_{\mathrm{I}})$
are the estimated azimuth AOD and the uncertainty of azimuth AOD of the path
between the UBS and Alice (Eve, IRS), respectively.
$\bar{\phi}_{\mathrm{U}}(\bar{\phi}_{\mathrm{E}},\bar{\phi}_{\mathrm{I}})$ and
$\Delta\phi_{\mathrm{U}}(\Delta\phi_{\mathrm{E}},\Delta\phi_{\mathrm{I}})$ are
the estimated elevation AOD and the uncertainty of elevation AOD of the path
between the UBS and Alice (Eve, IRS), respectively. $\Omega_{\mathrm{U}}$,
$\Omega_{\mathrm{E}}$ and $\Omega_{\mathrm{B}}$ are the sets containing all
the possible AOD uncertainties of the Alice, EVE and IRS, respectively. The
uncertainties of azimuth AOD and elevation AOD of Alice are bounded by the
maximum variation $\beta_{\mathrm{U}1}^{2}$ and $\beta_{\mathrm{U}2}^{2}$,
respectively, and the uncertainties of of Eve and IRS are represented in a
similar way. We note that
$\mathbf{h}_{\mathrm{U},\mathrm{L}}(\mathbf{h}_{\mathrm{E},\mathrm{L}},\mathbf{h}_{\mathrm{I},\mathrm{L}})$
is a nonlinear function with respect to
$\Delta\omega_{\mathrm{U}}(\Delta\omega_{\mathrm{E}},\Delta\omega_{\mathrm{I}})$
and
$\Delta\phi_{\mathrm{U}}(\Delta\phi_{\mathrm{E}},\Delta\phi_{\mathrm{I}})$. To
tackle this problem, we approximate
$\mathbf{h}_{\mathrm{U},\mathrm{L}}(\mathbf{h}_{\mathrm{E},\mathrm{L}},\mathbf{h}_{\mathrm{I},\mathrm{L}}^{(\text{D})})$
by applying Taylor expansion, i.e.,

$f(x+\Delta x,y+\Delta y)\approx f(x,y)+\Delta x\frac{\partial
f(x,y)}{\partial x}+\Delta y\frac{\partial f(x,y)}{\partial y}.$ (7)

Thus, each exponential term in (3) and (5) can be approximated as (8).

$\displaystyle\exp\left(-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(n_{x}\cos(\bar{\omega}+\Delta\omega)+n_{y}\sin(\bar{\phi}+\Delta\phi)\sin(\bar{\omega}+\Delta\omega)\right)\right)$
(8) $\displaystyle\approx\exp\left(-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(n_{x}\cos(\omega)+n_{y}\sin(\bar{\phi})\sin(\bar{\omega})\right)+\exp\left(-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(n_{x}\cos(\omega)+n_{y}\sin(\bar{\phi})\sin(\bar{\omega})\right)\right)\right.$
$\displaystyle\times\left(2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(n_{x}\sin(\bar{\omega})-n_{y}\sin(\bar{\phi})\cos(\bar{\omega})\right)\Delta\omega-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}n_{x}\cos(\bar{\phi})\sin(\bar{\omega})\Delta\phi\right).$

We define

$\displaystyle\mathbf{a}(\bar{\phi},\bar{\omega})$
$\displaystyle=\\!\sin(\bar{\omega})\left[0,2\pi
j\frac{b_{\mathrm{B}}}{\lambda},\ldots,2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(N_{x}-1\right)\right]$ (9)
$\displaystyle-\sin(\bar{\phi})\cos(\bar{\omega})\left[0,2\pi
j\frac{b_{\mathrm{B}}}{\lambda},\ldots,2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(N_{y}-1\right)\right],$
$\displaystyle\mathbf{b}(\bar{\phi},\bar{\omega})\\!=\\!\cos(\bar{\phi})\sin(\bar{\omega})\left[0,-2\pi
j\frac{b_{\mathrm{B}}}{\lambda},\ldots,-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(N_{y}\\!-\\!1\right)\right].$ (10)

Then, $\mathbf{h}_{\mathrm{U},\mathrm{L}},\mathbf{h}_{\mathrm{E},\mathrm{L}}$
and $\mathbf{h}_{\mathrm{I},\mathrm{L}}^{(\text{D})}$ in Eq. (3) and Eq. (5)
can be given as

$\displaystyle\mathbf{h}_{\mathrm{U},\mathrm{L}}$
$\displaystyle=\overline{\mathbf{h}}_{\mathrm{U},\mathrm{L}}+\Delta{\mathbf{h}}_{\mathrm{U},\mathrm{L}},$
(11) $\displaystyle\mathbf{h}_{\mathrm{E},\mathrm{L}}$
$\displaystyle=\overline{\mathbf{h}}_{\mathrm{E},\mathrm{L}}+\Delta{\mathbf{h}}_{\mathrm{E},\mathrm{L}},$
$\displaystyle\mathbf{h}_{\mathrm{I},\mathrm{L}}^{(\mathrm{D})}$
$\displaystyle=\overline{\mathbf{h}}_{\mathrm{I},\mathrm{L}}^{(\mathrm{D})}+\Delta{\mathbf{h}}_{\mathrm{I},\mathrm{L}}^{(\mathrm{D})},$

where

$\displaystyle\Delta{\mathbf{h}}_{\mathrm{U},\mathrm{L}}$
$\displaystyle\approx\overline{\mathbf{h}}_{\mathrm{U},\mathrm{L}}\odot\mathbf{a}\left(\bar{\phi}_{\mathrm{U}},\bar{\omega}_{\mathrm{U}}\right)\Delta\omega_{\mathrm{U}}+\overline{\mathbf{h}}_{\mathrm{U},\mathrm{L}}\odot\mathbf{b}\left(\bar{\phi}_{\mathrm{U}},\bar{\omega}_{\mathrm{U}}\right)\Delta\phi_{\mathrm{U}},$
(12) $\displaystyle\Delta{\mathbf{h}}_{\mathrm{E},\mathrm{L}}$
$\displaystyle\approx\overline{\mathbf{h}}_{\mathrm{E},\mathrm{L}}\odot\mathbf{a}\left(\bar{\phi}_{\mathrm{E}},\bar{\omega}_{\mathrm{E}}\right)\Delta\omega_{\mathrm{E}}+\overline{\mathbf{h}}_{\mathrm{E},\mathrm{L}}\odot\mathbf{b}\left(\bar{\phi}_{\mathrm{E}},\bar{\omega}_{\mathrm{E}}\right)\Delta\phi_{\mathrm{E}},$
$\displaystyle\Delta{\mathbf{h}}_{\mathrm{I},\mathrm{L}}^{(\mathrm{D})}$
$\displaystyle\approx\overline{\mathbf{h}}_{\mathrm{I},\mathrm{L}}^{(\mathrm{D})}\odot\mathbf{a}\left(\bar{\phi}_{\mathrm{I}},\bar{\omega}_{\mathrm{I}}\right)\Delta\omega_{\mathrm{I}}+\overline{\mathbf{h}}_{\mathrm{I},\mathrm{L}}^{(\mathrm{D})}\odot\mathbf{b}\left(\bar{\phi}_{\mathrm{I}},\bar{\omega}_{\mathrm{I}}\right)\Delta\phi_{\mathrm{I}},$

and
$\overline{\mathbf{h}}_{\mathrm{U},\mathrm{L}},\overline{\mathbf{h}}_{\mathrm{E},\mathrm{L}}$
and $\overline{\mathbf{h}}_{\mathrm{I},\mathrm{L}}^{(\mathrm{D})}$ are the
estimated channel vectors represented by

$\displaystyle\overline{\mathbf{h}}_{\mathrm{U},\mathrm{L}}=\left[0,e^{-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(\cos\bar{\omega}_{\mathrm{U}}+\sin\bar{\phi}_{\mathrm{U}}\sin\bar{\omega}_{\mathrm{U}}\right)},\cdots,\right.$
(13) $\displaystyle\left.e^{-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(\left(N_{x}-1\right)\cos\bar{\omega}_{\mathrm{U}}+\left(N_{y}-1\right)\sin\bar{\phi}_{\mathrm{U}}\sin\bar{\omega}_{\mathrm{U}}\right)}\right],$
$\displaystyle\overline{\mathbf{h}}_{\mathrm{E},\mathrm{L}}=\left[0,e^{-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(\cos\bar{\omega}_{\mathrm{E}}+\sin\bar{\phi}_{\mathrm{E}}\sin\bar{\omega}_{\mathrm{E}}\right)},\cdots,\right.$
$\displaystyle\left.e^{-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(\left(N_{x}-1\right)\cos\bar{\omega}_{\mathrm{E}}+\left(N_{y}-1\right)\sin\bar{\phi}_{\mathrm{E}}\sin\bar{\omega}_{\mathrm{E}}\right)}\right],$
$\displaystyle\overline{\mathbf{h}}_{\mathrm{I},\mathrm{L}}^{(\mathrm{D})}=\left[0,e^{-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(\cos\bar{\omega}_{\mathrm{I}}+\sin\bar{\phi}_{\mathrm{I}}\sin\bar{\omega}_{\mathrm{I}}\right)},\cdots,\right.$
$\displaystyle\left.e^{-2\pi
j\frac{b_{\mathrm{B}}}{\lambda}\left(\left(N_{x}-1\right)\cos\bar{\omega}_{\mathrm{I}}+\left(N_{y}-1\right)\sin\bar{\phi}_{\mathrm{I}}\sin\bar{\omega}_{\mathrm{I}}\right)}\right],$

where

$\displaystyle\sin\bar{\phi}_{\text{U}}\sin\bar{\omega}_{\text{U}}=\frac{y_{\text{U}}-y_{\text{A}}}{d_{\text{U}}},\cos\bar{\omega}_{\text{U}}=\frac{z_{\text{U}}-z_{\text{A}}}{d_{\text{U}}},$
(14)
$\displaystyle\sin\bar{\phi}_{\text{E}}\sin\bar{\omega}_{\text{E}}=\frac{y_{\text{U}}-y_{\text{E}}}{d_{\text{E}}},\cos\bar{\omega}_{\text{E}}=\frac{z_{\text{U}}-z_{\text{E}}}{d_{\text{E}}},$
$\displaystyle\sin\bar{\phi}_{\text{I}}\sin\bar{\omega}_{\text{I}}=\frac{y_{\text{U}}-y_{\text{I}}}{d_{\text{I}}},\cos\bar{\omega}_{\text{I}}=\frac{z_{\text{U}}-z_{\text{I}}}{d_{\text{I}}},$
$\displaystyle\sin\varphi_{\mathrm{B}}\sin\vartheta_{\mathrm{B}}\\!=\\!\frac{y_{\text{U}}-y_{\text{I}}}{d_{\text{I}}},\cos\varphi_{\mathrm{B}}\sin\vartheta_{\mathrm{B}}\\!=\\!\frac{x_{\text{U}}-x_{\text{I}}}{d_{\text{I}}}.$

Furthermore, the channel variations caused by UAV jittering
$\Delta\mathbf{h}_{\mathrm{U},\mathrm{L}},\Delta\mathbf{h}_{\mathrm{E},\mathrm{L}}$
and $\Delta\mathbf{h}_{\mathrm{I},\mathrm{L}}^{(\mathrm{D})}$ are given by

$\displaystyle\Delta\mathbf{h}_{\mathrm{U}}$
$\displaystyle=\sqrt{\frac{A_{\mathrm{L}}d_{\mathrm{U}}^{-\alpha_{\mathrm{L}}}K_{\mathrm{U}}}{1+K_{\mathrm{U}}}}\Delta\mathbf{h}_{\mathrm{U},\mathrm{L}}\triangleq\mathbf{a}_{\mathrm{U}}\Delta\omega_{\mathrm{U}}+\mathbf{b}_{\mathrm{U}}\Delta\phi_{\mathrm{U}},$
$\displaystyle\Delta\mathbf{h}_{\mathrm{E}}$
$\displaystyle=\sqrt{\frac{A_{\mathrm{L}}d_{\mathrm{E}}^{-\alpha_{\mathrm{L}}}K_{\mathrm{E}}}{1+K_{\mathrm{E}}}}\Delta\mathbf{h}_{\mathrm{E},\mathrm{L}}\triangleq\mathbf{a}_{\mathrm{E}}\Delta\omega_{\mathrm{E}}+\mathbf{b}_{\mathrm{E}}\Delta\phi_{\mathrm{E}},$
$\displaystyle\Delta\mathbf{H}_{\mathrm{I}}$
$\displaystyle=\mathbf{h}_{\mathrm{I},\mathrm{L}}^{(\mathrm{A})}\sqrt{\frac{A_{\mathrm{L}}d_{\mathrm{I}}^{-\alpha_{\mathrm{L}}}K_{\mathrm{I}}}{1+K_{\mathrm{I}}}}\Delta\mathbf{h}_{\mathrm{I},\mathrm{L}}^{(\mathrm{D})}\triangleq\mathbf{h}_{\mathrm{I},\mathrm{L}}^{(\mathrm{A})}\left(\mathbf{a}_{\mathrm{I}}\Delta\omega_{\mathrm{I}}+\mathbf{b}_{\mathrm{I}}\Delta\phi_{\mathrm{I}}\right).$
(15)

Thus, $\mathbf{h}_{\mathrm{U}},\mathbf{h}_{\mathrm{E}}$ and
$\mathbf{H}_{\mathrm{I}}$ in Eq. (1) and Eq. (2) can be rewritten as

$\displaystyle\mathbf{h}_{\mathrm{U}}$
$\displaystyle=\underbrace{\sqrt{\frac{A_{\mathrm{L}}d_{\mathrm{U}}^{-\alpha_{\mathrm{L}}}K_{\mathrm{U}}}{1+K_{\mathrm{U}}}}\overline{\mathbf{h}}_{\mathrm{U},\mathrm{L}}+\sqrt{\frac{A_{\mathrm{N}}d_{\mathrm{U}}^{-\alpha_{\mathrm{N}}}}{1+K_{\mathrm{U}}}}\mathbf{h}_{\mathrm{U},\mathrm{N}}}_{\overline{\mathbf{h}}_{\mathrm{U}}}+\Delta\mathbf{h}_{\mathrm{U}},$
(16) $\displaystyle\mathbf{h}_{\mathrm{E}}$
$\displaystyle=\underbrace{\sqrt{\frac{A_{\mathrm{L}}d_{\mathrm{E}}^{-\alpha_{\mathrm{L}}}K_{\mathrm{E}}}{1+K_{\mathrm{E}}}}\overline{\mathbf{h}}_{\mathrm{E},\mathrm{L}}+\sqrt{\frac{A_{\mathrm{N}}d_{\mathrm{E}}^{-\alpha_{\mathrm{N}}}}{1+K_{\mathrm{E}}}}\mathbf{h}_{\mathrm{E},\mathrm{N}}}_{\overline{\mathbf{h}}_{\mathrm{E}}}+\Delta\mathbf{h}_{\mathrm{E}},$
$\displaystyle\mathbf{H}_{\mathrm{I}}$
$\displaystyle=\underbrace{\sqrt{\frac{A_{\mathrm{L}}d_{\mathrm{I}}^{-\alpha_{\mathrm{L}}}K_{\mathrm{I}}}{1+K_{\mathrm{I}}}}\mathbf{h}_{\mathrm{I},\mathrm{L}}^{(\mathrm{A})}\overline{\mathbf{h}}_{\mathrm{I},\mathrm{L}}^{(\mathrm{D})}+\sqrt{\frac{A_{\mathrm{N}}d_{\mathrm{I}}^{-\alpha_{\mathrm{N}}}}{1+K_{\mathrm{I}}}}\mathbf{H}_{\mathrm{I},\mathrm{N}}}_{\overline{\mathbf{H}}_{\mathrm{I}}}+\Delta\mathbf{H}_{\mathrm{I}}.$

In order to evaluate the bounded uncertainty of the channel caused by UAV
jittering, we further derive the CSI error model of $\mathbf{h}_{\mathrm{U}}$
and $\mathbf{h}_{\mathrm{E}}$, which is given as

$\displaystyle\left\|\Delta\mathbf{h}_{\mathrm{U}}\right\|_{2}$
$\displaystyle\\!=\\!\left\|\mathbf{a}_{\mathrm{U}}\Delta\omega_{\mathrm{U}}\\!+\\!\mathbf{b}_{\mathrm{U}}\Delta\phi_{\mathrm{U}}\right\|_{2}\\!\leq\\!\left\|\mathbf{a}_{\mathrm{U}}\Delta\omega_{\mathrm{U}}\right\|_{2}\\!+\\!\left\|\mathbf{b}_{\mathrm{U}}\Delta\phi_{\mathrm{U}}\right\|_{2},$
(17) $\displaystyle\left\|\Delta\mathbf{h}_{\mathrm{E}}\right\|_{2}$
$\displaystyle\\!=\\!\left\|\mathbf{a}_{\mathrm{E}}\Delta\omega_{\mathrm{E}}\\!+\\!\mathbf{b}_{\mathrm{E}}\Delta\phi_{\mathrm{E}}\right\|_{2}\\!\leq\\!\left\|\mathbf{a}_{\mathrm{E}}\Delta\omega_{\mathrm{E}}\right\|_{2}\\!+\\!\left\|\mathbf{b}_{\mathrm{E}}\Delta\phi_{\mathrm{E}}\right\|_{2}.$

To further simplify the expression, by means of variable substitution
$\left\|\mathbf{a}_{\mathrm{U}}\right\|_{2}=a_{\mathrm{U}1},\left\|\mathbf{b}_{\mathrm{U}}\right\|_{2}=a_{\mathrm{U}2},\left\|\mathbf{a}_{\mathrm{E}}\right\|_{2}=a_{\mathrm{E}1},\left\|\mathbf{b}_{\mathrm{E}}\right\|_{2}=a_{\mathrm{E}2}$
and (6), (17) can be rewritten as

$\displaystyle\left\|\Delta\mathbf{h}_{\mathrm{U}}\right\|_{2}\leq\beta_{\mathrm{U}1}a_{\mathrm{U}1}+\beta_{\mathrm{U}2}a_{\mathrm{U}2}\triangleq\xi_{\mathrm{U},\mathrm{h}},$
(18)
$\displaystyle\left\|\Delta\mathbf{h}_{\mathrm{E}}\right\|_{2}\leq\beta_{\mathrm{E}1}a_{\mathrm{E}1}+\beta_{\mathrm{E}2}a_{\mathrm{E}2}\triangleq\xi_{\mathrm{E},\mathrm{h}}.$

Similarly, the CSI error model of $\mathbf{H}_{\mathrm{I}}$ is given as

$\displaystyle\left\|\Delta\mathbf{H}_{\mathrm{I}}\right\|_{F}$
$\displaystyle=\left\|\mathbf{H}_{\mathrm{I},\mathrm{L}}^{(\mathrm{A})}\left(\mathbf{a}_{\mathrm{B}}\Delta\omega_{\mathrm{B}}+\mathbf{b}_{\mathrm{B}}\Delta\phi_{\mathrm{B}}\right)\right\|_{F},$
(19)
$\displaystyle\leq\left|\Delta\omega_{\mathrm{B}}\right|\left\|\mathbf{H}_{\mathrm{I},\mathrm{L}}^{(\mathrm{A})}\right\|_{F}\left\|\mathbf{a}_{\mathrm{B}}\right\|_{2}\\!+\\!\left|\Delta\phi_{\mathrm{B}}\right|\left\|\mathbf{H}_{\mathrm{I},\mathrm{L}}^{(\mathrm{A})}\right\|_{F}\left\|\mathbf{b}_{\mathrm{B}}\right\|_{2}.$

In order to facilitate subsequent analysis, the channel UBS-IRS-Alice and UBS-
IRS-Eve are represented by the cascaded channel $\mathbf{G}_{\mathrm{U}}$ and
$\mathbf{G}_{\mathrm{E}}$, respectively. Denote by
$\mathbf{G}_{\mathrm{U}}=\operatorname{diag}\left(\mathbf{h}_{\mathrm{IU}}^{H}\right)\mathbf{H}_{\mathrm{I}}$
and
$\mathbf{G}_{\mathrm{E}}=\operatorname{diag}\left(\mathbf{h}_{\mathrm{IE}}^{H}\right)\mathbf{H}_{\mathrm{I}}$
the cascaded channel via IRS. The CSI error model of the cascaded channel
$\mathbf{G}_{\mathrm{U}}$ and $\mathbf{G}_{\mathrm{E}}$ is expressed as

$\displaystyle\left\|\Delta\mathbf{G}_{\mathrm{U}}\right\|_{F}\leq\beta_{\mathrm{B}1}g_{\mathrm{U}1}+\beta_{\mathrm{B}2}g_{\mathrm{U}2}\triangleq\xi_{\mathrm{U},\mathrm{G}},$
$\displaystyle\left\|\Delta\mathbf{G}_{\mathrm{E}}\right\|_{F}\leq\beta_{\mathrm{B}1}g_{\mathrm{E}1}+\beta_{\mathrm{B}2}g_{\mathrm{E}2}\triangleq\xi_{\mathrm{E},\mathrm{G}}.$
(20)

where

$\displaystyle
g_{\mathrm{U}1}=\left\|\operatorname{diag}\left(\mathbf{h}_{\mathrm{IU}}^{H}\right)\right\|_{F}\left\|\mathbf{H}_{\mathrm{I},\mathrm{L}}^{(\mathrm{A})}\right\|_{F}\left\|\mathbf{a}_{\mathrm{B}}\right\|_{2},$
(21) $\displaystyle
g_{\mathrm{U}2}=\left\|\operatorname{diag}\left(\mathbf{h}_{\mathrm{IU}}^{H}\right)\right\|_{F}\left\|\mathbf{H}_{\mathrm{I},\mathrm{L}}^{(\mathrm{A})}\right\|_{F}\left\|\mathbf{b}_{\mathrm{B}}\right\|_{2},$
$\displaystyle
g_{\mathrm{E}1}=\left\|\operatorname{diag}\left(\mathbf{h}_{\mathrm{IE}}^{H}\right)\right\|_{F}\left\|\mathbf{H}_{\mathrm{I},\mathrm{L}}^{(\mathrm{A})}\right\|_{F}\left\|\mathbf{a}_{\mathrm{B}}\right\|_{2},$
$\displaystyle
g_{\mathrm{E}2}=\left\|\operatorname{diag}\left(\mathbf{h}_{\mathrm{IE}}^{H}\right)\right\|_{F}\left\|\mathbf{H}_{\mathrm{I},\mathrm{L}}^{(\mathrm{A})}\right\|_{F}\left\|\mathbf{b}_{\mathrm{B}}\right\|_{2},$

###  II-C Signal Model

The signal received at Alice from both the UBS-Alice and UBS-IRS-Alice
channels is expressed as

$\displaystyle y_{\mathrm{U}}$
$\displaystyle=\left(\mathbf{h}_{\mathrm{IU}}^{H}\bm{\Theta}\mathbf{H}_{\mathrm{I}}+\mathbf{h}_{\mathrm{U}}^{H}\right)\mathbf{w}s+\mathbf{h}_{\mathrm{IU}}^{H}\bm{\Theta}\mathbf{n}_{\mathrm{I}}+n_{\mathrm{U}},$
(22)

and the signal received at Eve is

$\displaystyle y_{\mathrm{E}}$
$\displaystyle=\left(\mathbf{h}_{\mathrm{IE}}^{H}\bm{\Theta}\mathbf{H}_{\mathrm{I}}+\mathbf{h}_{\mathrm{E}}^{H}\right)\mathbf{w}s+\mathbf{h}_{\mathrm{IE}}^{H}\bm{\Theta}\mathbf{n}_{\mathrm{I}}+n_{\mathrm{E}},$
(23)

where
$\mathbf{n}_{\mathrm{I}}\sim\mathcal{C}\mathcal{N}\left(0,\sigma_{\mathrm{I}}^{2}\mathbf{I}_{M}\right)$
denotes the additive white Gaussian noise (AWGN) at the IRS.
$n_{\mathrm{U}}\sim\mathcal{C}\mathcal{N}\left(0,\sigma_{\mathrm{U}}^{2}\right),n_{\mathrm{E}}\sim\mathcal{C}\mathcal{N}\left(0,\sigma_{\mathrm{E}}^{2}\right)$
denotes the AWGN at the Alice and Eve, respectively. $s$ is the data symbol
vector with unit-power and $\mathbf{w}$ is the transmit beamformer. Denote by
$\mathbf{v}=\left[v_{1},\ldots,v_{M}\right]^{T}\in\mathbb{C}^{M\times 1}$ the
vector containing diagonal elements of matrix $\bm{\Theta}$ where
$v_{m}=\tau_{m}e^{j\theta_{m}},\forall m$. As such, the achievable rate of
Alice and Eve are respectively given by

$\displaystyle R_{\mathrm{U}}=\log_{2}\left(1+\gamma_{\mathrm{U}}\right),$
(24) $\displaystyle
R_{\mathrm{E}}=\log_{2}\left(1+\gamma_{\mathrm{E}}\right),$

where

$\displaystyle\gamma_{\mathrm{U}}$
$\displaystyle=\frac{\left|\left[\left(\overline{\mathbf{h}}_{\mathrm{U}}^{H}+\Delta\mathbf{h}_{\mathrm{U}}^{H}\right)+\mathbf{h}_{\mathrm{IU}}^{H}\bm{\Theta}\left(\overline{\mathbf{H}}_{\mathrm{I}}+\Delta\mathbf{H}_{\mathrm{I}}\right)\right]\mathbf{w}\right|^{2}}{\sigma_{\mathrm{I}}^{2}\left\|\mathbf{h}_{\mathrm{IU}}^{H}\bm{\Theta}\right\|^{2}+\sigma_{\mathrm{U}}^{2}}$
(25)
$\displaystyle=\frac{\left|\left[\left(\overline{\mathbf{h}}_{\mathrm{U}}^{H}+\Delta\mathbf{h}_{\mathrm{U}}^{H}\right)+\mathbf{v}^{H}\left(\overline{\mathbf{G}}_{\mathrm{U}}+\Delta\mathbf{G}_{\mathrm{U}}\right)\right]\mathbf{w}\right|^{2}}{\sigma_{\mathrm{I}}^{2}\left\|\mathbf{v}^{H}\operatorname{diag}(\mathbf{h}_{\mathrm{IU}}^{H})\right\|^{2}+\sigma_{\mathrm{U}}^{2}},$
$\displaystyle\gamma_{\mathrm{E}}$
$\displaystyle=\frac{\left|\left[\left(\overline{\mathbf{h}}_{\mathrm{E}}^{H}+\Delta\mathbf{h}_{\mathrm{E}}^{H}\right)+\mathbf{h}_{\mathrm{IE}}^{H}\bm{\Theta}\left(\overline{\mathbf{H}}_{\mathrm{I}}+\Delta\mathbf{H}_{\mathrm{I}}\right)\right]\mathbf{w}\right|^{2}}{\sigma_{\mathrm{I}}^{2}\left\|\mathbf{h}_{\mathrm{IE}}^{H}\bm{\Theta}\right\|^{2}+\sigma_{\mathrm{E}}^{2}}$
$\displaystyle=\frac{\left|\left[\left(\overline{\mathbf{h}}_{\mathrm{E}}^{H}+\Delta\mathbf{h}_{\mathrm{E}}^{H}\right)+\mathbf{v}^{H}\left(\overline{\mathbf{G}}_{\mathrm{E}}+\Delta\mathbf{G}_{\mathrm{E}}\right)\right]\mathbf{w}\right|^{2}}{\sigma_{\mathrm{I}}^{2}\left\|\mathbf{v}^{H}\operatorname{diag}(\mathbf{h}_{\mathrm{IE}}^{H})\right\|^{2}+\sigma_{\mathrm{E}}^{2}}.$

##  III Robust and Secure UAV Communication assisted by Active IRS

In order to ensure ASR while improving UAV’s power-efficiency, with the UAV
jittering model, we first formulate the transmission power minimization
problem under the worst-case secrecy rate constraints, UAV’s maximum
transmission power constraints, and the active IRS’s amplification power
constraint. Due to the coupling relationship of various optimization variables
and complex non-convex constraints, the proposed problem is non-convex and
difficult to solve. To solve this problem, we transform all the non-convex
constraints into tractable LMIs forms by S-procedure and Schur’s complement.
Then the AO strategy is utilized to decouple the parameters and transform the
original problems into two subproblems, i.e., passive beamforming and active
IRS’s reflecting coefficient optimization.

###  III-A Transmission Power Minimization Problem Formulation

In practice, the transmission power of UAVs is strictly limited which is
necessary to be reduce as much as possible while ensuring secure communication
requirements. Thus, we focus on reducing the transmission power of the active
IRS-assisted UAV system under the given ASR constraint, i.e., the worst-case
scenario where Alice reaches its required minimum data rate and Eve reaches
its maximum data rate. In order to ensure the secure transmission between the
UBS and Alice, the worst-case secrecy rate of the considered system should be
above 0, which is given by $R_{\mathrm{sec}}$

$\displaystyle
R_{\mathrm{sec}}=\left[\min_{\left\\{\Omega_{\mathrm{B}},\Omega_{\mathrm{U}}\right\\}}R_{\mathrm{U}}-\max_{\left\\{\Omega_{\mathrm{B}},\Omega_{\mathrm{E}}\right\\}}R_{\mathrm{E}}\right]^{+}.$
(26)

In this paper, we adopted an active IRS in which each reflecting element (RE)
is supported by a set of active-load impedances. In this way, active IRS
solves the double-fading attenuation problem not only by increasing the number
of REs, but also by amplifying the incident signal at each RE. Due to the
ability to amplify the incident signal, fewer number of REs is required to
achieve the target signal-to-noise ratio. That means the physical size of the
active IRS can be reduced, making it more suitable for UAV communication
limited by the payload. Furthermore, it offers more flexibility to reconfigure
the wireless propagation environment by optimizing the amplitudes of the
reflection coefficients instead of just phases, resulting in a more spectrum
and energy-efficient communication. Note that the active IRS amplifies both
the received signal and noise at each RE. Besides, the amplification power is
limited due to the overall power budget at the IRS [21]. Let $P_{\text{F}}$
denote the maximum amplification power of the active IRS, which is practically
much smaller than the conventional RF amplifier due to the limited
amplification power gain [11], [12]. Then we have

$\left\|\bm{\Theta}\mathbf{H}_{\mathrm{I}}\mathbf{w}\right\|^{2}+\sigma_{\mathrm{I}}^{2}\left\|\bm{\Theta}\right\|^{2}\leq
P_{\text{F}}.$ (27)

To minimize transmission power of UBS, the reflecting coefficient matrix
$\bm{\Theta}$ of the active IRS and the transmit beamformer $\mathbf{w}$ are
jointly optimized, and the corresponding optimization problem is formulated as
follows.

$\displaystyle(\mathrm{P}1):\min_{\bm{\Theta},\mathbf{w}}$ $\displaystyle\
\|\mathbf{w}\|^{2}$ s.t. $\displaystyle\|\mathbf{w}\|^{2}\leq
P_{\text{peak}},$ (28a)
$\displaystyle\min_{\left\\{\Omega_{\mathrm{B}},\Omega_{\mathrm{U}}\right\\}}R_{\mathrm{U}}\geq\eta_{\mathrm{U}},$
(28b)
$\displaystyle\max_{\left\\{\Omega_{\mathrm{B}},\Omega_{\mathrm{E}}\right\\}}R_{\mathrm{E}}\leq\eta_{\mathrm{E}},$
(28c)
$\displaystyle\left\|\bm{\Theta}\mathbf{H}_{\mathrm{I}}\mathbf{w}\right\|^{2}+\sigma_{\mathrm{I}}^{2}\left\|\bm{\Theta}\right\|^{2}\leq
P_{\text{F}},$ (28d) $\displaystyle|\bm{\Theta}[m,m]|\leq\tau_{\text{max}}$
(28e)

where $\mathbf{w}$ is constrained by a peak power limit $P_{\text{peak}}$
i.e., (28a), and $\tau_{\text{max}}$ is the maximum amplification factor at
the m-th reflecting element. Constraint (28b) ensures that the minimum data
rate of the legitimate channel is above $\eta_{\mathrm{U}}(>0)$, i.e., ensures
the effectiveness and reliability of the legitimate link. Constraint (28c)
ensures that the maximum eavesdropping data rate is below
$\eta_{\mathrm{E}}\left(0<\eta_{\mathrm{E}}<\eta_{\mathrm{U}}\right)$, i.e.,
ensures the security of the confidential message. (28b) and (28c) jointly
guarantee the secure transmission requirement of the confidential signal. As
mentioned before, (28d) is the amplification power constraint. It is observed
that the (28b) and (28c) are the constraints for lower bound and upper bound,
respectively. In addition, $\mathbf{w}$ and $\bm{\Theta}$ are coupled in
(28d). Thus, the optimization problem in (28) is non-convex and challenging to
solve. To cope with this difficulty, we propose an efficient algorithm based
on the AO method to optimize $\mathbf{w}$ and $\bm{\Theta}$ sequentially in an
iterative manner. Specifically, we divide problem P1 into two sub-problems:

1) The optimization of the transmit $\mathbf{w}$ under the given reflecting
coefficient matrix $\bm{\Theta}$;

2) The optimization of the reflecting coefficient matrix $\bm{\Theta}$ under
the given transmission power $\mathbf{w}$;

###  III-B Alternating Optimization Solution of the Problem

In this section, we alternately solve the two sub-problems to minimize the
transmission power at UBS. To deal with non-convex constraints, we first use
S-procedure to convert the worst-case secrecy rate constraints into LMIs.
Since the logarithmic function is a monotonically increasing function, by
considering the channel uncertainty in (LABEL:H_CSI) and (II-B) and denoting
$\mathcal{E}^{\text{Alice
}}\triangleq\left\\{\forall\left\|\Delta\mathbf{h}_{\text{U}}\right\|_{2}\leq\right.$
$\left.\xi_{\mathrm{U},\mathrm{h}},\forall\left\|\Delta\mathbf{G}_{\text{U}}\right\|_{F}\leq\xi_{\mathrm{U},\mathrm{G}}\right\\}$,
$\mathcal{E}^{\text{Eve
}}\triangleq\left\\{\forall\left\|\Delta\mathbf{h}_{\text{E}}\right\|_{2}\leq\xi_{\mathrm{E},\mathrm{h}},\right.$
$\left.\forall\left\|\Delta\mathbf{G}_{\text{E}}\right\|_{F}\leq\xi_{\mathrm{E},\mathrm{G}}\right\\}$,
constraints (28b) and (28c) can be rewritten as

$\left|\left((\overline{\mathbf{h}}_{\mathrm{U}}\\!+\\!\Delta\mathbf{h}_{\mathrm{U}})\\!+\\!\mathbf{v}^{H}(\overline{\mathbf{G}}_{\mathrm{U}}\\!+\\!\Delta\mathbf{G}_{\mathrm{U}})\right)\mathbf{w}\right|^{2}\\!\geq\\!\beta_{\text{Alice}}\left(2^{\eta_{\text{U}}}\\!-\\!1\right),\mathcal{E}^{\text{Alice}},$
(29)
$\left|\left(\left(\overline{\mathbf{h}}_{\mathrm{E}}\\!+\\!\Delta\mathbf{h}_{\mathrm{E}}\right)\\!+\\!\mathbf{v}^{H}\left(\overline{\mathbf{G}}_{\mathrm{E}}\\!+\\!\Delta\mathbf{G}_{\mathrm{E}}\right)\right)\mathbf{w}\right|^{2}\\!\leq\\!\beta_{\text{Eve}}\left(2^{\eta_{\mathrm{E}}}\\!-\\!1\right),\mathcal{E}^{\text{Eve}},$
(30)

where
$\beta_{\text{Alice}}=\sigma_{\mathrm{I}}^{2}\left\|\mathbf{v}^{H}\operatorname{diag}(\mathbf{h}_{\mathrm{IU}}^{H})\right\|^{2}+\sigma_{\mathrm{U}}^{2}$
and $\beta_{\text{Eve
}}=\sigma_{\mathrm{E}}^{2}\left\|\mathbf{v}^{H}\operatorname{diag}(\mathbf{h}_{\mathrm{IE}}^{H})\right\|^{2}+\sigma_{\mathrm{E}}^{2}$
is the interference-plus-noises power at Alice and Eve, respectively. In
particular, the linear approximation of the useful signal power in (29) is
given in the following lemma.

######  Lemma 1.

Let $\mathbf{w}^{(k)}$ and $\mathbf{v}^{(k)}$ be the optimal solutions
obtained at iteration $k$, then the left side of the inequality (29)
$\left|\left[\left(\overline{\mathbf{h}}_{\mathrm{U}}+\Delta\mathbf{h}_{\mathrm{U}}\right)^{H}+\mathbf{v}^{H}\left(\overline{\mathbf{G}}_{\mathrm{U}}+\Delta\mathbf{G}_{\mathrm{U}}\right)\right]\mathbf{w}\right|^{2}$
is lower bounded linearly at $\left(\mathbf{w}^{(k)},\mathbf{v}^{(k)}\right)$
as follows

$\displaystyle\mathbf{x}_{1}^{H}\widetilde{\mathbf{A}}\mathbf{x}_{1}+2\operatorname{Re}\left\\{\widetilde{\mathbf{a}}_{\mathrm{U}}^{H}\mathbf{x}_{1}\right\\}+\widetilde{a}_{\mathrm{U}},$
(31)

where

$\displaystyle\widetilde{\mathbf{A}}=\mathbf{C}+\mathbf{C}^{H}-\mathbf{Z},$
(32)
$\displaystyle\widetilde{\mathbf{a}}_{\mathrm{U}}=\mathbf{c}_{\mathrm{1U}}+\mathbf{c}_{\mathrm{2U}}-\mathbf{z}_{\mathrm{U}},$
$\displaystyle\widetilde{a}_{\mathrm{U}}=2\operatorname{Re}\left\\{c_{\mathrm{U}}\right\\}-z_{\mathrm{U}},$
$\displaystyle\mathbf{x}_{1}=\left[\Delta\mathbf{h}_{\mathrm{U}}^{H}\operatorname{vec}^{H}\left(\Delta\mathbf{G}_{\mathrm{U}}^{*}\right)\right]^{H}.$

Note that
$\mathbf{C},\mathbf{Z},\mathbf{c}_{\mathrm{1U}},\mathbf{c}_{\mathrm{2U}},\mathbf{z}_{\mathrm{U}},c_{\mathrm{U}}$,
and $z_{\mathrm{U}}$ are given by (III-B) at the bottom of the page,
respectively.

###### Proof.

Please refer to Appendix A. ∎

Thus, constraints (29) is equivalently rewritten as

$\displaystyle\mathbf{x}_{1}^{H}\widetilde{\mathbf{A}}\mathbf{x}_{1}+2\operatorname{Re}\left\\{\widetilde{\mathbf{a}}_{\mathrm{U}}^{H}\mathbf{x}_{1}\right\\}+\widetilde{a}_{\mathrm{U}}\geq\beta_{\text{Alice}}\left(2^{\eta_{\text{U}}}-1\right),\mathcal{E}^{\text{Alice}}.$
(33)

Similarly, constraint (30) can be equivalently rewritten as

$\displaystyle\mathbf{x}_{2}^{H}\widetilde{\mathbf{A}}\mathbf{x}_{2}+2\operatorname{Re}\left\\{\widetilde{\mathbf{a}}_{\mathrm{E}}^{H}\mathbf{x}_{2}\right\\}+\widetilde{a}_{\mathrm{E}}\leq\beta_{\text{Eve}}\left(2^{\eta_{\mathrm{E}}}-1\right),\mathcal{E}^{\text{Eve}},$
(34)

where

$\displaystyle\widetilde{\mathbf{a}}_{\mathrm{E}}=\mathbf{c}_{\mathrm{1E}}+\mathbf{c}_{\mathrm{2E}}-\mathbf{z}_{\mathrm{E}},$
(35)
$\displaystyle\widetilde{a}_{\mathrm{E}}=2\operatorname{Re}\left\\{c_{\mathrm{E}}\right\\}-z_{\mathrm{E}},$
$\displaystyle\mathbf{x}_{2}=\left[\Delta\mathbf{h}_{\mathrm{E}}^{H}\operatorname{vec}^{H}\left(\Delta\mathbf{G}_{\mathrm{E}}^{*}\right)\right]^{H},$

and
$\mathbf{c}_{\mathrm{1E}},\mathbf{c}_{\mathrm{2E}},\mathbf{z}_{\mathrm{E}},c_{\mathrm{E}}$,
and $z_{\mathrm{E}}$ are also given by (III-B), respectively.

######  Lemma 2.

(S-Procedure [33]): Let a function $f_{i}(\mathbf{x}),i\in$
$\\{1,2\\},\mathbf{x}\in\mathbb{C}^{N\times 1}$, be defined as

$\displaystyle
f_{i}(\mathbf{x})=\mathbf{x}^{H}\mathbf{B}_{i}\mathbf{x}+2\operatorname{Re}\left\\{\mathbf{b}_{i}^{\mathrm{H}}\mathbf{x}\right\\}+b_{i}.$
(36)

The condition $f_{1}(\mathbf{x})\geq 0\Rightarrow$ $f_{2}(\mathbf{x})\geq 0$
holds if and only if there exist $\varpi\geq 0$ such that

$\left[\begin{array}[]{ll}\mathbf{B}_{2}&\mathbf{b}_{2}\\\
\mathbf{b}_{2}^{H}&b_{2}\end{array}\right]-\varpi\left[\begin{array}[]{ll}\mathbf{B}_{1}&\mathbf{b}_{1}\\\
\mathbf{b}_{1}^{H}&b_{1}\end{array}\right]\succeq\mathbf{0}.$ (37)

Conversely, the condition $f_{1}(\mathbf{x})\leq 0\Rightarrow$
$f_{2}(\mathbf{x})\leq 0$ holds if and only if there exist $\psi\geq 0$ such
that

$\psi\left[\begin{array}[]{ll}\mathbf{B}_{1}&\mathbf{b}_{1}\\\
\mathbf{b}_{1}^{H}&b_{1}\end{array}\right]-\left[\begin{array}[]{ll}\mathbf{B}_{2}&\mathbf{b}_{2}\\\
\mathbf{b}_{2}^{H}&b_{2}\end{array}\right]\succeq\mathbf{0}.$ (38)

In order to transform the constraints (29) into LMIs by applying the Lemma 2,
$\mathcal{E}^{\text{Alice}}$ is rewritten as the following quadratic
expression

$\mathcal{E}^{\text{Alice}}\triangleq\left\\{\begin{array}[]{c}\mathbf{x}_{1}^{H}\left[\begin{array}[]{cc}\mathbf{I}_{N}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}\end{array}\right]\mathbf{x}_{1}-\xi_{\mathrm{U},h}^{2}\leq
0,\\\ \mathbf{x}_{1}^{H}\left[\begin{array}[]{cc}\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&\mathbf{I}_{MN}\end{array}\right]\mathbf{x}_{1}-\xi_{\mathrm{U},G}^{2}\leq
0.\end{array}\right.$ (39)

Then, after introducing $\varpi_{\mathrm{U1}}\geq 0$ and
$\varpi_{\mathrm{U2}}\geq 0$ as slack variables, constraints (33) can be
transformed by Lemma 1 into the following equivalent LMIs as

$\left[\begin{array}[]{cc}\widetilde{\mathbf{A}}+\left[\begin{array}[]{cc}\varpi_{\mathrm{U1}}\mathbf{I}_{N}&\mathbf{0}\\\
\mathbf{0}&\varpi_{\mathrm{U2}}\mathbf{I}_{MN}\end{array}\right]&\widetilde{\mathbf{a}}_{\mathrm{U}}\\\
\widetilde{\mathbf{a}}_{\mathrm{U}}^{\mathrm{H}}&V_{\mathrm{U}}\end{array}\right]\succeq\mathbf{0},$
(40)

where
$V_{\mathrm{U}}\\!=\\!\widetilde{a}_{\mathrm{U}}\\!-\\!\beta_{\text{Alice}}\left(2^{R_{\mathrm{U}}}\\!-\\!1\right)\\!-\\!\varpi_{\mathrm{U1}}\xi_{\mathrm{U},\mathrm{h}}^{2}\\!-\varpi_{\mathrm{U2}}\xi_{\mathrm{U},\mathrm{G}}^{2}$.
After introducing $\psi_{\mathrm{E1}}\geq 0$ and $\psi_{\mathrm{E2}}\geq 0$ as
slack variables, constraints (34) can be transformed into the following
equivalent LMIs as

$\left[\begin{array}[]{cc}\left[\begin{array}[]{cc}\psi_{\mathrm{E},\mathrm{h}}\mathbf{I}_{N}&\mathbf{0}\\\
\mathbf{0}&\psi_{\mathrm{E2}}\mathbf{I}_{MN}\end{array}\right]-\widetilde{\mathbf{A}}&-\widetilde{\mathbf{a}}_{\mathrm{E}}\\\
-{\widetilde{\mathbf{a}}_{\mathrm{E}}^{\mathrm{H}}}&V_{\mathrm{E}}\end{array}\right]\succeq\mathbf{0},$
(41)

where
$V_{\mathrm{E}}\\!=\\!\beta_{\text{Eve}}\left(2^{R_{\mathrm{E}}}\\!-1\right)\\!-\widetilde{a}_{\mathrm{E}}\\!-\psi_{\mathrm{E1}}\xi_{\mathrm{E},\mathrm{h}}^{2}\\!-\psi_{\mathrm{E2}}\xi_{\mathrm{E},\mathrm{G}}^{2}.$

Subsequently, we transeform (28d) into

$\displaystyle\left|\mathbf{v}^{H}\operatorname{diag}\left(\mathbf{H}_{\text{I}}\mathbf{w}\right)\right|^{2}+|\mathbf{v}|^{2}\sigma_{\text{I}}^{2}\leq
P_{\text{F}}.$ (42)

Then we adopt Schur’s complement [34] to equivalently convert (42) into matrix
inequalities as follows

$\displaystyle{\left[\begin{array}[]{cr}P_{F}-|\mathbf{v}|^{2}\sigma_{1}^{2}&\mathbf{v}^{H}\operatorname{diag}\left(\mathbf{H}_{\text{I}}\mathbf{w}\right)\\\
\operatorname{diag}\left(\mathbf{H}_{\text{I}}\mathbf{w}\right)^{H}\mathbf{v}&\mathbf{I}\end{array}\right]\succeq\mathbf{0}}.$
(43)

Based on the above discussions, for given $\mathbf{v}$, the subproblem of
$\mathbf{w}$ is given by

$\displaystyle\min_{\mathbf{w},\bm{\varpi}_{\mathrm{U1}},\atop\bm{\varpi}_{\mathrm{U2}},\bm{\psi}_{\mathrm{E1}},\bm{\psi}_{\mathrm{E2}}}\
$ $\displaystyle\|\mathbf{w}\|^{2}$ s.t.
$\displaystyle\eqref{C1},\eqref{LMI_Alice},\eqref{LMI_Eve},\eqref{Active_C}.$
(44)

Problem (III-B) is a convex problem and can be directly solved by the CVX
tool.

Then, for given $\mathbf{w}$, the subproblem of $\mathbf{v}$ is a feasibility-
check problem. According to [35] and in order to improve the converged
solution in the optimization of $\mathbf{v}$, the useful signal power
inequalities in (29) are modified by introducing slack variables
$\alpha_{\mathrm{U}}\geq 0$ and recast as

$\displaystyle\left|\left((\overline{\mathbf{h}}_{\mathrm{U}}+\Delta\mathbf{h}_{\mathrm{U}})+\mathbf{v}^{H}(\overline{\mathbf{G}}_{\mathrm{U}}+\Delta\mathbf{G}_{\mathrm{U}})\right)\mathbf{w}\right|^{2}$
$\displaystyle\geq\beta_{\text{Alice}}\left(2^{\eta_{\text{U}}}-1\right)+\alpha_{\text{U}},\mathcal{E}^{\text{Alice}}.$
(45)

Subsequently, the LMIs (40) are modified as

$\left[\begin{array}[]{cc}\widetilde{\mathbf{A}}+\left[\begin{array}[]{cc}\varpi_{\mathrm{U},\mathrm{h}}\mathbf{I}_{N}&\mathbf{0}\\\
\mathbf{0}&\varpi_{\mathrm{U},\mathrm{G}}\mathbf{I}_{MN}\end{array}\right]&\widetilde{\mathbf{a}}_{\mathrm{U}}\\\
\widetilde{\mathbf{a}}_{\mathrm{U}}^{\mathrm{H}}&V_{\mathrm{U}}-\alpha_{\text{U}}\end{array}\right]\succeq\mathbf{0},$
(46)

the Eve signal power inequalities in (30) are modified by introducing slack
variables $\alpha_{\mathrm{E}}\geq 0$ and recast as

$\displaystyle\left|\left(\left(\overline{\mathbf{h}}_{\mathrm{E}}+\Delta\mathbf{h}_{\mathrm{E}}\right)+\mathbf{v}^{H}\left(\overline{\mathbf{G}}_{\mathrm{E}}+\Delta\mathbf{G}_{\mathrm{E}}\right)\right)\mathbf{w}\right|^{2}$
$\displaystyle\leq\beta_{\text{Eve}}\left(2^{\eta_{\mathrm{E}}}-1\right)-\alpha_{\text{E}},\mathcal{E}^{\text{Eve}},$
(47)

and the LMIs (40) are modified as

$\left[\begin{array}[]{cc}\left[\begin{array}[]{cc}\varpi_{\mathrm{E},\mathrm{h}}\mathbf{I}_{N}&\mathbf{0}\\\
\mathbf{0}&\varpi_{\mathrm{E},\mathrm{G}}\mathbf{I}_{MN}\end{array}\right]-\widetilde{\mathbf{A}}&-\widetilde{\mathbf{a}}_{\mathrm{E}}\\\
-{\widetilde{\mathbf{a}}_{\mathrm{E}}^{H}}&V_{\mathrm{E}}-\alpha_{\text{E}}\end{array}\right]\succeq\mathbf{0}.$
(48)

Let $\mathbf{h}_{\text{g}}=\mathbf{H}_{\mathrm{I}}\mathbf{w}$, and
$\mathbf{F}=\operatorname{diag}\left(\left[\left|h_{\text{g1}}\right|^{2},\ldots,\left|h_{\text{gM}}\right|^{2}\right]\right)+\sigma_{2}^{2}\mathbf{I}_{\text{M}}$,
(27) can be expressed as $\mathbf{v}^{\mathrm{H}}\mathbf{Fv}\leq
P_{\text{F}}$. Thus, the sub-problem of $\mathbf{v}$ can be formulated as

$\displaystyle\max_{\mathbf{v},\bm{\varpi}_{\mathrm{U1}},\bm{\varpi}_{\mathrm{U2}},\atop\bm{\psi}_{\mathrm{E1}},\bm{\psi}_{\mathrm{E2}},\bm{\alpha}_{\mathrm{U}},\bm{\alpha}_{\mathrm{E}}}$
$\displaystyle\alpha_{\text{U}}+\alpha_{\text{E}}$ (49) s.t.
$\displaystyle\eqref{fc_LMI_Alice},\eqref{fc_LMI_Eve},$
$\displaystyle\mathbf{v}^{\mathrm{H}}\mathbf{Fv}\leq P_{\text{F}},$
$\displaystyle|\mathbf{v}[m]|\leq\tau_{\text{max}}$

Problem (49) is a convex optimization problem which also can be solved
efficiently and optimally with the tool CVX.

$\displaystyle\mathbf{C}=\left[\begin{array}[]{c}\mathbf{w}^{(k)}\\\
\mathbf{w}^{(k)}\otimes\mathbf{v}^{(k),*}\end{array}\right]\left[\mathbf{w}^{H}\mathbf{w}^{H}\otimes\mathbf{v}^{T}\right],$
$\displaystyle\mathbf{Z}=\left[\begin{array}[]{c}\mathbf{w}^{(k)}\\\
\mathbf{w}^{(k)}\otimes\mathbf{v}^{(k),*}\end{array}\right]\\!\left[\mathbf{w}^{(k),H}\mathbf{w}^{(k),H}\otimes\mathbf{v}^{(k),T}\right],$
$\displaystyle\mathbf{c}_{\mathrm{1U}}=\left[\begin{array}[]{c}\mathbf{w}\mathbf{w}^{(k),H}\left(\overline{\mathbf{h}}_{\mathrm{U}}+\overline{\mathbf{G}}_{\mathrm{U}}^{H}\mathbf{v}^{(k)}\right)\\\
\operatorname{vec}^{*}\left(\mathbf{v}\left(\overline{\mathbf{h}}_{\mathrm{U}}^{H}+\mathbf{v}^{(k),H}\overline{\mathbf{G}}_{\mathrm{U}}\right)\mathbf{w}^{(k)}\mathbf{w}^{H}\right)\end{array}\right],$
$\displaystyle\mathbf{c}_{\mathrm{2U}}=\left[\begin{array}[]{c}\mathbf{w}^{(k)}\mathbf{w}^{H}\left(\overline{\mathbf{h}}_{\mathrm{U}}+\overline{\mathbf{G}}_{\mathrm{U}}^{H}\mathbf{v}\right)\\\
\operatorname{vec}^{*}\left(\mathbf{v}^{(k)}\left(\overline{\mathbf{h}}_{\mathrm{U}}^{H}+\mathbf{v}^{H}\overline{\mathbf{G}}_{\mathrm{U}}\right)\mathbf{w}\mathbf{w}^{(k),H}\right)\end{array}\right],$
$\displaystyle\mathbf{z}_{\mathrm{U}}=\left[\begin{array}[]{c}\mathbf{w}^{(k)}\mathbf{w}^{(k),H}\left(\overline{\mathbf{h}}_{\mathrm{U}}+\overline{\mathbf{G}}_{\mathrm{U}}^{H}\mathbf{v}^{(k)}\right)\\\
\operatorname{vec}^{*}\left(\mathbf{v}^{(k)}\left(\overline{\mathbf{h}}_{\mathrm{U}}^{H}+\mathbf{v}^{(k),H}\overline{\mathbf{G}}_{\mathrm{U}}\right)\mathbf{w}^{(k)}\mathbf{w}^{(k),H}\right)\end{array}\right],$
$\displaystyle
c_{\mathrm{U}}=\left(\overline{\mathbf{h}}_{\mathrm{U}}^{H}+\mathbf{v}^{(k),H}\overline{\mathbf{G}}_{\mathrm{U}}\right)\mathbf{w}^{(k)}\mathbf{w}^{H}\left(\overline{\mathbf{h}}_{\mathrm{U}}+\overline{\mathbf{G}}_{\mathrm{U}}^{H}\mathbf{v}\right),$
$\displaystyle
z_{\mathrm{U}}=\left(\overline{\mathbf{h}}_{\mathrm{U}}^{H}+\mathbf{v}^{(k),H}\overline{\mathbf{G}}_{\mathrm{U}}\right)\mathbf{w}^{(k)}\mathbf{w}^{(k),H}\left(\overline{\mathbf{h}}_{\mathrm{U}}+\overline{\mathbf{G}}_{\mathrm{U}}^{H}\mathbf{v}^{(k)}\right),$
$\displaystyle\mathbf{c}_{\mathrm{1E}}=\left[\begin{array}[]{c}\mathbf{w}\mathbf{w}^{(k),H}\left(\overline{\mathbf{h}}_{\mathrm{E}}+\overline{\mathbf{G}}_{\mathrm{E}}^{H}\mathbf{v}^{(k)}\right)\\\
\operatorname{vec}^{*}\left(\mathbf{v}\left(\overline{\mathbf{h}}_{\mathrm{E}}^{H}+\mathbf{v}^{(k),H}\overline{\mathbf{G}}_{\mathrm{E}}\right)\mathbf{w}^{(k)}\mathbf{w}^{H}\right)\end{array}\right],$
$\displaystyle\mathbf{c}_{\mathrm{2E}}=\left[\begin{array}[]{c}\mathbf{w}^{(k)}\mathbf{w}^{H}\left(\overline{\mathbf{h}}_{\mathrm{E}}+\overline{\mathbf{G}}_{\mathrm{E}}^{H}\mathbf{v}\right)\\\
\operatorname{vec}^{*}\left(\mathbf{v}^{(k)}\left(\overline{\mathbf{h}}_{\mathrm{E}}^{H}+\mathbf{v}^{H}\overline{\mathbf{G}}_{\mathrm{E}}\right)\mathbf{w}\mathbf{w}^{(k),H}\right)\end{array}\right],$
$\displaystyle\mathbf{z}_{\mathrm{E}}=\left[\begin{array}[]{c}\mathbf{w}^{(k)}\mathbf{w}^{(k),H}\left(\overline{\mathbf{h}}_{\mathrm{E}}+\overline{\mathbf{G}}_{\mathrm{E}}^{H}\mathbf{v}^{(k)}\right)\\\
\operatorname{vec}^{*}\left(\mathbf{v}^{(k)}\left(\overline{\mathbf{h}}_{\mathrm{E}}^{H}+\mathbf{v}^{(k),H}\overline{\mathbf{G}}_{\mathrm{E}}\right)\mathbf{w}^{(k)}\mathbf{w}^{(k),H}\right)\end{array}\right],$
(50) $\displaystyle
c_{\mathrm{E}}=\left(\overline{\mathbf{h}}_{\mathrm{E}}^{H}+\mathbf{v}^{(k),H}\overline{\mathbf{G}}_{\mathrm{E}}\right)\mathbf{w}^{(k)}\mathbf{w}^{H}\left(\overline{\mathbf{h}}_{\mathrm{E}}+\overline{\mathbf{G}}_{\mathrm{E}}^{H}\mathbf{v}\right),$
$\displaystyle
z_{\mathrm{E}}=\left(\overline{\mathbf{h}}_{\mathrm{E}}^{H}+\mathbf{v}^{(k),H}\overline{\mathbf{G}}_{\mathrm{E}}\right)\mathbf{w}^{(k)}\mathbf{w}^{(k),H}\left(\overline{\mathbf{h}}_{\mathrm{E}}+\overline{\mathbf{G}}_{\mathrm{E}}^{H}\mathbf{v}^{(k)}\right).$

We summarize the proposed AO algorithm in Algorithm 1.

Algorithm 1 The proposed AO algorithm for secure and robust communication with
jittering UAV

Initialize: Set $k=0$, and initialize $\mathbf{v}^{(0)},\mathbf{w}^{(0)}$.
Compute the objective function value of problem (28) (i.e., the transmit
power) as $p(\mathbf{v}^{(0)},\mathbf{w}^{(0)})$. Give the error tolerance
$\epsilon$.
Repeat:

1\. Solve the problem (III-B) to obtain $\mathbf{w}^{(k+1)}$ for given
$\mathbf{v}^{(k)}$.

2\. Solve the problems (49) to obtain $\mathbf{v}^{(k+1)}$ for given
$\mathbf{w}^{(k+1)}$.

3\. With given $\mathbf{v}^{(k+1)}$ and $\mathbf{w}^{(k+1)}$, compute the
objective function value of problem (28) as
$p\left(\mathbf{v}^{(k+1)},\mathbf{w}^{(k+1)}\right)$.

4\. $k:=k+1$.

Until:
$\left|p\left(\mathbf{v}^{(k)},\mathbf{w}^{(k)}\right)\\!-\\!p\left(\mathbf{v}^{(k\\!-\\!1)},\mathbf{w}^{(k\\!-\\!1)}\right)\right|/p\left(\mathbf{v}^{(k\\!-\\!1)},\mathbf{w}^{(k\\!-\\!1)}\right)$
$<\epsilon$ or problem (49) becomes infeasible.

##  IV Simulation Results

In this section, the simulation results are presented to demonstrate the
secrecy performance of the proposed scheme. Note that transmission power in
Fig.3, Fig.4, Fig.5, and Fig.7 is optimized under the given ASR constraints to
ensure secure UAV communication requirements. Fig.6 analyzes the improvement
of the power efficiency of the proposed scheme at different ASR requirements.
In addition, we focus on analyzing the transmission power optimization with
reference to several important parameters, i.e., channel uncertainty caused by
UAV jittering, the number of RE, the power amplification characteristics of
active IRS, and UAV’s height. Without specification, the coordinations of UAV,
IRS, Alice and Eve are (10, 20, 10), (10, 0, 10), (20, 20, 0) and (10, 40, 0),
respectively. It is assumed that UBS are equipped with $N=2$ transmit
antennas, $P_{\text{F}}=10$ dBm, $\tau_{\text{max}}=30$ dB,
$\eta_{\mathrm{U}}=4.5$, $\eta_{\mathrm{E}}=1$, unless otherwise specified.
The Rician $K$-factors $K_{\mathrm{G}}$ and $K_{\mathrm{E}}$ are given by
$K_{*}=ae^{b\left(\frac{\pi}{2}-\phi_{*}\right)}$ with
$*=\\{\mathrm{U},\mathrm{E},\mathrm{I}\\}$ with environment parameters
$a=5,b=\frac{2}{\pi}\ln 3$ as in [36]. Meanwhile, the maximum transmission
power $P_{\text{peak}}=40$ dBm , the path loss factors $A_{\mathrm{L}}=-2.14$
dB, $A_{\mathrm{N}}=-3.14$ dB, the path loss exponents
$\alpha_{\mathrm{L}}=2.09,\alpha_{\mathrm{N}}=3.75$ [26], respectively. The
noise power are set to be $\sigma_{I}^{2}=\sigma_{U}^{2}=\sigma_{E}^{2}=-10$
dBm, unless otherwise specified.
${\beta_{\text{E}1}}={\beta_{\text{E}2}}=\frac{1}{2}{\beta_{\text{E}}}$ and
${\beta_{\text{U}1}}={\beta_{\text{U}2}}=\frac{1}{2}{\beta_{\text{U}}}$
measure the relative amount of CSI uncertainties. In addition, we set error
tolerance $\epsilon=10^{-4}$ in Algorithm 1. To demonstrate the advantage of
the proposed scheme (denoted as “Active IRS”), we also compare the results
with the benchmark schemes, e.i., similar scheme assisted by passive IRS
(denoted as “Passive IRS”).

Figure 3: Comparison between different $M$ versus the variations of AOD.

Figure 4: Comparison between different $N$ versus the variations of AOD.

Figure 5: Comparison between transmission power versus the active IRS’s
maximum amplification factor $\tau_{\text{max}}$.

Figure 6: Comparison between different ASR versus the active IRS’s maximum
amplification power $P_{\text{F}}$.

Figure 7: Comparison between different UAV altitude versus the active IRS’s
maximum amplification power $P_{\text{F}}$.

###  IV-A Impact of Channel Uncertainty

To evaluate the impact of channel uncertainty caused by UAV jittering for
active IRS-assisted UAV secure communication, Fig. 3(a) and Fig. 3(b) show the
transmission power w.r.t the variations of Alice’s AOD and Eve’s AOD (e.i.,
the ratio between the maximum variation and the elevation AOD) under different
number of elements $M$, respectively. Because the analysis of elevation angle
$\phi$ is similar to that of azimuth angle $\omega$, here we take elevation
angle as an example. Note that,
$\frac{\beta_{\mathrm{E}}}{\bar{\phi}_{\mathrm{E}}}=0.04$ for Fig. 3(a), and
$\frac{\beta_{\mathrm{U}}}{\bar{\phi}_{\mathrm{U}}}=0.02$ for Fig. 3(b). It
can be seen that the transmission power at UBS increases with the increase of
AOD uncertainty, which means that the more drastic of UAV jittering, the more
transmission power is required to satisfy the effectiveness of the legitimate
link and the security requirement of the confidential signal. In addition, the
active IRS solution can achieve a significant improvement of secrecy
performance with only a small number of elements, that is, reduce the UAV’s
transmission power under the same ASR. However, the equivalent noise due to
the CSI deviation will be amplified by active IRS, thus resulting in that the
secrecy performance gain from increasing the number of RE is not as
significant as with passive IRS. Furthermore, unlike the passive IRS solution,
the secrecy performance of active IRS solution will not continue to be
improved as $M$ increases. If the number of elements is too large, the secrecy
performance will decrease instead. Because active IRS is limited by the given
amplification gain $P_{\text{F}}$, the amplification factor will decrease as
the number of elements continue to increase. In conclusion, compared to
passive IRS, robust beamforming design is more important for active IRS-
assisted UAV secure communication.

Under the same settings as Figure. 3 and assuming $M$ = 10, Fig. 4(a) and Fig.
4(b) show the transmission power w.r.t the variations of Alice’s AOD and Eve’s
AOD under different number of transmitting antennas $N$, respectively.
Different from the variation of $M$, the transmitting power of both the active
IRS scheme and the passive IRS scheme continues to decrease as $N$ increases
as the amplified noise at the active IRS is independent of the $N$. Besides,
the secrecy performance gain brought by the increase of $N$ is more
significant than that of $M$, e.i., the transmission power can be greatly
reduced with a small increase in the number of transmitting antennas. This is
due to the fact that the antenna beams become more concentrated in the
targeted direction with more antennas equipped at the UBS, and thus less total
transmission power is required. However, for active IRS, when the channel
uncertainty increases, the secrecy performance gain will also gradually
decrease as $N$ increases due to the more severe noise caused by UAV jittering
will be amplified.

###  IV-B Impact of the Number of RE

Assuming that $\frac{\beta_{\mathrm{U}}}{\bar{\phi}_{\mathrm{U}}}=0.18$ and
$\frac{\beta_{\mathrm{E}}}{\bar{\phi}_{\mathrm{E}}}=0.04$, Fig. 5 compares the
transmission power of the IRS-assisted system in terms of the number of RE. In
this setup, the passive IRS can employ 90 REs at most to reflect the incident
signal. However, even with 90 REs, the transmission power with the passive IRS
is more than that with the active IRS equipped with $M^{*}$ (optimum $M$)
antennas, as the active IRS can directly amplify the incident signal. On the
contrary, the active IRS only takes few REs (15-20 or so) to reach its optimal
secrecy performance under different amplitude gains. With the less REs, the
active IRS is of smaller surface size and more suitable to the load-limited
UAV secure communication. Similar to Fig. 3, for given maximum amplification
gain $\tau_{\text{max}}$, it is observed that with the increase in $M$, the
transmission power for the active IRS first decreases and then increases. When
$M$ is larger (over 30 or 35), the transmission power of the active IRS
solution is higher than that of the passive IRS solution. Because the active
IRS-assisted system can benefit more from the increase in $M$ at first, but
when $M$ over its optimum value, the transmission power will increase, as more
REs reduce the power left for amplification. Besides, the active IRS can
reduce transmission power by increasing not only the number of REs but also
the maximum amplification gain $\tau_{\text{max}}$. For larger $M$, the gain
brought by higher $\tau_{\text{max}}$ will be more limited, as
$\tau_{\text{max}}$ is also constrained with less amplification power. It
alerts us that the tradeoff between the number of RE $M$ and the maximum
amplification gain $\tau_{\text{max}}$ is very important. Undoubtedly, under
the premise of meeting the same ASR, the proposed scheme with smaller surface
size and lower power consumption has wide application value.

###  IV-C Impact of the Maximum Amplification Gain and the Maximum
Amplification Power

To provide a basis for the design of active IRS for power optimization
problems under different ASR requirements. Fig. 6 show the transmission power
w.r.t the different ASR thresholds (Note that, $\eta_{\mathrm{E}}=1$) under
different maximum amplification power $P_{\text{F}}$. Assuming that
$\frac{\beta_{\mathrm{E}}}{\bar{\phi}_{\mathrm{E}}}=0.04$, and
$\frac{\beta_{\mathrm{U}}}{\bar{\phi}_{\mathrm{U}}}=0.06,M=20$. For a given
$\tau_{\text{max}}=30$ dB, it is observed that when the required ASR is small,
the IRS-assisted system with different amplification power budget has almost
the same transmit power, which implies that the amplification power budget
constraint is inactive for the weak transmit power. Therefore, for case with
low security rate requirements ($R_{\text{sec}}<6.5$ bps/Hz), only a small
amplification power $P_{\text{F}}$ needs to be reserved for the IRS. However,
as the required ASR increases, the curves with different $P_{\text{F}}$ are
distinguished from the other, and the increase of the transmission power
becomes dramatic. Because the constraint will be active as the transmission
power increases, resulting in less amplification gain at the active IRS. It
means that sufficient amplification power needs to be provided to meet the
requirement of high ASR. Meanwhile, the active IRS with larger
$\tau_{\text{max}}$ can provide more amplification gain with the same
$P_{\text{F}}$, thereby reducing the transmission power for the lower ASR
requirements. But when ASR gradually increase, the transmission power with
different $\tau_{\text{max}}$ tend to be consistent, because the gradually
increasing transmission power will reduce the actual amplification gain of the
active IRS, thus avoiding the difference of $\tau_{\text{max}}$. Therefore,
when designing the active IRS-assisted system, the IRS has to reconfigure its
maximum amplification gain according to the required ASR.

###  IV-D Impact of UAV’s Height

Fig.7 show the transmission power w.r.t the different UAV altitude under
different maximum amplification power $P_{\text{F}}$. Assuming that
$\frac{\beta_{\mathrm{E}}}{\bar{\phi}_{\mathrm{E}}}=0.04$, and
$\frac{\beta_{\mathrm{U}}}{\bar{\phi}_{\mathrm{U}}}=0.06,M=20$. The passive
IRS solution requires more transmit power when the UAV’s flight altitude
increases due to path loss while the active IRS scheme has almost the same
performance under different rated amplification power constraints, that is,
only a small amplification power is required to greatly reduce the
transmission power required by the UBS. Because the larger the distance
between the UBS and the IRS, the weaker the power of the incident signal, and
then, according to (27), the active IRS can provide more amplification gain to
compensate for the attenuation caused by the double-chain path loss.
Therefore, even for UAVs with high flying altitude, the transmission power at
UBS can also be greatly reduced, which shows active IRS solutions can also
provide better secrecy performance for long-distance UAV transmission.

##  V Conclusion

In this paper, we proposed a novel active IRS assisted robust secure
beamforming designing for UAV communication considering the impact of UAV
jittering, aming to reduce the transmission power at UBS while ensuring ASR.
Specifically, with appropriately designed reflecting coefficient, the active
IRS can create enhanced reflective paths for legitimate users by amplifying
the incident signal, while simultaneously reducing the eavesdropping of user’s
information by illegal eavesdroppers. Firstly, linear approximation applying
by Taylor expansion is used to characterize the impact of the variations of
elevation AOD and azimuth AOD on the channel response. Based on it, a power
minimization problem is formulated with constraints for the effectiveness of
legitimate link and the security requirement of confidential message, where
the precoding matrix and the active IRS’s reflecting coefficient were jointly
optimized. Secondly, S-procedure, Schur’s complement procedure, and slack
variables were adopted to transform the original highly non-convex problem
into a linear convex problem, and the AO algorithm was introduced to
alternately optimize the reflecting coefficient matrix and transmit beamformer
matrix. Simulation results showed that the proposed robust scheme assisted by
active IRS achieves superior performance compared with the passive IRS scheme,
even in the complex situation of UAV’s power limitation, severe jittering, and
high flight altitude. In general, it reveals the potential of active IRS in
secure UAV communication and the impact of the channel uncertainty caused by
UAV jittering.

$\displaystyle\left[\left(\overline{\mathbf{h}}^{H}+\Delta\mathbf{h}^{H}\right)+\mathbf{v}^{(k),H}\left(\overline{\mathbf{G}}+\Delta\mathbf{G}\right)\right]\mathbf{w}^{(k)}\mathbf{w}^{H}\left[\left(\overline{\mathbf{h}}+\Delta\mathbf{h}\right)+\left(\overline{\mathbf{G}}^{H}+\Delta\mathbf{G}^{H}\right)\mathbf{v}\right]$
(51) $\displaystyle=$
$\displaystyle\left(\overline{\mathbf{h}}^{H}+\mathbf{v}^{(k),H}\overline{\mathbf{G}}\right)\mathbf{w}^{(k)}\mathbf{w}^{H}\left(\overline{\mathbf{h}}+\overline{\mathbf{G}}^{H}\mathbf{v}\right)+\left(\overline{\mathbf{h}}^{H}+\mathbf{v}^{(n),H}\overline{\mathbf{G}}\right)\mathbf{w}^{(k)}\mathbf{w}^{H}\left(\Delta\mathbf{h}+\Delta\mathbf{G}^{H}\mathbf{v}\right)$
$\displaystyle+\left(\Delta\mathbf{h}^{H}+\mathbf{v}^{(k),H}\Delta\mathbf{G}\right)\mathbf{w}^{(k)}\mathbf{w}^{H}\left(\overline{\mathbf{h}}+\overline{\mathbf{G}}^{H}\mathbf{v}\right)+\left(\Delta\mathbf{h}^{H}+\mathbf{v}^{(k),H}\Delta\mathbf{G}\right)\mathbf{w}^{(k)}\mathbf{w}^{H}\left(\Delta\mathbf{h}+\Delta\mathbf{G}^{H}\mathbf{v}\right)$
$\displaystyle=$
$\displaystyle\left(\overline{\mathbf{h}}^{H}+\mathbf{v}^{(k),H}\overline{\mathbf{G}}\right)\mathbf{w}^{(k)}\mathbf{w}^{H}\left(\overline{\mathbf{h}}+\widehat{\mathbf{G}}^{H}\mathbf{v}\right)+\left(\overline{\mathbf{h}}^{H}+\mathbf{v}^{(k),H}\overline{\mathbf{G}}\right)\mathbf{w}^{(k)}\mathbf{w}^{H}\Delta\mathbf{h}$
$\displaystyle+\operatorname{vec}^{H}\left(\Delta\mathbf{G}\right)\operatorname{vec}\left(\mathbf{v}\left(\overline{\mathbf{h}}^{H}+\mathbf{v}^{(k),H}\overline{\mathbf{G}}\right)\mathbf{w}^{(k)}\mathbf{w}^{H}\right)+\Delta\mathbf{h}^{H}\mathbf{w}^{(k)}\mathbf{w}^{H}\left(\overline{\mathbf{h}}+\overline{\mathbf{G}}^{H}\mathbf{v}\right)+\Delta\mathbf{h}^{H}\mathbf{w}^{(k)}\mathbf{w}^{H}\Delta\mathbf{h}$
$\displaystyle+\operatorname{vec}^{H}\left(\mathbf{v}^{(k)}\left(\overline{\mathbf{h}}+\mathbf{v}^{H}\overline{\mathbf{G}}^{H}\right)\mathbf{w}\mathbf{w}^{(k),H}\right)\operatorname{vec}\left(\Delta\mathbf{G}\right)+\operatorname{vec}^{H}\left(\Delta\mathbf{G}\right)\left(\mathbf{w}^{*}\mathbf{w}^{(k),T}\otimes\mathbf{v}\right)\Delta\mathbf{h}^{*}$
$\displaystyle+\Delta\mathbf{h}^{T}\left(\mathbf{w}^{*}\mathbf{w}^{(k),T}\otimes\mathbf{v}^{(k),H}\right)\operatorname{vec}\left(\Delta\mathbf{G}\right)+\operatorname{vec}^{H}\left(\Delta\mathbf{G}\right)\left(\mathbf{w}^{*}\mathbf{w}^{(k),T}\otimes\mathbf{v}^{(k),H}\right)\operatorname{vec}\left(\triangle\mathbf{G}\right)$
$\displaystyle=$
$\displaystyle{\mathbf{x}}^{H}\mathbf{C}\mathbf{x}+\mathbf{c}_{1}^{H}\mathbf{x}+\mathbf{x}^{H}\mathbf{c}_{2}+c.$

##  Appendix A Proof of Lemma 1

Let $x$ be a complex scalar variable, we have the first-order Taylor
inequality

$|x|^{2}\geq 2\operatorname{Re}\left\\{x^{*,(k)}x\right\\}-x^{*,(k)}x^{(k)}$
(52)

for any fixed point $x^{(k)}$. By replacing $x$ and $x^{(k)}$ in (52) with
$\left(\mathbf{h}^{H}+\mathbf{v}^{H}\mathbf{G}\right)\mathbf{w}$ and
$\left(\mathbf{h}^{H}+\mathbf{v}^{(k),H}\mathbf{G}\right)\mathbf{w}^{(k)}$,
respectively, we have

$\displaystyle\left|\left(\mathbf{h}^{H}+\mathbf{v}^{H}\mathbf{G}\right)\mathbf{w}\right|^{2}$
$\displaystyle\geq
2\operatorname{Re}\left\\{\left(\mathbf{h}^{H}+\mathbf{v}^{(k),H}\mathbf{G}\right)\mathbf{w}^{(k)}\mathbf{w}^{H}\left(\mathbf{h}+\mathbf{G}^{H}\mathbf{v}\right)\right\\}$
(53)
$\displaystyle-\left(\mathbf{h}^{H}+\mathbf{v}^{(k),H}\mathbf{G}\right)\mathbf{w}^{(k)}\mathbf{w}^{(k),H}\left(\mathbf{h}+\mathbf{G}^{H}\mathbf{v}^{(k)}\right).$

The lower bound of (31) can be derived from (A) under the channel uncertainty.
In particular, we insert $\mathbf{h}=\overline{\mathbf{h}}+\Delta\mathbf{h}$
and $\mathbf{G}=\overline{\mathbf{G}}+\Delta\mathbf{G}$ into the first term on
the right hand side of (A), and then get (LABEL:Taylor_3) at the top of the
next page. With the similar mathematical transformations, the second terms on
the right hand side of (A) under the channel uncertainty can be expressed as

$\displaystyle\left(\mathbf{h}^{H}+\mathbf{v}^{H}\mathbf{G}\right)\mathbf{w}\mathbf{w}^{(k),H}\left(\mathbf{h}+\mathbf{G}^{H}\mathbf{v}^{(k)}\right)$
$\displaystyle=$
$\displaystyle\mathbf{x}^{H}\mathbf{C}^{H}\mathbf{x}+\mathbf{c}_{2}^{H}\mathbf{x}+\mathbf{x}^{H}\mathbf{c}_{1}+c^{*}$
$\displaystyle+\left(\mathbf{h}^{H}+\mathbf{v}^{(k),H}\mathbf{G}\right)\mathbf{w}^{(k)}\mathbf{w}^{(k),H}\left(\mathbf{h}+\mathbf{G}^{H}\mathbf{v}^{(k)}\right)$
$\displaystyle=$
$\displaystyle\mathbf{x}^{H}\mathbf{Z}\mathbf{x}+\mathbf{z}^{H}\mathbf{x}+\mathbf{x}^{H}\mathbf{z}+z.$
(54)

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