# User Cooperation for IRS-aided Secure SWIPT MIMO Systems

Gui Zhou  Cunhua Pan  Hong Ren  Kezhi Wang  Kok Keong Chai  and Kai-Kit Wong
(Corresponding author: Cunhua Pan) G. Zhou, C. Pan, and K.-K. Chai are with
the School of Electronic Engineering and Computer Science at Queen Mary
University of London, London E1 4NS, U.K. (e-mail: g.zhou, c.pan,
michael.chai@qmul.ac.uk). H. Ren is with the National Mobile Communications
Research Laboratory, Southeast University, Nanjing 210096, China.
(hren@seu.edu.cn). K. Wang is with Department of Computer and Information
Sciences, Northumbria University, UK. (e-mail: kezhi.wang@northumbria.ac.uk).
K.-K. Wong is with the Department of Electronic and Electrical Engineering,
University College London, London WC1E 6BT, U.K. (e-mail: kai-
kit.wong@ucl.ac.uk).

###### Abstract

In this paper, intelligent reflecting surface (IRS) is proposed to enhance the
physical layer security in the Rician fading channel where the angular
direction of the eavesdropper is aligned with a legitimate user. In this
scenario, we consider a two-phase communication system under the active
attacks and passive eavesdropping. Particularly, in the first phase, the base
station avoids direct transmission to the attacked user. While, in the second
phase, other users cooperate to forward signals to the attacked user with the
help of IRS and energy harvesting technology. Under the active attacks, we
investigate an outage constrained beamforming design problem under the
statistical cascaded channel error model, which is solved by using the
Bernstein-type inequality. As for the passive eavesdropping, an average
secrecy rate maximization problem is formulated, which is addressed by a low
complexity algorithm. Numerical results show that the negative effect of the
eavesdropper’s channel error is greater than that of the legitimate user.

###### Index Terms:

Intelligent reflecting surface (IRS), reconfigurable intelligent surface
(RIS), robust design, energy harvesting, physical layer security.

##  I Introduction

Communication security is widely regarded as one of the important issues in
wireless communications. Traditionally, security is enforced by imposing
cryptographic protocols in the application layer [1]. However, this upper
layer solution is not flexible as it requires complex key exchange protocols.
Fortunately, it is shown by Wyner [2] that secure communication can be
guaranteed by adopting advanced signal processing techniques developed in the
physical layer. In specific, these techniques exploit differences in channel
conditions and interference environment to enhance the received signal of
legitime users (LUs) and suppress the signal received by the eavesdropper
(ED).

In order to enhance physical layer security, intelligent reflecting surface
(IRS), a kind of passive metasurface, has emerged as a promising technique [3,
4, 5]. By changing the reflection direction of the incident signal, the IRS is
capable of reconfiguring the wireless channels, enhancing the signal from the
base station (BS) to LU and suppressing the radio frequency (RF) power leaked
to the ED [6, 7, 8, 9, 10]. Furthermore, the IRS can be readily coated on
existing buildings, such as the walls and ceilings, which reduces the cost and
complexity of deployment operations. Hence, IRS holds great promise for
security enhancement as it provides a cost-effective and energy-efficient
approach.

In general, ED works in two modes: active attacks and passive eavesdropping
[11, 12]. In an active attack, in order to mislead the BS to send signals to
the ED, the ED pretends to be a LU sending pilot signals to the BS during the
channel estimation procedure. Nonetheless, a passive attack is more
challenging to tackle since the passive ED can hide itself and its channel
state information (CSI) is not available at the BS.

Recently, the benefits of IRS in physical layer security under the active
attacks have been investigated in the existing literature [6, 7, 8, 9, 10].
The performance gains of IRS in terms of security capacity was first explored
in a simple model consisting of only one single-antenna LU and one single-
antenna ED in [6]. Closed-form solutions of the phase shifters of IRS were
obtained by leveraging the majorization-minimization (MM) technique in [6],
which had a better performance than the classical semidefinite relaxation
(SDR) method. The authors of [7] extended the results of [6] to a multiple-
input multiple-output (MIMO) system where artificial noise (AN) was introduced
to enhance the security performance. The results of [8] further showed that
the AN-aided system without an IRS outperforms the IRS-aided system without AN
when the IRS is surrounded by a large number of eavesdroppers. However, all
the above contributions were based on the assumption of perfect CSI of the
eavesdropping channels at the BS. This assumption is too strict and even
impractical. The reasons are twofold: 1) It is challenging to estimate the
IRS-related channels since IRS is passive and can neither send nor receive
pilot signals. 2) The pilot transmission from the ED to the BS may not be
continuous and the corresponding CSI at the BS may be outdated. To deal with
the imperfect CSI of the ED, robust transmission methods for secure
communication of IRS were proposed in [9, 10]. In particular, the authors of
[9] proposed a worst-case robust secure transmission strategy under the
assumption of imperfect CSI from the IRS to the ED. On the other hand, the
authors of [10] considered the more practical imperfect cascaded BS-IRS-ED
channel and proposed an outage constrained beamforming design method under the
statistical CSI error model. However, the imperfect CSI of both LU and ED was
not studied in [10].

To the best of our knowledge, all the existing contributions on the IRS-aided
security enhancement were developed under the active attacks, where the BS can
acquire the CSI of ED. There is no existing work studying the passive
eavesdropping in IRS-aided secure communication systems. In addition, even for
the imperfect CSI under the active attacks, the methods proposed in [9, 10]
are only applicable to small-size IRS (i.e., the number of the reflection
elements is less than 10) which can be observed from the numerical
simulations. The limitations for the research of small-size IRS lie in
twofold. The first is that some interesting observations can be found in the
robust design in an IRS-aided communication system only when the number of
reflection elements is large enough [13]. Secondly, IRS has advantages over
the conventional massive MIMO and relay in terms of energy efficiency only
when the number of IRS reflection elements is large [14].

Against the above background, this paper studies the IRS-aided secrecy
communication under the active attacks and passive eavesdropping. The
contributions of this paper are summarized as follows:

  * •

This paper proposes an IRS-aided two-phase secrecy communication scheme for a
scenario where the ED has a similar channel direction as a LU in order to
asquire high-quality eavesdropping information. In particular, in the
multicasting phase, the BS transmits signals to the LU with low transmission
power to reduce the information leakage to the ED. In the user cooperation
phase, other LUs forward the received signals to the attacked LU with the
assistance of IRS by using the energy harvested in the previous phase. In
addition, two models of ED are considered in this work, i.e. active attack and
passive eavesdropping.

  * •

In the presence of statistical CSI error under the active attack, we develop
an outage constrainted beamforming design that maximizes the secrecy rate
subject to the unit-modulus constraint, the energy harvesting constraint and
the secrecy rate outage probability constraint. Here, the outage probability
constraint guarantees the maximum secrecy rate of the system for secure
communication under a predetermined probability. By resorting to the
Bernstein-Type inequality (BTI) and convex approximations, the non-convexity
of constraints is addressed. Then, the active precoders and the passive
reflection beamforming are updated by using the proposed semidefinite
programming (SDP) and penalty convex-concave procedure (CCP) technique
respectively in an iterative manner.

  * •

For the passive ED case with only partial CSI, we maximize an average secrecy
rate subject to the unit-modulus constraint of the reflection beamforming and
the energy harvesting constraint. To address the numerical integration in the
objective function, an angular secrecy model, which is analytically non-
convex, is proposed. A low-complexity algorithm is proposed based on the MM-
based alternate optimization (AO) framework, where the precoders are updated
by solving a convex optimization problem and the reflection beamforming is
updated in a closed-form solution which is globally optimal.

  * •

The numerical results demonstrate that the level of the cascaded CSI error
plays a vital role in the IRS-aided secure communication systems. In
particular, at low error of cascaded CSI, the secrecy rate increases with the
number of elements at the IRS due to the increased beamforming gain. However,
at large level of cascaded CSI error, the secrecy rate decreases with the
number of elements at the IRS due to the increased channel estimation error.
Hence, whether to enable the IRS for enhancing the security capacity in the
communication systems depends on the level of the cascaded CSI error. In
addition, the IRS can enhance the average secrecy rate under the passive
eavesdropping.

The remainder of this paper is organized as follows. Section II introduces the
channel model and the system model. Outage constrained robust design problem
is formulated for the active eavesdropper model in Section III. Section IV
further investigates the average eavesdropping rate maximization problem under
the passive eavesdropping. Finally, Section V and Section VI show the
numerical results and conclusions, respectively.

Notations: The following mathematical notations and symbols are used
throughout this paper. Vectors and matrices are denoted by boldface lowercase
letters and boldface uppercase letters, respectively. The symbols
$\mathbf{X}^{*}$, $\mathbf{X}^{\mathrm{T}}$, $\mathbf{X}^{\mathrm{H}}$, and
$||\mathbf{X}||_{F}$ denote the conjugate, transpose, Hermitian (conjugate
transpose), Frobenius norm of matrix $\mathbf{X}$, respectively. The symbol
$||\mathbf{x}||_{2}$ denotes 2-norm of vector $\mathbf{x}$. The symbols
$\mathrm{Tr}\\{\cdot\\}$, $\mathrm{Re}\\{\cdot\\}$, $|\cdot|$,
$\lambda(\cdot)$, and $\angle\left(\cdot\right)$ denote the trace, real part,
modulus, eigenvalue, and angle of a complex number, respectively.
$\mathrm{diag}(\mathbf{x})$ is a diagonal matrix with the entries of
$\mathbf{x}$ on its main diagonal. $[\mathbf{x}]_{m}$ means the
$m^{\mathrm{th}}$ element of the vector $\mathbf{x}$. The Kronecker product
and the Hadamard product between two matrices $\mathbf{X}$ and $\mathbf{Y}$ is
denoted by $\mathbf{X}\otimes\mathbf{Y}$ and $\mathbf{X}\odot\mathbf{Y}$,
respectively. $\mathbf{X}\succeq\mathbf{Y}$means that $\mathbf{X}-\mathbf{Y}$
is positive semidefinite. Additionally, the symbol $\mathbb{C}$ denotes
complex field, $\mathbb{R}$ represents real field, and $j\triangleq\sqrt{-1}$
is the imaginary unit.

##  II System Model

Figure 1: Two-phase communication system

As shown in Fig. 1, we consider Rician wiretap channels where a BS with $N$
transmit antennas communicates with $K$ single-antenna LUs in the presence of
a single-antenna ED. An IRS with $M$ reflection elements is introduced to aid
the secure communication.

###  II-A Channel Model

Figure 2: Coordinates of communication nodes in the system

Define the set of all LUs as $\mathcal{K}=\\{1,2,...,K\\}$, and denote set
$\mathcal{K}_{-K}=\mathcal{K}/\\{K\\}$ and set
$\mathcal{K}_{+E}=\mathcal{K}\cup\\{E\\}$. By denoting
$\\{D_{i},\theta_{i}\\}_{\forall i\in\mathcal{\mathcal{K}}_{+E}}$ as the
distances and the azimuth angles respectively from the BS to the LUs and the
ED, as shown in Fig. 1, then the corresponding channels
$\\{\mathbf{g}_{i}\in\mathbb{C}^{N\times 1}\\}_{\forall
i\in\mathcal{\mathcal{K}}_{+E}}$ follow the Rician fading distribution [15]:

$\displaystyle\mathbf{g}_{i}=\sqrt{\varrho_{0}\left(\frac{D_{i}}{d_{0}}\right)^{-\alpha_{\mathrm{BS}}}}\Biggl{(}\sqrt{\frac{K_{\mathrm{BS}}}{1+K_{\mathrm{BS}}}}\mathbf{g}_{i}^{\mathrm{LOS}}+\sqrt{\frac{1}{1+K_{\mathrm{BS}}}}\mathbf{g}_{i}^{\mathrm{NLOS}}\Biggr{)},\forall
i\in\mathcal{K}_{+E},$ (1)

where $\varrho_{0}$ is the pathloss at the reference distance of $d_{0}$,
$\alpha_{BS}$ and $K_{BS}$ are the pathloss exponent and the Rician factor of
the BS-related links, respectively. It is assumed that the BS is equipped with
a uniform linear array (ULA). Then, the line-of-sight (LoS) component is given
by
$\mathbf{g}_{i}^{\mathrm{LOS}}=\left[1,e^{-\mathrm{j}\pi\sin\theta_{i}},\cdots,e^{-\mathrm{j}(N-1)\pi\sin\theta_{i}}\right]$,
and the non-LoS component is drawn from a Rayleigh fading, i.e.,
$\mathbf{g}_{i}^{\mathrm{NLOS}}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{N})$.

Furthermore, by denoting $\\{D_{\mathrm{IRS}},\theta_{\mathrm{IRS}}\\}$ as the
distance and the azimuth angle from the BS to the IRS, it is straightforward
to obtain the distances $\\{d_{\mathrm{IRS},i}\\}_{\forall
i\in\mathcal{\mathcal{K}}_{+E}}$ and the azimuth angles
$\\{\varphi_{i}\\}_{\forall i\in\mathcal{\mathcal{K}}_{+E}}$ from the IRS to
the LUs and the ED as shown in Fig. 2, i.e.,

$\displaystyle d_{\mathrm{IRS},i}$
$\displaystyle=\Bigl{[}(D_{\mathrm{IRS}}\cos\theta_{\mathrm{IRS}}-D_{i}\cos\theta_{i})^{2}+(D_{\mathrm{IRS}}\sin\theta_{\mathrm{IRS}}-D_{i}\sin\theta_{i})^{2}\Bigr{]}^{-1/2},$
$\displaystyle\sin\varphi_{i}$
$\displaystyle=\frac{1}{d_{\mathrm{IRS},i}}(D_{i}\sin\theta_{i}-D_{\mathrm{IRS}}\sin\theta_{\mathrm{IRS}}),$
$\displaystyle\cos\varphi_{i}$
$\displaystyle=\frac{1}{d_{\mathrm{IRS},i}}(D_{\mathrm{IRS}}\cos\theta_{\mathrm{IRS}}-D_{i}\cos\theta_{i}).$

The corresponding channels $\\{\mathbf{h}_{i}\in\mathbb{C}^{N\times
1}\\}_{\forall i\in\mathcal{\mathcal{K}}_{+E}}$ are given by

$\displaystyle\mathbf{h}_{i}$
$\displaystyle=\sqrt{\varrho_{0}\left(\frac{d_{\mathrm{IRS},i}}{d_{0}}\right)^{-\alpha_{\mathrm{IRS}}}}\Biggl{(}\sqrt{\frac{K_{\mathrm{IRS}}}{1+K_{\mathrm{IRS}}}}\mathbf{h}_{i}^{\mathrm{LOS}}+\sqrt{\frac{1}{1+K_{\mathrm{IRS}}}}\mathbf{h}_{i}^{\mathrm{NLOS}}\Biggr{)},\forall
i\in\mathcal{K}_{+E},$ (2)

where $\alpha_{\mathrm{IRS}}$ and $K_{\mathrm{IRS}}$ are the pathloss exponent
and the Rician factor of the IRS-related links, respectively. The non-LoS
component follows the distribution of
$\mathbf{h}_{i}^{\mathrm{NLOS}}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{M})$.
It is assumed that the IRS is an uniform plane array (UPA) with size of
$M=M_{{\rm x}}M_{{\rm y}}$, where $M_{{\rm x}}$ and $M_{{\rm y}}$ are the
number of reflection elements in x-axis and y-axis, respectively. Then, the
LoS component is written as

$\displaystyle\mathbf{h}_{i}^{\mathrm{LOS}}=[1,\cdots,e^{-\mathrm{j}\pi(x\cos\varphi_{i}\cos\phi+y\sin\varphi_{i}\cos\phi)\sin\theta_{i}},\cdots,e^{-\mathrm{j}\pi((M_{{\rm
x}}-1)\cos\varphi_{i}\cos\phi+(M_{y}-1)\sin\varphi_{i}\cos\phi)\sin\theta_{i}}],$

where $1\leq x\leq M_{{\rm x}}$, $1\leq y\leq M_{{\rm y}}$ and $\phi$ is the
elevation angle observed at the IRS side.

###  II-B Signal Transmission

To achieve high success rate of attack, the ED can locate the line between the
BS and legitimate user. In this situation, the signal received by the ED is
highly correlated with that of this user [11, 12], thus posing great threats
to the system. As shown in Fig. 1, we assume that the ED hides at the line
connecting the BS and one of the users, denoted as user $K$, which leads to
$\theta_{E}\approx\theta_{K}$,
$\mathbf{g}_{E}^{\mathrm{LOS}}\approx\mathbf{g}_{K}^{\mathrm{LOS}}$ and
$D_{E}\in(0,D_{K})$. When the Rician factor $K_{\mathrm{BS}}$ is sufficiently
large, the channel $\mathbf{g}_{E}$ is approximately equal to the channel
$\mathbf{g}_{K}$.

In order to achieve high-quality secure communication, the angle aware user
cooperation (AAUC) scheme [15] is adopted here. In particular, in the first
phase, the BS multicasts the common signal to all users except user $K$. In
the second phase, the helping users $(\forall k\in\mathcal{K}_{-K})$ forward
the decoded common signal to user $K$ via the IRS. In this work, in order to
implement the AAUC scheme without consuming extra energy, the LUs adopt the
hybrid information and energy harvesting receiving mode which splits the
received signal into two power streams with power splitting ratios $t_{k}$ and
$1-t_{k}$. The former is used for decoding the signal and the latter is for
energy harvesting.

####  II-B1 Multicasting Phase

In this phase, the BS multicasts the signal $s$ to the helping LUs through
beamforming vector ${\bf f}\in\mathbb{C}^{N\times 1}$ which is limited to the
maximum transmit power $P_{\max}$, i.e., $||{\bf f}||_{2}^{2}\leq P_{\max}$.
Since $\mathbf{g}_{E}\approx\mathbf{g}_{K}$, the beamforming ${\bf f}$ needs
to satisfy $|\mathbf{g}_{K}^{\mathrm{H}}{\bf f}|=0$ to ensure that
$|\mathbf{g}_{E}^{\mathrm{H}}{\bf f}|\approx 0$. Let
$\mathbf{Q}\in\mathbb{C}^{N\times\text{(}N-1)}$ be the orthogonal matrix which
spans the null space of $\mathbf{g}_{K}$ by using the QR decomposition, i.e.,
$\mathbf{Q}^{\mathrm{H}}\mathbf{Q}={\bf I}$. Then, we can design ${\bf
f}=\mathbf{Q}\mathbf{z}$, where $\mathbf{z}\in\mathbb{C}^{\text{(}N-1)\times
1}$ is a newly introduced variable. Therefore, the signal received by LU $k$
is given by $\mathbf{g}_{k}^{\mathrm{H}}\mathbf{Q}\mathbf{z}+n_{k}$, where
$n_{k}$ is the received noise with the noise power of $\sigma_{k}^{2}$. By
adopting the hybrid receiving mode and let
${\bf{t}}=[t_{1},...,t_{K-1}]^{{\rm{T}}}$ where $t_{k}$ is the power splitting
ratio of LU $k$, the achievable rate at LU $k\neq K$ is

$R_{k}\left({\bf
z},t_{k}\right)=\frac{1}{2}\log_{2}\left(1+\frac{t_{k}}{\sigma_{k}^{2}}\left|\mathbf{g}_{k}^{\mathrm{H}}\mathbf{Q}\mathbf{z}\right|^{2}\right),$
(3)

where the factor 1/2 is due to the two transmission phases. The harvested
power at LU $k\neq K$ is

$(1-t_{k})\left|\mathbf{g}_{k}^{\mathrm{H}}\mathbf{Q}\mathbf{z}\right|^{2}.$
(4)

####  II-B2 User Cooperation Phase

In this phase, the helping LUs $(\forall k\in\mathcal{K}_{-K})$ forward the
signal $s$ to LU $K$ through a beamforming vector ${\bf
w}\in\mathbb{C}^{(K-1)\times 1}=[w_{1},...,w_{K-1}]^{\mathrm{T}}$ by using the
power harvested in the multicasting phase. Since LU $K$ is randomly selected
by the ED and assume that there are many obstacles in the communication
environment, such as indoor applications, the direct links between the helping
LUs and the LU $K$ may be blocked. To address this issue, an IRS can be
installed on the building with a certain height, and thus the IRS is capable
of reflecting the signals forwarded by the helping LUs to LU $K$. Denote by
$\mathbf{e}$ the reflection coefficient vector of the IRS, where
$|e_{m}|^{2}=1,\forall m=1,\cdots,M$. Then, the signal received by LU $K$ is
given by

$\displaystyle y_{K}$
$\displaystyle=\mathbf{h}_{K}^{\mathrm{H}}\mathrm{diag}(\mathbf{e}^{*})\mathbf{H}_{\mathrm{IRS}}\mathbf{w}s+\sigma_{K}^{2}$
$\displaystyle=\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w}s+\sigma_{K}^{2},$

where
$\mathbf{H}_{\mathrm{IRS}}=\left[\mathbf{h}_{1}\text{,\thinspace\thinspace...,\thinspace\thinspace$\mathbf{h}_{K-1}$}\right]$,
$\mathbf{H}_{K}=\left[\mathbf{h}_{K}^{*}\odot\mathbf{h}_{1}\text{,\thinspace\thinspace...,\thinspace\thinspace$\mathbf{h}_{K}^{*}\odot\mathbf{h}_{K-1}$}\right]$
is the cascaded LU-IRS-LU (LIL) channel, and
$n_{K}\sim\mathcal{CN}(\mathbf{0},\sigma_{K}^{2})$ is the noise. The
corresponding achievable rate is

$R_{K}\left(\mathbf{w},\mathbf{e}\right)=\frac{1}{2}\log_{2}\left(1+\frac{1}{\sigma_{K}^{2}}\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w}\right|^{2}\right).$
(5)

On the other hand, the signal recieved by the ED is
$y_{E}=\mathbf{e}^{\mathrm{H}}\mathbf{H}_{E}\mathbf{w}s+n_{E}$, where
$\mathbf{H}_{E}=\left[\mathbf{h}_{E}^{*}\odot\mathbf{h}_{1}\text{,\thinspace\thinspace...,\thinspace\thinspace$\mathbf{h}_{E}^{*}\odot\mathbf{h}_{K-1}$}\right]$
is the cascaded LU-IRS-ED (LIE) channel, and
$n_{E}\sim\mathcal{CN}(\mathbf{0},\sigma_{E}^{2})$ is the received noise at
the ED.

The corresponding eavesdropping rate is

$R_{E}\left(\mathbf{w},\mathbf{e}\right)=\frac{1}{2}\log_{2}\left(1+\frac{1}{\sigma_{E}^{2}}\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{E}\mathbf{w}\right|^{2}\right).$
(6)

Finally, the secrecy rate of this system under the AAUC scheme can be
expressed as [12]:

$\left[\min_{\forall k\in\mathcal{K}}R_{k}-R_{E}\right]^{+}.$ (7)

In the following two sections, we consider the system design for two ED
models: the active eavesdropper model and the passive eavesdropper model.

##  III ED Model I: Active Eavesdropper Model

In this section, we consider the active attack case, in which the ED pretends
to be an LU sending pilot signals to the transmitters (including the BS and
the helping LUs) during the channel estimation procedure [11, 12]. It is
reasonable to assume that the BS is capable of addressing this attack by using
the multi-antenna technique, so as to obtain perfect CSI of the system.
Nevertheless, the signle-antenna helping LUs only have the imperfect CSI of LU
$K$ and the ED due to their limited anti-interference ability.

###  III-A Channel Uncertainties

Based on the above assumption, the cascaded channels can be modeled as

$\displaystyle\mathbf{H}_{K}$
$\displaystyle=\widehat{\mathbf{H}}_{K}+\bm{\bigtriangleup}_{K},\thinspace\thinspace\mathbf{H}_{E}=\widehat{\mathbf{H}}_{E}+\bm{\bigtriangleup}_{E},$
(8)

where $\widehat{\mathbf{H}}_{K}$ and $\widehat{\mathbf{H}}_{E}$ are the
estimated cascaded channels,
$\bm{\bigtriangleup}_{K}=[\bm{\bigtriangleup}_{1}^{K}\cdots\bm{\bigtriangleup}_{K-1}^{K}]$
and
$\bm{\bigtriangleup}_{E}=[\bm{\bigtriangleup}_{1}^{E}\cdots\bm{\bigtriangleup}_{K-1}^{E}]$
are the unknown cascaded channel errors. $\bm{\bigtriangleup}_{k}^{K}$ and
$\bm{\bigtriangleup}_{k}^{E}$ are the unknown cascaded LIL and LIE channel
error vectors at LU $k$, respectively.

According to [13], the robust beamforming under the statistical CSI error
model outperforms the bounded CSI error model in terms of the minimum transmit
power, convergence speed and computational complexity. In addition, the
statistical channel error model is more suitable to model the channel
estimation error when the channel estimation is based on the minimum mean sum
error (MMSE) method. Hence, we adopt the statistical model to characterize the
cascadepd CSI imperfection [13], i.e., each CSI error vector is assumed to
follow the circularly symmetric complex Gaussian (CSCG) distribution, i.e.,

$\displaystyle\bm{\bigtriangleup}_{k}^{K}$
$\displaystyle\sim\mathcal{CN}(\mathbf{0},\bm{\Sigma}_{k}^{K}),\bm{\Sigma}_{k}^{K}\succeq\mathbf{0},\forall
k\in\mathcal{K}_{-K},$ (9a) $\displaystyle\bm{\bigtriangleup}_{k}^{E}$
$\displaystyle\sim\mathcal{CN}(\mathbf{0},\bm{\Sigma}_{k}^{E}),\bm{\Sigma}_{k}^{E}\succeq\mathbf{0},\forall
k\in\mathcal{K}_{-K},$ (9b)

where $\bm{\Sigma}_{k}^{K}\in\mathbb{C}^{M\times M}$ and
$\bm{\Sigma}_{k}^{E}\in\mathbb{C}^{M\times M}$ are positive semidefinite error
covariance matrices. Note that the CSI error vectors of different LUs are
independent with each other. Therefore, we have

$\mathrm{vec}(\bm{\bigtriangleup}_{K})\sim\mathcal{CN}(\mathbf{0},\bm{\Sigma}_{K}),\thinspace\thinspace\mathrm{vec}(\bm{\bigtriangleup}_{E})\sim\mathcal{CN}(\mathbf{0},\bm{\Sigma}_{E}),$
(10)

where $\bm{\Sigma}_{K}$ and $\bm{\Sigma}_{E}$ are block diagonal matrices,
i.e.,
$\bm{\Sigma}_{K}=\mathrm{diag}(\bm{\Sigma}_{1}^{K},...,\bm{\Sigma}_{K-1}^{K})$
and
$\bm{\Sigma}_{E}=\mathrm{diag}(\bm{\Sigma}_{1}^{E},...,\bm{\Sigma}_{K-1}^{E})$.

###  III-B Outage constrained beamforming design

Under the statistical CSI error model, we develop a probabilistically robust
algorithm for the secrecy rate maximization problem, which is formulated as

$\displaystyle\max\limits_{R_{\mathrm{sec}},\mathbf{z},\mathbf{w},\mathbf{e},\mathbf{t}}$
$\displaystyle R_{\mathrm{sec}}$ (11a) s.t.
$\displaystyle\mathrm{Pr}\left\\{\min_{\forall k\in\mathcal{K}}R_{k}-R_{E}\geq
R_{\mathrm{sec}}\right\\}\geq 1-\rho$ (11b) $\displaystyle||{\bf
z}||_{2}^{2}\leq P_{\max}$ (11c) $\displaystyle\;|e_{m}|^{2}=1,1\leq m\leq M$
(11d) $\displaystyle 0\leq\mathbf{t}\leq 1$ (11e)
$\displaystyle|w_{k}|^{2}\leq(1-t_{k})\left|\mathbf{g}_{k}^{\mathrm{H}}\mathbf{Q}\mathbf{z}\right|^{2},\forall
k\in\mathcal{K}_{-K},$ (11f)

where $\rho\in(0,1]$ is the secrecy rate outage probability.

Problem (11) is difficult to solve due to the computationally intractable rate
outage probability constraint (11b), the non-convex unit-modulus constraint
(11d), and the non-convex power constraint (11f).

Firstly, we replace constraint (11b) with the development of a safe
approximation consisting of three steps in the following.

Step 1: Decouple the Probabilistic Constraint: First of all, based on the
independence between $\\{\mathbf{g}_{k}\\}_{\forall k\in\mathcal{K}_{-K}}$ and
$\mathbf{H}_{K}$, we have

$\displaystyle(\ref{eq:r-c})\Leftrightarrow$
$\displaystyle\prod_{k=1}^{K}\mathrm{Pr}\left\\{R_{k}-R_{E}\geq
R_{\mathrm{sec}}\right\\}\geq 1-\rho$ (12) $\displaystyle\Leftarrow$
$\displaystyle\mathrm{Pr}\left\\{R_{k}-R_{E}\geq R_{\mathrm{sec}}\right\\}\geq
1-\bar{\rho},\forall k\in\mathcal{K}_{K},$ (13)

where $\bar{\rho}=1-(1-\rho)^{1/K}$.

Step 2: Convenient Approximations: To address the non-concavity of
$R_{k}-R_{E},\forall k\in\mathcal{K}_{K}$, we need to construct a sequence of
surrogate functions of $\\{R_{i}\\}_{\forall i\in\mathcal{K}_{+E}}$. More
specifically, we need the following lemmas.

###### Lemma 1

[16] The quadratic-over-linear function $\frac{x^{2}}{y}$ is jointly convex in
$(x,y)$, and lower bounded by its linear first-order Tayler approximation
$\frac{2x^{(n)}}{y^{(n)}}x-(\frac{x^{(n)}}{y^{(n)}})^{2}y$ at fixed point
$(x^{(n)},y^{(n)})$.

By substituting $x$ with $\mathbf{g}_{k}^{\mathrm{H}}\mathbf{Q}\mathbf{z}$ and
$y$ with $1/t_{k}$, we utilize Lemma 1 to obtain a concave lower bound of rate
$R_{k}\left(\mathbf{z},t_{k}\right)$ for $\forall k\in\mathcal{K}_{-K}$. The
lower bound is given by

$\displaystyle\widetilde{R}_{k}(\mathbf{z},t_{k}|\mathbf{z}^{(n)},t_{k}^{(n)})=\frac{1}{2}\log_{2}\Biggl{(}1-\frac{t_{k}^{(n),2}}{\sigma_{k}^{2}t_{k}}\left|\mathbf{g}_{k}^{\mathrm{H}}\mathbf{Q}\mathbf{z}^{(n)}\right|^{2}+2t_{k}^{(n)}\mathrm{Re}\left\\{\frac{1}{\sigma_{k}^{2}}{\bf
z}^{(n),\mathrm{H}}\mathbf{Q}^{\mathrm{H}}\mathbf{g}_{k}\mathbf{g}_{k}^{\mathrm{H}}\mathbf{Q}\mathbf{z}\right\\}\Biggr{)}$
(14)

for any feasible solution $\\{\mathbf{z}^{(n)},t_{k}^{(n)}\\}$.

###### Lemma 2

The upper bound of rate $R_{E}\left(\mathbf{w},\mathbf{e}\right)$ is given by

$\displaystyle\widetilde{R}_{E}\left(\mathbf{w},\mathbf{e},a_{E}\right)$
$\displaystyle=\frac{a_{E}\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{E}\mathbf{w}\right|^{2}/\sigma_{E}^{2}+a_{E}-\ln
a_{E}-1}{2\ln 2},$

where $a_{E}$ is the auxiliary variable.

Proof: See Appendix A. $\blacksquare$

###### Lemma 3

The lower bound of rate $R_{K}\left(\mathbf{w},\mathbf{e}\right)$ is given by

$\displaystyle\widetilde{R}_{K}(\mathbf{w},\mathbf{e},a_{K},v)=\frac{1}{2\ln
2}(-a_{K}|v|^{2}|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w}|^{2}-\sigma_{K}^{2}a_{K}|v|^{2}+2a_{K}\text{Re}\left\\{v\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w}\right\\}-a_{K}+\ln
a_{K}+1),$

where $a_{K}$ and $v$ are the auxiliary variables.

Proof: See Appendix B. $\blacksquare$

For the convenience of derivations, we assume that
$\bm{\Sigma}_{k}^{K}=\varepsilon_{K,k}^{2}\mathbf{I}_{M}$ and
$\bm{\Sigma}_{k}^{E}=\varepsilon_{E,k}^{2}\mathbf{I}_{M}$, then
$\bm{\Sigma}_{K}=\bm{\Lambda}_{K}\otimes\mathbf{I}_{M}$ where
$\bm{\Lambda}_{K}=\mathrm{diag}(\varepsilon_{K,1}^{2},...,\varepsilon_{K,K-1}^{2})$,
and $\bm{\Sigma}_{E}=\bm{\Lambda}_{E}\otimes\mathbf{I}_{M}$ where
$\bm{\Lambda}_{E}=\mathrm{diag}(\varepsilon_{E,1}^{2},...,\varepsilon_{E,K-1}^{2})$.
Furthermore, the error vectors in (10) can be rewritten as
$\mathrm{vec}(\bm{\bigtriangleup}_{K})=\bm{\Sigma}_{K}^{\frac{1}{2}}\mathbf{i}_{K}$
where $\mathbf{i}_{K}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{M(K-1)})$, and
$\mathrm{vec}(\bm{\bigtriangleup}_{E})=\bm{\Sigma}_{E}^{\frac{1}{2}}\mathbf{i}_{E}$
where $\mathbf{i}_{E}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{M(K-1)})$. Define
$\mathbf{E}=\mathbf{e}\mathbf{e}^{\mathrm{H}}$ and
$\mathbf{W}=\mathbf{w}\mathbf{w}^{\mathrm{H}}$. Combining (14) with Lemma 2,
the secrecy rate outage probabilities for $\forall k\in\mathcal{K}_{-K}$ in
(13) are equivalent to

$\displaystyle\mathrm{Pr}\left\\{R_{k}-R_{E}\geq R_{\mathrm{sec}}\right\\}$
$\displaystyle\geq$
$\displaystyle\mathrm{Pr}\left\\{\widetilde{R}_{k}-\widetilde{R}_{E}\geq
R_{\mathrm{sec}}\right\\}$ $\displaystyle=$
$\displaystyle\mathrm{Pr}\Bigl{\\{}a_{E}\mathrm{Tr}\left(\mathbf{E}(\widehat{\mathbf{H}}_{E}+\bm{\bigtriangleup}_{E})\mathbf{W}(\widehat{\mathbf{H}}_{E}^{\mathrm{H}}+\bm{\bigtriangleup}_{E}^{\mathrm{H}})\right)-[2\ln
2(\widetilde{R}_{k}-R_{\mathrm{sec}})-a_{E}+\ln a_{E}+1]\sigma_{E}^{2}\leq
0\Bigr{\\}}$ $\displaystyle=$
$\displaystyle\mathrm{Pr}\left\\{\mathbf{i}_{E}^{\mathrm{H}}\mathbf{U}_{E}\mathbf{i}_{E}+2\mathrm{Re}\left\\{\mathbf{u}_{E}^{\mathrm{H}}\mathbf{i}_{E}\right\\}+u_{k}\leq
0\right\\},$ (15)

where

$\displaystyle\mathbf{U}_{E}$
$\displaystyle=a_{E}\bm{\Sigma}_{E}^{\frac{1}{2}}(\mathbf{W}^{\mathrm{T}}\otimes\mathbf{E})\bm{\Sigma}_{E}^{\frac{1}{2}},$
(16a) $\displaystyle\mathbf{u}_{E}$
$\displaystyle=a_{E}\bm{\Sigma}_{E}^{\frac{1}{2}}\mathrm{vec}(\mathbf{E}\mathbf{\widehat{H}}_{E}\mathbf{W}),$
(16b) $\displaystyle u_{k}$
$\displaystyle=a_{E}\mathrm{Tr}\left(\mathbf{E}\mathbf{\widehat{H}}_{E}\mathbf{W}\widehat{\mathbf{H}}_{E}^{\mathrm{H}}\right)-[(\widetilde{R}_{k}-R_{\mathrm{sec}})2\ln
2-a_{E}+\ln a_{E}+1]\sigma_{E}^{2}.$ (16c)

Combining Lemma 2 with Lemma 3, the secrecy rate outage probability for LU $K$
in (13) is equivalent to

$\displaystyle\mathrm{Pr}\left\\{R_{K}-R_{E}\geq R_{\mathrm{sec}}\right\\}$
$\displaystyle\geq$
$\displaystyle\mathrm{Pr}\left\\{\widetilde{R}_{K}-\widetilde{R}_{E}\geq
R_{\mathrm{sec}}\right\\}$ $\displaystyle=$
$\displaystyle\mathrm{Pr}\Bigl{\\{}a_{K}|v|^{2}\mathrm{Tr}\left(\mathbf{E}(\widehat{\mathbf{H}}_{K}+\bm{\bigtriangleup}_{K})\mathbf{W}(\widehat{\mathbf{H}}_{K}^{\mathrm{H}}+\bm{\bigtriangleup}_{K}^{\mathrm{H}})\right)-2a_{K}\mathrm{Re}\left\\{v\mathbf{e}^{\mathrm{H}}(\widehat{\mathbf{H}}_{K}+\bm{\bigtriangleup}_{K})\mathbf{w}\right\\}+$
$\displaystyle\frac{a_{E}}{\sigma_{E}^{2}}\mathrm{Tr}\left(\mathbf{E}(\widehat{\mathbf{H}}_{E}+\bm{\bigtriangleup}_{E})\mathbf{W}(\widehat{\mathbf{H}}_{E}^{\mathrm{H}}+\bm{\bigtriangleup}_{E}^{\mathrm{H}})\right)-c\leq
0\Bigr{\\}}$ $\displaystyle=$
$\displaystyle\mathrm{Pr}\left\\{\mathbf{i}^{\mathrm{H}}\mathbf{U}_{K}\mathbf{i}+2\mathrm{Re}\left\\{\mathbf{u}_{K}^{\mathrm{H}}\mathbf{i}\right\\}+u_{K}\leq
0\right\\},$ (17)

where

$\displaystyle\mathbf{i}$
$\displaystyle=[\begin{array}[]{cc}\mathbf{i}_{K}^{\mathrm{H}},&\mathbf{i}_{E}^{\mathrm{H}}\end{array}]^{\mathrm{H}},$
(18b) $\displaystyle\mathbf{U}_{K}$
$\displaystyle=\mathrm{diag}\Bigl{\\{}a_{K}|v|^{2}\bm{\Sigma}_{K}^{\frac{1}{2}}(\mathbf{W}^{\mathrm{T}}\otimes\mathbf{E})\bm{\Sigma}_{K}^{\frac{1}{2}},\
\frac{a_{E}}{\sigma_{E}^{2}}\bm{\Sigma}_{E}^{\frac{1}{2}}(\mathbf{W}^{\mathrm{T}}\otimes\mathbf{E})\bm{\Sigma}_{E}^{\frac{1}{2}}\Bigr{\\}},$
(18c) $\displaystyle\mathbf{u}_{K}$
$\displaystyle=[a_{K}|v|^{2}\mathrm{vec}^{\mathrm{H}}(\mathbf{E}\mathbf{\widehat{H}}_{K}\mathbf{W})\bm{\Sigma}_{K}^{\frac{1}{2}}-a_{K}v\mathrm{vec}^{\mathrm{H}}(\mathbf{e}\mathbf{w}^{\mathrm{H}})\bm{\Sigma}_{K}^{\frac{1}{2}},\
\frac{a_{E}}{\sigma_{E}^{2}}\mathrm{vec}^{\mathrm{H}}(\mathbf{E}\mathbf{\widehat{H}}_{E}\mathbf{W})\bm{\Sigma}_{E}^{\frac{1}{2}}]^{\mathrm{H}},$
(18d) $\displaystyle u_{K}$
$\displaystyle=a_{K}|v|^{2}\mathrm{Tr}\left(\mathbf{E}\mathbf{\widehat{H}}_{K}\mathbf{W}\widehat{\mathbf{H}}_{K}^{\mathrm{H}}\right)+\frac{a_{E}}{\sigma_{E}^{2}}\mathrm{Tr}\left(\mathbf{E}\mathbf{\widehat{H}}_{E}\mathbf{W}\widehat{\mathbf{H}}_{E}^{\mathrm{H}}\right)\
-2a_{K}\mathrm{Re}\left\\{v\mathbf{e}^{\mathrm{H}}\mathbf{\widehat{H}}_{K}\mathbf{w}\right\\}-c,$
(18e) $\displaystyle c$ $\displaystyle=\ln a_{E}+\ln
a_{K}-a_{E}-a_{K}-2R_{\mathrm{sec}}\ln 2\ -\sigma_{K}^{2}a_{K}|v|^{2}+2.$
(18f)

Now, substituting (15) and (17) into (13), then (13) is replaced by

$\displaystyle\mathrm{Pr}\left\\{\mathbf{i}_{E}^{\mathrm{H}}\mathbf{U}_{E}\mathbf{i}_{E}+2\mathrm{Re}\left\\{\mathbf{u}_{E}^{\mathrm{H}}\mathbf{i}_{E}\right\\}+u_{k}\leq
0\right\\}\geq 1-\bar{\rho},\forall k\in\mathcal{K}_{-K},$ (19a)
$\displaystyle\mathrm{Pr}\left\\{\mathbf{i}^{\mathrm{H}}\mathbf{U}_{K}\mathbf{i}+2\mathrm{Re}\left\\{\mathbf{u}_{K}^{\mathrm{H}}\mathbf{i}\right\\}+u_{K}\leq
0\right\\}\geq 1-\bar{\rho}.$ (19b)

Step 3: A Bernstein-Type Inequality-Based Safe Approximation: The outage
probabilities in (19) are characterized by quadratic inequalities, which can
be safely approximated by using the following lemma.

###### Lemma 4

(Bernstein-Type Inequality) [17] Assume
$f(\mathbf{x})=\mathbf{x}^{\mathrm{H}}\mathbf{U}\mathbf{x}+2\mathrm{Re}\\{\mathbf{u}^{\mathrm{H}}\mathbf{x}\\}+u$,
where $\mathbf{U}\in\mathbb{H}^{n\times n}$, $\mathbf{u}\in\mathbb{C}^{n\times
1}$, $u\in\mathbb{R}$ and $\mathbf{x}\in\mathbb{C}^{n\times
1}\sim\mathcal{CN}(\mathbf{0},\mathbf{I})$. Then for any $\rho\in[0,1]$, the
following approximation holds:

$\displaystyle\mathrm{Pr}\\{\mathbf{x}^{\mathrm{H}}\mathbf{U}\mathbf{x}+2\mathrm{Re}\\{\mathbf{u}^{\mathrm{H}}\mathbf{x}\\}+u\leq
0\\}\geq 1-\rho$ $\displaystyle\Rightarrow$
$\displaystyle\mathrm{Tr}\left\\{\mathbf{U}\right\\}+\sqrt{2\ln(1/\rho)}x-\ln(\rho)\lambda_{\max}^{+}(\mathbf{U})+u\leq
0$ $\displaystyle\Rightarrow$
$\displaystyle\left\\{\begin{array}[]{c}\mathrm{Tr}\left\\{\mathbf{U}\right\\}+\sqrt{2\ln(1/\rho)}x-\ln(\rho)y+u\leq
0\\\ \sqrt{||\mathbf{U}||_{F}^{2}+2||\mathbf{u}||_{2}^{2}}\leq x\\\
y\mathbf{I}-\mathbf{U}\succeq\mathbf{0},y\geq 0,\end{array}\right.$ (23)

where $\lambda_{\max}^{+}(\mathbf{U})=\max(\lambda_{\max}(\mathbf{U}),0)$. $x$
and $y$ are slack variables. $\blacksquare$

Before using Lemma 4, we need the following simplified derivations for LU $k$,
$\forall k\in\mathcal{K}_{-K}$, i.e.,

$\displaystyle\mathrm{Tr}\left\\{\mathbf{U}_{E}\right\\}=\mathrm{Tr}\left\\{a_{E}\bm{\Sigma}_{E}^{\frac{1}{2}}(\mathbf{W}^{\mathrm{T}}\otimes\mathbf{E})\bm{\Sigma}_{E}^{\frac{1}{2}}\right\\}$
$\displaystyle\ \ \ \ \ \ \ \ \ \
=\mathrm{Tr}\left\\{a_{E}(\mathbf{W}^{\mathrm{T}}\otimes\mathbf{E})(\bm{\Lambda}_{E}\otimes\mathbf{I}_{M})\right\\}$
$\displaystyle\ \ \ \ \ \ \ \ \ \
=a_{E}M\mathrm{Tr}\left\\{\bm{\Lambda}_{E}\mathbf{W}\right\\},$ (24a)
$\displaystyle||\mathbf{U}_{E}||_{F}^{2}=a_{E}^{2}M^{2}||\bm{\Lambda}_{E}\mathbf{W}||_{F}^{2},$
(24b)
$\displaystyle||\mathbf{u}_{E}||_{2}^{2}=a_{E}^{2}\mathrm{vec}^{\mathrm{H}}(\mathbf{E}\mathbf{\widehat{H}}_{E}\mathbf{W})(\bm{\Lambda}_{E}\otimes\mathbf{I}_{M})\mathrm{vec}(\mathbf{E}\mathbf{\widehat{H}}_{E}\mathbf{W})$
$\displaystyle\ \ \ \ \ \ \
=a_{E}^{2}M||\bm{\Lambda}_{E}^{\frac{1}{2}}\mathbf{W}\mathbf{\widehat{H}}_{E}^{\mathrm{H}}\mathbf{e}||_{2}^{2},$
(24c)
$\displaystyle\lambda_{\max}(\mathbf{U}_{E})=\lambda_{\max}(a_{E}\bm{\Sigma}_{E}^{\frac{1}{2}}(\mathbf{W}^{\mathrm{T}}\otimes\mathbf{E})\bm{\Sigma}_{E}^{\frac{1}{2}})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
=\lambda_{\max}(a_{E}(\bm{\Lambda}_{E}\mathbf{W}^{\mathrm{T}}\otimes\mathbf{E}))$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
=a_{E}\lambda_{\max}(\bm{\Lambda}_{E}\mathbf{W})\lambda(\mathbf{E})=a_{E}M\mathbf{w}^{\mathrm{H}}\bm{\Lambda}_{E}\mathbf{w}.$
(24d)

By substituting (24) into (23) and introducing slack variables
$\\{x_{E},y_{E}\\}$, the constraints for $\forall k\in\mathcal{K}_{-K}$ in
(19a) are transformed into the following deterministic form:

$\displaystyle\textrm{BTI}_{1}\triangleq\left\\{\begin{array}[]{c}\mathrm{Tr}\left\\{\mathbf{U}_{E}\right\\}+\sqrt{2\ln(1/\bar{\rho})}x_{E}-\ln(\bar{\rho})y_{E}\\\
+u_{k}\leq 0,\forall k\in\mathcal{K}_{-K}\\\
\left\|\begin{array}[]{c}a_{E}M\mathrm{vec}(\bm{\Lambda}_{E}\mathbf{W})\\\
\sqrt{2M}a_{E}\bm{\Lambda}_{E}^{\frac{1}{2}}\mathbf{W}\mathbf{\widehat{H}}_{E}^{\mathrm{H}}\mathbf{e}\end{array}\right\|\leq
x_{E}\\\ y_{E}\geq
a_{E}M\mathbf{w}^{\mathrm{H}}\bm{\Lambda}_{E}\mathbf{w}.\end{array}\right.$
(31)

On the other hand, the simplified derivations for LU $K$ are given by

$\displaystyle\mathrm{Tr}\left\\{\mathbf{U}_{K}\right\\}=a_{K}|v|^{2}M\mathrm{Tr}\left\\{\bm{\Lambda}_{K}\mathbf{W}\right\\}+\frac{a_{E}}{\sigma_{E}^{2}}M\mathrm{Tr}\left\\{\bm{\Lambda}_{E}\mathbf{W}\right\\},$
(32a)
$\displaystyle||\mathbf{U}_{K}||_{F}^{2}=a_{K}^{2}|v|^{4}M^{2}||\bm{\Lambda}_{K}\mathbf{W}||_{F}^{2}+\frac{a_{E}^{2}}{\sigma_{E}^{4}}M^{2}||\bm{\Lambda}_{E}\mathbf{W}||_{F}^{2},$
(32b)
$\displaystyle||\mathbf{u}_{K}||^{2}=M||\bm{\Lambda}_{K}^{\frac{1}{2}}\left(a_{K}|v|^{2}\mathbf{W}\mathbf{\widehat{H}}_{K}^{\mathrm{H}}\mathbf{e}-a_{K}v\mathbf{w}\right)||_{2}^{2}+\frac{a_{E}^{2}}{\sigma_{E}^{4}}M||\mathbf{e}^{\mathrm{H}}\mathbf{\widehat{H}}_{E}\mathbf{W}\bm{\Lambda}_{E}^{\frac{1}{2}}||_{2}^{2},$
(32c)
$\displaystyle\lambda_{\max}(\mathbf{U}_{K})=\max\left\\{a_{K}|v|^{2}M\mathbf{w}^{\mathrm{H}}\bm{\Lambda}_{K}\mathbf{w},\thinspace\thinspace\thinspace\frac{a_{E}}{\sigma_{E}^{2}}M\mathbf{w}^{\mathrm{H}}\bm{\Lambda}_{E}\mathbf{w}\right\\}.$

By substituting the above equations into (23) and introducing slack variables
$\\{x_{K},y_{K}\\}$, the constraint for LU $K$ in (19b) is transformed into
the following deterministic form:

$\displaystyle\textrm{BTI}_{2}\triangleq\left\\{\begin{array}[]{c}\mathrm{Tr}\left\\{\mathbf{U}_{K}\right\\}+\sqrt{2\ln(1/\bar{\rho})}x_{K}-\ln(\bar{\rho})y_{K}+u_{K}\leq
0\\\
\left\|\begin{array}[]{c}a_{K}|v|^{2}M\mathrm{vec}(\bm{\Lambda}_{K}\mathbf{W})\\\
a_{E}M\mathrm{vec}(\bm{\Lambda}_{E}\mathbf{W})/\sigma_{E}^{2}\\\
\sqrt{2M}\bm{\Lambda}_{K}^{\frac{1}{2}}\left(a_{K}|v|^{2}\mathbf{W}\mathbf{\widehat{H}}_{K}^{\mathrm{H}}\mathbf{e}-a_{K}v\mathbf{w}\right)\\\
\sqrt{2M}a_{E}\bm{\Lambda}_{E}^{\frac{1}{2}}\mathbf{W}\mathbf{\widehat{H}}_{E}^{\mathrm{H}}\mathbf{e}/\sigma_{E}^{2}\end{array}\right\|\leq
x_{K}\\\ y_{K}\geq\lambda_{\max}(\mathbf{U}_{K}),y_{K}\geq
0.\end{array}\right.$ (40)

Then, to handle the non-convex power constraint (11f), we replace the right
hand side of (11f) with its linear lower bound

$\displaystyle\Xi({\bf z},t_{k})=$ $\displaystyle
2(1-t_{k}^{(n)})\mathrm{Re}\left\\{{\bf
z}^{(n),\mathrm{H}}\mathbf{Q}^{\mathrm{H}}\mathbf{g}_{k}\mathbf{g}_{k}^{\mathrm{H}}\mathbf{Q}\mathbf{z}\right\\}-\frac{(1-t_{k}^{(n)})^{2}\left|\mathbf{g}_{k}^{\mathrm{H}}\mathbf{Q}\mathbf{z}^{(n)}\right|^{2}}{(1-t_{k})}$
(41)

at feasible point $\\{\mathbf{z}^{(n)},t_{k}^{(n)}\\}$ by adopting the same
first-order Taylor approximation used in Lemma 1.

Therefore, based on (31), (40) and (41) and denoting
$\mathbf{x}=[x_{E},x_{K}]^{\mathrm{T}}$and
$\mathbf{y}=[y_{E},y_{K}]^{\mathrm{T}}$, the approximation problem of Problem
(11) is given by

$\displaystyle\max\limits_{R_{\mathrm{sec}},\mathbf{z},\mathbf{w},\mathbf{e},\mathbf{t},a_{K},a_{E},v,\mathbf{x},\mathbf{y}}$
$\displaystyle R_{\mathrm{sec}}$ (42a) s.t.
$\displaystyle(\ref{eq:BTI-1}),(\ref{eq:BTI-2}),(\ref{eq:p-c})-(\ref{eq:sp-c}),$
(42b) $\displaystyle|w_{k}|^{2}\leq\Xi({\bf z},t_{k}),\forall
k\in\mathcal{K}_{-K}.$ (42c)

For given $\\{\mathbf{e},a_{K},a_{E},v\\}$, we introduce a new variable
$\mathbf{W}=\mathbf{w}\mathbf{w}^{\mathrm{H}}$ with
$\mathrm{rank}(\mathbf{W})=1$. However, different from the general
semidefinite programming (SDP), $\mathbf{w}$ and $\mathbf{W}$, here, coexist
in (LABEL:eq:v) and (32c). Therefore, the semidefinite relaxation (SDR)
technique is not applicable here. In order to handle this problem, we assume
$\mathbf{w}$ and $\mathbf{W}$ are two different variables. If
$\mathrm{Tr}\left\\{\mathbf{W}\right\\}=\lambda_{\max}(\mathbf{W})$, then we
have $\mathrm{rank}(\mathbf{W})=1$. If the obtained $\mathbf{W}$ is not rank
one, we will have
$\mathrm{Tr}\left\\{\mathbf{W}\right\\}-\lambda_{\max}(\mathbf{W})>0$.
Therefore, we constrain
$\mathrm{Tr}\left\\{\mathbf{W}\right\\}-\lambda_{\max}(\mathbf{W})$ less than
a very small real positive number threshold $\varepsilon$ to guarantee the
rank-1 condition of $\mathbf{W}$, yielding the surrogate constraint of rank-1
constraint as

$\displaystyle\mathrm{Tr}\left\\{\mathbf{W}\right\\}-\lambda_{\max}(\mathbf{W})\leq\varepsilon.$
(43a)

When $\mathrm{rank}(\mathbf{W})\approx 1$, the relationship between
$\mathbf{w}$ and $\mathbf{W}$ is given by the following constraint:

$\displaystyle-\varepsilon\leq||\mathbf{w}||^{2}-\mathrm{Tr}\left\\{\mathbf{W}\right\\}\leq\varepsilon.$
(44a)

As for constraint (43a), since $\lambda_{\max}(\mathbf{W})$ is a convex
function of $\mathbf{W}$ [16], the left hand side of (43a) is concave, which
is the difference between a linear function and a convex function. Hence, we
need to construct a convex approximation of constraint (43a). To address this
issue, we introduce the following lemma.

###### Lemma 5

Denote by $\mathbf{v}_{max}$ the eigenvector corresponding to the maximum
eigenvalue of a matrix $\mathbf{V}$, we have

$\displaystyle\mathrm{Tr}\left\\{\mathbf{v}_{max}\mathbf{v}_{max}^{\mathrm{H}}(\mathbf{Z}-\mathbf{V})\right\\}$
$\displaystyle=\mathbf{v}_{max}^{\mathrm{H}}\mathbf{Z}\mathbf{v}_{max}-\mathbf{v}_{max}^{\mathrm{H}}\mathbf{V}\mathbf{v}_{max}$
$\displaystyle=\mathbf{v}_{max}^{\mathrm{H}}\mathbf{Z}\mathbf{v}_{max}-\lambda_{\max}(\mathbf{V})$
$\displaystyle\leq\lambda_{\max}(\mathbf{Z})-\lambda_{\max}(\mathbf{V})$

for any Hermitian matrix $\mathbf{Z}$. $\blacksquare$

Let $\mathbf{d}_{\mathbf{W}}^{(n)}$ be the eigenvector corresponding to the
maximum eigenvalue of the feasible point $\mathbf{W}^{(n)}$. Then, by using
Lemma 5, the surrogate convex constraint of (43a) is given by

$\displaystyle\mathrm{Tr}\left\\{\mathbf{W}\right\\}-\lambda_{\max}(\mathbf{W}^{(n)})-\mathrm{Tr}\left\\{\mathbf{d}_{\mathbf{W}}^{(n)}\mathbf{d}_{\mathbf{W}}^{(n),\mathrm{H}}(\mathbf{W}-\mathbf{W}^{(n)})\right\\}\leq\varepsilon.$
(45)

Now, we consider constraint (44a). By appying the first-order Tayler
approximation to $||\mathbf{w}||^{2}$, we obtain the following convex
approximation of the constraint in (44a) as

$\displaystyle||\mathbf{w}||^{2}-\mathrm{Tr}\left\\{\mathbf{W}\right\\}$
$\displaystyle\leq\varepsilon,$ (46a) $\displaystyle
2\mathrm{Re}\left\\{\mathbf{w}^{(n),\mathrm{H}}\mathbf{w}\right\\}-||\mathbf{w}^{(n)}||^{2}-\mathrm{Tr}\left\\{\mathbf{W}\right\\}$
$\displaystyle\geq-\varepsilon.$ (46b)

Finally, the subproblem w.r.t.,
$\\{\mathbf{z},\mathbf{w},\mathbf{W},\mathbf{t}\\}$ is formulated as

$\displaystyle\max\limits_{R_{\mathrm{sec}},\mathbf{z},\mathbf{w},\mathbf{W},\mathbf{t},\mathbf{x},\mathbf{y}}$
$\displaystyle R_{\mathrm{sec}}$ (47a) s.t.
$\displaystyle(\ref{eq:BTI-1}),(\ref{eq:BTI-2}),(\ref{eq:p-c}),(\ref{eq:sp-c}),$
(47b) $\displaystyle(\ref{eq:ph2}),(\ref{Pro:W-c-1}),(\ref{Pro:W-c-2})$ (47c)
$\displaystyle\mathbf{W}\succeq\mathbf{0}.$ (47d)

Problem (47) is an SDP and can be solved by the CVX tool [18].

For given $\\{\mathbf{w},a_{K},a_{E},v\\}$, Problem (42) with optimization
variable $\mathbf{e}$ can be solved by applying the penalty CCP [13, 19, 20]
to relax the unit-modulus constraint (11d). Comparing with the SDR technique,
the penalty CCP method is capable of finding a feasible solution to meet
constraint (11d). In particular, the constraint of (11d) can be relaxed by

$\displaystyle|e_{m}^{[j]}|^{2}-2\mathrm{Re}(e_{m}^{\mathrm{H}}e_{m}^{[j]})\leq
b_{m}-1,1\leq m\leq M,$ (48a) $\displaystyle|e_{m}|^{2}\leq 1+b_{M+m},1\leq
m\leq M,$ (48b)

where $e_{m}^{[j]}$ is any feasible solution and
$\mathbf{b}=[b_{1},...,b_{2M}]^{\mathrm{T}}$ is slack vector variable. The
proof of (48) can be found in [13, 19]. Following the penalty CCP framework,
the subproblem for optimizing $\mathbf{e}$ is formulated as

$\displaystyle\max\limits_{R_{\mathrm{sec}},\mathbf{e},\mathbf{x},\mathbf{y}}$
$\displaystyle R_{\mathrm{sec}}-\lambda^{[j]}||\mathbf{b}||_{1}$ (49a) s.t.
$\displaystyle(\ref{eq:BTI-1}),(\ref{eq:BTI-2}),(\ref{Pro:e-c-1}).$ (49b)

Problem (49) is an SDP and can be solved by the CVX tool. The algorithm for
finding a feasible solution of $\mathbf{e}$ is summarized in Algorithm 1.

Algorithm 1 Penalty CCP optimization for reflection beamforming optimization

0: Initialize $\mathbf{e}^{[0]}$, $\gamma^{[0]}>1$, and set $j=0$.

1: repeat

2: if $j<J_{max}$ then

3: Update $\mathbf{e}^{[j+1]}$ by solving Problem (49);

4: Update $\lambda^{[j+1]}=\min\\{\gamma\lambda^{[j]},\lambda_{max}\\}$;

5: $j=j+1$;

6: else

7: Initialize with a new random $\mathbf{e}^{[0]}$, set up $\gamma^{[0]}>1$
again, and set $j=0$.

8: end if

9: until $||\mathbf{b}||_{1}\leq\chi$ and
$||\mathbf{e}^{[j]}-\mathbf{e}^{[j-1]}||_{1}\leq\nu$.

10: Output $\mathbf{e}^{(n+1)}=\mathbf{e}^{[j]}$.

In addition, Problem (42) is convex in
$\\{R_{\mathrm{sec}},v,\mathbf{x},\mathbf{y}\\}$ for given
$\\{\mathbf{z},\mathbf{w},\mathbf{e},\mathbf{t},a_{K},a_{E},\\}$, and convex
in $\\{R_{\mathrm{sec}},a_{K},a_{E},\mathbf{x},\mathbf{y}\\}$ for given
$\\{\mathbf{z},\mathbf{w},\mathbf{e},\mathbf{t},v,\\}$. Finally, Problem (42)
is addressed under the alternate optimization (AO) framework containing four
subproblems. The convergence of the AO framework can be guaranteed due to the
fact that each subproblem can obtain a non-decreasing sequence of objective
function values.

##  IV ED Model II: Passive Eavesdropper Model

In this section, we focus on the transmission design for the passive attack,
which is more practical and more challenging to address, since the passive ED
can hide itself and its CSI is not known [11, 12]. The authors of [15]
proposed to exploit the angular information of the ED to combat its passive
attack, which is also applicable here. In this section, the cascaded LIL
channel $\mathbf{H}_{K}$ and the channel $\mathbf{H}_{\mathrm{IRS}}$ are
assumed to be perfect, which is reasonable due to the fact that the pilot
information for channel estimation for LUs is known at the BS.

###  IV-A Average eavesdropping rate Maximization

The signal received by the ED is formulated as

$\displaystyle y_{E}$
$\displaystyle=\mathbf{h}_{E}^{\mathrm{H}}\mathrm{diag}(\mathbf{e}^{*})\mathbf{H}_{\mathrm{IRS}}\mathbf{w}s+\sigma_{E}^{2}.$

Since the ED is passive, we can only detect the activity of the ED on the line
segment between the BS and LU $K$ without knowing its exact location. This
detection of a passive attack is based on spectrum sensing [21]. Hence, the
average eavesdropping rate is considered which can be computed as follows [22,
23, 15]:

$\displaystyle
R_{E}^{av}\left(\mathbf{w},\mathbf{e}\right)=\frac{1}{D_{K}}\int_{0}^{D_{K}}\mathbb{E}_{\\{\mathbf{h}_{E}\\}}\Biggl{[}\frac{1}{2}\log_{2}\biggl{(}1+\frac{1}{\sigma_{E}^{2}}\left|\mathbf{h}_{E}^{\mathrm{H}}\mathrm{diag}(\mathbf{e}^{*})\mathbf{H}_{\mathrm{IRS}}\mathbf{w}\right|^{2}\biggr{)}\Biggr{]}\mathrm{d}_{D_{E}}.$
(50)

With (50), we formulate the following optimization problem:

$\displaystyle\max\limits_{\mathbf{z},\mathbf{w},\mathbf{e},\mathbf{t}}$
$\displaystyle\left\\{\min_{\forall
k\in\mathcal{K}}R_{k}-R_{E}^{av}\left(\mathbf{w},\mathbf{e}\right)\right\\}$
(51a) s.t. $\displaystyle(\ref{eq:p-c})-(\ref{eq:ph-c}).$ (51b)

The main challenge to solve Problem (51) is from the average eavesdropping
rate containing the integration over $D_{E}$ and the expectation over
$\mathbf{h}_{E}$. To address this issue, we use Jensen’s inequality to
construct an upper bound of $R_{E}^{av}\left(\mathbf{w},\mathbf{e}\right)$
given by

$\displaystyle R_{E}^{up}\left(\mathbf{w},\mathbf{e}\right)$
$\displaystyle=\frac{1}{2}\log_{2}\left(1+\frac{\int_{0}^{D_{K}}\mathbb{E}_{\\{\mathbf{h}_{E}\\}}\left[\left|\mathbf{h}_{E}^{\mathrm{H}}\mathrm{diag}(\mathbf{e}^{*})\mathbf{H}_{\mathrm{IRS}}\mathbf{w}\right|^{2}\right]\mathrm{d}_{D_{E}}}{\sigma_{E}^{2}D_{K}}\right)$
$\displaystyle=\frac{1}{2}\log_{2}\left(1+\frac{1}{\sigma_{E}^{2}}\mathbf{w}^{\mathrm{H}}\mathbf{H}_{\mathrm{IRS}}^{\mathrm{H}}\mathrm{diag}(\mathbf{e})\mathbf{R}_{E}\mathrm{diag}(\mathbf{e}^{*})\mathbf{H}_{\mathrm{IRS}}\mathbf{w}\right),$
(52)

where
$\mathbf{R}_{E}=\frac{1}{D_{K}}\int_{0}^{D_{K}}\mathbb{E}_{\\{\mathbf{h}_{E}\\}}[\mathbf{h}_{E}\mathbf{h}_{E}^{\mathrm{H}}]\mathrm{d}_{D_{E}}$
which can be computed via one-dimension integration.

According to (2), we define

$\displaystyle\overline{\mathbf{h}}_{E}$
$\displaystyle=\sqrt{\varrho_{0}\left(\frac{d_{\mathrm{IRS},E}}{d_{0}}\right)^{-\alpha_{\mathrm{IRS}}}\frac{K_{\mathrm{IRS}}}{1+K_{\mathrm{IRS}}}}\mathbf{h}_{E}^{\mathrm{LOS}},$
(53a) $\displaystyle\mathbf{R}_{E}$
$\displaystyle=\varrho_{0}\left(\frac{d_{\mathrm{IRS},E}}{d_{0}}\right)^{-\alpha_{\mathrm{IRS}}}\frac{1}{1+K_{\mathrm{IRS}}}\mathbf{I}_{M},$
(53b)

where $\overline{\mathbf{h}}_{E}$ describes the LoS component and is the mean
of channel $\mathbf{h}_{E}$. Moreover, $\mathbf{R}_{E}$ is a positive semi-
definite covariance matrix representing the spatial correlation
characteristics of the NLoS component. Therefore, we have
$\mathbf{h}_{E}\sim\mathcal{CN}(\overline{\mathbf{h}}_{E},\mathbf{R}_{E})$
[24], and further get

$\displaystyle\mathbb{E}_{\\{\mathbf{h}_{E}\\}}[\mathbf{h}_{E}\mathbf{h}_{E}^{\mathrm{H}}]$
$\displaystyle=\left[\mathbf{R}_{E}+\overline{\mathbf{h}}_{E}\overline{\mathbf{h}}_{E}^{\mathrm{H}}\right]$
$\displaystyle=\varrho_{0}\left(\frac{d_{\mathrm{IRS},E}}{d_{0}}\right)^{-\alpha_{\mathrm{IRS}}}\cdot\left[\frac{1}{1+K_{\mathrm{IRS}}}\mathbf{I}_{M}+\frac{K_{\mathrm{IRS}}}{1+K_{\mathrm{IRS}}}\mathbf{h}_{E}^{\mathrm{LOS}}(\mathbf{h}_{E}^{\mathrm{LOS}})^{\mathrm{H}}\right].$

###  IV-B Proposed Algorithm

By replacing $R_{E}^{av}\left(\mathbf{w},\mathbf{e}\right)$ with
$R_{E}^{up}\left(\mathbf{w},\mathbf{e}\right)$ in the objective function of
Problem (51), we have

$\displaystyle\max\limits_{\mathbf{z},\mathbf{w},\mathbf{e},\mathbf{t}}$
$\displaystyle\left\\{\min_{\forall k\in\mathcal{K}}R_{k}-R_{E}^{up}\right\\}$
(54a) s.t. $\displaystyle(\ref{eq:p-c})-(\ref{eq:ph-c}).$ (54b)

Problem (54) is still difficult to solve due to the non-convex constraints and
objective function, as well as the coupled variables $\mathbf{w}$ and
$\mathbf{e}$. Hence, we propose an MM-based AO method to update $\mathbf{w}$
and $\mathbf{e}$ iteratively. More specifically, by first fixing $\mathbf{e}$,
the non-concave objective function w.r.t., $\\{{\bf
z},\mathbf{w},\mathbf{t}\\}$ is replaced by its customized concave surrogate
function and then solved by the CVX. $\\{{\bf z},\mathbf{w},\mathbf{t}\\}$ are
then fixed and the closed-form solution of $\mathbf{e}$ can be found by
constructing an easy-to-solve surrogate objective function w.r.t $\mathbf{e}$.

The surrogate functions of $R_{k}\left(\mathbf{z},t_{k}\right)$ for $\forall
k\in\mathcal{K}_{-K}$ are given by
$\widehat{R}_{k}(\mathbf{z},t_{k}|\mathbf{z}^{(n)},t_{k}^{(n)})=\widetilde{R}_{k}(\mathbf{z},t_{k}|\mathbf{z}^{(n)},t_{k}^{(n)})$
given in (14), and those of $R_{K}\left(\mathbf{w},\mathbf{e}\right)$ and
$R_{E}\left(\mathbf{w},\mathbf{e}\right)$ are given in the following lemma by
using the first-order Taylor approximation.

###### Lemma 6

Let $\\{\mathbf{w}^{(n)},\mathbf{e}^{(n)}\\}$ be any feasible solution, then
$R_{K}\left(\mathbf{w},\mathbf{e}\right)$ is lower bounded by a concave
surrogate function
$\widehat{R}_{K}\left(\mathbf{w},\mathbf{e}|\mathbf{w}^{(n)},\mathbf{e}^{(n)}\right)$
defined by

$\displaystyle\widehat{R}_{K}(\mathbf{w},\mathbf{e}|\mathbf{w}^{(n)},\mathbf{e}^{(n)})$
$\displaystyle=\frac{1}{2}\log_{2}\Biggl{(}1-\frac{q_{K}^{(n)}}{\sigma_{K}^{2}}+2\mathrm{Re}\left\\{\frac{q_{K}}{\sigma_{K}^{2}}\right\\}\Biggr{)},$
(55)

where
$q_{K}^{(n)}=\left|\mathbf{e}^{(n),\mathrm{H}}\mathbf{H}_{K}\mathbf{w}^{(n)}\right|^{2}$
and
$q_{K}=\mathbf{e}^{(n),\mathrm{H}}\mathbf{H}_{K}\mathbf{w}^{(n)}\mathbf{w}^{\mathrm{H}}\mathbf{H}_{K}^{\mathrm{H}}\mathbf{e}$.

Meanwhile, $R_{E}\left(\mathbf{w},\mathbf{e}\right)$ is upper bounded by a
convex surrogate function
$\widehat{R}_{E}\left(\mathbf{w},\mathbf{e}|\mathbf{w}^{(n)},\mathbf{e}^{(n)}\right)$
given by

$\displaystyle\widehat{R}_{E}^{up}\left(\mathbf{w},\mathbf{e}|\mathbf{w}^{(n)},\mathbf{e}^{(n)}\right)=\frac{1}{2}\log_{2}\left(1+\frac{q_{E}^{(n)}}{\sigma_{E}^{2}}\right)+\frac{q_{E}-q_{E}^{(n)}}{2(\sigma_{E}^{2}+q_{E}^{(n)})\ln
2},$ (56)

where
$q_{E}^{(n)}=\mathbf{w}^{(n),\mathrm{H}}\mathbf{H}_{\mathrm{IRS}}^{\mathrm{H}}\mathrm{diag}(\mathbf{e}^{(n)})\mathbf{R}_{E}\mathrm{diag}(\mathbf{e}^{(n),*})\mathbf{H}_{\mathrm{IRS}}\mathbf{w}^{(n)}$
and
$q_{E}=\mathbf{w}^{\mathrm{H}}\mathbf{H}_{\mathrm{IRS}}^{\mathrm{H}}\mathrm{diag}(\mathbf{e})\mathbf{R}_{E}\mathrm{diag}(\mathbf{e}^{*})\mathbf{H}_{\mathrm{IRS}}\mathbf{w}$.

Furthermore, we have the following proposition.

###### Proposition 1

The functions $\\{\widehat{R}_{k},\widehat{R}_{K},\widehat{R}_{E}^{up}\\}$
preserve the first-order property of functions $\\{R_{k},R_{K},R_{E}\\}$,
respectively. Let’s take $\widehat{R}_{K}$ and $R_{K}$ as an example

$\displaystyle\nabla_{\mathbf{w}}\widehat{R}_{K}(\mathbf{w},\mathbf{e}^{(n)}|\mathbf{w}^{(n)},\mathbf{e}^{(n)})|_{\mathbf{w}=\mathbf{w}^{(n)}}=$
$\displaystyle\nabla_{\mathbf{w}}R_{K}\left(\mathbf{w},\mathbf{e}^{(n)}\right)|_{\mathbf{w}=\mathbf{w}^{(n)}},$
$\displaystyle\nabla_{\mathbf{e}}\widehat{R}_{K}(\mathbf{w}^{(n)},\mathbf{e}|\mathbf{w}^{(n)},\mathbf{e}^{(n)})|_{\mathbf{e}=\mathbf{e}^{(n)}}=$
$\displaystyle\nabla_{\mathbf{e}}R_{K}\left(\mathbf{w}^{(n)},\mathbf{e}\right)|_{\mathbf{e}=\mathbf{e}^{(n)}}.$

Proof: See Appendix B in [15]. $\blacksquare$

Giving $\mathbf{e}$ and combining (14), (55), (56) and (42c), the subproblem
of optimizing $\\{{\bf z},\mathbf{w},\mathbf{t}\\}$ is formulated as

$\displaystyle\max\limits_{\mathbf{z},\mathbf{w},\mathbf{t}}$
$\displaystyle\left\\{\min_{\forall
k\in\mathcal{K}}\widehat{R}_{k}-\widehat{R}_{E}^{up}\right\\}$ (57a) s.t.
$\displaystyle(\ref{eq:p-c}),(\ref{eq:sp-c}),(\ref{eq:ph2}).$ (57b)

Introducing auxiliary variable $r$, we can transform Problem (57) into

$\displaystyle\max\limits_{\mathbf{z},\mathbf{w},\mathbf{t},r}$
$\displaystyle\left\\{r-\widehat{R}_{E}^{up}\right\\}$ (58a) s.t.
$\displaystyle(\ref{eq:p-c}),(\ref{eq:sp-c}),(\ref{eq:ph2})$ (58b)
$\displaystyle\widehat{R}_{k}\geq r,\forall k\in\mathcal{K},$ (58c)

which is convex and can be solved by using CVX.

Giving $\\{{\bf z},\mathbf{w},\mathbf{t}\\}$ and combining (14), (55) and
(56), the subproblem w.r.t., $\mathbf{e}$ is formulated as

$\displaystyle\max\limits_{\mathbf{e}}$ $\displaystyle\left\\{\min_{\forall
k\in\mathcal{K}}\widehat{R}_{k}-\widehat{R}_{E}^{up}\right\\},\textrm{s.t.}(\ref{eq:e-c}).$
(59)

Problem (59) can be addressed by transforming it into an SOCP under the
penalty CCP method mentioned in Section III-B. However, the penalty CCP method
needs to solve a series of SOCP problems which incurs a high computational
complexity. In the following, we aim to derive a low-complexity algorithm
containing the closed-form solution of $\mathbf{e}$.

Denote by
$\mathcal{R}=\min_{k=1}^{K-1}\left\\{R_{k}\left(\mathbf{z},t_{k}\right)\right\\}$
that is a constant, then the subproblem of Problem (54) corresponding to the
optimization of $\mathbf{e}$ is given by

$\displaystyle\max\limits_{\mathbf{e}}$
$\displaystyle\left\\{\min\\{\mathcal{R},R_{K}\left(\mathbf{e}\right)\\}-R_{E}^{up}\left(\mathbf{e}\right)\right\\},\textrm{s.t.}(\ref{eq:e-c}).$
(60)

Before solving Problem (60), we first consider the following two subproblems:

$\displaystyle\mathcal{P}1:\thinspace\thinspace\thinspace\thinspace$
$\displaystyle\min\limits_{\mathbf{e}}R_{E}^{up}\left(\mathbf{e}\right),\textrm{s.t.}(\ref{eq:e-c}),R_{K}\left(\mathbf{e}\right)\geq\mathcal{R}.$
(61) $\displaystyle\mathcal{P}2:\thinspace\thinspace\thinspace\thinspace$
$\displaystyle\max\limits_{\mathbf{e}}R_{K}\left(\mathbf{e}\right)-R_{E}^{up}\left(\mathbf{e}\right),\textrm{s.t.}(\ref{eq:e-c}),R_{K}\left(\mathbf{e}\right)\leq\mathcal{R}.$
(62)

Denote the solutions to $\mathcal{P}1$ and $\mathcal{P}2$ as
$\mathbf{e}_{1}^{\\#}$ and $\mathbf{e}_{2}^{\\#}$, respectively. In addition,
let us denote the objective function value of Problem (60) as
$obj(\mathbf{e})$, which is a function of $\mathbf{e}$. If
$obj(\mathbf{e}_{1}^{\\#})\geq obj(\mathbf{e}_{2}^{\\#})$, then the optimal
solution of Problem (60) is given by $\mathbf{e}_{1}^{\\#}$. Otherwise, the
optimal solution is $\mathbf{e}_{2}^{\\#}$.

The solutions to Problem $\mathcal{P}1$ and Problem $\mathcal{P}2$ are given
in the following lemma.

###### Lemma 7

The optimal solution of $\mathcal{P}1$ is given by

$\mathbf{e}_{1}^{\\#}=\exp\\{\mathrm{j}\arg((\lambda_{\max}(\mathbf{A}_{E})\mathbf{I}-\mathbf{A}_{E}+\varrho_{1}^{opt}\mathbf{A}_{K})\mathbf{e}^{(n)})\\},$
(63)

where
$\mathbf{A}_{E}=\text{($\mathbf{H}_{\mathrm{IRS}}$$\mathbf{w}\mathbf{w}^{\mathrm{H}}\mathbf{H}_{\mathrm{IRS}}^{\mathrm{H}}$)}\odot(\mathbf{R}_{E}^{\mathrm{T}}/\sigma_{E}^{2})$,
$\mathbf{A}_{K}=\mathbf{H}_{K}\mathbf{w}\mathbf{w}^{\mathrm{H}}\mathbf{H}_{K}^{\mathrm{H}}/\sigma_{K}^{2}$
and $\varrho_{1}^{opt}$ is the price introduced by the price mechanism [25].

The optimal solution of $\mathcal{P}2$ is given by

$\mathbf{e}_{2}^{\\#}=\exp\\{\mathrm{j}\arg(\mathbf{c}+\varrho_{2}^{opt}(\lambda_{\max}(\mathbf{A}_{K})\mathbf{I}-\mathbf{A}_{K})\mathbf{e}^{(n)})\\},$
(64)

where $\varrho_{2}^{opt}$ is the price and

$\displaystyle\mathbf{c}=\frac{1+d_{K}^{(n)}}{(1+d_{E}^{(n)})^{2}}(\lambda_{\max}(\mathbf{A}_{E})\mathbf{I}-\mathbf{A}_{E})\mathbf{e}^{(n)}+\frac{\mathbf{A}_{K}\mathbf{e}^{(n)}}{1+d_{E}^{(n)}},$
$\displaystyle
d_{K}^{(n)}=\mathbf{e}^{(n),\mathrm{H}}\mathbf{A}_{K}\mathbf{e}^{(n)},\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace
d_{E}^{(n)}=\mathbf{e}^{(n),\mathrm{H}}\mathbf{A}_{E}\mathbf{e}^{(n)}.$

Proof: See Appendix C. $\blacksquare$

Since $\mathbf{e}_{1}^{\\#}$ and $\mathbf{e}_{2}^{\\#}$ are the globally
optimal solutions of $\mathcal{P}1$ and $\mathcal{P}2$, respectively, hence
$\mathbf{e}^{\\#}$ is the globally optimal solution of Problem (60). The
optimal price parameter can be obtained by using the bisection search method
detailed in [25].

According to Proposition 1 and Theorem 1 in [26], the sequence
$\\{\mathbf{z}^{(n)},\mathbf{w}^{(n)},\mathbf{t}^{(n)},\mathbf{e}^{(n)}\\}_{n=1,2,3,\ldots}$
obtained in each iteration is guaranteed to converge to the set of stationary
points of Problem (54).

##  V Numerical results and discussions

In this section, we provide numerical results to evaluate the performance of
the proposed schemes. The results are obtained by using a computer with a 1.99
GHz i7-8550U CPU and 16 GB RAM. The simulated system setup is measured with
polar coordinates: The BS is located at (0 m, 0 m) and the IRS is placed at
(50 m, 0 m) with elevation angle $\phi=\frac{2\pi}{3}$; $K$ LUs are randomly
and uniformly distributed in an area with
$D_{k}\sim\mathcal{U}(20\thinspace\mathrm{m},\thinspace
40\thinspace\mathrm{m})$ and
$\theta_{k}\sim\mathcal{U}(-\frac{\pi}{2},\thinspace\frac{\pi}{2})$ for
$\forall k\in\mathcal{K}$, where $\mathcal{U}$ is the uniform distribution.
The ED is located at $(D_{E},\theta_{K})$ with $D_{E}\in(0,D_{K})$. The
pathloss at the distance of 1 m is $-30$ dB, the pathloss exponents are set to
$\alpha_{\mathrm{BS}}=\alpha_{\mathrm{IRS}}=2.2$ and the Rician factor is 5.
The transmit power budget at the BS is $P_{\max}=30$ dBm and the noise powers
are $\\{\sigma_{i}^{2}=-100\thinspace\thinspace\mathrm{dBm}\\}_{\forall
i\in\mathcal{K}_{+E}}$. For the statistical CSI error model, the variances of
{$\bm{\bigtriangleup}_{i}^{K},\bm{\bigtriangleup}_{i}^{E}\\}_{\forall
i\in\mathcal{K}_{-K}}$ are defined as
$\\{\varepsilon_{K,i}^{2}=\delta_{K}^{2}||\mathbf{h}_{K}^{*}\odot\mathbf{h}_{i}||_{2}^{2},\varepsilon_{E,i}^{2}=\delta_{E}^{2}||\mathbf{h}_{E}^{*}\odot\mathbf{h}_{i}||_{2}^{2}\\}_{\forall
i\in\mathcal{K}_{-K}}$, where $\delta_{K}\in[0,1)$ and $\delta_{E}\in[0,1)$
measure the relative amount of CSI uncertainties. In addition, the outage
probability of secrecy rate is $\rho=0.05$.

###  V-A Robust secrecy rate in ED Model I

In order to verify the performance of the proposed outage constrained
beamforming in the AAUC, the case of $N=8$ and $K=5$ is simulated. For
comparison, we also consider the “Non-robust” as the benchmark scheme, in
which the estimated cascaded LIL and LIE channels are naively regarded as
perfect channels, resulting in the following problem

$\displaystyle\max\limits_{\mathbf{z},\mathbf{w},\mathbf{e},\mathbf{t}}$
$\displaystyle\left\\{\min_{\forall k\in\mathcal{K}}R_{k}-R_{E}\right\\}$
(66a) s.t. $\displaystyle(\ref{eq:p-c})-(\ref{eq:ph-c}),$ (66b)

where
$R_{K}\left(\mathbf{e}\right)=\frac{1}{2}\log_{2}(1+|\mathbf{e}^{\mathrm{H}}\widehat{\mathbf{H}}_{K}\mathbf{w}|^{2}/\sigma_{K}^{2})$
and
$R_{E}\left(\mathbf{e}\right)=\frac{1}{2}\log_{2}(1+|\mathbf{e}^{\mathrm{H}}\widehat{\mathbf{H}}_{E}\mathbf{w}|^{2}/\sigma_{E}^{2})$.
Problem (66) can be solved by using the proposed low complexity algorithm used
to solve Problem (51).

Fig. 3 investigates the feasibility rate and the maximum secrecy rate versus
the distance of the ED, in which the coordinate of X-axis is set to the ratio
of $D_{E}/D_{K}$. The feasibility rate is defined as the ratio of the number
of channel realizations that have a feasible solution to the outage
constrained problem of (11) to the total number of channel realizations. It is
observed from Fig. 3(a) that the closer the ED is to LU $K$, the lower the
feasibility rate will be, which means that the location of the eavesdropper
imposes a great threat to the security system. From Fig. 3(b), we can see that
the secrecy rate drops fast when the ED approaches LU $K$, and this secrecy
rate reduces to zero when the channel error is large. At this situation, the
whole system is no longer suitable for secure communication.

(a) Feasibility rate (b) Secrecy rate

Figure 3: Performance versus $D_{E}/D_{K}$ under $N=8$, $M=32$ and $K=5$.

Next, the performance versus the size of the IRS, i.e., $M$, is verified in
Fig. 4. Assume that the ED is located at $D_{E}/D_{K}=0.5$. In Fig. 4(a), the
feasibility rate decreases rapidly with $M$ when the level of channel
uncertainty is high. In Fig. 4(b), the case of $\delta_{K}=\delta_{E}=0$ is
regarded as the perfect cascaded CSI case, and its maximum secrecy rate
increases with $M$, which is consistent with that of Fig. 6 in [14]. The
performance of $\delta_{K}=\delta_{E}=0$ can be used as the performance upper
bound of the proposed outage constrained beamforming method. The maximum
secrecy rate under small channel uncertainty levels, e.g.,
$(\delta_{K}=\\{0.1,0.2\\},\delta_{E}=0)$ or $(\delta_{K}=0,\delta_{E}=0.1)$,
also increases with $M$. In addition, it is observed that black lines of
$\delta_{E}=0$ are higher than blue lines of $\delta_{K}=0$, which means that
the negative impact of cascaded LIL channel error on secrecy rate is smaller
than that of the cascaded LIE channel error. Furthermore, when
$\delta_{\mathrm{E}}$ increases to $0.1$ or larger, secrecy rate starts to
decrease at large $M$. The reason is that increasing $M$ can not only enhance
the secrecy rate due to its increased beamforming gain, but also increase the
cascaded channel estimation error.

Therefore, at small channel uncertainty level, especially that of the ED, the
benefits brought by the increased $M$ outweighs its drawbacks, and vice versa.
This observation provides useful guidance to carefully chose the size of the
IRS according to the level of the channel uncertainty.

(a) Feasibility rate (b) Secrecy rate

Figure 4: Performance versus $M$ under $N=8$ and $K=5$.

###  V-B Average secrecy rate in ED Model II

This subsection evaluates the performance of the proposed scheme under the
passive attack. In order to evaluate the performance of the proposed low
complexity algorithm, we consider two benchmark algorithms which are given by:
1) The “SOCP” based scheme, i.e., the CVX is used to solve the SOCP version of
Problem (59). 2) The “Random” phase scheme, in which the phases of the
reflection elements are randomly generated.

Under the case of $N=8$ and $K=5$, the average secrecy rate versus $M$ is
shown in Fig. 5(a). It can be seen that the performance of the proposed
algorithm is almost the same as that of the SOCP algorithm, and both of them
outperform the scheme with random phases. Moreover, increasing the size of the
IRS can significantly increase the average secrecy rate of the system.
Accordingly, Fig. 5(b) describes the CPU time required for these three
algorithms. The simulation setup is the same as that of Fig. 5(a). It is
observed from Fig. 5(b) that the proposed algorthm with closed-form solution
requires much less CPU running time than the SOCP scheme. In addition, the CPU
running time of the SCOP-based algorithm increases significantly with the size
of the IRS, but the proposed algorithm is not sensitive to the size of the
IRS. This is because the computational complexity of the SOCP depends on $M$,
while that of the closed-form solution does not.

(a) Average secrecy rate (b) Average CPU time

Figure 5: Performance versus $M$ under $N=8$ and $K=5$.

Finally, Fig. 6 illustrates the performance and the complexity versus the
number of users in the case of $M=64$. Again, the proposed algorithm can
achieve the same performance of the SOCP-based scheme with less CPU running
time. However, the average secrecy rate is saturated when the number of users
is greater than 7.

(a) Average secrecy rate (b) Average CPU time

Figure 6: Performance versus $K$ under $M=64$.

##  VI Conclusions

In this work, we have proposed a two-phase IRS-aided communication system to
realize the secure communication under the active attacks and passive
eavesdropping. In order to address the cascaded channel error caused by the
active attacks, we maximize the secrecy rate subject to secrecy rate outage
probability constraints, which has been tackled by using the Bernstein-Type
inequality. For the case of the partial CSI of the ED, average secrecy rate
maximization problem was considered, which is addressed by the proposed low-
complexity algorithm. It was shown that the negative effect of the ED’s
channel error is greater than that of the LU. In addition, the number of
elements on the IRS has a negative impact on system performance when the
channel error is large. This conclusion provides an engineering insight for
the careful selection of the number of elements at the IRS.

##  Appendix A The proof of Lemma 2

To begin, we introduce the following lemma.

###### Lemma 8

Let $x\geq 0$ be a positive real number, and consider the function
$g_{1}(a,x)=-ax+\ln a+1$, then

$\ln x^{-1}=\max_{a\geq 0}g_{1}(a,x).$

By appying Lemma 8, we can construct an upper bound of rate
$R_{E}\left(\mathbf{w},\mathbf{e}\right)$ as

$\displaystyle R_{E}\left(\mathbf{w},\mathbf{e}\right)$
$\displaystyle=\frac{-\ln(1+\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{E}\mathbf{w}\right|^{2}/\sigma_{E}^{2})^{-1}}{2\ln
2}$ $\displaystyle\overset{\mathrm{(a)}}{=}\frac{-\max_{a_{E}\geq
0}g_{1}(a_{E},1+\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{E}\mathbf{w}\right|^{2}/\sigma_{E}^{2})}{2\ln
2}$ $\displaystyle=\frac{\min_{a_{E}\geq
0}-g_{1}(a_{E},1+\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{E}\mathbf{w}\right|^{2}/\sigma_{E}^{2})}{2\ln
2}$
$\displaystyle\leq\frac{-g_{1}(a_{E},1+\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{E}\mathbf{w}\right|^{2}/\sigma_{E}^{2})}{2\ln
2},\textrm{for any }a_{E}\geq 0$
$\displaystyle=\frac{a_{E}\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{E}\mathbf{w}\right|^{2}/\sigma_{E}^{2}+a_{E}-\ln
a_{E}-1}{2\ln 2}$
$\displaystyle=\widetilde{R}_{E}\left(\mathbf{w},\mathbf{e},a_{E}\right),$
(67)

where the equality (a) is from Lemma 8.

Hence, the proof is completed.

##  Appendix B The proof of Lemma 3

To prove Lemma 3, we first introduce the following lemma.

###### Lemma 9

Let $v$ be a complex number, and consider the function
$g_{2}(v,x)=(|x|^{2}+\sigma^{2})|v|^{2}-2\text{Re}\left\\{vx\right\\}+1$, then

$\frac{\sigma^{2}}{|x|^{2}+\sigma^{2}}=\min_{v}g_{2}(a,x).$

By appying Lemma 9, we can construct a lower bound of rate
$R_{K}\left(\mathbf{w},\mathbf{e}\right)$ as

$\displaystyle R_{K}\left(\mathbf{w},\mathbf{e}\right)$
$\displaystyle=\frac{\ln\left(1-\frac{\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w}\right|^{2}}{\sigma_{K}^{2}+\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w}\right|^{2}}\right)^{-1}}{2\ln
2}$ $\displaystyle\overset{\mathrm{(a)}}{=}\frac{\max_{a_{K}\geq
0}g(a_{K},1-\frac{\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w}\right|^{2}}{\sigma_{K}^{2}+\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w}\right|^{2}})}{2\ln
2}$
$\displaystyle\geq\frac{g(a_{K},1-\frac{\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w}\right|^{2}}{\sigma_{K}^{2}+\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w}\right|^{2}})}{2\ln
2},\textrm{for any }a_{K}\geq 0$
$\displaystyle=\frac{-a_{K}\left(\frac{\sigma_{K}^{2}}{\sigma_{K}^{2}+\left|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w}\right|^{2}}\right)+\ln
a_{K}+1}{2\ln 2}$
$\displaystyle\overset{\mathrm{(b)}}{=}\frac{-a_{K}\left(\min_{v}g_{2}(v,\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w})\right)+\ln
a_{K}+1}{2\ln 2}$
$\displaystyle=\frac{a_{K}\left(\max_{v}-g_{2}(v,\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w})\right)+\ln
a_{K}+1}{2\ln 2}$
$\displaystyle\geq\frac{a_{K}\left(-g_{2}(v,\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w})\right)+\ln
a_{K}+1}{2\ln 2},\textrm{for any }v\geq 0$ $\displaystyle=\frac{1}{2\ln
2}(-a_{K}|v|^{2}|\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w}|^{2}-\sigma_{K}^{2}a_{K}|v|^{2}+2a_{K}\text{Re}\left\\{v\mathbf{e}^{\mathrm{H}}\mathbf{H}_{K}\mathbf{w}\right\\}-a_{K}+\ln
a_{K}+1)$
$\displaystyle=\widetilde{R}_{K}\left(\mathbf{w},\mathbf{e},a_{K},v\right)$
(68)

where Equality (a) is from Lemma 8, and Equality (b) is from Lemma 9.

Hence, the proof is completed.

##  Appendix C The proof of Lemma 7

To begin with, we solve $\mathcal{P}1$: $\mathcal{P}1$ is equivalent to

$\displaystyle\min\limits_{\mathbf{e}}$
$\displaystyle\thinspace\thinspace\mathbf{e}^{\mathrm{H}}\mathbf{A}_{E}\mathbf{e}$
(69a) $\displaystyle\mathrm{s.t.}$ $\displaystyle(\ref{eq:e-c}),$ (69b)
$\displaystyle\mathbf{e}^{\mathrm{H}}\mathbf{A}_{K}\mathbf{e}\geq
e^{2\mathcal{R}}-1.$ (69c)

Step 1: Construct a surrogate problem: Under the MM algorithm framework [27],
we have the following lemma.

###### Lemma 10

[28, 29] Given $\mathbf{A}\succeq\mathbf{A}_{0}$ and $\mathbf{x}$, then
quadratic function $\mathbf{x}^{\mathrm{H}}\mathbf{A}_{0}\mathbf{x}$ is
majorized by
$\mathbf{x}^{\mathrm{H}}\mathbf{A}\mathbf{x}-2\mathrm{Re}\\{\mathbf{x}^{(n),\mathrm{H}}(\mathbf{A}-\mathbf{A}_{0})\mathbf{x}\\}+\mathbf{x}^{(n),\mathrm{H}}(\mathbf{A}-\mathbf{A}_{0})\mathbf{x}^{(n)}$
at $\mathbf{x}^{(n)}$.

By adopting Lemma 10 and setting
$\mathbf{A}=\lambda_{\max}(\mathbf{A}_{E})\mathbf{I}$ for simplicity, the
quadratic objective function in (69a) is majorized by

$\displaystyle 2\lambda_{\max}(\mathbf{A}_{E})M$
$\displaystyle-2\mathrm{Re}\\{\mathbf{e}^{(n),\mathrm{H}}(\lambda_{\max}(\mathbf{A}_{E})\mathbf{I}-\mathbf{A}_{E})\mathbf{e}\\}-\mathbf{e}^{(n),\mathrm{H}}\mathbf{A}_{E}\mathbf{e}^{(n)}$
(70)

at feasible point $\mathbf{e}^{(n)}$. To deal with the non-convex constraint
(69c), we replace $\mathbf{e}^{\mathrm{H}}\mathbf{A}_{K}\mathbf{e}$ with its
linear lower bound, resulting in the following equivalent constraint

$(\ref{eq:r-c1})\Rightarrow
2\mathrm{Re}\\{\mathbf{e}^{(n),\mathrm{H}}\mathbf{A}_{K}\mathbf{e}\\}\geq
e^{2\mathcal{R}}-1+\mathbf{e}^{(n),\mathrm{H}}\mathbf{A}_{K}\mathbf{e}^{(n)}.$
(71)

Step 2: Closed-form solution :

By omitting the constant, Problem (69) then becomes

$\displaystyle\max\limits_{\mathbf{e}}$ $\displaystyle\
2\mathrm{Re}\\{\mathbf{e}^{(n),\mathrm{H}}(\lambda_{\max}(\mathbf{A}_{E})\mathbf{I}-\mathbf{A}_{E})\mathbf{e}\\},$
(72a) s.t. $\displaystyle\ (\ref{eq:e-c}),(\ref{eq:r-c2}).$ (72b)

According to [25], we introducing a price mechanism for solving Problem (72),
i.e.,

$\displaystyle\max\limits_{\mathbf{e}}$ $\displaystyle\thinspace\thinspace
2\mathrm{Re}\\{\mathbf{e}^{(n),\mathrm{H}}(\lambda_{\max}(\mathbf{A}_{E})\mathbf{I}-\mathbf{A}_{E})\mathbf{e}\\}+\varrho_{1}2\mathrm{Re}\\{\mathbf{e}^{(n),\mathrm{H}}\mathbf{A}_{K}\mathbf{e}\\}$
$\displaystyle\mathrm{s.t.}$ (11d).

where $\varrho_{1}$ is a nonnegative price. Then, the globally optimal
solution is given by

$\mathbf{e}_{1}^{\\#}(\varrho_{1}^{opt})=\exp\\{\mathrm{j}\arg((\lambda_{\max}(\mathbf{A}_{E})\mathbf{I}-\mathbf{A}_{E}+\varrho_{1}\mathbf{A}_{K})\mathbf{e}^{(n)})\\}.$

The optimal $\varrho_{1}^{opt}$ is determined by using the bisection search
method, the detailed information about which can be found in [25].

Then, we solve $\mathcal{P}2$: $\mathcal{P}2$ is equivalent to

$\displaystyle\max\limits_{\mathbf{e}}$ $\displaystyle\
\frac{1+\mathbf{e}^{\mathrm{H}}\mathbf{A}_{K}\mathbf{e}}{1+\mathbf{e}^{\mathrm{H}}\mathbf{A}_{E}\mathbf{e}},$
(73a) $\displaystyle\mathrm{s.t.}$ $\displaystyle\ (\ref{eq:e-c}),$ (73b)
$\displaystyle\ \mathbf{e}^{\mathrm{H}}\mathbf{A}_{K}\mathbf{e}\leq
e^{2\mathcal{R}}-1.$ (73c)

Step 1: Construct a surrogate problem: Under the MM algorithm framework, we
construct a linear lower bound of the objective funcion in (73a) as

$\displaystyle\frac{1+\mathbf{e}^{\mathrm{H}}\mathbf{A}_{K}\mathbf{e}}{1+\mathbf{e}^{\mathrm{H}}\mathbf{A}_{E}\mathbf{e}}$
$\displaystyle\overset{\mathrm{(a)}}{\geq}\frac{2\mathrm{Re}\\{1+d_{K}\\}}{1+d_{E}^{(n)}}-\frac{1+d_{K}^{(n)}}{(1+d_{E}^{(n)})^{2}}\left(1+\mathbf{e}^{\mathrm{H}}\mathbf{A}_{E}\mathbf{e}\right)$
$\displaystyle\overset{\mathrm{(b)}}{\geq}\frac{2\mathrm{Re}\\{1+d_{K}\\}}{1+d_{E}^{(n)}}-\frac{1+d_{K}^{(n)}}{(1+d_{E}^{(n)})^{2}}-\frac{1+d_{K}^{(n)}}{(1+d_{E}^{(n)})^{2}}(2\lambda_{\max}(\mathbf{A}_{E})M$
$\displaystyle\
-2\mathrm{Re}\\{\mathbf{e}^{(n),\mathrm{H}}(\lambda_{\max}(\mathbf{A}_{E})\mathbf{I}-\mathbf{A}_{E})\mathbf{e}\\}-d_{E}^{(n)})$
$\displaystyle=2\mathrm{Re}\\{\mathbf{c}^{\mathrm{H}}\mathbf{e}\\}+\mathrm{const},$

where $d_{K}=\mathbf{e}^{(n),\mathrm{H}}\mathbf{A}_{K}\mathbf{e}$.
$\\{d_{K}^{(n)},d_{E}^{(n)},\mathbf{c}\\}$ are defined in Lemma 7. Inequality
(a) is due to Lemma 1, and inequality (b) is from Lemma 10. By using Lemma 10
again, the convex constraint (73c) can be replaced by an easy-to-solve form as

$\displaystyle(\ref{eq:r-c2-1})\Rightarrow
2\mathrm{Re}\\{\mathbf{e}^{(n),\mathrm{H}}(\lambda_{\max}(\mathbf{A}_{K})\mathbf{I}-\mathbf{A}_{K})\mathbf{e}\\}$
$\displaystyle\geq-2\lambda_{\max}(\mathbf{A}_{K})M-e^{2\mathcal{R}}+1-\mathbf{e}^{(n),\mathrm{H}}\mathbf{A}_{K}\mathbf{e}^{(n)}.$
(74)

Step 2: Closed-form solution: By omitting the constant, Problem (73) is then
equivalent to

$\displaystyle\max\limits_{\mathbf{e}}$ $\displaystyle\thinspace\thinspace
2\mathrm{Re}\\{\mathbf{c}^{\mathrm{H}}\mathbf{e}\\},$ (75a)
$\displaystyle\mathrm{s.t.}$ $\displaystyle\ (\ref{eq:e-c}),(\ref{eq:ff-1}).$
(75b)

By using the price mechanism, Problem (73) is reformulated as

$\displaystyle\max\limits_{\mathbf{e}}$ $\displaystyle\thinspace\thinspace
2\mathrm{Re}\\{\mathbf{c}^{\mathrm{H}}\mathbf{e}\\}+\varrho_{2}2\mathrm{Re}\\{\mathbf{e}^{(n),\mathrm{H}}(\lambda_{\max}(\mathbf{A}_{K})\mathbf{I}-\mathbf{A}_{K})\mathbf{e}\\}$
$\displaystyle\mathrm{s.t.}$ $\displaystyle(\ref{eq:e-c}).$

where $\varrho_{2}$ is a nonnegative price. The globally optimal solution is
given by
$\mathbf{e}_{2}^{\\#}(\varrho_{2}^{opt})=\exp\\{\mathrm{j}\arg(\mathbf{c}+\varrho_{2}(\lambda_{\max}(\mathbf{A}_{K})\mathbf{I}-\mathbf{A}_{K})\mathbf{e}^{(n)})\\}$
where the optimal $\varrho_{2}^{opt}$ is determined by using the bisection
search method.

Hence, the proof is completed.

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