# Covert Transmission Assisted by Intelligent Reflecting Surface

Jiangbo Si, Zan Li, _Senior Member, IEEE_ , Yan Zhao, Julian Cheng, _Senior
Member, IEEE_ , Lei Guan, Jia Shi, and Naofal Al-Dhahir, _Fellow, IEEE_
Jiangbo Si, Zan Li, Yan Zhao, Lei Guan, and Jia Shi are with the Integrated
Service Networks Lab of Xidian University, Xi’an, 710071, China. Jiangbo Si is
also a visiting scholar at the University of Texas at Dallas. (e-mail:
jbsi@xidian.edu.cn).Julian Cheng is with the School of Engineering, The
University of British Columbia, Kelowna, BC V1V 1V7, Canada (e-mail:
julian.cheng@ubc.ca).Naofal Al-Dhahir is with the Department of Electrical and
Computer Engineering, The University of Texas at Dallas, Richardson, TX 75080
USA.(e-mail: aldhahir@utdallas.edu).

###### Abstract

Covert transmission is studied for an intelligent reflecting surface (IRS)
aided communication system, where Alice aims to transmit messages to Bob
without being detected by the warden Willie. Specifically, an IRS is used to
increase the data rate at Bob under a covert constraint. For the considered
model, when Alice is equipped with a single antenna, the transmission power at
Alice and phase shifts at the IRS are jointly optimized to maximize the covert
transmission rate with either instantaneous or partial channel state
information (CSI) of Willie’s link. In addition, when multiple antennas are
deployed at Alice, we formulate a joint transmit beamforming and IRS phase
shift optimization problem to maximize the covert transmission rate. One
optimal algorithm and two low-complexity suboptimal algorithms are proposed to
solve the problem. Furthermore, for the case of imperfect CSI of Willie’s
link, the optimization problem is reformulated by using the triangle and the
Cauchy-Schwarz inequalities. The reformulated optimization problems are solved
using an alterative algorithm, semidefinite relaxation (SDR) and Gaussian
randomization techniques. Finally, simulations are performed to verify our
analysis. The simulation results show that an IRS can degrade the covert
transmission rate when Willie is closer to the IRS than Bob.

###### Index Terms:

Covert transmission, imperfect CSI, intelligent reflecting surface (IRS),
joint beamforming and phase shift optimization.

##  I Introduction

Exploiting reflecting surface is an energy and spectrum efficient technique to
enhance the transmission rate and reliability of wireless communications.
Different from the existing beamforming, multi-input multi-output and relay
techniques, intelligent reflecting surface (IRS) reflects the received signals
without dedicated energy and additional signal processing functions, such as
encoding, decoding, and modulation. The reflection is software controlled and
realized by adjusting the amplitudes and phase shifts of the IRS elements. By
proactively adjusting the amplitudes and phase shifts, the desired signals can
be strengthened significantly and the undesired signals can be suppressed or
cancelled effectively. Due to these advantages, IRS can be deployed densely in
future cellular networks to improve the transmission coverage and reliability.

Recently, there have been several works focusing on IRS-aided transmission. In
[1, 2], with or without channel knowledge, IRS was employed to improve single-
input single-output (SISO) system performance. In a multi-input single-output
(MISO) system, transmit beamforming, maximal ratio transmission (MRT), and IRS
phase shifts were adapted according to only the channel state information
(CSI) of the line of sight (LoS) components. The authors of [3] extended the
MISO system to the multi-input multi-output (MIMO) system, where the IRS
reflecting elements and the precoder at the transmitter were alternatively
optimized to minimize the symbol error rate. Moreover, the ergodic capacity
and optimal phase shifts were computed based on the statistical CSI in [4, 5].
These research results show that IRS can boost the received signal-to-noise
ratio (SNR) and enable ultra reliable communications at extremely low SNR.

In addition, IRS can be applied in multi-user systems. For example, in multi-
user MISO downlink communications, transmit beamforming and IRS phase shifts
were jointly optimized according to the instantaneous CSI of all links in [6],
where zero-forcing beamforming was applied at the transmitter to eliminate the
multiuser interference and the majorization-minimization algorithm was
proposed to maximize the total transmission rate. For the same scenario, the
authors in [7] studied the reflect beamforming design to maximize the minimum
signal-to-interference-plus-noise ratio (SINR) with only the channel’s
statistical CSI. For the case of broadcasting messages to multiple users, the
transmission beamforming vector at the base station (BS) and the phase shifts
at the IRS were jointly optimized to minimize the total transmission power
under quality of service (QOS) constraints at each user [8]. In addition,
joint active beamforming at the transmitter and passive beamforming at the IRS
was optimized by an iterative method in [9], where beamforming was designed by
applying the well-known minimum mean squared error (MMSE) criterion to cope
with the multiuser interference. Subsequently, since the continuous phase
shift is costly to be realized in practice, joint continuous transmit
beamforming at the AP and discrete phase shifts reflect beamforming at the IRS
was addressed in [10]. It was shown that discrete IRS phase shifts incurred a
constant performance loss that depends only on the number of phase shift
levels. Moreover, the IRS was used to assist the downlink transmission to
cell-edge users by mitigating the inter-cell interference [11]. All of the
above studies show that the IRS can improve the transmission rate and decrease
the energy consumption in multiuser systems.

Furthermore, IRS was also deployed to enhance the secrecy transmission
performance. For an IRS-aided multi-input single-output single-antenna
eavesdropper (MISOSE) system, the transmit beamforming vector at Alice and
phase shifts at the IRS were jointly optimized for secrecy rate maximization
in [12] or transmission power minimization in [13], where an alterative
optimization algorithm and semidefinite program (SDP) were used. The block
coordinate descent (BCD) and minorization maximization (MM) techniques were
also applied to solve the same optimization problem [14]. When multiple
eavesdroppers and multiple legitimate users coexist in a MISO downlink
communication system, joint active beamforming at the BS and passive
beamforming at the IRS were optimized to maximize the minimum secrecy rate
[15]. The authors of [16, 17] extended the optimization problem to multi-input
multi-output multi-antenna eavesdropper (MIMOME) systems. In addition, a
beamforming and artificial noise (AN) scheme was proposed in an IRS-aided
system to enhance secrecy performance [18].

Covert transmission is an emerging and cutting-edge communication security
technique, which guarantees a negligible detection probability at a warden
[19]. Covert transmission can protect confidential information from detection
and can be used in systems with high security requirements, such as finance,
national security, and military. There are many papers focusing on covert
transmission [20, 21, 22, 23]. However, few works have investigated IRS-aided
covert transmission. In [24], the authors introduced the IRS technique in
covert transmission, and outlined the possible applications of IRS in covert
transmission systems, but they did not investigate IRS-aided covert
transmission techniques or quantify their performance. For an IRS-aided covert
transmission, on the one hand, the covert signal received at the legitimate
receiver should be strengthened by IRS reflection. On the other hand, the
covert signal received at the warden should be weakened by IRS reflection.
Hence, it is challenging to achieve a trade-off between strengthening the
legitimate receiver’s link and weakening the warden’s link. This paper fills
this gap and provides insights on IRS design in covert transmission. In
addition, considering that multiple antennas is an adopted technology in the
fifth-generation wireless (5G) systems, we consider the IRS-aided covert
transmission with either a single antenna or multiple antennas at the
transmitter. Furthermore, the covert transmission rate of an IRS-aided system
is investigated under different kinds of CSI: instantaneous CSI of Willie’s
link, partial CSI of Willie’s link, and imperfect CSI of Willie’s link. The
main contributions of this paper are summarized as follows:

  * 1.

For the first time, the performance of IRS-aided covert transmission is
comprehensively investigated. IRS is used to strengthen the signal power
received at Bob and to weaken the signal power received at Willie. When Alice
is equipped with a single antenna, a joint transmission power and phase shifts
optimization problem is formulated to maximize the covert transmission rate.
According to partial CSI of Willie’s link, the optimal transmission power
satisfying the covert constraint is first obtained. Then, the triangle
inequality is used to derive the optimal phase shifts at the IRS. When full
instantaneous CSI of Willie’s link is available, the coupled transmission
power and phase shifts are alternatively optimized for covert rate
maximization. Analysis and simulation results show that IRS is not necessarily
beneficial to covert transmission, even if the partial or instantaneous CSI of
Willie’s link are available at Alice.

  * 2.

When Alice is equipped with multiple antennas, a joint transmit beamforming
and phase shifts optimization problem is formulated to maximize the covert
transmission rate. When only partial CSI of Willie’s link is available,
optimal transmit beamforming, namely MRT, is applied at Alice. Then, SDP is
used to calculate the optimal IRS phase shifts. When all the channels’
instantaneous CSI are available, optimal transmit beamforming with given phase
shifts and optimal phase shifts with given transmit beamforming are performed
iteratively until the accuracy of the covert transmission rate is satisfied.
Moreover, to decrease the algorithm’s complexity, two suboptimal algorithms
are proposed to minimize the signal power received at Willie. Simulation
results show that the covert transmission rate without IRS assistance can be
greater than that with IRS assistance. Moreover, although increasing the
number of IRS elements can improve the covert transmission performance, the
amount of increase is affected significantly by the channel quality of the
IRS-to-Bob links.

  * 3.

Considering that Willie is often unfriendly to Alice, we investigate the
impact of imperfect CSI of Willie’s link on IRS-aided covert transmission. By
using the triangle and the Cauchy-Schwarz inequalities, we reformulate the
optimization problem under the cases of imperfect CSI of the Alice-to-Willie
link, imperfect CSI of the Alice-to-IRS link, imperfect CSI of the IRS-to-
Willie link, and imperfect CSI of both the Alice-to-Willie and the IRS-to-
Willie links. The alternative optimization algorithm, semidefinite relaxation
(SDR), and Gaussian randomization techniques are used to solve the
optimization problem and compute the maximum achievable covert transmission
rate. We find that imperfect CSI of the Alice-to-Willie link degrades the
covert transmission rate significantly.

The remainder of this paper is organized as follows. Section II describes the
system model. A joint transmission power and phase shifts optimization problem
is proposed and solved with either partial CSI or instantaneous CSI of
Willie’s link in Section III. In Section IV, when Alice is equipped with
multiple antennas, the transmit beamforming and IRS phase shifts are jointly
optimized under both power and covert constraints. An optimal algorithm and
two suboptimal algorithms are proposed to solve the problem. In Section V, the
optimization problem is reformulated and solved with imperfect CSI of Willie’s
link. Numerical results are presented and discussed in Section VI. Finally, we
conclude the paper in Section VII.

_Notations_ \- $\left(\cdot\right)^{T}$ and $\left(\cdot\right)^{H}$ denote
the transpose and conjugate transpose, respectively; The operator
$\left|\cdot\right|$ denotes the absolute value;
$\left\|{\left.\cdot\right\|}\right.$ denotes the Frobenius norm;
$f_{\upsilon}\left(\cdot\right)$ is the probability density function (PDF) of
random variable (RV) $\upsilon$; $F_{\upsilon}\left(\cdot\right)$ is the
cumulative distribution function (CDF) of RV $\upsilon$; $\mathbb{E}{(\cdot)}$
and $Var{(\cdot)}$ denote the expected value and the variance of a RV,
respectively; $Re{(\cdot)}$ and $Im{(\cdot)}$ respectively denote the real
part and the imaginary part of a complex RV; $\exp(\sigma^{2})$ denotes
exponential distribution having mean $\sigma^{2}$; $arg\left(\bf{x}\right)$
stands for the phase vector of $\bf{x}$. $Tr\left(\bf{X}\right)$ denotes the
trace of matrix $\bf{X}$. ${\lambda_{\min}}\left({\bf{X}}\right)$ and
${\lambda_{\max}}\left({\bf{X}}\right)$, respectively, denote the minimum and
maximum eigenvalues of matrix ${{{\bf{X}}}}$; ${{{\mathbb{C}}}^{m\times n}}$
denotes the ${m\times n}$ complex number domain; ${\bf{I}}_{M}$ denotes the
$M\times M$ identity matrix;
${\bf{x}}\sim\mathcal{CN}\left({{\bf{\Lambda}},{\bf{\Delta}}}\right)$ denotes
the circular symmetric complex Gaussian vector with mean vector $\bf{\Lambda}$
and covariance matrix ${\bf{\Delta}}$. In addition, the symbol notations are
given in Table I.

TABLE I: Summary of Notations Notation | Description
---|---
${\bf{h}}_{a,j}$ |  Instantaneous CSI between Alice and node $j$, $j\in\\{{Bob(b),Willie(w),IRS(s)}\\}$
${\bf{\tilde{h}}}_{a,j}$ |  Imperfect CSI between Alice and node $j$, $j\in\\{{Willie(w),IRS(s)}\\}$
${\Delta\bf{{h}}}_{a,j}$ |  Estimation error of ${\bf{h}}_{a,j}$
${\bf{g}}_{s,j}$ |  Instantaneous CSI between IRS and node $j$, $j\in\\{{Bob(b),Willie(w)}\\}$
${\bf{\tilde{g}}}_{s,j}$ |  Imperfect CSI between IRS and node $j$. $j\in\\{{Willie(w)}\\}$
${\Delta\bf{{g}}}_{s,j}$ |  Estimation error of ${\bf{g}}_{s,j}$
${\zeta_{i,j}}$ |  The norm-bound of channel estimation errors between node $i$ and node $j$
${\bf{\Theta}}$ | The diagonal reflecting matrix at the IRS
$\bf{v}$ | The IRS phase shifts vector
$\theta_{i}$ |  The phase angle of the $i$-th IRS element
$\bf{w}$ | Transmit beamforming vector at Alice
$\sigma^{2}_{j}$ |  The noise covariance at node $j$, $j\in\\{{Bob(b),Willie(w)}\\}$
$\rho$ | The noise uncertainty coefficient at Willie
$\lambda$ | The detection threshold at Willie
$\Gamma$ | The required accuracy for the covert transmission rate
$P_{D}$ | The transmission power at Alice without IRS assistance
$P_{I}$ | The transmission power at Alice with IRS assistance
${P_{\max}}$ | The transmission power constraint at Alice
$\xi$ | The conditional detection error probability at Willie
$\kappa$ | A required threshold of detection error probability for covert transmission
$\eta$ | The maximum allowed received power at Wille to satisfy covert constraint

##  II System Model

Figure 1: Covert transmission with IRS assistance

As shown in Fig. 1, Alice wishes to transmit messages to Bob without being
detected by Willie, who is a warden and tries to detect whether there exists a
transmission from Alice or not. Moreover, Alice resorts to IRS with $N$
reflecting elements for improving the covert transmission rate. Both Bob and
Willie are equipped with a single antenna, while $M$ antennas are deployed at
Alice. The instantaneous CSI of the Alice $\rightarrow$ Bob and Alice
$\rightarrow$ Willie links are, respectively, denoted by
${\bf{h}}_{a,b}\in\mathbb{C}^{M\times
1}\sim\mathcal{CN}\left({{\bf{0}},\sigma^{2}_{a,b}{{\bf{I}}_{N}}}\right)$ and
${\bf{h}}_{a,w}\in\mathbb{C}^{M\times
1}\sim\mathcal{CN}\left({{\bf{0}},\sigma^{2}_{a,w}{{\bf{I}}_{N}}}\right)$. In
addition, ${{\bf{h}}_{a,s}}\in\mathbb{C}^{N\times
M}\sim\mathcal{CN}\left({{\bf{0}},\sigma^{2}_{a,s}{{\bf{I}}_{N}}}\right)$,
${\bf{g}}_{s,b}\in\mathbb{C}^{N\times
1}\sim\mathcal{CN}\left({{\bf{0}},\sigma^{2}_{s,b}{{\bf{I}}_{N}}}\right)$, and
${\bf{g}}_{s,w}\in\mathbb{C}^{N\times
1}\sim\mathcal{CN}\left({{\bf{0}},{\sigma^{2}_{s,w}}{{\bf{I}}_{N}}}\right)$
denote the instantaneous CSI of the Alice $\rightarrow$ IRS, IRS $\rightarrow$
Bob, and IRS $\rightarrow$ Willie links, respectively. The diagonal reflection
matrix for the IRS is
${\bf{\Theta}}=diag\left({{e^{j{\theta_{1}}}},{e^{j{\theta_{2}}}},\cdots,{e^{j{\theta_{N}}}}}\right)$,
where the amplitude of each element is assumed to be equal to $1$. The maximum
transmission power at Alice is $P_{max}$.

For covert transmission, we assume that Willie employs an energy detection
method to determine whether there exists a transmission from Alice. Due to the
electromagnetic environment variations, there is noise uncertainty at Willie,
which affects the covert rate significantly. When the bounded uncertainty
model is applied, the PDF of the noise power at Willie is given by [25, 26]

$\displaystyle\
{f_{{\sigma^{2}}_{w}}}\left(x\right)=\left\\{\begin{array}[]{l}\frac{1}{{2\ln\left(\rho\right)x}},\frac{1}{\rho}{{\tilde{\sigma}}^{2}}_{w}\leq
x\leq\rho{{\tilde{\sigma}}^{2}}_{w}\\\
\begin{array}[]{*{20}{c}}{0,}&{}\hfil&{\begin{array}[]{*{20}{c}}{}\hfil&{otherwise}\end{array}}\end{array}\end{array}\right.$
(5)

where $\rho$ is the noise uncertainty coefficient, and
${\tilde{\sigma}^{2}}_{w}$ denotes the noise power without noise uncertainty,
i.e. $\rho=1$. The noise power at Willie satisfies the following inequality

$\displaystyle\
\frac{1}{\rho}{\tilde{\sigma}^{2}}_{w}\leq{\sigma^{2}}_{w}\leq\rho{\tilde{\sigma}^{2}}_{w}.$
(6)

In addition, since Bob does not detect the signal, we assume that the noise
power at Bob is the additive noise (AWGN) with zero mean and fixed variance
$\sigma^{2}_{b}$. The covert transmission is evaluated under two events:
$H_{0}$ and $H_{1}$, where $H_{0}$ denotes the null hypothesis that Alice does
not transmit messages to Bob, and $H_{1}$ denotes the alternative hypothesis
that Alice transmits messages to Bob. In this paper, we assume that the
transmission is covert when the conditional detection error probability (DEP),
which is defined as the summation of the miss detection probability under
$H_{1}$ and the false alarm probability under $H_{0}$, is larger than a
predetermined threshold $1-\kappa$ [26, 27].

##  III Single Antenna at Alice

In this section, we investigate IRS-aided covert transmission with a single
antenna deployed at Alice ($M=1$). The covert transmission rate for an IRS
assisted system is studied when either the instantaneous or partial CSI of
Willie’s link is known to Alice. Under both cases, considering that Bob is
always friendly to Alice, we assume that the full instantaneous CSI of Bob’s
link is perfectly known at Alice.

###  III-A With Partial CSI of Willie’s Link

Note that without a covert constraint, the transmission rate at Alice is given
by

$\displaystyle\ {R_{a}}=\log
2\left({1+\frac{{{P_{I}}\left|{{h^{H}}_{a,b}+{{\bf{g}}^{H}}_{s,b}{\bf{\Theta}}{{\bf{h}}_{a,s}}}\right|}}{{{\sigma^{2}}_{b}}}}\right)$
(7)

Neglecting the logarithmic function and constant variables of (7), the
optimization problem to maximize the covert constraint is formulated as

$\displaystyle\mathop{\max}\limits_{P,\bf{\Theta}}{P_{I}\left|{{h^{H}_{a,b}}+{{\bf{g}}^{H}_{s,b}}{\bf{\Theta}}{{\bf{h}}_{a,s}}}\right|^{2}}$
(8a) $\displaystyle{s.t.}\quad{P_{I}\leq{P_{\max}}}$ (8b)
$\displaystyle\quad\quad{{P_{FA}}+{P_{MD}}\geq 1-\kappa}$ (8c)

where the transmission power at Alice $P_{I}$ and the diagonal phase shifts
matrix $\bf{\Theta}$ are jointly optimized for covert rate maximization.
Equation (8a) denotes the optimization goal: maximizing the received signal
power at Bob, (8b) denotes the total transmit power constraint at Alice, and
the covert transmission constraint is given in (8c), where ${P_{FA}}$ and
${P_{MD}}$ denote, respectively, the false alarm probability and the miss
detection probability, and will be defined shortly. When only partial CSI of
Willie’s link is available, the IRS phase shifts are optimized according to
instantaneous CSI of Bob’s link. Then, optimal phase shifts for Bob’s link are
random for Willie’s link, and the covert constraint is only determined by the
transmission power $P_{I}$. Hence, to solve optimization problem (8a), we
first derive the maximum permitted transmission power, denoted by $P^{*}_{I}$,
which is determined by the covert constraint.

Covert transmission is often evaluated under two events: $H_{0}$ and $H_{1}$.
For the two events, the average signal power received at Willie is given by

$\displaystyle\
{T_{w}}=\left\\{{\begin{array}[]{*{20}{c}}{\sigma_{w}^{2}}&{{H_{0}}}\\\
{P_{I}{\left|{{h^{H}_{a,w}}+{{\bf{g}}^{H}_{s,w}}\bf{\Theta}}{{\bf{h}}_{a,s}}\right|^{2}}+\sigma_{w}^{2}}&{{H_{1}}}.\end{array}}\right.$
(11)

According to (11), the probability of missed detection ${P_{MD}}$ and the
probability of false alarm ${P_{FA}}$ are, respectively, given by

$\displaystyle\
{P_{FA}}=\Pr\left({{T_{w}}\geq\lambda|{H_{0}}}\right)=\Pr\left({{\sigma^{2}}_{w}\geq\lambda|{H_{0}}}\right)$
(12)

and

$\displaystyle\
{P_{MD}}=\Pr\left({\left.{{T_{w}}\leq\lambda}\right|{H_{1}}}\right)=\Pr\left({P_{I}{{\left|{{h^{H}_{a,w}}+{{\bf{g}}^{H}_{s,w}}\bf{\Theta}}{{\bf{h}}_{a,s}}\right|}^{2}}+{\sigma^{2}}_{w}\leq\lambda|{H_{1}}}\right)$
(13)

where $\lambda$ is the detection threshold at Willie. Then, under the
assumption that the apriori probability of hypotheses either $H_{0}$ or
$H_{1}$ being true is 0.5 [22], the conditional DEP is

$\displaystyle\ \xi$ $\displaystyle={P_{MD}}+{P_{FA}}$
$\displaystyle=1-\Pr\left({\lambda-
P_{I}{{\left|{{h^{H}_{a,w}}+{{\bf{g}}^{H}_{s,w}}\bf{\Theta}}{{\bf{h}}_{a,s}}\right|}^{2}}<{\sigma^{2}}_{w}<\lambda}\right).$
(14)

It is noted that when $\lambda<\frac{1}{\rho}{\tilde{\sigma}^{2}}_{w}$,
${P_{FA}}=1$ and ${P_{MD}}=0$. By contrast, when
$\lambda>{\rho}{\tilde{\sigma}^{2}}_{w}$, ${P_{FA}}=0$ and ${P_{MD}}$
increases with $\lambda$. Thus, from Willie’s perspective, to minimize the
DEP, the optimal detection threshold $\lambda$ should satisfy the following
constraint

$\displaystyle\
\frac{1}{\rho}{\tilde{\sigma}^{2}}_{w}\leq\lambda\leq{\rho}{\tilde{\sigma}^{2}}_{w}.$
(15)

Let
$z={{\left|{{h_{a,w}}+{{\bf{h}}_{a,s}}{\bf{\Theta}}{{\bf{g}}_{s,w}}}\right|}^{2}}$,
then, the average DEP conditioned on $\lambda$ is written as

$\displaystyle\
\bar{\xi}\left(\lambda\right)=1-\int_{\max\left({\lambda-{{zP_{I}}},\frac{1}{\rho}{{\tilde{\sigma}}^{2}}_{w}}\right)}^{\min\left({\lambda,\rho{{\tilde{\sigma}}^{2}}_{w}}\right)}{{f_{{{\sigma^{2}_{w}}}}}\left(x\right)}dx=1-\int_{\max\left({\lambda-{{zP_{I}}},\frac{1}{\rho}{{\tilde{\sigma}}^{2}}_{w}}\right)}^{\lambda}{{f_{{{\sigma^{2}_{w}}}}}\left(x\right)}dx.$
(16)

Differentiating $\bar{\xi}\left(\lambda\right)$ with respect to (w.r.t.)
$\lambda$, we obtain

$\displaystyle\
{{2\ln\left(\rho\right)}}\frac{{\partial\left({\bar{\xi}\left(\lambda\right)}\right)}}{{\partial\lambda}}=\left\\{\begin{array}[]{l}\begin{array}[]{*{20}{c}}{-\frac{1}{\lambda},}&{}\hfil&{\lambda\leq
zP_{I}+\frac{{{{\tilde{\sigma}}^{2}}_{w}}}{\rho}}\end{array}\\\
\begin{array}[]{*{20}{c}}{-\frac{1}{\lambda}+\frac{1}{{\lambda-
zP_{I}}},}&{}\hfil&{\lambda>zP_{I}+\frac{{{{\tilde{\sigma}}^{2}}_{w}}}{\rho}}.\end{array}\end{array}\right.$
(21)

Hence, when ${\lambda\leq zP_{I}+\frac{{{{\tilde{\sigma}}^{2}}_{w}}}{\rho}}$,
$\frac{{\partial\left({\bar{\xi}\left(\lambda\right)}\right)}}{{\partial\lambda}}<0$,
and when ${\lambda>zP_{I}+\frac{{{{\tilde{\sigma}}^{2}}_{w}}}{\rho}}$,
$\frac{{\partial\left({\bar{\xi}\left(\lambda\right)}\right)}}{{\partial\lambda}}>0$.
Then, according to (15), to minimize the DEP, the optimum detection threshold
$\lambda^{*}$ is given by

$\displaystyle\
{\lambda^{*}}=\min\left({zP_{I}+\frac{{{{\tilde{\sigma}}^{2}}_{w}}}{\rho},\rho{{\tilde{\sigma}}^{2}}_{w}}\right).$
(22)

Substituting (22) into (16), we obtain the minimum DEP conditioned on $z$ as

$\displaystyle\
\bar{\xi}\left(\lambda\right){\rm{=}}\left\\{\begin{array}[]{l}{\rm{1-}}\frac{1}{{{\rm{2}}\ln\left(\rho\right)}}\ln\left({1+\rho
z\frac{P_{I}}{{{{\tilde{\sigma}}^{2}}_{w}}}}\right),z\leq\frac{{{{\tilde{\sigma}}^{2}}_{w}}}{P_{I}}\left({\rho-\frac{1}{\rho}}\right)\\\
\begin{array}[]{*{20}{c}}0&{}\hfil&{}\hfil&{\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil\end{array}}&{}\hfil\end{array}}&{}\hfil&{,z>\frac{{{{\tilde{\sigma}}^{2}}_{w}}}{P_{I}}\left({\rho-\frac{1}{\rho}}\right)}.\end{array}}\end{array}\end{array}\right.$
(29)

Then, averaging $\bar{\xi}\left(\lambda\right)$ w.r.t. $z$, we can obtain the
average minimum DEP as

$\displaystyle\ \xi$
$\displaystyle=\int_{0}^{\frac{{{{\tilde{\sigma}}^{2}}_{w}}}{P_{I}}\left({\rho-\frac{1}{\rho}}\right)}{\bar{\xi}\left(\lambda\right)}{f_{{z}}}\left(z\right)dz$
$\displaystyle{{=}}\int_{\rm{0}}^{\frac{{{{\tilde{\sigma}}^{2}}_{w}}}{{{P_{I}}}}\left({\rho-\frac{1}{\rho}}\right)}{\left({1-\frac{1}{{2\ln\left(\rho\right)}}\ln\left({1+\rho
z\frac{{{P_{I}}}}{{{{\tilde{\sigma}}^{2}}_{w}}}}\right)}\right)}\frac{{\exp\left({-\frac{z}{{{\sigma^{2}}_{s,w}+N{\sigma^{2}}_{a,s}{\sigma^{2}}_{s,w}}}}\right)}}{{\left({{\sigma^{2}}_{s,w}+N{\sigma^{2}}_{a,s}{\sigma^{2}}_{s,w}}\right)}}dz$
$\displaystyle=1-\frac{1}{{2\ln\left(\rho\right)}}\int_{0}^{\frac{{{{\tilde{\sigma}}^{2}}_{w}}}{{{P_{I}}}}\left({\rho-\frac{1}{\rho}}\right)}{\exp\left({-\frac{z}{{{\sigma^{2}}_{s,w}+N{\sigma^{2}}_{a,s}{\sigma^{2}}_{s,w}}}}\right)\frac{{\rho\frac{{{P_{I}}}}{{{{\tilde{\sigma}}^{2}}_{w}}}}}{{1+\rho
z\frac{{{P_{I}}}}{{{{\tilde{\sigma}}^{2}}_{w}}}}}}dz$
$\displaystyle=1-\frac{1}{{2\ln\left(\rho\right)}}\int_{1}^{{\rho^{2}}}{\exp\left({-\frac{{\left({y-1}\right)}}{{{{\bar{\gamma}}^{I}}_{w}\rho}}}\right)\frac{1}{y}}dy$
$\displaystyle=1-\frac{{\exp\left({\frac{1}{{{{\bar{\gamma}}^{I}}_{w}\rho}}}\right)}}{{2\ln\left(\rho\right)}}\left({\int_{-\infty}^{{\rho^{2}}}{\exp\left({-\frac{y}{{{{\bar{\gamma}}^{I}}_{w}\rho}}}\right)\frac{1}{y}}dy-\int_{-\infty}^{1}{\exp\left({-\frac{y}{{{{\bar{\gamma}}^{I}}_{w}\rho}}}\right)\frac{1}{y}}dy}\right)$
$\displaystyle=1-\frac{{\exp\left({\frac{1}{{\rho\bar{\gamma}^{I}_{w}}}}\right)}}{{2\ln\left(\rho\right)}}\left({Ei\left({-\frac{{\rho}}{{\bar{\gamma}^{I}_{w}}}}\right)-Ei\left({-\frac{1}{{\rho\bar{\gamma}^{I}_{w}}}}\right)}\right)$
(30)

where
$\bar{\gamma}^{I}_{w}={\frac{{\left({{\sigma^{2}}_{s,w}{\rm{+}}N{\sigma^{2}}_{a,s}{\sigma^{2}}_{s,w}}\right)P_{I}}}{{{{\tilde{\sigma}}^{2}}_{w}}}}$
denotes the average SNR at Willie with IRS assistance, $y=1+\rho
z\frac{{{P_{I}}}}{{{{\tilde{\sigma}}^{2}}_{w}}}$,
$Ei\left(x\right)=\int_{-\infty}^{x}{\exp\left(t\right){t^{-1}}}dt$, and
${f_{\rm{z}}}\left(z\right)$ denotes the PDF of $z$, which is derived in
Appendix A. According to (8c) and (III-A), the maximum allowed transmission
power $\tilde{P_{I}}$ can be computed numerically or using a look-up table at
Matlab. In addition, considering the transmission power constraint in (8b),
the optimum transmission power is given by
$P^{*}_{I}=\min\left({P_{max},\tilde{P_{I}}}\right)$. After the optimal
transmission power is obtained, we derive the optimal phase shifts by
maximizing Bob’s link power gain, given by
${\left|{{h^{H}_{a,b}}+{{\bf{g}}^{H}_{s,b}}{\bf{\Theta}}{{\bf{h}}_{a,s}}}\right|^{2}}$.
Let
${{\bf{v}}^{H}}=\left[{{e^{j{\theta_{1}}}},{e^{j{\theta_{2}}}},\cdots,{e^{j{\theta_{N}}}}}\right]$,
then,
${{\bf{g}}^{H}_{s,b}}{\bf{\Theta}}{{\bf{h}}_{a,s}}={{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){{\bf{h}}_{a,s}}$.
Thus, Bob’s link power gain satisfies the following inequality

$\displaystyle\
{\left|{{h^{H}_{a,b}}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){{\bf{h}}_{a,s}}}\right|^{2}}\leq\left(\left|{h^{H}_{a,b}}\right|+\left|{{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){{\bf{h}}_{a,s}}}\right|\right)^{2}.$
(31)

From (31), we find that when equality holds, ${h^{H}_{a,b}}$ and
${{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){{\bf{h}}_{a,s}}$ have
the same angle, and Bob’s link power gain is maximized. Thus, the optimal IRS
phase shifts are given by

$\displaystyle\
{\theta^{*}_{i}}=arg\left(h_{a,b}\right)-{arg({{g}^{H}_{{s_{i}},b}})}-{arg({h_{a,{s_{i}}}})},1\leq
i\leq N.$ (32)

At the optimal ${\theta^{*}_{i}}$, the SNR at Bob is given by

$\displaystyle\
{\gamma^{\prime}}_{IRS}=\frac{{{P^{*}_{I}}{{\left({\sum\limits_{i=1}^{N}{\left|{{h_{a,{s_{i}}}}}\right|\left|{{g_{{s_{i}},b}}}\right|}+\left|{{h_{a,b}}}\right|}\right)}^{2}}}}{{{\sigma^{2}_{b}}}}.$
(33)

_Discussion:_ For comparison, we study the covert transmission rate without
IRS assistance. For this case, $z$ is reduced to
$z={{\left|{{h_{a,w}}}\right|}^{2}}$, and the PDF of $z$ is
${f_{z}}\left(z\right)=\frac{1}{{{\sigma^{2}}_{a,w}}}\exp\left({-\frac{z}{{{\sigma^{2}}_{a,w}}}}\right)$.
Thus, the DEP without IRS aid is given by

$\displaystyle\ \xi$
$\displaystyle=1-\frac{{\exp\left({\frac{1}{{\rho\bar{\gamma}^{d}_{D}}}}\right)}}{{2\ln\left(\rho\right)}}\left({Ei\left({-\frac{{\rho}}{\bar{\gamma}^{d}_{D}}}\right)-Ei\left({-\frac{{1}}{{\rho\bar{\gamma}^{d}_{D}}}}\right)}\right)$
(34)

where
$\bar{\gamma}^{d}_{D}={\frac{{{{\sigma^{2}}_{s,w}}P_{D}}}{{{{\tilde{\sigma}}^{2}}_{w}}}}$
denotes the average SNR at Willie without IRS assistance. According to the
covert constraint and (34), the allowed transmission power $\tilde{P_{D}}$ can
be calculated. Then, considering the maximum power constraint, the optimal
direct transmission power is
$P^{*}_{D}=\min\left({P_{max},\tilde{P_{D}}}\right)$. Thus, without IRS
assistance, the SNR at Bob is given by

$\displaystyle\
{\gamma^{\prime}}_{dir}=\frac{{P^{*}_{D}{{\left({\left|{{h_{a,b}}}\right|}\right)}^{2}}}}{{{\sigma^{2}_{b}}}}.$
(35)

Compared (33) to (35), we find that the IRS strengthens Bob’s link power gain
since
${{\left({\sum\limits_{i=1}^{N}{\left|{{h_{a,{s_{i}}}}}\right|\left|{{g_{{s_{i}},b}}}\right|}+\left|{{h_{a,b}}}\right|}\right)}^{2}}\geq\left|{{h_{a,b}}}\right|^{2}$.
However, the signal power received at Willie is also strengthened due to IRS
assistance, which can be seen from (81). As a result, $P^{*}_{D}\geq
P^{*}_{I}$. Hence, when only partial CSI of Willie’s link is available, it is
not necessary that the covert rate of IRS aided system is greater than that
without IRS assistance. Only when the covert constraint is loose, or Willie’s
link quality is poor, does IRS become beneficial to covert transmission
performance. In these cases, the covert constraint has little impact on the
transmission power at Alice, and the effect of Bob’s link dominates the covert
transmission rate.

###  III-B With Instantaneous CSI of Willie’s Link

In practice, Alice, Bob and Willie can be in the same network, where Alice
wishes to send confidential messages to Bob, and assumes Willie to be an
untrusted or suspicious node. For this scenario, Willie can be friendly to
Alice, and Alice has perfect knowledge of the instantaneous CSI of Willie’s
link. When Willie is friendly to Alice, several methods can be used to
estimate the IRS link. The active channel sensors can be inserted into the
array of IRS passive elements to sense channel information [28]. In addition,
the active sensors can work in the channel sensing mode for channel estimation
and in the reflection mode for electromagnetic reflection. This two-function
method at active sensors increases the hardware complexity and requires extra
energy and time to transfer the estimated channel information towards the
controller. Moreover, the cascaded IRS channels can be decomposed into a
series of sub-channels, which can be realized by turning on one IRS element
and turning off the other IRS elements [29]. Finally, structure-learning based
CSI acquisition can also be used by exploiting strong structural features,
such as sparsity and low-rank. With this structural information, the
estimation of a cascaded channel can be performed by utilizing advanced signal
processing tools such as compressed sensing, sparse matrix factorization, and
low-rank matrix recovery algorithms [30, 31, 32]. Moreover, the covert rate
with perfect instantaneous CSI of Willie’s link can be an upper bound on that
with partial or imperfect CSI of Willie’s link. Hence, it is necessary to
investigate the covert rate with perfect CSI of Willie’s link. In this case,
the transmission power at Alice and IRS phase shifts are changed continuously
according to both Bob’s link and Willie’s link, which is different from the
partial CSI case. To be undetected by Willie, the DEP should be larger than
$1-\kappa$. According to (29), the signal power received at Willie should be
less than the covert transmission threshold
$\eta=\min\left({{{\tilde{\sigma}}^{2}}_{w}\left({\rho-\frac{1}{\rho}}\right),\frac{{\left({\rho^{2\kappa}-1}\right){{\tilde{\sigma}}^{2}}_{w}}}{\rho}}\right)$.
Therefore, the optimization problem for covert rate maximization is given by

$\displaystyle\mathop{\max}\limits_{{\bf{v}},P_{I}}\quad{P_{I}}{\left|{{h^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){{\bf{h}}_{a,s}}}\right|^{2}}$
$\displaystyle s.t.\quad{P_{I}}\leq{P_{\max}}$
$\displaystyle\quad\quad{{P_{I}}{{\left|{{h^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){{\bf{h}}_{a,s}}}\right|}^{2}}\leq\eta}$
(36)

where the transmission power $P_{I}$ and phase shifts vector $\bf{v}$ are
coupled. It is challenging to derive the exact expressions for the optimal
$P_{I}$ and ${\bf{v}}$. Hence, we use an alterative algorithm to solve this
optimization problem. For a given phase shifts vector ${\bf{v}}$, to satisfy
the covert constraint, the optimal transmission power is given by

$\displaystyle\
P^{*}_{I}=\min\left({{P_{\max}},\frac{\eta}{{{{\left|{{h^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){{\bf{h}}_{a,s}}}\right|}^{2}}}}}\right).$
(37)

Then, for a given $P_{I}$, the optimization problem (III-B) is rewritten as

$\displaystyle\mathop{\max}\limits_{\bf{V}}\quad{P_{I}}\left({Tr\left({{{\bf{T}}{{}_{a,b}\bf{V}}}}\right)+{{\left|{{h_{a,b}}}\right|}^{2}}}\right)$
$\displaystyle{s.t.}\quad{{P_{{I}}}\left({Tr\left({{\bf{T}}_{a,w}{\bf{V}}}\right)+{{\left|{{h_{a,w}}}\right|}^{2}}}\right)\leq\eta}$
$\displaystyle\quad\quad\quad{{{\bf{V}}_{n,n}}=1},{{\bf{V}}\succeq
0},{rank}\left({\bf{V}}\right)=1$ (38)

where ${\bf{V}}={\bf{\bar{v}}}{{{\bf{\bar{v}}}}^{H}}$,
${{{\bf{\bar{v}}}}^{H}}=t^{H}\left[{{{\bf{v}}^{H}},1}\right]$, and $t$ is an
auxiliary variable satisfying $\left|t\right|=1$. In addition,
${{\bf{T}}_{a,j}}$ $(j\in(b,w))$ is given by

$\displaystyle\
{{\bf{T}}_{a,j}}{{=}}\left[\begin{array}[]{l}diag\left({{{\bf{g}}^{H}}_{s,j}}\right){{\bf{h}}_{a,s}}{{\bf{h}}^{H}}_{a,s}diag\left({{{\bf{g}}_{s,j}}}\right),diag\left({{{\bf{g}}^{H}}_{s,j}}\right){{\bf{h}}_{a,s}}{{{h}}_{a,j}}\\\
{{{h}}^{H}}_{a,j}{{\bf{h}}^{H}}_{a,s}diag\left({{{\bf{g}}_{s,j}}}\right),{\bf{0}}\end{array}\right].$
(41)

The optimization problem in (III-B) is non-convex due to the constraint of
$rank({\bf{V}})=1$. Successive convex approximation (SCA) solves the second-
order cone programming (SOCP) problem by replacing the non-convex constraint
with a convex constraint [33, 8]. However, it is challenging to find a convex
restriction to replace the non-convex constraint in (III-B). By contrast, SDR
can be used to solve the SOCP by relaxing the non-convex constraint. Hence, we
use SDR to solve the optimization problem. We omit the constraint
$rank({\bf{V}})=1$, and write the optimization problem as follows

$\displaystyle\mathop{\max}\limits_{\bf{V}}\quad{P_{I}}\left({Tr\left({{{\bf{T}}{{}_{a,b}\bf{V}}}}\right)+{{\left|{{h_{a,b}}}\right|}^{2}}}\right)$
$\displaystyle{s.t.}\quad{{P_{{I}}}\left({Tr\left({{\bf{T}}_{a,w}{\bf{V}}}\right)+{{\left|{{h_{a,w}}}\right|}^{2}}}\right)\leq\eta}$
$\displaystyle\quad\quad\quad{{{\bf{V}}_{n,n}}=1},{{\bf{V}}\succeq 0}.$ (42)

It is clear that (III-B) is a convex problem and can be solved by the SDP
technique. The optimization tool, such as CVX in Matlab, can be used to
compute the optimal ${\bf{V}}^{*}$. Then, the Gaussian randomization technique
is used to calculate the optimal IRS phase shifts vector satisfying the
constraint of $rank\left({{\bf{V}}^{*}}\right)=1$. In Gaussian randomization,
we first calculate the eigen-decomposition of
${\bf{V}}^{*}={\bf{X}\Sigma\bf{X}}^{H}$ and make
${\bf{\bar{v^{\prime}}}}_{l}=\arg\left({\bf{X}}{\bf{\Sigma}}^{1/2}{\bf{e}}_{l}\right)$,
where ${\bf{e}}_{l}(1\leq l\leq L)$ is a vector of zero-mean, unit-variance
complex circularly Gaussian random variables, and $L$ is the number of
randomizations. Let
${\bf{V}}^{\prime}_{l}={\bf{\bar{v^{\prime}}}}_{l}{\bf{\bar{v^{\prime}}}}^{H}_{l}$,
and search the ${\bf{V}}^{\prime}_{l}$ achieving the largest covert
transmission rate as well as satisfying the covert constraint in (III-B).
Finally, according to the corresponding ${\bf{\bar{v}}}^{\prime}_{l}$, the
optimal $\bf{v}^{*}$ is computed as
${\bf{v}^{*}}=\frac{{[{\bf{\bar{v}}}^{\prime}_{l}]}_{\left(1:N\right)}}{{{\bf{\bar{v}}}^{\prime}_{l}}\left(N+1\right)}$,
where $[{\bf{\bar{v}}}^{\prime}_{l}]_{(1:N)}$ denotes the elements of Row $1$
to Row $N$ in ${\bf{\bar{v}}}^{\prime}_{l}$. Then, $P^{*}_{I}$ and
${\bf{v}^{*}}$ are iteratively calculated until the desired accuracy of covert
transmission is satisfied.

For this alterative algorithm, assume that at the end of the $t$-th iteration,
the calculated optimal power and phase shifts vector are ${{P_{I}}}^{t}$ and
${\bf{v}}^{t}$. Then, in the $t+1$ iteration, we can calculate the optimal
${{P_{I}}}^{t+1}$ given ${\bf{v}}^{t}$ and the optimal ${\bf{v}}^{t+1}$ given
${{P_{I}}}^{t+1}$. Assume that the objective function is
$f\left({{P_{I}}},{\bf{v}}\right)$. It is obvious that
$f\left({{P^{t}}_{I},{{\bf{v}}^{t}}}\right)\leq
f\left({{P^{t+1}}_{I},{{\bf{v}}^{t}}}\right)\leq
f\left({{P^{t+1}}_{I},{{\bf{v}}^{t+1}}}\right)$. This guarantees that the
alternating algorithm is non-decreasing. Since there are $N+1$ variables and 1
linear constraint in the optimization problem (III-B), the complexity of
calculating the phase shifts vector is
$O\left({{{\left({{{\left({N+1}\right)}^{{2}}}{{+1}}}\right)}^{{{3}}{{.5}}}}}\right)$
for each iteration [34]. In addition, the complexity of calculating the
transmission power can be neglected according to (37). Hence, using this
alternating algorithm, the complexity of (III-B) is
$O\left({{{\left({{{\left({N+1}\right)}^{{2}}}{{+1}}}\right)}^{{{3}}{{.5}}}}}\right)$
for each iteration. To compare the covert rate with IRS assistance to that
with direct transmission, we need to derive the SNR at Bob under the covert
constraint. When IRS is used to forward the messages, it is challenging to
derive a closed-form expression for SNR at Bob. Hence, with IRS assistance, we
alternatively investigate an upper bound on the SNR at Bob. It is noted that a
lower-bound on Willie’s link power gain is

$\displaystyle\
{\left|{{h^{H}_{a,w}}+{{\bf{g}}^{H}_{s,w}}\bf{\Theta}}{{\bf{h}}_{a,s}}\right|^{2}}$
$\displaystyle={{{\bf{\bar{v}}}}^{H}}{{\bf{T}}_{a,w}}{\bf{\bar{v}}}+{\left|{{h^{H}}_{a,w}}\right|^{2}}{\geq}{\lambda_{\min}}\left({{{\bf{T}}_{a,w}}}\right)\left({{\rm{N+1}}}\right){\rm{+}}\left|{{h^{H}}_{a,w}}\right|^{2}$
(43)

where ${\lambda_{\min}}\left({{{\bf{T}}_{a,w}}}\right)$ is the minimum
eigenvalue of ${{{\bf{T}}_{a,w}}}$, and the equality in the right side of (25)
holds when ${\bf{\bar{v}}}$ is the eigenvector corresponding to the minimum
eigenvalue ${\lambda_{\min}}\left({{{\bf{T}}_{a,w}}}\right)$. In addition, an
upper bound on Bob’s link power gain is given by

$\displaystyle\
{\left|{{h^{H}_{a,b}}+{{\bf{g}}^{H}_{s,b}}\bf{\Theta}}{{\bf{h}}_{a,s}}\right|^{2}}\leq{{{{\left({\left|{{h_{a,b}}}\right|+\sum\limits_{k=1}^{N}{\left|{{h_{a,{s_{k}}}}}\right|}\left|{{g_{{s_{k}},b}}}\right|}\right)}^{2}}}}$
(44)

where the equality holds when $\arg(h^{H}_{a,b})$ is equal to
$\arg({{\bf{g}}^{H}_{s,b}}\bf{\Theta}{{\bf{h}}_{a,s}})$. Thus, we can obtain
the following upper bound on the SNR at Bob

$\displaystyle\
{\gamma^{u}}_{IRS}=\frac{{\eta\left({\left|{{h_{a,b}}}\right|+\sum\limits_{k=1}^{N}{\left|{{h_{a,{s_{k}}}}}\right|}\left|{{g_{{s_{k}},b}}}\right|}\right)^{2}}}{\sigma^{2}_{b}{\left({\lambda_{\min}}\left({{{\bf{T}}_{a,w}}}\right)\left({{\rm{N+1}}}\right){\rm{+}}\left|{{h^{H}}_{a,w}}\right|^{2}\right)}}.$
(45)

By contrast, the SNR at Bob without IRS is given by
${\gamma_{dir}}=\frac{{\eta{{\left|{{{\rm{h}}^{H}}_{a,b}}\right|}^{\rm{2}}}}}{\sigma^{2}_{b}{{{\left|{{h_{a,w}}}\right|}^{2}}}}$.
Then, if ${\gamma^{u}}_{IRS}<{\gamma_{dir}}$, the covert transmission rate
without IRS is greater than that with IRS. Note that if
${\lambda_{\min}}\left({{{\bf{T}}_{a,w}}}\right)>0$, compared to the direct
transmission, Willie’s link power gain with IRS is strengthened. To satisfy
the covert constraint, the transmission power is reduced compared to direct
transmission. As a result, the covert transmission rate without IRS can
outperform that with IRS. By contrast, if
${\lambda_{\min}}\left({{{\bf{T}}_{a,w}}}\right)<0$, the covert transmission
rate with IRS is greater than that without IRS. Hence, it is not necessary
that the IRS is beneficial to the covert transmission rate for the single
transmit antenna case. Many factors, such as the predetermined covert
threshold, the relationship between Willie’s link power gain and Bob’s link
power gain, and the number of IRS elements, determine whether the transmission
with IRS outperforms that without IRS or not, even if the full instantaneous
CSIs of all links are perfectly known at Alice.

##  IV Multiple Antennas at Alice

When there are multiple antennas at Alice, transmit beamforming technology can
be used to achieve a higher covert performance. In this section, we
investigate joint transmit beamforming and phase shift optimization for covert
transmission rate maximization. Next, we discuss the two cases of partial CSI
and instantaneous CSI of Willie’s link.

###  IV-A With Partial CSI of Willie’s Link

In the case of partial CSI of Willie’s link, only the statistical information,
i.e., mean and variance, of the Alice-to-Willie and IRS-to-Willie channels are
available, and the transmit beamforming vector at Alice and phase shifts
vector at the IRS are jointly optimized and updated according to the
instantaneous CSI of Bob’s link. Thus, the transmit beamforming and phase
shifts vectors are random for Willie’s link. For the case of random
beamforming towards Willie, the received SNR at Willie has the same expression
as that with a single antenna. As a result, the maximum transmission power
with multiple antennas at Alice has the same expression as that with a single
antenna at Alice. After the transmission power is determined, the transmit
beamforming and the phase shifts vectors, which are coupled, are jointly
optimized for the covert rate maximization using an alterative algorithm. For
a given phase shifts vector ${\bf{v}}$, the optimal beamforming vector for
MISO systems is MRT, i.e.
${\bf{{\bf{w}}}^{{*}}}{\rm{=}}\sqrt{P_{I}}\frac{{{{\left({{\bf{h}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right)}^{H}}}}{{\left\|{{\bf{h}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right\|}}\in\mathbb{C}^{M\times
1}$. When ${\bf{{\bf{w}}}^{{*}}}$ and ${P^{*}_{I}}$ are determined, the
optimization problem is reduced to maximizing Bob’s link power gain, and it is
given by

$\displaystyle\mathop{\max}\limits_{\bf{v}}{\left\|{{{\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right)}}}\right\|^{2}}=\mathop{\max}\limits_{\bf{v}}\left({{\bf{v}}^{H}}{diag\left({{{\bf{g}}^{H}}_{s,b}}\right){{\bf{h}}_{a,s}}}{{{\bf{h}}^{H}_{a,s}}diag\left({{{\bf{g}}}_{s,b}}\right)}{\bf{v}}\right.$
$\displaystyle\quad\quad\quad\left.+{{\bf{h}}^{H}}_{a,b}{{{{\bf{h}}^{H}_{a,s}}diag\left({{{\bf{g}}}_{s,b}}\right)}}{\bf{v}}+{{\bf{v}}^{H}}{diag\left({{{\bf{g}}^{H}}_{s,b}}\right){{\bf{h}}_{a,s}}}{{\bf{h}}_{a,b}}+{\left\|{{{\bf{h}}^{H}}_{a,b}}\right\|^{2}}\right).$
(46)

Similar to (III-B), this optimization problem can be rewritten as

$\displaystyle{\mathop{\max}\limits_{\bf{V}}}\quad{Tr\left({{\bf{R_{a,b}V}}}\right)+{\left\|{{{\bf{h}}^{H}}_{a,b}}\right\|^{2}}}$
(47a) $\displaystyle{s.t.}\quad{{{\bf{V}}_{n,n}}=1,}n=1,\cdots N+1$ (47b)
$\displaystyle{{\bf{V}}\succeq 0},{Rank}\left({\bf{V}}\right)=1$ (47c)

where
${{\bf{R}}_{a,b}}=\left[{{\bf{\Phi}}_{a,b}}{{\bf{\Phi}}^{H}}_{a,b},{{\bf{\Phi}}_{a,b}}{\alpha^{H}_{a,b}};{\alpha_{a,b}}{{\bf{\Phi}}^{H}}_{a,b},\bf{0}\right]$,
${{\bf{\alpha}}_{a,b}}={{\bf{h}}^{H}}_{a,b}{\bf{{\bf{w}}}}$, and
${{\bf{\Phi}}_{a,b}}=diag\left({{{\bf{g}}^{H}}_{s,b}}\right){{\bf{h}}_{a,s}}{\bf{{\bf{w}}}}$.
The optimal ${\bf{v}}^{*}$ in (IV-A) can be calculated using the CVX and
Gaussian randomization [9], and the maximum covert transmission rate is
obtained by substituting ${\bf{v}}^{*}$ into (IV-A). For this case, no
iteration is required. But, it is challenging to derive an exact SNR
expression at Bob. Alternatively, we obtain the following upper bound on Bob’s
link power gain

$\displaystyle{\mathbb{E}}\left({{{\left\|{{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right\|}^{2}}}\right)$
$\displaystyle\leq
M\left({{\sigma^{2}}_{a,b}+N\left({1+\frac{{\left({N-1}\right){\pi^{2}}}}{{16}}}\right){\sigma^{2}}_{a,s}{\sigma^{2}}_{s,b}+\frac{{N{\pi^{\frac{3}{2}}}{\sigma_{a,b}}{\sigma_{a,s}}{\sigma_{s,b}}}}{4}}\right).$
(48)

The proof of (IV-A) can be found in [1, 8]. In (IV-A), equality holds when
$N\gg 1$, and
$\arg({{\bf{h}}^{H}}_{a,b})=\arg({{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s})$.

###  IV-B With Instantaneous CSI of Willie’s Link

When the instantaneous CSI of Willie’s link is perfectly known at Alice, to
maximize the covert transmission rate, the transmit beamforming and phase
shifts vectors are continuously adjusted according to the instantaneous CSIs
of both Willie’s and Bob’s links. In this case, the optimization problem is
formulated as follows

$\displaystyle\mathop{\max}\limits_{\bf{v},{\bf{w}}}{\left|{\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right)\bf{{\bf{w}}}}\right|^{2}}$
$\displaystyle
s.t.\quad{Tr\left({{\bf{w}}{{\bf{w}}^{H}}}\right)}\leq{P_{\max}}$
$\displaystyle{{{\left|{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)\bf{{\bf{w}}}}\right|}^{2}}\leq\eta}.$
(49)

For this optimization problem, we first present an alternating algorithm to
maximize the covert rate. Then, we propose two suboptimal algorithms to reduce
the complexity. The details of these algorithms are discussed next.

####  IV-B1 Alternating algorithm

For a given phase shifts vector $\bf{v}$, we need to optimize the beamforming
vector $\bf{{\bf{w}}}$. For the MISO system with covert constraint, we can
resort to the traditional MISO cognitive system, which has the interference
constraint at the primary user and has a similar optimization objective
function. Thus, by leveraging results from MISO cognitive systems in [35, 36],
we can derive the optimal beamforming vector as
${\bf{{\bf{w}}^{*}}}=P_{max}{{\bf{\bar{\bf{w}}}}}$, where
${{\bf{\bar{\bf{w}}}}}$ denotes the unit-norm beamforming vector and is given
by

$\displaystyle{{\bf{\bar{\bf{w}}}}}=A\frac{{\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right)^{H}}}{{\left\|{{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right\|}}$
$\displaystyle+B\frac{{\left({{\bf{I}}_{M}-\frac{{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)^{H}{{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)}}}}{{{{\left\|{{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right\|}^{2}}}}}\right)\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right)^{H}}}{{\left\|{\left({{\bf{I}}_{M}-\frac{{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)^{H}{{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)}}}}{{{{\left\|{{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right\|}^{2}}}}}\right){{\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right)}^{H}}}\right\|}}$
(50)

where

$\displaystyle\
\left({A,B}\right)=\left\\{\begin{array}[]{l}\left({1,0}\right),\eta\geq{P_{\max}}\left({\left\|{{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right\|\cos{\Omega}}\right)^{2}\\\
\left({\frac{\tau}{{\cos{\Omega}}},\sqrt{1-{\tau^{2}}}-\tau\tan{\Omega}}\right),otherwise\end{array}\right.$
(53)

$\tau=\sqrt{\frac{\eta}{{{P_{\max}}{{\left\|{{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right\|}^{2}}}}}$,
and
$\cos{\Omega}=\frac{{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right){{\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right)}^{H}}}}{{\left\|{{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right\|\left\|{{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right\|}}$.
For a given $\bf{{\bf{w}}}$ , similar to (III-B), the optimization problem
(IV-B) is rewritten as

$\displaystyle{\mathop{\max}\limits_{\bf{v}}}\quad{{{\left|{{\alpha_{a,b}}}\right|}^{2}}+Tr\left({{{\bf{R}}_{a,b}}{\bf{V}}}\right)}$
$\displaystyle{s.t.}\quad\quad{{{\left|{{\beta_{a,w}}}\right|}^{2}}+Tr\left({{{\bf{R}}_{a,w}}{\bf{V}}}\right)\leq\eta}$
$\displaystyle\quad\quad{{{\bf{V}}_{n,n}}=1},{{\bf{V}}\succeq
0},rank\left({\bf{V}}\right)=1$ (54)

where${{\bf{R}}_{a,w}}=\left[{{\bf{\Psi}}_{a,w}}{{\bf{\Psi}}^{H}}_{a,w},{{\bf{\Psi}}_{a,w}}{\beta^{H}_{a,w}};{\beta_{a,w}}{{\bf{\Psi}}^{H}}_{a,w},\bf{0}\right]$,
${\beta_{a,w}}={{\bf{h}}^{H}}_{a,w}{\bf{{\bf{w}}}}$, and
${{\bf{\Psi}}_{a,w}}=diag\left({{{\bf{g}}^{H}}_{s,w}}\right){{\bf{h}}_{a,s}}{\bf{{\bf{w}}}}$.
When neglecting the constraint $rank\left({\bf{V}}\right)=1$, problem (IV-B1)
can be solved using the software package CVX. Then, Gaussian randomization can
be used to construct a rank-one solution. Finally, iterations for transmit
beamforming and phase shift vectors optimization are performed for covert rate
maximization. The complexity of solving (IV-B1) is
$O\left({{{\left({{{\left({N+1}\right)}^{\rm{2}}}{\rm{+1}}}\right)}^{{\rm{3}}{\rm{.5}}}}}\right)$
for each iteration. In the alternating algorithm, for a given beamforming
vector $\bf{w}$, it is possible that there is no available phase shifts vector
satisfying the covert constraint. Hence, we first obtain the optimal $\bf{w}$
for a given phase shifts vector. The order for the alterative algorithm cannot
be changed. In practice, the number of iterations is determined by the covert
transmission rate accuracy. For example, when the gap between the covert
transmission rate for the $(i-1)$-th iteration and for the $i$-th iteration is
less than a predetermined threshold $\Gamma$, the alterative algorithm ends at
the $i$-th iteration.

####  IV-B2 Zero-forcing Beamforming

To decrease the complexity of the alterative algorithm, we propose a
suboptimal algorithm, where zero-forcing beamforming is used at Alice, and no
legitimate signal can be received at Willie. Thus, Alice can transmit messages
with the maximum power, namely, $P^{*}_{I}=P_{max}$, and the covert constraint
is always satisfied. Since it is challenging to first derive the zero-forcing
beamforming, and then achieve optimal phase shifts, we give a alterative
algorithm with zero-beamforming deployed at Alice. Specifically, for a given
beamforming vector $\bf{w}$, the phase shifts vector ${\bf{v}}$ is optimized
for covert rate maximization, and the optimization problem is written as

$\displaystyle\
\mathop{\max}\limits_{\bf{v}}\left|{\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right){\bf{w}}}\right|^{2}.$
(55)

Since
$\left|{\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right){\bf{w}}}\right|\leq\left|{{{\bf{h}}^{H}}_{a,b}{\bf{w}}}\right|+\left|{{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}{\bf{w}}}\right|$,
to maximize the covert transmission rate, the following equality should be
satisfied

$\displaystyle\
\arg\left({{{\bf{h}}^{H}}_{a,b}{\bf{w}}}\right)=\arg\left({{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}}_{s,b}}\right){{\bf{h}}_{a,s}}{\bf{w}}}\right).$
(56)

According to (56), we can obtain the optimal ${\theta_{i}}^{*}$ $\left(1\leq
i\leq N\right)$ as

$\displaystyle\
{\theta_{i}}^{*}=\arg\left({{{\bf{h}}^{H}}_{a,b}{\bf{w}}}\right)-\arg\left({{g^{H}}_{{s_{i}},b}}\right)-\arg\left({{{\bf{h}}}_{a,{s_{i}}}{\bf{{\bf{w}}}}}\right).$
(57)

Conditioned on the derived phase shift ${\theta_{i}}^{*}$, the zero-forcing
transmit beamforming vector is

$\displaystyle\
{{\bf{{\bf{w}}}}}{{=}}{P_{max}}\frac{{\left({{\bf{I}}_{M}-\frac{{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right){{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)}^{H}}}}{{{{\left\|{{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right\|}^{2}}}}}\right){{\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right)}^{H}}}}{{\left\|{\left({{\bf{I}}_{M}-\frac{{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{g}}^{H}}_{s,w}\Theta{{\bf{h}}_{a,s}}}\right){{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)}^{H}}}}{{{{\left\|{{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right\|}^{2}}}}}\right){{\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right)}^{H}}}\right\|}}$
(58)

where
${{\bf{I}}_{M}-\frac{{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)^{H}{{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)}}}}{{{{\left\|{{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right\|}^{2}}}}}$
is a Hermitian orthogonal projection matrix. We iterate between (57) and (58)
for covert rate maximization. Since the exact analytical expression of phase
shifts and transmit beamforming vectors can be obtained, this algorithm has a
low complexity $O\left({M+N}\right)$ for each iteration.

####  IV-B3 Minimizing Willie’s Link Power Gain

Minimizing the channel quality of Willie’s link is an effective way to improve
the covert transmission rate. Next, we adjust the IRS phase shifts vector to
minimize Willie’s link power gain. Since the phase shifts and transmit
beamforming vectors are coupled, we solve the optimization problem by the
iteration method. For a given phase shifts vector $\bf{v}$, the optimal
beamforming vector $\bf{w}$ can be obtained according to (IV-B1). Then, for a
given beamforming vector $\bf{w}$, we minimize Willie’s link power gain by
adjusting the phase shifts vector $\bf{v}$. Note that the signal power
received at Willie is expressed as

$\displaystyle\
{\left|{{{\bf{\beta}}_{a,w}}}\right|^{2}}+{{\bf{v}}^{H}}{{\bf{\Psi}}_{a,w}}{\beta_{a,w}}+{\beta_{a,w}}{{\bf{\Psi}}^{H}}_{a,w}{\bf{v}}+{{\bf{v}}^{H}}{{\bf{\Psi}}_{a,w}}{{\bf{{{\Psi}}}}^{H}}_{a,w}{\bf{v}}$
$\displaystyle\leq
N{\lambda_{\max}}\left({{{\bf{\Psi}}_{a,w}}{{\bf{\Psi}}^{H}}_{a,w}}\right)+2{\mathop{\rm
Re}\nolimits}\left({{{\bf{v}}^{H}}{{\bf{\Psi}}_{a,w}}{\beta_{a,w}}}\right)+{\left|{{\beta_{a,w}}}\right|^{2}}$
(59)

where the equality holds when $\bf{v}$ is the eigenvector corresponding to the
maximum eigenvalue
${\lambda_{\max}}\left({{{\bf{\Psi}}_{a,w}}{{\bf{\Psi}}^{H}}_{a,w}}\right)$.
To minimize the Willie’s link power gain, we can minimize its upper bound.
According to (IV-B3), the optimal phase shifts vector is calculated by
minimizing ${\mathop{\rm
Re}\nolimits}\left({{{\bf{v}}^{H}}{{\bf{\Psi}}_{a,w}}{\beta_{a,w}}}\right)$,
which is given by

$\displaystyle{\mathop{\rm
Re}\nolimits}\left({{{\bf{v}}^{H}}{{\bf{\Psi}}_{a,w}}{\beta_{a,w}}}\right)=\sum\limits_{i=1}^{N}{\left|{{g^{H}}_{{s_{i}},w}}\right|}\left|{{{\bf{h}}_{a,{s_{i}}}}{{{\bf{w}}}}}\right|\left|{{{\bf{h}}^{H}}_{a,w}{{{\bf{w}}}}}\right|$
$\displaystyle\quad\quad\quad\quad\times\cos\left({{\theta_{i}}+\arg\left({{g^{H}}_{{s_{i}},w}}\right)+\arg\left({{{\bf{h}}_{a,{s_{i}}}}{{{\bf{w}}}}}\right)-\arg\left({{{\bf{h}}^{H}}_{a,w}{{{\bf{w}}}}}\right)}\right).$
(60)

When
$\cos\left({{\theta_{i}}+\arg\left({{g^{H}}_{{s_{i}},w}}\right)+\arg\left({{{\bf{h}}_{a,{s_{i}}}}{{{\bf{w}}}}}\right)-\arg\left({{{\bf{h}}^{H}}_{a,w}{{{\bf{w}}}}}\right)}\right)=-1$,
the upper bound on Willie’s link power gain is minimized. Thus, the optimal
${\theta_{i}}^{*}$ is given by

$\displaystyle\
{\theta_{i}}^{*}=\pi+\arg\left({{{\bf{h}}^{H}}_{a,w}{\bf{w}}}\right)-\arg\left({{g^{H}}_{{s_{i}},w}}\right)-\arg\left({{{\bf{h}}}_{a,{s_{i}}}{{{\bf{w}}}}}\right).$
(61)

Then, perform iterations between (IV-B1) and (61) until the accuracy of the
covert transmission rate is satisfied. Note that in this suboptimal algorithm,
we should first calculate the optimal beamforming vector for a given $\bf{v}$.
Then, the optimal phase shifts vector is calculated for a given beamforming
vector. This is because for a random $\bf{w}$, there might not exist a phase
shifts vector $\bf{v}$ satisfying the covert constraint. In addition, Since
the exact expressions of phase shifts and transmit beamforming vectors are
given by (IV-B1) and (61), respectively, the complexity for each iteration is
$O\left({M+N}\right)$.

##  V Covert transmission with imperfect CSI

Since Willie can be unfriendly to Alice and deliberately hides himself, Alice
cannot obtain the full instantaneous CSI of Willie’s link perfectly. Hence, we
investigate the covert transmission with imperfect CSI of Willie’s link. In
the following, we will investigate the covert transmission with imperfect CSI
of the Alice-to-Wilie link, the IRS-to-Willie link, and the Alice-to-IRS link,
respectively. In addition, the impact of imperfect cascaded CSI of Willie’s
link on covert transmission is omitted due to space limitation 111If the
estimated cascaded CSI of the Alice-IRS-Willie and Alice-IRS-Bob links are
independent, our proposed algorithm can handle the case with imperfect
cascaded CSI of Alice-IRS-Willie link; otherwise, it is challenging to obtain
the maximum covert rate using our proposed algorithm..

###  V-A Imperfect CSI of the Alice-to-Willie Link

Assume that the channels between Alice and Willie are subject to error,
denoted by ${{\Delta}}{\bf{h}}_{a,w}$, which is norm-bounded, i.e.,
$\|{{\Delta}}{\bf{h}}_{a,w}\|\leq\zeta_{a,w}$. Then, the estimated Alice-to-
Willie link is expressed as
${\tilde{\bf{h}}}_{a,w}{=}{\bf{h}}_{a,w}+{{\Delta}}{\bf{h}}_{a,w}$. In this
case, the signal power received at Willie should be less than $\eta$, and the
covert constraint is rewritten as

$\displaystyle\
{{{\left|{\left({{{\bf{h}}^{H}}_{a,w}+{{\Delta}}{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)\bf{{\bf{w}}}}\right|}^{2}}\leq\eta}.$
(62)

The following inequalities hold

$\displaystyle\left|{\left({{{\bf{h}}^{H}}_{a,w}+\Delta{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right){\bf{w}}}\right|$
$\displaystyle\quad\overset{(a)}{\leq}\left|{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right){\bf{w}}}\right|{{+}}\left|{\Delta{{\bf{h}}^{H}}_{a,w}{\bf{w}}}\right|$
$\displaystyle\quad\overset{(b)}{\leq}\left|{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right){\bf{w}}}\right|{{+}}\left\|{\Delta{{\bf{h}}^{H}}_{a,w}}\right\|\left\|{\bf{w}}\right\|$
$\displaystyle\quad{\leq}\left|{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right){\bf{w}}}\right|{\rm{+}}\zeta_{a,w}\left\|{\bf{w}}\right\|$
(63)

where (a) is obtained by using the triangle inequality and (b) is obtained by
applying the Cauchy-Schwarz inequality [37]. Specifically, the equality of (a)
holds when
$\arg\left({\left({{\bf{h}}_{a,w}^{H}+{{\bf{v}}^{H}}diag\left({{\bf{g}}_{s,w}^{H}}\right){{\bf{h}}_{a,s}}}\right){\bf{w}}}\right)=\arg\left({\Delta{\bf{h}}_{a,w}^{H}{\bf{w}}}\right)$,
and the equality of (b) holds when ${\Delta{\bf{h}}_{a,w}^{H}}$ and $\bf{w}$
are linear correlation. Then, according to (V-A), the signal power received at
Willie satisfies the following inequalities

$\displaystyle\
{\left|{\left({{{\bf{h}}^{H}}_{a,w}+\Delta{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right){\bf{w}}}\right|^{\rm{2}}}\leq{{\bf{w}}^{H}}{{{\bf{\tilde{U}}}}_{a,w}}{\bf{w}}$
(64)

where
${{{\bf{\tilde{U}}}}_{a,w}}={{{\bf{U}}}_{a,w}}{{\rm{+}}\left({{\zeta_{a,w}^{\rm{2}}}{\rm{+}}2\zeta_{a,w}\left\|{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)}\right\|}\right){\bf{I}}_{M}}$,
and
${{{\bf{U}}}_{a,w}}={{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)}^{H}}\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)$.
Since the phase shifts vector $\bf{v}$ and transmit beamforming vector
$\bf{w}$ are coupled, we still use the alternating algorithm to solve the
optimization problem. First, for a given phase shifts vector $\bf{v}$, let
${\bf{W}}={\bf{w}}{{\bf{w}}^{H}}$, then, the optimization problem is
formulated as

$\displaystyle\mathop{\max}\limits_{{\bf{w}}}\quad
Tr\left({{{\bf{U}}_{a,b}}{\bf{W}}}\right)$ $\displaystyle
s.t.\quad{Tr\left({{\bf{W}}}\right)\leq{P_{\max}}}$ $\displaystyle\quad\quad
Tr\left({{{\bf{\tilde{U}}}_{a,w}}{\bf{W}}}\right)\leq\eta,{\bf{W}}\succeq
0,rank\left({{\bf{W}}}\right)=1$ (65)

where
${{\bf{U}}_{a,b}}={\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,b}}}\right){\bf{h}}_{a,s}}\right)^{H}}\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{v}}^{H}}diag\left({{{\bf{g}}^{H}_{s,w}}}\right){\bf{h}}_{a,s}}\right)$.
When neglecting the constraint $rank\left({{\bf{W}}}\right)=1$, SDP can be
used to solve the optimization problem. Moreover, Gaussian randomization is
used to obtain the optimal $\bf{w}$ satisfying the rank 1 constraint. After
the beamforming vector $\bf{w}$ is obtained, we derive next the phase shifts
vector with a fixed $\bf{w}$. For a given beamforming vector $\bf{w}$, the
optimization problem is reformulated as

$\displaystyle{\mathop{\max}\limits_{\bf{v}}}\quad{{{\left|{{\alpha_{a,b}}}\right|}^{2}}+Tr\left({{{\bf{R}}_{a,b}}{\bf{V}}}\right)}$
$\displaystyle
s.t.\quad{{{\left|{{\alpha_{a,w}}}\right|}^{2}}+Tr\left({{{\bf{R}}_{a,w}}{\bf{V}}}\right)\leq{\left({\sqrt{\eta}-\zeta_{a,w}\left\|{\bf{w}}\right\|}\right)^{2}}}$
$\displaystyle\quad\quad{{{\bf{V}}_{n,n}}=1},{{\bf{V}}\succeq
0},rank\left({\bf{V}}\right)=1$ (66)

where the covert constraint is obtained using (V-A). Similar to (IV-B1), the
optimal phase shifts vector can be obtained using the SDP and Gaussian
randomization techniques. The complexity of (V-A) is
$O\left({{{\left({{M^{\rm{2}}}{\rm{+1}}}\right)}^{{\rm{3}}{\rm{.5}}}}}\right)$
for each iteration, and the complexity of (V-A) is
$O\left({{{\left({{{\left({N+1}\right)}^{\rm{2}}}{\rm{+1}}}\right)}^{{{3}}{{.5}}}}}\right)$
for each iteration. Then, we iterate between (V-A) and (V-A) until the gap of
covert transmission rate between the _i_ -th and $(i+1)$-th iteration is less
than a predetermined threshold $\Gamma$. The detail of the algorithm is
presented in the following table. In the Algorithm 1, we should first derive
the beamforming vector with a fixed phase shift. If we first derive the phase
shifts with a fixed beamforming vector, the covert constraint cannot be
satisfied.

Algorithm 1 Maximize the Covert Transmit Rate With imperfect CSI of Alice-to-
Willie link

1: Initialization: $N$, $M$, $\eta$, $P_{\max}$.

2: Randomly set the phase shift $\bf{v}$, let $f_{0}{(\bf{w},\bf{v})}$=0, and
set $i=1$.

3: For a given $\bf{v}$, calculate the optimal $\bf{W}$ according to (V-A).

4: Perform Gaussian randomization for $\bf{W}$, and obtain the optimal
transmit beamforming vector ${\bf{w}}^{*}$.

5: For the given ${\bf{w}}^{*}$, calculate the optimal $\bf{V}$ according to
(V-A).

6: Perform Gaussian randomization for $\bf{V}$, and obtain the optimal phase
shifts vector ${\bf{v}}^{*}$.

7: Calculate the covert transmit rate $f_{i}{(\bf{w}^{*},\bf{v}^{*})}$.

8: If $f_{i}{(\bf{w}^{*},\bf{v}^{*})}-f_{i-1}{(\bf{w},\bf{v})}\leq\Gamma$,
break; else $i=i+1$, update $\bf{w}=\bf{w}^{*}$ and $\bf{v}=\bf{v}^{*}$, and
repeat step 3 until the accuracy of the covert rate is satisfied.

###  V-B Imperfect CSI of the IRS-to-Willie Link

Assume that the estimated CSI of the IRS-to-Willie links is imperfect, and
subject to error, denoted by ${{\Delta}}{\bf{g}}_{s,w}$, which is norm-
bounded, i.e., $\|{{\Delta}}{\bf{g}}_{s,w}\|\leq\zeta_{s,w}$. The estimated
IRS-to-Willie link is
${{\bf{\tilde{g}}}^{H}}_{s,w}={{{\bf{g}}^{H}}_{s,w}+\Delta{{\bf{g}}^{H}}_{s,w}}$.
Similar to (V-A) and (64), the signal power received at Willie satisfies the
following inequality

$\displaystyle\
{\left|{\left({{{\bf{h}}^{H}}_{a,w}+\left({{{\bf{g}}^{H}}_{s,w}+\Delta{{\bf{g}}^{H}}_{s,w}}\right){\bf{\Theta}}{{\bf{h}}_{a,s}}}\right){\bf{w}}}\right|^{2}}\leq{{\bf{w}}^{H}}{{{\bf{\hat{U}}}}_{a,w}}{\bf{w}}$
(67)

where
${{{\bf{\hat{U}}}}_{a,w}}={{{\bf{U}}}_{a,w}}{+\left({{\zeta^{2}}_{s,w}{{\left\|{{\bf{\Theta}}{{\bf{h}}_{a,s}}}\right\|}^{2}}+2{\zeta_{s,w}}\left\|{{\bf{\Theta}}{{\bf{h}}_{a,s}}}\right\|\left\|{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{g}}^{H}}_{s,w}{\bf{\Theta}}{{\bf{h}}_{a,s}}}\right)}\right\|}\right){\bf{I}}_{M}}$.
Then, for a given phase shifts vector $\bf{v}$, the optimization problem is
given by

$\displaystyle\mathop{\max}\limits_{{\bf{w}}}\quad
Tr\left({{{\bf{{U}}}_{a,b}}{\bf{W}}}\right)$ $\displaystyle
s.t.\quad{Tr\left({{\bf{W}}}\right)\leq{P_{\max}}}$ $\displaystyle\quad\quad
Tr\left({{{\bf{\hat{U}}}_{a,w}}{\bf{W}}}\right)\leq\eta,{\bf{W}}\succeq
0,rank\left({{\bf{W}}}\right)=1.$ (68)

For (V-B), neglect the rank-1 constraint, and SDP can be used to obtain the
optimal $\bf{W}$. Then, Gaussian randomization is applied to obtain the
optimal beamforming vector $\bf{w}$. For the calculated $\bf{w}$, we optimize
the phase shifts for covert transmission rate maximization. Note that for the
imperfect ${{\bf{g}}^{H}}_{s,w}$, the following inequalities are satisfied

$\displaystyle\left|{\left({{{\bf{h}}^{H}}_{a,w}+\left({{{\bf{g}}^{H}}_{s,w}+\Delta{{\bf{g}}^{H}}_{s,w}}\right){\bf{\Theta}}{{\bf{h}}_{a,s}}}\right){\bf{w}}}\right|\overset{(a)}{\leq}\left|{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{g}}^{H}}_{s,w}{\bf{\Theta}}{{\bf{h}}_{a,s}}}\right){\bf{w}}}\right|+{\zeta_{s,w}}\left\|{{\bf{\Theta}}{{\bf{h}}_{a,s}}{\bf{w}}}\right\|$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\overset{(b)}{\leq}\left|{{{\bf{h}}^{H}}_{a,w}{\bf{w}}+{{\bf{v}}^{\rm{H}}}{\rm{diag}}\left({{{\bf{g}}^{H}}_{s,w}}\right){{\bf{h}}_{a,s}}{\bf{w}}}\right|+{\zeta_{s,w}}\left\|{{{\bf{h}}_{a,s}}{\bf{w}}}\right\|$
(69)

where (a) is obtained by applying the triangle and the Cauchy-Schwarz
inequalities; (b) holds due to
$\left\|{{\bf{\Theta}}{{\bf{h}}_{a,s}}{\bf{w}}}\right\|{{=}}\left\|{{{\bf{h}}_{a,s}}{\bf{w}}}\right\|$.
Then, according to (V-B), the received signal power at Willie is given by

$\displaystyle{\left|{\left({{{\bf{h}}^{H}}_{a,w}+\left({{{\bf{g}}^{H}}_{s,w}+\Delta{{\bf{g}}^{H}}_{s,w}}\right){\bf{\Theta}}{{\bf{h}}_{a,s}}}\right){\bf{w}}}\right|^{2}}\leq
Tr\left({{{{\bf{R}}_{a,w}}}{\bf{V}}}\right)+\Lambda_{s,w}$ (70)

where
$\Lambda_{s,w}={\left|{{{\bf{h}}^{H}}_{a,w}{\bf{w}}}\right|^{2}}+{\zeta^{\rm{2}}}_{s,w}{\left\|{{{{\bf{h}}_{a,s}}{\bf{w}}}}\right\|^{\rm{2}}}{\rm{+2}}{\zeta_{s,w}}\left\|{{{{\bf{h}}_{a,s}}{\bf{w}}}}\right\|\left({\left|{{{\bf{h}}^{H}}_{a,w}{\bf{w}}}\right|{\rm{+}}\|{{{\bf{g}}^{H}}_{s,w}}\|\left\|{{\bf{h}}_{a,s}}{\bf{w}}\right\|}\right)$.
Therefore, for a given $\bf{w}$, the optimization problem is expressed as

$\displaystyle{\mathop{\max}\limits_{\bf{v}}}\quad{Tr\left({{{\bf{R}}_{a,b}}{\bf{V}}}\right)}$
$\displaystyle{s.t.}\quad
Tr\left({{{{\bf{R}}_{a,w}}}{\bf{V}}}\right)\leq\eta-\Lambda_{s,w}$
$\displaystyle\quad\quad{{{\bf{V}}_{n,n}}=1},{{\bf{V}}\succeq
0},rank\left(\bf{V}\right)=1.$ (71)

Similarly, SDP and Gaussian randomization are used to solve the problem. Eqs.
(V-B) and (V-B) are iteratively solved until the gap of covert transmission
rate between the _i_ -th and the (_i_ +1)-th iteration is less than a
predetermined threshold $\Gamma$.

###  V-C Imperfect CSI of the Alice-to-IRS Link

When the Alice-to-IRS link is not estimated perfectly, both Bob’s and Willie’s
links are affected. Assume that the estimated CSI
${{\bf{\tilde{h}}}}^{H}_{a,s}={{{\bf{h}}^{H}}_{a,s}+\Delta{{\bf{h}}^{H}}_{a,s}}$,
where the estimation error $\Delta{{\bf{h}}^{H}}_{a,s}$ is norm-bounded as
$\|\Delta{{\bf{h}}^{H}}_{a,s}\|\leq\zeta_{a,s}$. For a given phase shifts
vector $\bf{v}$, the received signal power at Willie and Bob are,
respectively, expressed as

$\displaystyle\
{\left|{\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{g}}^{H}}_{s,b}{\bf{\Theta}}\left({{{\bf{h}}_{a,s}}+\Delta{{\bf{h}}_{a,s}}}\right)}\right){\bf{w}}}\right|^{2}}$
$\displaystyle\geq{\left({\left|{\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{g}}^{H}}_{s,b}{\bf{\Theta}}{{\bf{h}}_{a,s}}}\right){\bf{w}}}\right|-\left|{{{\bf{g}}^{H}}_{s,b}{\bf{\Theta}}\Delta{{\bf{h}}_{a,s}}{\bf{w}}}\right|}\right)^{2}}$
$\displaystyle\geq{{\bf{w}}^{H}}{{{\bf{\bar{U}}}}_{a,b}}{\bf{w}}$ (72)

and

$\displaystyle\
{\left|{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{g}}^{H}}_{s,w}{\bf{\Theta}}\left({{{\bf{h}}_{a,s}}+\Delta{{\bf{h}}_{a,s}}}\right)}\right){\bf{w}}}\right|^{2}}$
$\displaystyle\leq{\left|{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{g}}^{H}}_{s,w}{\bf{\Theta}}{{\bf{h}}_{a,s}}}\right){\bf{w}}}\right|^{2}}+{\zeta^{2}}_{a,s}{\left\|{{{\bf{g}}^{H}}_{s,w}{\bf{\Theta}}}\right\|^{2}}{\left\|{\bf{w}}\right\|^{2}}$
$\displaystyle\quad+2{\zeta_{a,s}}\left\|{{{\bf{g}}^{H}}_{s,w}{\bf{\Theta}}}\right\|\left\|{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{g}}^{H}}_{s,w}{\bf{\Theta}}{{\bf{h}}_{a,s}}}\right)}\right\|{\left\|{\bf{w}}\right\|^{2}}$
$\displaystyle\leq{{\bf{w}}^{H}}{{{\bf{\bar{U}}}}_{a,w}}{\bf{w}}$ (73)

where
${{{\bf{\bar{U}}}}_{a,w}}={{{\bf{{U}}}}_{a,w}}+\left({{\zeta^{2}}_{a,s}{{\left\|{{{\bf{g}}^{H}}_{s,w}{\bf{\Theta}}}\right\|}^{2}}+2{\zeta_{a,s}}\left\|{{{\bf{g}}^{H}}_{s,w}{\bf{\Theta}}}\right\|\left\|{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{g}}^{H}}_{s,w}{\bf{\Theta}}{{\bf{h}}_{a,s}}}\right)}\right\|}\right){\bf{I}}_{M}$,
and
${{{\bf{\bar{U}}}}_{a,b}}={{{\bf{U}}}_{a,b}}+\left({{\zeta^{2}}_{a,s}{{\left\|{{{\bf{g}}^{H}}_{s,b}{\bf{\Theta}}}\right\|}^{2}}-2{\zeta_{a,s}}\left\|{{{\bf{g}}^{H}}_{s,b}{\bf{\Theta}}}\right\|\left\|{\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{g}}^{H}}_{s,b}{\bf{\Theta}}{{\bf{h}}_{a,s}}}\right)}\right\|}\right){\bf{I}}_{M}$.
Thus, according to the lower bound on the signal power received at Bob, (V-C)
and the upper bound on the signal power received at Willie given in (V-C), for
a given phase shift vector, the optimization problem is given by

$\displaystyle\mathop{\max}\limits_{{\bf{W}}}Tr\left({{{\bf{\bar{U}}}_{a,b}}{\bf{W}}}\right)$
$\displaystyle s.t.\quad{Tr\left({{\bf{W}}}\right)\leq{P_{\max}}}$
$\displaystyle\quad\quad
Tr\left({{{\bf{\bar{U}}}_{a,w}}{\bf{W}}}\right)\leq\eta,{\bf{W}}\succeq
0,rank\left({{\bf{W}}}\right)=1.$ (74)

SDP and Gaussian randomization are performed to calculate the optimal
$\bf{w}$. For the calculated $\bf{w}$, the received signal power at Bob and
Willie satisfy the following inequalities

$\displaystyle{\left|{\left({{{\bf{h}}^{H}}_{a,b}+{{\bf{g}}^{H}}_{s,b}{\bf{\Theta}}\left({{{\bf{h}}_{a,s}}+\Delta{{\bf{h}}_{a,s}}}\right)}\right){\bf{w}}}\right|^{2}}\geq{{\bf{\bar{v}}}^{H}}{{\bf{R}}_{a,b}}{\bf{\bar{v}}}{\rm{+}}{\zeta^{2}}_{a,s}{\left\|{\bf{w}}\right\|^{2}}{\left\|{diag\left({{{\bf{g}}^{H}}_{s,b}}\right)}\right\|^{\rm{2}}}{\rm{+}}{\left|{{{\bf{h}}^{H}}_{a,b}{\bf{w}}}\right|^{\rm{2}}}$
$\displaystyle\quad\quad\quad\quad\quad\quad-2{\zeta_{a,s}}\left\|{{{{\bf{g}}^{H}}_{s,b}}}\right\|\left\|{\bf{w}}\right\|\left({\left|{{{\bf{h}}^{H}}_{a,b}{\bf{w}}}\right|+\|{{{\bf{g}}^{H}}_{s,b}}\|\left\|{{{\bf{h}}_{a,s}}{\bf{w}}}\right\|}\right)$
(75)

and

$\displaystyle\
{\left|{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{g}}^{H}}_{s,w}{\bf{\Theta}}\left({{{\bf{h}}_{a,s}}+\Delta{{\bf{h}}_{a,s}}}\right)}\right){\bf{w}}}\right|^{2}}\leq{{\bf{\bar{v}}}^{H}}{{\bf{R}}_{a,w}}{\bf{\bar{v}}}+{\Lambda_{a,s}}$
(76)

where
${\Lambda_{a,s}}={\zeta^{2}}_{a,s}{\left\|{\bf{w}}\right\|^{2}}{\left\|{{{\bf{g}}^{H}}_{s,w}}\right\|^{\rm{2}}}{\rm{+}}{\left|{{{\bf{h}}^{H}}_{a,w}{\bf{w}}}\right|^{\rm{2}}}\\\
+2{\zeta_{a,s}}\left\|{{{\bf{g}}^{H}}_{s,w}}\right\|\left\|{\bf{w}}\right\|\left({\left|{{{\bf{h}}^{H}}_{a,w}{\bf{w}}}\right|+\|{{{\bf{g}}^{H}}_{s,w}}\|\left\|{{{\bf{h}}_{a,s}}{\bf{w}}}\right\|}\right)$.
Then, for a beamforming vector $\bf{w}$, the optimization problem is rewritten
as

$\displaystyle{\mathop{\max}\limits_{\bf{V}}}\quad{Tr\left({{{\bf{R}}_{a,b}}{\bf{V}}}\right)}$
$\displaystyle s.t.\quad
Tr\left({{{{\bf{R}}_{a,w}}}{\bf{V}}}\right)\leq\eta-\Lambda_{a,s}$
$\displaystyle\quad\quad{{{\bf{V}}_{n,n}}=1},{{\bf{V}}\succeq
0},rank\left({\bf{V}}\right)=1.$ (77)

SDR and Gausssian randomization are used to solve (V-C). Finally, Eqs. (V-C)
and (V-C) are iteratively solved until the gap of covert transmission rate
between the _i_ -th and (_i_ +1)-th iteration is less than a predetermined
threshold $\Gamma$.

###  V-D Imperfect CSI of Both the IRS-to-Willie and Alice-to-Willie Links

When both the Alice-to-Willie and IRS-to-Willie links are imperfect, i.e.,
$\|{{\Delta}}{\bf{h}}_{a,w}\|\leq\zeta_{a,w}$, and
$\|{{\Delta}}{\bf{g}}_{s,w}\|\leq\zeta_{s,w}$, the maximum covert transmission
rate can also be derived iteratively. For a given phase shifts vector
$\bf{v}$, the optimization problem is formulated as

$\displaystyle\mathop{\max}\limits_{{\bf{w}}}\quad
Tr\left({{{\bf{U}}_{a,b}}{\bf{W}}}\right)$ $\displaystyle
s.t.\quad{Tr\left({{\bf{W}}}\right)\leq{P_{\max}}}$ $\displaystyle\quad\quad
Tr\left({{{\bf{\breve{U}}}_{a,w}}{\bf{W}}}\right)\leq\eta,{\bf{W}}\succeq
0,rank\left({{\bf{W}}}\right)=1$ (78)

where
${{\bf{\breve{U}}}_{a,w}}=\left({2\left({{\zeta_{s,w}}\left\|{{{\bf{h}}_{a,s}}}\right\|+{\zeta_{a,w}}}\right)\left\|{\left({{{\bf{h}}^{H}}_{a,w}+{{\bf{v}}^{H}}diag({{\bf{g}}^{H}}_{s,w}){{\bf{h}}_{a,s}}}\right)}\right\|+{{\left({{\zeta_{s,w}}\left\|{{{\bf{h}}_{a,s}}}\right\|+{\zeta_{a,w}}}\right)}^{2}}}\right){{\bf{I}}_{M}}\\\
+{{\bf{U}}_{a,w}}$. Then, problem in (V-D) can be solved by SDR and Gaussian
randomization. For a given beamforming vector $\bf{w}$, the optimization
problem is rewritten as

$\displaystyle{\mathop{\max}\limits_{\bf{v}}}\quad{Tr\left({{{\bf{R}}_{a,b}}{\bf{V}}}\right)}$
$\displaystyle{s.t.}\quad
Tr\left({{{{\bf{R}}_{a,w}}}{\bf{V}}}\right)\leq\eta-\Lambda_{a,w}$
$\displaystyle\quad\quad{{{\bf{V}}_{n,n}}=1},{{\bf{V}}\succeq
0},rank\left(\bf{V}\right)=1.$ (79)

where
${\Lambda_{a,w}}={\left|{{{\bf{h}}^{H}}_{a,w}{\bf{w}}}\right|^{{2}}}{{+}}{\left({{\zeta_{s,w}}\left\|{{{\bf{h}}_{a,s}}{\bf{w}}}\right\|+{\zeta_{a,w}}\left\|\bf{w}\right\|}\right)^{\rm{2}}}\\\
{\rm{+2}}\left({{\zeta_{s,w}}\left\|{{{\bf{h}}_{a,s}}{\bf{w}}}\right\|+{\zeta_{a,w}}\left\|{\bf{w}}\right\|}\right)\left({\left|{{{\bf{h}}^{H}}_{a,w}{\bf{w}}}\right|+\left\|{{{\bf{g}}^{H}}_{s,w}}\right\|\left\|{{{\bf{h}}_{a,s}}{\bf{w}}}\right\|}\right)$.
Similarly, problem (V-D) is solved by SDR and Gaussian randomization. Eqs.
(V-D) and (V-D) are iteratively solved until the gap of the covert
transmission rate between the _i_ -th and (_i_ +1)-th iterations is less than
a predetermined threshold $\Gamma$.

##  VI Numerical Results

In this section, numerical results for covert transmission with IRS assistance
are presented. Moreover, covert transmission without IRS is simulated as a
benchmark. We set ${P_{max}}=10$ dBm,
${{{{\tilde{\sigma}}^{2}}_{w}}}={{{{\sigma}^{2}}_{b}}}=-90$ dBm, and
$\Gamma=10^{-4}$. The small-scale fading of all channels follows the Rayleigh
fading model. The path loss model is
$PL=\left({P{L_{0}}-10{{\log}_{10}}{{\left({\frac{d}{{{d_{0}}}}}\right)}^{\mu}}}\right)$
dBm, where $P{L_{0}}=-30$ $\mathbf{dB}$, is the path loss with reference
distance ${d_{0}}=1$ m, $\mu$ is the path loss exponent, and $d$ is the
distance between the transmitter and receiver. The average iteration number
for the optimal algorithm is about 30, and the number of Gaussian
randomizations is 1000. The simulation model is presented in Fig. 2, where
Alice and the IRS lie along the same line, and the horizontal distance between
Alice and the IRS is ${\bar{d}_{A,S}}$. The horizontal and vertical distances
between Alice and Willie are denoted as ${{\bar{d}}_{A,W}}$ and $h_{w}$,
respectively. The distance between Alice and Willie is
${{{d}}_{A,W}}=\sqrt{\left({{\bar{d}}^{2}_{A,W}}+h^{2}_{w}\right)}$. The
horizontal and vertical distances between Alice and Bob are denoted as
$\bar{d}_{A,B}$ and ${{{h}}_{B}}$, respectively. The distance between Alice
and Bob is
${{{d}}_{A,B}}=\sqrt{\left({{\bar{d}}^{2}_{A,B}}+h^{2}_{B}\right)}$. The path
loss exponents of the Alice-to-IRS, the Alice-to-Bob, the Alice-to-Willie, the
IRS-to-Bob and the IRS-to-Willie links are $\mu_{AI}$, $\mu_{AB}$, $\mu_{AW}$,
$\mu_{IB}$, and $\mu_{IW}$, respectively.

Figure 2: The simulation model. Figure 3: The covert transmission rate against
${\bar{d}}_{A,W}$ with partial and instantaneous CSI of Willie’s link,
$\rho=3$ $\mathbf{dB}$, $\kappa=10^{-2}$, $N=10$, $h_{W}=5$ m, $h_{B}=3$ m,
$\mu_{A,B}=2.5$, $\mu_{A,S}=\mu_{S,B}=2$, $\mu_{A,W}=\mu_{S,W}=2.5$, and
${\bar{d}}_{A,B}={d}_{A,S}=40$ m.

We investigate the covert transmission with and without IRS assistance in Fig.
3. When only the partial CSI of Willie’s link available, we find that the
covert transmission with IRS is not necessarily better than that without IRS
whether single or multiple antennas are deployed at Alice. When Willie is
close to the IRS, the allowed transmission power is constrained significantly
and less than that without IRS. Although Bob’s link power gain is strengthened
with IRS assistance, Willie’s link power gain is also strengthened even if the
phase shifts vector is random for Willie’s link in this case. When Willie is
far away from the IRS, the covert transmission rate with IRS outperforms that
without IRS. In this case, the impact of the IRS on Bob’s link power gain is
larger than that on Willie’s link power gain. For comparison, the covert rate
with instantaneous CSI of Willie’s link is also illustrated in Fig. 3. We can
see that even if $M$=1, the covert rate with instantaneous CSI of Willie’s
link is much greater than that with partial CSI of Willie’s link, which is
consistent with our intuition. Hence, obtaining the instantaneous CSI of
Willie’s link is an important and effective way to improve the covert rate in
practice. Moreover, when instantaneous CSI of all links are available, and the
channel quality of Bob’s link is comparable to that of Willie’s link, the
covert rate with IRS assistance is greater than that without IRS assistance.

For the case of a single antenna at Alice and full instantaneous CSI of
Willie’s link available, Fig. 4 shows the covert transmission rate with and
without IRS assistance. We find that the covert rate without IRS is greater
than that with IRS. It is because the channel quality of the IRS-to-Bob link
is much better than that of the IRS-to-Willie link, i.e., $\mu_{S,B}=4.5$ and
$\mu_{S,W}=1.5$. In this case, although the signal power received at Bob is
strengthened by the IRS, the signal power received at Willie is also
strengthened by the IRS, which leads to the transmission power decrease at
Alice. Moreover, under the case of IRS assistance, our proposed algorithm
outperforms significantly the random phase shifts at IRS in terms of covert
rate. This demonstrates the advantage of the proposed optimal algorithm.

Figure 4: The covert transmission rate against ${\bar{d}}_{A,B}$ with
instantaneous CSI of Willie’s link, $N=4$, $M=1$, $\rho=3$ $\mathbf{dB}$,
$h_{B}=10$ m, $h_{W}=3$ m, $\mu_{A,B}=2$, $\mu_{A,W}=4.5$, $\mu_{S,B}=4.5$,
$\mu_{S,W}=1.5$, $\mu_{A,S}=2$, and ${{d}}_{A,S}={d}_{A,W}=60$ m.

The covert rate against ${\bar{d}}_{A,S}$ is plotted when transmit beamforming
is applied at Alice in Fig. 5, where the distance between the IRS and Willie
is fixed with $h_{W}$. The distance between the IRS and Bob is
$\sqrt{\left(\left({{\bar{d}}_{A,B}}+{{\bar{d}}_{A,S}}\right))^{2}+h^{2}_{B}\right)}$,
where Bob lies on the left side of Alice. We can see that the covert rate with
IRS is greater than that without IRS when ${\bar{d}}_{A,S}$ is small, but the
covert rate with IRS is less than that without IRS when ${\bar{d}}_{A,S}$ is
large. When the IRS is close to Alice and Bob, the effect of IRS on Bob’s link
dominates the covert performance. However, when the IRS is far away from Alice
and Bob, the effect of IRS on Willie’s link dominates the covert transmission.
This demonstrates that the IRS is not necessarily beneficial to the covert
transmission rate even if the transmit beamforming and phase shift are jointly
optimized. In addition, we can see that our proposed optimal algorithm is
better than the optimal transmit beamforming and random phase shift algorithm,
which demonstrate the advantage of our proposed algorithm.

Figure 5: The covert transmission rate against ${\bar{d}}_{A,S}$ with
instantaneous CSI of Willie’s link, ${\bar{d}}_{A,S}={\bar{d}}_{A,W}$,
$\kappa=0.01$, $\mu_{A,B}=\mu_{A,S}=\mu_{S,W}=2$, $\mu_{A,W}=\mu_{S,B}=4$,
$\rho=3$ $\mathbf{dB}$, ${\bar{d}}_{A,B}=200$ m, $M=5$, $N=10$, $h_{B}=200$ m,
and $h_{W}=5$ m. Figure 6: The covert transmission rate against
${\bar{d}}_{A,B}$ with different algorithm, $\rho=5$ $\mathbf{dB}$,
$\kappa=10^{-2}$, $M=5$, $N=20$, $h_{W}=5$ m, $h_{B}=20$ m, $\mu_{A,B}=3$,
$\mu_{A,S}=\mu_{S,B}=2$, $\mu_{A,W}=4$, $\mu_{S,W}=2$, and
${\bar{d}}_{A,W}={d}_{A,S}=40$ m. Figure 7: The covert transmission rate
against $N$ with instantaneous CSI of Willie’s link, $\rho=5$ $\mathbf{dB}$,
$\kappa=10^{-2}$, $M=5$, $h_{W}=5$ m, $h_{B}=20$ m, $\mu_{A,B}=3$,
$\mu_{A,S}=\mu_{S,B}=2$, $\mu_{A,W}=4$, $\mu_{S,W}=2$, ${\bar{d}}_{A,B}=60$ m,
and ${\bar{d}}_{A,W}={d}_{A,S}=40$ m.

In Fig. 6 and Fig. 7, the covert transmission rates for the optimal and
suboptimal algorithms proposed in this paper are plotted against
$\bar{d}_{A,B}$ and against $N$. In terms of the covert transmission rate, the
optimal algorithm is better than the algorithm minimizing Willie’s link power
gain, because the optimal algorithm achieves a trade off between maximizing
Bob’s link and minimizing Willie’s link power gains. The zero-forcing
algorithm is the worse than the optimal algorithm and the suboptimal algorithm
minimizing the Willie’s link, because the diversity order of zero-forcing
algorithm decreases to $M-1$, while the diversity order of the optimal
algorithm and the algorithm minimizing Willie’s link power gain is $M$. In
addition, we can see that our proposed optimal algorithm is better than the
optimal transmit beamforming and random phase shift algorithm.

Figure 8: The impact of imperfect CSI of the Willie’s link on the covert
transmission rate, ${\bar{d}}_{A,B}=60$ m, ${\bar{d}}_{S,B}=20$ m, $\rho=3$
$\mathbf{dB}$,, $\kappa=0.01$, $N=20$, $\mu_{A,B}=\mu_{S,B}=2$,
$\mu_{A,S}=\mu_{A,W}=\mu_{S,W}=3$, ${\zeta}_{A,W}=5*{10}^{-9}$,
${\zeta}_{A,S}={\zeta}_{S,W}=5*{10}^{-6}$, $M=6$, $N=20$, and
$d_{A,S}=d_{A,W}=d_{S,W}=40$ m.

The covert transmission rates with imperfect CSI of Willie’s link are shown in
Fig. 8, where the distance between Alice, Willie, and the IRS is equal, i.e.
$d_{A,S}=d_{A,W}=d_{S,W}=40$m. We find that the imperfect CSI of Willie’s link
leads to degradation of the covert transmission rate. In addition, we can see
that in this scenario the covert rates decrease with the vertical distance
$h_{B}$. Moreover, the imperfect CSI of both Alice-to-Willie and IRS-to-Willie
links has the most serious effect on the covert rate. The imperfect CSI of
Alice-to-Willie link has the second most serious effect on covert rate. This
is because the Alice-to-Willie link is an independent part of Willie’s link.
By contrast, the combination of the Alice-to-IRS link and the IRS-to-Willie
link is an independent part of Willie’s link. In addition, since the Alice-to-
IRS link affects both Willie’s link and Bob’s link power gains, the impact of
the Alice-to-IRS link on the covert rate is lower than that of the IRS-to-
Willie link.

##  VII Conclusion

In this paper, we investigated the covert transmission rate in an IRS-aided
system, where an energy detection method is applied at Willie. We found that
the IRS does not always benefit the covert transmission. This conclusion is
different from the multiuser system and secrecy transmission system where the
IRS always improves performance. The reason is that in a covert transmission
system, the signal power received at Willie should be less than a
predetermined threshold, which limits the transmitter and IRS abilities to
improve the quality of Bob’s link. Hence, when we use IRS in covert
transmission, the distances relationship between Bob, IRS, and Willie should
be carefully considered. Moveover, although in this paper the proposed
algorithm is used to handle the case of a single warden, the proposed
algorithm can also handle the case of multiple wardens.

##  Appendix A The PDF of $z$

The complex-valued variables ${h_{a,{s_{i}}}}$ and ${g_{{s_{i}},w}}$ can be
expressed as ${h_{a,{s_{i}}}}{\rm{=}}{A_{i}}+j{B_{i}}$, and
${g_{{s_{i}},w}}={C_{i}}+j{D_{i}}$, where
${A_{i}}=Re\left({h_{a,{s_{i}}}}\right)$ and
${B_{i}}=Im\left({h_{a,{s_{i}}}}\right)$ follow the Gaussian distribution with
mean zero and variance $\frac{{\sigma^{2}}_{a,s}}{2}$, while
${C_{i}}=Re\left({h_{{s_{i}},w}}\right)$ and
${D_{i}}=Im\left({h_{{s_{i}},w}}\right)$ follow the Gaussian distribution with
mean zero and variance $\frac{{\sigma^{2}}_{s,w}}{2}$. Then, we can obtain

$\displaystyle\
{h_{a,{s_{i}}}}{g_{{s_{i}},w}}={A_{i}}{C_{i}}-{B_{i}}{D_{i}}+j\left({{B_{i}}{C_{i}}+{A_{i}}{D_{i}}}\right).$
(80)

In (80), we have
${\mathbb{E}}\left({{A_{i}}{C_{i}}}\right){\rm{=0}},{\mathbb{E}}\left({{{\left({{A_{i}}{C_{i}}}\right)}^{\rm{2}}}}\right){\rm{=}}\frac{1}{4}{\sigma^{2}}_{a,s}{\sigma^{2}}_{s,w}$,
${\mathbb{E}}\left({{B_{i}}{D_{i}}}\right){\rm{=0}},{\mathbb{E}}\left({{{\left({{B_{i}}{D_{i}}}\right)}^{\rm{2}}}}\right){\rm{=}}\frac{1}{4}{\sigma^{2}}_{a,s}{\sigma^{2}}_{s,w}$,
${\mathbb{E}}\left({{B_{i}}{C_{i}}}\right){\rm{=0}},{\mathbb{E}}\left({{{\left({{B_{i}}{C_{i}}}\right)}^{\rm{2}}}}\right){\rm{=}}\frac{1}{4}{\sigma^{2}}_{a,s}{\sigma^{2}}_{s,w}$,
and
${\mathbb{E}}\left({{A_{i}}{D_{i}}}\right){\rm{=0}},{\mathbb{E}}\left({{{\left({{A_{i}}{D_{i}}}\right)}^{\rm{2}}}}\right){\rm{=}}\frac{1}{4}{\sigma^{2}}_{a,s}{\sigma^{2}}_{s,w}$.
Thus, when $N\rightarrow\infty$, applying the central limit theorem, we can
obtain that
$\sum\limits_{i=1}^{N}{{e^{j{\theta_{i}}}}{h_{a,{s_{i}}}}{g_{{s_{i}},w}}}$
follows the complex Gaussian distribution with zero mean and variance
$N{\sigma^{2}}_{a,s}{\sigma^{2}}_{s,w}$. Since ${h_{a,w}}$ follows the complex
Gaussian distribution with zero mean and variance ${\sigma^{2}}_{a,w}$,
${h_{a,w}}+\sum\limits_{i=1}^{N}{{e^{j{\theta_{i}}}}{h_{a,{s_{i}}}}{g_{{s_{i}},w}}}$
follows the complex Gaussian distribution with zero mean and variance
$N{\sigma^{2}}_{a,s}{\sigma^{2}}_{s,w}+{\sigma^{2}}_{a,w}$. Thus, the PDF of
$z={\left|{{h_{a,w}}{\rm{+}}\sum\limits_{i=1}^{N}{{e^{j{\theta_{i}}}}{h_{a,{s_{i}}}}{g_{{s_{i}},w}}}}\right|^{2}}$
is exponential distributed, and it is given by

$\displaystyle\
{f_{z}}\left(z\right)=\frac{{\rm{1}}}{{\left({{\sigma^{2}}_{s,w}{\rm{+}}N{\sigma^{2}}_{a,s}{\sigma^{2}}_{s,w}}\right)}}\exp\left({-\frac{z}{{\left({{\sigma^{2}}_{s,w}{\rm{+}}N{\sigma^{2}}_{a,s}{\sigma^{2}}_{s,w}}\right)}}}\right).$
(81)

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