arXiv:2011.03726v2 [cs.IT] 11 Jan 2022 1 Intelligent Reflecting Surface (IRS)-Aided Covert Wireless Communications with Delay Constraint Xiaobo Zhou, Member, IEEE, Shihao Yan, Member, IEEE, Qingqing Wu, Member, IEEE, Feng Shu, Member, IEEE, and Derrick Wing Kwan Ng, Fellow, IEEE Abstract—This work examines the performance gain achieved by deploying an intelligent reflecting surface (IRS) in covert communications. To this end, we formulate the joint design of the transmit power and the IRS reflection coefficients by taking into account the communication covertness for the cases with global channel state information (CSI) and without a warden’s instantaneous CSI. For the case of global CSI, we first prove that perfect covertness is achievable with the aid of the IRS even for a single-antenna transmitter, which is impossible without an IRS. Then, we develop a penalty successive convex approximation (PSCA) algorithm to tackle the design problem. Considering the high complexity of the PSCA algorithm, we further propose a low-complexity two-stage algorithm, where analytical expressions for the transmit power and the IRS’s reflection coefficients are derived. For the case without the warden’s instantaneous CSI, we first derive the covertness constraint analytically facilitating the optimal phase shift design. Then, we consider three hardware- related constraints on the IRS’s reflection amplitudes and deter- mine their optimal designs together with the optimal transmit power. Our examination shows that significant performance gain can be achieved by deploying an IRS into covert communications. Index Terms—Intelligent reflecting surface, covert communi- cations, reflection beamforming, transmit power design. I. I NTRODUCTION To meet the ever-increasing demand for high-data rate applications and massive connections in wireless networks, multiple advanced technologies, such as massive multiple- input multiple-out (MIMO), millimeter wave (mmWave), and ultra-dense network (UDN), have been advocated [1], [2]. However, these technologies generally suffer from high energy consumption or high hardware complexity, due to the use of a large number of power-hungry radio frequency (RF) chains. As a remedy, intelligent reflecting surface (IRS) is emerging as a promising solution to improving the spectral and energy efficiency effectively [3]. Specifically, IRS is a planar surface consisting of a large number of re-configurable and low-cost passive reflecting elements, each of which is able to reflect the incident signals with controllable amplitudes and phase shifts. X. Zhou is with the Institute of Intelligent Agriculture, School of Informa- tion and Computer, Anhui Agricultural University, Hefei 230036, China, and also with the School of Physics and Electronic Engineering, Fuyang Normal University, Fuyang 236037, China (e-mail: [email protected]). S. Yan and D. W. K. Ng are with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, 2052, Australia (email: [email protected]; [email protected]). Q. Wu is with the State Key Laboratory of Internet of Things for Smart City and Department of Electrical and Computer Engineering, University of Macau, Macao, 999078, China (e-mail:[email protected]). F. Shu is with the School of Information and Communication Engineering, Hainan University, Hainan 570228, China (e-mail: [email protected]). Thus, IRS can customize the propagation environment from the transmitter to receiver to achieve various design objectives (e.g., signal enhancement, interference suppression). Due to the aforementioned advantages, IRS has been investigated in various application scenarios, e.g., single-user systems [4], [5], multi-user systems [6]–[11], and wireless information and power transfer systems [12], [13], and it has been considered as a promising technology for enabling the sixth-generation (6G) wireless networks [14], [15]. Recently, considering the increasing concerns on security issues in wireless communications, several recent works ad- dressed communication security in the context of IRS-assisted wireless networks from the perspective of physical layer security, e.g., [16]–[23]. In general, the secrecy performance of IRS-assisted networks can be improved by properly designing the IRS reflection coefficients to simultaneously enhance the received signal strength at desired users and weaken them at eavesdroppers. For example, as shown in [16], by jointly optimizing the transmit beamforming together with the reflect beamforming, physical layer security is guaranteed in IRS- assisted network, even if the eavesdropping channel quality is higher than that of the legitimate channel. Along this direction, an alternative optimization algorithm based on semidefinite programming (SDP) relaxation technique was proposed to determine the secure transmit beamforming and reflecting phase shifts in [17]. In addition, the authors of [18] tackled the question whether and when artificial noise (AN) is beneficial to the physical layer security in IRS-assisted wireless commu- nication systems. Meanwhile, MIMO wiretap channels were considered in [19]–[21] for optimizing the transmit covariance matrix and IRS phase shifts, and the channel imperfectness on multiuser multiple-input-single-output (MISO) and MIMO wireless secure communications were considered in [22] and [23], respectively. The aforementioned physical layer security technologies focus on protecting the content of the transmitted message against eavesdropping. However, these technologies cannot alleviate privacy issues posed by discovering the presence of the transmitter or transmissions. Fortunately, the emerging and cutting-edge covert communication technology, which aims at hiding the existence of a wireless transmission, is able to preserve such a high-level security and privacy [24]. In general, a positive covert transmission rate can be achieved when the warden (Willie) has various uncertainties, e.g., noise uncertainty [25] and channel uncertainty [26]. In particular, the fundamental limits of covert communication in additive white Gaussian noise (AWGN) channels was established in
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Abstract—This work examines the performance gain achievedby deploying an intelligent reflecting surface (IRS) in covertcommunications. To this end, we formulate the joint design ofthe transmit power and the IRS reflection coefficients by takinginto account the communication covertness for the cases withglobal channel state information (CSI) and without a warden’sinstantaneous CSI. For the case of global CSI, we first provethat perfect covertness is achievable with the aid of the IRS evenfor a single-antenna transmitter, which is impossible without anIRS. Then, we develop a penalty successive convex approximation(PSCA) algorithm to tackle the design problem. Considering thehigh complexity of the PSCA algorithm, we further propose alow-complexity two-stage algorithm, where analytical expressionsfor the transmit power and the IRS’s reflection coefficients arederived. For the case without the warden’s instantaneous CSI, wefirst derive the covertness constraint analytically facilitating theoptimal phase shift design. Then, we consider three hardware-related constraints on the IRS’s reflection amplitudes and deter-mine their optimal designs together with the optimal transmitpower. Our examination shows that significant performance gaincan be achieved by deploying an IRS into covert communications.
Index Terms—Intelligent reflecting surface, covert communi-cations, reflection beamforming, transmit power design.
I. INTRODUCTION
To meet the ever-increasing demand for high-data rate
applications and massive connections in wireless networks,
multiple advanced technologies, such as massive multiple-
input multiple-out (MIMO), millimeter wave (mmWave), and
ultra-dense network (UDN), have been advocated [1], [2].
However, these technologies generally suffer from high energy
consumption or high hardware complexity, due to the use of
a large number of power-hungry radio frequency (RF) chains.
As a remedy, intelligent reflecting surface (IRS) is emerging
as a promising solution to improving the spectral and energy
efficiency effectively [3]. Specifically, IRS is a planar surface
consisting of a large number of re-configurable and low-cost
passive reflecting elements, each of which is able to reflect the
incident signals with controllable amplitudes and phase shifts.
X. Zhou is with the Institute of Intelligent Agriculture, School of Informa-tion and Computer, Anhui Agricultural University, Hefei 230036, China, andalso with the School of Physics and Electronic Engineering, Fuyang NormalUniversity, Fuyang 236037, China (e-mail: [email protected]).
S. Yan and D. W. K. Ng are with the School of Electrical Engineering andTelecommunications, University of New South Wales, Sydney, 2052, Australia(email: [email protected]; [email protected]).
Q. Wu is with the State Key Laboratory of Internet of Things for SmartCity and Department of Electrical and Computer Engineering, University ofMacau, Macao, 999078, China (e-mail:[email protected]).
F. Shu is with the School of Information and Communication Engineering,Hainan University, Hainan 570228, China (e-mail: [email protected]).
Thus, IRS can customize the propagation environment from
the transmitter to receiver to achieve various design objectives
(e.g., signal enhancement, interference suppression). Due to
the aforementioned advantages, IRS has been investigated in
various application scenarios, e.g., single-user systems [4],
[5], multi-user systems [6]–[11], and wireless information and
power transfer systems [12], [13], and it has been considered
as a promising technology for enabling the sixth-generation
(6G) wireless networks [14], [15].
Recently, considering the increasing concerns on security
issues in wireless communications, several recent works ad-
dressed communication security in the context of IRS-assisted
wireless networks from the perspective of physical layer
security, e.g., [16]–[23]. In general, the secrecy performance of
IRS-assisted networks can be improved by properly designing
the IRS reflection coefficients to simultaneously enhance the
received signal strength at desired users and weaken them
at eavesdroppers. For example, as shown in [16], by jointly
optimizing the transmit beamforming together with the reflect
beamforming, physical layer security is guaranteed in IRS-
assisted network, even if the eavesdropping channel quality is
higher than that of the legitimate channel. Along this direction,
an alternative optimization algorithm based on semidefinite
programming (SDP) relaxation technique was proposed to
determine the secure transmit beamforming and reflecting
phase shifts in [17]. In addition, the authors of [18] tackled the
question whether and when artificial noise (AN) is beneficial
to the physical layer security in IRS-assisted wireless commu-
nication systems. Meanwhile, MIMO wiretap channels were
considered in [19]–[21] for optimizing the transmit covariance
matrix and IRS phase shifts, and the channel imperfectness
on multiuser multiple-input-single-output (MISO) and MIMO
wireless secure communications were considered in [22] and
[23], respectively.
The aforementioned physical layer security technologies
focus on protecting the content of the transmitted message
against eavesdropping. However, these technologies cannot
alleviate privacy issues posed by discovering the presence of
the transmitter or transmissions. Fortunately, the emerging and
cutting-edge covert communication technology, which aims
at hiding the existence of a wireless transmission, is able
to preserve such a high-level security and privacy [24]. In
general, a positive covert transmission rate can be achieved
when the warden (Willie) has various uncertainties, e.g., noise
uncertainty [25] and channel uncertainty [26]. In particular,
the fundamental limits of covert communication in additive
white Gaussian noise (AWGN) channels was established in
Fig. 7. Received SNR at Bob and transmit power at Alice versus IRShorizontal location for covertness level ǫ = 0.1, where σ2
b= −90 dBm.
but is independent of IRS phase shifts. As such, the average
covertness constraint Pa
σ2w
(
χrw
∑Nn=1 ρ
2n|harn |2 + χaw
)
≤ ǫ
is harder to be satisfied as N increases. Interestingly, we also
observe that Pa is higher in the system without IRS relative
to that in the system with IRS, which is completely different
from the case with Willie’s instantaneous CSI.
In Fig. 7, we plot Bob’s SNR and Alice’s transmit power
Pa achieved by different solutions versus the IRS horizontal
location. We first observe from Fig. 7(a) that the optimal IRS
horizontal location is close to the LHS of Bob (denoted by
the verticle dashed line in this figure). This is due to the fact
that satisfying the covertness constraint becomes challenging
as IRS moves closer to Willie. We note that the optimal
IRS horizontal location is to balance the transmission quality
and communication covertness, since it not only guarantees a
small reflection path loss from IRS to Bob, but also makes
the covertness constraint easier to be satisfied. We also note
that this observation is different from the case with Willie’s
instantaneous CSI, in which the covertness constraint becomes
easier to be satisfied as the IRS moves closer to Willie. In
addition, we observe from Fig. 7(b) that the transmit power
Pa achieved by our developed solutions decreases as the IRS
moves closer to Willie. This is attributed to the fact that the
covertness constraint becomes harder to be satisfied as the
reflect-path gain from IRS to Willie becomes larger.
VI. CONCLUSION
This paper tackled covert communication system designs
by considering the assistance of an IRS in the cases with
global CSI and without Willie’s instantaneous CSI. For the
case with global CSI, we proved that the perfect covertness
can be achieved if the channel quality of the reflected path is
higher than that of the direct path. Then, we developed a PSCA
algorithm and a low-complexity two-stage algorithm to jointly
design the IRS’s reflection coefficients and Alice’s transmit
power. For the case without the Willie’s instantaneous CSI,
our analysis showed that the phase shift of each IRS element is
13
independent of the covertness constraint, based on which the
optimal phase shifts and reflection amplitudes together with
the Alice’s transmit power are determined. Our examinations
showed that deploying an IRS is able to enhance the reflection
signal at Bob and deteriorate the detection performance at
Willie by properly designing the reflection coefficients, so as
to improve covert communication performance. Interestingly,
it was revealed that the optimal horizontal location of the
IRS is between Bob and Willie for the case with global
CSI, while it is close to the LHS of Bob for the case
without Willie’s instantaneous CSI. To further unleash the
potential of IRS-aided covert communications, one challenge
to be addressed in future works is how to obtain accurate
CSI covertly without using the traditional pilot-based channel
estimation methods. Addressing this challenge may call for
new emerging techniques (e.g., machine learning) to conduct
passive channel estimation (e.g., based on three-dimensional
images).
APPENDIX A
PROOF OF THEOREM 1
We first note that the perfect covertness implies the required
covertness level ǫ = 0. In addition, we observe from (P1)
that D(P0|P1) in covertness constraint (8b) monotonically
increases with the received energy at Willie (i.e., Pa|vHa +
haw|2), and Pa|vHa + haw|2 = 0 means Willie cannot
detect any transmission. Thus, under the perfect covertness
constraint, problem (P1) can be reformulated into problem
(P1′). We note that problem (P1′) is not always feasible when
the transmit power Pa is non-zero. In addition, one can verify
that Pa = Pmax is the optimal solution to problem (P1′)when Ew , |vH
a+haw|2 = 0. In the following, we focus on
deriving the condition of Ew = 0. To this end, we first recall
that vn = ρne−jθn , ∀n, it follows that
Ew =
∣
∣
∣
∣
∣
N∑
n=1
ρn|an|ej(arg(an)+θn) + |haw|ej arg(haw)
∣
∣
∣
∣
∣
2
. (49)
We note that Ew=0 only when the signs of the real and imag-
inary part of∑N
n=1ρn|an|ej(arg(an)+θn) and |haw|ej arg(haw)
are opposite. We also note that if each summation term
ρn|an|ej(arg(an)+θn) has the same phase for coherent combin-
ing, the synthesized∑N
n=1 ρn|an|ej(arg(an)+θn) achieves the
largest modulus, which is given by∑N
n=1 ρn|an|. As a result,
ifN∑
n=1
ρn|an| ≥ |haw|, (50)
we can always adjust reflection amplitude ρn and reflection
phase shift θn such that Ew = 0. We note that the maximum
value of∑N
n=1 ρn|an| is∑N
n=1 |an| due to ρn ∈ [0, 1], ∀n.
Following above discussions, we can conclude that the perfect
covertness can be achieved with non-zero transmit power if
and only if∑N
n=1 |an| ≥ |haw|. This completes the proof of
Theorem 1.
APPENDIX B
PROOF OF LEMMA 1
We first note that the second term on the LHS of (16) is
a linear function of the concerned optimization variable W,
while the right-hand side (RHS) of (16) is a constant. As such,
we only need to prove that the first term on the LHS of (16)
is convex with respect to W, which is redefined as
f(W) =
(
1 +Tr(AW)
σ2w
)
ln
(
1 +Tr(AW)
σ2w
)
. (51)
To proceed, we first note that x ln(x) is a convex function
of x for x > 0 [46]. Then, following the fact that A � 0and W � 0, we have that f(W) is convex with respect to
W since the affine transformation of x ln(x) is convex with
respect to x. This completes the proof of Lemma 1.
APPENDIX C
PROOF OF THEOREM 2
As per (34), in order to determine EX [D(P0|P1)], we have
to derive the distribution of X . To this end, we first rewrite
hHrwΘhar + haw as
hHrwΘhar + haw =
N∑
n=1
h∗rwn
ρnejθnharn + haw, (52)
where hrwnand harn are the n-th element of hrw and
har, respectively. As a result, hHrwΘhar + haw follows the
distribution CN (0, δ), where δ , χrw
∑Nn=1 ρ
2n|harn |2+χaw.
Then, the probability density function (pdf) of X denoted as
fX(x) is an exponential distribution with parameter δ−1. As
such, EX [D(P0|P1)] can be rewritten as
EX [D(P0|P1)]
=
∫ ∞
0
L
[
ln
(
1 +Pax
σ2w
)
− Pax
Pax+ σ2w
]
fX(x)dx,
= L
(
1 +σ2w
δPa
)
eσ2w
δPa E1
(
σ2w
δPa
)
− L. (53)
Following (53), the covertness constraint EX [D(P0|P1)] ≤2ǫ2 can be equivalently rewritten as
g
(
σ2w
δPa
)
≤ 1 +2ǫ2
L, (54)
where g(
σ2
w
δPa
)
,(
1 +σ2
w
δPa
)
eσ2w
δPa E1
(
σ2
w
δPa
)
. We note that
an exact analytical expression for g(
σ2
w
δPa
)
is mathematically
intractable, since it involves an exponential integral function.
To overcome this difficulty, we next show that g(
σ2
w
δPa
)
is
a monotonically decreasing function ofσ2
w
δPa. To this end, we
first define g(x) = (1 + x)exE1(x), where x ≥ 0. Then, the
first derivative of g(x) with respect to x is given by
dg(x)
dx= (x+ 2)exE1(x)−
x+ 1
x
= (x+ 2)ex(∫ ∞
x
e−t
tdt− s(x)
)
, (55)
where s(x) = x+1x(x+2)ex . We note that the first derivative of
s(x) with respect to x is given by
ds(x)
dx=
e−x(
−2− x(x+ 2)2)
x2(x+ 2)2. (56)
14
Considering thatds(x)dx
is a continuous function of x for x > 0,
we have∫ ∞
x
e−t(
−2− t(t+ 2)2)
t2(t+ 2)2dt = s(∞)− s(x)
= −s(x). (57)
As per (57), we can rearrange (55) as
dg(x)
dx= (x+ 2)ex
(
∫ ∞
x
e−t
t+
e−t(
−2− t(t+ 2)2)
t2(t+ 2)2dt
)
= (x+ 2)ex∫ ∞
x
−2e−t
t2(t+ 2)2dt. (58)
Following the fact that the integrand −2e−t
t2(t+2)2 is always less
than or equal to 0, we havedg(x)dx
≤ 0 must hold for x > 0.
As a result, g(x) is a monotonically decreasing function of x
for x > 0.
Following the above fact, we have g(
σ2
w
δPa
)
given in (54)
as a monotonically increasing function of δPa
σ2w
. Then, the
covertness constraint (54) can be equivalently rewritten as that
given in (35), which completes the proof of Theorem 2.
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