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Robust Probabilistic-Constrained Optimization for IRS-Aided MISO Communication Systems

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Abstract

Taking into account imperfect channel state information, this letter formulates and solves a joint active/passive beamforming optimization problem in multiple-input single-output systems with the support of an intelligent reflecting surface. In particular, we introduce an optimization problem to minimize the total transmit power subject to maintaining the users’ signal-to-interference-plus-noise-ratio coverage probability above a predefined target. Due to the presence of probabilistic constraints, the proposed optimization problem is non-convex. To circumvent this issue, we first recast the proposed problem in a convex form by adopting the Bernstein-type inequality, and we then introduce a converging alternating optimization approach to iteratively find the active/passive beamforming vectors. In particular, the transformed robust optimization problem can be effectively solved by using standard interior-point methods. Numerical results demonstrate the effectiveness of jointly optimizing the active/passive beamforming vectors.
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Le, Tuan Anh ORCID: https://orcid.org/0000-0003-0612-3717, Trinh, Van Chien and Di Renzo,
Marco (2020) Robust probabilistic-constrained optimization for IRS-aided MISO communication
systems. IEEE Wireless Communications Letters . ISSN 2162-2337 (Accepted/In press)
(doi:10.1109/LWC.2020.3016592)
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1
Robust Probabilistic-Constrained Optimization for
IRS-Aided MISO Communication Systems
Tuan Anh Le, Senior Member IEEE, Trinh Van Chien, and Marco Di Renzo, Fellow IEEE
Abstract—Taking into account imperfect channel state infor-
mation, this letter formulates and solves a joint active/passive
beamforming optimization problem in multiple-input single-
output systems with the support of an intelligent reflecting
surface. In particular, we introduce an optimization problem to
minimize the total transmit power subject to maintaining the
users’ signal-to-interference-plus-noise-ratio coverage probability
above a predefined target. Due to the presence of probabilistic
constraints, the proposed optimization problem is non-convex. To
circumvent this issue, we first recast the proposed problem in a
convex form by adopting the Bernstein-type inequality, and we
then introduce a converging alternating optimization approach
to iteratively find the active/passive beamforming vectors. In
particular, the transformed robust optimization problem can be
effectively solved by using standard interior-point methods. Nu-
merical results demonstrate the effectiveness of jointly optimizing
the active/passive beamforming vectors.
Index Terms—6G wireless, intelligent reflecting surface.
I. INTRODUCTION
Intelligent reflecting surface (IRS), i.e., a programmable pla-
nar array of passive reflecting elements, has been identified as
an energy-efficient technology for improving the performance
of wireless communication systems [1]–[4]. An IRS-aided
communication system comprises an IRS, a base station (BS)
and the mobile users. The location of the IRS is usually chosen
to assist the BS communicating with the users. An IRS can
also extend the communication range where there is no direct
link between the BS and the users. Each element of the IRS
can be independently controlled to produce a desired phase
shift on the impinging electromagnetic waves. A careful design
of the phase shifts of all the elements of the IRS helps to
improve the channel propagation conditions [5]. Moreover, the
combination of transmit beamforming and phase shift designs
has been proved to offer higher spectral and energy efficiencies
compared to conventional beamforming methods [6].
Manuscript received March 24, 2020; revised July 09, 2020 and August
07, 2020; accepted August 07, 2020. Date of publication XXX; date of
current version XXX. The work of M. Di Renzo was supported in part
by the European Commission through the H2020 ARIADNE Project under
Grant 871464. The associate editor coordinating the review of this letter and
approving it for publication was Dr. J. Xu. (Corresponding author: Trinh Van
Chien.)
T. A. Le is with the Department of Design Engineering & Mathematics,
Faculty of Science and Technology, Middlesex University, The Burroughs,
Hendon, London, NW4 4BT, U. K. Email: t.le@mdx.ac.uk.
T. V. Chien was with the School of Electronics and Telecommunications,
Hanoi University of Science and Technology, 100000 Hanoi, Vietnam. He
is now with the Interdisciplinary Centre for Security, Reliability and Trust
(SnT), University of Luxembourg, L-1855 Luxembourg, Luxembourg. Email:
vanchien.trinh@uni.lu.
M. Di Renzo is with Universit´
e Paris-Saclay, CNRS, CentraleSup´
elec,
Laboratoire des Signaux et Syst`
emes, 91192 Gif-sur-Yvette, France. Email:
marco.direnzo@centralesupelec.fr.
The performance of an IRS-aided communication system
heavily depends on the accuracy of the channel state informa-
tion (CSI), i.e., the CSI between the BS and the IRS, as well
as the CSI between the IRS and the mobile users. Most of
the works on IRS-aided systems are based on a perfect CSI
assumption, see e.g., [7] and references therein. Unfortunately,
due to the random nature of wireless systems, obtaining
accurate CSI is, in practice, a challenging task. Therefore, the
robust optimization of the active/passive beamforming against
imprecise estimates of the CSI is highly desirable in order
to fully realize the potential of IRS-aided communication
systems.
The first attempt to tackle imperfect estimates of the CSI
in IRS-aided systems can be found in [8]. In [8], the transmit
beamforming vectors at the BS and the phase shifts at the IRS
are jointly designed so as to minimize the total transmit power
subject to the worst-case quality of service (QoS) constraints,
i.e., maintaining the required achievable rate for every user
under all possible cases of CSI errors. To that end, an ellipsoid
approach was adopted to capture the channel uncertainties,
where the Frobenious norm of the channel error vector is
assumed to be confined within a given radius of the uncertainty
region. This method results in a conservative approach since it
allocates excessive system resources to satisfy rarely occurring
worst-case events.
In this letter, we tackle the problem of allocating excessive
system resources by allowing that the QoS constraints can be
violated with a certain probability. In particular, we formulate
an optimization problem to jointly design the active/passive
beamforming vectors at the BS and the IRS. The proposed
problem minimizes the total transmit power while maintain-
ing the users’ signal-to-interference-plus-noise-ratio (SINR)
coverage probability above a predefined value. Since the
considered probabilistic constraints are not convex, we adopt
the Bernstein-type inequality [9] to transform the formulated
problem into a linear matrix inequality (LMI) form, i.e., into
a convex optimization problem. In addition, we introduce
an alternating optimization approach to iteratively find the
active/passive beamforming vectors. Our proposed problem
differs from that in [8] for two main reasons. First, the QoS
constraints are formulated in a probabilistic form while those
of [8] are not. Second, unlike [8], the considered problem does
not rely on the assumption of bounded channel uncertainty
region.
The following notation is adopted in this letter. Bold
lower/upper case letters denote vectors/matrices; k·kand k·k𝐹
denote the Euclidean norm and the Frobenius norm, respec-
tively; (·)𝐻,(·)𝑇and (·)denote the complex conjugate
transpose operator, the transpose operator and the complex
2
conjugate operator, respectively; tr (·)denotes the trace of a
matrix; X0denotes the positive semidefinite condition; I𝑎
denotes an 𝑎×𝑎identity matrix; diag (x)denotes a diagonal
matrix whose diagonal entries are the elements of the vector
x; diag (X)denotes a vector comprising the diagonal elements
of matrix X;1𝑁denotes an 𝑁×1vector of all unity elements;
CN,·) denotes a circularly symmetric complex Gaussian
distribution; vec(·) denotes the vectorization operator; Pr (·)
denotes the probability of an event; denotes the Kronecker
product; |·|and arg(·) denote the absolute value and the
phase of a complex-valued scalar, respectively; E[·] denotes
the expectation of a random variable.
II. SY ST EM MO DE L
We consider an IRS-aided communication system that con-
sists of an 𝑀-antenna BS serving 𝐾single-antenna users. It is
assumed that there is no direct communication link from the
BS to the users due to the presence of blockages, e.g., high
buildings. To overcome this issue, an IRS with 𝑁reflective
elements is deployed to assist the communication between
the BS and the users. Let H=[h1, . . . , h𝑁] C𝑀×𝑁and
g𝑘=[𝑔𝑘1, . . . , 𝑔𝑘 𝑁 ]𝑇C𝑁×1denote the channel coefficients
between the BS and the IRS, as well as between the IRS and
the 𝑘-th user, respectively. We assume that the instantaneous
CSI is not known, and that the channel is estimated from the
uplink training data. Therefore, the system operates under a
time division duplexing protocol and by leveraging the channel
reciprocity of the downlink and uplink channels.1
A. Uplink training phase
In the uplink training phase, 𝑁time slots are dedicated to
the channel estimation by utilizing 𝐾orthogonal pilot signals
{𝜓
𝜓
𝜓1, . . . , 𝜓
𝜓
𝜓𝐾}, each comprising 𝜏𝑝symbols with 𝜏𝑝𝐾. In
each time slot, one reflective element is chosen to only reflect
the training pilot signal without introducing any phase shift
while the other 𝑁1elements do not reflect the incident
signal.2The pilot signal reflected by the 𝑛-th reflective element
Y𝑝
𝑛C𝑀×𝜏𝑝received at the BS is
Y𝑝
𝑛=h𝑛
𝐾
Õ
𝑡=1
𝜌𝑡𝑔𝑡𝑛 𝜓
𝜓
𝜓𝐻
𝑡+N𝑝
𝑛,(1)
where 𝜌𝑡is the pilot power of each symbol and N𝑝
𝑛
C𝑀×𝜏𝑝is the additive noise whose elements are distributed
as CN(0, 𝜎2). The channel between the 𝑘-th user and the
BS that receives the pilot via the 𝑛-th reflective element is
estimated from
y𝑝
𝑘𝑛 =Y𝑝
𝑛𝜓
𝜓
𝜓𝑘
=𝜌𝑘𝜏𝑝h𝑛𝑔𝑘𝑛 +˜
n𝑘𝑛 ,(2)
1Although channel reciprocity is a widely adopted assumption in wireless
communications, it is affected by some practical realization difficulties, which
include the non-symmetric characteristics of the RF front-end circuitry at the
receiver and transmitter.
2The activation of all IRS elements by using reflection patterns based on
orthogonal designs, e.g., reflection patterns based on the discrete Fourier
transform [10], [11], can provide higher antenna gains and reduce channel
estimation errors.
where ˜
n𝑘𝑛 =N𝑝
𝑛𝜓
𝜓
𝜓𝑘∼ CN(0, 𝜏𝑝𝜎2I𝑀). By utilizing least
squares estimation methods [12], the estimated channel be-
tween the 𝑘-th user and the BS assisted by the 𝑛-th reflective
element can be formulated as
˜
g𝑘𝑛 =1
𝜌𝑘𝜏𝑝
y𝑝
𝑘𝑛
=h𝑛𝑔𝑘𝑛 +1
𝜌𝑘𝜏𝑝
˜
n𝑘𝑛 .
(3)
Given the matrix e
G𝑘=[˜
g𝑘1, . . . , ˜
g𝑘 𝑁 ] C𝑀×𝑁, the exact
channel G𝑘between the BS and the 𝑘-th user is
G𝑘=e
G𝑘+E𝑘,(4)
where E𝑘C𝑀×𝑁is the channel estimation error matrix
whose individual elements are distributed as CN(0,𝜎2
𝜌𝑘𝜏𝑝). In
the sequel, we use the channel estimates in (3) to evaluate
the impact of channel uncertainty on the spectral efficiency of
each user.
B. Downlink data transmission
Let w𝑘C𝑀×1and 𝑥𝑘, with E[|𝑥𝑘|2]=1, be the active
beamforming vector and the data symbol for the 𝑘-th user,
respectively. In the downlink phase, each reflective element
introduces a reflection coefficient to assist the communication
between the BS and the users. Let 𝜃
𝜃
𝜃=[𝜃1, 𝜃2, . . . , 𝜃 𝑁]𝑇
be the coefficients, i.e., the passive beamforming vector, intro-
duced by the IRS where |𝜃𝑛| 1and arg(𝜃𝑛) ∈ [−𝜋, 𝜋)are
the amplitude and the phase shift, respectively, 𝑛=1, . . . , 𝑁
[7].3The signal received by the 𝑘-th user can be written as
𝑦𝑘=g𝐻
𝑘diag(𝜃
𝜃
𝜃)𝐻H𝐻
𝐾
Õ
𝑡=1
w𝑡𝑥𝑡+𝑛𝑘,(5)
where 𝑛𝑘∼ CN(0, 𝜎2)is the additive noise at the 𝑘-th user.
Denoting G𝐻
𝑘=diag(g
𝑘)H𝐻C𝑁×𝑀, we can rewrite (5) as
𝑦𝑘=𝜃
𝜃
𝜃𝐻G𝐻
𝑘
𝐾
Õ
𝑡=1
w𝑡𝑥𝑡+𝑛𝑘.(6)
Using (4) and (6), the exact SINR at the 𝑘-th user, which
is denoted by Γ𝑘({w𝑘}, 𝜃
𝜃
𝜃), can be written as4
Γ𝑘({w𝑘}, 𝜃
𝜃
𝜃)=|𝜃
𝜃
𝜃𝐻e
G𝐻
𝑘+E𝐻
𝑘w𝑘|2
𝐾
Í
𝑡=1,𝑡 𝑘|𝜃
𝜃
𝜃𝐻e
G𝐻
𝑘+E𝐻
𝑘w𝑡|2+𝜎2
𝑘
,(7)
where {w𝑘}={w1,w2, . . . , w𝐾}is the set of active beam-
forming vectors.
3In this letter, the amplitude and phase of the reflection coefficient of each
reflecting element of the IRS can be independently optimized. Depending
on the practical implementation of the reflecting elements, it may not be
possible to optimize the amplitude and the phase of each reflecting element
independently of each other. This is shown in e.g., [7] for a specific
implementation of the reflecting elements of the IRS. The case study in [7]
constitutes a valuable and promising generalization of the problem formulation
analyzed in this letter, which is postponed to future research due to space
limitations.
4In (7), the exact SINR is considered. This implies that the estimation error
E𝐻
𝑘appears in both the numerator and denominator of (7). In [13], on the
other hand, the estimation error E𝐻
𝑘is regarded as an additional interference
term, and, therefore, it appears in the denominator of the corresponding SINR.
3
III. PROP OS ED OPTIMIZATION PROBLEM
We define the coverage probability of a user as the proba-
bility that its SINR is greater than a predefined threshold. Our
aim is to jointly design the active/passive beamforming vectors
so as to minimize the total transmit power while guaranteeing
that the coverage probability of each user is greater than a
predefined target. This problem can be formulated as follows
minimize
{w𝑘},𝜃
𝜃
𝜃
𝐾
Õ
𝑘=1
w𝐻
𝑘w𝑘
subject to Pr (Γ𝑘({w𝑘}, 𝜃
𝜃
𝜃)𝛾𝑘)1𝜌𝑘,𝑘,
|𝜃𝑛| ≤ 1,𝑛,
(8)
where 𝛾𝑘and 𝜌𝑘are the required minimum SINR to be in
coverage and the predefined outage target, respectively, at the
𝑘-th user. Hereafter, unless otherwise stated, we assume 𝑘
{1, . . . , 𝐾 }and 𝑛∈ {1, . . . , 𝑁 }. Problem (8) is not convex
due to the robust probabilistic SINR constraints. To proceed
further, using (7) we recast the event Γ𝑘({w𝑘}, 𝜃
𝜃
𝜃)𝛾𝑘as
|𝜃
𝜃
𝜃𝐻e
G𝐻
𝑘+E𝐻
𝑘w𝑘|2
𝛾𝑘
𝐾
Õ
𝑡=1,
𝑡𝑘𝜃
𝜃
𝜃𝐻e
G𝐻
𝑘+E𝐻
𝑘w𝑡
2
𝜎2
𝑘.(9)
Defining e𝑘=vec (E𝑘),K𝑘=𝜃
𝜃
𝜃𝜃
𝜃
𝜃𝐻𝑇B𝑘,W𝑘=w𝑘w𝐻
𝑘and
B𝑘=W𝑘
𝛾𝑘Í𝐾
𝑡=1,𝑡 𝑘W𝑡, we can rewrite (9) as
𝜃
𝜃
𝜃𝐻e
G𝐻
𝑘+E𝐻
𝑘B𝑘e
G𝑘+E𝑘𝜃
𝜃
𝜃𝜎2
𝑘,(10)
tr e
G𝐻
𝑘+E𝐻
𝑘B𝑘e
G𝑘+E𝑘𝜃
𝜃
𝜃𝜃
𝜃
𝜃𝐻𝜎2
𝑘,(11)
vec e
G𝑘+E𝑘𝐻
vec B𝑘e
G𝑘+E𝑘𝜃
𝜃
𝜃𝜃
𝜃
𝜃𝐻𝜎2
𝑘,(12)
vec e
G𝑘+E𝑘𝐻
K𝑘vec e
G𝑘+E𝑘𝜎2
𝑘,(13)
e𝐻
𝑘K𝑘e𝑘+2Re ne𝐻
𝑘K𝑘vec e
G𝑘o𝑑𝑘.(14)
To obtain (11) from (10), the identity a𝐻Ba =Tr Baa𝐻is
used, and to obtain (13) from (12), the identity vec (AYB)=
B𝑇Avec (Y)is used. By defining the following function
that accounts for the channel estimation error of the 𝑘-th user
𝑓(e𝑘)=e𝐻
𝑘K𝑘e𝑘+2Re ne𝐻
𝑘K𝑘vec e
G𝑘o,(15)
problem (8) can be equivalently formulated as
minimize
{W𝑘},𝜃
𝜃
𝜃tr 𝐾
Õ
𝑘=1
W𝑘!
subject to Pr (𝑓(e𝑘)𝑑𝑘)1𝜌𝑘,𝑘,
rank(W𝑘)=1,𝑘 ,
|𝜃𝑛| ≤ 1,𝑛.
(16)
In (16), we have introduced the set of matrices {W𝑘}=
{W1,W2, . . . , W𝐾}and 𝑑𝑘=𝜎2
𝑘vec e
G𝑘𝐻
K𝑘vec e
G𝑘.
We note that problem (16) is still non-convex due to the
inherent non-convexity of the probabilistic SINR and the rank
constraints. We first handle the probabilistic SINR constraints
by introducing the following lemma.
Lemma 1. For any 𝜂𝑘>0, the statement below holds true:
Pr (𝑓(e𝑘) ≥ Υ(𝜂𝑘))1𝑒𝜂𝑘,(17)
where Υ(𝜂𝑘)is defined as
Υ(𝜂𝑘)=p2𝜂𝑘s𝜎4
𝜌2
𝑘𝜏2
𝑝kKkk2
𝐹+2𝜎2
𝜌𝑘𝜏𝑝kK𝑘vec(e
G𝑘)k2
+𝜎2
𝜌𝑘𝜏𝑝
tr (K𝑘)𝜂𝑘𝜎4
𝜌2
𝑘𝜏2
𝑝
𝜆+(K𝑘)
(18)
and 𝜆+(K𝑘)=min{𝜆min (K𝑘),0}where 𝜆min (K𝑘)is the
minimum eigenvalue of K𝑘.
Proof. The proof is obtained by applying the Bernstein-
type inequality [9] to the Gaussian random variable e𝑘
CN 0,𝜎2
𝜌𝑘𝜏𝑝I𝑀 𝑁 with the positive definite matrix 𝜎2
𝜌𝑘𝜏𝑝K𝑘
and the vector 𝜎
𝜌𝑘𝜏𝑝K𝑘vec(e
G𝑘)in a standard quadratic
form.
By setting 𝜂𝑘=ln 𝜌𝑘and using Lemma 1, the probabilis-
tic constraint in (16) can be rewritten as
p2 ln(1/𝜌𝑘)s𝜎4
𝜌2
𝑘𝜏2
𝑝kK𝑘k2
𝐹+2𝜎2
𝜌𝑘𝜏𝑝kK𝑘vec(e
G𝑘)k2
+𝜎2
𝜌𝑘𝜏𝑝
tr (K𝑘)ln(1/𝜌𝑘)𝜎4
𝜌2
𝑘𝜏2
𝑝
𝜆+(K𝑘) ≥ 𝑑𝑘.
(19)
We introduce two auxiliary variables 𝜇𝑘and 𝜈𝑘to equivalently
cast (19) as the following series of inequalities
𝜎2
𝜌𝑘𝜏𝑝
tr (K𝑘)p2 ln(1/𝜌𝑘)𝜇𝑘ln(1/𝜌𝑘)𝜎4
𝜌2
𝑘𝜏2
𝑝
𝜈𝑘𝑑𝑘,(20)
s𝜎4
𝜌2
𝑘𝜏2
𝑝kKkk2
𝐹+2𝜎2
𝜌𝑘𝜏𝑝kK𝑘vec(e
G𝑘)k2𝜇𝑘,(21)
𝜈𝑘I𝑀 𝑁 +K𝑘0,(22)
𝜈𝑘0.(23)
While the constraints (20), (22), and (23) adhere to the
standard form of a semi-definite program with respect to
K𝑘, we can reformulate (21) in a standard second-order-cone
(SOC) constraint as
2𝜎
𝜌𝑘𝜏𝑝K𝑘vec e
G𝑘
𝜎2
𝜌𝑘𝜏𝑝vec (K𝑘)𝜇𝑘.(24)
To proceed further, we introduce a new variable 𝚯=𝜃
𝜃
𝜃𝜃
𝜃
𝜃𝐻and
recast (16) as
minimize
{W𝑘},𝚯,{𝜇𝑘},{𝜈𝑘}tr 𝐾
Õ
𝑘=1
W𝑘!
subject to (20),(22),(23),(24),𝑘,
W𝑘0,𝑘,
rank(W𝑘)=1,𝑘 ,
diag(𝚯) 1𝑁,
𝚯0,
rank(𝚯)=1.
(25)
4
By analyzing the constraints (20), (22) and (24), we observe
that they are non-convex with respect to {W𝑘}and 𝚯. These
constraints, in fact, depend on K𝑘, which is a function of {W𝑘}
and 𝚯. To overcome this issue, we use alternating optimization
so as to obtain a fixed-point solution of problem (25). In partic-
ular, starting from an initial value of the reflection coefficients
Θ
Θ
Θ(0)by replacing W𝑘,𝜇𝑘, and 𝜈𝑘, respectively, with W(𝑖)
𝑘,
𝜇(𝑖)
𝑘, and 𝜈(𝑖)
𝑘,𝑘, we solve the following subproblem during
the 𝑖-th iteration
minimize
{W(𝑖)
𝑘},{𝜇(𝑖)
𝑘},{𝜈(𝑖)
𝑘}
tr 𝐾
Õ
𝑘=1
W(𝑖)
𝑘!
subject to (20),(22),(23),(24),𝑘,
W(𝑖)
𝑘0,𝑘,
rank(W(𝑖)
𝑘)=1,𝑘.
(26)
Problem (26) is still non-convex due to the rank-one constraint
on W(𝑖)
𝑘. However, problem (26) has a structure similar to that
in [14]. Hence, we can exploit the same methodology as in
[14, Theorem 2] to show that its optimal solution is rank-
one if problem (26) is feasible. Consequently, problem (26) is
equivalent to the following LMI form
minimize
{W(𝑖)
𝑘},{𝜇(𝑖)
𝑘},{𝜈(𝑖)
𝑘}
tr 𝐾
Õ
𝑘=1
W(𝑖)
𝑘!
subject to (20),(22),(23),(24),𝑘,
W(𝑖)
𝑘0,𝑘,
(27)
which yields the optimal solution {W(𝑖)
𝑘},{𝜇(𝑖)
𝑘}and {𝜈(𝑖)
𝑘}for
a given matrix Θ
Θ
Θ(𝑖1). Since (27) is a semidefinite program,
its optimal solution is obtained in polynomial time by using
general purpose interior-point toolboxes, e.g., CVX [15]. After
obtaining {W(𝑖)
𝑘},{𝜇(𝑖)
𝑘}and {𝜈(𝑖)
𝑘}from the solution of (27),
the matrix Θ
Θ
Θ(𝑖)is attained by solving the following feasibility
problem
find 𝚯(𝑖)
subject to (20),(22),(24),𝑘,
diag(𝚯(𝑖)) 1𝑁,
𝚯(𝑖)0,
rank 𝚯(𝑖)=1.
(28)
Problem (28) is non-convex due to the rank-one constraint on
𝚯(𝑖). To tackle this problem, we propose to solve the following
problem
minimize
𝚯(𝑖)tr(𝚯(𝑖))
subject to (20),(22),(24),𝑘,
diag(𝚯(𝑖)) 1𝑁,
𝚯(𝑖)0,
(29)
whose global optimum can be obtained by using CVX.
Exploiting the same methodology as in [14, Theorem 2],
we can show that problem (29) yields a rank-one optimal
solution. Therefore, we can conclude that problem (29) is an
approximation of problem (28), i.e., every feasible solution of
Algorithm 1 Alternating optimization to solve (25)
Input: Channel estimate matrices e
G𝑘,𝑘; Initial phase shift
coefficients 𝚯(0); Tolerance value 𝛿; Set 𝑖=0and initial the
cost function 𝐶(0)=0.
1. 𝑖=𝑖+1; Update {W(𝑖)
𝑘},{𝜇(𝑖)
𝑘}, and {𝜈(𝑖)
𝑘}for all 𝑘 ,
by solving problem (27) where every 𝜃(𝑖1)
𝑛is computed
from the previous iteration.
2. Update Θ
Θ
Θ(𝑖)
𝑘by solving problem (29) where {W(𝑖)
𝑘},
{𝜇(𝑖)
𝑘}, and {𝜈(𝑖)
𝑘}are obtained from Step 1.
3. Compute the cost function 𝐶(𝑖)=tr Í𝐾
𝑘=1W(𝑖)
𝑘.
4. Check the stopping criterion: If 𝐶(𝑖)𝐶(𝑖1)𝛿
Stop. Otherwise, repeat Steps 13.
Output: The fixed point solution: {W(𝑖)
𝑘}and Θ
Θ
Θ(𝑖).
(29) is also feasible for (28), see e.g. [14, Section III.B] and
references therein.
By direct inspection, subproblems (27) and (29) are convex
and their feasible domains are convex sets. Therefore, the pro-
posed iterative algorithm converges to a fixed-point solution
[16] when the subproblems (27) and (29) are feasible. The
proposed alternating optimization method is summarized in
Algorithm 1.5
Finally, the active/passive beamforming vectors w𝑘and 𝜃
𝜃
𝜃
are, respectively, as follows q𝜆(𝑤)
𝑘z(𝑤)
𝑘and 𝜆(𝜃)z(𝜃), where
𝜆(𝑤)
𝑘and 𝜆(𝜃)are the non-zero eigenvalues, and z(𝑤)
𝑘and z(𝜃)
are the corresponding eigenvectors of the rank-one fixed-point
solution W(𝑖)
𝑘and Θ
Θ
Θ(𝑖), respectively [14, Section IV.A].
IV. NUMERICAL RES ULT S
We consider a MISO system in the absence of light-of-
sight so that there is no direct path from the BS to the users.
The 3GPP Urban Micro channel model is used [17]. In the
coverage area, the users are randomly distributed, but the
minimum distance to the IRS is 10 m. The distance between
the BS and the IRS is 80 m. The large-scale fading coefficient
of the point-to-point link between the BS and the IRS is
1. There are 15 elements at the IRS. The noise variance is
96 dBm. The bandwidth is 10 MHz. The large-scale fading
coefficient between the user 𝑘and the IRS is defined as
𝛽𝑘[dB]=15.126 log10 (𝑓𝑐) − 37.6 log10 (𝑑𝑘/1m),(30)
where 𝑓𝑐=3GHz is the carrier frequency, 𝑑𝑘is the distance
between the user 𝑘and the IRS (𝑑𝑘10 m). The SINR
requirement of each user is 4dB and the outage probability
is 0.1.
Fig. 1 illustrates the convergence of the proposed algorithm
as a function of the number of users. We observe that Algo-
rithm 1 converges to a fixed-point solution after less than 5
iterations in all tested cases. These numerical results confirm
the statement about the convergence of the proposed algorithm.
5Although global optimality can be obtained by solving subproblems (27)
and (29) in each iteration, the global optimal solution of the original problem
(25) may not be attained due to the inherent non-convexity of (25). In fact, the
proposed algorithm yields a suboptimal solution to the original non-convex
problem.
5
12345
0
2
4
6
8
10
12
14
Fig. 1: Convergence of Algorithm 1 for a different number of
users.
K=2
K=4
Fig. 2: Transmit power consumption per user [mW].
Fig. 2 illustrates the transmit power per user of the proposed
approach, i.e., Algorithm 1, which is denoted as “Robust
probabilistic-constrained”, and the benchmark method in [8],
which is denoted as “Robust beamforming design”. It can be
observed that the proposed approach consumes 66% and 28%
less power than the benchmark method in [8] when the system
serves 2users and 4users, respectively. This is due to the fact
that the benchmark method in [8] allocates extra resources to
protect rarely occurring worst-case events while the proposed
approach allows the QoS constraints to be violated with some
non-zero probabilities.
Due to the increase of the mutual interference, Fig. 2
shows that the system must allocate more power to each user
when the number of coexisting users increases. For example,
the proposed approach requires approximately 0.06 mW in
order to guarantee the SINR requirements with an outage
probability of 0.1when 2users are in the coverage area. On
the other hand, the power allocated to each user increases up
to 0.14 mW if 4users need to be served.
V. CONCLUSION
We have formulated and solved a robust probabilistic-
constrained optimization problem for IRS-aided MISO com-
munication systems in order to tackle imperfect estimates of
the CSI. The optimal beamforming vectors at the BS and the
reflecting elements at the IRS are iteratively computed via
a converging alternating optimization algorithm. Numerical
results reveal a fast convergent behavior of the proposed
algorithm, i.e., within a few iterations. The results confirm
the superior performance of the proposed approach compared
with a benchmark method.
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We consider a fading channel in which a multi-antenna transmitter communicates with a multi-antenna receiver through a reconfigurable intelligent surface (RIS) that is made of N reconfigurable passive scatterers impaired by phase noise. The beamforming vector at the transmitter, the combining vector at the receiver, and the phase shifts of the N scatterers are optimized in order to maximize the signal-to-noise-ratio (SNR) at the receiver. By assuming Rayleigh fading (or line-of-sight propagation) on the transmitter-RIS link and Rayleigh fading on the RIS-receiver link, we prove that the SNR is a random variable that is equivalent in distribution to the product of three (or two) independent random variables whose distributions are approximated by two (or one) gamma random variables and the sum of two scaled non-central chi-square random variables. The proposed analytical framework allows us to quantify the robustness of RIS-aided transmission to fading channels. For example, we prove that the amount of fading experienced on the transmitter-RIS-receiver channel linearly decreases with N1. This proves that RISs of large size can be effectively employed to make fading less severe and wireless channels more reliable.
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Reconfigurable intelligent surfaces (RISs) are an emerging transmission technology for application to wireless communications. RISs can be realized in different ways, which include (i) large arrays of inexpensive antennas that are usually spaced half of the wavelength apart; and (ii) metamaterial-based planar or conformal large surfaces whose scattering elements have sizes and inter-distances much smaller than the wavelength. Compared with other transmission technologies, e.g., phased arrays, multi-antenna transmitters, and relays, RISs require the largest number of scattering elements, but each of them needs to be backed by the fewest and least costly components. Also, no power amplifiers are usually needed. For these reasons, RISs constitute a promising software-defined architecture that can be realized at reduced cost, size, weight, and power (C-SWaP design), and are regarded as an enabling technology for realizing the emerging concept of smart radio environments (SREs). In this paper, we (i) introduce the emerging research field of RIS-empowered SREs; (ii) overview the most suitable applications of RISs in wireless networks; (iii) present an electromagnetic-based communication-theoretic framework for analyzing and optimizing metamaterial-based RISs; (iv) provide a comprehensive overview of the current state of research; and (v) discuss the most important research issues to tackle. Owing to the interdisciplinary essence of RIS-empowered SREs, finally, we put forth the need of reconciling and reuniting C. E. Shannon’s mathematical theory of communication with G. Green’s and J. C. Maxwell’s mathematical theories of electromagnetism for appropriately modeling, analyzing, optimizing, and deploying future wireless networks empowered by RISs.
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Prior studies on intelligent reflecting surface (IRS) have mostly assumed perfect channel state information (CSI) available for designing the IRS passive beamforming as well as the continuously adjustable phase shift at each of its reflecting elements, which, however, have simplified two challenging issues for implementing IRS in practice, namely, its channel estimation and passive beamforming designs both under the constraint of discrete phase shifts. To address them, we consider in this paper an IRS-aided single-user communication system and design the IRS training reflection matrix for channel estimation as well as the passive beamforming for data transmission, both subject to the new constraint of discrete phase shifts. We show that the training reflection matrix design with discrete phase shifts greatly differs from that with continuous phase shifts, and the corresponding passive beamforming design should take into account the correlated IRS channel estimation errors due to discrete phase shifts. Moreover, a novel hierarchical training reflection design is proposed to progressively estimate IRS elements’ channels over multiple time blocks by exploiting the IRS-elements grouping and partition. Based on the resolved IRS channels in each block, we further design the progressive passive beamforming at the IRS with discrete phase shifts to improve the achievable rate for data transmission over the blocks. Extensive numerical results are presented, which demonstrate the significant performance improvement of proposed channel estimation and passive beamforming designs as compared to various benchmark schemes.
Article
In the intelligent reflecting surface (IRS)-enhanced wireless communication system, channel state information (CSI) is of paramount importance for achieving the passive beamforming gain of IRS, which, however, is a practically challenging task due to its massive number of passive elements without transmitting/receiving capabilities. In this letter, we propose a practical transmission protocol to execute channel estimation and reflection optimization successively for an IRS-enhanced orthogonal frequency division multiplexing (OFDM) system. Under the unit-modulus constraint, a novel reflection pattern at the IRS is designed to aid the channel estimation at the access point (AP) based on the received pilot signals from the user, for which the channel estimation error is derived in closed-form. With the estimated CSI, the reflection coefficients are then optimized by a low-complexity algorithm based on the resolved strongest signal path in the time domain. Simulation results corroborate the effectiveness of the proposed channel estimation and reflection optimization methods.
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IRS is a new and revolutionizing technology that is able to significantly improve the performance of wireless communication networks, by smartly reconfiguring the wireless propagation environment with the use of massive low-cost passive reflecting elements integrated on a planar surface. Specifically, different elements of an IRS can independently reflect the incident signal by controlling its amplitude and/or phase and thereby collaboratively achieve fine-grained 3D passive beamforming for directional signal enhancement or nulling. In this article, we first provide an overview of the IRS technology, including its main applications in wireless communication, competitive advantages over existing technologies, hardware architecture as well as the corresponding new signal model. We then address the key challenges in designing and implementing the new IRS-aided hybrid (with both active and passive components) wireless network, as compared to the traditional network comprising active components only. Finally, numerical results are provided to show the great performance enhancement with the use of IRS in typical wireless networks.