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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 ]𝑇∈C𝑁×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 𝜏𝑝≥𝐾. In

each time slot, one reﬂective element is chosen to only reﬂect

the training pilot signal without introducing any phase shift

while the other 𝑁−1elements do not reﬂect the incident

signal.2The pilot signal reﬂected by the 𝑛-th reﬂective element

Y𝑝

𝑛∈C𝑀×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 E𝑘∈C𝑀×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 w𝑘∈C𝑀×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)H𝐻∈C𝑁×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)≥1−𝜌𝑘,∀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,

𝑡≠𝑘𝜃

𝜃

𝜃𝐻e

G𝐻

𝑘+E𝐻

𝑘w𝑡

2

≥𝜎2

𝑘.(9)

Deﬁning e𝑘=vec (E𝑘),K𝑘=𝜃

𝜃

𝜃𝜃

𝜃

𝜃𝐻𝑇⊗B𝑘,W𝑘=w𝑘w𝐻

𝑘and

B𝑘=W𝑘

𝛾𝑘−Í𝐾

𝑡=1,𝑡 ≠𝑘W𝑡, we can rewrite (9) as

𝜃

𝜃

𝜃𝐻e

G𝐻

𝑘+E𝐻

𝑘B𝑘e

G𝑘+E𝑘𝜃

𝜃

𝜃≥𝜎2

𝑘,(10)

⇔tr e

G𝐻

𝑘+E𝐻

𝑘B𝑘e

G𝑘+E𝑘𝜃

𝜃

𝜃𝜃

𝜃

𝜃𝐻≥𝜎2

𝑘,(11)

⇔vec e

G𝑘+E𝑘𝐻

vec B𝑘e

G𝑘+E𝑘𝜃

𝜃

𝜃𝜃

𝜃

𝜃𝐻≥𝜎2

𝑘,(12)

⇔vec e

G𝑘+E𝑘𝐻

K𝑘vec e

G𝑘+E𝑘≥𝜎2

𝑘,(13)

⇔e𝐻

𝑘K𝑘e𝑘+2Re ne𝐻

𝑘K𝑘vec e

G𝑘o≥𝑑𝑘.(14)

To obtain (11) from (10), the identity a𝐻Ba =Tr Baa𝐻is

used, and to obtain (13) from (12), the identity vec (AYB)=

B𝑇⊗Avec (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)≥1−𝜌𝑘,∀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

𝑘−vec 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))≥1−𝑒−𝜂𝑘,(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)𝜇𝑘−ln(1/𝜌𝑘)𝜎4

𝜌2

𝑘𝜏2

𝑝

𝜈𝑘≥𝑑𝑘,(20)

s𝜎4

𝜌2

𝑘𝜏2

𝑝kKkk2

𝐹+2𝜎2

𝜌𝑘𝜏𝑝kK𝑘vec(e

G𝑘)k2≤𝜇𝑘,(21)

𝜈𝑘I𝑀 𝑁 +K𝑘0,(22)

𝜈𝑘≥0.(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(𝑖−1). 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(𝑖−1)

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(𝑖−1)≤𝛿→

Stop. Otherwise, repeat Steps 1−3.

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]=−15.1−26 log10 (𝑓𝑐) − 37.6 log10 (𝑑𝑘/1m),(30)

where 𝑓𝑐=3GHz is the carrier frequency, 𝑑𝑘is the distance

between the user 𝑘and the IRS (𝑑𝑘≥10 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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