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1

Robust Beamforming Design for Intelligent

Reﬂecting Surface Aided MISO Communication

Systems

Gui Zhou, Cunhua Pan, Hong Ren, Kezhi Wang, Marco Di Renzo, Fellow, IEEE, and

Arumugam Nallanathan, Fellow, IEEE

Abstract—Perfect channel state information (CSI) is chal-

lenging to obtain due to the limited signal processing capability

at the intelligent reﬂection surface (IRS). This is the ﬁrst work

to study the worst-case robust beamforming design for an

IRS-aided multiuser multiple-input single-output (MU-MISO)

system under the assumption of imperfect CSI. We aim for min-

imizing the transmit power while ensuring that the achievable

rate of each user meets the quality of service (QoS) requirement

for all possible channel error realizations. With unit-modulus

and rate constraints, this problem is non-convex. The imperfect

CSI further increases the difﬁculty of solving this problem.

By using approximation and transformation techniques, we

convert this problem into a squence of semideﬁnite program

(SDP) subproblems that can be efﬁciently solved. Numerical

results show that the proposed robust beamforming design can

guarantee the required QoS targets for all the users.

Index Terms—Intelligent reﬂecting surface (IRS), large in-

telligent surface (LIS), robust design, imperfect channel state

information (CSI), semideﬁnite program (SDP).

I. INTRODUCTION

Intelligent reﬂecting surface (IRS) has recently been pro-

posed as a cost-effective and energy-efﬁcient high data rate

communication technology due to the rapid development

of radio frequency (RF) micro-electro-mechanical systems

(MEMS) as well as the abundant applications of the pro-

grammable and reconﬁgurable metasurfaces [1]. It consists

of an passive array structure that is capable of adjusting the

phase of each passive element on the surface continuously or

discretely with low power consumption [2], [3]. The beneﬁts

of IRS in enhancing the spectral and energy efﬁciency have

been demonstrated in various schemes (e.g., [4]–[9]) by the

joint design of active precoder at the base station (BS) and

passive reﬂection beamforming at the IRS.

However, all the existing contributions on IRS are based

on the assumption of perfect channel state information (CSI)

at the BS, which is too idealistic in IRS communications.

For the imperfect CSI, the authors in [10] studied the impact

of the channel error by adopting the performance analysis

G. Zhou, C. Pan, H. Ren and A. Nallanathan are with the School

of Electronic Engineering and Computer Science at Queen Mary Uni-

versity of London, London E1 4NS, U.K. (e-mail: g.zhou, c.pan, h.ren,

a.nallanathang@qmul.ac.uk). K. Wang is with Department of Com-

puter and Information Sciences, Northumbria University, UK. (e-mail:

kezhi.wang@northumbria.ac.uk). M. Di Renzo is with Universit´

e Paris-

Saclay, CNRS, CentraleSup´

elec, Laboratoire des Signaux et Systemes, Gif-

sur-Yvette, France (e-mail: marco.direnzo@centralesupelec.fr).

technique in an uplink MISO system. There are three types

of channels in an IRS-aided system: the direct channel from

the BS to the user, the indirect channel from the BS to the

IRS and the reﬂection channel from the IRS to the user.

The ﬁrst one can be obtained with high accuracy by using

conventional channel estimation methods. The accurate CSI

of the latter two, however, are challenging to obtain in

practice due to the fact that the reﬂective elements at the IRS

are passive and have limited signal processing capability.

Fortunately, the location of the IRS is ﬁxed and is usually

installed in the building facades, ceilings, walls, etc. In

this case, the indirect channel can be accurately estimated

through calculating the angles of arrival and departure, which

vary slowly. In contrast, the reﬂection channel is more

challenging to acquire as the locations of users are changing

and their environmental conditions are varying.

Against the above background, this paper investigates the

robust active precoder and passive reﬂection beamforming

design for an IRS-aided downlink multiple-user multiple-

input single-output (MU-MISO) system based on the as-

sumption of imperfect reﬂection channel. An ellipsoid

model of the reﬂection channel uncertainties are adopted.

To the best of our knowledge, this is the ﬁrst work to study

the worst-case robust beamforming design problem in IRS-

aided wireless systems. The contributions of this paper are

as follows: 1) We aim to minimize the transmit power of

the BS through the joint design of an active precoder at the

BS and a passive beamforming at the IRS while ensuring

that each user’s QoS target can be achieved for all possible

channel error realizations. This problem is non-convex and

difﬁcult to solve due to the unit-modulus constraints and the

imperfect CSI. 2) To address this problem, we propose an

iterative algorithm based on approximation transformations

and a convex–concave procedure (CCP). Speciﬁcally, to han-

dle the non-convex rate expression and CSI uncertainties, we

ﬁrst approximately linearize the rates by using the ﬁrst-order

Taylor expansion, and then transform the resultant semi-

inﬁnite constraints into linear matrix inequalities (LMIs).

The non-convex unit-modulus constraints of the reﬂection

beamforming are handled by the penalized CCP [11]. 3)

Numerical results conﬁrm the effectiveness of the proposed

algorithms in guaranteeing the QoS targets of all users.

2

BS

IRS

IRS controller

Wireless control link

ireless control link

Fig. 1: An IRS-aided multiuser communication system.

II. SY ST EM MO DE L

A. Signal Transmission Model

We consider an IRS-aided MISO broadcast (BC) com-

munication system shown in Fig. 1, in which there is a

BS equipped with Ntransmit antennas serving Ksingle-

antenna users. Denote by s= [s1,· · · , sK]T∈CK×1the

Gaussian data symbols, in which each element is an inde-

pendent random variable with zero mean and unit variance,

i.e., E[ssH] = I. Denote by F=[f1,· · · ,fK]∈CN×Kthe

corresponding precoding vectors for the users. Then, the

transmit signal at the BS is x=Fs, the transmit power

of which is E{Tr xxH}=||F||2

F.

In the MISO BC system, we propose to employ an IRS

with the goal of enhancing the received signal strength of

the users by reﬂecting signals from the BS to the users. It

is assumed that the IRS has Mpassive reﬂection elements

e= [e1,· · · , eM]T∈CM×1, the modulus of each element

is |em|2= 1,1≤m≤M. Then, the reﬂection beamforming

at the IRS is modeled as a diagonal matrix E=ιdiag(e)∈

CM×Mwhere ι∈[0,1] indicates the reﬂection efﬁciency.

The channels from the BS to user k, from the BS to the IRS,

and from the IRS to user kare denoted by hd,k ∈CN×1,

Hdr ∈CM×N, and hr,k ∈CM×1, respectively.

The BS is responsible for designing the reﬂection beam-

forming at the IRS and sending it to the IRS controller [4].

Let us deﬁne the set of all users as K={1,2, ..., K}, then

the received signal of the users is

yk= (hH

d,k +hH

r,kEHdr )Fs +nk,∀k∈K,(1)

where nkis the received noise at user k, which is an additive

white Gaussian noise (AWGN) with distribution CN(0, σ2

k).

The achievable data rate (bit/s/Hz) at user kis given by

Rk(F,e) = log21 + hH

d,k +hH

r,kEHdr fk2/βk(2)

where βk=||(hH

d,k +hH

r,kEHdr )F−k||2

2+σ2

k,∀k∈Krep-

resent the interference-plus-noises (INs) term with F−k=

[f1,· · · ,fk−1,fk+1,· · · ,fK].

In the IRS-aided communication system, there are three

types of channels: the direct channel from the BS to the

user, i.e., hd,k, the indirect channel from the BS to the IRS,

i.e., Hdr, and the reﬂection channel from the IRS to the

user, i.e., hr,k. As mentioned in the introduction section, the

reﬂection channel is much more challenging to obtain than

the other two channels. Hence, in this paper, we assume that

the third type of channel is imperfect. The reﬂection channel

{hr,k}∀k∈Kcan be modeled as {hr,k =b

hr,k +4k}∀k∈K,

where {b

hr,k}∀k∈Kdenote the contaminated channel vectors

and {4k}∀k∈Kdenote the corresponding channel error

vectors. In this paper, we adopt the channel error bounded

model, i.e., {||4k||2≤εk}∀k∈K, where εkis the radius of

the uncertainty region known by the BS.

B. Problem Formulation

With imperfect CSI, we aim to minimize the total trans-

mit power via the joint design of the precoding matrix

Fand the reﬂection vector eunder the worst-case QoS

constraints, i.e., ensuring that the achievable rate of each

user is above a threshold for all possible channel error

realizations. Mathematically, the worst-case robust design

problem is formulated as

min

F,e||F||2

F(3a)

s.t. Rk(F,e)≥rk,∀ k4kk2≤εk,∀k∈K,(3b)

|em|2= 1,1≤m≤M. (3c)

Constraints (3b) are the minimum QoS targets for each

user, while constraints (3c) correspond to the unit-modulus

requirements of the reﬂection elements at the IRS.

III. ROBUST BEAMFORMING DESIGN

Problem (3) is a non-convex problem and the main chal-

lenge lies in the non-convex QoS constraints (3b) over the

CSI uncertainty regions and the non-convex unit-modulus

constraints (3c). Since variables Fand eare coupled, we

propose an alternate optimization (AO) method to solve

Problem (3).

A. Problem Transformation

To start with, the non-convexity of constraints (3b) can

be addressed by ﬁrstly treating the INs β= [β1, ..., βK]Tas

auxiliary variables. Hence, constraints (3b) are rewritten as

hH

d,k +hH

r,kEHdr fk2≥βk(2rk−1),

∀ k4kk2≤εk,∀k∈K,(4a)

hH

d,k +hH

r,kEHdr F−k

2

2+σ2

k≤βk,

∀ k4kk2≤εk,∀k∈K.(4b)

We ﬁrst handle the inﬁnite inequalities in (4a), which are

non-convex. Speciﬁcally, the left hand side (LHS) of (4a) is

approximated as its lower bound, as shown below..

Lemma 1 Let f(n)

kand E(n)be the optimal solutions

obtained at iteration n, then the linear lower bound of

|(hH

d,k +hH

r,kEHdr )fk|2in (4a) at (f(n)

k,E(n)) is

hH

r,kXkhr,k +hH

r,kxk+xH

khr,k +ck,(5)

3

where

Xk=EHdrfkfH,(n)

kHH

drEH,(n)+E(n)Hdr f(n)

kfH

kHH

drEH

−E(n)Hdrf(n)

kfH,(n)

kHH

drEH,(n),

xk=EHdrfkfH,(n)

khd,k +E(n)Hdrf(n)

kfH

khd,k

−E(n)Hdrf(n)

kfH,(n)

khd,k,

ck=hH

d,k(fkfH,(n)

k+f(n)

kfH

k−f(n)

kfH,(n)

k)hd,k.

Proof :Let abe a complex scalar variable. By applying

Appendix B of [12], we have the inequality

|a|2≥a∗,(n)a+a∗a(n)−a∗,(n)a(n)(6)

for any ﬁxed a(n). Then, (5) is obtained by replac-

ing aand a(n)with (hH

d,k +hH

r,kEHdr )fkand (hH

d,k +

hH

r,kE(n)Hdr )f(n)

k, respectively. The proof is complete.

With hr,k =b

hr,k +4kand Lemma 1, the inequality (4a)

is reformulated as

4hH

r,kXk4hr,k + 2Re n(xH

k+b

hH

r,kXk)4hr,k o+dk

≥βk(2rk−1),∀ k4kk2≤εk,∀k∈K,(7)

where dk=b

hH

r,kXkb

hr,k +xH

kb

hr,k +b

hH

r,kxk+ck.

In order to tackle the CSI uncertainties, the S-Procedure

in [13] is used to transform (7) into equivalent LMIs as

"$kIM+Xk(xH

k+b

hH

r,kXk)H

(xH

k+b

hH

r,kXk)dk−βk(2rk−1) −$kε2

k#0,

∀k∈K,(8)

where $= [$1, ..., $K]T≥0are slack variables.

Now, we consider the uncertainties in {4k}∀k∈Kof (4b).

To this end, we ﬁrst adopt Schur’s complement [14] to

equivalently recast (4b) as

βk−σ2

ktH

k

tkI0,∀ k4kk2≤εk,∀k∈K,(9)

where tk= ((hH

d,k +hH

r,kEHdr )F−k)H.

Then, by using Nemirovski lemma [15] and introducing

the slack variables ξ= [ξ1, ..., ξK]T≥0, (9) is rewritten as

βk−σ2

k−ξkb

tH

k01×M

b

tkI(K−1) εk(EHdrF−k)H

0M×1εkEHdrF−kξkIM

0,

∀k∈K,(10)

where b

tk= ((hH

d,k +b

hH

r,kEHdr )F−k)H.

With (8) and (10), we obtain the following approximated

reformulation of Problem (3) as

min

F,e,β,$,ξ||F||2

F(11a)

s.t.(8),(10),(3c),(11b)

$≥0,ξ≥0.(11c)

It is difﬁcult to optimize the variables Fand esi-

multaneously as they are coupled in the LMIs (8) and

(10). Therefore, the AO method is adopted to solve the

subproblems corresponding to different sets of variables

iteratively. Speciﬁcally, for a given reﬂection beamforming

e, the subproblem of Problem (11) corresponding to the

precoder Fis formulated as

F(n+1) = arg min

F,β,$,ξ||F||2

F(12a)

s.t.(8),(10),(11c), (12b)

where F(n+1) is the optimal solution obtained in the (n+1)-

th iteration. Problem (12) is a semideﬁnite program (SDP)

and can be solved by using the CVX tool.

On the other hand, for a given precoding matrix F,

the subproblem of Problem (11) corresponding to eis a

feasibility-check problem. According to the Problem (P4’) in

[16], the converged solution in the optimization of ecan be

improved by introducing slack variables a= [a1, ..., aK]T

which are interpreted as the “signal-to-interference-plus-

noise ratio (SINR) residual” of users. Please refer to [16]

about the theory of “SINR residual”. Thus, the feasibility-

check problem of eis formulated as follows

max

e,a,β,$,ξ||a||1(13a)

s.t.Modiﬁed-(8),(10),(3c),(11c),(13b)

a≥0,(13c)

where the Modiﬁed-(8) constraints are LMIs obtained from

(8) by replacing βk(2rk−1) with βk(2rk−1) + akfor

∀k∈K.

However, the above problem cannot be solved directly

due to the non-convex constraint (3c). In addition, the

semideﬁnite relaxation (SDR) method used in [16] cannot

always guarantee a feasible solution due to the fact that the

QoS constraints may be violated when the SDR solution

is not rank one. To handle this issue, we apply the penalty

CCP [11] which is capable of ﬁnding a feasible solution that

meets the unit-modulus constraint and the QoS constraints.

In particular, the constraints |em|2= 1,1≤m≤Mcan be

equivalently rewritten as 1≤ |em|2≤1,1≤m≤M. The

non-convex parts of these constraints are again linearized

by |e[t]

m|2−2Re(eH

me[t]

m)≤ −1,1≤m≤Mat ﬁxed e[t]

m.

Following the penalty CCP framework, we impose the use

of slack variables b= [b1, ..., b2M]Tover the equivalent

constraints of the unit-modulus constraints, which yields

max

e,a,b,

β,$,ξ

||a||1−λ[t]||b||1(14a)

s.t.Modiﬁed-(8),(10),(11c),(13c),(14b)

|e[t]

m|2−2Re(eH

me[t]

m)≤bm−1,1≤m≤M(14c)

|em|2≤1 + bM+m,1≤m≤M(14d)

b≥0,(14e)

where λ[t]is the regularization factor to scale the impact of

the penalty term ||b||1, which controls the feasibility of the

constraints. At low λ, Problem (14) targets to maximize the

“SINR residual”, while Problem (14) seeks for a feasible

point rather than optimizing the “SINR residual” at high λ.

4

Problem (14) is an SDP and can be solved by using the

CVX tool. The algorithm for ﬁnding a feasible solution of e

is summarized in Algorithm 1. Some points are emphasized

as follows: a) The maximum value λmax is imposed to avoid

numerical problems, that is, a feasible solution may not be

found when the iteration converges under increasing large

values of λ[t];b) The stopping criteria ||b||1≤χguarantees

the unit-modulus constraints in the original Problem (13)

to be met for a sufﬁciently low χ;c) The stopping criteria

||e[t]−e[t−1]||1≤νcontrols the convergence of Algorithm 1;

d) As mentioned in [11], a feasible solution for Problem (14)

may not be feasible for Problem (13). Hence, the feasibility

of Problem (13) is guaranteed by imposing a maximum

number of iterations Tmax and, in case it is reached, we

restart the iteration based on a new initial point.

Algorithm 1 Penalty CCP optimization for reﬂection beam-

forming optimization

Initialize: Initialize e[0],γ[0] >1, and set t= 0.

1: repeat

2: if t<Tmax then

3: Update e[t+1] from Problem (14);

4: λ[t+1] = min{γλ[t], λmax };

5: t=t+ 1;

6: else

7: Initialize with a new random e[0], set γ[0] >1, and

t= 0.

8: end if

9: until ||b||1≤χand ||e[t]−e[t−1]||1≤ν.

10: Output e(n+1) =e[t].

B. Algorithm Description

Algorithm 2 summarizes the AO method for solving

Problem (11).

Algorithm 2 AO algorithm for Problem (11)

Initialize: Initialize e(0) and F(0), and set n= 0.

1: repeat

2: Update F(n+1) from Problem (12) with given e(n);

3: Update e(n+1) from Problem (13) with given F(n+1);

4: n←n+ 1;

5: until The objective value ||F(n+1)||2

Fconverges.

a) Convergence analysis: The convergence of Algo-

rithm 2 can be guaranteed. In particular, denoting the objec-

tive value of Problem (12) as F(F,e), it follows that

F(F(n),e(n))≥F(F(n),e(n+1))≥F(F(n+1) ,e(n+1)).

The above equality holds true because the objective value of

Problem (12) is independent of e, and also e(n+1) is feasible

for Problem (12) if it is a feasible solution for Problem

(13). The above inequality follows from the globally optimal

solution F(n+1) of Problem (12) for a given e(n+1). Hence,

the sequence {F(F(n),e(n))}is non-increasing and the

algorithm is guaranteed to converge.

b) Initial point: As for the method of initialization,

e(0) can be chosen as a full-1 vector for simplicity. Inspired

by [17], the initial point F(0) can be chosen as the optimal

solution to the following optimization problem

min

F,ϕX

k∈K

(ϕk−1)2(15a)

s.t. Rk(F,e)≥ϕkrk,∀k∈K(15b)

ϕk≥0,∀k∈K,(15c)

where ϕ= [ϕ1, ..., ϕK]Tis an auxiliary variable vector.

Problem (15) is guaranteed to be feasible since at least

{ϕk= 0,∀k∈K,F(n)=0}is a feasible solution.

Problem (15) can also be solved by reformulating it into an

alternative optimization problem that is similar to Problems

(12). Denote by {ϕ(opt)

k}∀k∈Kthe solution of Problem (15),

then the corresponding optimal precoding matrix can be used

as the initial point for Algorithm 2 if ϕ(opt)

k= 1,∀k∈K.

IV. NUMERICAL RESULTS AND DISCUSSIONS

In this section, numerical results are provided to evaluate

the performance of the proposed algorithm. We consider that

the BS is equipped with N= 6 transmit antennas serving

K= 4 users with the assistance of an IRS. The number of

the reﬂection elements is M= 16. We assume a rectangular

coordinate to discribe the system, i.e., the locations of the

BS and IRS are (0 m, 0 m) and (50 m, 10 m) respectively,

and users are distributed randomly on a circle centered at

(70 m, 0 m) with radius 5 m.

The large-scale path loss is PL = −30−10αlog10 (d)dB,

where αis the path loss exponent and dis the link length

in meters. The path loss exponents for the BS-IRS link, BS-

user link, and the IRS-user link are equal to αBI = 2.2

[18], αBU = 4 and αIU = 2, respectively. The small-

scale fading of the channels [Hdr,{hd,k ,b

hr,k}∀k∈K]follows

a Rician distribution with Ricean factor 5. The line-of-

sight (LoS) components are deﬁned by the product of the

steering vectors of the transmitter and receiver and the non-

LoS components are drawn from a Rayleigh fading. The

CSI error bounds are deﬁned as εk=δ||b

hr,k||2,∀k∈K,

where δ∈[0,1) accounts for the relative amount of CSI

uncertainties. The power of the AWGN at all users is set to

−100 dBm and the target rates of all users are the same,

i.e., r1=... =rK=r. The IRS and benchmark schemes

considered are the following: 1) “IRS, ι= 1(or 0.5)”. 2)

“Non-robust IRS”, in which the channel estimation error is

ignored when designing the beamformings. 3) “Non IRS”, in

which there is no IRS in the MU-MISO system. 4) “Relay”,

in which a full-duplex relay is located at the same place of

the IRS. The numbers of transmit and receive antennas at

the relay are both M.

Firstly, Fig. 2 shows the total transmit power and energy

efﬁciency versus the channel uncertainty level δwhen r= 4

bit/s/Hz. It is observed from Fig. 2(a) that the required

transmit power of the robust IRS beamforming is higher

than other schemes. This is the price to pay to have a

robust design and to employ passive reﬂection elements. In

5

0 0.01 0.02 0.03 0.04 0.05

20

21

22

23

24

25

Total transmit power (dBm)

(a) The transmit power

0 0.01 0.02 0.03 0.04 0.05

8

10

12

14

16

Energy efficiency (bit/J/Hz)

(b) The energy efﬁciency

Fig. 2: Performance versus the channel uncertainty level δ

under N= 6,M= 16,K= 4 and r= 4 bit/s/Hz.

0.01 0.02 0.03 0.04 0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Outage probability

Fig. 3: Outage probability of rate versus the channel uncer-

tainty level δunder N= 6,M= 16 and K= 4.

any case, it is less than the “Non IRS” case. The energy

efﬁciency (EE) reported in Fig. 2(b) is deﬁned as the ratio

between the smallest achievable rate among the users and the

total power consumption. The total power consumption of

the IRS schemes is equal to ||F||2

F+NPactive +M Ppassive

and that of the relay scheme is equal to ||F||2

F+Prelay +

(N+ 2M)Pactive, where Pr elay is the relay transmit power.

We set the circuit power consumption of the active antennas

to Pactive = 10 mW and that of the passive antennas

as Pactive = 5 mW [19]. Fig. 2(b) illustrates the high

EE performance of the IRS-aided systems compared with

the relay system for the reason of the low circuit power

consumption of the passive elements in the IRS. In addition,

from Fig. 2(a) and Fig. 2(b) we come to the conclusion

that only when the reﬂection efﬁciency of the reﬂecting

metasurfaces is high (ιis nearly 1) and the estimation error

of the reﬂection channel is small (δis less than 0.03), the

IRS can show its advantages of enhancing the spectral and

energy efﬁciency.

Fig. 3 shows the outage probability of rate for the nonro-

bust design. Here, outage probability refers to the probability

that the target rate of at least one user is not satisﬁed. It

is observed that when the beamforming design ignores the

channel error, the target rate of at least one user is frequently

not met, especially at high value of ror δ. However, our

adopted worst-case robust design method can guatantee no

outage happens.

V. CONCLUSIONS

In this paper, we considered the robust beamforming

design for the IRS-aided MU-MISO system when the CSI

is imperfect. The CSI uncertainties were addressed by us-

ing approximation and transformation techniques, and the

non-convex unit-modulus constraints were solved under the

penalty CCP framework. Numerical results demonstrated the

robustness of our proposed algorithm.

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