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Abstract and Figures

Perfect channel state information (CSI) is challenging to obtain due to the limited signal processing capability at the intelligent reflection surface (IRS). In this paper, we study the robust beamforming design for the IRS-aided multiple-user multiple-input single-output (MU-MISO) systems under the assumption of imperfect CSI. We aim for minimizing the transmit power while ensuring the achievable rate of each user above a threshold for all possible channel error realizations. With unit-modulus constraint and rate constraints, this problem is non-convex and the CSI uncertainties additionally increase the difficulty of solving this problem. Using certain approximation and transformation techniques, we convert this problem into a squence of semidefinite programming (SDP) subproblems. Numerical results show that the proposed robust beamforming design can guarantee the required quality of service (QoS) for all the users. Index Terms-Intelligent reflecting surface (IRS), large intelligent surface (LIS), robust design, imperfect channel state information (CSI), semidefinite programming (SDP).
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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]TCK×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]TCM×1, the modulus of each element
is |em|2= 1,1mM. 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,kK,(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 fk2k(2)
where βk=||(hH
d,k +hH
r,kEHdr )Fk||2
2+σ2
k,kKrep-
resent the interference-plus-noises (INs) term with Fk=
[f1,· · · ,fk1,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}kKcan be modeled as {hr,k =b
hr,k +4k}kK,
where {b
hr,k}kKdenote the contaminated channel vectors
and {4k}kKdenote the corresponding channel error
vectors. In this paper, we adopt the channel error bounded
model, i.e., {||4k||2εk}kK, 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,∀ k4kk2εk,kK,(3b)
|em|2= 1,1mM. (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 fk2βk(2rk1),
∀ k4kk2εk,kK,(4a)
hH
d,k +hH
r,kEHdr Fk
2
2+σ2
kβk,
∀ k4kk2εk,kK.(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
kf(n)
kfH,(n)
k)hd,k.
Proof :Let abe a complex scalar variable. By applying
Appendix B of [12], we have the inequality
|a|2a,(n)a+aa(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(2rk1),∀ k4kk2εk,kK,(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)dkβk(2rk1) $kε2
k#0,
kK,(8)
where $= [$1, ..., $K]T0are slack variables.
Now, we consider the uncertainties in {4k}kKof (4b).
To this end, we first adopt Schur’s complement [14] to
equivalently recast (4b) as
βkσ2
ktH
k
tkI0,∀ k4kk2εk,kK,(9)
where tk= ((hH
d,k +hH
r,kEHdr )Fk)H.
Then, by using Nemirovski lemma [15] and introducing
the slack variables ξ= [ξ1, ..., ξK]T0, (9) is rewritten as
βkσ2
kξkb
tH
k01×M
b
tkI(K1) εk(EHdrFk)H
0M×1εkEHdrFkξkIM
0,
kK,(10)
where b
tk= ((hH
d,k +b
hH
r,kEHdr )Fk)H.
With (8) and (10), we obtain the following approximated
reformulation of Problem (3) as
min
F,e,β,$,ξ||F||2
F(11a)
s.t.(8),(10),(3c),(11b)
$0,ξ0.(11c)
It is difficult to optimize the variables Fand esi-
multaneously as they are coupled in the LMIs (8) and
(10). Therefore, the AO method is adopted to solve the
subproblems corresponding to different sets of variables
iteratively. Specifically, for a given reflection beamforming
e, the subproblem of Problem (11) corresponding to the
precoder Fis formulated as
F(n+1) = arg min
F,β,$,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,$,ξ||a||1(13a)
s.t.Modified-(8),(10),(3c),(11c),(13b)
a0,(13c)
where the Modified-(8) constraints are LMIs obtained from
(8) by replacing βk(2rk1) with βk(2rk1) + akfor
kK.
However, the above problem cannot be solved directly
due to the non-convex constraint (3c). In addition, the
semidefinite relaxation (SDR) method used in [16] cannot
always guarantee a feasible solution due to the fact that the
QoS constraints may be violated when the SDR solution
is not rank one. To handle this issue, we apply the penalty
CCP [11] which is capable of finding a feasible solution that
meets the unit-modulus constraint and the QoS constraints.
In particular, the constraints |em|2= 1,1mMcan be
equivalently rewritten as 1≤ |em|21,1mM. The
non-convex parts of these constraints are again linearized
by |e[t]
m|22Re(eH
me[t]
m)≤ −1,1mMat fixed e[t]
m.
Following the penalty CCP framework, we impose the use
of slack variables b= [b1, ..., b2M]Tover the equivalent
constraints of the unit-modulus constraints, which yields
max
e,a,b,
β,$,ξ
||a||1λ[t]||b||1(14a)
s.t.Modified-(8),(10),(11c),(13c),(14b)
|e[t]
m|22Re(eH
me[t]
m)bm1,1mM(14c)
|em|21 + bM+m,1mM(14d)
b0,(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||1χguarantees
the unit-modulus constraints in the original Problem (13)
to be met for a sufficiently low χ;c) The stopping criteria
||e[t]e[t1]||1νcontrols 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||1χand ||e[t]e[t1]||1ν.
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: nn+ 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
kK
(ϕk1)2(15a)
s.t. Rk(F,e)ϕkrk,kK(15b)
ϕk0,kK,(15c)
where ϕ= [ϕ1, ..., ϕK]Tis an auxiliary variable vector.
Problem (15) is guaranteed to be feasible since at least
{ϕk= 0,kK,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}kKthe 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,kK.
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 = 3010αlog10 (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}kK]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,kK,
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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