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IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 5, NO. 4, DECEMBER 2021 2027
Robust Active and Passive Beamformer Design for
IRS-Aided Downlink MISO PS-SWIPT With a
Nonlinear Energy Harvesting Model
Shayan Zargari , Shahrokh Farahmand , Bahman Abolhassani , and Chintha Tellambura ,Fellow, IEEE
Abstract—This paper optimizes the energy consumption of the
downlink of a multiple-antenna base station (BS) transmitting to
several single-antenna users. The BS utilizes simultaneous wire-
less information and power transfer (SWIPT) while receivers
apply power-splitting (PS) with a nonlinear energy harvesting
model leading to PS-SWIPT. We use an intelligent reflecting sur-
face (IRS) and propose a joint design to optimize the active data
and the BS’s energy beamformers, IRS’s passive beamformers,
and the receivers’ PS ratios under perfect and imperfect CSI
availability. In particular, the total BS transmit power is mini-
mized while guaranteeing a minimum rate and harvested energy
for each receiver. We apply the block coordinate descent (BCD)
method to optimize active and passive beamformers iteratively.
We enforce the rank-one constraint and solve the corresponding
optimization problem via successive convex approximation (SCA)
for accurate semidefinite relaxations. Furthermore, we propose
a worst-case robust design for the imperfect CSI case and refor-
mulate this problem with infinitely many constraints. With the
BCD method, the problem is iteratively solved via semidefinite
programming (SDP) and a second sub-problem with a linear
objective and quadratic matrix inequalities, which is also solved
via SCA. Numerical results show significant improvements (e.g.,
30% decrease in transmit power) than those of no-IRS and IRS
with random phase shifts.
Index Terms—Intelligent reflecting surface, simultaneous wire-
less information and power transfer, energy harvesting, imperfect
channel state information.
I. INTRODUCTION
MASSIVE growth in wireless mobile devices and their
ever-increasing data demands constantly challenge ser-
vice providers to design more energy- and spectral-efficient
systems. Among the newest innovative technologies, the intel-
ligent reflecting surface (IRS) has attracted a significant
amount of attention in academia and industry as a novel, cost-
effective concept [1], [2]. IRS’s fundamental principle is to
employ adjustable panels on the surface of outdoor buildings
Manuscript received April 2, 2021; revised June 2, 2021 and June 16, 2021;
accepted June 24, 2021. Date of publication June 30, 2021; date of current
version November 22, 2021. The editor coordinating the review of this article
was G. Yu. (Corresponding author: Chintha Tellambura.)
Shayan Zargari and Chintha Tellambura are with the Department of
Electrical and Computer Engineering, University of Alberta, Edmonton,
AB T6G 2R3, Canada (e-mail: zargari@ualberta.ca; ct4@ualberta.ca).
Shahrokh Farahmand and Bahman Abolhassani are with the School
of Electrical Engineering, Iran University of Science and Technology,
Tehran 1684613114, Iran (e-mail: shahrokhf@iust.ac.ir; abolhassani@
iust.ac.ir).
Digital Object Identifier 10.1109/TGCN.2021.3093825
to reflect the base station’s received signals (BS) with pas-
sive elements [3] at any desired phase/direction. Compared to
the typical relaying schemes, the IRS requires no active cir-
cuits and power sources. To be more specific, an IRS produces
directional beams by adjusting low-cost passive reconfigurable
elements to steer the signals towards the desired receiver [4].
This directionality enhances both the reliability and coverage
of wireless communication systems [5].
However, the deployment of the IRS presents many chal-
lenges, and researchers have extensively addressed them.
Specifically, [6] studied sum-rate maximization where the
transmit power of the BS and the IRS reflection coefficients are
jointly optimized. Reference [7] considered a single-antenna
user, where spectral efficiency (SE) was maximized by opti-
mizing the beamforming weights of the BS and phase shifts of
the IRS, respectively. Minimizing total BS transmit power was
studied in [8], which jointly designed beamforming weights
and reflection coefficients. In [9] a simple single-user IRS
system was considered where the transmit power subject
to signal-to-noise ratio (SNR) constraint was minimized by
joint optimization of discrete phase shifts and the continu-
ous transmit beamforming at the IRS and access point (AP),
respectively. Reference [10] has proposed a joint design of
the active beamforming at the AP and phase shifts at the IRS
by considering the amplitude variation. Specifically, under the
assumption of the practical IRS channel estimation method,
the achievable rate of each user was first derived. Then, for
single user and multi-user scenarios, the rate and weighted
sum-rate maximization problems subjected to channel state
information (CSI) errors were formulated. The work in [11]
investigated the physical layer security of the IRS systems
under the assumption of imperfect channel state information
corresponding to multi-antenna potential eavesdroppers. In
particular, the system sum-rate was maximized by jointly opti-
mizing the beamformers/artificial noise covariance matrix at
the AP and reflecting elements at the IRS.
Besides high SE requirements, the development of an
energy-efficient (EE) system is also of chief interest. To
ensure high EE, energy harvesting (EH) from ambient radio-
frequency (RF) signals is a candidate solution [12], [13]. For
instance, each user can extract both energy and information
from the received signal by utilizing the simultaneous wire-
less information and power transfer (SWIPT) technique.
For instance, [12] studied a multiple-input single-output
(MISO) SWIPT system, where the weighted sum-power at
2473-2400 c
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2028 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 5, NO. 4, DECEMBER 2021
EH receivers (EHRs) was maximized. Besides, [13] inves-
tigated a downlink (DL) MISO SWIPT-aided system where
all users could perform information decoding (ID) and EH
simultaneously based on the power splitting (PS) structure.
The total transmit power was minimized by jointly optimizing
the beamforming weights and PS ratios at the BS and each
user, respectively. The joint design of transmit beamforming
and the phase shifts at the BS and the IRS, respectively, in
an IRS-aided massive multiple-input multiple-output (MIMO)
system, was investigated in [14] by using a deep reinforce-
ment learning approach. The concept of SWIPT has also been
incorporated into IRS-aided settings to provide more energy-
efficient solutions. Reference [15] considered two groups of
receivers performing either EH or ID. Specifically, EHRs’
minimum received energy was maximized via jointly design-
ing beamforming weights of information and energy beams at
the BS and phase shifts at the IRS. Weighted sum-harvested-
power maximization at all EHRs and total transmit power
minimization were studied by [16] and [17], respectively. They
jointly optimized the active beamformers at the BS and pas-
sive reflecting elements at the IRS. Previous works [15]–[17]
assumed that the BS perfectly knows the CSI. However,
practical systems may violate this assumption. Thus, [19]
studied performance analysis of an IRS-aided SWIPT system
with a single-antenna BS under imperfect CSI, where the
achievable data rate of the designed system was analyzed.
Reference [20] studied an IRS-aided MIMO system under
imperfect CSI, where the average sum data rate was max-
imized by jointly optimizing the beamforming weights and
phase shifts at the BS and the IRS, respectively. Reference [21]
studied a robust beamforming design based on an imperfect
cascaded channel link BS-to-IRS-to-receivers to minimize the
total transmit power. However, [21] neither considered SWIPT
nor PS-SWIPT specifically.
This paper proposes two DL MISO IRS-aided SWIPT trans-
mission strategies for a wireless network with co-located
EH/ID receivers capable of performing simultaneous EH and
ID via PS architecture under perfect and imperfect CSI. We
aim to minimize the total BS transmit power subject to a min-
imum prescribed rate and EH constraints at each receiver. We
enumerate our contributions as follows.
1) We exploit a new nonlinear EH (NLEH) model based on
the Gaussian error function, which can perform differ-
ently and even closer to actual EH modules than other
nonlinear models in the literature. This model dramati-
cally enhances IRS-enabled SWIPT design practicality
compared to [15]–[18] where a linear EH model was
considered. In particular, this NLEH model uses four
parameters, which can be obtained via a simple best-fit
search through measured data [23], [24].
2) In [15]–[17] separate EH and ID receivers were consid-
ered. Each receiver either asked for minimum energy
to be harvested or minimum SINR to be supported.
However, we study the performance gain of PS-SWIPT
where each receiver simultaneously demands EH and
SINR constraints to be satisfied. Thus, our model is more
general and subsumes these references as special cases.
However, the cost of our design is more complex, as an
additional PS ratio needs to be optimized per receiver
to guarantee the simultaneous EH and ID constraints.
3) Our primary aim is to provide an energy-efficient IRS-
aided PS-SWIPT system by minimizing the total trans-
mit power as an objective function subject to rate and
harvested energy constraints. However, in [15], [16],
maximization of the minimum harvested power and
weighted sum harvested power were studied, respec-
tively. Their formulations lead to different optimization
problems. As for [17], their objective is the same as ours,
but we further assume PS-SWIPT with simultaneous
EH/ID QoS constraints per receiver and imperfect CSI
on IRS-receivers links, which [16] does not consider.
4) Previous works in IRS-aided SWIPT systems either
have not considered imperfect CSI [15]–[17], have
not followed a model-based approach to the imper-
fect CSI challenge [14], or have considered a different
optimization problem with a sum-rate objective and
without SWIPT [20]. Thus, our formulated optimization
problem of robust IRS-enabled PS-SWIPT is novel and
has not been considered before.
5) Upon applying Schur complement and generalized
S-procedure, we end up with a set of nonlinear matrix
inequalities (MIs). We hasten to add that even applying
the block coordinate descent (BCD) method to itera-
tively solve for active and passive beamformers, as is the
usual technique in current literature, does not transform
these constraints into LMIs. Indeed, IRS phase shifts
appear quadratically in positive semi-definite (PSD) con-
straints. A novel contribution of our work is to reformu-
late these nonlinear MIs into linear matrix inequalities
(LMIs).
6) For perfect and imperfect CSI scenarios, we provide
convergence and complexity analysis of our proposed
algorithms. Furthermore, we conduct extensive sim-
ulations to evaluate the proposed approaches’ con-
vergence rate and performance gains versus existing
methods.
Notation: Vectors and matrices are expressed by boldface
lower case letters aand capital letters A, respectively. For
a square matrix A,AH,AT,Tr(A), and Rank(A)are
Hermitian (conjugate transpose), transpose, trace, and rank
of a matrix, respectively. A positive semidefinite is indi-
cated by A0.IMdenotes the M-by-Midentity matrix.
diag(v)returns a square diagonal matrix with the elements
of vector von the main diagonal. The Euclidean norm of
a complex vector, the absolute value of a complex scalar
and the nuclear norm of a square matrix are denoted by
·,|·|, and ·
∗respectively. (.)crepresents conjugation
operator. The distribution of a circularly symmetric complex
Gaussian (CSCG) random vector with mean μand covari-
ance matrix Cis denoted by ∼CN(μ,C). The expectation
operator is denoted by E[·].CM×Nrepresents M×Ndimen-
sional complex matrices and HMdenotes M×Mdimensional
complex Hermitian matrices. ∇xf(x)denotes the gradient of
function fwith respect to vector x, and Ois the big-O nota-
tion. Finally, erf(x)= 2
√πx
0e−t2dt denotes the well-known
error function.
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ZARGARI et al.: ROBUST ACTIVE AND PASSIVE BEAMFORMER DESIGN FOR IRS-AIDED DOWNLINK MISO PS-SWIPT 2029
Fig. 1. A multiuser MISO IRS-assisted SWIPT system.
II. SYSTEM MODEL
A downlink (DL) scenario with an M-antenna base station
(BS) and Ksingle antenna receivers is considered (Fig. 1).
Besides, an IRS with Nelements is considered to enhance
the communication link. Furthermore, receiving nodes per-
form power splitting (PS) to divide the received signal power
into information decoding (ID) and energy harvesting (EH)
modules. Each time-frequency resource block is used simul-
taneously for transmission to all Kreceivers.
A. Transmission Scheme
The signal transmitted by the BS can be expressed as
x=K
i=1 wisi+ve,where wi∈CM×1and siare the
beamforming weights and data symbol intended for user i,
respectively. We assume that si,∀i,are independent random
variables with zero-mean and unit-variance. As well, veis a
dedicated energy beam to enable the users perform EH. It
is assumed that ve∼CN(0,Ve)is drawn according to a
Gaussian pseudo-random sequence with mean 0and covari-
ance matrix Ve∈HM,Ve0. The reflection-coefficients
matrix at the IRS is given by Θ=diag(ejθ1,...,ejθN). Here,
θnindicates the phase of each reflecting unit. Moreover, IRS
is assumed to be passive leading to unit amplitude element
gains. The flat-fading channel links from the BS-to-IRS, IRS-
to-user k, and BS-to-user kare denoted by H∈CN×M,
hH
r,k∈CN×1, and hH
b,k∈CM×1, respectively. Accordingly,
the received signal at user k={1,2,...,K}can be written as
rk=
K
i=1
hH
kwisi+hH
kvE+zk,∀k∈K,(1)
where hH
k=hH
r,kΘH +hH
b,kdenotes the aggregate channel
gain from the BS to user kand zk∼CN(0,σ
2
k)represents the
received complex Gaussian noise at user k. The signal received
by every user is split into two steams, where ρkportion is
utilized for ID, and (1 −ρk)remaining portion is used for the
EH. The received signals for the ID and EH modules are given
by yID
k=√ρkrk+nkand yEH
k=√1−ρkrk, respectively.
Thus, the PS ratio at user kis given by ρk∈(0,1), and
nk∼CN(0,δ
2
k)is the additional noise introduced by the
signal processing circuits for the ID module. Therefore, the
received signal-to-interference-plus-noise ratio (SINR) at user
kcan be expressed as
SINRk=hH
kwk
2
K
i=1
i=khH
kwi
2+σ2
k+δ2
k
ρk
,∀k∈K,(2)
where it is assumed that veis known to all users. Hence, the
interference induced by the energy beam can be eliminated
before decoding the desired user’s data.
B. Non-Linear Energy Harvesting Model
Practical EH circuits have two main limitations. First, the
harvested power saturates as the input power grows large.
Second, as long as the input power falls below the sensi-
tivity level of the EH circuit, harvested power is zero. To
capture these nonlinear effects, we employ a recently proposed
nonlinear EH (NLEH) model [22]. Specifically, we have
PNL
EH =Pmaxerf a((PL
k−Pse)+b)−erf(ab)
1−erf(ab) +
.(3)
Here, PNL
EH represents the harvested power, which is the output
of the EH module. PL
kis the input power to the EH module,
which is linearly proportional to the received power at user k.
Pmax denotes the maximum possible harvested power, which
specifies the saturation level of the EH module. Pse denotes the
input power sensitivity level, below which the output power is
zero. Other model parameters are a>0 and b>0. When PL
k
grows large in (3), the harvested power PNL
EH saturates at Pmax.
When PL
k≤Pse, output power is PNL
EH =0. Hence, both
aforementioned effects are well captured through this model.
By applying a best-fit match with experimental data, a,b, and
Pmax can be obtained [23], [24]. For comparative purposes,
we also consider the standard linear EH model. We hasten
to add that this model fits poorly with the measured data of
actual EH circuits. For instance, it does not represent the satu-
ration effect or the input sensitivity level. Nevertheless, it has
been widely used in the literature and represents the harvested
power at the k-th user by
PL
k=ηk(1 −ρk)K
i=1 hH
kwi
2+Tr(HH
kVe),∀k∈K,
(4)
where ηk∈(0,1] is the energy conversion efficiency and
Hk=hkhH
k,∀k∈K. It is notable that for the utilized NLEH
model, we have ηk=1for all users.
III. PERFECT CSI AT THE IRS
Two strategies exist for IRS-involved channel acquisition.
If the IRS components have RF chains, standard channel
estimation techniques can be employed. For instance, the time-
division duplex (TDD) mode allows for channel estimation
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2030 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 5, NO. 4, DECEMBER 2021
by exploiting channel reciprocity. In contrast, when the IRS
components have no RF chains, uplink pilots enable esti-
mating the aggregate channel at the BS. We can then adopt
a codebook-based passive beamforming approach, where the
IRS exploits it to select the best beam based on the received
pilot sequences from the BS or users [25], [26]. Channel esti-
mation for multiuser scenario with IRS was also investigated
in [27], [28], and [29]. The work in [27] proposed a novel
three-phase pilot-based channel estimation algorithm, which
exploits correlations between various users’ channels to reduce
CSI estimation overhead. Specifically, in the first and second
phases, links of the BS-to-IRS-to-user k, and BS-to-user kare
estimated. Then, in the third phase, the channels of BS-to-IRS-
to- other users are estimated by utilizing the strong correlation
between an individual user and remaining K−1 users-to-
IRS channels leading to diminished overhead. Reference [28]
proposed a two-phase channel estimation for the uplink chan-
nels. Specifically, the proposed approach can mitigate error
propagation effects that were present in [27] and improve the
channel estimation performance without the need to increase
the required overhead. The work in [29] proposed two iterative
channel estimation algorithms based on the alternating least
squares (ALS) and vector approximate message passing for
an IRS-empowered multi-user MISO system. In particular,
they proved that the proposed ALS algorithm could achieve
Cramér-Rao Lower Bound (CRLB). Also, system performance
in terms of the sum-rate by using the estimated channels meets
that of perfect CSI knowledge. Following these references,
we assume that one can accurately estimate IRS-to-users, BS-
to-users, and BS-to-IRS channels. Thus, perfect CSI for all
channel gains is available at the BS and is utilized in the
optimal design.
A. Optimization Problem
In this subsection, we aim to minimize the BS’s total trans-
mit power by jointly designing the BS’s beamforming weights,
PS ratios, and passive reflecting element phases at the IRS. The
optimization problem is formulated as
(P1) : minimize
{wk,ρk}K
k=1,Θ,Ve
K
k=1 wk2
2+Tr(Ve),(5a)
s.t.hH
kwk
2
K
i=1
i=khH
kwi
2+σ2
k+δ2
k
ρk
≥γk,
∀k∈K,(5b)
PNL
EH ≥ek,∀k∈K,(5c)
0<ρ
k<1,∀k∈K,(5d)
|φn|=1,∀n,(5e)
Ve0,(5f)
where γkand ekin (5b) and (5c) denote the minimum SINR
and harvested power requirements at user k, respectively. A
minimum SINR constraint translates directly to a minimum
data rate constraint if we add one to both sides of (5b) and
take logarithms. Constraints (5d) and (5e) indicate trivial but
important PS ratio and unit-module requirements, respectively.
Finally, (5f) implies that Veneeds to be a positive semidefinite
(PSD) matrix. To reformulate (5c) in a tractable form, we first
rewrite the inverse function of (3) as follows
PL
kPNL
EH =Pse −b+1
aerf−1
×PNL
EH
Pmax [1 −erf(ab)] + erf(ab),(6)
Thus, problem (P1) can be recast as
(P2) : minimize
ρk,wk,Θ,Ve
K
k=1 wk2
2+Tr(Ve),(7a)
s.t.(5b),(5d)-(5f),(7b)
K
i=1 hH
kwi
2+Tr(HkVe)≥PL
k(ek)
1−ρk
,
∀k∈K,(7c)
where PL
k(ek)is fixed and known since both ekand inverse
function are known. It is worth noting that any NLEH model
can be used in (3) as long as it is strictly increasing and thus
one-to-one. Subsequently, (6) is replaced with the correspond-
ing inverse function. Hence, the proposed reformulation is very
general indeed.
Problem (P1) is non-convex and challenging to obtain
the optimal global solution. Therefore, we use the BCD
approach, which works as follows. Let us consider minimiz-
ing a real-valued continuously differentiable function fof n
real variables, subject to some constraints. This method par-
titions the variables into N>1 blocks. At each iteration, f
is minimized for one of the variable blocks while the other
variables are held fixed. Thus, BCD methods perform gradi-
ent steps along with alternating subgroups of variables. This
is in contrast to full gradient descent, where a gradient step
updates variables simultaneously. Following this approach, we
break the problem into two simpler sub-problems (e.g., N=2).
The first sub-problem can be a semidefinite program (SDP),
while successive convex approximation (SCA) is exploited to
solve the second sub-problem iteratively. In the first BCD sub-
problem, we fix the phase shifts at the IRS and optimize the
beamforming weights at the BS and users’ PS ratios. We fix
BS’s beamformers and users’ PS ratios and optimize over the
phase shifts at the IRS in the second sub-problem.
B. Optimizing wk,ρkand VeWith Given Θ
We define Wk=wkwH
k,∀k∈Kand transform (P1) into
the following SDP:
(P3) : minimize
{Wk,ρk}K
k=1,Ve
K
k=1
Tr(Wk)+Tr(Ve),(8a)
s.t.Tr(HkWk)
γk−
K
i=1
i=k
Tr(HkWi)≥σ2
k
+δ2
k
ρk
,∀k,(8b)
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ZARGARI et al.: ROBUST ACTIVE AND PASSIVE BEAMFORMER DESIGN FOR IRS-AIDED DOWNLINK MISO PS-SWIPT 2031
K
i=1
Tr(HkWi)+Tr(HkVe)
≥PL
k(ek)
1−ρk
,∀k,(8c)
0≤ρk≤1,∀k∈K,(8d)
Wk0,Ve0,∀k∈K,(8e)
where the rank-one constraint, i.e., Rank(Wk)≤1,∀k∈K,
is dropped to make (P3) a convex problem. To obtain (P3), we
have applied semi-definite relaxation (SDR). Objective (8a) is
linear while constraints (8b–8d) are ordinary (not PSD) con-
vex constraints. Specifically, the left-hand side of constraints
(8b) and (8c) are linear, while the right-hand sides are both
quasi-convex functions of ρk, indicating that these constraints
are convex. Constraints in (8e) are convex PSD constraints.
Accordingly, (P3) is a standard semi-definite program (SDP)
that can be solved to global optimality by employing well-
known software such as CVX in polynomial time. Also, it
was shown by [30] that the optimal solution to (P3) satisfies
rank-one constraints for all Wk’s and hence the relaxation is
tight and produces the globally optimal solution with rank-
one constraint included. Optimal solution for W∗
kin (P3) can
be obtained by applying an interior-point algorithm [31] such
as CVX [32]. Then, the optimal beamforming weights, w∗
k,
are easily obtained by performing eigenvalue decomposition
(EVD) for W∗
k. According to [13], the complexity of solving
(P3) is O(√KM (K3M2+K2M3)).
C. Optimizing ΘWith Given wk,ρkand Ve
Next, we optimize the phase shifts at the IRS by fixing
the obtained optimal solution of (P3), i.e., {wk,ρ
k}K
k=1,Ve
in the previous step. It is notable that the objective func-
tion is independent of Θ, so the problem is transformed into
a feasibility check problem. However, the problem is still
non-convex due to constraint (5e). Hence, we first define
θ:=(φ1,φ
2,...,φ
N)H∈CN×1and ˜
θ:=[θT1]T∈
C(N+1)×1, respectively. We further define Q:=(
˜
θ˜
θH)c∈
C(N+1)×(N+1). Accordingly, the following equivalent forms
hold:
hH
r,kΘH +hH
b,kwi
2≡TrQLkWiLH
k=Tr
ˆ
Ai,k,
(9)
TrhH
r,kΘH +hH
b,kVehH
r,kΘH +hH
b,kH
≡TrQLkVeLH
k=Tr
ˆ
Bk,(10)
where Lk=[GT
khc
b,k]Tand Gk=diag(hH
r,k)H,∀k∈
K. Consequently, the feasibility problem can be mathemati-
cally formulated as
(P4) : Find Q,(11a)
s.t.
Trˆ
Ak,k
γk−
K
i=1
i=k
Trˆ
Ai,k≥σ2
k+δ2
k
ρ∗
k
,
∀k∈K,(11b)
Algorithm 1 Iterative Successive Convex Approximation
(SCA) Algorithm
Input: Set number of iterations j=0, maximum number of
iterations Jmax, and initialize Q(0) .
1: repeat
2: Calculate ˜
Ψ(Q)according to (14).
3: Solve problem (P6)/(P7) to obtain {Q(j)}.
4: j←j+1;
5: until j=Jmax
Output: return solution {Q∗}.
K
i=1
Trˆ
Ai,k+Tr
ˆ
Bk≥PL
k(ek)
1−ρ∗
k
,
∀k∈K,(11c)
Diag(Q)=1N,Q0.(11d)
where the rank-one constraint, Rank(Q)≤1is dropped to
relax the problem. It is observed that problem (P4) is an SDP
that can be solved efficiently by using CVX [32]. However, the
solution may not be rank-one. To enforce the rank-one con-
straint, we adopt the penalty-based method [33]. First, it can
be observed that an equivalent form for the rank-one constraint
can be expressed as
||Q||∗−||Q||2≤0.(12)
Take into account that the inequality ||Q||∗=iσi≥
||Q||2=max
i{σi}holds for any given Q∈Hm×n, where
σiis the i-th singular value of Q. In particular, the equality
holds if and only if Qhas rank-one. As a result, we propose
to solve the following optimization problem
(P5) : minimize
Q||Q||∗−||Q||2,(13a)
s.t.(11b)-(11d) .(13b)
Problem (P5) is not convex as it’s objective function is the
difference of two convex functions (DC). By using the SCA
technique, we approximate Ψ(Q)=||Q||2with its first order
Taylor series expansion, which is a global lower bound as
Ψ(Q)is convex. Accordingly, the Taylor-series expansion can
be written as
Ψ(Q)≥Ψ(Qi)+Tr
∇H
QΨQiQ−Qi˜
Ψ(Q).
(14)
Here, ∇QΨ(Qi)is given by
∇Q
Qi
2=∇QuH
1Qiu1=∇QTrQiu1uH
1=u1uH
1,
where u1is the eigenvector corresponding to the largest
eigenvalue of Qi.
Then, the optimization problem (P5) can be restated as
(P6) : minimize
Q||Q||∗−˜
Ψ(Q),(15a)
s.t.(11b)-(11d) .(15b)
Problem (P6) is convex and thus can be efficiently solved by
solvers such as CVX [32]. The iterative SCA algorithm cor-
responding to (P6) is summarized in Algorithm 1. The overall
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2032 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 5, NO. 4, DECEMBER 2021
Algorithm 2 Block Coordinate Descent (BCD) Algorithm
Input: Set iteration number i=0 and initialize the phase
shifts as Q=Q(0).
1: While problem (P6)/(P7) are feasible do
2: Solve problem (P3) for given Q(i), and obtain the
optimal solutions as {W(i)
k,V(i)
e,ρ
(i)
k}.
3: Solve problem (P6)/(P7) for given {W(i)
k,V(i)
e,ρ
(i)
k}
and according to Algorithm 1, denote the solution as
Q(i+1).IfQ(i+1) is not rank-one, utilize the Gaussian
randomization to obtain a feasible rank-one solution.
4: Set i←i+1;
5: end while
Output: return solutions {w∗
k,Q∗,V∗
e,ρ
∗
k}.
BCD algorithm to solve (P2) is presented in Algorithm 2.
While (P6) is endowed with an objective function, it is implic-
itly a feasibility problem as any Qthat has rank-one and
satisfies the constraints is an optimal solution. Thus, we can
enforce an optimal solution that optimizes SINR and harvested
power margins while satisfying the rank-one constraint. For
this purpose, by proposing two new slack variables, τkand
ψkas the “SINR residual” and “EH residual,” respectively,
we propose to solve the following sub-problem at every SCA
iteration:
(P7) : minimize
Q,{τk,ψk}K
k=1 ||Q||∗−˜
Ψ(Q)−
K
k=1
(λ1τk+λ2ψk),
(16a)
s.t.
Trˆ
Ak,k
γk−
K
i=1
i=k
Tr(ˆ
Ai,k)≥σ2
k+δ2
k
ρ∗
k
+τk,∀k,(16b)
K
i=1
Trˆ
Ai,k+Tr
ˆ
Bk≥PL
k(ek)
1−ρ∗
k
+ψk,∀k,(16c)
(11d),τ
k,ψ
k≥0,∀k∈K,(16d)
where λ1and λ2are positive constants. It should be men-
tioned that the feasible set for both problems (P4) and (P7)
is the same, but problem (P7) is more efficient in terms of
convergence [34]. The iterative SCA of Algorithm 1 prov-
ably converges to a stationary point/local optimum of (P5),
see [35, Th. 1]. Furthermore, the computational complexity
of solving problem (P7) is O((N+1)
6). Due to coupled
constraints between first block variables wk,ρ
kand second
block variables Θ, BCD method may not converge to a local
optimum. However, it has two desirable properties. First, it is
guaranteed to generate a non-increasing sequence of objective
function values. Second, it is guaranteed to converge. Please
check Theorem 1 for details.
Remark 1: Practical IRS phase shifts can assume discrete
values only. However, solving the corresponding optimization
problem becomes considerably more challenging as integer
constraints are non-convex and non-differentiable. Because
this work is the first study of the robust active and pas-
sive beamformer design for an IRS-aided MISO PS-SWIPT,
we only investigate the continuous phase shifts at the IRS.
One contribution of our proposed algorithm is to provide a
performance benchmark for the more practical case of discrete
phase shifts.
Remark 2: For future work, we elaborate on two possible
extensions of our proposed scheme to accommodate discrete
phase shifts. First, a sub-optimal solution is to solve the con-
tinuous phase shift problem and then round (or quantize) each
phase shift to the nearest feasible point. The second direc-
tion pertains to extending the single-user approach in [9] to
our multiple-user scenario. To be more specific, [9] aimed to
minimize the BS transmit power subject to a minimum signal-
to-noise ratio (SNR) constraint on the single receiver. Due
to the single-user assumption, the optimal linear beamformer
was maximum ratio transmission (MRT). Upon replacing
the general beamformer with MRT, the discrete phase shifts
were optimized iteratively based on the coordinate descent
algorithm. The same idea can extend our work. However,
beamforming in our setup is non-trivial because we have
multiple users and EH constraints in addition to SINR con-
straints. Furthermore, our reasonable assumption of imperfect
CSI also complicates the beamformer design. Hence, beam-
formers and PS ratios should be optimized iteratively with the
discrete phase shifts in a joint BCD fashion. Both approaches
reach sub-optimal solutions and can not achieve a global
minimum.
Remark 3: For the relaxed sub-problem (P3), optimal val-
ues for {Wk}K
k=1 has been proven to achieve rank-one and
thus their relaxation is tight [30]. Therefore, the active beam-
former design for sub-problem (P2) can be solved to global
optimality. This is not the case for (P5) which is non-convex,
and thus, its iterative solution obtained via SCA in (P7) does
not achieve the global optimum of (P5). We should mention
that (P7) is convex and evidently can be solved to global opti-
mality. However, the global optimums of SCA iterations do
not converge to the global optimum of (P5) but to a sub-
optimal solution. Subsequently, the overall BCD converges to
a sub-optimal solution of (P2), and neither global nor local
optimality is claimed.
Theorem 1: Algorithm 2 iterations yield a non-increasing
sequence of objective values with guaranteed convergence.
Proof: See Appendix A.
IV. IMPERFECT CSI AT TH E IRS
Here, we assume that the IRS structure lacks any RF
chains and the capability to estimate its channels accu-
rately. Unfortunately, the IRS-assisted PS-SWIPT system’s
performance profoundly depends on the accurate knowledge
of aggregate channel gains. Thus, we design the system with
explicit consideration of errors in channel gain estimates. To
capture this effect, we assume that the IRS position is fixed,
so the channel gains between the BS and the IRS can be esti-
mated with satisfactory accuracy by utilizing the angles of
arrival and departure. Similarly, the channel gains between
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ZARGARI et al.: ROBUST ACTIVE AND PASSIVE BEAMFORMER DESIGN FOR IRS-AIDED DOWNLINK MISO PS-SWIPT 2033
the BS and each user can be estimated accurately by con-
ditional channel estimation methods [25]–[29]. However, the
channel gains between the IRS and users vary fast since the
users are mobile. Due to the lack of RF chains and power-
ful signal processors at the IRS, the corresponding channel
estimations are error-prone. We utilize a deterministic error
model to represent CSI uncertainty. Specifically, the channel
gain between the IRS and user kis assumed to belong to the
following set:
Ωk=ˆ
hr,k+Δhr,k|ΔhH
r,kΔhr,k≤2
k,∀k∈K.(17)
In particular, ˆ
hr,kand Δhr,kdenote the acquired channel esti-
mation and the uncertainty of user k’s actual channel gain from
the IRS, respectively. In addition, kis the size of the channel
uncertainty region which is considered to be a ball of radius
kcentered around ˆ
hr,k.
A. Optimization Problem
To incorporate the effect of imperfect CSI at the IRS in
our design, we formulate the worst-case robust optimization
problem as
(P10) : minimize
{wk,ρk}K
k=1,Θ,Ve
K
k=1 wk2
2+Tr(Ve),(18a)
s.t.min
hr,k∈ΩkhH
kwk
2
K
i=1
i=khH
kwi
2+σ2
k+δ2
k
ρk
≥γk,∀k,(18b)
min
hr,k∈Ωk
K
i=1 hH
kwi
2+Tr(HkVe)
≥PL
k(ek)
1−ρk
,∀k,(18c)
(5d),(5e),(5f) .(18d)
Setting aside the non-convex nature of the original (P1),
(P10) must deal with infinitely many inequalities in con-
straints (18b) and (18c) as well as unit-module constraints,
i.e., |φn|2=1in (5f). To address these challenges, we fol-
low the following approach. First, we reformulate (P10) into
a non-convex problem with only a finite number of con-
straints by applying two lemmas. The first one is Schur’s
complement and the second one is a generalized version of S-
procedure. We add that due to coupling between optimization
variables, i.e., ρkand wk. Thus, we first need to exploit
Schur’s complement to rewrite the rate and EH constraints
in terms of linear matrix inequalities for wk’s and ρk’s. By
replacing hr,k=ˆ
hr,k+Δhr,k, we end up with an infinite
number of nonlinear matrix inequalities regarding Δhr,k,wk,
and ρk. Then, we apply S-procedures to replace the infinite
number of matrix inequalities (MIs) with a finite number of
nonlinear MIs.
B. Reformulation of (P10) With Finitely Many Constraints
Let us first define Wk=wkwH
k, and rewrite con-
straint (18b), ∀k∈K, as follows
hkHWk
γk
hk−
K
i=1
i=k
hH
kWihk−σ2
k−δ2
k
ρk≥0,
∀hr,k∈Ωk,(19)
where (19) can be further expressed as
hkHZkhk−σ2
k−δ2
k
ρk≥0,∀hr,k∈Ωk,∀k∈K,(20)
where Zk=Wk
γk−i=k,i∈K Wi,∀k∈K. Consequently,
we adopt the following Lemma to transfer (20) into a linear
matrix inequality (LMI) form.
Lemma 1 (Schur Complement [31]): Suppose X∈Snis
partitioned as
X=AB
BTC0,(21)
where A∈Sk. The Schur complement of X(with respect to
A)isS=C−BTA−1B. The PSD inequality in (21) holds
when A0and S0.
By substituting hr,k=ˆ
hr,k+Δhr,kinto (20) and applying
Lemma 1, we obtain
ρkδk
δkT0,∀hr,k∈Ωk,∀k∈K,(22)
where B:=ΘH and
T:=ˆ
hH
r,kBZkBHˆ
hr,k+ˆ
hH
r,kBZkBHΔhr,k
+ΔhH
r,kBZkBHˆ
hr,k+ΔhH
r,kBZkBHΔhr,k(25)
+ˆ
hH
r,kBZkhb,k+ΔhH
r,kBZkhb,k+hH
b,kZkBHˆ
hr,k
+hH
b,kZkBHΔhr,k+hH
b,kZkhb,k−σ2
k.(26)
Still (22) includes infinitely many constraints, one for each
element of Ωk. To obtain the finite MI form of (22), we employ
[36, Th. 4.2] which is stated as the following lemma.
Lemma 2 [36, Th. 4.2]: If there exists PSD matrices Di
0,i={1,2}then the following quadratic matrix inequality
(QMI) constraint
A1A2+A3X
(A2+A3X)HA4+A5X+(A5X)H+XHA6X0,
∀X:Tr
DiXXH≤1,i={1,2},
is equivalent to the following LMI with ∃ti≥0,∀i,
⎡
⎣
A1A2A3
AH
2A4A5
AH
3AH
5A6⎤
⎦−t1⎡
⎣
00 0
0I 0
00−D1⎤
⎦
−t2⎡
⎣
00 0
0I 0
00−D2⎤
⎦0.(27)
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2034 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 5, NO. 4, DECEMBER 2021
Based on Lemma 2, by setting
X=Δhr,k,D1=1
2
k
I,D2=0,A1=ρk,A2=δ2
k,
A3=0,A5=ˆ
hH
r,kBZkBH+hH
b,kZkBH,A6=BZkBH
A4=ˆ
hH
r,kBZkBHˆ
hr,k+ˆ
hH
r,kBZkhb,k+hH
b,kZkBHˆ
hr,k
+hH
b,kZkhb,k−σ2
k,(28)
the finite MI form of (22) can be obtained which is given
by (23), shown at the bottom of the page, for k≥0,k∈K.
Accordingly, (18b) is reformulated into a finite number of MI
constraints. Similarly, for (18c) we adopt Schur complement
to rewrite it as follows:
⎡
⎣
1−ρkPL
k(ek)
PL
k(ek)hH
kYhk⎤
⎦0,∀hr,k∈Ωk,∀k∈K,(29)
where Y=iwiwH
i+Ve. By substituting hr,k=ˆ
hr,k+
Δhr,kinto (29), we obtain
⎡
⎣
1−ρkPL
k(ek)
PL
k(ek)R⎤
⎦0,∀hr,k∈Ωk,k∈K,(30)
where
R=ˆ
hH
r,kBYBHˆ
hr,k+ˆ
hH
r,kBYBHΔhr,k
+ΔhH
r,kBYBHˆ
hr,k+ΔhH
r,kBYBHΔhr,k(31)
+ˆ
hH
r,kBYhb,k+ΔhH
r,kBYhb,k+hH
b,kYBHˆ
hr,k
+hH
b,kYBHΔhr,k+hH
b,kYhb,k.(32)
Subsequently, we apply Lemma 2 by setting
X=Δhr,k,D1=1
2
k
,D2=0,A1=1−ρk,
A3=0,A5=ˆ
hH
r,kBYBH+hH
b,kYBH,A6=BYBH
A4=ˆ
hH
r,kBYBHˆ
hr,k+ˆ
hH
r,kBYhb,k+hH
b,kYBHˆ
hr,k
+hH
b,kYhb,k,A2=PL
k(ek).(33)
Therefore, the finite MI form of (30) is represented by (24),
shown at the bottom of the page, where tk≥0,k∈K. Finally,
problem (P10) is reformulated as
(P11) : minimize
Θ,{Wk,ρk,k,tk}K
k=1,Ve
K
k=1
Tr(Wk)+Tr(VE),
(34a)
s.t.(18d),(23),(24),(34b)
k≥0,tk≥0,∀k∈K,
(34c)
Wk0,Rank(Wk)≤1,
∀k∈K.(34d)
Problem (P11) is non-convex but has only a finite number of
constraints. In the following, the BCD method is adopted to
solve problem (P11) iteratively.
C. Optimizing wk,ρkand VeWith Given Θ
The sub-problem for beamforming weights and PS ratios
can be written as
(P12) : minimize
{Wk,ρk,k,tk}K
k=1,Ve
K
k=1
Tr(Wk)+Tr(Ve),(35a)
s.t.(5d),(5f),(23),(24),(35b)
k≥0,tk≥0,∀k∈K,(35c)
Wk0,∀k∈K,(35d)
where Rank(Wk)≤1,∀k∈Kis dropped. It is observed that
(P12) is a SDP problem which can be solved via an interior-
point method [31]. It is noteworthy that if the relaxation is
tight, i.e., Rank(W∗
k)=1,∀k∈K, the optimal solution can
be found by applying EVD for W∗
k. In contrast, a Gaussian
randomization procedure can be employed to find an approxi-
mate solution [37]. If more processing power is available at the
BS, one can approximately enforce the rank-one constraint for
Wk’s by adding the penalty term K
k=1 ϑ(Wk∗−Wk2)
to the objective in (P12) and apply the SCA technique to solve
the main problem as in the perfect CSI case. By doing so, we
arrive at the following objective function:
(P13) : minimize
{Wk}K
k=1,s
F{Wk}K
k=1,s,(36a)
s.t.(35b)-(35d) .(36b)
where
F{Wk}K
k=1,s=
K
k=1
Tr(Wk)+Tr(Ve)
+ϑ
K
k=1
[Tr(Wk)−λmax(Wk)].(37)
⎡
⎢
⎣
ρkδk0
δkˆ
hH
r,kBZkBHˆ
hr,kˆ
hH
r,kBZkBH
0(ˆ
hH
r,kBZkBH)HBZkBH⎤
⎥
⎦+⎡
⎢
⎣
00 0
0ˆ
hH
r,kBZkhb,k+hH
b,kZkBHˆ
hr,k+hH
b,kZkhb,k−σ2
k−khH
b,kZkBH
0(hH
b,kZkBH)Hk
ε2
k
I⎤
⎥
⎦0.
(23)
⎡
⎢
⎢
⎢
⎣
1−ρkPL
k(ek)0
PL
k(ek)ˆ
hH
r,kBYBHˆ
hr,kˆ
hH
r,kBYBH
0(ˆ
hH
r,kBYBH)HBYBH
⎤
⎥
⎥
⎥
⎦
+⎡
⎢
⎣
00 0
0ˆ
hH
r,kBYhb,k+hH
b,kYBHˆ
hr,k+hH
b,kYhb,k−tkhH
b,kYBH
0(hH
b,kYBH)Htk
ε2
k
I⎤
⎥
⎦0.
(24)
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ZARGARI et al.: ROBUST ACTIVE AND PASSIVE BEAMFORMER DESIGN FOR IRS-AIDED DOWNLINK MISO PS-SWIPT 2035
Algorithm 3 The Cost Function-Based Algorithm for (P14)
Input: Set iterative number i=1, the tolerance error , cost
factor ϑ>0, and W(0)
k.
1: While Tr(W(i+1)
k)−λmax(W(i)
k)≤do
2: Solve (P14) and obtain the solutions as
{W(i+1)
k,s(i+1)}.
3: if W(i+1)
k=W(i)
k
4: Set ϑ=2ϑ;
5: end
6: Set i←i+1;
7: end while
Output: return solutions {w∗
k,s∗}.
It should be mentioned that Tr(Wk)=Wk∗and
λmax(Wk)=Wk2. Here, ϑ>0is the penalty term
weight corresponding to the initial objective function and
s={Ve,{ρk,
k,tk}K
k=1}is the set of optimization variables.
Note that we set small value for ϑin the beginning to find
a feasible set of Wk’s. Then, it is increased to enforce rank-
one on those feasible Wk’s. Since λmax(Wk)is convex, its
negative is concave and thus (P13) is non-convex. We replace
λmax(Wk)with its first order Taylor series which is a global
lower bound. Then, (P13) can be approximated as
(P14) : minimize
Wk,s
˜
F{Wk}K
k=1,s,(38a)
s.t.(35a)-(35d) .(38b)
where ˜
F({Wk}K
k=1,s)at i-th iteration is given by
˜
F{Wk}K
k=1,s:=
K
k=1
Tr(Wk)+Tr(Ve)
+ϑ
K
k=1Tr(Wk)−λmaxWi
k
−wi
k
HWk−Wi
kwi
k.
(39)
In (39), wi
kis the eigenvector corresponding to the maximum
eigenvalue of Wi
k. An iterative SCA-type algorithm denoted
by Algorithm 3, is used to solve (P14).
D. Optimizing ΘWith Given wk,ρkand Ve
Similar to problem (P7), we introduce two slack variables,
νkand μkas the “SINR residual” and “EH residual” to
obtain better convergence. Therefore, the IRS phase shifts
optimization sub-problem can be formulated as
(P15) : maximize
u,{νk,μk,k,tk}K
k=1
K
k=1
(νk+Υ
kμk),(40a)
s.t.Modified–(23), Modified–(24),
(5d),(5f),(40b)
νk,μ
k≥0,∀k∈K,(40c)
|φn|2=1,∀n,(40d)
where u=[φ1,φ
2,...,φ
N]Hand B=diag(u)H.
Furthermore, Modified–(23) and Modified–(24) are obtained
from (23) and (24) by substituting γkwith γk+νkand
PL
k(ek)with PL
k(ek)+μk,∀k∈K, respectively. Nevertheless,
(P15) is non-convex with respect to (40d) which is an unit-
modulus constraints. In contrast to the SDR approach which
may not guarantee a rank-one solution, we apply the penalty
convex-concave procedure (CCP) to obtain a sub-optimal
solution with approximate rank-one property [38]–[40]. This
penalty approach finds a feasible solution while satisfying the
unit-modules constraints. Specifically, |φn|2=1,∀ncan be
equivalently modified as 1≤|φn|2≤1,∀n. The left-hand-
side (LFS) of this new constraint is non-convex. Hence, we
replace |φn|2with a global lower bound as follows:
φn−φ(i)
n
2≥0−→ |φn|2≥2Reφ∗
nφ(i)
n−φ(i)
n
2
Upon replacing |φn|2in 1≤|φn|2with the aforementioned
lower bound, we arrive at |φ(i)
n|2−2Re(φH
nφ(i)
n)≤−1,∀n.
The resultant sub-problem is given by
(P16) : maximize
u,{νk,μk,k,tk}K
k=1
K
k=1
(νk+Υ
kμk),(41a)
s.t.Modified–(23), Modified–(24),
(5d),(5f), (40c),(41b)
φ(l)
n
2−2Re
φH
nφ(l)
n
≤−1,∀n,(41c)
|φn|2≤1,∀n.(41d)
Constraints (23) and (24) in (P16) are non-convex due to the
quadratic forms of B. However, the first term of (23) and (24)
can be rewritten as
⎡
⎢
⎢
⎣
00 0
0ˆ
hH
r,kBZkBHˆ
hr,kˆ
hH
r,kBZkBH
0ˆ
hH
r,kBZkBHHBZkBH
⎤
⎥
⎥
⎦
=⎡
⎣
0
ˆ
hH
r,kB
B⎤
⎦(Zk)0B
Hˆ
hr,kBH=˜
BZk˜
BH,∀k∈K,
(42)
and
⎡
⎢
⎢
⎣
00 0
0ˆ
hH
r,kBYBHˆ
hr,kˆ
hH
r,kBYBH
0ˆ
hH
r,kBYBHHBYBH
⎤
⎥
⎥
⎦
=⎡
⎣
0
ˆ
hH
r,kB
B⎤
⎦(Y)0B
Hˆ
hr,kBH=˜
BY ˜
BH,∀k∈K,
(43)
respectively. The quadratic forms of ˜
Bare replaced by their
lower bounds derived from the following Lemma.
Lemma 3: Let ˜
B[l]be fixed and known. For example, it
can be the optimal value for ˜
Bobtained at iteration l, then
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2036 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 5, NO. 4, DECEMBER 2021
Algorithm 4 The Penalty Convex-Concave Procedure (CCP)
Optimization Algorithm
Input: Set l=0, maximum number of iterations Lmax,u0,
and B0.
1: While u(l)−u(l−1)1≤1do
2: if l≤Lmax then
3: Calculate (44) and (45).
4: Solve problem (P16) to obtain u(l+1).
5: Set l=l+1;
6: else
7: Set l=0, initialize u0, and B0with new random
values.
8: end if
9: end while
Output: return solution u∗.
Algorithm 5 The Robust BCD Algorithm
Input: Set t=0and u=u(0) .
1: While problem (P16) is feasible do
2: Solve problem (P14) according to Algorithm 4 for given
u(t), and obtain optimal solutions as {W(t)
k,V(t)
e,ρ
(t)
k}.
3: Solve problem (P16) to obtain u(t+1) for given
{W(t)
k,V(t)
e,ρ
(t)
k}.
4: Set t=t+1;
5: end while
Output: return solutions {w∗
k,ρ
∗
k,u∗,V∗
e}.
the linear lower bounds (in the PSD sense) of BZkBHand
BYBHare given by
˜
BZk˜
BH˜
BZk˜
BH,[l]+˜
B[l]Zk˜
BH−˜
B[l]Zk˜
BH,[l],(44)
˜
BY ˜
BH˜
BY ˜
BH,[l]+˜
B[l]Y˜
BH−˜
B[l]Y˜
BH,[l],(45)
respectively.
Proof: We apply the following series of inequalities
˜
B−˜
B[l]Zk˜
B−˜
B[l]H0
⇒˜
BZk˜
BH−˜
BZk˜
BH,[l]−˜
B[l]Zk˜
BH+˜
B[l]Zk˜
BH,[l]0
⇒˜
BZk˜
BH˜
BZk˜
BH,[l]+˜
B[l]Zk˜
BH−˜
B[l]Zk˜
BH,[l].
Similarly,
˜
B−˜
B[l]Y˜
B−˜
B[l]H0
⇒˜
BY ˜
BH−˜
BY ˜
BH,[l]+˜
B[l]Y˜
BH−˜
B[l]Y˜
BH,[l]0
⇒˜
BY ˜
BH˜
BY ˜
BH,[l]+˜
B[l]Y˜
BH−˜
B[l]Y˜
BH,[l].
For any fixed PSD matrices Zkand Y, we can obtain (44)
and (45).
Substituting these new linear lower bounds in (23) and (24),
(P16) becomes a convex optimization problem. The main goal
of (P16) is to obtain a feasible solution rather than just max-
imizing SINR and EH. The details of solving problem (P16)
is summarized in Algorithm 4, where ul−u(l−1)1≤
1controls the convergence of the algorithm. In the end,
the robust beamforming problem (P11) can be solved iter-
atively by employing (P14) and (P16) which is given in
TAB LE I
SIMULATION PARAMETERS
Algorithm 5. Furthermore, the convergence properties of the
robust BCD algorithm are similar to its non-robust counterpart.
Additionally, the approximate complexity of Problem (P14)
and (P16) and O([K(2N+4+K)]1/2M[M2K3+MK 3((N+
2)2+(N+2)
2)+K2((N+2)
3+(N+2)
3+K2)]) and
O([K(MN +1+K)+K+2N]1/2K[K2+K2((N+2)2+(N+
2)2)+K((N+2)3+(N+2)3+K2+KN )]), respectively [21].
Remark 4: Our proposed robust BCD method converges
to a sub-optimal solution of (P10). We can not claim even
local optimality for either the first sub-problem or the sec-
ond one. Indeed, the SDR in (P12) may not be tight. The
same goes for (P15), which is solved via SCA. Therefore,
there is no guarantee for even local optimality of the proposed
solution to (P10). Our approach only provides a sub-optimal
solution.
Theorem 2: Our proposed SCA iterations for solving (P14)
converges. Similarly, the SCA iterations for solving (P15)
converges as well. Subsequently, our proposed robust BCD
converges to a sub-optimal solution of (P11). Furthermore,
SCA for both sub-problems (P14) and (P15) and the overall
BCD for rank-relaxed (P11) possess the desirable property of
decreasing the objective function with every iteration.
Proof: See Appendix B.
V. NUMERICAL RESULTS
Simulation parameters are given in Table I. We consider
a two-dimensional plane where the BS and the IRS are
located at coordinates (3.5 m, 0 m) and (0 m, 5 m), respec-
tively. The single-antenna users are distributed randomly and
uniformly in a circle of radius 2.5 m centered at (3.5 m,
5 m). We consider the path-loss model L(d)=C0(d
D0)α
where C0=(
4π
λc)2is the path loss at the reference distance
D0=1m,dis the link distance, and αdenotes the path loss
exponent.
For all the involved channel links, a Rician fading model
with a dominant line-of-sight (LoS) path is considered as
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ZARGARI et al.: ROBUST ACTIVE AND PASSIVE BEAMFORMER DESIGN FOR IRS-AIDED DOWNLINK MISO PS-SWIPT 2037
follows:
hH
i,k=κ
1+κhLoS
i,k+1
1+κhNLoS
i,k,i∈{b,r},(46)
H=κ
1+κHLoS +1
1+κHNLoS.(47)
Here, (46) indicates the channel model between the BS-to-
users and the IRS-to-users, while (47) denotes the channel
model between the BS and the IRS. Besides, κis the Rician
factor assumed to be the same for all channels. Furthermore,
LoS and non-line-of-sight (NLoS) superscript indexes denote
the LoS component and the Rayleigh fading component with
zero-mean and unit-variance. In particular, a uniform linear
antenna array (ULA) model is adopted for the LoS compo-
nent, for instance, hLoS
i,k=[1,ejθk,ej2θk,...,ej(M−1)θk]T
with θk=−2πxsin(φk)
λ, where λdenotes the carrier wave-
length and x=λ/2is the distance between two consecutive
antennas. Similarly, HLoS can be defined as
HLoS =⎡
⎢
⎢
⎢
⎣
1ejθk··· ej(M−1)θk
e−jθk1··· ej(M−2)θk
.
.
.··· ....
.
.
e−j(N−1)θke−j(N−2)θk··· 1
⎤
⎥
⎥
⎥
⎦
.
(48)
The CSI error bounds are represented as k=δˆ
hr,k2,∀k,
where δ∈[0,1) indicates the relative value of CSI uncer-
tainties. A normalized maximum channel estimation error of
δ=5%is assumed. We compare our proposed robust BCD
(Algorithm 5) against the following benchmarks.
1) Robust BCD method (Algorithm 5) with linear energy
harvesting (LEH) model [15]–[17], [19].
2) Random phase shifts at the IRS [17], [34]. In this case,
only the beamformers and PS ratios are designed via
solving the corresponding sub-problem.
3) Robust BCD method (Algorithm 5) with only a NLOS
Rayleigh fading channel component for H[16].
4) BCD method with perfect CSI (Algorithm 2) [15]–[17].
5) Without IRS [12], [13], [30].
A. Convergence of the Robust BCD
Fig. 2 depicts the convergence behavior of Algorithm 5.
Here, we analyze convergence for different values of the
prescribed minimum rate, i.e., RI=log
2(1+γk)=5 Mbps
and Rk=4 Mbps. As can be observed, Algorithm 5 con-
verges quickly after approximately 6 to 7 iterations, which
implies that a solution meeting all the users’ QoS constraints
is obtained fast.
B. Transmit Power Versus Number of Antennas at BS
Next, we look at the minimum required transmit power at
the BS versus the number of BS antennas, M. As per Fig 3,
we observe that transmit power decreases as the number of BS
antennas increases in all proposed schemes, which is the typ-
ical behavior when adding to BS antennas. It shows that the
employment of large or even massive antennas is beneficial
Fig. 2. Transmit power versus the number of iterations for the proposed
robust BCD method (Algorithm 5) for different values of target rate, Rk.
Fig. 3. Transmit power versus the number of antennas at the BS, M,for
different schemes.
for MISO IRS-aided SWIPT systems in decreasing the trans-
mit power even further. The proposed robust scheme performs
worse than Algorithm 2, which assumes that CSI is perfectly
known. On the other hand, robust BCD guarantees the min-
imum required SINR and rate for all devices even if their
channel estimates to IRS are erroneous. However, Algorithm 2
may fail to satisfy all rate requirements when errors are present
in channel estimates. When we select IRS’s phase shifts ran-
domly, the transmit power increases considerably compared to
the optimal scheme. This increase occurs because, without IRS
phase-shift design, IRS’s reflected signal is not guaranteed to
be aligned with the directly received signal from the BS.
We also observe that the NLEH model suggests worse
system performance compared to the LEH model. Of course,
the LEH model is unrealistic. The NLEH model is more prac-
tical and is a better model for the EH device behavior. Besides,
multipath fading affects the system’s performance. To flesh this
out, we investigate the effect of Rayleigh fading. When the
BS-to-IRS link is NLOS only, i.e., a Rayleigh fading channel
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2038 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 5, NO. 4, DECEMBER 2021
Fig. 4. Transmit power versus the target rate, Rk, for different schemes.
for H, the BS requires more transmit power to meet the QoS
requirements of each user. Finally, this figure illustrates how an
IRS improves the performance of SWIPT systems. The global
optimum of PS-SWIPT with perfect CSI is still considerably
larger than cases with IRS despite possible impairments such
as NLOS BS-IRS link and error in BS-to-IRS link. Curiously,
random IRS phase shifts outperform no IRS scenario. Thus,
even a non-intelligent IRS is better than no IRS.
C. Transmit Power Versus Target Rate
Fig. 4 illustrates the total transmit power versus an equal
minimum data rate requirements for all users. We observe
that the transmit power is a monotonically increasing func-
tion of the minimum required data rate in all cases. This
increase happens because a higher transmit power is nec-
essary if users’ minimum SINR requirement becomes more
stringent. Comparative performance patterns are similar to
Fig. 3. Transmit power for random IRS phase shifts is higher
than the robust BCD since, in this case, the IRS does not
optimize its phase shifts to align passive beams toward the
intended user. Also, when the NLOS Rayleigh fading chan-
nel model is adopted, more transmit power is required than
the Rician model. Additionally, robust BCD with the NLEH
model requires more transmit power compared to the LEH
model. If the transmit power limits to 20 dBW, we observe
that a joint BS-IRS design with Algorithm 2 under perfect CSI
can support rates as high as 6.5 Mbps, which is more than a
three-fold improvement over no-IRS case supporting 2 Mbps.
Under imperfect CSI, rates as high as 5.7 can be supported,
which is just less than a three-fold improvement.
D. Sum-Power Received at Users Versus Distance
In Fig. 5, we plot the received sum-power at receivers ver-
sus the distance between BS and IRS for different schemes.
Specifically, we increase the distance between IRS and BS by
moving the IRS away while BS and users’ locations remain
fixed. In both perfect and imperfect CSI scenarios, we observe
that by deploying the IRS around SWIPT-based receivers, the
Fig. 5. Sum-power received at users versus the distance between the BS and
the IRS, d0(m), with fixed N=60 for different schemes.
Fig. 6. Transmit power versus the number of reflecting elements at the IRS,
N, for different schemes.
proposed design significantly improves the receivers’ received
sum-power. Also, the proposed scheme in the perfect CSI case
with the LEH model outperforms other benchmark methods.
However, as the distance between BS and IRS increases, the
harvested power diminishes due to the increased path-loss phe-
nomenon. Additionally, one can observe that the sum received
power at each user decreases by adopting the Rayleigh fading
channel model due to NLOS. In contrast, using a Rician model
can facilitate the EH process for each user. This figure sug-
gests that to reap the IRS benefits fully, we should deploy it
either close to BS or mobile users.
E. Transmit Power Versus Number of Reflecting Elements
Finally, in Fig. 6, we investigate the impact of the num-
ber of IRS elements Non system performance. As expected,
the BS transmit power decreases by increasing the number
of IRS’s elements for perfect and imperfect CSI scenarios.
This decrease highlights the importance of utilizing an IRS
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ZARGARI et al.: ROBUST ACTIVE AND PASSIVE BEAMFORMER DESIGN FOR IRS-AIDED DOWNLINK MISO PS-SWIPT 2039
even with low-cost passive reflecting elements. While the over-
all performance trends resemble those of previous figures,
several interesting new observations can be made. The first
observation deals with the rate with which various approaches
improve as Nincreases. As expected, transmit power is con-
stant when no IRS is present. However, random phase-shift
selection reduces transmit power at a close-to-linear rate, while
joint designs reduce transmit power faster than linear as the
curves suggest. Even if we treat the reduction of transmit
power for joint designs to be linear, its slope is significantly
higher than the random phase shifts. Secondly, comparing
Fig. 6 with Fig. 3 reveals that a BS with five antennas aided by
an IRS with ten elements provides almost similar performance
to a stand-alone BS with 12 antennas. This observation sug-
gests that an IRS can compensate for a small number of
BS antennas and thus significantly improve device-to-device
communication when space limitation prevents incorporating
many antennas at the transmitter. This observation points to an
alternative path in implementing massive MIMO in practice.
VI. CONCLUSION
This paper investigated the minimization of BS total trans-
mit power for a downlink multiuser IRS-aided PS-SWIPT
scenario. The formulated problem involved joint optimization
of active beamformers at the BS, passive beamformers at the
IRS, and PS ratios at the receiving users. Simultaneously, a
minimum prescribed rate and harvested energy requirements
were enforced for all the users. The BCD method was applied
to iteratively optimize both active and passive beamformers
when perfect CSI is available. A non-convex penalty term was
added to the objective and later the main problem was solved
iteratively via the SCA technique to enforce rank-one for the
passive beamformer’s SDR-based optimum solution. A robust
problem was proposed for an imperfect CSI scenario, which
guaranteed the QoS constraints for all possible channel errors
of the limited Euclidean norm. The problem was recast into a
convex objective with a finite number of nonlinear MIs. The
BCD was utilized to solve the ensuing problem by iterating
between an SDP and a QMI-constrained with a convex objec-
tive sub-problem. The previously proposed penalty term was
also applied. Numerical results corroborated that the submitted
designs perform significantly better than no IRS and IRS with
random phase shifts. Finally, IRS-aided communication poten-
tials such as the potential to mimic a virtual massive MIMO
for a BS with a limited number of antennas were revealed
through simulations and further discussed.
APPENDIX A
PROOF OF THEOREM 1
We consider {w(i+1)
k,ρ
(i+1)
k}K
k=1,V(i+1)
eas the optimal
solution of (P3) given Θ(i). Furthermore let Θ(i+1) be the
optimal solution of (P7), given {w(i+1)
k,ρ
(i+1)
k}K
k=1,V(i+1)
e.
If we define the objective function of P1 as
fΘ,{wk,ρ
k}K
k=1,Ve=
K
k=1 wk2.
Then, we will have
fΘ(i+1),w(i+1)
k,ρ
(i+1)
kK
k=1,V(i+1)
e
=fΘ(i),w(i+1)
k,ρ
(i+1)
kK
k=1,V(i+1)
e
≤fΘ(i),w(i)
k,ρ
(i)
kK
k=1,V(i)
e.(49)
Equality arises because the objective is not a function of Θ.
So, as long as a feasible Θis selected, equality remains valid.
Inequality comes from the fact that for given Θ(i),thesolu-
tions {w(i+1)
k,ρ
(i+1)
k}K
k=1,V(i+1)
eare optimal. As for the
convergence, we have a sequence of non-increasing objective
values, which are bounded below by zero. Note that the objec-
tive function can not become negative. Hence the sequence of
objective values is guaranteed to converge.
APPENDIX B
PROOF OF THEOREM 2
Constraints in (P14) are convex, and the objective function
is the difference of convex functions. Given that λmax(Wk)
is convex, its negative is concave. Hence, −K
k=1 λmax(Wk)
is also concave. It is well-known that the derivative of a con-
cave function evaluated at an arbitrary point is a global upper
bound on the function itself. At step of the SCA, we lin-
earize the concave function around the obtained optimum at
the previous step −1. Given that linear approximation is tight
at the expansion point, which is the optimum at −1, the fact
that a different point is selected as the iteration optimum
means that objective upper bound and thus objective itself
can be further reduced by switching to a new feasible point.
Subsequently, the SCA objective is non-increasing at every
iteration. Given that objective in (P14) is bounded below as
it is always positive, SCA iterations converge. As for (P15),
optimum at SCA iteration −1is a feasible point at SCA
iteration . Hence, when the optimum at step is different
from the optimum at the previous step, the objective has been
further reduced. Given that the objective of (P15) is always
positive and bounded below, SCA iterations converge. Finally,
using the same argument as in Appendix A, BCD on (P11)
reduces the objective at every iteration because SCA iterations
for each sub-problem reduce the objective. Since the objective
in (P11) is bounded below by zero, robust BCD converges to
a sub-optimal solution.
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Shayan Zargari received the B.Sc. degree in elec-
tronic engineering from the Azad University South
Tehran Branch in 2018, and the M.Sc. degree in
telecommunication engineering Iran University of
Science and Technology in 2020. He is currently
pursuing the Ph.D. degree in communications and
signal processing with the University of Alberta,
Edmonton, AB, Canada. From 2019 until 2020,
he was a Visiting Researcher with the Electronics
Research Institute, Sharif University of Technology,
Tehran, Iran. His research interests include convex
and non-convex optimization, intelligent reflecting surface, resource alloca-
tion in wireless communication, and green communication. He also served
as a Reviewer for the IEEE Globecom Conference in 2020 and 2021. He
served as a Reviewer for several IEEE journals, e.g., IEEE TRANSACTIONS
ON VEHICULAR TECHNOLOGY and IEEE WIRELESS COMMUNICATIONS
LETTERS.
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ZARGARI et al.: ROBUST ACTIVE AND PASSIVE BEAMFORMER DESIGN FOR IRS-AIDED DOWNLINK MISO PS-SWIPT 2041
Shahrokh Farahmand was born in Tehran, Iran,
in 1980. He received the B.Sc. degree in elec-
trical engineering from the Sharif University of
Technology in 2003, and the M.Sc. and Ph.D.
degrees in communications and signal processing
from the University of Minnesota at Twin Cities,
in 2006 and 2011, respectively. From 2011 to 2014,
he was with Iran Research Organization for Science
and Technology, where he held a research faculty
position. Since 2018, he has been with the School
of Electrical Engineering, Iran University of Science
and Technology, where he is currently an Assistant Professor. His general
interests include applications of statistical signal processing, optimization, and
machine learning in communications and networking. His current focus is on
the Internet of Things, intelligent reflecting surfaces, massive MIMO, and
ultra-wideband impulse radio.
Bahman Abolhassani was born in Tehran, Iran.
He received the B.S. degree in electrical engineer-
ing from Iran University of Science and Technology
(IUST), Tehran, and the M.S. and Ph.D. degrees
in electrical engineering from the University of
Saskatchewan, Saskatoon, SK, Canada. He was an
Instrumentation Engineer with the College of Water
and Power Technology, Iranian Ministry of Energy,
for three years. He worked as a Communication
System Engineer in a number of private and govern-
ment companies. He joined the School of Electrical
Engineering, IUST, where he is currently an Associate Professor. He served
as the Dean of the School of Electrical Engineering and an Associate Dean
for Research. He also served as a Sessional Lecturer with the University of
Saskatchewan. His research interests are in the fields of wireless communi-
cation systems, network planning, spread spectrum, cognitive radio networks,
resource allocation, VANETs, and optimization of large systems.
Chintha Tellambura (Fellow, IEEE) received the
B.Sc. degree in electronics and telecommunications
from the University of Moratuwa, Sri Lanka, the
M.Sc. degree in electronics from the Kings College,
University of London, and the Ph.D. degree in elec-
trical engineering from the University of Victoria,
Canada.
He was with Monash University, Australia, from
1997 to 2002. Since 2002, he has been with the
Department of Electrical and Computer Engineering,
University of Alberta, where he is currently a Full
Professor. He has authored or coauthored over 560 journal and conference
papers, with an h-index of 77 (Google Scholar). He has supervised or co-
supervised over 70 M.Sc., Ph.D., and PDF trainees. His current research
interests include future wireless networks and machine learning algorithms.
He received the best paper awards from the IEEE International Conference
on Communications in 2012 and 2017. He is the winner of the presti-
gious McCalla Professorship and the Killam Annual Professorship from the
University of Alberta. He served as an Editor for the IEEE TRANSACTIONS
ON COMMUNICATIONS from 1999 to 2012 and IEEE TRANSACTIONS ON
WIRELESS COMMUNICATIONS from 2001 to 2007. He was an Area Editor
of Wireless Communications Systems and Theory from 2007 to 2012. He was
elected as a fellow of The Canadian Academy of Engineering in 2017.
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