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ARTICLE OF THE IEEE SYSTEMS JOURNAL 1
Joint Power Allocation and Power Splitting for
MISO-RSMA Cognitive Radio Systems with SWIPT
and Information Decoder Users
Mario R. Camana, Carla E. Garcia, and Insoo Koo
Abstract—In this paper, we investigate the rate-splitting multiple
access (RSMA) framework in a multi-user multiple-input single-
output (MISO) underlay cognitive radio system with power-
splitting secondary users and information secondary users by
using simultaneous wireless information and power transfer
(SWIPT). The RSMA framework is a novel multiple access
method based on linearly precoded rate-splitting at the secondary
base station, and successive interference cancellation (SIC) at
the secondary receivers, which has shown better performance
compared with the traditional space-division multiple access
(SDMA) method. In particular, the objective is to minimize the
transmit power at the secondary base station subject to the
constraints of minimum energy harvesting, minimum data rate,
and a maximum power level for allowable interference with the
primary users. The challenging problem under consideration
is non-convex, and the solution relies on dividing it into outer
and inner optimization problems. Then, we propose a particle
swarm optimization–based algorithm to solve the outer problem,
with the inner problem managed by the semidefinite relaxation
method. In addition, we present a comparative scheme based
on the successive convex approximation method and a scheme
based on SDMA. Finally, simulation results confirm the superior
performance of the proposed approach in comparison with the
other schemes.
Index Terms—rate splitting, SWIPT, semidefinite relaxation,
cognitive radio system, particle swarm optimization.
I. Introduction
I
N recent years, the challenges of fifth-generation (5G)
technology, such as heterogeneity of applications, the
increasing number of devices, and a scarcity of radio resources,
have led to the development of novel multiple access techniques.
In particular, the rate-splitting (RS) transmission strategy was
proposed to improve the data rate and robustness for multiple
access in downlink wireless systems, compared with traditional
multiple access techniques [1]. RS is based on the idea of
splitting the user messages into common and private parts at
the transmitter, with successive interference cancellation (SIC)
This work was supported by the National Research Foundation of Korea
(NRF) through the Korean Government and the Ministry of Science and ICT
(MSIT) under Grant NRF-2018R1A2B6001714.
Mario R. Camana is with the School of Electrical and Computer
Engineering, University of Ulsan, Ulsan 680-749, South Korea (e-mail:
mario camana@hotmail.com).
Carla E. Garcia ias with the School of Electrical and Computer En-
gineering, University of Ulsan, Ulsan 680-749, South Korea (e-mail:
carli.garcia27@hotmail.com).
Insoo Koo is with the School of Electrical and Computer Engineering,
University of Ulsan, Ulsan 680-749, South Korea (e-mail: iskoo@ulsan.ac.kr)
(Corresponding Author).
Date of publication November 4, 2020
doi.org/10.1109/JSYST.2020.3032725
at the receiver. The common parts of the user messages are
grouped together and encoded by using a shared codebook,
with this common group being the first one decoded by all
users. The private parts of the user messages are encoded
independently using a private codebook, and the common
and private messages are transmitted simultaneously by using
linear precoding. At the receiver, the common message is
decoded by treating all the private messages as interference,
and the SIC procedure is applied to remove the contribution
of the common message. Then, each receiver decodes its own
private message by considering the other private messages
as interference. Consequently, RS can cover offtwo extreme
cases: In the first case, we treat all the interference from other
messages as noise, and in the second case we totally decode the
interference through SIC, because we can adjust the message
split ratio and the power assigned to the private and common
messages.
To overcome the scarcity of radio resources and enhance
spectrum efficiency, a candidate technique is cognitive radio
(CR), which allows secondary users to share a spectrum
band with primary users (PUs) provided that interference
with the primary users by the secondary transmitters is
acceptable [2]. The massive number of devices and the new
applications and services expected in 5G networks created
a need to develop green technology and a mechanism to
reduce power consumption. In this sense, simultaneous wireless
information and power transfer (SWIPT) enables a transmitter
to simultaneously provide information and power [3]. SWIPT
is based on a trade-offbetween information rate and energy
harvesting (EH) at the receiver, and the power-splitting (PS)
architecture was selected for the approach proposed in this
paper. The PS user divides the received signal into two different
power streams, where one stream is used for the information
decoder (ID) module and the other stream for the EH module,
based on the PS ratio.
In the literature, the authors in [4] proposed SWIPT in
a multi-user multiple-input single-output (MISO) downlink
system by studying the minimum transmit power optimization
problem at the base station (BS) under the constraints of signal-
to-interference-plus-noise ratio (SINR) and harvested power
at the EH module. The non-convex problem was solved by
applying the semi-definite relaxation (SDR) technique. Based
on the same system model, the authors in [5] investigated
minimization of the transmit power subject to the constraints
of mean square error (MSE) and EH. The solution was based on
an iterative algorithm and the SDR method. In [6], the authors
ARTICLE OF THE IEEE SYSTEMS JOURNAL 2
presented a SWIPT system model composed of a single-antenna
BS that co-exists with power beacon equipment to transmit
information and energy to a PS user. The aforementioned works
did not consider a CR system and a RS-based multiple access
method, which permits to enhance spectrum efficiency and
improve the quality-of-service (QoS).
CR in a MISO NOMA network with SWIPT was investigated
in [2], where the authors formulated a power minimization
problem subject to the constraints of minimum SINR, minimum
harvested power, and a maximum level for interference with
the PUs. The near-optimal solution was based on SCA and
SDR algorithms. In [7], a multi-user MISO SWIPT cognitive
radio system with space-division multiple access (SDMA) was
formulated to study the max-min fair secondary users’ harvested
energy problem and the problem considering the trade-off
between interference power to the PUs and EH by the secondary
users. A MISO CR system on downlink with SWIPT was
considered in [8], where a cognitive BS transmits information
and energy to a set of secondary PS users and a set of secondary
EH users, which are considered eavesdroppers. The objective
of the authors was to solve the worst-case transmit power
problem and the max-min fairness EH problem with imperfect
channel state information at the transmitter (CSIT). However,
these works did not investigate the impact of the RS approach.
The rate-splitting multiple access (RSMA), introduced in
[9], is based on linearly precoded RS that applies the SIC
procedure at the receiver to decode part of the interference
while considering the remaining part as noise. In [9], a multi-
user (MU) MISO system was used to maximize the weighted
sum rate (WSR) subject to the maximum transmit power at the
BS and QoS for the users. The problem was solved with the
weighted minimum mean squared error (WMMSE) method, and
simulation results showed that RSMA outperformed NOMA
and SDMA in several scenarios. In [10], the authors presented
a discussion of RSMA as a candidate solution to deal with
the requirements and challenges of future services in 6G. The
authors presented relevant papers to support the superiority of
RSMA compared with existing multiple access schemes such
as SDMA, NOMA and orthogonal multiple access (OMA). RS
is a robust scheme compared with SDMA and NOMA to deal
with the problem of imperfect CSIT, which produces rate loss
and is caused by channel estimation error, latency, mobility,
and so on [11], [12], [13]. In addition, the robustness of RSMA
with imperfect CSIT is useful for reaching high performance
in low-latency services and achieving the best performance
compared with other multiple access techniques in the presence
of interference [10], [12]. The superior performance of RSMA
with perfect and imperfect CSIT was also proved in scenarios
with different channel strengths and directions and with
underload and overload conditions [9], [14], [15]. With respect
to the complexity at the receivers, the one-layer RS uses only
one layer of SIC which provides a lower complexity compared
with K−1 layers of SIC used in NOMA.
The performance of RS in a MU-MISO system was investi-
gated based on the maximization of the ergodic sum-rate in
[11], and based on the maximization of the minimum data rate
in [13]. The numerical simulations demonstrated the superiority
of RS compared with the traditional SDMA scheme. In [15], the
authors considered a MISO broadcast channel composed of two
users with the objective to prove that RS is a flexible scheme to
deal with diverse user channel strengths and directions, where
SDMA, NOMA and OMA are analytically demonstrated to
be special cases of RS. An analysis of the energy efficiency
of RSMA compared with SDMA and NOMA was presented
in [16] considering a multi-user MISO system. The trade-off
optimization problem between EE and spectral efficiency (SE)
was studied in [17] while considering a downlink multi-user
MISO system with RSMA. Through simulations, the authors
demonstrated the superior performance of RSMA versus SDMA
with respect to SE, EE and their trade-off. RS with imperfect
CSIT in a Massive MIMO system was studied in [12], in which
a 2-layer RS scheme is proposed by consisting of two types
of common messages, one to be decoded by all the users and
the other to be decoded by a subset of users. However, the
aforementioned papers did not consider the SWIPT system
and CR networks along with the novel multiple access method
based on RS.
In [18], RS was used in a multi-user MISO SWIPT
system, where the BS simultaneously serves several information
receivers and several energy-harvesting receivers. The authors
proposed maximization of the WSR, subject to the constraints
of total harvested energy and maximum transmit power. The
solution was based on the SCA and WMMSE algorithms. Based
on the numerical results, the authors emphasized the superior
performance of the RS strategy, compared with multi-user linear
precoding and NOMA. However, contrary to the proposed
system model in this paper, the authors did not study a CR
system. Moreover, they did not consider the case when the user
is capable to receive information and energy simultaneously.
A MU-MISO SWIPT system with RSMA was proposed in
[19], with the aim being to minimize the total transmit power
under QoS constraints. The solution was based on the particle
swarm optimization (PSO) algorithm with SDR. In [20], the RS
strategy was implemented on a MU-MISO interference channel
with SWIPT, consisting of several multi-antenna transmitters
and a set of single-antenna PS users. A robust transmit power
minimization problem was proposed under the constraints of
QoS and EH requirements. The problem was solved with a
gradient-based scheme along with a semi-definite programming
(SDP) problem with rank relaxation. However, the latter two
references did not investigate an underlying CR system with
different types of secondary users.
Therefore, motivated by improvement in the performance
of the RSMA technique, in the paper we propose a multi-user
MISO SWIPT cognitive radio system with RSMA in which a
secondary BS (SBS) simultaneously transmits information and
energy to a set of secondary users implementing a PS structure
(the PS receivers) and a group of secondary users that only
decode information (the ID receivers) subject to a maximum
power threshold for interference with the primary users. Our
objective is to obtain the precoder vectors, common rates, and
PS ratio variables that achieve the minimum transmit power at
the SBS. To the best of our knowledge, the proposed system
model with RSMA and the problem formulation have not been
addressed in the literature. The main contributions of this paper
are summarized as follows:
ARTICLE OF THE IEEE SYSTEMS JOURNAL 3
•
The transmit power minimization problem is studied under
the constraints of a minimum data rate at the secondary
receivers, minimum EH at the PS receivers, a maximum
power level for allowable interference with the PUs, and
maximum available power at the SBS.
•
The non-convex optimization problem is converted into
bilevel programming in which the outer optimization
problem is solved with a PSO-based algorithm, and the
inner optimization problem is managed with an algorithm
based on SDR. A Gaussian randomization method is used
to obtain the approximate optimal solution, and a linear
problem is proposed to guarantee the feasibility of the
solutions.
•
A comparative approach is proposed based on the SCA
technique along with the PSO algorithm. Initial feasible
points in the SCA method are guaranteed by relaxing the
constraints with slack variables, and a penalty function
is added to make the slack variables zero as iterations
progress.
•
The performance advantages of the proposed scheme are
presented in comparison with the SCA, equal power split-
ting (EPS), and SDMA techniques. Numerical simulations
confirm the superiority of the proposed approach based
on RSMA in reducing the total transmit power at the SBS
in comparison with the EPS and SDMA techniques while
reaching results similar to the SCA method but with lower
computational time.
The rest of the paper is organized as follows. Section II
presents the system model. Section III describes the problem
formulation, the solution, and the comparative schemes. Section
IV illustrates the simulation results. Finally, the conclusions
are presented in Section V.
II. The System Model
We consider a multi-user MISO SWIPT cognitive radio
system with RSMA involving
K
PS users,
M
ID users, and
L
PUs, as illustrated in Fig. 1. The SBS is equipped with
N≥
2
antennas, while the
K
PS receivers,
M
ID receivers, and
L
PUs
each have a single antenna. The baseband equivalent channels
between the SBS and the
k
-th PS receiver, between the SBS
and the
m
-th ID receiver, and between the SBS and the
l
-th PU
receiver are denoted by
hk∈CN×1
,
gm∈CN×1
, and
ql∈CN×1
,
respectively.
In this paper, we use the one-layer RS strategy [9] in which
the receivers only need to apply one SIC procedure. We denote
as
mPS
k
the message for the
k
-th PS user, and we divide
that message into a common component,
mc,PS
k
, and a private
component,
mp,PS
k
,
∀k
. In the same way, we denote the message
intended for the
m
-th ID user as
mID
m
, which is divided into a
common part, mc,ID
m, and a private part, mp,ID
m,∀m.
As illustrated in Fig. 2, the common messages for all PS and
ID receivers, i.e.
nmc,PS
1, ...., mc,PS
K,mc,ID
1, ...., mc,ID
M,o
are jointly
encoded into a common stream,
sc
, while the private messages
for each
k
-th PS receiver and
m
-th ID receiver are encoded into
the private streams,
nsPS
1, ..., sPS
Ko
and
nsID
1, ..., sI D
Mo
, respectively,
with
En|sc|2o
=1,
EsPS
k
2
=1, and
EsID
m
2
=1. The
precoders for streams
sc
,
sPS
k
, and
sID
m
are denoted as
pc∈CN×1
,
Fig. 1. The multi-user MISO SWIPT underlay CR system.
pk∈CN×1
, and
wm∈CN×1
, respectively. Thus, the transmit
signal at the SBS can be expressed as
x=pcsc+
K
X
k=1
pksPS
k+
M
X
m=1
wmsID
m.(1)
The SINR of common stream
sc
at the ID module of the
k-th PS user is given by
SINRPS
c,k=θkhH
kpc
2
θk K
P
i=1hH
kpi
2+
M
P
m=1hH
kwm
2+σ2
k!+δ2
k
,∀k,(2)
where
nPS
k∼ CN
(0
, σ2
k
) and
nID
m∼ CN
(0
, α2
m
) are the additive
white Gaussian noise at the
k
-th PS receiver and
m
-th ID
receiver, respectively;
vk∼ CN
(0
, δ2
k
) is the additive circuit
noise at the ID module of the
k
-th PS receiver, and
θk∈(0,1)
represents the ratio for power splitting.
After decoding common stream
sc
, the SIC procedure is
applied, and the
k
-th PS receiver can decode its message by
considering as noise the other private symbols. Then, the SINR
is given by
SINRPS
k=hH
kpk
2
K
P
i=1,i,khH
kpi
2+
M
P
m=1hH
kwm
2+σ2
k+δ2
k
θk
,∀k.(3)
The corresponding instantaneous achievable rates of
sc
and
sPS
k
at the
k
-th PS receiver are denoted as
RPS
c,k
=
log21+SINRPS
c,kand RPS
k=log21+SINRPS
k, respectively.
For the
m
-th ID user, the SINR of common stream
sc
is
expressed as follows:
SINRID
c,m=gH
mpc
2
K
P
k=1gH
mpk
2+
M
P
j=1gH
mwj
2+α2
m
,∀m.(4)
After decoding
sc
and performing the SIC procedure, the
SINR for decoding the private
sID
m
at the
m
-th ID receiver is
given by
SINRID
m=gH
mwm
2
K
P
k=1gH
mpk
2+
M
P
j=1,j,mgH
mwj
2+α2
m
,∀m.(5)
ARTICLE OF THE IEEE SYSTEMS JOURNAL 4
m
sc
enc ode
enc ode
enc ode
p1
pK
hk
PS
ID
EH
k k
k
dec ode
SIC
dec ode sk
^
^
1
pc
sc
...
...
...
...
...
...
p,PS
mK
p,PS
m
enc ode
enc ode
w1
wM
1
...
...
...
p,ID
mM
p,ID
1
p,PS s11
PS
sK
PS
sID
sM
s1
ID
s1
m1
c,ID
m1
c, mM
c,ID
mc,
m1
c,PS
m1
c,PS mK
c,PS
mc,PS
...
gm
ID de cod e
SIC
dec ode sm
^
^
sc
PS
ID
...
k
PS
m
ID
Fig. 2. RSMA scheme in the proposed system model.
The corresponding instantaneous achievable rates of com-
mon stream
sc
and the private stream,
sID
m
, at the
m
-th
ID receiver are denoted as
RID
c,m
=
log21+SINRID
c,m
and
RID
m=log21+SINRID
m, respectively.
To ensure that common stream
sc
is successfully decoded
during the SIC procedure by all the PS and ID receivers, the
achievable super-common rate,
Rc
, should not exceed any
RPS
c,k
and any
RID
c,m
, i.e.
Rc
=
min nRPS
c,1, ..., RPS
c,K,RID
c,1, ..., RID
c,M,o
. Rate
Rc
is shared among the receivers based on the rates of the
common streams of all PS and ID users,
CPS
k
being the
k
-th PS
receiver’s portion of the common rate, and
CID
m
being the
m
-th
ID receiver’s portion of the common rate [9]. Then, achievable
super-common rate Rcis expressed as
Rc=
K
X
k=1
CPS
k+
M
X
m=1
CID
m.(6)
The harvested energy at the EH module of the
k
-th PS
receiver is given by
EPS
k=ηk(1−θk)
hH
kpc
2+
K
X
i=1hH
kpi
2
+
M
X
m=1hH
kwm
2+σ2
k
,∀k,(7)
where
ηk∈(0,1]
is the energy-harvesting efficiency at the
k
-th
PS receiver.
In addition, the interference power (IP) of the link between
the SBS and the l-th PU receiver is given by
IPl=qH
lpc
2+
K
X
k=1qH
lpk
2+
M
X
m=1qH
lwm
2,∀l.(8)
III. Problem Formulation and Solution
A. Problem Formulation
Our aim is minimization of total transmit power at the SBS
under the constraints of a minimum information rate for the
PS and ID users, minimum EH at the PS receivers, maximum
power available at the SBS, and maximum power permitted for
interference with the PUs. The transmit power minimization
problem can be mathematically formulated as follows:
min
pc,{pk,wm,CPS
k,CID
m,θk}∥pc∥2+
K
X
k=1∥pk∥2+
M
X
m=1∥wm∥2(9a)
s.t. CPS
k+RPS
k≥γPS
k,∀k(9b)
CID
m+RID
m≥γID
m,∀m(9c)
K
X
i=1
CPS
i+
M
X
m=1
CID
m≤RPS
c,k,∀k(9d)
K
X
k=1
CPS
k+
M
X
j=1
CID
j≤RID
c,m,∀m(9e)
CPS
k≥0,CID
m≥0,∀k,∀m(9f)
EPS
k≥εk,∀k(9g)
∥pc∥2+
K
X
k=1∥pk∥2+
M
X
m=1∥wm∥2≤Pmax (9h)
IPl≤ψl,∀l(9i)
0< θk<1,∀k,(9j)
where
Pmax
represents the maximum available power at the
SBS,
ψl
is the maximum power threshold for interference with
the
l
-th PU,
εk
is the minimum harvested energy needed at the
k
-th PS receiver, and
γPS
k
and
γID
m
are the required rates at the
k
-th PS user and the
m
-th ID user, respectively. The constraints
(9d) and (9e) are used to ensure that the common stream
sc
is
successfully decoded by all the PS and ID receivers.
Problem (9) is non-convex and cannot be solved directly.
So, we convert problem (9) to a bilevel programming problem
in which
CPS
k
and
CID
m
represent the upper-level variables, as
follows:
min
{CPS
k,CID
m}ψCPS
k,CID
m (10a)
ψCPS
k,CID
m=min
pc,{pk,wm,θk}∥pc∥2+
K
X
k=1∥pk∥2+
M
X
m=1∥wm∥2
(10b)
s.t. (9b), (9c), (9d), (9e), (9g),
(9h), (9i), (9j),
where
ψCPS
k,CID
m
represents the inner minimization problem
with variables
pc,{pk,wm, θk}
, and the outer minimization
problem is represented by
min{CPS
k,CID
m}ψCPS
k,CID
m
with
variables nCPS
k,CID
mounder constraint (9f).
First, the near-optimal values of the common rates,
nCPS
k,CID
mo
, are obtained with a PSO-based algorithm in the
outer minimization problem. Second, for any given
nCPS
k,CID
mo
,
ARTICLE OF THE IEEE SYSTEMS JOURNAL 5
the inner optimization problem of (10b) is solved with the
SDR technique. In particular, the proposed approach is an iter-
ative method that starts with upper-level variables
nCPS
k,CID
mo
obtained by the PSO algorithm, which are the input data
to obtain the solution of inner minimization problem (10b).
Next, based on the solution of inner optimization problem
(10b), we update variables
nCPS
k,CID
mo
in the PSO algorithm
explained in Subsection III.B. Then, we again use updated
variables
nCPS
k,CID
mo
to solve inner problem (10b), repeating the
aforementioned process until convergence. Inner optimization
problem (10b) can be equivalently transformed into:
min
pc,{pk,wm,θk}∥pc∥2+
K
X
k=1∥pk∥2+
M
X
m=1∥wm∥2(11a)
subject to:
hH
kpk
2
K
P
i=1,i,khH
kpi
2+
M
P
m=1hH
kwm
2+σ2
k+δ2
k
θk
≥ϕPS
k,∀k(11b)
gH
mwm
2
K
P
k=1gH
mpk
2+
M
P
j=1,j,mgH
mwj
2+α2
m
≥ϕID
m,∀m(11c)
hH
kpc
2
K
P
i=1hH
kpi
2+
M
P
m=1hH
kwm
2+σ2
k+δ2
k
θk
≥ς, ∀k(11d)
gH
mpc
2
K
P
k=1gH
mpk
2+
M
P
j=1gH
mwj
2+α2
m
≥ς, ∀m(11e)
and (9g), (9h), (9i) and (9j),
where
ς
=
2
K
P
i=1
CPS
i+
M
P
m=1
CID
m−
1,
ϕPS
k
=
max n0,2γPS
k−CPS
k−1o
, and
ϕID
m
=
max n0,2γID
m−CID
m−1o
. In this paper, we propose an SDR-
based solution to problem (11) in Subsection III.C, and we
provide a comparative approach with an SCA-based scheme
in Subsection III.D, and a baseline solution with SDMA in
Subsection III.E.
B. PSO-based algorithm to solve outer problem (10)
In this paper, we consider a PSO-based algorithm [21] to
solve outer optimization problem (10), obtaining the approxi-
mately optimal
k
-th PS and
m
-th ID receiver’s portion of the
common rate variables, i.e.
nCPS
k,CID
mo
. PSO is a metaheuristic
algorithm widely used in the literature with high performance
and low computational complexity [19], [21], [22].
The number of particles to be used and the maximum number
of iterations are represented by
Imax
and
S
, respectively. We
define the velocity and position of the
n
particles as vectors
vn
and
xn
, respectively. In addition, we denote as p
n
best
the best
local position for each particle
n
, and
gbest
is the global best
position among all the particles considered in the swarm.
The objective is to minimize
f(xn)
, which is obtained by
solving inner optimization problem (11) considering the particle
position
xn
=
nCPS
1n, ..., CPS
Kn,CID
1n, ..., CI D
Mno
as the values for
the common rate variables. In addition, the proposed PSO
algorithm considers a fixed parameter for the inertia weight,
w
, and acceleration coefficients
c1
and
c2
are obtained through
a uniform distribution.
We define the maximum values for
CPS
k
and
CID
m
as
CPS
k,max
and
CID
m,max
, respectively. First, the constraint (9h) is used to
consider the upper limit of the power variables
pc,{pk}
. Then,
we use the constraint (9h), the Cauchy–Schwarz inequality and
the constraint (11d) to obtain the following:
CPS
k,max ≤ϑ1,(12)
where ϑ1=min1≤i≤Klog21+∥hi∥2Pmax
σ2
i+δ2
i.
Next, according to constraints (11e) and (9h),
CPS
k,max
can be
given as follows:
CPS
k,max ≤ϑ2,(13)
where
ϑ2
=
min1≤j≤Mlog21+∥gj∥2Pmax
α2
j
. In addition, the
value of
CID
m,max
follows the same aforementioned procedure in
(12) and (13).
Now, we use the expressions
ς
=
2
K
P
i=1
CPS
i+
M
P
m=1
CID
m−
1 and
ϕPS
k
=
max n0,2γPS
k−CPS
k−1o
to complete the definition for the
limit value of
CPS
k,max
. Then, we separate the analysis into two
cases, one for
CPS
k< γPS
k
, and the other when
CPS
k≥γPS
k
. In
the first case, the value of
ϕPS
k
is always larger than zero, and
then, the values that
CPS
k
can be assigned are
h0, γPS
k
. For the
second case,
ϕPS
k
becomes zero, and then, any value of variables
pc,{pk,wm, θk}
always satisfies constraint (11b). In addition,
we can use constraint (11d) to see that decreasing the value of
ς
leads to a reduction of
hH
kpc
2
, which matches the objective
of transmit power minimization. Then, a minimum
ς
can be
achieved when
CPS
k
is
γPS
k
in the second case, (
CPS
k≥γPS
k
).
Therefore, the first case and second case lead to the conclusion
of a maximum value of CPS
kdefined by γPS
k.
Furthermore, by following the aforementioned procedure,
we use
ς
=
2
K
P
i=1
CPS
i+
M
P
m=1
CID
m−
1,
ϕID
m
=
max n0,2γID
m−CID
m−1o
, and
constraints (11c) and (11e) to define the maximum value of
CID
mas γID
m.
Finally, by combining the expressions obtained for the
maximum user common rates, we define the following:
CPS
k,max =min γPS
k, ϑ1, ϑ2(14a)
CID
m,max =min γID
m, ϑ1, ϑ2.(14b)
On the other hand, the minimum values for
CPS
k
and
CID
m
are defined by
CPS
k,min
and
CID
m,min
, respectively. First, based on
constraint (9b), we obtain the following
CPS
k,min =max
0, γPS
k−log2
1+∥hk∥2Pmax
σ2
k+δ2
k
.(15)
Second, the expression of
CID
m,min
is based on constraint (9c),
as follows:
CID
m,min =max (0, γI D
m−log2 1+∥gm∥2Pmax
α2
m!).(16)
The overall description of the proposed PSO-based algorithm
is illustrated in Table I.
ARTICLE OF THE IEEE SYSTEMS JOURNAL 6
TABLE I
The proposed PSO-based scheme to solve problem (9) by using problem (10).
1: data: S,Imax,c1,c2,w,CPS
k,min,CI D
m,min,CPS
k,max,CI D
m,max,
vmax, and {xn},n=1, ..., S.
2: Start index i=1, set initial particle’s velocity, vn=0,∀n, and
initialize positions xn=nCPS
1n, ..., CPS
Kn,CID
1n, ..., CID
Mnoselected
from the ranges hCPS
k,min,CPS
k,maxiand hCI D
m,min,CI D
m,maxi.
3: Solve inner minimization problem (11) to evaluate f(xn).
4: Initialize the position of the best particle: gbest =arg min
1≤n≤N
f(xn).
5: Initialize pn
best with pn
best =xn,∀n.
6: while i≤Imax
7: For n=1,...,S,do
8: Calculate the random coefficients: rn
1,rn
2∼U(0,1).
9: Update the velocity of particle n:
vn←wvn+c1rn
1pn
best −xn+c2rn
2(gbest −xn)
10: Restrict vnwith [−vmax, vmax ].
11: Update the position of particle n:xn←xn+vn.
12: Restrict the position of particle xnbased on (14a), (14b),
(15), and (16).
13: Calculate f(xn)and obtain pc,{pk,wm, θk}by solving inner
optimization problem (11) when xnis the common rate
variables.
14: Update the local best position of particle n:
if f(xn)<fpn
best then pn
best ←xn
end if
15: Revise the global best position:
if f(xn)<f(gbest )then
gbest ←xn,p∗
c,np∗
k,w∗
m, θ∗
ko←pcn,{pk,wm, θk}n
end if
16: end for
17: Make: i←i+1.
18: end while
19: results: f(gbest )is the minimum transmit power of problem (9)
with the common rates variables nCPS
1n, ..., CPS
Kn,CID
1n, ..., CID
Mno
=gbest , optimal precoders vectors p∗
c,np∗
k,w∗
m,o, and optimal
power split ratios nθ∗
ko.
C. SDR-based approach to solving problem (11)
In this subsection, we apply the SDR technique [23] to find a
solution to optimization problem (11). We define the following:
Pc
=
pc
p
H
c
,
Pk
=
pk
p
H
k
,
Wm
=
wm
w
H
m
,
Hk
=
hk
h
H
k
,
Gm
=
gm
g
H
m
,
and
Ql
=
ql
q
H
l
. By using the properties
x
=
Tr (x)
,
xH
x=
∥x∥2
, and
Tr (AB)
=
Tr (BA)
, we can define
∥pc∥2
=
Tr (Pc)
,
∥pk∥2
=
Tr (Pk)
,
∥wm∥2
=
Tr (Wm)
,
hH
kpk
2
=
Tr
(
HkPk
), and
gH
mwm
2
=
Tr (GmWm)
. In addition, we have the equivalence
P=p
pH⇔
P
≻
0, and
rank (P)
=1. Furthermore, we do not
consider the constraint (9h) since it is not necessary to solve
the problem (11) with SDR. Therefore, problem (11) can be
reformulated in the following equivalent form:
min
Pc,{Pk,Wm,θk}Tr (Pc)+
K
X
k=1
Tr (Pk)+
M
X
m=1
Tr (Wm)(17a)
subject to:
−Tr (HkPk)+
K
X
i,k
Tr (HkPi)ϕPS
k+
M
X
m=1
Tr (HkWm)ϕPS
k
+σ2
kϕPS
k+ϕPS
kδ2
k
θk≤0,∀k(17b)
−Tr (GmWm)+
K
X
k=1
Tr (GmPk)ϕID
m+
M
X
j,m
Tr GmWjϕID
m
+α2
mϕID
m≤0,∀m(17c)
−Tr (HkPc)
ς+
K
X
i=1
Tr (HkPi)+
M
X
m=1
Tr (HkWm)
+σ2
k+δ2
k
θk≤0,∀k(17d)
−Tr (GmPc)
ς+
K
X
k=1
Tr (GmPk)+
M
X
j=1
Tr GmWj+α2
m≤0,∀m
(17e)
−Tr (HkPc)−
K
X
i=1
Tr (HkPi)−
M
X
m=1
Tr (HkWm)
−σ2
k+εk
ηk(1−θk)≤0,∀m(17f)
Tr (QlPc)+
K
X
k=1
Tr (QlPk)+
M
X
m=1
Tr (QlWm)−ψl≤0,∀l(17g)
Pc,Pk,Wm⪰0,∀k,∀m(17h)
rank (Pc),rank (Pk),rank (Wm)=1,∀k,∀m(17i)
0< θk<1,∀k.(17j)
By removing the rank constraints in problem (17), we obtain
the SDP problem called (17)-SDR, which is convex and can
be solved through the convex optimization toolbox CVX [24].
Note that the both functions
1
θk
and
1
1−θk
are convex over
θk
with 0
< θk<
1 since these functions satisfy the second-
order convexity condition [25]. The SDP problem obtained
from (17)-SDR is composed of Υ =
K
+
M
+1 matrix
variables with size
N×N
and T =3
K
+2
M
+
L
+1 linear
constraint variables, which leads to computational complexity
of
O√ΥNΥ3N6+ ΥTN2log 1
/
ζ
with a solution accuracy
of
ζ >
0 [7], [19], [26]. If the optimal matrix solutions to
problem (17) are rank one, then the optimal precoders are
defined as the optimal solution to problem (17). Otherwise, we
can apply the Gaussian randomization method [26], [19].
To describe the Gaussian randomization method, let us
denote P
∗
c,nP∗
k,W∗
m, θ∗
ko
as the optimal solution to problem
(11). First, we realize the eigen-decomposition of precoder
matrices P
∗
c
=
UcΛc
U
H
c
,P
∗
k
=
Uk,PS Λk,PS
U
H
k,PS
, and W
∗
m
=
Um,ID Λm,ID
U
H
m,ID
. Next, we define the candidate precoder
vectors
pc
=
Uc
Λ
1/2
cvc
,
pk
=
Uk,PS
Λ
1/2
k,PS vk,PS
, and
wm
=
Um,ID
Λ
1/2
m,ID vm,ID
, where the components of
vc
,
vk,PS
, and
vm,ID
follow a complex circularly symmetric Gaussian distribution
with unit variance and zero mean. To guarantee feasible
solutions, we introduce the scalar factors,
zc
,
nzPS
k,zID
mo
as results
of the following optimization problem:
min
zc,{zPS
k,zID
m}zc∥pc∥2+
K
X
k=1
zPS
k∥pk∥2+
M
X
m=1
zID
m∥wm∥2(18a)
subject to:
−zPS
khH
kpk
2+ϕPS
k
K
X
i=1,i,k
zPS
ihH
kpi
2+
M
X
m=1
zID
mhH
kwm
2
+ϕPS
kσ2
k+ϕPS
kδ2
k
θk≤0,∀k(18b)
ARTICLE OF THE IEEE SYSTEMS JOURNAL 7
−zID
mgH
mwm
2+ϕID
m
K
X
k=1
zPS
kgH
mpk
2+
M
X
j=1,j,m
zID
jgH
mwj
2
+ϕID
mα2
m≤0,∀m(18c)
−zchH
kpc
2+ς
K
X
i=1
zPS
ihH
kpi
2+
M
X
m=1
zID
mhH
kwm
2
+ςσ2
k+ςδ2
k
θk≤0,∀k(18d)
−zcgH
mpc
2+ς
K
X
k=1
zPS
kgH
mpk
2+
M
X
j=1
zID
jgH
mwj
2
+ςα2
m≤0,∀m(18e)
εk
ηk(1−θk)−zchH
kpc
2−
K
X
i=1
zPS
ihH
kpi
2
−
M
X
m=1
zID
mhH
kwm
2−σ2
k≤0,∀k(18f)
zc∥pc∥2+
K
X
k=1
zPS
k∥pk∥2+
M
X
m=1
zID
m∥wm∥2≤Pmax (18g)
zcqH
lpc
2+
K
X
k=1
zPS
kqH
lpk
2+
M
X
m=1
zID
mqH
lwm
2≤ψl,∀l(18h)
zc,zPS
k,zID
m≥0,∀k,∀m.(18i)
Problem (18) is a type of linear program, and can be
solved using Matlab’s CVX toolbox. Problem (18) is com-
posed of Υ =
K
+
M
+1 nonnegative real variables
and T =3
K
+2
M
+
L
+1 linear inequality constraints,
in which the computational complexity can be given as
O√ΥΥ3+ ΥTlog 1
/
ν
with a solution accuracy of
ν >
0
[26]. The general description of the proposed algorithm based
on SDR and Gaussian randomization is illustrated in Table
II. In addition, the total computational complexity of the
proposed scheme based on PSO and SDR algorithms is
OImaxS√ΥNΥ3N6+ ΥTN2+Numrand √ΥΥ3+ ΥT.
Finally, we analyse the feasibility of the problem (9), which
can be verified by solving the following problem:
find pc,npk,wm,CPS
k,CID
m, θko(19a)
s.t (9b), (9c), (9d), (9e), (9f), (9g),
(9h), (9i), (9j).(19b)
Problem (19) can be solved by taking a similar procedure
that we propose for the problem (9). In the paper, we analyze
the maximum possible values for the requirements of data rate
γPS
kand γID
m; and required EH εkas following:
First, the maximum value for the requirement of data rate
at the
k
-th PS user,
γPS
k
, and at the
m
-th ID user,
γID
m
, can be
evaluated based on (9b), (9c), (9h), (12b), (13b) and by using
the Cauchy–Schwarz inequality such that we have
γPS
k,max ≤min (ϑ1, ϑ2)+log2
1+∥hk∥2Pmax
σ2
k+δ2
k
.(20)
γID
m,max ≤min (ϑ1, ϑ2)+log2 1+∥gm∥2Pmax
α2
m!.(21)
TABLE II
Proposed algorithm based on SDR and Gaussian randomization to solve
problem (11).
1: data: Number of randomizations Numr and,Ob jF min =Pmax,
matrix precoders from (17)-SDR problem P∗
c,nP∗
k,W∗
mo.
2: Realize the eigen-decomposition of P∗
c,nP∗
k,W∗
mo
3: For i=1 : Numrand
4: Generate the candidate precoder vectors: pi
c=UcΛ1/2
cvi
c,
pi
k=Uk,PS Λ1/2
k,PS vi
k,PS ,∀k
and wi
m=Um,ID Λ1/2
m,ID vi
m,ID ,∀m
6: Get zi
c,nzi
k,PS ,zi
m,ID oby solving problem (18).
7: Obtain the feasible solutions to problem (11).
pi
c=qzi
cUcΛ1/2
cvi
c,pi
k=qzi
k,PS Uk,PS Λ1/2
k,PS vi
k,PS ,∀k
wi
m=qzi
m,ID Um,ID Λ1/2
m,ID vi
m,ID ,∀m
8: Define ObjF i=
pi
c
2+
K
P
k=1
pi
k
2+
M
P
m=1
wi
m
2.
9: if Ob jFi<Ob jF min then
Ob jFmin =Ob jF i,p∗
c=pi
c,np∗
k=pi
k,w∗
m=wi
mo
end if
10: end for
20: outputs: p∗
c,np∗
k,w∗
mo.
Next, the maximum possible value for the required EH
εk
can be evaluated based on (9g) and (9h) as follows:
εmax
k≤ηk∥hk∥2Pmax +σ2
k.(22)
Therefore, the problem (9) is infeasible if any of the
conditions (20), (21), and (22) are not satisfied. However, the
fulfillment of these conditions is not sufficient to guarantee the
feasibility. A complete analysis of the feasibility conditions is
a complex problem such that we leave it as a future work.
D. Comparison approach based on SCA to solve inner problem
(11)
In this subsection, we consider a comparison technique to
solve problem (11) based on the SCA technique [27] to obtain
the convex approximation of non-convex constraints (11b),
(11c), (11d), (11e), and (9g). The SCA is an iterative algorithm
in which the objective in each iteration is to replace the non-
convex function with an upper convex approximation function.
In each constraint, we separate the convex and concave parts,
where the latter is approximated with a convex function.
First, let us denote
˜
pk,˜
pc,˜
wm
as the initial feasible points
for precoders
pk,pc,wm
, respectively, and define
pk
=
˜
pk
+ ∆
pk
,
where ∆
pk
represents the difference of the variable
pk
between
two successive iterations of the SCA algorithm [28]. Second,
we insert pk=˜
pk+ ∆pkinto hH
kpk
2as follows:
hH
kpk
2=pH
khkhH
kpk(23a)
hH
kpk
2=(˜
pk+ ∆pk)HhkhH
k(˜
pk+ ∆pk).(23b)
Then, by dropping expression ∆p
H
khkhH
k
∆
pk
, we get the
following:
hH
kpk
2≥˜
pH
khkhH
k˜
pk+2Rn˜
pH
khkhH
k∆pko(24)
Following the aforementioned procedure, we can transform
the non-convex expressions
−hH
kpc
2,−gH
mpc
2,−gH
mwm
2
and
ARTICLE OF THE IEEE SYSTEMS JOURNAL 8
−hH
kwm
2
. One important aspect in the SCA technique is the
initial feasible point, which needs to be selected carefully to
avoid infeasible solutions. To overcome this issue, we use slack
variables
st
with
t
=1
, ...,
3
K
+2
M
+
L
+1, which allows us to
ensure feasibility in optimization problem (11) [29]. Therefore,
we can reformulate inner optimization problem (11) as follows:
min
pc,{pk,wm,θk,st}∥pc∥2+
K
X
k=1∥pk∥2+
M
X
m=1∥wm∥2+β
3K+2M+L+1
X
t=1
st
(25a)
subject to:
ϕPS
k
K
X
i=1,i,khH
kpi
2+
M
X
m=1hH
kwm
2+σ2
k+δ2
k
θk
−˜
pH
khkhH
k˜
pk−2Rn˜
pH
khkhH
k∆pko≤sk,∀k(25b)
ϕID
m
K
X
k=1gH
mpk
2+
M
X
j=1,j,mgH
mwj
2+α2
m
−˜
wH
mgmgH
m˜
wm
−2Rn˜
wH
mgmgH
m∆wmo≤sK+m,∀m(25c)
ς
K
X
i=1hH
kpi
2+
M
X
m=1hH
kwm
2+σ2
k+δ2
k
θk
−˜
pH
chkhH
k˜
pc
−2Rn˜
pH
chkhH
k∆pco≤sK+M+k,∀k(25d)
ς
K
X
k=1gH
mpk
2+
M
X
j=1gH
mwj
2+α2
m
−˜
pH
cgmgH
m˜
pc
−2Rn˜
pH
cgmgH
m∆pco≤s2K+M+m,∀m(25e)
εk
ηk(1−θk)−˜
pH
chkhH
k˜
pc−2Rn˜
pH
chkhH
k∆pco
−
K
X
i=1˜
pH
ihkhH
k˜
pi+2Rn˜
pH
ihkhH
k∆pio
−
M
X
m=1˜
wH
mhkhH
k˜
wm+2Rn˜
wH
mhkhH
k∆wmo
−σ2
k≤s2K+2M+k,∀k(25f)
∥pc∥2+
K
X
k=1∥pk∥2+
M
X
m=1∥wm∥2−Pmax ≤s3K+2M+1(25g)
qH
lpc
2+
K
X
k=1qH
lpk
2+
M
X
m=1qH
lwm
2−ψl≤s3K+2M+1+l,∀l
(25h)
∆pk=pk−˜
pk,∆pc=pc−˜
pc,∆wm=wm−˜
wk,∀k,∀m(25i)
0< θk<1,∀k(25j)
st≥0,∀t,(25k)
where
β≫
1 is a factor to make the slack variables approach
zero, and allows the solution to problem (25) to be a feasible
solution to inner optimization problem (11). Problem (25) is
convex, and we can solve it with the CVX toolbox in Matlab
[24]. The iterative algorithm based on the SCA method is
described in Table III. It is noteworthy that after each iteration
of the SCA algorithm the initial points
˜
pk,˜
pc,˜
wm
are updated
based on the current solution of the convex problem. In addition,
through simulations, we observed that power constraint (25g)
has an important effect on reducing the number of iterations
TABLE III
Comparative SCA algorithm to solve problem (11) based on problem (25).
1: data: channels hk,gm,ql, minimum total rate of users γPS
k, γID
m,
common rates CPS
k,CID
m, minimum EH, εk, maximum transmit
power Pmax, maximum iterations Qmax , and tolerance κ.
2: Initialize counter i=0 and variables pc,i,pk,i,wm,i.
3: Initialize the value of the objective function as Obj Fi≫Pmax.
4: repeat
5: Solve problem (25) by using ˜
pc=pc,i,˜
pk=pk,i, and
˜
wm=wm,i. Define the solution pc,{pk,wm, θk}to problem (25)
as p∗
c,np∗
k,w∗
m, θ∗
ko.
6: Set pc,i+1=p∗
c,pk,i+1=p∗
k,wm,i+1=w∗
kand update
counter i←i+1.
7: Evaluate Ob jFi=
pc,i
2+
K
P
k=1
pk,i
2+
M
P
m=1
wm,i
2
8: until |Ob jFi−1−Ob jFi|
Ob jFi−1< κ or i≥Qmax
9: outputs: p∗
c,np∗
k,w∗
k, θ∗
ko.
needed for the convergence of the SCA algorithm because it
provides a limit for the initial values of the slack and power
variables. Problem (25) includes Υ =
K
+
M
+1 variables of
size
N
, and B =4
K
+2
M
+
L
+1 real variables, which leads to
a computational complexity of
O(B+ ΥN)3.5log 1
/
ζ′
with
a solution accuracy of
ζ′>
0 in each iteration [29]. Therefore,
the total computational complexity of the PSO-SCA based
algorithm can be given as OImaxS Qmax(B+ ΥN)3.5.
E. Comparison method based on SDMA
In this subsection, we explain the conventional approach
using the SDMA framework applied to the proposed system
model. In SDMA, message mPS
kfor the k-th PS and message
mID
m
for the
m
-th ID user are encoded into streams
ˆsPS
k
and
ˆsID
m
, respectively. The precoder vectors for streams
ˆsPS
k
and
ˆsID
m
are defined as ˆ
pk∈CN×1and ˆ
wm∈CN×1, respectively.
The SINR at the k-th PS receiver is given by
SINRPS
k,S D =hH
kˆ
pk
2
K
P
i=1,i,khH
kˆ
pi
2+
M
P
m=1hH
kˆ
wm
2+σ2
k+δ2
k
θk
,∀k.(26)
The SINR at the m-th ID user can be expressed as
SINRID
m,S D =gH
mˆ
wm
2
K
P
k=1gH
mˆ
pk
2+
M
P
j=1,j,mgH
mˆ
wj
2+α2
m
,∀m.(27)
The instantaneous achievable rates of
ˆsPS
k
and
ˆsID
m
are
denoted as
RPS
k,S DM A
=
log21+SINRPS
k,S D
and
RID
m,S DM A
=
log21+SINRID
m,S D, respectively.
The energy harvested at the k-th PS user is given by
EPS
k,S DM A =ηk(1−θk)
K
X
i=1hH
kˆ
pi
2+
M
X
m=1hH
kˆ
wm
2+σ2
k
,∀k.
(28)
The interference power at the
l
-th PU user is formulated as
IPl,S DM A =
K
X
k=1qH
lˆ
pk
2+
M
X
m=1qH
lˆ
wm
2,∀l.(29)
Then, the optimization problem of the transmit power at the
SBS, considering the SDMA method, subject to the constraints
ARTICLE OF THE IEEE SYSTEMS JOURNAL 9
of minimum information rate, minimum energy harvested, and
maximum power allowed for interference with the PUs is
expressed as follows:
min
{ˆ
pk,ˆ
wm,θk}
K
X
k=1∥ˆ
pk∥2+
M
X
m=1∥ˆ
wm∥2(30a)
s.t. RPS
k,S DM A ≥γPS
k,∀k(30b)
RID
m,S DM A ≥γI D
m,∀m(30c)
EPS
k,S DM A ≥εk,∀k(30d)
IPl,S DM A ≤ψl,∀l(30e)
0< θk<1,∀k.(30f)
Problem (30) is non-convex, and it needs to be reformulated
following a procedure similar to the one in Section III.C and
further can be solved with the SDR algorithm.
IV. Simulation Results
In this section, we present a performance analysis of the
algorithm based on the PSO and SDR methods (PSO-SDR)
for the proposed system model. The system parameters used in
the simulation are
K
=2
,N
=8
,M
=2
,L
=2
, δ2
k
=
σ2
k
=
α2
m
=
−
60
dBm
,
γPS
k
=
γID
m
=
γ, εk
=
ε
,
ηk
=1
, ψl
=
−
60
dBm
and
Pmax
=40
dBm
. The simulations were carried out on a
computer with 16GB of RAM and an Intel Core i7-6700K
CPU. The channel model for the PS receivers is modeled using
Rician fading, where the signal attenuation was 40 dB and the
channel vectors were the following:
hk=rKR
1+KR
hLOS
k+r1
1+KR
hNLOS
k,(31)
where
KR
denotes the Rician factor equal to 5dB, h
NLOS
k
is the
Rayleigh fading component that follows a circularly symmetric
complex Gaussian random variable considering a covariance
of -40 dB and zero mean. The line-of-sight (LOS) component
is modeled as follows [30]:
hLOS
k=10−4h1e−jπsin(ϕk)e−j2πsin(ϕk)... e−j(N−1)πsin(ϕk)iT,(32)
where the angles from the SBS to the PS receivers are
ϕPS
1
=
30o
and
ϕPS
1
=
50o
, and the angles from the SBS to the ID
receivers are
ϕID
1
=
−25o
and
ϕID
1
=
−60o
. The model of
the channels for ID users,
gm
, follows the same procedure
discussed before, using their respective angles. For channels
ql
from the SBS to the primary users, we used a Rayleigh
fading model with power attenuation of 60dB. We compared
the proposed PSO-SDR algorithm with a scheme based on
SCA and PSO (PSO-SCA) and with an equal power-splitting
(EPS) ratio scheme based on fixing the power split ratios of
the PS users to 0.5, i.e.
θk
=0
.
5. In addition, we compared the
proposed RSMA-based approach with one based on SDMA.
The parameters selected for the PSO algorithm were
S
=10,
w
=0
.
7,
c1
=1
.
494 and
c2
=1
.
494. Fig. 3 illustrates the
convergence behavior of the PSO algorithm, where we plot the
value of the objective function of problem (9) versus the number
of iterations in PSO with a EH requirement of
ε
=
−
15
dBm
and
minimum data rates at the PS and ID receivers of
γ
=7
,
6
,
5
bits/s/Hz. We see that the algorithm achieves a stable value
1 4 7 10 13 16 19 22 25 28 30
Iteration index in PSO
25
25.5
26
26.5
27
27.5
28
28.5
29
29.5
Objective function (dBm)
=7 (bits/s/Hz)
=6 (bits/s/Hz)
=5 (bits/s/Hz)
Fig. 3. PSO convergence behavior under different minimum data rates.
22.6622
22.6624
22.6626
22.6628 =7 (bits/s/Hz)
=5 (bits/s/Hz)
0 10 20 30 40 50 60 70 80 90 100
Number of randomizations
22.129
22.1295
22.13
22.1305
Total transmit power (dBm)
Fig. 4. Gaussian randomization performance in SDR.
for the total transmit power from iteration index 10, and we
can select the maximum number of iterations as Imax =15.
In order to analyze the convergence behavior of the Gaussian
randomization procedure in the SDR-based technique, we
present in Fig. 4 the objective function of problem (9) versus
the number of randomizations,
Numrand
, with a minimum EH
of
ε
=
−
18
dBm
and minimum data rates at the PS and ID
receivers of
γ
=7 and
γ
=5 bits/s/Hz. We observe that the
total transmit power presents steady behavior from the 20
randomizations. Then, we selected
Numrand
=25 for the rest
of the simulations. It is worth highlighting that the difference
between the total transmit power in the first and the 25th
randomizations was around 0.02%, which permits to reduce
the number of randomizations even more if we want to decrease
the computational time.
Next, we present the convergence of the SCA algorithm
described in Table III. Fig. 5 illustrates objective function (9a)
versus the number of iterations of the SCA algorithm with a
minimum EH requirement of
ε
=
−
15
dBm
and minimum data
rates of
γ
=7
,
6
,
5 bits/s/Hz. We observe that the algorithm
converges to steady behavior at around eight iterations, the
first five iterations being where the objective value faced the
bigger changes. Based on this observation, we selected 15
as the maximum number of iterations,
Qmax
. In addition, Fig.
ARTICLE OF THE IEEE SYSTEMS JOURNAL 10
2 4 6 8 10 12 14 16 18 20
Number of iterations in SCA
24
26
28
30
32
34
36
Objective function (dBm)
=7 (bits/s/Hz)
=6 (bits/s/Hz)
=5 (bits/s/Hz)
456
25
25.5
26
26.5
27
Fig. 5. SCA convergence performance with different minimum data rates in
terms of the total transmit power at the SBS.
4 6 8 10 12 14 16 18 20
Number of iterations in SCA
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Relative change of the objective function
=7 (bits/s/Hz)
=6 (bits/s/Hz)
=5 (bits/s/Hz)
7 8 10 12 14 15
0
1
2
310-3
Fig. 6. SCA convergence performance with different minimum data rates in
terms of the relative change of the objective function.
6 portrays the relative change in the total transmit power at
the SBS based on the number of iterations to determine an
appropriate value for the tolerance parameter
κ
. Similar to Fig.
5, we notice that convergence is achieved after around eight
iterations, and we defined
κ
=0
.
0005 as the target value for
the relative change of the objective function.
In the following figures, we compare the proposed PSO-SDR
approach with the PSO-SCA and EPS schemes and with the
SDMA technique. Fig. 7 illustrates the total transmit power at
the SBS as a function of minimum EH,
ε
, with minimum data
rates of
γ
=3 and
γ
=6 bits/s/Hz. We can see that for both
minimum rate requirements, the proposed scheme reaches a
lower total transmit power compared with the traditional SDMA
technique. The reason is that RSMA implements SIC in the
ID module of the PS receiver, which allows eliminating the
interference of the common symbol, thus increasing the SINR.
In the PS receivers, an increase in SINR produces a decrease
in the value of the power split variables, i.e. more power is
available at the EH module, reducing the total transmit power
at the SBS. On the other hand, SDMA does not implement SIC,
and considers all the other messages as interference, which
affects the SINR, leading to an increase in the total transmit
power. The superiority of the PSO-SDR scheme over the EPS
-26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6
Harvested energy, (dBm)
15
20
25
30
35
40
45
Total transmit power (dBm)
PSO-SDR with =3 (bits/s/Hz)
PSO-SCA with =3 (bits/s/Hz)
EPS with =3 (bits/s/Hz)
SDMA with =3 (bits/s/Hz)
PSO-SDR with =6 (bits/s/Hz)
PSO-SCA with =6 (bits/s/Hz)
EPS with =6 (bits/s/Hz)
SDMA with =6 (bits/s/Hz)
-9.04 -9 -8.96
31
31.1
31.2
Fig. 7. Performance comparison of the proposed scheme PSO-SDR versus
PSO-SCA, EPS and SDMA according to the minimum EH at the PS receivers.
method is because the proposed scheme optimizes the value
of the power split variables, leading to an optimal selection to
trade offthe received SINR and EH at the PS users. For the
EPS technique, the power-splitting ratios had a fixed value of
0.5, which does not permit efficient use of the received energy.
Note that both PSO-SDR and PSO-SCA approaches achieved
similar results but with a difference in computational com-
plexity. We compare the computational time of PSO-SDR
and PSO-SCA based on the inner optimization problem (10)
since the outer optimization problem is the same for both
schemes. Table IV shows the computational time of the SDR
and SCA techniques according to the number of antennas in
the SBS. In the case of SDR, we show the results for two
values of the number of randomizations of
Numrand
=25 and
Numrand
=10. In addition, we also include the computational
time for solving the problem (17)-SDR. In the case of SCA,
we include the computational time for solving the problem
(25) and we consider two values for the relative change of
the objective function;
κ
=0
.
0005 and
κ
=0
.
001. According
to the Table IV, we can observe that the algorithms require
more computational time as the number of antennas increases
because of the increase in the dimension of the precoder
variables. The SDR technique achieves a lower computational
time than SCA, where the case with
Numrand
=10 provides
the lowest computational cost. The reason is because the SCA
algorithm solves the problem (25) in each iteration with a
maximum number of iterations
Qmax
and the tolerance value
κ
, which leads to a higher total computational complexity
compared with the SDR scheme, which only needs to solve
one SDP problem along with linear optimization problems for
the Gaussian randomization technique. Moreover, it is worth
highlighting that the computational time in the proposed PSO-
SDR can be further reduced by selecting a lower number of
randomizations with a small reduction in performance.
Fig. 8 shows the total transmit power in the SBS as a
function of the minimum data rate at the PS and ID users,
with a minimum EH of
ε
=
−
15 dBm and
ε
=
−
18 dBm.
For the data rate requirements of 11 and 12 bits/s/Hz, we
have increased the maximum number of iterations for the PSO
ARTICLE OF THE IEEE SYSTEMS JOURNAL 11
TABLE IV
Computational time comparison (sec)between the SDR and SCA techniques.
Number of
Antennas
SDR SCA
Numrand κ
25 10 P. (17)-
SDR 0.0005 0.001 P. (25)
6 6.03 2.65 0.59 15.13 14.11 1.87
8 6.14 2.71 0.62 15.55 14.62 1.89
12 6.51 2.99 0.87 16.36 15.12 1.97
15 6.81 3.58 1.25 18.66 17.60 2.01
30 7.92 4.74 2.40 21.64 19.68 2.24
40 10.14 7.53 5.08 23.87 22.45 2.56
1 2 3 4 5 6 7 8 9 10 11 12
Data rate, PS,ID (bits/s/Hz)
22
24
26
28
30
32
34
36
Total transmit power (dBm)
PSO-SDR with =-15 (dBm)
PSO-SCA with =-15 (dBm)
EPS with =-15 (dBm)
SDMA with =-15 (dBm)
PSO-SDR with =-18 (dBm)
PSO-SCA with =-18 (dBm)
EPS with =-18 (dBm)
SDMA with =-18 (dBm)
Fig. 8. Performance comparison among the proposed scheme PSO-SDR,
PSO-SCA, EPS and SDMA according to the minimum data rate at the PS
and ID receivers.
algorithm by
Imax
=25. It is observed that the proposed PSO-
SDR scheme outperforms the other SDMA and EPS schemes
and has results similar to PSO-SCA. The one-layer of SIC at
the users in RSMA permits to eliminate the interference of
the common stream and treats as noise the interference from
the other private streams. Subsequently, the achievable data
rate is increased at the PS and ID receivers, which reduces
the required transmit power at the SBS. The RSMA scheme
is a more general approach that has SDMA as a special case,
which was investigated in [9], [15]. The latter statement is
supported as following: since the SDMA scheme is obtained
by setting to zero the precoder vector of the common stream,
pc
, in the RSMA approach, each user message is directly
encoded into a private stream. Then, SDMA can not decode
the interference from other user’s messages and these are treated
as noise, which results in reducing the achievable rate at the
users and increasing the required transmit power at the SBS.
For higher data rates, infeasible issues may exist for some
channel realizations under the considered scenarios since it has
to satisfy a maximum power constraint and a maximum power
threshold for interference with the PU users, which limits the
achievable rate of the users.
Fig. 9 shows the transmit power as a function of the
number of antennas at the SBS. We can see that transmit
power is reduced as we increase the number of antennas,
since we take advantage of the extra degrees of freedom with
more antennas. In addition, we can see that the proposed
PSO-SDR schemes outperform the other comparative EPS,
6 8 10 12 14 16 18 20 22 24 26 28
Number of transmitting antennas
15
20
25
30
35
40
45
Total transmit power (dBm)
PSO-SDR with =-12 (dBm)
PSO-SCA with =-12 (dBm)
EPS with =-12 (dBm)
SDMA with =-12 (dBm)
PSO-SDR with =-18 (dBm)
PSO-SCA with =-18 (dBm)
EPS with =-18 (dBm)
SDMA with =-18 (dBm)
Fig. 9. Performance comparison of the proposed scheme PSO-SDR versus
PSO-SCA, EPS and SDMA according to the number of antennas at the SBS.
SCA, and SDMA schemes. Finally, we can conclude that the
RSMA approach provides a significant improvement over the
conventional SDMA technique, where RSMA can allow a
reduction of around 4dBm to 6dBm in the total transmit power
at the SBS, in comparison with the SDMA framework.
V. Conclusion
In this paper, we proposed a multi-user MISO SWIPT
cognitive radio system with RSMA, consisting of several
PS and ID receivers. We investigated total transmit power
minimization at the SBS under the constraints of a minimum
information rate at the PS and ID receivers, minimum required
EH at the PS users, maximum available power at the SBS,
and a maximum power level permitted for interference with
the primary users. The proposed approximate optimal solution
is based on the PSO algorithm and the SDR technique with
Gaussian randomization. The numerical simulations confirm the
superiority of the proposed technique based on SDR, compared
with the EPS and SDMA schemes, and obtains a reduction
in computational time, compared to the SCA method for the
studied scenarios. In addition, the proposed PSO-SDR method
can reach a stable condition and converges after 10 iterations
of the PSO algorithm. Finally, we confirmed the advantage of
the novel RSMA method in comparison with the traditional
SDMA technique, where the reduction of the transmit power
at the SBS permitted by the RSMA method is around 4dBm to
6dBm compared with SDMA in the studied scenario varying
the EH requirement.
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Mario R. Camana received the B.E. degree in
electronics and telecommunications engineering from
the Escuela Polit
´
ecnica Nacional (EPN), Quito, in
2016. He is currently a graduate research student
at the School of Electrical Engineering, University
of Ulsan, Ulsan, South Korea. His research interests
include machine learning, optimizations, and MIMO
communications.
Carla E. Garcia received the B.S degree in elec-
tronics and telecommunications engineering from the
Escuela Polit
´
ecnica Nacional (EPN), Quito, in 2016.
She is currently a graduate research student at the
School of Electrical Engineering, University of Ulsan,
Ulsan, South Korea. Her main research interests are
machine learning, MIMO communications, NOMA
and optimizations.
Insoo Koo received the BE from Kon-Kuk Uni-
versity, Seoul, Korea, in 1996, and an MSc and
a PhD from the Gwangju Institute of Science and
Technology (GIST), Gwangju, Korea, in 1998 and
2002, respectively. From 2002 to 2004, he was with
the Ultrafast Fiber-Optic Networks Research Center,
GIST, as a Research Professor. In 2003, he was a
Visiting Scholar with the Royal Institute of Science
and Technology, Stockholm, Sweden. In 2005, he
joined the University of Ulsan, Ulsan, Korea, where
he is currently a Full Professor. His current research
interests include spectrum sensing issues for CRNs, channel and power
allocation for cognitive radios (CRs) and military networks, SWIPT MIMO
issues for CRs, MAC and routing protocol design for UW-ASNs, and relay
selection issues in CCRNs.
