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JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012 1
A SCA and Relaxation Based Energy Efficiency
Optimization for Multi-User IRS-Assisted NOMA
Networks
Tianqi Wang, Fang Fang, Member, IEEE and Zhiguo Ding, Fellow, IEEE
Abstract—Recently, the fifth-generation (5G) wireless network
has been widely deployed throughout the world. The application
value of 5G has also gradually attracted attention in academia,
industry, and even medicine. Non-orthogonal multiple access
(NOMA) and intelligent reflecting surface (IRS) are very
advanced technology in 5G network. [1] [2] the high spectral
efficiency allows NOMA to produce higher data rate than
orthogonal multiple access (OMA). Besides, Intelligent reflecting
surface (IRS) is a type of smart equipment which can tune
the amplitude and phase shift of the incident signals and
then improve the performance of a network. In this paper,
we will mainly build the IRS-Assisted NOMA network and
design an algorithm to achieve the trade-off between sum data
rate and power under a multiple-input single-output (MISO)
downlink circumstance. [2] [3] Besides, the successive convex
approximation (SCA) and relaxation will be also applied in this
paper.
Keywords: 5G, IRS, NOMA, MISO, Convex Optimization
I. INTRODUCTION
Nowadays, although there has been a great progress in
5G technology, 5G is still facing some technical challenges.
Firstly, the millimeter wave (mmWave) produced by 5G Base
Station (BS) is hard to penetrate the obstructions due to
its physical characteristic. To guarantee the wide coverage
of 5G signal, humans have to build more hardware with
high complexity and cost of hardware as well as the power
consumption. Based on this, it is imperative to study how to
find out an energy-saving and low- cost solution for the future
5G/B5G wireless network.
Faced with the high power consumption issue of 5G networks,
various advanced technologies are applied to achieve the high
data rate as well as low power consumption requirement.
None orthogonal multiple access (NOMA) has been treated
as the key technology in the 5G network because of its high
spectral efficiency, which produces a higher data rate. [4] [5]
Intelligent reflected surface (IRS) has also been considered as
an innovative and potential technology. [6] [7] IRS consists
of various passive reflection elements, these elements are able
to modify the amplitude as well as phase shift of the incident
T. Wang is with the School of Electrical and Electronic Engineer-
ing, University of Manchester, Manchester, UK, e-mail: (tianqi.wang-
6@postgrad.manchester.ac.uk).
F. Fang with Department of Engineering, Durham University, Durham, UK,
email:(fang.fang@durham.ac.uk)
Z. Ding is with the School of Electrical and Electronic Engineering, Univer-
sity of Manchester, Manchester, UK, e-mail:(zhiguo.ding@manchester.ac.uk)
NO
MA
NOMA
NOMANO
MA
NOMABS
U1
UK
IRS
………
U2
Fig. 1: MISO IRS-NOMA Downlink System
signal to achieve the 3D reflected beamforming. Secondly, IRS
is also a low power consumption device and it is convenient
to be equipped in the service dead zone. [7]
Since blindly pursuing data rate of a network is not an
economical and efficient scenario in modern 5G network, [8]
For achieving the trade-off scenario between the data rate and
power consumption, we introduce a power efficiency variable
which is the ratio between the total data rate and power
consumption. [9] Recently, some work has been completed
to demonstrate the feasibility of power efficiency optimization
with two users IRS-assisted NOMA system. [10] [11] In this
paper, different from the above work, we are aimed to extend
the above work to multi-user case and design an algorithm to
obtain the maximum energy efficiency in a downlink MISO
IRS-assisted NOMA network. Meanwhile, the direct link
between BS and users will also be taken into consideration.
II. SYSTEM MODEL
In this paper, inspired by [10] [12], we consider a downlink
MISO IRS-NOMA network with a single base station (BS)
and multiple single-antenna users. As is shown in Fig. 1, The
BS is equipped with M antennas and the number of users is
K while the IRS consists of N passive reflecting elements
to assist the transmission between the BS and users. What is
worth mentioning is that the number of users should be greater
than the number of antennas. Otherwise, there is no need to
apply NOMA technique. Besides, another difference from the
work in [10] is that BS can communicate with users through
both IRS and direct link between the BS and users in this
model, G ∈ CN×M and hHk,r ∈ C1×N represent the channel
gains from BS to IRS and IRS to users respectively. The direct
channel gain from the BS to users is denoted by hHk,d ∈ C1×M
, which k represents the kth user and k = 1, · · · ,K. Next we
JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012 2
denote Θ = diag(β1ejθ1 , · · · , βNejθN ) as the phase reflection
matrix, where βn is the amplitude reflection coefficient and θn
is the reflection phase shift. In this model, all the amplitude
reflection coefficients are assumed to be fixed, meaning that
βn = 1,∀n. Consider the channel strength among these users
is that |hH1 |, · · · , |hHK |, where |hHk | = |hHk,rΘG + hHk,d|. In
terms of the above descriptions, the received signal at the kth
user can be written as [13] [14]
K∑
k=1
hHk wksk + nk (1)
Where wk ∈ CN×1 represents the beamforming vector from
the BS to the kth user. nk ∼ CN (0, σ2) represents the additive
white Gaussian noise. In downlink IRS-Assisted NOMA sys-
tem, the effect of successive interference cancellation (SIC)
is highly related to the decoding order, meaning that when
decoding Uk, the interference from the users who has been
decoded before could be cancelled out. In fact, the decoding
order depends on the users’ capability. In this model, con-
sider the decoding order of the users is (U1, · · · , UK−1, UK),
the signal-interference-plus-noise ratio(SINR) when decoding
Uk’s signal at Ui (k = 1, · · · ,K and i = 1, · · · ,K, k ≤ i, k
and i cannot be equal to K simultaneously) can be shown as
SINRk,i =
|hHi wk|2
K∑
l=k+1
|hHi wl|2 + σ2
, (2)
Due to the fact that the UK is the last decoded user, it is able
to eliminate the interference from all the users. Therefore, the
SINR of UK can be expressed as
SINRK =
|hHKwK |2
σ2
(3)
In this case, we could define the SINR of the Uk as
SINRk = min{SINRk,i} (4)
Then, the data rate which the kth user is able to achieve can
be written as
Rk = log2(1 + SINRk) (5)
III. PROBLEM FORMULATION
The objective of the problem is to maximize the power
efficiency variable which is the ratio between sum data rate
and power consumption. Initially, we define the internal circuit
power consumption of the BS is Pc. and set a parameter
η ∈ [0, 1] to indicate the power amplifier coefficient at the
BS. Besides, Some constraints should also be taken into
consideration. Firstly, the data rate for each user has to be
no less than a minimum data rate Rk,min, k = 1, · · · ,K to
satisfy the quality of service (QoS) requirement. Secondly, the
total power consumption of beamforming at the BS for users is
not supposed to surpass a maximum threshold Pmax. Besides,
the reflection phase shift should be in the range of [0, 2pi].
After these, the power efficiency optimization problem in this
network can be formulated as
max
Θ,w1,··· ,wK
K∑
k=1
Rk
1
η
K∑
k=1
‖wk‖2 + Pc
(6a)
s. t. Rk ≥ Rk,min, k = 1, · · · ,K (6b)
K∑
k=1
|wk|2 ≤ Pmax (6c)
θn ∈ [0, 2pi], n = 1, · · · , N (6d)
However, (P6) is non-convexity problem due to the fact that
the Rk is not convex with not only Θ and w. Besides, until
now, QoS requirement (6b) is also not convex, making this
problem more complicated. In order to find out the optimal
solution of this problem, we have to modify it to a convex
form.
IV. PROPOSED OPTIMIZATION METHODOLOGY
A. Beamforming Optimization
In this section, we mainly focus on the design of a feasible
algorithm which can effectively find the suboptimal solution
of (P6). Although applying the Karush-Kuhn-Tucker (KKT)
conditions directly is quite straightforward to solve the non-
convex problems, it is not a wise strategy under this circum-
stance due to the fact that plenty of constraints caused by
multiple users can still bring great complexity if solving it with
the KKT conditions. Thus, other methods such as introducing
variables to slack this non-convex problem and successive
convex algorithm (SCA) are suggested to be utilized in the
algorithm design to get the suboptimal solution.
Obviously, the objective function (6a) of (P6) is not convex
with respect to the variables Θ and w. Besides, due to
the fact that the vector wk still has the phase shift, there
is no doubt that the complexity of the problem could be
enhanced if solving the joint optimization problem with w
and Θ. Therefore, in this paper, the strategy to reduce the
complexity is that fix the Θ vector temporarily and only
consider the impact of beamforming on the power-efficiency
problem. Therefore, In this case, a slack variable α could be
introduced to the (P6) and the optimization problem can be
transferred as
max
w1,··· ,wK ,α
α (7a)
s. t.
K∑
k=1
Rk
1
η
K∑
k=1
‖wk‖2 + Pc
≥ α (7b)
(6b), (6c) (7c)
Although the objective function of (P7) has been transferred
to affine, the fact that the constraint (7b) is not a convex set still
exists. In order to deal with it, we have to introduce another
slack variable to further transfer this non-convex constraint to
another form. Select a variable ρ such that
K∑
k=1
Rk ≥ αρ, (8)
JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012 3
Therefore, the constraint (7b) will be replaced by
1
η
K∑
k=1
‖wk‖2 + Pc ≤ ρ (9)
Till now, based on the constraints transformation above, it is
not hard to find that the constraint (8) is still non-convex
set. Next step, we firstly create a new variable vector γ =
[γ1, · · · , γK ]T to slack (8), which is
K∑
k=1
log2(γk) ≥ αρ (10a)
1 + SINRk ≥ γk (10b)
Subsequently, we apply another variable vector named δ =
[δ1, · · · , δK ]T to replace the constraint (10a) such that
K∑
k=1
δk ≥ αρ (11a)
γk ≥ 2δk (11b)
At this point, we can summarize the relationship between these
slack variables introduced based on (8), (10a), (11a) and (11b),
which is
K∑
k=1
log2(1 + SINRk) ≥
K∑
k=1
log2(γk) ≥
K∑
k=1
δk ≥ αρ (12)
Considering the expression of Signal-interference-noise ratio
(SINR) is a form of fraction, it is alternative to introduce a
series of variables with the form of Ωk,i, which k ≤ i, k, i =
1, · · · ,K to deal with the SINR expression such that
|hHi wk|2 ≥ (γk − 1)Ωk,i (13a)
K∑
l=k+1
|hHi wl|2 + σ2 ≤ Ωk,i (13b)
However, (13a) and (13b) are not applicable for the situation
that i and k are equal to K simultaneously. When both of i and
k are equal to K, (13a) and (13b) are supposed to be written
as
|hHKwK |2 ≥ (γK − 1)ΩK,K (14a)
σ2 ≤ ΩK,K (14b)
In previous work of [10], to deal with the left hand side of
non-convex form of (13a), an arbitrary phase rotation of the
beamforming vector was introduced to make the imaginary
part of hHi wk equal to zero only considering two users
in the model. However, this method cannot be applied in
this paper because for multiple users case, once an arbitrary
phase rotation of wk is calculated with a given i to satisfy
={hHi wk}=0, we cannot guarantee that this arbitrary phase
rotation can still make the imaginary part equal to zero with
another given i. Therefore, At this stage, we apply several
variables in the form of wRek,i and w
Im
k,i , which makes
wRek,i = <{hHi wk} (15a)
wImk,i = ={hHi wk} (15b)
Although (11a) and (14a) are still not convex sets, instead
of applying new slack variables again, other techniques are
feasible to make them convex. In this paper, we use successive
convex approximation (SCA) to deal with this type non-convex
constraints. SCA algorithm is actually a very useful mathemat-
ical tool in solving the non-convex problem. The main idea
of SCA is to find out the problem whose optimal value is
close to that of the original problem with several iterations.
Specifically, we select the First-order Taylor approximation to
extend the non-convex functions to affine. As the number of
iterations grows, the result will be more and more approached
to the optimal. [15] In this paper, the constraints (11a), (13a)
can become affine when applying SCA such that
K∑
k=1
δk ≥ α(t)ρ(t) + ρ(t)(α− α(t)) + α(t)(ρ− ρ(t)), (16)
((wRek,i)
(t))2 + ((wImk,i )
(t))2 + 2(wRek,i)
(t)(wRek,i − (wRek,i)(t))
+ 2(wImk,i )
(t)(wImk,i − (wImk,i )(t)) ≥
((γk)
(t) − 1)(Ωk,i)(t) + (Ωk,i)(t)(γk − (γk)(t))
+ ((γk)
(t) − 1)(Ωk,i − (Ωk,i)(t))
(17)
t represents the tth iteration. Then, we can explore the
convexity of (6b). Apparantly, it is obvious to figure that (6b)
is nonconvex in terms of the expression directly. Now, we
rewrite this constraint as
|hHi wk|2
K∑
l=k+1
|hHi wl|2 + σ2
≥ 2Rk,min − 1, (18)
Which Rk,min is constant. After that, this constraint can be
reformulated with SCA as
((wRek,i)
(t))2 + ((wImk,i )
(t))2 + 2(wRek,i)
(t)(wRek,i − (wRek,i)(t))
+ 2(wImk,i )
(t)(wImk,i − (wImk,i )(t)) ≥
(2Rk,min − 1)(
K∑
l=k+1
|hH,i wl|2 + σ2)
(19)
Which k + 1 ≤ l ≥ K Obviously, the constraint (19) is
a convex set because it satisfies the form of Second-Order-
Cone (SOC). Now, after the above discussion, the final convex
problem can be formulated as
max
w1,··· ,wK ,α,ρ,γ,δ,Ω,wRe,wIm
α (20a)
s. t. (6c), (9), (11b), (15a), (15b), (16), (17), (19) (20b)
Where t represents tth iteration of the first-order Taylor
extension.
B. Phase Shift Optimization
The following problem is about the phase shift optimization
in this model. In previous sections, we have solved the beam-
forming optimization with the fixed phase shift. Since now the
beamforming has been optimized by the existing algorithm,
meaning that the beamforming vectors w are fixed now.
JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012 4
Therefore, the denominator of the objective function (6a) is a
constant and the power-efficiency problem can be transferred
to a sum-rate optimization problem which be formulated as
max
Θ
K∑
k=1
Rk (21a)
s. t. Rk ≥ Rk,min, k = 1, · · · ,K (21b)
θn ∈ [0, 2pi], n = 1, · · · , N (21c)
In the work of [11], the phase shift optimization is dealt with
semi-definite relaxation (SDR). In this paper, we still optimize
it through SCA. Similar to (P6), (21a) is still not a convex
problem because Rk is not convex with respect to Θ. To deal
with it, we can transform the problem formulation by splitting
the variable θn from the received signal expression of decoded
Uk at Ui. Define a variable vector v = [v1, · · · , vN ]H =
[ejθ1 , · · · , ejθN ]H and fixed vectors ak,i = diag(hHi,r)Gwfk
which k ≤ i ≤ K and wfk denotes the optimized beamforming
vector, which is fixed. Thus, the SINR for the kth user when
decoding at ith user (k ≤ i) can be expressed as
SINRk,i =
|vHak,i + hHi,dwfk |2
K∑
l=k+1
|vHal,i + hHi,dwfl |2 + σ2
(22)
For the case of k = i = K, the SINR expression can be
written as
SINRK =
|vHaK,K + hHK,dwfK |2
σ2
(23)
Similar to process of beamforming optimization, we define a
slack variable q such that
K∑
k=1
Rk ≥ q (24)
After this, we further introduce a new vector λ =
[λ1, · · · , λK ], which
K∑
k=1
log2(λk) ≥ q (25a)
1 + SINRk ≥ λk (25b)
After this, we apply another variable vector named µ =
[µ1, · · · , µK ] such that,
K∑
k=1
µk ≥ q (26a)
λk ≥ 2µk (26b)
After the above processes, applying the similar method to deal
with the SINR part, several variable ϕk,i which has the same
purpose with Ωk,i are applied as
|vHak,i + hHi,dwfk |2 ≥ (λk − 1)ϕk,i (27a)
K∑
l=k+1
|vHal,i + hHi,dwfl |2 + σ2 ≤ ϕk,i (27b)
Fig. 2: Energy efficiency versus the number of reflecting
elements
For the situation of i = k = K, the expressions should be
written as
|vHaK,K + hHK,dwfK |2 ≥ (λK − 1)ϕK,i (28a)
σ2 ≤ ϕK,K (28b)
Then, next step is to apply first Taylor to make (27a) convex.
Introduce a series of variables vRek,i and v
Im
k,i , such that
vRek,i = <{(v)Hak,i + hHi,dwfk} (29a)
vImk,i = ={(v)Hak,i + hHi,dwfk} (29b)
Then, extend (29a) and (29b) with SCA,
((vRek,i)
(t))2 + ((vImk,i )
(t))2 + 2(vRek,i)
(t)(vRek,i − (vRek,i)(t))
+ 2(vImk,i )
(t)(vImk,i − (vImk,i )(t)) ≥
((λk)
(t) − 1)(ϕk,i)(t) + (ϕk,i)(t)(λk − (λk)(t))
+ ((λk)
(t) − 1)(ϕk,i − (ϕk,i)(t))
(30)
To deal with the QoS contraint, we can still apply the same
technique to make it become a second-order cone expression,
((vRek,i)
(t))2 + ((vImk,i )
(t))2 + 2(vRek,i)
(t)(vRek,i − (vRek,i)(t))
+ 2(vImk,i )
(t)(vImk,i − (vImk,i )(t)) ≥
(2Rk,min − 1)(
K∑
l=k+1
|(v)Hal,i + hHi,dwfl |2
+ σ2)
(31)
After the transformation of (P20), the final form of this
phase shift optimization can be expressed as
max
v,q,λ,µ,ϕ,vRe,vIm
q (32a)
s. t. (21c), (26b), (29a), (29b), (30), (31) (32b)
V. SIMULATION RESULT
In this section, we will mainly show some simulation
results about the algorithm to illustrate the performance of the
JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012 5
Fig. 3: Energy efficiency versus internal circuit power
joint optimization scheme in this paper. Firstly, consider the
distance between BS and IRS is 10 m and users are located
in the area that is 16-20 m away from both BS and IRS.
The channels from BS to IRS and from IRS to users are
Racian fading channels. The LoS component GLoS and nLoS
component GnLoS are both set to 2. The channels from BS
to users obey Rayleigh distribution. The path loss exponents
from BS to IRS, IRS to users and BS to users are all set
to 2.5. Without the loss of generality, the minimal SINR
requirement for all users and Pmax are set to -10 dB and -20
dB respectively. Besides, the AWGN noise σ2 is -60 dB and
the power amplifier coefficient η is set to 0.8. The parameters
mentioned above are all fixed during the whole process of this
simulation.
In Fig. 2, we present the effect of the number of reflecting
elements on energy efficiency value. Under this circumstance,
the internal circuit power consumption Pc is set to -20 dB.
For comprehensively proving the rationality of the result, we
select 4 different cases, which are shown in the Fig. 2. Firstly,
it is obvious to find that the value of energy efficiency value
increases as the number of elements grows no matter which
case is selected. Besides, initially, energy efficiency increases
very rapidly. However, the curve for each case is approached
to saturation gradually as the number of elements grows.
In Fig. 3, we show the effect of the internal circuit consump-
tion on energy efficiency value. We set the number of reflecting
elements is 10 and select the same 4 cases. Then, discover the
value of energy efficiency when the circuit power is from -20
dB to 10 dB respectively. From the curve, it can be found that
the energy efficiency value decreases as the internal circuit
power increases. Besides, we can see that when Pc rises from
-20 dB to -10 dB, there is a sharp decrease in the energy
efficiency value. The curve becomes flatter and flatter and
tends to zero as the Pc further rises.
VI. CONCLUSION
In this paper, we mainly try to find an optimization method
to enhance the value of energy efficiency in a MISO IRS-
Assisted NOMA system with multi-user. Although it is hard
to find out the optimal solution of the original problem, we
can still relax the original problem and get the suboptimal
value of it. Except for problem relaxation, the successive
convex approximation is still a very useful mathematical tool
to deal with the non-convex functions. Besides, the joint
optimization method can effectively decrease the complexity
of the problem. In this paper, assuming the phase shift is
fixed, we firstly optimize beamforming. After obtaining the
beamforming vectors, we then optimize phase shift. as the
number of iterations grows, the value obtained can be more
and more approached to the optimal. Next, it is also available
for researchers to extend this network to a MIMO IRS-
Assisted NOMA system and try to find out some new methods
and algorithms to solve it.
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学霸联盟
A SCA and Relaxation Based Energy Efficiency
Optimization for Multi-User IRS-Assisted NOMA
Networks
Tianqi Wang, Fang Fang, Member, IEEE and Zhiguo Ding, Fellow, IEEE
Abstract—Recently, the fifth-generation (5G) wireless network
has been widely deployed throughout the world. The application
value of 5G has also gradually attracted attention in academia,
industry, and even medicine. Non-orthogonal multiple access
(NOMA) and intelligent reflecting surface (IRS) are very
advanced technology in 5G network. [1] [2] the high spectral
efficiency allows NOMA to produce higher data rate than
orthogonal multiple access (OMA). Besides, Intelligent reflecting
surface (IRS) is a type of smart equipment which can tune
the amplitude and phase shift of the incident signals and
then improve the performance of a network. In this paper,
we will mainly build the IRS-Assisted NOMA network and
design an algorithm to achieve the trade-off between sum data
rate and power under a multiple-input single-output (MISO)
downlink circumstance. [2] [3] Besides, the successive convex
approximation (SCA) and relaxation will be also applied in this
paper.
Keywords: 5G, IRS, NOMA, MISO, Convex Optimization
I. INTRODUCTION
Nowadays, although there has been a great progress in
5G technology, 5G is still facing some technical challenges.
Firstly, the millimeter wave (mmWave) produced by 5G Base
Station (BS) is hard to penetrate the obstructions due to
its physical characteristic. To guarantee the wide coverage
of 5G signal, humans have to build more hardware with
high complexity and cost of hardware as well as the power
consumption. Based on this, it is imperative to study how to
find out an energy-saving and low- cost solution for the future
5G/B5G wireless network.
Faced with the high power consumption issue of 5G networks,
various advanced technologies are applied to achieve the high
data rate as well as low power consumption requirement.
None orthogonal multiple access (NOMA) has been treated
as the key technology in the 5G network because of its high
spectral efficiency, which produces a higher data rate. [4] [5]
Intelligent reflected surface (IRS) has also been considered as
an innovative and potential technology. [6] [7] IRS consists
of various passive reflection elements, these elements are able
to modify the amplitude as well as phase shift of the incident
T. Wang is with the School of Electrical and Electronic Engineer-
ing, University of Manchester, Manchester, UK, e-mail: (tianqi.wang-
6@postgrad.manchester.ac.uk).
F. Fang with Department of Engineering, Durham University, Durham, UK,
email:(fang.fang@durham.ac.uk)
Z. Ding is with the School of Electrical and Electronic Engineering, Univer-
sity of Manchester, Manchester, UK, e-mail:(zhiguo.ding@manchester.ac.uk)
NO
MA
NOMA
NOMANO
MA
NOMABS
U1
UK
IRS
………
U2
Fig. 1: MISO IRS-NOMA Downlink System
signal to achieve the 3D reflected beamforming. Secondly, IRS
is also a low power consumption device and it is convenient
to be equipped in the service dead zone. [7]
Since blindly pursuing data rate of a network is not an
economical and efficient scenario in modern 5G network, [8]
For achieving the trade-off scenario between the data rate and
power consumption, we introduce a power efficiency variable
which is the ratio between the total data rate and power
consumption. [9] Recently, some work has been completed
to demonstrate the feasibility of power efficiency optimization
with two users IRS-assisted NOMA system. [10] [11] In this
paper, different from the above work, we are aimed to extend
the above work to multi-user case and design an algorithm to
obtain the maximum energy efficiency in a downlink MISO
IRS-assisted NOMA network. Meanwhile, the direct link
between BS and users will also be taken into consideration.
II. SYSTEM MODEL
In this paper, inspired by [10] [12], we consider a downlink
MISO IRS-NOMA network with a single base station (BS)
and multiple single-antenna users. As is shown in Fig. 1, The
BS is equipped with M antennas and the number of users is
K while the IRS consists of N passive reflecting elements
to assist the transmission between the BS and users. What is
worth mentioning is that the number of users should be greater
than the number of antennas. Otherwise, there is no need to
apply NOMA technique. Besides, another difference from the
work in [10] is that BS can communicate with users through
both IRS and direct link between the BS and users in this
model, G ∈ CN×M and hHk,r ∈ C1×N represent the channel
gains from BS to IRS and IRS to users respectively. The direct
channel gain from the BS to users is denoted by hHk,d ∈ C1×M
, which k represents the kth user and k = 1, · · · ,K. Next we
JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012 2
denote Θ = diag(β1ejθ1 , · · · , βNejθN ) as the phase reflection
matrix, where βn is the amplitude reflection coefficient and θn
is the reflection phase shift. In this model, all the amplitude
reflection coefficients are assumed to be fixed, meaning that
βn = 1,∀n. Consider the channel strength among these users
is that |hH1 |, · · · , |hHK |, where |hHk | = |hHk,rΘG + hHk,d|. In
terms of the above descriptions, the received signal at the kth
user can be written as [13] [14]
K∑
k=1
hHk wksk + nk (1)
Where wk ∈ CN×1 represents the beamforming vector from
the BS to the kth user. nk ∼ CN (0, σ2) represents the additive
white Gaussian noise. In downlink IRS-Assisted NOMA sys-
tem, the effect of successive interference cancellation (SIC)
is highly related to the decoding order, meaning that when
decoding Uk, the interference from the users who has been
decoded before could be cancelled out. In fact, the decoding
order depends on the users’ capability. In this model, con-
sider the decoding order of the users is (U1, · · · , UK−1, UK),
the signal-interference-plus-noise ratio(SINR) when decoding
Uk’s signal at Ui (k = 1, · · · ,K and i = 1, · · · ,K, k ≤ i, k
and i cannot be equal to K simultaneously) can be shown as
SINRk,i =
|hHi wk|2
K∑
l=k+1
|hHi wl|2 + σ2
, (2)
Due to the fact that the UK is the last decoded user, it is able
to eliminate the interference from all the users. Therefore, the
SINR of UK can be expressed as
SINRK =
|hHKwK |2
σ2
(3)
In this case, we could define the SINR of the Uk as
SINRk = min{SINRk,i} (4)
Then, the data rate which the kth user is able to achieve can
be written as
Rk = log2(1 + SINRk) (5)
III. PROBLEM FORMULATION
The objective of the problem is to maximize the power
efficiency variable which is the ratio between sum data rate
and power consumption. Initially, we define the internal circuit
power consumption of the BS is Pc. and set a parameter
η ∈ [0, 1] to indicate the power amplifier coefficient at the
BS. Besides, Some constraints should also be taken into
consideration. Firstly, the data rate for each user has to be
no less than a minimum data rate Rk,min, k = 1, · · · ,K to
satisfy the quality of service (QoS) requirement. Secondly, the
total power consumption of beamforming at the BS for users is
not supposed to surpass a maximum threshold Pmax. Besides,
the reflection phase shift should be in the range of [0, 2pi].
After these, the power efficiency optimization problem in this
network can be formulated as
max
Θ,w1,··· ,wK
K∑
k=1
Rk
1
η
K∑
k=1
‖wk‖2 + Pc
(6a)
s. t. Rk ≥ Rk,min, k = 1, · · · ,K (6b)
K∑
k=1
|wk|2 ≤ Pmax (6c)
θn ∈ [0, 2pi], n = 1, · · · , N (6d)
However, (P6) is non-convexity problem due to the fact that
the Rk is not convex with not only Θ and w. Besides, until
now, QoS requirement (6b) is also not convex, making this
problem more complicated. In order to find out the optimal
solution of this problem, we have to modify it to a convex
form.
IV. PROPOSED OPTIMIZATION METHODOLOGY
A. Beamforming Optimization
In this section, we mainly focus on the design of a feasible
algorithm which can effectively find the suboptimal solution
of (P6). Although applying the Karush-Kuhn-Tucker (KKT)
conditions directly is quite straightforward to solve the non-
convex problems, it is not a wise strategy under this circum-
stance due to the fact that plenty of constraints caused by
multiple users can still bring great complexity if solving it with
the KKT conditions. Thus, other methods such as introducing
variables to slack this non-convex problem and successive
convex algorithm (SCA) are suggested to be utilized in the
algorithm design to get the suboptimal solution.
Obviously, the objective function (6a) of (P6) is not convex
with respect to the variables Θ and w. Besides, due to
the fact that the vector wk still has the phase shift, there
is no doubt that the complexity of the problem could be
enhanced if solving the joint optimization problem with w
and Θ. Therefore, in this paper, the strategy to reduce the
complexity is that fix the Θ vector temporarily and only
consider the impact of beamforming on the power-efficiency
problem. Therefore, In this case, a slack variable α could be
introduced to the (P6) and the optimization problem can be
transferred as
max
w1,··· ,wK ,α
α (7a)
s. t.
K∑
k=1
Rk
1
η
K∑
k=1
‖wk‖2 + Pc
≥ α (7b)
(6b), (6c) (7c)
Although the objective function of (P7) has been transferred
to affine, the fact that the constraint (7b) is not a convex set still
exists. In order to deal with it, we have to introduce another
slack variable to further transfer this non-convex constraint to
another form. Select a variable ρ such that
K∑
k=1
Rk ≥ αρ, (8)
JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012 3
Therefore, the constraint (7b) will be replaced by
1
η
K∑
k=1
‖wk‖2 + Pc ≤ ρ (9)
Till now, based on the constraints transformation above, it is
not hard to find that the constraint (8) is still non-convex
set. Next step, we firstly create a new variable vector γ =
[γ1, · · · , γK ]T to slack (8), which is
K∑
k=1
log2(γk) ≥ αρ (10a)
1 + SINRk ≥ γk (10b)
Subsequently, we apply another variable vector named δ =
[δ1, · · · , δK ]T to replace the constraint (10a) such that
K∑
k=1
δk ≥ αρ (11a)
γk ≥ 2δk (11b)
At this point, we can summarize the relationship between these
slack variables introduced based on (8), (10a), (11a) and (11b),
which is
K∑
k=1
log2(1 + SINRk) ≥
K∑
k=1
log2(γk) ≥
K∑
k=1
δk ≥ αρ (12)
Considering the expression of Signal-interference-noise ratio
(SINR) is a form of fraction, it is alternative to introduce a
series of variables with the form of Ωk,i, which k ≤ i, k, i =
1, · · · ,K to deal with the SINR expression such that
|hHi wk|2 ≥ (γk − 1)Ωk,i (13a)
K∑
l=k+1
|hHi wl|2 + σ2 ≤ Ωk,i (13b)
However, (13a) and (13b) are not applicable for the situation
that i and k are equal to K simultaneously. When both of i and
k are equal to K, (13a) and (13b) are supposed to be written
as
|hHKwK |2 ≥ (γK − 1)ΩK,K (14a)
σ2 ≤ ΩK,K (14b)
In previous work of [10], to deal with the left hand side of
non-convex form of (13a), an arbitrary phase rotation of the
beamforming vector was introduced to make the imaginary
part of hHi wk equal to zero only considering two users
in the model. However, this method cannot be applied in
this paper because for multiple users case, once an arbitrary
phase rotation of wk is calculated with a given i to satisfy
={hHi wk}=0, we cannot guarantee that this arbitrary phase
rotation can still make the imaginary part equal to zero with
another given i. Therefore, At this stage, we apply several
variables in the form of wRek,i and w
Im
k,i , which makes
wRek,i = <{hHi wk} (15a)
wImk,i = ={hHi wk} (15b)
Although (11a) and (14a) are still not convex sets, instead
of applying new slack variables again, other techniques are
feasible to make them convex. In this paper, we use successive
convex approximation (SCA) to deal with this type non-convex
constraints. SCA algorithm is actually a very useful mathemat-
ical tool in solving the non-convex problem. The main idea
of SCA is to find out the problem whose optimal value is
close to that of the original problem with several iterations.
Specifically, we select the First-order Taylor approximation to
extend the non-convex functions to affine. As the number of
iterations grows, the result will be more and more approached
to the optimal. [15] In this paper, the constraints (11a), (13a)
can become affine when applying SCA such that
K∑
k=1
δk ≥ α(t)ρ(t) + ρ(t)(α− α(t)) + α(t)(ρ− ρ(t)), (16)
((wRek,i)
(t))2 + ((wImk,i )
(t))2 + 2(wRek,i)
(t)(wRek,i − (wRek,i)(t))
+ 2(wImk,i )
(t)(wImk,i − (wImk,i )(t)) ≥
((γk)
(t) − 1)(Ωk,i)(t) + (Ωk,i)(t)(γk − (γk)(t))
+ ((γk)
(t) − 1)(Ωk,i − (Ωk,i)(t))
(17)
t represents the tth iteration. Then, we can explore the
convexity of (6b). Apparantly, it is obvious to figure that (6b)
is nonconvex in terms of the expression directly. Now, we
rewrite this constraint as
|hHi wk|2
K∑
l=k+1
|hHi wl|2 + σ2
≥ 2Rk,min − 1, (18)
Which Rk,min is constant. After that, this constraint can be
reformulated with SCA as
((wRek,i)
(t))2 + ((wImk,i )
(t))2 + 2(wRek,i)
(t)(wRek,i − (wRek,i)(t))
+ 2(wImk,i )
(t)(wImk,i − (wImk,i )(t)) ≥
(2Rk,min − 1)(
K∑
l=k+1
|hH,i wl|2 + σ2)
(19)
Which k + 1 ≤ l ≥ K Obviously, the constraint (19) is
a convex set because it satisfies the form of Second-Order-
Cone (SOC). Now, after the above discussion, the final convex
problem can be formulated as
max
w1,··· ,wK ,α,ρ,γ,δ,Ω,wRe,wIm
α (20a)
s. t. (6c), (9), (11b), (15a), (15b), (16), (17), (19) (20b)
Where t represents tth iteration of the first-order Taylor
extension.
B. Phase Shift Optimization
The following problem is about the phase shift optimization
in this model. In previous sections, we have solved the beam-
forming optimization with the fixed phase shift. Since now the
beamforming has been optimized by the existing algorithm,
meaning that the beamforming vectors w are fixed now.
JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012 4
Therefore, the denominator of the objective function (6a) is a
constant and the power-efficiency problem can be transferred
to a sum-rate optimization problem which be formulated as
max
Θ
K∑
k=1
Rk (21a)
s. t. Rk ≥ Rk,min, k = 1, · · · ,K (21b)
θn ∈ [0, 2pi], n = 1, · · · , N (21c)
In the work of [11], the phase shift optimization is dealt with
semi-definite relaxation (SDR). In this paper, we still optimize
it through SCA. Similar to (P6), (21a) is still not a convex
problem because Rk is not convex with respect to Θ. To deal
with it, we can transform the problem formulation by splitting
the variable θn from the received signal expression of decoded
Uk at Ui. Define a variable vector v = [v1, · · · , vN ]H =
[ejθ1 , · · · , ejθN ]H and fixed vectors ak,i = diag(hHi,r)Gwfk
which k ≤ i ≤ K and wfk denotes the optimized beamforming
vector, which is fixed. Thus, the SINR for the kth user when
decoding at ith user (k ≤ i) can be expressed as
SINRk,i =
|vHak,i + hHi,dwfk |2
K∑
l=k+1
|vHal,i + hHi,dwfl |2 + σ2
(22)
For the case of k = i = K, the SINR expression can be
written as
SINRK =
|vHaK,K + hHK,dwfK |2
σ2
(23)
Similar to process of beamforming optimization, we define a
slack variable q such that
K∑
k=1
Rk ≥ q (24)
After this, we further introduce a new vector λ =
[λ1, · · · , λK ], which
K∑
k=1
log2(λk) ≥ q (25a)
1 + SINRk ≥ λk (25b)
After this, we apply another variable vector named µ =
[µ1, · · · , µK ] such that,
K∑
k=1
µk ≥ q (26a)
λk ≥ 2µk (26b)
After the above processes, applying the similar method to deal
with the SINR part, several variable ϕk,i which has the same
purpose with Ωk,i are applied as
|vHak,i + hHi,dwfk |2 ≥ (λk − 1)ϕk,i (27a)
K∑
l=k+1
|vHal,i + hHi,dwfl |2 + σ2 ≤ ϕk,i (27b)
Fig. 2: Energy efficiency versus the number of reflecting
elements
For the situation of i = k = K, the expressions should be
written as
|vHaK,K + hHK,dwfK |2 ≥ (λK − 1)ϕK,i (28a)
σ2 ≤ ϕK,K (28b)
Then, next step is to apply first Taylor to make (27a) convex.
Introduce a series of variables vRek,i and v
Im
k,i , such that
vRek,i = <{(v)Hak,i + hHi,dwfk} (29a)
vImk,i = ={(v)Hak,i + hHi,dwfk} (29b)
Then, extend (29a) and (29b) with SCA,
((vRek,i)
(t))2 + ((vImk,i )
(t))2 + 2(vRek,i)
(t)(vRek,i − (vRek,i)(t))
+ 2(vImk,i )
(t)(vImk,i − (vImk,i )(t)) ≥
((λk)
(t) − 1)(ϕk,i)(t) + (ϕk,i)(t)(λk − (λk)(t))
+ ((λk)
(t) − 1)(ϕk,i − (ϕk,i)(t))
(30)
To deal with the QoS contraint, we can still apply the same
technique to make it become a second-order cone expression,
((vRek,i)
(t))2 + ((vImk,i )
(t))2 + 2(vRek,i)
(t)(vRek,i − (vRek,i)(t))
+ 2(vImk,i )
(t)(vImk,i − (vImk,i )(t)) ≥
(2Rk,min − 1)(
K∑
l=k+1
|(v)Hal,i + hHi,dwfl |2
+ σ2)
(31)
After the transformation of (P20), the final form of this
phase shift optimization can be expressed as
max
v,q,λ,µ,ϕ,vRe,vIm
q (32a)
s. t. (21c), (26b), (29a), (29b), (30), (31) (32b)
V. SIMULATION RESULT
In this section, we will mainly show some simulation
results about the algorithm to illustrate the performance of the
JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012 5
Fig. 3: Energy efficiency versus internal circuit power
joint optimization scheme in this paper. Firstly, consider the
distance between BS and IRS is 10 m and users are located
in the area that is 16-20 m away from both BS and IRS.
The channels from BS to IRS and from IRS to users are
Racian fading channels. The LoS component GLoS and nLoS
component GnLoS are both set to 2. The channels from BS
to users obey Rayleigh distribution. The path loss exponents
from BS to IRS, IRS to users and BS to users are all set
to 2.5. Without the loss of generality, the minimal SINR
requirement for all users and Pmax are set to -10 dB and -20
dB respectively. Besides, the AWGN noise σ2 is -60 dB and
the power amplifier coefficient η is set to 0.8. The parameters
mentioned above are all fixed during the whole process of this
simulation.
In Fig. 2, we present the effect of the number of reflecting
elements on energy efficiency value. Under this circumstance,
the internal circuit power consumption Pc is set to -20 dB.
For comprehensively proving the rationality of the result, we
select 4 different cases, which are shown in the Fig. 2. Firstly,
it is obvious to find that the value of energy efficiency value
increases as the number of elements grows no matter which
case is selected. Besides, initially, energy efficiency increases
very rapidly. However, the curve for each case is approached
to saturation gradually as the number of elements grows.
In Fig. 3, we show the effect of the internal circuit consump-
tion on energy efficiency value. We set the number of reflecting
elements is 10 and select the same 4 cases. Then, discover the
value of energy efficiency when the circuit power is from -20
dB to 10 dB respectively. From the curve, it can be found that
the energy efficiency value decreases as the internal circuit
power increases. Besides, we can see that when Pc rises from
-20 dB to -10 dB, there is a sharp decrease in the energy
efficiency value. The curve becomes flatter and flatter and
tends to zero as the Pc further rises.
VI. CONCLUSION
In this paper, we mainly try to find an optimization method
to enhance the value of energy efficiency in a MISO IRS-
Assisted NOMA system with multi-user. Although it is hard
to find out the optimal solution of the original problem, we
can still relax the original problem and get the suboptimal
value of it. Except for problem relaxation, the successive
convex approximation is still a very useful mathematical tool
to deal with the non-convex functions. Besides, the joint
optimization method can effectively decrease the complexity
of the problem. In this paper, assuming the phase shift is
fixed, we firstly optimize beamforming. After obtaining the
beamforming vectors, we then optimize phase shift. as the
number of iterations grows, the value obtained can be more
and more approached to the optimal. Next, it is also available
for researchers to extend this network to a MIMO IRS-
Assisted NOMA system and try to find out some new methods
and algorithms to solve it.
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