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arXiv:2007.11704v1 [eess.SP] 22 Jul 2020
1
Analysis and Optimization for IRS-Aided Multi-pair
Communications Relying on Statistical CSI
Zhangjie Peng, Tianshu Li, Cunhua Pan, Member, IEEE, Hong Ren Member, IEEE,
Wei Xu, Senior Member, IEEE, and Marco Di Renzo Fellow, IEEE
Abstract—In this paper, we investigate an intelligent reflecting
surface (IRS) assisted multi-pair communication system, in which
multiple pairs of users exchange information via an IRS. We
derive an approximate expression for the achievable rate when
only statistical channel state information (CSI) is available.
Then, a genetic algorithm (GA) is proposed to solve the rate
maximization problem. In particular, both the scenarios of
continuous phase shift (CPS) and discrete phase shift (DPS)
are considered. Simulation results verify the correctness of our
derived results and show that the proposed GA method has
almost the same performance as the globally optimal solution
obtained by the exhaustive search method. In addition, three
bits for discretization are capable of achieving a large portion of
the achievable rate for the CPS case.
Index Terms—Intelligent reflecting surface (IRS), statistical
channel state information (CSI), reconfigurable intelligent surface
(RIS) , multi-pair communication, genetic algorithm (GA)
I. INT RO DUC TI ON
Recently, with the development of meta-surfaces [1], a new
transmission technique named intelligent reflecting surface
(IRS), which is capable of reconfiguring the channel into a
desirable form by carefully turning its phase shifts [2], has
been widely regarded as a promising technology for future
6G wireless communications [2]. Due to its appealing features
of low cost and low power consumption, IRS has attracted
extensive research attention [3]–[6].
Some initial efforts have been devoted to the study of
various IRS-assisted communication systems such as physical
layer security in [7], [8], multicell networks in [9], full duplex
system in [10], mobile edge computing in [11], and wireless
power transfer in [12]. However, to the best of our knowledge,
a paucity of contributions studied the IRS-assisted multi-pair
communication system, which is a typical communication
scenario due to the rapid increase of the number of machine
devices in future networks. On the other hand, the direct
communication links of the multi-pair communications may be
blocked in both indoor and outdoor applications. In particular,
the direct signals may be readily blocked by the trees and
Z. Peng and T. Li are with the College of Information, Mechanical and
Electrical Engineering, Shanghai Normal University, Shanghai 200234, China
(e-mail: pengzhangjie@shnu.edu.cn, 1000479056@smail.shnu.edu.cn).
C. Pan and H. Ren are with the School of Electronic Engineering and
Computer Science at Queen Mary University of London, London E1 4NS,
U.K. (e-mail: c.pan,h.ren@qmul.ac.uk).
W. Xu is with National Mobile Communications Research Laboratory,
Southeast University, Nanjing 210096, China (e-mail: wxu@seu.edu.cn).
M. Di Renzo is with Universit´e Paris-Saclay, CNRS, CentraleSup´elec,
Laboratoire des Signaux et Systemes, Gifsur-Yvette, France (e-mail:
marco.direnzo@centralesupelec.fr).
IRS
1
UA
UAi
U
AK
1a
g
ai
g
aK
g
1b
g
bi
g
bK
g
1
UB
UBi
UBK
Fig. 1. System model for IRS-assisted multi-pair communication.
huge buildings in outdoor scenarios. In indoor scenarios, the
signals can be blocked by thick walls, especially in high-
frequency mmWave communication systems. We intend to
deploy an IRS between two sides of communication devices
to address the above issue. Compared with the existing papers
on relay-assisted multi-pair communications in [13]–[15], the
IRS has some appealing advantages. In specific, IRS requires
low power consumption and will not introduce noise since it
is passive and does not have signal processing units such as
amplifier and radio frequency chains. In addition, the transmis-
sion between two sides can take place at the same time, while
the relay needs two time slots for the whole transmission.
This means the IRS can save transmission time, which is very
suitable for ultra-reliable low latency communication.
Against the above background, we study the transmission
design for an IRS-assisted multi-pair communication system.
Unlike most of the existing papers in [7]–[12] where in-
stantaneous channel state information (CSI) is assumed, we
consider the statistical CSI [16] that is much easier to obtain
since it varies much slowly. Specifically, our contributions are
threefold: 1) We derive the achievable rate; 2) We propose the
genetic algorithm (GA) method to solve the phase shift opti-
mization problem , where both the cases of continuous phase
shift (CPS) and discrete phase shift (DPS) are considered; 3)
Extensive simulation results are provided to demonstrate the
correctness of our derived results, and also show that three
bits are enough to discretize the phase shifts, which provides
useful engineering design insights for IRS-assisted systems.
The rest of the paper is organized as follows. In Section
II, we introduce the IRS-assisted multi-pair communication
system model. We derive the achievable rate in Section III
2
and optimize the phase shift in Section IV. Numerical results
are provided to demonstrate the correctness of our analysis in
Section V. In the end, we draw conclusions in Section VI.
II. SY S TE M MODE L
We consider an IRS-assisted multi-pair communication sys-
tem, where Kpairs of users exchange information via an IRS,
as shown in Fig. 1. The IRS consists of Lreflective elements,
which are capable of customizing the channel environment by
turning the phase shifts. The phase shift matrix Θis given by
Θ=diag(ejθ1,···, ejθℓ,···, ej θL), where θℓis the phase
shift of the ℓth reflective element. We denote the ith single-
antenna transmitter as UAi and the ith single-antenna receiver
as UBi , for i= 1, ..., K .
The channel between UAi and the IRS and that between the
IRS and UBi can be written as
gai =√αaihai ,(1)
gbi =√αbihbi ,(2)
where αai and αbi denote the large-scale fading coefficients,
and gi∈CL×1and hi∈CL×1denote the fast fading vectors.
All of the channels follow Rician fading, thus the vectors hai
and hbi can be expressed as
hai =rεi
εi+ 1hai +r1
εi+ 1 ˜
hai,(3)
hbi =sβi
βi+ 1 hbi +r1
βi+ 1 ˜
hbi,(4)
where εidenotes the Rician factor, ˜
hai ∈CL×1and ˜
hbi ∈
CL×1both denote the non-line-of-sight channel vector, each
element of which follows independent and identically distri-
bution of CN(0,1), and hai ∈CL×1and hbi ∈CL×1denote
the line-of-sight channel vector. In particular, hai and hbi can
be expressed as
hai = [1, ej2πd
λsinςi,···, ej2πd
λ(L−1)sinςi]T,(5)
hbi = [1, ej2πd
λsinϕi,···, ej2πd
λ(L−1)sinϕi]T,(6)
where ϕiand ςirepresent the ith pair of users’ AoA (angle
of arrival) and AoD (angle of departure), respectively. For
convenience, we will set d=λ
2in the rest of this paper.
We assume the availability of statistical CSI at UAi , for
i= 1,···, K. The statistical CSI can be readily obtained
since it varies much slowly than the instantaneous CSI. The
signal received at UBi is given by
yi=gT
biΘ
K
X
j=1
√pjgaj xj+ni
=√pigT
biΘgai xi
|{z }
Desired signal
+
K
X
j=1,j6=i
√pjgT
biΘgaj xj
|{z }
Inter-user interference
+ni
|{z}
Noise
,(7)
where xj∼ CN (0,1) represents the signal UAj transmits, pj
denotes the transmission power at UAj , and ni∼ CN(0, σ2
i)
is the additive white Gaussian noise (AWGN) of UBi , for
i= 1,···, K.
From (7), it is readily seen that yiconsists of three parts:
the desired signal which UB i wants to receive, the interference
produced by other multi-pair users and the noise. Furthermore,
the signal-to-interference plus noise ratio (SINR) for UB i is
given by
γi=piαbiαai hT
biΘhai 2
K
P
j=1,j6=ipjαbiαaj hT
biΘhaj 2+σ2
i
.(8)
Hence, the average achievable rate for UBi can be expressed
as
Ri=E{log2(1 + γi)}.(9)
Therefore, the sum achievable rate can be written as
C=
K
X
i=1
Ri.(10)
III. ACHI EVAB LE RATE ANA LYSIS
To analyze the performance of IRS-assisted multi-pair com-
munication systems, we first introduce Theorem 1.
Theorem 1. The achievable rate of the ith receiver can be
approximated as
Ri≈log2
1 + piαbiαai
εiβiΩi,i+L(εi+βi)+L
(εi+1)(βi+1)
K
P
j=1,j6=i
pjαbiαaj
εiβjΩi,j +L(εi+βj)+L
(εi+1)(βj+1) +σ2
i
,
(11)
where Ωi,i and Ωi,j are defined as
Ωi,i =L+ 2 X
1≤m<n≤L
cos[θn−θm+ (n−m)π(sinϕi+sinςi)],
(12)
Ωi,j =L+ 2 X
1≤m<n≤L
cos[θn−θm+ (n−m)π(sinϕi+sinςj)].
(13)
Proof: Using Lemma 1 in [17], Riin (9) can be approxi-
mated as
Ri≈log2
1 +
piαbiαai EnhT
biΘhai 2o
K
P
j=1,j6=ipjαbiαbj EnhT
biΘhaj 2o+σ2
i
.
(14)
The ℓth element of hai and hbi can be written as follows:
[hai]ℓ=rεi
εi+ 1 ej(ℓ−1)πsinϕi+r1
εi+ 1 (sℓi +jtℓi ),(15)
and
[hbi]ℓ=sβi
βi+ 1ej(ℓ−1)πsinςi+r1
βi+ 1 (uℓi +jvℓi ),(16)
3
θ
...
θ
...
θ
...
Initial
population
Saved group
θ
...
θ
...
θ
...
Yes
No
Evaluation
Eq. (23)
Next generation
Evaluation
Eq. (23)
and Sort Selection
Function
and
Crossover
Function
Mutation
Function
Current generation
Mutable group
Unnamed group
Termination
Criterion
Reached?
Output the
best individual
in current
population
θ
Elites
Fig. 2. Genetic algorithm structure.
where sℓi ∼ N(0,1/2) and tℓi ∼ N(0,1/2) denote the
independent real and imaginary parts of [˜
hai]ℓ, respectively.
uℓj ∼ N(0,1/2) and vℓj ∼ N(0,1/2) denote the independent
real and imaginary parts of [˜
hbi]ℓ, respectively.
By substituting (15) and (16) into
hT
biΘhaj =
L
X
ℓ=1
[hbi]ℓθℓ[haj ]ℓ,(17)
we can obtain the real and imaginary parts, shown at the
bottom of the next page.
As we know
EnhT
biΘhaj 2o=E(hT
biΘhaj )2
real + (hT
biΘgaj )2
imag .
(20)
Substituting (18) and (19) into (20) and removing the terms
with zero value, we can rewrite (20) as
EnhT
biΘhaj 2o=εiβjΩi,j +L(εi+βj) + L
(εi+ 1)(βj+ 1) .(21)
By substituting (21) into (14), we arrive at the final result
in (11). ✷
Substituting (11) into (10), we obtain the sum achievable
rate. According to Theorem 1, when αbi,αai , AoA, AoD,
εi, and σiremain unchanged, the sum achievable rate is
determined by the number of pairs of users K, transmission
power pi, phase shift matrix Θand the number of reflective
elements L.
IV. PHA SE SHI FT OP T IM IZATI ON
To obtain the maximal sum achievable rate, we solve the
phase shift optimization problem, where both the scenarios of
CPS and DPS are taken into account.
(hT
iΘgj)real =1
p(εi+ 1)(βj+ 1)
L
X
ℓ=1 pεiβj{cos[θℓ+ (ℓ−1)π(sinϕi+sinςj)] }
+1
p(εi+ 1)(βj+ 1)
L
X
ℓ=1
√εi{cosθℓ[cos[(ℓ−1)πsinϕi]uℓj −sin[(ℓ−1)πsinϕi]vℓj ]
−sinθℓ[cos[(ℓ−1)πsinϕi]uℓj +sin[(ℓ−1)πsinϕi]vℓj ]}
+pβj{cosθℓ[cos[(ℓ−1)πsinςj]sℓi −sin[(ℓ−1)πsinςj]tℓi]
−sinθℓ[cos[(ℓ−1)πsinςj]tℓi +sin[(ℓ−1)πsinςj]sℓi]}
+cosθℓ(sℓiuℓj −tℓi vℓj )−sinθℓ(sℓi vℓj +tℓiuℓj )(18)
(hT
iΘgj)imag =1
p(εi+ 1)(βj+ 1)
L
X
ℓ=1 pεiβj{sin[θℓ+ (ℓ−1)π(sinϕi+sinςj)] }
+1
p(εi+ 1)(βj+ 1)
L
X
ℓ=1
√εi{sinθℓ[cos[(ℓ−1)πsinϕi]uℓj −sin[(ℓ−1)πsinϕi]vℓj ]
+cosθℓ[cos[(ℓ−1)πsinϕi]uℓj +sin[(ℓ−1)πsinϕi]vℓj ]}
+pβj{cosθℓ[sin[(ℓ−1)πsinςj]sℓi −sin[(ℓ−1)πsinςj]tℓi ]
+cosθℓ[cos[(ℓ−1)πsinςj]tℓi +sin[(ℓ−1)πsinςj]sℓi]}
+sinθℓ(sℓiuℓj −tℓi vℓj ) + cosθℓ(sℓi vℓj +tℓiuℓj )(19)
4
A. Optimal CPS Design
To begin with, we consider the CPS case. The optimization
problem is formulated as
max
Θ
K
X
i=1
log2(1 + γi)(22a)
s.t. θℓ∈[0,2π)∀ℓ= 1,···, L. (22b)
The computational complexity of solving Problem (22) by
using conventional optimization methods is prohibitively high,
since the data rate expression is a complex expression of the
optimization variables. As a result, a GA method is proposed.
The algorithm structure is shown in Fig.2. We denote each in-
dividual as a 1 ×L phase shift vector θ=[θ1,···, θℓ,···, θL]
and θℓas its ℓth gene. First, Ntindividuals are generated, by
setting each genes randomly distributed within [0,2π), known
as the initial population (with population size of Nt= 100).
Besides, the evaluation of each individual is evaluated by the
fitness function:
f(θ) = 1
PK
i=1 log2(1 + γi).(23)
We make a list sorted by fitness function value (lowest to
highest). The individuals who achieve better evaluation have
lower fitness function values, and appear higher on the list.
Second, first Nsindividuals (Ns= 50) in the list are placed in
the saved group and then pass into the unnamed group directly.
Then, the selection function is capable of obtaining one candi-
date from Ntindividuals. We can repeat the selection function
procedure until the number of candidates satisfies the request.
The crossover function operates on two candidates (obtained
by selection function), and generates two crossover children.
The number of children generated by crossover function is
Nc, which satisfies Nt=Ns+Nc. Then, we place the children
in the unnamed group. Third, elites are the individuals who
have lowest fitness function value in the unnamed group (Ns
saved individuals and Ncchildren), with elites size of Ne=
1. Except for elites, the rest of individuals are placed in the
mutable group. Thus, this group contains (Np-Ne)individuals,
and (Np-Ne)Lgenes. Each gene is capable of mutating to a
random number between 0 and 2πwith mutation rate Pm=
0.1 by utilizing mutation function. Thus, we have produced
the current generation population, with elites and mutated
individuals, on each iteration. Fourth, repeat the iterative
method mentioned above until the termination criterion is
reached. Finally, the individual with the lowest fitness function
value in the current generation is chosen as the output of the
algorithm. The selection function and crossover function are
illustrated as follows.
Algorithm 1: Selection Function
1Generate a random number rbetween 0 and Nt;
2Take the [r]th individual in the list ([r]denotes the
nearest maximum integer to r).
1) Selection function: The selection function is proposed to
obtain candidates for crossover function. The individuals with
Algorithm 2: Crossover Function
1Get θ1= [θ(1)
1,···, θ(1)
ℓ,···, θ(1)
L]and
θ2= [θ(2)
1,···, θ(2)
ℓ,···, θ(2)
L]selected by Algorithm 1;
2Identify a crossover point ℓ′∈[1, L]randomly;
3Crossover θ1and θ2at crossover point ℓ′;
4Obtain two children
θc1 = [θ(1)
1,···, θ(1)
ℓ′, θ(2)
ℓ′+1,···, θ(2)
L]and
θc2 = [θ(2)
1,···, θ(2)
ℓ′, θ(1)
ℓ′+1,···, θ(1)
L].
higher fitness function values will have a lower probability to
be selected. The selection function is described in Algorithm
1.
2) Crossover Function: The crossover function operates on
θ1and θ2and generates two children θc1 and θc2 . The
crossover function is derived under a single-point crossing
algorithm and described in Algorithm 2.
B. Optimal DPS Design
In real scenarios, the IRS only has a limited number of phase
shifts. We assume that each reflective element is encoded with
Bbits, and thus 2Bvalues of phase shifts can be chosen to
enhance the signal reflected by the IRS [18]. We denote the
DPS matrix as ˆ
Θ=diag(ejˆ
θ1,···, ejˆ
θℓ,···, ejˆ
θL), where ˆ
θℓ
is the DPS of the ℓth reflective element. Replacing the CPSs
in Problem (22) by the DPSs, the optimization problem for
the DPS scenario can be formulated as
max
ˆ
Θ
K
X
i=1
log2(1 + γi)(24a)
s.t. ˆ
θℓ∈ {0,2π/2B,···,2π(2B−1)/2B}
∀ℓ= 1,···, L. (24b)
It is observed that Problem (24) is similar to Problem (22).
Accordingly, the above GA method proposed for the CPS
scenario can also be extended to this DPS scenario. For this
DPS scenario, we only need to replace CPSs with DPSs in the
above GA method.
V. NUM ER ICA L RES ULTS
We evaluate the impact of different factors on the sum
achievable rate. We assume that the Rician factor is εi= 10,
the noise power is σ2
i= 1, and the transmission power is
denoted as SNR = pi, for i= 1, ..., K. Furthermore, the other
parameters are summarized in Table I.
In Fig. 3, we draw the sum rate versus SNR for analytical
and Monte-Carlo simulation results with two bits by the
proposed GA method. We observe that the analytical results
are consistent with the Monte-Carlo simulation results, which
verifies the correctness of the derivations. In addition, we
can also find that the sum rate increases with the number of
reflective elements L, since more energy is reflected.
Fig. 4 depicts the sum rate versus SNR with two bits
for various schemes. As expected, the proposed GA and
exhaustive search method achieve higher sum rate than that of
random method. It is interesting to observe that the proposed
5
TABLE I
PAR AM ET ER S FO R SI M UL ATI ON
No. of pairs AoA AoD αai and αbi
15.5629 1.1450 0.0023
25.6486 0.6226 0.0285
33.9329 3.0773 0.0025
40.8663 1.2142 0.0012
51.3685 5.6290 0.0550
61.1444 0.6226 0.0141
−15 −10 −5 0 5 10 15 20 25
0
1
2
3
4
5
6
7
8
SNR (dB)
Sum Rate (bits/s/Hz)
L = 16,K = 6
L = 8,K = 6
simulation
Fig. 3. Sum rate versus SNR with B= 2 and GA.
GA method has almost the same performance as the globally
optimal solution obtained by the exhaustive search method.
Fig. 5 shows the sum rate versus the number of coding
bits for the scenarios of CPS and DPS with SNR = 20 dB.
The sum rate by using DPSs increases rapidly when Bis
small, while the curve gradually becomes saturated when B
is large. It is well known using a large number of coding bits
to control the phase shifts incurs high hardware cost and power
consumption. The figure shows that three bits for discretization
can achieve a large portion of the sum rate, which provides
useful engineering design insights for IRS-assisted systems.
VI. CO NCL US I ON
In this paper, we investigated IRS-assisted communications
for multiple pairs of users. We derived the approximate ex-
pression for the achievable rate. We proposed the GA method
to achieve the maximal achievable rate by optimizing the
phase shifts, where both the scenarios of CPS and DPS were
considered. Simulation results verified the correctness of our
derivations.
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