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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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