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30th Telecommunications forum TELFOR 2022 Serbia, Belgrade, November 15-16, 2022.

Secrecy Performance of RIS-enabled Linear

Interference Alignment Multi-User MIMO Network

Hakan Alakoca and Lutﬁye Durak-Ata

Information and Communications Research Group, Informatics Institute,

Istanbul Technical University, Istanbul, Turkey

{alakoca, durakata}@itu.edu.tr

Abstract—In this study, an reconﬁgurable intelligent surface

(RIS)-enabled linear interference alignment (IA) is examined

for multi-user multiple-input multiple-output (MU-MIMO) in a

presence of an eavesdropper. Modiﬁed precoder and postcoder

matrix designs are introduced to minimize interference for legit-

imate users which is used as ensuring secure communications.

According to our ﬁndings, the secrecy sum rate of the IA-based

RIS-enabled MU-MIMO outperforms that without IA case due

to the intensive interference of the communication environment.

Index Terms—interference alignment, multi-user MIMO, re-

conﬁgurable intelligent surfaces, secure communications

I. INTRODUCTION

Reconﬁgurable intelligent surfaces (RIS) is most promising

solution in next generation networks for enhancing quality of

service of the communication units by altering electromagnetic

property of the signals. The advantage of the interference

alignment (IA) is still critical to multi-user multiple input

multiple output (MU-MIMO) communications thanks to the

its efﬁcient usage of limited resources. An overview of the an

RIS architecture covering with recent hardware designs pre-

sented in [1]. Also, potential smart environment utilizing RIS-

enabled networks are studied in [2] considering reliability and

vulnerability scenarios. As a secure communication aspect,

an electromagnetic interference based vulnerability scenario

on RIS-enabled communications are given in [3]. In addition,

secrecy outage probability of the RIS-enabled secure wireless

communications in a presence of eavesdropper model covered

in [4].

Fundamental functioning mechanisms and advantages de-

scribed intelligent radio environments and differences of relay-

enabled wireless networks indicated in [5]. A real-time 2x2

MIMO-QAM based transmission system design demonstrated

in [6]. In [7] introduces a new RIS-enabled MIMO design

compared to VBLAST and Alamouti scheme. To cope with

training overhead problem in MU-MIMO enabling RIS net-

works, authors proposed matrix calibration based channel

estimation method in [8]. Similarly, work in [9] different

channel estimation performances are proposed and compared

for an RIS-enabled MIMO communication system. A novel

phase shift design matrices with low complexity adaptation

for an RIS-enabled MIMO scenario given in [10]. A precoding

optimization scheme given in [11] for an RIS-enabled MIMO

system to reduce SER.

IA enables a speciﬁc multi-access technique as an alter-

native to FDMA, TDMA, and CDMA, reducing interference

This work has been supported by The Scientiﬁc and Technological Research

Council of Turkey (TUBITAK) under Project 120E307.

The work of Hakan Alakoca was supported by the Research Fund of the

Istanbul Technical University under Project MDK-2022-43782.

effects due to the sharing of the same time or frequency

spectrum. The fundamental principles of IA presented in [12].

In [13] zero-forcing method for multi-user MIMO channels

is presented. An adaptive beamformer design based on LS

and MMSE presented in [14]. Sung et al. [15] encounter an

interference channel using an iterative method that compares

conventional precoding and decoding schemes. However, up-

dating the channel state information requirements to comprise

the precoding and post-coding matrix is the major drawback

of this technique. In [16], authors recently applied a closed-

form solution and an alignment chain method to MU-MIMO in

the presence of interference. Work in the [17] block-structured

Riemannian pursuit proposed an IA-based RIS communication

system under MU-MIMO to achieve the maximum degree of

freedom (DoF).

The main contributions of this study are listed below.

•C1) The interference impact due to the simultaneous use

of resources in MU-MIMO in RIS-enabled communica-

tions is reduced by jointly operated IA-based precoding

and postcoding schemes.

•C2) An RIS-enabled linear IA-based precoder design

for minimizing reﬂected interference from RIS in MU-

MIMO communications is presented.

•C3) Modiﬁed precoder and postcoder matrix methods are

also given to interference minimization for the legitimate

users in an RIS-enabled MIMO while ensuring secure

communications.

•C4) Secrecy sum rate enhancement owing to IA in an

RIS-enabled MU-MIMO communications is examined

via numerical results.

The organization of this paper is indicated as follows. Signal

model of the proposed scenario is given in Section II. Precoder

design for the an RIS-enabled linear IA is presented in

Section III. Section IV introduces the algorith of the precoder

matrix design along with secrecy rate. In Section V, we cover

numerical ﬁndings and comparisons. Lastly, we conclude the

paper in Section VI.

II. SI GNA L MOD EL

The signal model of the an RIS-enabled network with

the MU-MIMO channel in a presence of an eavesdropper is

illustrated as in Fig. 1. Let ibe denoted for the transceiver

user index, in addition to Ntand Nrindicated as the number

of transmitted and received antennas, respectively. Each user

sends its desired symbols to the destination represented as

xi∈Cds×1. The transmitted signals are independent of each

other and have unit variance indicated via E[xixiH] = Ids)

where dsis given a number of data streams from each user.

The MIMO channel coefﬁcients between BS-to-RIS and the

Recongurable Intelligent Surface

Microcontroller

Eavesdropper

Blockage

Fig. 1. System model of the an RIS-enabled multi-user MIMO

network in a presence of an eavesdropper.

reﬂecting sum of the MIMO channel coefﬁcients between RIS-

to-UE are assigned as Hi∈CN×Ntand Ti∈CNr×N,

respectively. Here, Ti, is the joint signal and the interference

channel represented as Ti=Gii +PK

j=1,j=iGij . Here, Gii ,

Gij and Kare denoted as the corresponding channel coef-

ﬁcients between RIS-to-associated UE’s, interference channel

coefﬁcients between RIS-to-interferer UE’s and the number

of users, respectively. All channel coefﬁcients are assumed to

be independent identical distributed complex Gaussian random

variables with zero mean and unit variances. We also note that

direct link is not available between transmitters and receivers.

All CSI can be accessible by each individual node on the

network. RIS, which is connected to the microcontroller ele-

ment, is capable of changing the electromagnetic property of

the incident signal with the surface element N. The estimated

symbols ˆ

xi∈Cds×1at the i-th receiver node in the presence

of an interference reﬂection channel can be expressed as

ˆ

xi=ρiUH

iGiiΦHiVixi

| {z }

desired signal

+

K

X

j=1,j=i

ρjUH

iGij ΦHjVjxj

| {z }

reﬂected interference

+UH

ini

| {z }

noise

,

(1)

where, the noise component which is constituted via inde-

pendent and identically distributed complex Gaussian noise

vector at i-th receiver node, is indicated as ni∈CNr×1

with zero mean and σ2

Bvariance. It is also noted that, power

components of the received signal ρuwhere u∈ {i, j}

is determined as ρu=√Pud−v/2

uT X d−v/2

uRX . Here, Pu,duT X ,

duRX and vis given as transmit power of u-th user, dis-

tance between u-th transmitter to RIS, distance between RIS

to u-th receiver, and path-loss exponent, respectively. Also,

UH

iGiiΦHiVixiis desired signal component coming from

i-th user. PK

j=1,j=iUH

iGij ΦHjVjxjis sum of interference

terms coming from j-th users. Interfence suppression oper-

ation is applied to transmitted signals as Vi∈CNt×dsis

denoting as precoding matrix for i-th user. At the receiver

node postcoding is applied at the end of the RF-chain via

post processing matrix to minimize interference terms as,

UH

i∈Cds×Nr. We assume K= 3 due to compact closed-

form solution existence IA-based network. On the other hand,

precoding and postcoding distribution is not available to the

eavesdropper node. Hence, received signals of the eavesdrop-

per for i-th interested user can be also expressed as,

yEi=ρiGeiΦHixi

| {z }

target signal

+

K

X

j=1,j=i

ρjGej ΦHjxj

| {z }

reﬂected interference

+ne

|{z}

noise

.(2)

Here, Gei and Gej are denoted as CSI between signal of

interest for i-th user and interference sources coming from j-

th users, respectively. Also, neis presented as Gaussian noise

components at the eavesdropper node with zero mean and σ2

E

variance.

III. LINEAR IA AND PRECODER DESIGN

Basic principle of IA is based on spanning interference

signals in the same direction. Precoding and postcoding ma-

trices are applied to transmitted and received signals when the

CSI is available where both transmitter and receiver nodes.

Compound MIMO channel matrix from i-th transmitter to j-

th receiver are represented as Ωij =GijΦHj. In order to

serve multiple users at the same time and frequency resources,

phase-shift matrix of RIS, Φis selected as DFT matrix.

Similar strategy is also applied in [15]. Besides, the following

conditions must be satisﬁed to eliminate the total interference

terms via linear interference alignment as,

UH

jΩij Vjxj= 0,(3)

Rank(UH

jΩij Vjxj) = ds.(4)

Interfering signals are perfectly eliminated by the IA, which

is collect them into the single signal subspace via precoding

matrix. In order to apply linear IA, Ntand Nrshould be

selected as equally, with M∈2ξwhere ξis an integer. Size

of data stream is restricted as M

2×1to achieve maximum

degree of freedom for all users. The following conditions must

be met to ensure that the signals in the receiving node are not

affected by the interference. For K= 3 user paired network

of compound matrices can be presented as,

span(Ω12V2) = span(Ω13 V3),

span(Ω21V1) = span(Ω23 V3),(5)

span(Ω31V1) = span(Ω32 V2).

Here, span(ν) refers to the spanning vector from νspanning

space.

span(V1) = span(ζV1).(6)

Cascaded CSI terms are utilized to constitute modiﬁed

precoder matrices to minimize interference terms. The pseu-

docode of the modiﬁed precoding matrix for an RIS-enabled

IA is presented in Algorithm 1. Here, Vz,i is the modiﬁed

precoder matrix, Qiis an orthonormal basis matrix, and Ziis

the power coefﬁcient matrix. The equivalent expression of the

effective channel matrix is given as, Fij =Ωij Qjbetween

j-th receiver and i-th transmitter.

IV. POS TC OD ER MATRIX DE SI GN A ND SE CR EC Y RATE

In this section, postcoding matrix design scheme as well

as secrecy rate of the determined system are provided. The

modiﬁed precoding matrix from MMSE decoder Vi∈CNt×ds

can be obtained as Vi=QiZi. The closed form solution is

not efﬁcient due to massive computation power requirement to

ﬁnd power coefﬁcient matrix for all users, Zi. Instead power

Algorithm 1 Algorithm for precoding matrix design for an

RIS-enabled linear IA

Input: Ωij for {i, j} ∈ {1,2,3}

Output: Precoding matrix, effective channel matrix

1: Calculate spanning vectors and ζ

ζ←Ω−1

31 Ω32Ω−1

12 Ω13Ω−1

23 Ω21.

2: Compute individual precoding matrix

V1←eig(ζ, M

2){V1=c1c2. . . c M

2where

c1, c2. . . c M

2components are eigenvectors of ζ}

V2←Ω−1

32 Ω31V1

V3←Ω−1

32 Ω21V1

3: for i←1to 3do

4: Perform QR decomposition for each individual user

QiRi=QR(Vi)

5: Setting precoders for each individual users

Vz,i←QiZi

6: for j←1to 3do

7: Calculate effective channel matrix for postcoding

Fij ←Ωij Qj

8: end for

9: end for

10: return Vz,i,Fij for {i, j } ∈ {1,2,3}

terms are determined via equal power distribution which is

expressed as Zi=ρiX. Where ρiwith ρi=√Pid−v/2

T X d−v/2

RX

is an average received of each data stream and Xis an unitary

matrix. In addition, Piis determined as Pi= 2PTi/M where

PTiis the transmit power of i-th user. The estimated symbols

can be given as

ˆ

xi=UH

iΩiiQiZixi+UH

i

K

X

j=i

Ωij QjZjxj+UH

ini.(7)

In order to express a solution, post-coding matrix is deﬁned

via UH

i=˜

UH

i¯

UH

i. Primary condition to obtain ˜

UHis given

as

˜

Ui[¯

Ui(

K

X

j=i

Fij ZjZH

jFH

ij +σ2

BI)−1¯

UH

i]˜

UH

i=IM

2,(8)

and

˜

Ui¯

UiFiiZiZH

iFH

ii ¯

UH

i˜

UH

i=λi.(9)

Here λiis a diagonal matrix. Postcoder matrix design pro-

cedure using (8) and (9) is presented in Algorithm 2. After

following these steps, modiﬁed precoder and postcoder ma-

trices will be provided for K= 3 multi-user MIMO com-

munications. Secrecy sum rate of the predetermined system is

indicated as,

Rsec =

K

X

i=1

Ri−RE.(10)

Here, individual rate expression of each i-th user and eaves-

dropper rate are provided as

Ri= log2|I+FiipiFH

ii (

K

X

j=i

Fij pjFH

ij +σ2

BI)−1|,(11)

Algorithm 2 Algorithm for MMSE-based modiﬁed precoder

and postcoder matrices for an RIS-enabled linear IA

Input: Fij ,Qi,σ2

Bfor {i, j}∈{1,2,3}

Output: Modiﬁed precoding and postcoding matrices

1: for i←1to 3do

2: Compute left side of the postcoding matrix, ¯

UH

i

¯

UH

i←PK

j=uFij ZjZH

jFH

ij +σ2

BI−1HFiiZi

3: Calculate Liusing Cholesky decomposition of Zi,

Li=Chol(Zi)

4: Compute block channel matrix

Bi←L−1

iUiFii

5: Obtain SVD of the block chanel matrix

SiΛiDH

i= SVD(Bi)

6: Compute right side of the postcoding matrix, ˜

UH

i

˜

UH

i←SH

iL−1

i

7: Compute MMSE based modiﬁed postcoder

UH

i←SH

iL−1

i¯

Ui

8: Compute MMSE based modiﬁed precoder

Vi←√piQiDi.

9: end for

10: return UH

i,Vifor i∈ {1,2,3}

and

RE= log2|I+GeiΦHipi(Gei ΦHi)H(12)

×(

K

X

j=i

Gej ΦHjpj(Gej ΦHj)H+σ2

EI)−1|,

respectively.

V. NUMERICAL RESULTS

In this section, we investigate the impact of IA on an RIS-

enabled MU-MIMO communication environment considering

the perspective of PLS through numerical ﬁndings. In our

simulations, there are K= 3 transceiver pairs that are

communicating with each other using the same time/frequency

resources in the presence of an eavesdropper. The number of

antennas for each pair and eavesdropper is selected equally

as M. The node distances of the legitimate pairs are also

determined as du=duT X =duRX where u∈ {1,2,3}. In

addition, dEand σ2

E/σ2

Bare indicated as the distance between

RIS and the eavesdropper node and the ratio of noise variances

of the eavesdropper to the legitimate receiver, respectively.

The performance of the secrecy sum rate for RIS-enhanced

MU-MIMO networks is depicted with surface elements for

varying values of Mand σ2

E/σ2

Bin Fig. 2 under N= 32,

du= 3 and dE= 5. It is obvious that the performance of

the secrecy sum rate is outperforms owing to IA-based for

RIS-enhanced MU-MIMO communication environment. For

example, the difference in secrecy sum rate between IA and

without IA less than 20 dB SNR with M= 8 is given as

approximately 37 bit/s/Hz. According to the ﬁndings, increas-

ing Msigniﬁcantly improves the performance of the secrecy

sum rate for the existence of IA and without IA operations, as

expected. For example, secrecy sum rate performances under

10 dB SNR obtained as 7.6bit/s/Hz and 26.2bit/s/Hz, while

using M= 2 and M= 8, respectively. In addition, it is also

observed that σ2

B/σ2

E, obviously outperforms the performance

of the secrecy sum rate.

IA-based secrecy sum rate for varied parameters Nand

dEin RIS-enhanced MU-MIMO network communications

are presented in Fig. 3 under M= 4, SNR = 10 dB

and σ2

E/σ2

B= 10 dB. Increasing number of RIS elements

accelerate secrecy sum rate for IA-based evaluation. However,

the secrecy sum rate without IA in an RIS-enabled MU-MIMO

reaches saturation after N= 20 for the cases of dE= 1

m and dE= 10 m. It is also obvious that reducing the

distances dEalso degrades the performance of the secrecy

sum rate. Although keeping dE= 1 m, which is closer to

the RIS units than the legitimate receiver nodes, du= 3 m,

an improvement in secrecy sum rate performance could be

observed approximately 20 bit / s / Hz under N= 100 as a

result of the precoding and postcoding operations of IA.

-20 -10 0 10 20 30

SNR [dB]

0

10

20

30

40

50

60

70

80

90

Secrecy Sum Rate [bit/s/Hz]

Fig. 2. Impact of IA-based on secrecy sum rate performance

for RIS-enhanced MU-MIMO network under N= 32,du= 3,

and dE= 5 for varying Mand σ2

B/σ2

Evalues.

20 40 60 80 100 120

0

5

10

15

20

25

30

35

Secrecy Sum Rate [bit/s/Hz]

Fig. 3. Impact of IA-based on the secrecy sum rate per-

formance for the varied Nand dEin the RIS-enhanced

MU-MIMO network under M= 4, SNR = 10 dB, and

σ2

E/σ2

B= 10 dB.

VI. CONCLUSION

In this study, secure communications of an RIS-enabled

MU-MIMO in the presence of an eavesdropper have been

considered using the IA approach. To reduce interference, a

linear IA-based precoder design has been utilized to make an

RIS-enabled MU-MIMO network. Additionally, modiﬁed pre-

coder and postcoder matrix algorithms have also been provided

to mitigate reﬂected interference terms. Our numerical results

have been demonstrated that IA-based an RIS-enabled MU-

MIMO surpasses its counterpart in terms of secrecy sum rate.

This is due to the interference signals coming from legitimate

user links being severely degraded to the eavesdropper’s SNR

while legitimate users combating interference with the aid of

IA.

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