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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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