Dual Gradient Descent EMF-Aware MU-MIMO Beamforming in RIS-Aided 6G Networks Yi Yu Rita Ibrahim and Dinh-Thuy Phan-Huy

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Dual Gradient Descent EMF-Aware MU-MIMO

Beamforming in RIS-Aided 6G Networks

Yi Yu, Rita Ibrahim and Dinh-Thuy Phan-Huy

Orange Labs

92320, Chˆ

atillon, France

Email: yu.yi@orange.com, rita.ibrahim@orange.com, dinhthuy.phanhuy@orange.com

Abstract—Reconﬁgurable Intelligent Surface (RIS) is one of

the key technologies for the upcoming 6th Generation (6G)

communications, which can improve the signal strength at the

receivers by adding artiﬁcial propagation paths. In the context

of Downlink (DL) Multi-User Multiple-Input Multiple-Output

(MU-MIMO) communications, designing an appropriate Beam-

forming (BF) scheme to take full advantage of this reconﬁgured

propagation environment and improve the network capacity is a

major challenge. Due to the spatial dimension provided by MIMO

systems, independent data streams can be transmitted to multiple

users simultaneously on the same radio resources. It is important

to note that serving the same subset of users over a period of time

may lead to undesired areas where the average Electromagnetic

Field Exposure (EMFE) exceeds regulatory limits. To address

this challenge, in this paper, we propose a Dual Gradient

Descent (Dual-GD)-based Electromagnetic Field (EMF)-aware

MU-MIMO BF scheme that aims to optimize the overall capacity

under EMFE constraints in RIS-aided 6G cellular networks.

Index Terms—Dual gradient descent, EMF exposure, MU-

MIMO, RIS, Reinforcement learning, 6G networks.

I. INTRODUCTION

6G has enormous commercial potential and is attracting

attention from both academia and industry [1]. Various innova-

tive technologies are being extensively studied for application

in the 6G era. One of these hot topics is the Reconﬁgurable

Intelligent Surface (RIS), which is essentially a large array

of low-cost passive components that performs phase shift of

incident waves to reﬂect them in the desired direction [2]. In

this way, additional propagation paths can be artiﬁcially added

to reconﬁgure the propagation environment and improve the

link budgets between transmitters and receivers.

Meanwhile, in the 6G era, operators will continue to

leverage the Multi-User Multiple-Input Multiple-Output (MU-

MIMO) technology [3] with massive MIMO (M-MIMO) an-

tennas to meet increasing data rate demands. Downlink (DL)

MU-MIMO technology enables efﬁcient spatial multiplexing

by applying appropriate Beamforming (BF) weights that direct

signals to target devices and mitigate or eliminate the inﬂuence

of interfering data streams.

However, sometimes the radiation patterns generated by DL

MU-MIMO BF may produce some undesired areas of strong

Electromagnetic Field Exposure (EMFE). The International

Commission on Non-Ionizing Radiation Protection (ICNIRP)

[4] has speciﬁed the average limits of human exposure to Elec-

tromagnetic Field (EMF) for a given time period [5]. These

EMFE limits are habitually respected due to some averaging

factors met in the network such as scheduling decision, trafﬁc

demand, users’ spatial distribution etc. However, respecting

EMFE limits becomes more challenging [6] when the same

subset of users is served for long periods of time, such as in

ﬁxed wireless access use cases.

Therefore, it is crucial to deploy in the network an efﬁcient

EMF-aware MU-MIMO BF that meets the high requirements

of 6G networks. Dual Gradient Descent (Dual-GD) is an itera-

tive Reinforcement Learning (RL) algorithm, which can cope

with optimization problems under multiple linear inequality

constraints. This is suitable for our problem: designing a MU-

MIMO BF scheme that maximizes the network capacity under

maximum transmit power constraint and EMFE constraints

on all the observation points. The key idea of the Dual-GD

technique is transforming the original constrained optimization

problem into a Lagrange dual function which can be optimized

iteratively. This algorithm involves an alternation between

maximizing the Lagrangian function with respect to the primal

variables and decrementing the Lagrange multipliers by theirs

gradients. By repeating this iteration, we can gradually adjust

the Lagrangian multiplier corresponding to each constraint

according to its impact on the optimization objective, and the

solution will converge.

In [7]–[10], the authors have proposed different EMF-aware

BF schemes in RIS-aided Single-User MIMO (SU-MIMO)

networks. In [11], we have focused on the RIS-aided MU-

MIMO scenario and proposed two EMF-aware BF schemes:

(i) ”reduced” EMF-aware BF which consists of decreasing the

overall transmit power until the EMFE limits are fulﬁlled and

(ii) ”enhanced” EMF-aware BF with a per-layer power control

mechanism. In this paper, we reﬁne these ﬁndings and propose

a novel Dual-GD based EMF-aware MU-MIMO BF scheme

that enhances furthermore the network capacity while strictly

satisfying EMFE constraints.

The rest of the paper is organized as follows. A 6G MU-

MIMO RIS-aided network model is deﬁned in section II. The

”reference” BF scheme (i.e. without any EMFE constraint) is

presented in section III. We describe the details of the Dual-

GD EMF-aware BF scheme in the section IV. Then, in section

V, the performance of the Dual-GD BF scheme in terms of

DL channel capacity and power efﬁciency is evaluated and

compared with other MU-MIMO BF schemes (i.e. ”reference”,

”reduced” and ”enhanced”). Finally, the paper is concluded in

section VI.

arXiv:2210.00766v1 [cs.IT] 3 Oct 2022

II. SYSTEM MODEL

In this section, we consider the DL MU-MIMO communica-

tions of a RIS-aided cellular network. As shown in Fig. 1, we

assume a single cell scenario with Ldifferent User Equipment

(UEs). The Base Station (BS) is equipped with a 2D antenna

array of Mtransmitting antenna elements. According to the

3GPP standard [12], the BS is modeled by a uniform rectangu-

lar panel array, with NHthe number of columns and NVthe

number of antenna elements with the same polarization in each

column. We assume that the antenna panel is dual polarized

(i.e. P= 2). So M=NHNVP. Both the horizontal dH

and vertical dVantenna spacing are equal to 0.5λ, where λ

indicates the wavelength of the carrier frequency. Each UE is

equipped with Nreceiving antenna elements spaced by 0.5λ.

The total number of received antennas is thus Nt=LN .

Assume that Sscatterers and ZRISs are randomly dis-

tributed in the given cell space. Each RIS is equipped with

a linear array of Kelements spaced by 0.5λ. Both scatterers

and RISs are assumed far from the BS and the UEs, therefore

for simplicity, we consider the far-ﬁeld calculation method, i.e.

the electromagnetic waves propagate at the speed of light and

electric and magnetic ﬁelds are mutually perpendicular [5].

We consider an Orthgonal Frequency Division Multiplexing

(OFDM) waveform and random Rayleigh fading. The network

adopts Time Division Duplex (TDD) mode and thus the chan-

nel reciprocity is feasible. With MU-MIMO, multiple streams

are sent from the BS to distinct active UEs simultaneously.

These streams are spatially multiplexed by using appropriate

BF schemes. In our work, an adapted channel inversion BF

is applied: Zero Forcing (ZF) precoding adapted to multiple

receiving antennas scenario.

In this paper, the main focus is on the BF at the BS side.

The joint optimization of the BS and RIS BF weights is the

subject of our future work. Here, we assume that the RISs

are randomly distributed as reﬂective surfaces to work on

transmitting the incident signal to a speciﬁc UE. The reﬂection

weights at the RIS side are selected based on the following

procedure:

•Each UE sends some pilots which allow the RIS to

estimate the UE-to-RIS channels;

•Based on the UE-to-RIS channel estimation, each RIS

computes the BF reﬂection weight wz∈CK×1;

•The weight wzis multiplied by a reﬂection amplitude

rris, where 0≤rris ≤1is a constant value depending

on the hardware structure of the RIS. Here we set rris =

1/K;

•Then each RIS applies these reﬂection weights and

freezes;

Once the RISs are conﬁgured, the UE sends pilots again in

such a way that the BS can estimate the DL channel taking into

account the RIS conﬁguration and determine the appropriate

BF weight to be used for data transmission.

In order to satisfy the EMF exposure compliance, a safety

circle of radius Rcentered at the BS is deﬁned. Outside

this safety circle, the received power at any location within

Fig. 1. A MU-MIMO RIS-aided Network Model

the observation range should not exceed a given threshold

EMFth ∈R+. The safety circle, also known as the exclusion

zone, is guaranteed to be closed to the public.

In our network model, there are three different kinds of

propagation paths:

1) m→Ul

ndenotes direct Line of Sight (LoS) propagation

from the mth BS antenna element to the nth antenna

element of the lth UE.

2) m→s→Ul

nindicates the path from the mth antenna

element of the BS to the nth antenna element of the lth

UE through scatterer s.

3) m→Rz

k→Ul

nis the path from mth BS antenna

element to the nth antenna element of the lth UE,

through kth antenna element of the zth RIS.

with 16m6M,16n6N,16l6L,16k6K,

16s6Sand 16z6Z.

According to the 3GPP standardization [12], the 3D antenna

radiation pattern of each antenna element in the horizontal cut

is generated as:

AdB (θ= 90◦, φ) = −min (12 φ

φ3dB 2

, Amax),(1)

with φ3dB = 65◦,Amax = 30 dB and φ∈[−180◦,180◦]is

the azimuth angle.

In case of polarized antennas, the polarization is modeled

as angle-independent in both azimuth and elevation. In the

horizontal polarization, the antenna element ﬁeld component

is given by:

Fθ,φ =pABeam (θ, ϕ) sin(ζ),(2)

with ζ= +/−45◦being the polarization slant angle corre-

sponds to a pair of cross-polarized antenna elements. For the

detailed calculation of the 3D radiation pattern ABeam (θ, ϕ)

of the entire antenna array, please refer to Appendix A.

The propagation channel Hl∈CN×Mbetween the BS and

a given UE lthrough the considered scatterers and RISs is

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DualGradientDescentEMF-AwareMU-MIMOBeamforminginRIS-Aided6GNetworksYiYu,RitaIbrahimandDinh-ThuyPhan-HuyOrangeLabs92320,Chatillon,FranceEmail:yu.yi@orange.com,rita.ibrahim@orange.com,dinhthuy.phanhuy@orange.comAbstractRecongurableIntelligentSurface(RIS)isoneofthekeytechnologiesfortheupcoming6thGen...

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Dual Gradient Descent EMF-Aware MU-MIMO Beamforming in RIS-Aided 6G Networks Yi Yu Rita Ibrahim and Dinh-Thuy Phan-Huy

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