1 Nonlocal Reconfigurable Intelligent Surfaces for Wireless Communication

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Nonlocal Reconﬁgurable Intelligent Surfaces for

Wireless Communication:

Modeling and Physical Layer Aspects

Amine Mezghani, Faouzi Bellili, and Ekram Hossain

University of Manitoba, Winnipeg, MB, R3T 5V6, Canada.

Emails: {amine.mezghani,faouzi.bellili,ekram.hossain}@umanitoba.ca

Abstract—Conventional Reconﬁgurable intelligent surfaces

(RIS) for wireless communications have a local position-

dependent (phase-gradient) scattering response on the surface.

We consider more general RIS structures, called nonlocal (or

redirective) RIS, that are capable of selectively manipulate the

impinging waves depending on the incident angle. Redirective

RIS have nonlocal wavefront-selective scattering behavior and

can be implemented using multilayer arrays such as metal-

enses. We demonstrate that this more sophisticated type of

surfaces has several advantages such as: lower overhead through

coodebook-based reconﬁgurability, decoupled wave manipula-

tions, and higher efﬁciency in multiuser scenarios via multi-

functional operation. Additionally, redirective RIS architectures

greatly beneﬁt form the directional nature of wave propagation at

high frequencies and can support integrated fronthaul and access

(IFA) networks most efﬁciently. We also discuss the scalability

and compactness issues and propose efﬁcient nonlocal RIS archi-

tectures such as fractionated lens-based RIS and mirror-backed

phase-masks structures that do not require additional control

complexity and overhead while still offering better performance

than conventional local RIS.

Index Terms—Reconﬁgurable Intelligent Surfaces, local and

nonlocal metasurfaces, Directional communication, angular ﬁl-

tering/conversion, RF routing, Control overhead, Retroreﬂective

channel estimation, Integrated (in-band) Fronthaul and Access,

Physically secured communication.

I. INTRODUCTION

The network capacity and coverage of previous cellular

systems have been expanded mainly by deploying more access

points and adding more frequency bands. Despite the rapid

technological progress, a cost-effective way to deliver ubiqui-

tous multi-Gpbs mobile data transmissions anywhere/anytime

remains however still a challenging problem, since all current

cellular technologies become very capital intensive beyond a

certain average rate per user threshold [1].

Higher network capacity and peak data rates are theoret-

ically conceivable at the mmWave and THz spectrum, due

to the availability of larger bandwidths and the realizability

of more directional transmissions in 3D. The main challenge

for these spectrum parts is, however, the reduced propagation

range and exacerbated blockage issues in addition to the radio

frequency (RF) power limitations, especially in the uplink. As

a consequence, mmWave/TeraHz mobile access intrinsically

requires small-cell deployment with a line-of-sight (LOS) path

or strong mirror-like reﬂective paths and is currently targeting

ultra-high-density areas. Another issue is the trafﬁc variation

and the strongly asymmetric performance between the uplink

and the downlink, thereby limiting the deployment scenarios.

It becomes clear then that more innovations in network

design and communication techniques are needed to enable

wider deployment of mmWave/TeraHz technologies. In fact, a

simple expansion by adding radio nodes with more backhaul

infrastructure signiﬁcantly increases cost and complexity while

potentially leading to an underutilized system [1] due to the

stronger impact of time-variant user distribution and trafﬁc

ﬂuctuations on dense networks. It is also worth mentioning that

user-centric generalized coverage by multiple base stations is

also critical even in lower frequencies. In addition, controlled

cell range extension for network resilience (to node failures for

instance) or mobile broadband coverage in rural areas is also

an open problem even at sub-6GHz frequencies necessitating

concepts for low-cost ﬂexible network edge components. This

calls for innovation in network design that could enable an

access point to multiplex across different areas, optimizing

both spatial coverage and temporal utilization of the network.

To address these issues, the use of reconﬁgurable intelligent

surfaces (RIS) [2], [3], also known as smart passive and active

(network-controlled) repeaters, is being widely investigated in

academia and industry as a potentially low-cost, low-power

stopgap solution for mitigating the coverage issue of future

networks. While densiﬁcation is unavoidable for widespread

service availability, another equally important goal is to reduce

the number of required sophisticated and energy-intensive

nodes/cells in the network and distribute them sparsely and

ﬂexibly (closer to existing ﬁber points of presence) so as

to make mobile mmWave/TeraHz communication economi-

cally viable. To be considered as a cheaper, competitive, and

economically scalable alternative to small cells for network

expansion, RIS-based edge nodes scattered around a central

baseband node have to meet the following criteria [3]

•Low-power/quasi-passive (e.g., solar-powered), low-cost,

low-maintenance, lightweight, and easy-to-deploy and

operate nodes for instance on an aerial platform

•Full-duplex operation, i.e., on-channel manipulation (as

opposed to standard relays) and with very low latency

(data forwarding with no processing delay)

•Restricted RF and physical layer capabilities (transparent

layer-0 nodes with only control channel receiver and no

transmitter chain)

•In-band or out-of-band IoT-dedicated remote control link

(e.g., via a small control bandwidth partition), which can

arXiv:2210.05928v2 [cs.IT] 3 Apr 2024

High site/Gateway (multifaceted)

Ultra-high capacity

RIS

Fig. 1: Streetlight-mounted RIS nodes assisting an ultra-high

capacity cell site.

be decoded without demodulating the entire waveform).

Furthermore, the form factor and aesthetic aspects of future

wireless equipment will also be critical to not further impact

the urban landscape and to keep wireless radiations and

deployments at levels that are acceptable by the general public

while meeting the levels already imposed by regulations.

In this context, new antenna designs and deployment strate-

gies are being considered to enable multipurpose reconﬁg-

urable intelligent active/passive surfaces/scatterers (RIS) that

can favorably alter wave propagation. These antennas are

ultrathin surfaces with subwavelength (i.e., microscopic) struc-

tures, also known as metasurfaces with 2D applied/induced

magnetic and electric currents along the surface [2]. In the

context of passive metasurfaces, the subwavelength structure

offers a ﬂexible and efﬁcient wavefront manipulation such as

beam-steering of the incident wave by optimizing the surface

impedance distribution. A popular use case for RIS is when

there are obstructions between the transmitter and the receiver

that lead to a communication failure. By bending/deﬂecting RF

beams in 3D, one can easily navigate around the obstructions

along appropriate trajectories (c.f. Fig. 1). For sub-6 GHz

frequencies, RIS can be used to eliminate the coverage gap in

rural areas by closing multiple links to far-away base stations

on the edge of the network that are generally underutilized as

shown in Fig. 2.

While the majority of prior work has focused on phased

reﬂective1RIS architecture with local wave manipulation [2],

recent research papers have considered nonlocal designs such

as directive reconﬁgurable surfaces [4], [5], layered RIS with

multi-plane wave conversion [6], [7], as well RIS with non-

diagonal phase-shift matrices [8]. However, the existing RIS-

based directive communication designs rely, for instance, on

the use of separate reception and re-transmission arrays that

are connected in a back-to-back manner, thereby resulting in

a very spatially selective wave manipulation. In contrast to

1For simplicity, transmissive RIS conﬁgurations are also categorized as

being reﬂective throughout this paper.

RIS

Rural area

Suburban area Suburban area

Fig. 2: Closing the coverage gap in rural areas with RIS-

enabled multiple data pipes that exploit the extra capacity of

lightly loaded base stations.

reﬂective conﬁgurations, redirective systems are very common

in practice and are known as bent-pipe (through) relays in

satellite communications and as (network-controlled) smart

repeaters in the 5G industry [9], [10]. From the metamaterial

perspective, redirective antenna systems can be regarded as

nonlocal metasurfaces due to their angle-dependent processing

[11]. Nonlocal RIS design offers additional degrees of freedom

to design wave manipulations that are not possible with

conventional local designs. However, designs based on layered

RIS [6], [7] suffer from higher complexity, control overhead,

and losses as compared with the conventional local design.

Local metasurfaces only provide pointwise scattering re-

sponses and have restricted functionalities due to the local

power conservation requirement. By contrast, nonlocal meta-

surfaces allow for more general responses that are angular (or

wave vector) dependent [11]–[13]. Throughout the paper, RIS

design that is implemented using local metasurfaces is also

described as reﬂective, whereas nonlocal metasurface-based

design is also termed redirective RIS.

The question of which type of RIS scattering, conﬁguration,

and controllability to use for cost-effective mmWave//TeraHz

networks with low control and training overhead remains,

however, an open research question. This paper is an attempt

to provide a systematic comparison of several aspects of these

two different types of RIS deployment. While some ques-

tions remain open (regarding compactness and scalability),

our conclusion is quite in favor of network conﬁgurations

with redirective RIS-based quasi-passive nodes, which provide

better scaling in terms of aperture, bandwidth, control, and

estimation overhead. We also show that surprisingly simple

salable nonlocal RIS designs that do not require additional

complexity or overhead as compared to local designs are

potentially possible while offering better performance.

Notation: Vectors and matrices are denoted by lower- and

upper-case italic boldface letters. The operators (•)T,(•)H,

tr(•), and (•)∗stand for transpose, Hermitian (i.e., conjugate

transpose), trace, and complex conjugate, respectively. The

identity matrix is denoted as Iand the size will be understood

from the context. All vectors are in column-wise orientation

by default and xiis the i-th element of x. The i-th column

of a matrix Xis denoted as xiwhile [X]i,jstands for its

(ith, jth) entry. We represent the Kronecker and Hadamard

product of vectors and matrices by the operators "⊗" and "⊙",

respectively. Additionally, both Diag(X)and Diag(x)return

2−θ

bα,1

aα,1 bα,m

aα,m

aβ(θ,φ)

bβ(θ,φ)

Fig. 3: Antenna scattering with maccessible ports.

a diagonal matrix by, respectively, keeping only the diagonal

elements in X, or placing the vector xon its main diagonal.

Finally, jis the imaginary unit (i.e., j2=−1), and the notation

≜is used for deﬁnitions.

II. PHYSICALLY-CONSISTENT MODELING OF RIS

The modeling of antenna arrays or reconﬁgurable surfaces

is becoming critical for the design and optimization of fu-

ture wireless systems [2], [3]. In this section, we provide a

general modeling methodology for antenna arrays as well as

reconﬁgurable surfaces based on the linear scattering theory.

Contrary to prior work [14], we adapt the linear scattering

description rather than the impedance description since it is

easier to measure and gives a more informative and intuitive

interpretation of the model. In particular, we show that the

conventional model that ignores the mutual coupling effects

can be regarded as a ﬁrst-order approximation of the physically

consistent model.

A. Linear Scattering Model

As shown in Fig 3, the electromagnetic properties of an

antenna array or a scatterer are described by its radiat-

ing/receiving patterns, the space-side scattering pattern, as

well as the electrical multi-port behavior of its terminals

[15] (assuming lumped elements connected to the structure).

For simplicity, we treat the electromagnetic ﬁeld, which is

generally a vectorial complex quantity, as a complex-valued

scalar quantity (e.g., considering a single polarization). At

each mth accessible port, the forward and backward traveling

(voltage) wave phasors along the antenna feed line are denoted

by aα,mand bα,m, respectively. At a distant sphere, the angular

transmission characteristic of a certain polarization when the

antenna port mis connected to a wave-source amplitude aα,m

(in [√W]) at port mis deﬁned as (also known as complex

pattern)

sm(θ,φ) = lim

r→∞ −jrejkr E(m)(θ,φ,r)

aα,mp1/η0,(1)

where η0is the characteristic impedance of free space and

E(m)(θ,φ,r)is the corresponding generated electrical far-

ﬁeld. These element-embedded complex patterns, sm(θ,φ),

are stacked together to form the receiving/transmitting charac-

teristic vector sαβ (θ,φ). Assuming single polarization hemi-

spherical radiation, the general antenna array can now be char-

acterized by the linear scattering description for the terminal

side

bα=Sααaα+Zπ

0Zπ

−π

sαβ (θ,φ)aβ(θ,φ)sinθdφdθ,(2)

where aα≜[aα,1,. . . ,aα,M]Tand Sαα is the scattering

matrix . For the space-side, the angular spectra of the outgoing

propagating wave phasors bβ(θ,φ)are expressed as a linear

functional of the incoming one aβ(θ,φ)as well as the port

incident phasor

bβ(θ,φ) = sαβ (θ,φ)Taα+

Zπ

0Zπ

−π

sββ (θ,φ,θ′,φ′)aβ(θ′,φ′)sinθ′dφ′dθ′,

(3)

where sββ (θ,φ,θ′,φ′)is the wave back-scattering character-

istic which is in general hard to determine experimentally.

Ideally, sββ (θ,φ,θ′,φ′)is effectively negligible or is just

considered as part of the ﬁxed (direct) communication channel.

In the context of RIS deployment, the antenna ports are

terminated by loads characterized by the load scattering matrix

SL, i.e., bα=SLaα. Consequently, we get the total scattering

bβ(θ,φ) = sαβ (θ,φ)T(S−1

L−Sαα)−1×

Zπ

0Zπ

−π

sαβ (θ′,φ′)aβ(θ′,φ′)sinθ′dφ′dθ′+

Zπ

0Zπ

−π

sββ (θ,φ,θ′,φ′)aβ(θ′,φ′)sinθ′dφ′dθ′

| {z }

Residual scattering

(4)

For passive RIS, the spectral radius of this scattering opera-

tor (aβ(θ,φ)−→ bβ(θ,φ)) is less or equal to one. That is, the

scattered power is less than or equal to the impinging power

regardless of the impinging wave

Zπ

0Zπ

−π

|bβ(θ,φ)|2sinθdφdθ≤Zπ

0Zπ

−π

|aβ(θ,φ)|2sinθdφdθ.

(5)

If the antenna is lossless then we have equality and the

operator is unitary with all eigenvalues having unit absolute

values. A simplistic approach that is commonly used in prior

work is to use the Neumann series approximation

(S−1

L−Sαα)−1=∞

k=0

(SLSαα)kSL=SL+SLSααSL+. . . ,

(6)

thus, bβ(θ,φ)main-reﬂ. ≈sαβ (θ,φ)TSL×

Zπ

0Zπ

−π

sαβ (θ′,φ′)aβ(θ′,φ′)sinθ′dφ′dθ′,

(7)

which neglects the residual backscattering and the high-order

multiple reﬂections between the antenna and load given by (6).

This applies for Sαα =0(or ρ(SLSαα)≪1), i.e., perfectly

matched array (or high losses). In general, the exact expression

might give better results since (S−1

L−Sαα)−1might have

singular values larger than 1 as opposed to SL. Nevertheless,

the main reﬂection approximation in (7) has a more tractable

mathematical structure for optimizing the reactive load SL,

the reason for which it is widely used in the literature.

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1NonlocalReconfigurableIntelligentSurfacesforWirelessCommunication:ModelingandPhysicalLayerAspectsAmineMezghani,FaouziBellili,andEkramHossainUniversityofManitoba,Winnipeg,MB,R3T5V6,Canada.Emails:{amine.mezghani,faouzi.bellili,ekram.hossain}@umanitoba.caAbstract—ConventionalReconfigurableintelligents...

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