Content uploaded by Konstantinos Ntontin

Author content

All content in this area was uploaded by Konstantinos Ntontin on Mar 23, 2021

Content may be subject to copyright.

1

Reconﬁgurable Intelligent Surface Optimal Placement in

Millimeter-Wave Networks

Konstantinos Ntontin, Member, IEEE, Alexandros–Apostolos A. Boulogeorgos, Senior Member, IEEE,

Dimitrios Selimis, Graduate Student Member, IEEE, Fotis Lazarakis,

Angeliki Alexiou, Member, IEEE, and Symeon Chatzinotas, Senior Member, IEEE

This work discusses the optimal reconﬁgurable intelligent surface placement in highly-directional millimeter wave links. In

particular, we present a novel system model that takes into account the relationship between the transmission beam footprint at

the RIS plane and the RIS size. Subsequently, based on the model we derive the end-to-end expression of the received signal power

and, furthermore, provide approximate closed-form expressions in the case that the RIS size is either much smaller or at least equal

to the transmission beam footprint. Moreover, building upon the expressions, we derive the optimal RIS placement that maximizes

the end-to-end signal-to-noise ratio. Finally, we substantiate the analytical ﬁndings by means of simulations, which reveal important

trends regarding the optimal RIS placement according to the system parameters.

Index Terms—Optimal placement, Reconﬁgurable intelligent surfaces, Signal-to-noise ratio analysis.

NOMENCLATURE

B5G Beyond the Fifth Generation

FNBW First-Null Beamwidth

HPBW Half-Power Beamwidth

LoS Line-of-Sight

mmWave Millimeter Wave

NLoS Non-Line-of-Sight

PIN Positive-Intrinsic-Negative

RF Radio-Frequency

RIS Reconﬁgurable Intelligent Surface

RU Reﬂection Unit

RX Receiver

SNR Signal-to-Noise-Ratio

TX Transmitter

I. INTRODUCTION

A. Background and Related Works

Increasing data-rate demands have led current mobile-access

networks relying on sub-6 GHz bands reach their limits in

terms of available bandwidth. This bottleneck created the need

to consider beyond-6 GHz bands for mobile-access networks.

K. Ntontin is with the Interdisciplinary Centre for Security, Reliability

and Trust (SnT) – University of Luxembourg, L-1855 Luxembourg and the

Wireless Communications Laboratory of the Institute of Informatics and

Telecommunications, National Centre for Scientiﬁc Research–“Demokritos,”

Athens, Greece. E-mail: kostantinos.ntontin@uni.lu.

A.-A. A. Boulogeorgos and A. Alexiou are with the Department of

Digital Systems, University of Piraeus Piraeus 18534 Greece. E-mails:

al.boulogeorgos@ieee.org, alexiou@unipi.gr.

D. Selimis and F. Lazarakis are with the Wireless Communications

Laboratory of the Institute of Informatics and Telecommunications, Na-

tional Centre for Scientiﬁc Research–“Demokritos,” Athens, Greece. E-mails:

{dselimis, ﬂaz}@iit.demokritos.gr.

S. Chatzinotas is with the Interdisciplinary Centre for Security, Reliability

and Trust (SnT) – University of Luxembourg, L-1855 Luxembourg. E-mail:

symeon.chatzinotas@uni.lu.

This work was supported by the European Commission’s Horizon 2020

research and innovation programme (ARIADNE) under grant agreement No.

871464 and the Luxembourg National Research Fund (FNR) under the CORE

project RISOTTI.

The associate editor coordinating the review process and accepting it for

publication was Prof. Emil Bj¨

ornson.

Currently, bands in the lower-end of the millimeter wave

(mmWave) spectrum are used for point-to-point and point-

to-multipoint line-of-sight (LoS) wireless backhaul/fronthaul

and ﬁxed-wireless access networks [1]. Such deployments

span the 30-100 GHz operational frequency range. However,

the expected migration of future mobile-access networks to

the 30-100 GHz band pushes the corresponding wireless

backhaul/fronthaul links towards the beyond-100 GHz bands.

Due to this, backhauling/fronthauling transceiver equipment

vendors have performed LoS trials in the D-band (130−174.8

GHz), which showcase the potential of using it in such

deployments [2]. Apart from LoS, street-level deployments in

dense urban scenarios necessitate devising non-LoS (NLoS)

solutions since LoS links may not always be available. How-

ever, despite the fact that according to measurements [3], [4]

NLoS communication through scattering and reﬂection from

objects in the radio path is feasible in the 30-100 GHz range,

the higher propagation loss of beyond-100 GHz bands is likely

to challenge this assumption.

The conventional approach of counteracting NLoS links is

by providing alternative LoS routes through relay nodes [5].

Although this is a well-established method to increase the

coverage when the signal quality of the direct links is low, it is

argued that it cannot constitute a viable approach for massive

deployment, especially for mmWave networks. This is due to

the increased power consumption of the active radio-frequency

(RF) components in high frequencies that relays need to

be equipped with [6]. Apart from relaying, communication

through passive non-reconﬁgurable specular reﬂectors, such

as dielectric mirrors, has been proposed as another alternative.

Such an approach has the potential to be notably more cost

efﬁcient compared with relaying and has been documented at

both mmWave and beyond-100 GHz bands [6], [7]. Due to

the highly dynamic nature of blockage at high frequencies

together with the trafﬁc conditions, which may necessitate

fast rerouting of information within a network, it would be

desirable that such reﬂectors can change the angle of departure

of the waves so that they direct the beams towards different

routes. However, passive reﬂectors are incapable of supporting

2

the aforementioned functionality since the conventional Snell’s

law applies. Furthermore, even by enabling this functionality

by means of mechanical steering of the passive reﬂectors, the

resulting latency would substantially compromise the desired

reliability. Based on the above, an intriguing question that

arises is the following: Would it be possible to deploy reconﬁg-

urable reﬂectors that can arbitrarily steer the impinging beam

based on dynamic blockage and trafﬁc conditions and without

compromizing the desired latency? The answer is afﬁrmative

by considering the reconﬁgurable intelligent surface (RIS)

paradigm.

RISs are two-dimensional structures of dielectric material,

which embed tunable reﬂection units (RUs) [8]–[12]. They

constitute a substantially different technology than active re-

laying, due to the absence of bulky and power-hungry analog

electronic components, such as power ampliﬁers. Additionally,

their operation, in contrast with active relaying, does not

require dividers and combiners, which can incur high insertion

losses. By individually tuning the phase response of each

individual RIS element, the reﬂected signals can constructively

aggregate at a particular focal point, such as the receiver. Such

a tuning can be enabled by electronic phase-switching com-

ponents, such as positive-intrinsic-negative (PIN) diodes, RF-

microelectromechanical systems, and varactor diodes, that are

introduced between adjacent elements [13]. Hence, RISs offer

an alternative-to-relaying method for large-scale beamforming

without the incorporation of high power consuming electronics

and insertion losses involved by the additional circuitry. In

practice, the RIS element phase shift can be controlled by a

central controller through programmable software [13].

Recognizing the unprecedented features that RISs can bring

to beyond the ﬁfth generation (B5G) wireless systems, a great

amount of research effort has been put on analyzing, designing

and optimizing RIS-aided wireless systems [13]–[20], as well

as comparing them with their predecessors, i.e., relaying-aided

ones [11], [21]–[23]. In more detail, in [13] and [14] the au-

thors introduced the idea of employing an RIS in order to mit-

igate the impact of blockage and steer the transmission beam

towards the desired direction. Likewise, in [15] the authors

presented the optimization framework for the maximization of

the reception power in a RIS-aided system, assuming that all

the RIS area can be used to reﬂect the induced electromagnetic

wave. Moreover, in [16] the authors studied the asymptotic

uplink ergodic capacity performance of an RIS-aided wireless

system, while in [17] the coverage of a downlink RIS-assisted

network was studied, assuming that the entire RIS area can

be used, and a strategy for maximizing the cell coverage by

optimizing the RIS orientation and horizontal distance was

proposed. In [18], the RIS empowered holographic multiple-

input-multiple-output architecture is introduced, whereas in

[19] the joint design of transmit beamforming matrix at the

base station and the phase shift matrix at the RIS in a

multiuser multiple-input-multiple-output setup is investigated.

In addition, [20] provides an extensive survey on RIS-related

works in various domains. Finally, several works consider RISs

operating as reﬂectors and show that sufﬁciently large RISs

can outperform conventional active relays either is terms of

rate or energy efﬁciency [11],[21]–[23].

B. Motivation, Novelty, and Contribution

All the presented RIS-related works consider the case of

the entire RIS area being illuminated by the transmitted

beam. However, due to the highly directional transmissions in

mmWave networks and the low manufacturing cost of RISs,

which make them suitable, as it is envisioned, to cover a big

portion of the facades of large structures, such as buildings, it

is expected that in many cases only a part of the total RIS area

is going to be illuminated. Based on this, our work is motivated

by the need to answer the question of what the optimal RIS

placement policy is, which can be seen as a network planning

question, in the two cases of the RIS area being smaller and

larger than the transmitted beam footprint. Summarizing, the

technical contribution of the paper is as follows1:

•We present a system model for RIS-aided highly direc-

tional mmWave links of ﬁxed topology, such as wireless

backhaul/fronthaul links, and use electromagnetic theory

to evaluate the received power in the general case of

an RIS of arbitrary size. Of note, the presented ana-

lytical methodology can also ﬁnd application in mobile

mmWave networks, as it is elaborated in Section IV.

•Based on the resulting received-power expression, we

provide approximate closed-form expressions for the

cases in which the transmission footprint at the RIS plane

is either much larger or smaller than the RIS. According

to the expressions, we evaluate the end-to-end SNR for

both cases.

•We use the closed-form SNR expressions to analytically

extract the policies for the optimum RIS placement that

results in SNR maximization.

•Finally, we provide an extensive simulation campaign

in various scenarios in order to validate the analytical

results and, furthermore, to provide design guidelines to

the system designer from a practical point of view.

Organization: The rest of this contribution is structured

as follows: In Section II, the system model is presented.

In Section III, we ﬁrstly provide an expression for the

transmission beam footprint at the RIS plane. Subsequently,

based on it, through electromagnetic theory we compute the

end-to-end received power. In addition, we provide closed-

form approximate expressions for the received power and,

correspondingly, for the end-to-end SNR, in the two cases of

the RIS being either much smaller than then transmission beam

footprint or larger. Finally, in the same section by leveraging

the analytical SNR expressions we mathematically compute

the optimal RIS placement that maximizes the SNR. Extensive

numerical results that validate the analytical outcomes together

with a discussion of how the system designer can use the

presented results are provided in Section IV. Finally, Section V

concludes this work by highlighting the most important ﬁnd-

ings and remarks.

Notation: For the convenience of the readers, recurrent

parameters and symbols with their meaning are presented in

Table I.

3

TABLE I

RECURRENT PARAMETERS AND SYMBOLS.

Parameter/Symbol Meaning

fCarrier frequency

λWavelength

PtTransmit power

WSignal bandwidth

FdB Noise ﬁgure

N0Thermal noise power

hsRIS height with respect to the ground

ht/hrTX/RX height with respect to the ground

Dt/DrTX/RX antenna diameter

φ0TX antenna FNBW

φHP BW TX antenna HPBW

et/erTX/RX antenna aperture efﬁciency

Gmax

t/Gmax

rTX/RX antenna gain at the boresight

Gt,n/Gr,n TX/RX antenna gain with respect to the nth RU.

ΓAmplitude reﬂection coefﬁcient of the RUs

dx,dyx-axis and y-axis length, respectively, of the RUs

α,βRadii of the TX beam elliptic footprint

Eccentricity of TX beam elliptic footprint

r1

Distance between the center of the TX antenna and the center of

the TX footprint at the RIS plane

r2

Distance between the center of the TX footprint at the

RIS plane and the center of the RX antenna

r1,n Distance between the center of the TX antenna and the nth RU

r2,n Distance between the nth RU and the center of the RX antenna

r1,h TX-RIS horizontal distance

r∗

1,h Optimal RIS horizontal distance

rhTX-RX horizontal distance

θi,θr

Electromagnetic-wave incidence and departure angles, respectively,

with respect to the RIS center

θi,n,θr,n

Electromagnetic-wave incidence and departure angles, respectively,

with respect to the nth RU

SsRIS area

SiArea of the TX beam elliptic footprint corresponding to the FNBW

SHP BW Area of the TX beam elliptic footprint corresponding to the HPBW

PRReceived power

ρSNR

(r1,h,ys,hs)

hthr

hs

r1,h

φ0

θi

θr

α

β

TX RX

(0,0,0)

A

B

rh

Blocker

r1

C

ys

Fig. 1. System model.

II. SY S TE M MODE L

As illustrated in Fig. 1, we consider a ﬁxed-topology street-

level scenario, in which a TX communicates with a RX

through an RIS. r1,h,r2,h , and rhare the horizontal TX-

RIS, RIS-RX, and TX-RX distances, respectively, while ht,

hs, and hrare the TX, RIS, and RX heights, respectively. θi

and θrare the incidence and departure angles, respectively,

1This work constitutes an extension of [24].

of the electromagnetic wave with respect to the center of

the illuminated area. The considered TX-RIS and RIS-RX

blockage-free links are established in a mmWave band and

constitute an alternative path to the direct TX-RX link that is

assumed to be blocked. To countermeasure the high pathloss

in this band, both the TX and RX are equipped with highly

directional parabolic antennas with diameters Dtand Dr,

respectively. As a result, for Dt, Dr>> λ, where λrepresents

the wavelength, their power radiation patterns Et(φ)and

Er(φ), respectively, are given by [25]

Em(φ) = 2λ

πDm

J1πDmsin(φ)

λ

sin (φ),0≤φ < π/2(1)

for m∈ {t, r}.φis measured from the broadside direction

(φ= 0) and J1(·)is the ﬁrst-order Bessel function of the

ﬁrst kind. Hence, their gains, denoted by Gt(φ)and Gr(φ),

respectively, are given by

Gm(φ) = em4πE2

m(φ)

R2π

0Rπ

2

0E2

m(φ) sin (φ)dφdθ (2)

= 4em

J1πDmsin(φ)

λ

sin (φ)

2

, m ∈ {t, r},

where etand erdenote the aperture efﬁciencies of the TX

and RX antennas, respectively. Consequently, their maximum

gain, denoted by Gmax

m, that is obtained for φ= 0 is given by

Gmax

m=emπDm

λ2

, m ∈ {t, r}.(3)

Note that this type of antennas has been extensively used

for wireless backhaul/fronthaul scenarios (see e.g., [26] and

reference therein), due to their capability to support pencil-

beamforming transmissions. Under such highly-directional

transmissions, the three-dimensional antenna pattern can be

modeled as a cone for half-power beamwidths (HPBWs),

which we denote by φHP B W , smaller than approximately

15◦[27, Ch. 12]. Furthermore, we assume that the TX and

RX antennas can be mechanically steered, both in azimuth

and elevation, towards the desired angle of transmission and

reception, respectively, and they are pointing towards the

center of the illuminated RIS region.

As far as the channel model is concerned, the assumption

of mmWave links means that in the general case besides

the direct LoS component several distinguishable multipath

components also arrive at the RX either at the same or at

different time instants depending on whether a narrowband

or wideband model applies, respectively [28]. However, when

highly-directional antennas are employed at both the TX and

RX sides, in the case of a wideband channel, for instance,

that corresponds to common bandwidths at mmWave bands,

there is virtually no delay spread according to real-world

measurements [29]. Consequently, due to the considered ﬁxed-

topology scenario of this work with pencil-beam deployed

antennas and the fact that the RISs are deployed in elevated

positions with respect to the TX and RX positions so to ensure

strong direct TX-RIS and RIS-RX LoS conditions, we assume

free-space propagation for both the TX-RIS and RIS-RX links.

4

Remark 1: Since we only consider free-space propagation

in this work, the outcomes could potentially apply also to

sub-6 GHz links. However, we emphasize the mmWave case

from a practical viewpoint since street-level implementation of

transceiver nodes and RISs that can enable highly directional

transmissions could be much more feasible in mmWave bands.

This is attributed to the smaller packaging space needed in

mmWave bands to achieve the same antenna gain compared

with their sub-6 GHz counterparts. Consequently, we reckon

our work as much more tailored to mmWave bands under

practical deployment considerations.

The RIS acts as a beamformer, which by adjusting the phase

response of the RUs is capable of steering the beam at θr,

which is the RX direction. It consists of Nx×NyRUs of size

dx×dyand a controller that has perfect knowledge of the TX

and RX positions. Each RU is an electrically-small low-gain

element embedded on a substrate, with power radiation pattern

that can be expressed as in [30]

Gs(θ) = 4cos (θ),0≤θ < π/2.(4)

Regarding the pattern of (4), it is reported that it is suitable

for sub-wavelength RISs and it yields a good matching with

respect to measurements conducted [31], [30].

Due to the the fact that the TX-RIS and RIS-RX links are

directional LoS links, they are deterministic. Moreover, it is

assumed that the transmission power is Ptand that the received

signal is subject to additive white Gaussian noise with power

N0=−174 + 10 log10 (W) + FdB,(5)

where FdB is the noise ﬁgure in dB and Wis the transmission

bandwidth [32].

III. RIS ILL UM I NATE D AR EA, SNR, A ND OPTIMAL RIS

PL ACEM ENT

In this section, we ﬁrstly derive the illuminated RIS area.

Subsequently, we compute the received power assuming an

RIS of arbitrary size. Moreover, we provide approximate

expressions of the received power in the two cases of the RIS

area being either much smaller or larger than the TX beam

footprint. Finally, based on the corresponding approximate

expressions, we analytically derive the optimal RIS placement

for both cases.

A. RIS’s Illuminated Area

Since the main lobe of the TX antenna has a conical shape,

its footprint in the RIS plane is an ellipse, according to the

conic-section theory [33].

Lemma 1: Under the pencil-beam transmission assumption,

at least 97% of the transmit energy is located within the ﬁrst-

null beamwidth (FNBW) of the TX beam, which we denote

by φ0.

Proof: The proof is provided in Appendix A.

According to Lemma 1, almost all of the transmit energy for

pencil-beam transmissions is within the main lobe. Therefore,

without loss of generality, we approximate the illuminated

area at the RIS plane by the footprint corresponding to the

particular lobe.

Lemma 2: The two radii of the illuminated elliptic area at the

RIS plane that corresponds to the FNBW can be obtained as

α=

sin φ0

2

cos θi+φ0

2r1(6)

and

β=αp1−2,(7)

where

=sin (θi)

cos φ0

2.(8)

Moreover, r1denotes the distance between the center of the

TX and the center of the TX footprint at the RIS plane, while

θiis the incident angle at the RIS center with respect to its

broadside direction.

Proof: The proof is provided in Appendix B.

Based on (6) and (7), the TX main lobe footprint at the RIS

plane can be evaluated as

Si=παβ. (9)

The power that is reﬂected by the RIS is the one that

falls within

S= min (Si, Ss),(10)

where Ssdenotes the RIS area. If Ss≤Si, only part of the

power that falls within Sican be reﬂected towards the RX;

thus, beam waste occurs. On the other hand, if Ss> Sionly

part of the RIS is used to reﬂect the incident electromagnetic

wave2.

Finally, we note that in the case Ss> Sithere are some

RUs in the perimeter of the ellipse that are partly illuminated,

which would pose a challenge regarding how to adjust the

amplitude and phase response of the particular RUs. However,

by taking into account that the Ss> Sicase would correspond

to an illuminated RIS region of a relatively large size, ignoring

those elements in the RU response adjustment process is not

expected to have a notable effect on the resulting end-to-end

performance.

B. End-to-end SNR

The following proposition returns a tight approximation for

the received power.

Proposition 1: By adjusting the phase response of each

of the RUs in a way that the received reﬂected signals are

co-phased at the RX, which means that the received power,

denoted by PR, is maximized for certain r1,h and it can be

evaluated as3

PR=λ

4π4

PtΓ2

M

X

n=1 sGt,nGr,n Gs(θi,n)Gs(θr,n )

r2

1,nr2

2,n

2

,(11)

2In the Ss> Sicase, (10) corresponds to a very tight approximation due

to the fact that under pencil-beam transmissions at least 97% of the impinging

power is included within Si, according to Lemma 1.

3(11) holds under the assumption of negligible mutual coupling among the

RUs. Based on antenna theory, such an assumption approximately holds for

adjacent RU distance equal to λ/2.

5

where Γis the amplitude reﬂection coefﬁcient that we consider

is the same of all RUs, Mis the number of illuminated RUs

that are included within S. In the Ss> Sicase, we consider

that only the RUs that correspond to the FNBW are activated

since that region contains at least 97% of the impinging power,

according to Lemma 1. Consequently, it holds that

M=S

dxdy

.(12)

r1,n and r2,n are the distances between the centers of the TX

and RX antennas and the nth RU, respectively. In addition,

Gt,n and Gr,n represent the TX and RX antenna gains

corresponding to the same RU, respectively. Finally, Gs(θi,n )

is the gain of the nth RU towards the TX antenna and Gs(θr,n)

is its corresponding gain towards the RX antenna.

Proof: The proof is provided in Appendix C.

In the special case in which Si>> Ss, Proposition 2 that

follows presents a simpliﬁed closed-form expression for the

received power.

Proposition 2: If Si>> Ss, (11) is reduced to

PR=λ

4π4PtΓ2(Ss)2Gmax

tGmax

rGs(θi)Gs(θr)

d2

xd2

yr2

1r2

2

,(13)

where r2denotes the distance between the center of the TX

footprint at the RIS plane and the center of the RX.

Proof: We consider that Si>> Ssholds under far-ﬁeld

conditions, which means that the TX gain, RX gain, incident

and departure RU gains together with the corresponding TX-

RIS and RIS-RX distances are approximately independent of

n.

Lemma 3: Under the pencil-beam transmission assumption,

the amount of energy included within the FNBW can be

tightly approximated to a level of at least 97% by the amount

of energy included within a step function with magnitude

Gmax

tin the interval h−φHP BW

2,φHP BW

2i. Furthermore, the

particular amount of energy within the step function is at least

equal to 94% of the total impinging transmit energy at the RIS

plane.

Proof: The proof is provided in Appendix D.

In the special case in which Ss≥Si. Proposition 3 that

follows presents a simpliﬁed closed-form expression for the

received power.

Proposition 3: If: i) Ss≥Si; ii) Gr,n is aproximately

independent of nand equal to Gmax

r; and iii) the incident

and departure RU gains together with the corresponding TX-

RIS and RIS-RX distances are approximately independent of

n,PRis tightly approximated, by at least 94% accuracy, as

PR≈λ

4π4PtΓ2(SHP BW )2Gmax

tGmax

rGs(θi)Gs(θr)

d2

xd2

yr2

1r2

2

,(14)

where SHP B W is the HPBW footprint of the main lobe on

the RIS. SHP B W can be computed by the same process used

in the computation of Si, where in (6), (7), and (8), φ0should

be replaced by φHP B W .

Proof: The proof of Proposition 3 is a direct result of

Lemma 3.

In addition, by replacing SHP B W in (14) with its correspond-

ing expression, PRis further given by (16) at the top of the

following page.

We note that the referred 94% minimum approximation

accuracy is achieved for the maximum HPBW of 15◦needed

for the transmission to be considered as pencil beam, based on

Lemma 3. The smaller the HPBW is, the higher the accuracy

becomes since more energy is included within the main-lobe

region deﬁned by the HPBW.

Remark 2: Although the condition Si>> Sscan ensure

that the RIS is located in the Fraunhofer region of both the TX

and RX antennas, which means that the impinging on the RIS

electromagnetic wave can be considered as a plane wave, this

does not necessarily hold in the Ss≥Sicase. In such a case,

the requirements that the RX antenna gain is approximately

constant over the illuminated RIS region and the incident and

departure RU gains together with the corresponding TX-RIS

and RIS-RX distances are approximately independent of n,

under which (14) holds, could be valid even if the phase of

the impinging wave notably varies over the surface4. Based

on this, for the Ss≥Sicase the ”far” condition in which

the independency of the RX antenna gain, incident RU gain,

departure RU gain, TX-RIS distance, and RIS-RX distance

with respect to nholds, is not necessarily equivalent to the

Fraunhofer region, as it is also noted in [34] and [35].

Finally, the end-to-end SNR, which we denote by ρ, is given

by dividing PRwith N0, i.e.

ρ=PR

N0

.(15)

C. Optimal RIS Placement

r1,r2,θi, and θrcan be expressed as

r1=qr2

1,h +y2

s+ (hs−ht)2,(17)

r2=q(rh−r1,h)2+y2

s+ (hs−hr)2,(18)

θi= tan−1

qr2

1,h + (hs−ht)2

ys

(19)

and

θr= tan−1

q(r1,h −rh)2+ (hs−hr)2

ys

,(20)

for ys>0. Next, for the Si>> Ssand Ss≥Sicases we

determine the r1,h that maximizes the end-to-end SNR.

1) Si>> Sscase

Proposition 4: The optimum TX-RIS horizontal distance,

denoted by r∗

1,h, that maximizes the end-to-end SNR can be

obtained by taking the 1st derivative of ρwith respect to r1,h

and setting it equal to 0. This yields

a(1)r3

1,h +b(1)r2

1,h +c(1)r1,h +d(1) = 0,(21)

4The phase of an impinging wave on a surface should not vary by more

than π

8so that the wave is considered planar over the surface.

6

PR≈λ

4π4PtΓ2Gmax

tGmax

rGs(θi)Gs(θr)

d2

xd2

yr1

r22

π2sin4φHP BW

2

cos4φHP BW

2+θi

1−sin2(θi)

cos2φHP BW

2

.(16)

where

a(1) = 6,(22)

b(1) =−9rh,(23)

c(1) = 3 2y2

s+r2

h+ (hs−ht)2+ (hs−hr)2,(24)

d(1) =−3rhy2

s+ (hs−ht)2.(25)

Proof: The proof is provided in Appendix E.

2) Ss≥Sicase

Proposition 5: By by taking the 1st derivative of ρwith

respect to r1,h and setting it equal to 0, r∗

1,h can approximated

by (26), given at the top of the next page.

Proof: The proof is provided in Appendix F.

The tightness of (26) with respect to the exact value of r∗

1,h

is validated in Section IV by means of simulations.

IV. NUMERICAL RES ULTS & DESIGN GUIDELINES

The aim of this section is twofold: i) to validate, by means

of simulations, Proposition 2, Proposition 3, and the analytical

frameworks for the computation of r∗

1,h based on (21) and

(26);) and ii) to provide design guidelines based on the

resulting trends for various conﬁgurations.

A. Results

We consider the parameters of Table II.

TABLE II

PARAMETER VALUES USED IN THE SIMULATION.

Parameter Value

f140 GHz

Pt1W

W2GHz

FdB 10 dB

dx,dyλ/2

hs12 m

Dt15 cm

et,er0.7

Γ 0.9

1) Validation of Proposition 2

As far as the validation of Proposition 2 is concerned, in

Fig. 2 we depict the exact, based on (11), and closed-form,

based on (13), ρvs. r1,h curves for Ss= 0.012 m2,Dr= 3

cm, rh= 30 m, and different values of ht,hr, and ys. We note

that for the considered value of Dtit holds that φHP BW =

1.25◦, which means that a highly pencil beam transmission is

enacted. In addition, for the examined scanning range of r1,h

the minimum value of Siis equal to 0.09 m2and 0.69 m2

for ys= 5 m and ys= 15 m, respectively. Hence, it holds

that Si>> Ssthroughout the considered r1,h range in both

cases.

As we observe from Fig. 2, there is a relatively good match

between the exact and closed-form expressions of ρ, which

validates (13). In addition, we observe that for ys= 5 mρis

maximized when the RIS is closer to the TX, closer to the RX,

and either closer to the TX or the RX for ht> hr,ht< hr,

and ht=hr, respectively. In addition, ρis minimized if the

RIS is placed close to the middle of the TX-RX distance.

However, there is a relatively small variation among the two

local optima of ρand the local minimum near the middle

of the TX-RX distance. On the other hand, for ys= 15 m

we observe that the optimal placement of the RIS has moved

closer to the middle of the TX-RX distance in the ht> hr

and ht< hrcases. This indicates that the higher ysis, the

less pronounced the effect of the height difference between

the TX and RX antennas is on the optimal RIS placement.

Consequently, ρis maximized when the RIS is placed near to

the middle of the TX-RX distance, as we observe from Fig. 2.

To further validate the observed trends mentioned in the

previous paragraph, in Fig. 3 we depict the ρvs. r1,h curves

for a much larger rhdistance and 3 values of ys. As we

observe from Fig. 3, for small values of ysthe variation

among the two local optima of ρand its local minimum is

notable. In particular, a 12.5 dB difference is observed between

the value that maximizes ρand the one that minimizes it

near the middle of the TX-RX distance. Furthermore, Fig. 3

again reveals that the two local optima of ρdo not differ

substantially in value. As ysincreases, the corresponding

difference diminishes. More speciﬁcally, as ysincreases (21)

moves from having 3 real roots, which results in 2 local optima

and 1 local minimum, to having only 1 real root for sufﬁciently

large ys.

The trends of Fig. 3 can be validated by observing how

the discriminant of the 3rd degree polynomial of (21), which

we denote by ∆, varies with respect to ys. Regarding this, in

Fig. 4 we depict ∆with respect to ys. As we observe from

Fig. 4, as ysincreases ∆moves from positive values, which

means that (21) has 3 real roots, to negative values, which

means that only one real root exists.

Finally, in order to examine the effect of Sson the optimal

RIS placement, in Fig. 5 we compare the exact r∗

1,h vs.

yscurves, based on (11), with the closed-form analytically

obtained one, based on (21), for rh= 80 m and 3 values of

Ss. Although there is a relatively close match of the obtained

by simulations exact r∗

1,h values with the analytical one for

Ss= 0.012 m2and Ss= 0.026 m2, there is a substantial

discrepancy between them in the Ss= 0.046 m2case for

low-to-moderate ysvalues. This is justiﬁed by the fact that

for the particular ysvalues it does not hold that the minimum

value of Siin in the considered scanning range of r1,h is much

larger than Ss. For instance, for ys= 10 m the minimum value

of Si, which occurs for r1,h = 0, is equal to 0.15 m2. Hence,

it does not hold that Si>> Ssthroughout the r1,h scanning

range, which results in (13) providing inaccurate results. To

7

r∗

1,h ≈

−(hs−ht)2+r2

h+ (hs−hr)2+r(hs−ht)2−r2

h−(hs−hr)22+ 4r2

hy2

s+ (hs−ht)2

2rh

.(26)

-20 -10 0 10 20 30 40 50

35

40

45

50

55

60

65

(a) ht= 6 m, hr= 3 m.

-20 -10 0 10 20 30 40 50

35

40

45

50

55

60

65

(b) ht= 3 m, hr= 6 m.

-20 -10 0 10 20 30 40 50

35

40

45

50

55

60

65

(c) ht= 3 m, hr= 3 m.

Fig. 2. ρvs. r1,h for Dr= 3 cm, rh= 30 m, and Ss= 0.012 m2.

-20 0 20 40 60 80 100

30

35

40

45

50

(a) ht= 6 m, hr= 3 m.

-20 0 20 40 60 80 100

25

30

35

40

45

50

(b) ht= 3 m, hr= 6 m.

Fig. 3. ρvs. r1,h for Dr= 3 cm, rh= 80 m, and Ss= 0.012 m2.

5 10 15 20 25 30 35 40 45 50 55 60

-5

-4

-3

-2

-1

0

1

2

3

Fig. 4. Discriminant of (21) for ht= 6 m, hr= 3 m, Dr= 3 cm, and

rh= 80 m.

8

5 10 15 20 25 30 35 40 45 50 55 60

0

10

20

30

40

50

60

70

80

90

Fig. 5. r∗

1,h vs. ysfor ht= 6 m, hr= 3 m, Dr= 3 cm, and rh= 80 m.

-20 0 20 40 60 80 100

44

46

48

50

52

54

56

58

60

Fig. 6. ρvs. r1,h for ht= 6 m, hr= 3 m, Dr= 3 cm, rh= 80 m,

ys= 10 m, and Ss= 0.046 m2.

substantiate the latter, in Fig. 6 we illustrate the ρvs. r1,h

curve for Ss= 0.046 m2and ys= 10 m. As we observe from

Fig. 6, there is a substantially discrepancy between the exact

and the closed-form ρcurves for r1,h = 0. As aforementioned,

in the particular position Siexhibits its smallest value, equal

to 0.15 m2, and, hence, it holds that Ss/Si= 0.31, which

is smaller than 1, but not notably smaller. The latter is the

requirement for (13) to hold.

2) Validation of Proposition 3

As far as the validation of Proposition 3 is concerned, in

Fig. 7 we depict the ρvs. r1,h curves for rh= 20 m, ys=

10 m, and different Drunder the assumption that Ss> Si

throughout the examined r1,h range. In particular, Sitakes

its largest value for r1,h = 40 m, which is equal to 7.97

m2. Outdoor objects where such large RIS surfaces can be

mounted are, for instance, the facades of buildings. As we see

from Fig. 7, the RIS should be placed closer to the RX so that

ρis maximized.

In addition, from Fig. 7 we further observe that the higher

Dris, the larger the deviation of the exact value of ρ, based

on (11), with its closed-form counterpart, based on (14), is for

-20 -10 0 10 20 30 40

60

65

70

75

80

85

90

95

100

105

(a) ht= 6 m, hr= 3 m.

-20 -10 0 10 20 30 40

65

70

75

80

85

90

95

100

105

(b) ht= 3 m, hr= 6 m.

Fig. 7. ρvs. r1,h for rh= 20 m, ys= 10 m, and Ss> Si.

-20 -10 0 10 20 30 40

0

50

100

150

200

Fig. 8. max{Gr,n}

min{Gr,n}vs. r1,h for ht= 6 m, hr= 3 m, rh= 20 m, ys= 10

m, and Ss> Si.

9

-20 -10 0 10 20 30 40

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Fig. 9. r1

max{r1,n}and r2

max{r2,n}vs. r1,h for ht= 6 m, hr= 3 m,

Dr= 5 cm, rh= 20 m, ys= 10 m, and Ss> Si.

-20 -10 0 10 20 30 40

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Fig. 10. max Gs(θi,n )

Gs(θi)and max Gs(θr,n)

Gs(θr)vs. r1,h for ht= 6 m,

hr= 3 m, Dr= 5 cm rh= 20 m, ys= 10 m, and Ss> Si.

4 6 8 10 12 14 16 18 20 22 24 26

20

25

30

35

40

45

Fig. 11. r∗

1,h vs. ysfor ht= 6 m, hr= 3 m, rh= 20 m, Dr= 1 cm,

and Ss> Si.

larger r1,h values. This discrepancy is attributed to the fact

that as Drincreases the footprint of the main lobe of the RX

antenna on the RIS reduces and, consequently, Gr,n cannot

be considered approximately constant for the illuminated RIS

region, as it is required so that (14) holds with high accuracy.

This is more pronounced for larger r1,h values due to the

larger illuminated region on the RIS and its closer distance

with respect to the RX antenna.

To substantiate our claim about the larger variation of Gr,n

across the illuminated RIS region as Drincreases, in Fig. 8

we illustrate how the ratio max{Gr,n}

min{Gr,n}varies with respect to

r1,h for 3 values of Dr. As we observe, for Dr= 1 cm Gr,n

remains almost constant across the illuminated RIS region,

but it starts varying as Drincreases. In particular, it varies

substantially for Dr= 5 cm in the r1,h >10 m region, which

justiﬁes the increasing discrepancy in the particular region

between the exact and closed-form results as r1,h increases.

On the other hand, from Fig. 9, which depicts the ratios

r1

max{r1,n}and r2

max{r2,n}with respect to r1,h , we see that

throughout the r1,h scanning range there are small variations

of the distances between the TX (RX) antenna center and the

nth RU, which indicates that r1,n and r2,n can be closely ap-

proximated by r1and r2, respectively, for each n. Furthermore,

small variations are also observed for Gs(θi,n)and Gs(θr,n)

with respect to Gs(θi)and Gs(θr), respectively, across the

r1,h scanning range, as Fig. 10 depicts. Consequently, from

Fig. 8, Fig. 9, and Fig. 10, we conclude that the variations of

Gr,n over the illuminated RIS region are the reason for the

large discrepancy between the exact numerical and theoretical

results in the r1,h >10 m region for Dr= 5 cm that is

observed in Fig. 7.

Finally, to validate the close match of r∗

1,h obtained by (26)

with the exact value obtained by simulations, in Fig. 11 we

depict the variation of both the analytically obtained r∗

1,h and

the one by simulations with respect to ysfor Dr= 1 cm. As

we see, there is a relatively close match of the two curves,

which validates (26). In addition, the higher ysis the larger

r∗

1,h becomes.

B. Design Guidelines

Let us now recapitulate on the main outcomes regarding

the Si>> Ssand Si≤Sscases, according to the presented

numerical results, that can be used by the system designer

so that the optimal RIS placement is achieved based on the

system parameters.

1) Si>> Sscase

•For relatively small values of yscompared with rh, there

are two positions, one close to the TX and the other close

to the RX, that locally maximize ρ, and one position

close to the middle of the TX-RX distance that minimizes

it. Between the two local optima, the one closer to the

TX is the global optimum for ht> hr, whereas it is

the opposite for ht< hr. Moreover, the difference in

magnitude between the two optima is relatively small for

small values of rh, but it becomes notable for relatively

large values. This indicates that, from a practical point of

view, in the former case the RIS can be placed either

10

closer to the TX or RX with small difference in the

resulting SNR, but in the latter case the SNR can notably

differ and, hence, the optimal placement provided by the

analytical model should be followed.

•As ysincreases, the magnitude difference between the

optima and the minimum substantially reduces and the

location of the optima move closer to the middle of the

TX-RX distance. Furthermore, there is a threshold ys

value above which there is only one real value of r1,h that

maximizes ρthat is close to the middle of the TX-RX

distance. This means that under such conditions the RIS

can be practically placed close to the middle of the TX-

RX distance regardless of whether ht> hror hr< ht.

2) Si≤Sscase

•Regardless of whether ht> hror hr< ht, there is only

one optimum of ρwith respect to r1,h that is located

closer to the RX. In addition, a substantially higher ρis

achieved by placing the RIS at the particular point rather

than close to the TX or the middle of the TX-RX distance.

•Drshould be notably smaller than Dtso that (14) holds.

Under such a condition, Gr,n is approximately constant

throughout the illuminated RIS region and approximately

equal to Gmax

r. Equivalently, the RX antenna aperture

needs to be sufﬁciently small so that the footprint of the

highly directional departing beam from the RIS on the

RX antenna is larger than the aperture of the latter.

Finally, we would like to note that the same analytical

methodology for extracting the optimal RIS placement in

ﬁxed-topology scenarios can be used for extracting the optimal

placement also in mobile scenarios. In the latter case, due

to fading that would likely arise as the user is moving, the

analysis should consider the statistical average power effect of

the fading process on the RIS-RX channel that would depend

on the elevation of the RIS with respect to the ground.

V. CONCLUSIONS

This work has been motivated by the need to answer

the question of where an RIS that aids a highly-directional

mmWave TX-RX link of ﬁxed topology under blockage should

be placed, so as to maximize the SNR performance. Based on

this, we have ﬁrstly computed the end-to-end received power

and SNR under an RIS of arbitrary size. Subsequently, we

provided closed-form approximate expressions for the cases

of the RIS being either much smaller or larger than the

transmit beam footprint at the RIS plane. Finally, based on the

resulting SNR expressions, we analytically derived the optimal

horizontal RIS placement that maximizes the end-to-end SNR.

The analytical outcomes have been validated by an extensive

simulation campaign in various scenarios, which reveal that:

i) when the transmission beam footprint at the RIS plane is

much larger than the RIS size, the optimal RIS placement

is either close to the TX, RX, or the middle of the TX-RX

horizontal distance, depending on the system parameters; ii)

when the footrpint is equal to or smaller than the RIS size,

the optimal RIS placement is close to the RX. Such outcomes

can be readily used by the system designer to properly deploy

RISs in a way that the system performance is maximized.

ACK NO WL EDG EM E NT S

The authors would like to cordially thank the associate

editor and anonymous reviewers, whose comments and sug-

gestions have led to a substantial improvement of this work.

APPENDICES

APPENDIX A

PROO F OF LEM MA 1

Based on (1), for the total normalized transmit power, which

we denote by Ptot

dish, it holds that

Ptot

dish =Zπ

2

−π

2

E2(φ)dφ. (27)

As far as φ0is concerned, which encompasses the main lobe, it

can be computed by ﬁnding the points for which πDtsin(φ)

λ=

3.83 since J1(3.83) = 0. Hence, it holds that

φ0= 2 sin−13.83λ

πDt= 2 sin−11.22λ

Dt.(28)

Consequently, the normalized transmit power that is corre-

sponding to the main lobe, which we denote by Pmain lobe

dish ,

is given by

Pmain lobe

dish =Zφ0

2

−φ0

2

E2(φ)dφ. (29)

Let us denote the ratio of the normalized power inside the main

lobe over the total normalized transmit power as a function of

φHP B W by ρmain lobe (φH P BW ). Consequently, it holds that

ρmain lobe (φHP B W ) = Pmain lobe

dish /P tot

dish and for φHP B W <

π

12 it holds that ρmain lobe (φHP B W )> ρmain lobe π

12 .

π

12 corresponds to the 15◦minimum HPBW value upper

limit for which the transmission can be considered as pencil

beam according to our assumptions. ρmain lobe (φHP B W )>

ρmain lobe π

12 holds due to the fact that as φHP B W decreases

the main lobe becomes sharper, which means that more

power is concentrated inside the main lobe compared with the

φHP B W =π

12 case. Hence, ρmain lobe (φHP B W )increases.

As far as the value of the term Dt

λfor which φHP B W =π

12

is achieved, it holds that

E2π

24 = 0.5⇒

2λ

πDt

J1πDtsin(π

24 )

λ

sin π

24

2

= 0.5

⇒Dt

λ= 3.94.(30)

For Dt

λ= 3.94 under which φHP B W =π

12 is achieved, it

holds that

ρmain lobe π

12 =Rsin−1(1.22

3.94 )

−sin−1(1.22

3.94 )2

π3.94

J1(π3.94 sin(φ))

sin(φ)2

dφ

Rπ

2

−π

22

π3.94

J1(π3.94 sin(φ))

sin(φ)2

dφ

= 0.97.(31)

By taking into account that ρmain lobe (φHP B W )>

ρmain lobe π

12 for φHP B W <π

12 = 15◦, the proof of

Lemma 1 is concluded.

11

APPENDIX B

PROO F OF LEMMA 2

By applying the law of sines in the ABC triangle, we obtain

sin φ0

2

α=

sin ∧

C

r1

,(32)

where

∧

C denotes the angle of corner C and can be calculated as

∧

C=π

2−φ0

2−θi.(33)

By substituting (33) into (32), we obtain (6).

The eccentricity of the elliptical footprint can be evaluated

as

=sin (θi)

sin π

2−φ0

2,(34)

or equivalently as in (8). Finally, βcan be obtained as in (7).

This concludes the proof.

APPENDIX C

PROO F OF PROPO SI TIO N 1

The incident electric ﬁeld on the nth RU of the RIS

illuminated area can be obtained as

En=Ene−j2π

λr1,n no, n = 1,2, ..., M, (35)

where Enis the amplitude of the incident wave and nois a

unitary vector that is perpendicular to the 2D plane that the

electric ﬁeld lies on [36, Example 11-3]. It holds that

En=s2ηPtGt,n

4πr2

1,n

,(36)

where ηis the free-space impedance. As a consequence, the

power density at the nth RU of the RIS can be expressed as

Pn=E2

n

2η(37)

or, with the aid of (36), as

Pn=PtGt,n

4πr2

1,n

,(38)

where ηis the free-space impedance. Thus, the incident power

at the nth RU can be evaluated as

Pi,n =PnAn,(39)

where Anstands for the effective aperture of the nth illumi-

nated RU and can be obtained as

An=λ2

4πGs(θi,n).(40)

By substituting (38) and (40) into (39), we obtain

Pi,n =λ

4π2PtGt,nGs(θi,n )

r2

1,n

.(41)

As a result and due to the energy conservation law, the

reﬂected power density by the nth RU, which is captured by

the RX antenna, can be expressed as

Pr,n =Pi,nΓ2Gs(θr,n)

4πr2

2,n

.(42)

By substituting (41) into (42), we obtain

Pr,n =λ2

(4π)3

PtΓ2Gt,nGs(θi,n )Gs(θr,n)

r2

1,nr2

2,n

.(43)

The power captured by the receiver from the nth RU is given

by

PR,n =Pr,n

λ2

4πGr,n

=λ

4π4PtΓ2Gt,nGr,nGs(θi,n)Gs(θr,n)

r2

1,nr2

2,n

.(44)

Moreover, the corresponding electric ﬁeld observed at the

receiver from the nth RU can be evaluated as

ER,n =ER,ne−jθn+2π(r1,n +r2,n )

λao,(45)

where aois a unitary vector perpendicular to the 2D plane

that the reﬂected electric ﬁeld lies on and θnis adjustable

the phase response of the nth RU. Additionally, ER,n can be

computed as

ER,n =p2ηPR,n ,(46)

which, by employing (43), can be rewritten as

ER,n =s2ηλ

4π4PtΓ2Gt,nGr,nGs(θi,n)Gs(θr,n)

r2

1,nr2

2,n

.

(47)

From (45), the aggregated electric ﬁeld at the RX can be

written as

ER=

M

X

n=1

ER,n.(48)

Consequently, by employing (45) and (47) it holds that

ER=λ

4π2

p2ηPtΓ

M

X

n=1 sGt,nGr,nGs(θi,n)Gs(θr,n)

r2

1,nr2

2,n

×e−jθn+2π(r1,n+r2,r )

λao,(49)

Hence, at the RX the received power can be obtained as

PR=|ER|2

2η,(50)

which, with the aid of (49), can be rewritten as

PR=λ

4π4

PtΓ2

×

M

X

n=1 sGt,nGr,n Gs(θi,n)Gs(θr,n )

r2

1,nr2

2,n

e

−j θn+2π(r1,n+r2,n )

λ!

2

.

(51)

12

By assuming that the optimal phase shift is induced by each

RU so as to maximize PR, i.e.,

θn=−2π(r1,n +r2,n)

λ,(52)

the received power can be rewritten as

PR=λ

4π4

PtΓ2

M

X

n=1 sGt,nGr,n Gs(θi,n)Gs(θr,n )

r2

1,nr2

2,n

2

,(53)

which concludes the proof.

APPENDIX D

PROO F OF LEMMA 3

Let us denote the ratio of the normalized power included

within a step function in the interval h−φHP BW

2,φHP BW

2i

over the normalized power included within the FNBW by

κ(φHP B W ). It holds that

κ(φHP BW ) = φHP B W

Pmain lobe

dish

=φHP BW

R

φ0

2

−φ0

22λ

πDt

J1πDtsin(φ)

λ

sin(φ)2

dφ

.

(54)

For the minimum HPBW of value of 15◦required for

the transmit beam to be considered pencil beam, it holds

that κπ

12 = 0.97. Moreover, κ(φHP BW )is a monoton-

ically increasing function as φHP B W ↓since the smaller

the beamwidth is the sharper the main lobe becomes. Con-

sequently, it can be more accurately approximated by a

step function with magnitude equal to Gmax

tin the interval

h−φHP BW

2,φHP BW

2i.

In addition, by denoting the ratio of the normalized power

included within the step function over the total normalized

power that impinges on the RIS by µ(φHP BW ), it holds that

µ(φHP BW ) = φHP B W

Ptot

dish

=φHP BW

Rπ

2

−π

22λ

πDt

J1πDtsin(φ)

λ

sin(φ)2

dφ

.

(55)

For the minimum HPBW of value of 15◦it holds that

µπ

12 = 0.94. Moreover, for the same reason as in the

κ(φHP B W )case µ(φH P BW )is a monotonically increasing

function as φHP B W ↓.

APPENDIX E

PROO F OF PROPO SI TIO N 4

The proof begins by plugging (13) into (15) and rewriting

ρas

ρ=λ

4π416PtΓ2(Ss)2Gmax

tGmax

r

d2

xd2

yN0

F(r1,h)

G(r1,h),(56)

where

F(r1,h)= cos

tan−1

qr2

1,h + (hs−ht)2

ys

×cos

tan−1

q(r1,h −rh)2+ (hs−hr)2

ys

(57)

and

G(r1,h)= r2

1,h +y2

s+ (hs−ht)2

×(rh−r1,h)2+y2

s+ (hs−hr)2.(58)

From (56), we observe that the end-to-end SNR depends

on r1,h through the ratio F(r1,h )

G(r1,h). Hence, the optimum r1,h

that maximizes ρcan be obtained by evaluating the roots of

the ﬁrst derivative of F(r1,h )

G(r1,h)with respect to r1,h . The ﬁrst

derivative of F(r1,h )

G(r1,h)can be obtained as in (59), given at the

top of the following page. Consequently, r∗

1,h is obtained as

one of the solutions of a(1)r3

1,h +b(1)r2

1,h +c(1)r1,h +d(1) = 0.

APPENDIX F

PROO F OF PROPO SI TIO N 5

The proof begins by plugging (16) into (15) and rewriting

ρas

ρ≈λ

4π416PtΓ2Gmax

tGmax

rπ2sin4φHP BW

2

d2

xd2

yN0

×F(r1,h)H(r1,h ),(60)

where

H(r1,h) = r2

1,h +y2

s+ (hs−ht)2

(rh−r1,h)2+y2

s+ (hs−hr)2

1−sin2(θi)

cos2φHP BW

2

cos4φHP BW

2+θi.

(61)

Hence, the optimum r1,h that maximizes ρcan be obtained by

evaluating the roots of the ﬁrst derivative of F(r1,h)H(r1,h)

with respect to r1,h. Before computing the corresponding

derivative, we simplify things by taking into that for pencil-

beam transmissions it holds that φHP B W << 1. Hence,

considering that cos φHP BW

2≈1and cos φHP BW

2+θi≈

cos (θi),H(r1,h)can be approximated as

H(r1,h)≈r2

1,h +y2

s+ (hs−ht)2

(rh−r1,h)2+y2

s+ (hs−hr)2

1

cos2(θi).(62)

As a result, the ﬁrst derivative of F(r1,h)H(r1,h )can

be approximated by (63) given at the top of the next

page. Consequently, r1,h is obtained as one of the two

solutions of rhr2

1,h +(hs−ht)2−r2

h−(hs−hr)2r1,h −

rhy2

s+ (hs−ht)2= 0. As it is shown in Section IV,

between its two real roots the one that maximizes ρis the

one closer to RX, given by (26).

(64)

REFERENCES

[1] “Mobile backhaul options: Spectrum analysis and recommendations,”

ABI Research, Tech. Rep., Sep. 2018.

[2] “Millimetre wave transmission (mwt); analysis of spectrum, license

schemes and network scenarios in the d-band,” ETSI, Tech. Rep., Aug.

2018.

13

dF(r1,h)

G(r1,h)

dr1,h

=−6r3

1,h −9rhr2

1,h + 3 2y2

s+r2

h+ (hs−ht)2+ (hs−hr)2r1,h −3rhy2

s+ (hs−ht)2

× y2

s+r2

1,h + (hs−ht)2y2

s+ (rh−r1,h)2+ (hs−hr)2

(hs−ht)2(hs−hr)2!−1

2

×y2

s+r2

1,h + (hs−ht)2y2

s+ (rh−r1,h)2+ (hs−hr)2−2.(59)

d(F(r1,h)H(r1,h ))

dr1,h

≈ −

3q(r2

1,h +y2

s+ (hs−ht)2rhr2

1,h +(hs−ht)2−r2

h−(hs−hr)2r1,h −rhy2

s+ (hs−ht)2

(rh−r1,h)2+y2

s+ (hs−hr)25

2

.(63)

[3] M. R. Akdeniz, Y. Liu, M. K. Samimi, S. Sun, S. Rangan, T. S.

Rappaport, and E. Erkip, “Millimeter Wave Channel Modeling and

Cellular Capacity Evaluation,” IEEE J. Sel. Areas Commun., vol. 32,

no. 6, pp. 1164–1179, June 2014.

[4] T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N.

Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter Wave

Mobile Communications for 5G Cellular: It Will Work!” IEEE Access,

vol. 1, pp. 335–349, May 2013.

[5] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity

in wireless networks: Efﬁcient protocols and outage behavior,” IEEE

Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004.

[6] W. Khawaja, O. Ozdemir, Y. Yapici, F. Erden, and I. Guvenc, “Coverage

Enhancement for NLOS mmwave Links Using Passive Reﬂectors,” IEEE

Open Journal of the Communications Society, vol. 1, Jan. 2020.

[7] M. T. Barros, R. Mullins, and S. Balasubramaniam, “Integrated Terahertz

Communication With Reﬂectors for 5g Small-Cell Networks,” IEEE

Trans. Veh. Tech., vol. 66, no. 7, pp. 5647–5657, July 2017.

[8] Q. Wu and R. Zhang, “Beamforming optimization for intelligent reﬂect-

ing surface with discrete phase shifts,” in IEEE International Conference

on Acoustics, Speech and Signal Processing (ICASSP), 2019, pp. 7830–

7833.

[9] ——, “Towards smart and reconﬁgurable environment: Intelligent re-

ﬂecting surface aided wireless network,” IEEE Commun. Mag., vol. 58,

no. 1, Nov. 2020.

[10] O. ¨

Ozdogan, E. Bj¨

ornson, and E. G. Larsson, “Intelligent reﬂecting

surfaces: Physics, propagation, and pathloss modeling,” IEEE Wirel.

Commun. Lett., vol. 9, no. 5, May 2020.

[11] A.-A. A. Boulogeorgos and A. Alexiou, “Performance Analysis of

Reconﬁgurable Intelligent Surface-Assisted Wireless Systems and Com-

parison With Relaying,” IEEE Access, vol. 8, May 2020.

[12] S. Kisseleff, W. A. Martins, H. Al-Hraishawi, S. Chatzinotas, and

B. Ottersten, “Reconﬁgurable Intelligent Surfaces for Smart Cities:

Research Challenges and Opportunities,” IEEE Open Journal of the

Communications Society, vol. 1, Nov. 2020.

[13] E. Basar, M. D. Renzo, J. de Rosny, M.-S. Alouini, and R. Zhang,

“Wireless Communications Through Reconﬁgurable Intelligent Sur-

faces,” IEEE Access, vol. 7, pp. 116 753–116 773, Aug. 2019.

[14] L. Bariah, L. Mohjazi, S. Muhaidat, P. C. Sofotasios, G. K. Kurt,

H. Yanikomeroglu, and O. A. Dobre, “A prospective look: Key enabling

technologies, applications and open research topics in 6g networks,”

IEEE Access, vol. 8, 2020.

[15] Q. Wu and R. Zhang, “Intelligent reﬂecting surface enhanced wireless

network via joint active and passive beamforming,” IEEE Trans. Wireless

Commun., vol. 18, no. 11, pp. 5394–5409, Nov. 2019.

[16] M. Jung, W. Saad, Y. Jang, G. Kong, and S. Choi, “Performance analysis

of large intelligent surfaces (LISs): Asymptotic data rate and channel

hardening effects,” IEEE Trans. Wireless Commun., vol. 19, no. 3, pp.

2052–2065, Mar. 2020.

[17] S. Zeng, H. Zhang, B. Di, Z. Han, and L. Song, “Reconﬁgurable intelli-

gent surface (RIS) assisted wireless coverage extension: RIS orientation

and location optimization,” IEEE Commun. Lett., pp. 1–1, Nov. 2020.

[18] C. Huang, S. Hu, G. C. Alexandropoulos, A. Zappone, C. Yuen,

R. Zhang, M. D. Renzo, and M. Debbah, “Holographic MIMO Surfaces

for 6G Wireless Networks: Opportunities, Challenges, and Trends,”

IEEE Wireless Communications, vol. 27, no. 5, Oct. 2020.

[19] C. Huang, R. Mo, and C. Yuen, “Reconﬁgurable Intelligent Surface

Assisted Multiuser MISO Systems Exploiting Deep Reinforcement

Learning,” IEEE Journal on Selected Areas in Communications, vol. 38,

no. 8, pp. 1839–1850, Aug. 2020.

[20] M. Di Renzo, A. Zappone, M. Debbah, M. S. Alouini, C. Yuen, J. de

Rosny, and S. Tretyakov, “Smart Radio Environments Empowered by

Reconﬁgurable Intelligent Surfaces: How It Works, State of Research,

and The Road Ahead,” IEEE Journal on Selected Areas in Communi-

cations, vol. 38, no. 11, Nov. 2020.

[21] M. D. Renzo, K. Ntontin, J. Song, F. H. Danufane, X. Qian, F. Lazarakis,

J. D. Rosny, D.-T. Phan-Huy, O. Simeone, R. Zhang, M. Debbah,

G. Lerosey, M. Fink, S. Tretyakov, and S. Shamai, “Reconﬁgurable in-

telligent surfaces vs. relaying: Differences, similarities, and performance

comparison,” IEEE Open Journal of the Communications Society, vol. 1,

pp. 798–807, Jun. 2020.

[22] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and

C. Yuen, “Reconﬁgurable Intelligent Surfaces for Energy Efﬁciency in

Wireless Communication,” IEEE Trans. Wirel. Commun., vol. 18, no. 8,

pp. 4157–4170, Aug. 2019.

[23] E. Bj¨

ornson, O. ¨

Ozdogan, and E. G. Larsson, “Intelligent Reﬂecting

Surface vs. Decode-and-Forward: How Large Surfaces Are Needed to

Beat Relaying?” IEEE Wirel. Commun. Lett., vol. 9, no. 2, pp. 244–248,

Feb. to appear.

[24] K. Ntontin, D. Selimis, A.-A. A. Boulogeorgos, A. Alexandridis, A. Tso-

lis, V. Vlachodimitropoulos, and F. Lazarakis, “Optimal Reconﬁgurable

Intelligent Surface Placement in Millimeter-Wave Communications,” in

European Conference on Antennas and Propagation (EuCAP), 2021.

[25] H. Anderson, Fixed broadband wireless system design. Chichester,

West Sussex, England Hoboken, NJ: John Wiley & Sons, 2003.

[26] D. Schulz, V. Jungnickel, C. Alexakis, M. Schlosser, J. Hilt,

A. Paraskevopoulos, L. Grobe, P. Farkas, and R. Freund, “Robust optical

wireless link for the backhaul and fronthaul of small radio cells,” Journal

of Lightwave Technology, vol. 34, no. 6, Mar. 2016.

[27] S. Silver, Microwave antenna theory and design. London, UK: P.

Peregrinus on behalf of the Institution of Electrical Engineers, 1984.

[28] I. A. Hemadeh, K. Satyanarayana, M. El-Hajjar, and L. Hanzo,

“Millimeter-Wave Communications: Physical Channel Models, Design

Considerations, Antenna Constructions, and Link-Budget,” IEEE Com-

munications Surveys Tutorials, vol. 20, no. 2, pp. 870–913, 2018.

[29] T. S. Rappaport, G. R. MacCartney, M. K. Samimi, and S. Sun, “Wide-

band millimeter-wave propagation measurements and channel models

for future wireless communication system design,” IEEE Transactions

on Communications, vol. 63, no. 9, Sep. 2015.

[30] W. Tang, X. Chen, M. Z. Chen, J. Y. Dai, Y. Han, M. D. Renzo, S. Jin,

Q. Cheng, and T. J. Cui, “Path Loss Modeling and Measurements for

Reconﬁgurable Intelligent Surfaces in the Millimeter-Wave Frequency

Band,” arXiv:1906.09490.

[31] W. Tang, M. Z. Chen, X. Chen, J. Y. Dai, Y. Han, M. Di Renzo,

Y. Zeng, S. Jin, Q. Cheng, and T. J. Cui, “Wireless communications with

reconﬁgurable intelligent surface: Path loss modeling and experimental

measurement,” IEEE Transactions on Wireless Communications, vol. 20,

no. 1, Jan. 2021.

[32] K. Ntontin, M. Di Renzo, and C. Verikoukis, “On the Feasibility of Full-

Duplex Relaying in Multiple-Antenna Cellular Networks,” IEEE Trans.

Commun., vol. 65, no. 5, pp. 2234–2249, May 2017.

14

[33] D. Hilbert and S. Cohn-Vossen, Geometry and the imagination. New

York: Chelsea Publishing Company, 1952.

[34] S. W. Ellingson, “Path Loss in Reconﬁgurable Intelligent Surface-

Enabled Channels,” arXiv:1912.06759.

[35] E. Bj¨

ornson and L. Sanguinetti, “Power scaling laws and near-ﬁeld

behaviors of massive mimo and intelligent reﬂecting surfaces,” IEEE

Open Journal of the Communications Society, vol. 1, Sep. 2020.

[36] C. Balanis, Advanced engineering electromagnetics. Hoboken, N.J:

John Wiley & Sons, 2012.

Konstantinos Ntontin (S’12-M’14) is currently a

research associate of the SIGCOM Research Group

at SnT, University of Luxembourg. In the past, he

held research associate positions at the Electronic

Engineering and Telecommunications department of

the University of Barcelona and at the Informatics

and Telecommunications department of the Univer-

sity of Athens. In addition, he held an internship

position at Ericsson Eurolab Gmbh, Germany. He

received the Diploma in Electrical and Computer

Engineering in 2006, the M. Sc. Degree in Wireless

Systems in 2009, and the Ph. D. degree in 2015 from the University of Patras,

Greece, the Royal Institute of Technology (KTH), Sweden, and the Technical

University of Catalonia (UPC), Spain, respectively. His research interests are

related to the physical layer of wireless telecommunications with focus on

performance analysis in fading channels, MIMO systems, array beamforming,

transceiver design, and stochastic modeling of wireless channels.

Alexandros-Apostolos A. Boulogeorgos (S’11,

M’16, SM’19) was born in Trikala, Greece in 1988.

He received the Electrical and Computer Engineer-

ing (ECE) diploma degree and Ph.D. degree in

Wireless Communications from the Aristotle Uni-

versity of Thessaloniki (AUTh) in 2012 and 2016,

respectively.

From November 2012, he has been a member of

the wireless communications system group of AUTh,

working as a research assistant/project engineer in

various telecommunication and networks projects.

During 2017, he joined the information technologies institute, while from

November 2017, he has joined the Department of Digital Systems, University

of Piraeus, where he conducts research in the area of wireless communications.

Moreover, from October 2012 until September 2016, he was a teaching

assistant at the department of ECE of AUTh, whereas, from February 2017,

he serves as an adjunct lecturer at the Department of Informatics and

Telecommunications Engineering of the University of Western Macedonia and

as an visiting lecturer at the Department of Computer Science and Biomedical

Informatics of the University of Thessaly.

Dr. Boulogeorgos has authored and co-authored more than 65 technical

papers, which were published in scientiﬁc journals and presented at prestigious

international conferences. Furthermore, he has submitted two (one national

and one European) patents. Likewise, he has been involved as member of

Technical Program Committees in several IEEE and non-IEEE conferences

and served as a reviewer in various IEEE journals and conferences. Dr.

Boulogeorgos was awarded with the “Distinction Scholarship Award” of the

Research Committee of AUTh for the year 2014 and was recognized as an

exemplary reviewer for IEEE Communication Letters for 2016 (top 3% of

reviewers). Moreover, he was named a top peer reviewer (top 1% of reviewers)

in Cross-Field and Computer Science in the Global Peer Review Awards 2019,

which was presented by the Web of Science and Publons. His current research

interests spans in the area of wireless communications and networks with

emphasis in high frequency communications, optical wireless communications

and communications for biomedical applications. He is a Senior Member of

the IEEE and a member of the Technical Chamber of Greece. He is currently

an Editor for IEEE Communications Letters, and an Associate Editor for the

Frontier In Communications And Networks.

Dimitrios G. Selimis (S’20) received his Diploma

degree in Electrical and Computer Engineering

from the University of Patras, Greece, in 2018.

He received his master degree in Modern Wireless

Communications from University of Peloponnese in

2019. His master thesis focused on the performance

analysis of Spatial Modulation-MIMO systems for

several fading scenarios. He is currently a PhD can-

didate at University of Peloponnese in collaboration

with the National Centre for Scientiﬁc Research-

“Demokritos”. His current research interests are re-

lated to the physical layer of wireless communications with focus on statistical

modeling of wireless channels.

Fotis I. Lazarakis received his diploma in Physics

in 1990, from Department of Physics, Aristotle

University of Thessaloniki, Greece, and his Ph.D in

Mobile Communications, in 1997, from Department

of Physics, National and Kapodistrian University of

Athens, Greece, holding at the same time a schol-

arship from National Center for Scientiﬁc Research

”Demokritos” (NCSRD), Institute of Informatics and

Telecommunications (IIT). From 1999 to 2002 he

was with Telecommunications Laboratory, National

Technical University of Athens, as a senior research

associate. In 2003 he joined NCSRD, Institute of Informatics and Telecom-

munications as a Researcher and since 2013 is a Research Director. He has

been involved in a number of national and international projects, acting

as a Project Manager to several of those. His research interests include

WLANs, 5G and beyond, propagation models and measurements, fading

channel characteristics and capacity, diversity techniques, MIMO antennas

and systems, radio resource management and performance evaluation of

mobile/wireless networks. Dr. Lazarakis has authored or co-authored more

than 100 journal and conference papers and he is co-owner of a patent.

Angeliki Alexiou is a professor at the department

of Digital Systems, ICT School, University of Pi-

raeus. She received the Diploma in Electrical and

Computer Engineering from the National Techni-

cal University of Athens in 1994 and the PhD

in Electrical Engineering from Imperial College of

Science, Technology and Medicine, University of

London in 2000. Since May 2009 she has been

a faculty member at the Department of Digital

Systems, where she conducts research and teaches

undergraduate and postgraduate courses in the area

of Broadband Communications and Advanced Wireless Technologies. Prior to

this appointment she was with Bell Laboratories, Wireless Research, Lucent

Technologies, (later Alcatel-Lucent, now NOKIA), in Swindon, UK, ﬁrst as

a member of technical staff (January 1999-February 2006) and later as a

Technical Manager (March 2006-April 2009). Professor Alexiou is a co-

recipient of Bell Labs President’s Gold Award in 2002 for contributions

to Bell Labs Layered Space-Time (BLAST) project and the Central Bell

Labs Teamwork Award in 2004 for role model teamwork and technical

achievements in the IST FITNESS project. Professor Alexiou is the Chair

of the Working Group on Radio Communication Technologies and of the

Working Group on High Frequencies Radio Technologies of the Wireless

World Research Forum. She is a member of the IEEE and the Technical

Chamber of Greece. Her current research interests include radio interface

for 5G systems and beyond, MIMO and high frequencies (mmWave and

THz wireless) technologies, cooperation, coordination and efﬁcient resource

management for Ultra Dense wireless networks and machine-to-machine

communications, ‘cell-less’ architectures based on virtualization and extreme

resources sharing and machine learning for wireless systems. She is the project

coordinator of the H2020 TERRANOVA project (ict-terranova.eu) and the

technical manager of H2020 ARIADNE project (ict-ariadne.eu).

15

Symeon Chatzinotas (S’06–M’09–SM’13) is cur-

rently Full Professor / Chief Scientist I and Co-Head

of the SIGCOM Research Group at SnT, University

of Luxembourg. In the past, he has been a Visiting

Professor at the University of Parma, Italy and he

was involved in numerous Research and Develop-

ment projects for the National Center for Scientiﬁc

Research Demokritos, the Center of Research and

Technology Hellas and the Center of Communica-

tion Systems Research, University of Surrey. He

received the M.Eng. degree in telecommunications

from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 2003,

and the M.Sc. and Ph.D. degrees in electronic engineering from the University

of Surrey, Surrey, U.K., in 2006 and 2009, respectively. He was a co-recipient

of the 2014 IEEE Distinguished Contributions to Satellite Communications

Award, the CROWNCOM 2015 Best Paper Award and the 2018 EURASIC

JWCN Best Paper Award. He has (co-)authored more than 400 technical

papers in refereed international journals, conferences and scientiﬁc books.

He is currently in the editorial board of the IEEE Open Journal of Vehicular

Technology and the International Journal of Satellite Communications and

Networking.