# Reconfigurable Intelligent Surface Optimal Placement in Millimeter-Wave
Networks

Konstantinos Ntontin, , Alexandros–Apostolos A. Boulogeorgos, ,
Dimitrios Selimis, , Fotis Lazarakis,
Angeliki Alexiou, , and Symeon Chatzinotas 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:
<EMAIL_ADDRESS>A. Boulogeorgos and A. Alexiou are with the
Department of Digital Systems, University of Piraeus Piraeus 18534 Greece.
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>D. Selimis and F.
Lazarakis are with the Wireless Communications Laboratory of the Institute of
Informatics and Telecommunications, National Centre for Scientific
Research–“Demokritos,” Athens, Greece. E-mails: $\left\\{\text{dselimis,
<EMAIL_ADDRESS>Chatzinotas is with the Interdisciplinary
Centre for Security, Reliability and Trust (SnT) – University of Luxembourg,
L-1855 Luxembourg. E-mail<EMAIL_ADDRESS>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 Björnson.

###### Abstract

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

###  I-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. 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. However,
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 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 reconfigurable 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, 9, 10, 11, 12]. They constitute a
substantially different technology than active relaying, 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 components, 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, 14,
15, 16, 17, 18, 19, 20], as well as comparing them with their predecessors,
i.e., relaying-aided ones [11, 21, 22, 23]. In more detail, in [13] and [14]
the authors introduced the idea of employing an RIS in order to mitigate 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, 22, 23].

###  I-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 follows111This work constitutes an extension of [24].:

  * •

We present a system model for RIS-aided highly directional 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 analytical 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 findings and
remarks.

_Notation_ : For the convenience of the readers, recurrent parameters and
symbols with their meaning are presented in Table I.

TABLE I: Recurrent parameters and symbols.

Parameter/Symbol | Meaning
---|---
$f$ | Carrier frequency
$\lambda$ | Wavelength
$P_{t}$ | Transmit power
$W$ | Signal bandwidth
$\mathcal{F}_{{\rm{dB}}}$ | Noise figure
$N_{0}$ | Thermal noise power
$h_{s}$ | RIS height with respect to the ground
$h_{t}$/$h_{r}$ | TX/RX height with respect to the ground
$D_{t}$/$D_{r}$ | TX/RX antenna diameter
$\phi_{0}$ | TX antenna FNBW
$\phi_{HPBW}$ | TX antenna HPBW
$e_{t}/e_{r}$ | TX/RX antenna aperture efficiency
$G_{t}^{max}/G_{r}^{max}$ | TX/RX antenna gain at the boresight
$G_{t,n}/G_{r,n}$ | TX/RX antenna gain with respect to the $n_{th}$ RU.
$\Gamma$ | Amplitude reflection coefficient of the RUs
$d_{x}$, $d_{y}$ | x-axis and y-axis length, respectively, of the RUs
$\alpha$, $\beta$ | Radii of the TX beam elliptic footprint
$\epsilon$ | Eccentricity of TX beam elliptic footprint
$r_{1}$ |  Distance between the center of the TX antenna and the center of the TX footprint at the RIS plane
$r_{2}$ |  Distance between the center of the TX footprint at the RIS plane and the center of the RX antenna
$r_{1,n}$ |  Distance between the center of the TX antenna and the $n_{th}$ RU
$r_{2,n}$ |  Distance between the $n_{th}$ RU and the center of the RX antenna
$r_{1,h}$ | TX-RIS horizontal distance
$r_{1,h}^{*}$ | Optimal RIS horizontal distance
$r_{h}$ | TX-RX horizontal distance
$\theta_{i}$, $\theta_{r}$ |  Electromagnetic-wave incidence and departure angles, respectively, with respect to the RIS center
$\theta_{i,n}$, $\theta_{r,n}$ |  Electromagnetic-wave incidence and departure angles, respectively, with respect to the $n_{th}$ RU
$S_{s}$ | RIS area
$S_{i}$ | Area of the TX beam elliptic footprint corresponding to the FNBW
$S_{HPBW}$ | Area of the TX beam elliptic footprint corresponding to the HPBW
$P_{R}$ | Received power
$\rho$ | SNR

##  II System Model

Figure 1: System model.

As illustrated in Fig. 1, we consider a fixed-topology street-level scenario,
in which a TX communicates with a RX through an RIS. $r_{1,h}$, $r_{2,h}$, and
$r_{h}$ are the horizontal TX-RIS, RIS-RX, and TX-RX distances, respectively,
while $h_{t}$, $h_{s}$, and $h_{r}$ are the TX, RIS, and RX heights,
respectively. $\theta_{i}$ and $\theta_{r}$ are the incidence and departure
angles, respectively, 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 $D_{t}$ and $D_{r}$, respectively. As a
result, for $D_{t},D_{r}>>\lambda$, where $\lambda$ represents the wavelength,
their power radiation patterns $E_{t}\left(\phi\right)$ and
$E_{r}\left(\phi\right)$, respectively, are given by [25]

$\displaystyle E_{m}\left(\phi\right)=\frac{2\lambda}{\pi
D_{m}}\frac{J_{1}\left(\frac{\pi
D_{m}\sin\left(\phi\right)}{\lambda}\right)}{\sin\left(\phi\right)},\quad
0\leq\phi<\pi/2$ (1)

for $m\in\\{t,r\\}$. $\phi$ is measured from the broadside direction
($\phi=0$) and $J_{1}\left(\cdot\right)$ is the first-order Bessel function of
the first kind. Hence, their gains, denoted by $G_{t}\left(\phi\right)$ and
$G_{r}\left(\phi\right)$, respectively, are given by

$\displaystyle G_{m}\left(\phi\right)$ $\displaystyle=\frac{e_{m}4\pi
E_{m}^{2}\left(\phi\right)}{\int_{0}^{2\pi}\int_{0}^{\frac{\pi}{2}}E_{m}^{2}\left(\phi\right)\sin\left(\phi\right)d\phi
d\theta}$ (2) $\displaystyle=4e_{m}\left(\frac{J_{1}\left(\frac{\pi
D_{m}\sin\left(\phi\right)}{\lambda}\right)}{\sin\left(\phi\right)}\right)^{2},\quad
m\in\\{t,r\\},$

where $e_{t}$ and $e_{r}$ denote the aperture efficiencies of the TX and RX
antennas, respectively. Consequently, their maximum gain, denoted by
$G_{m}^{max}$, that is obtained for $\phi=0$ is given by

$\displaystyle G_{m}^{max}=e_{m}\left(\frac{\pi
D_{m}}{\lambda}\right)^{2},\quad m\in\\{t,r\\}.$ (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
$\phi_{HPBW}$, 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.

###### 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 $\theta_{\text{r}}$, which is the RX
direction. It consists of $N_{x}\times N_{y}$ RUs of size $d_{x}\times d_{y}$
and 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]

$\displaystyle G_{s}\left(\theta\right)=4{\cos\left(\theta\right)},\quad
0\leq\theta<\pi/2.$ (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 $P_{t}$ and that the received signal is subject to additive white
Gaussian noise with power

$\displaystyle N_{0}=-174+10\log_{10}\left(W\right)+\mathcal{F}_{{\rm{dB}}},$
(5)

where $\mathcal{F}_{{\rm{dB}}}$ is the noise figure in dB and $W$ is the
transmission bandwidth [32].

##  III RIS Illuminated area, SNR, and Optimal RIS placement

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.

###  III-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 $\phi_{0}$.

###### 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

$\displaystyle\alpha=\frac{\sin\left(\frac{\phi_{0}}{2}\right)}{\cos\left(\theta_{i}+\frac{\phi_{0}}{2}\right)}r_{1}$
(6)

and

$\displaystyle\beta=\alpha\sqrt{1-\epsilon^{2}},$ (7)

where

$\displaystyle\epsilon=\frac{\sin\left(\theta_{i}\right)}{\cos\left(\frac{\phi_{0}}{2}\right)}.$
(8)

Moreover, $r_{1}$ denotes the distance between the center of the TX and the
center of the TX footprint at the RIS plane, while $\theta_{i}$ is 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

$\displaystyle S_{i}=\pi\alpha\beta.$ (9)

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

$\displaystyle S=\min\left(S_{i},S_{s}\right),$ (10)

where $S_{s}$ denotes the RIS area. If $S_{s}\leq S_{i}$, only part of the
power that falls within $S_{i}$ can be reflected towards the RX; thus, beam
waste occurs. On the other hand, if $S_{s}>S_{i}$ only part of the RIS is used
to reflect the incident electromagnetic wave222In the $S_{s}>S_{i}$ case, (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
$S_{i}$, according to Lemma 1..

Finally, we note that in the case $S_{s}>S_{i}$ there 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 $S_{s}>S_{i}$ case
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.

###  III-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 $P_{R}$, is maximized for certain $r_{1,h}$ and it can be
evaluated as333(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 $\lambda/2$.

$\displaystyle P_{R}$
$\displaystyle=\left(\frac{\lambda}{4\pi}\right)^{4}P_{t}\Gamma^{2}\left|\sum_{n=1}^{M}\sqrt{\frac{G_{t,n}G_{r,n}G_{s}{\left(\theta_{i,n}\right)}G_{s}{\left(\theta_{r,n}\right)}}{r_{1,n}^{2}r_{2,n}^{2}}}\right|^{2},$
(11)

where $\Gamma$ is the amplitude reflection coefficient that we consider is the
same of all RUs, $M$ is the number of illuminated RUs that are included within
$S$. In the $S_{s}>S_{i}$ case, 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

$M=\frac{S}{d_{x}d_{y}}.$ (12)

$r_{1,n}$ and $r_{2,n}$ are the distances between the centers of the TX and RX
antennas and the $n_{th}$ RU, respectively. In addition, $G_{t,n}$ and
$G_{r,n}$ represent the TX and RX antenna gains corresponding to the same RU,
respectively. Finally, $G_{s}{\left(\theta_{i,n}\right)}$ is the gain of the
$n_{th}$ RU towards the TX antenna and $G_{s}{\left(\theta_{r,n}\right)}$ is
its corresponding gain towards the RX antenna.

###### Proof:

The proof is provided in Appendix C. ∎

In the special case in which $S_{i}>>S_{s}$, Proposition 2 that follows
presents a simplified closed-form expression for the received power.

###### Proposition 2

If $S_{i}>>S_{s}$, (11) is reduced to

$\displaystyle P_{R}$
$\displaystyle=\left(\frac{\lambda}{4\pi}\right)^{4}\frac{P_{t}\Gamma^{2}\left(S_{s}\right)^{2}G_{t}^{max}G_{r}^{max}G_{s}{\left(\theta_{i}\right)}G_{s}{\left(\theta_{r}\right)}}{d_{x}^{2}d_{y}^{2}r_{1}^{2}r_{2}^{2}},$
(13)

where $r_{2}$ denotes the distance between the center of the TX footprint at
the RIS plane and the center of the RX.

###### Proof:

We consider that $S_{i}>>S_{s}$ holds 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 $G_{t}^{max}$
in the interval $\left[-\frac{\phi_{HPBW}}{2},\frac{\phi_{HPBW}}{2}\right]$.
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 $S_{s}\geq S_{i}$. Proposition 3 that follows
presents a simplified closed-form expression for the received power.

###### Proposition 3

If: i) $S_{s}\geq S_{i}$; ii) $G_{r,n}$ is aproximately independent of $n$ and
equal to $G_{r}^{max}$; and iii) the incident and departure RU gains together
with the corresponding TX-RIS and RIS-RX distances are approximately
independent of $n$, $P_{R}$ is tightly approximated, by at least $94\%$
accuracy, as

$\displaystyle P_{R}$
$\displaystyle\approx\left(\frac{\lambda}{4\pi}\right)^{4}\frac{P_{t}\Gamma^{2}\left(S_{HPBW}\right)^{2}G_{t}^{max}G_{r}^{max}G_{s}{\left(\theta_{i}\right)}G_{s}{\left(\theta_{r}\right)}}{d_{x}^{2}d_{y}^{2}r_{1}^{2}r_{2}^{2}},$
(14)

where $S_{HPBW}$ is the HPBW footprint of the main lobe on the RIS. $S_{HPBW}$
can be computed by the same process used in the computation of $S_{i}$, where
in (6), (7), and (8), $\phi_{0}$ should be replaced by $\phi_{HPBW}$.

###### Proof:

The proof of Proposition 3 is a direct result of Lemma 3. ∎

In addition, by replacing $S_{HPBW}$ in (14) with its corresponding
expression, $P_{R}$ is 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 $S_{i}>>S_{s}$ can 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 $S_{s}\geq S_{i}$ case. 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 surface444The phase of an impinging wave on a
surface should not vary by more than $\frac{\pi}{8}$ so that the wave is
considered planar over the surface.. Based on this, for the $S_{s}\geq S_{i}$
case 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 $n$ holds, 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 $\rho$, is given by dividing
$P_{R}$ with $N_{0}$, i.e.

$\displaystyle\rho=\frac{P_{R}}{N_{0}}.$ (15)

$\displaystyle P_{R}$
$\displaystyle\approx\left(\frac{\lambda}{4\pi}\right)^{4}\frac{P_{t}\Gamma^{2}G_{t}^{max}G_{r}^{max}G_{s}{\left(\theta_{i}\right)}G_{s}{\left(\theta_{r}\right)}}{d_{x}^{2}d_{y}^{2}}\left(\frac{r_{1}}{r_{2}}\right)^{2}\pi^{2}\frac{\sin^{4}\left(\frac{\phi_{HPBW}}{2}\right)}{\cos^{4}\left(\frac{\phi_{HPBW}}{2}+\theta_{i}\right)}\left(1-\frac{\sin^{2}\left(\theta_{i}\right)}{\cos^{2}\left(\frac{\phi_{HPBW}}{2}\right)}\right).$
(16)

###  III-C Optimal RIS Placement

$r_{1}$, $r_{2}$, $\theta_{i}$, and $\theta_{r}$ can be expressed as

$\displaystyle r_{1}$
$\displaystyle=\sqrt{r_{{1},{h}}^{2}+y_{s}^{2}+\left(h_{s}-h_{t}\right)^{2}},$
(17) $\displaystyle r_{2}$
$\displaystyle=\sqrt{\left(r_{h}-r_{1,{h}}\right)^{2}+y_{s}^{2}+\left(h_{s}-h_{r}\right)^{2}},$
(18) $\displaystyle\theta_{i}$
$\displaystyle=\tan^{-1}\left(\frac{\sqrt{r_{{1},{h}}^{2}+\left(h_{s}-h_{t}\right)^{2}}}{y_{s}}\right)$
(19)

and

$\displaystyle\theta_{r}$
$\displaystyle=\tan^{-1}\left(\frac{\sqrt{\left(r_{{1},{h}}-r_{h}\right)^{2}+\left(h_{s}-h_{r}\right)^{2}}}{y_{s}}\right),$
(20)

for $y_{s}>0$. Next, for the $S_{i}>>S_{s}$ and $S_{s}\geq S_{i}$ cases we
determine the $r_{1,h}$ that maximizes the end-to-end SNR.

####  III-C1 $S_{i}>>S_{s}$ case

###### 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
$\rho$ with respect to $r_{1,h}$ and setting it equal to 0. This yields

$\displaystyle
a^{\left(1\right)}r_{1,h}^{3}+b^{\left(1\right)}r_{1,h}^{2}+c^{\left(1\right)}r_{1,h}+d^{\left(1\right)}=0,$
(21)

where

$\displaystyle a^{\left(1\right)}=6,$ (22) $\displaystyle
b^{\left(1\right)}=-9r_{h},$ (23) $\displaystyle
c^{\left(1\right)}=3\left(2y_{s}^{2}+r_{h}^{2}+(h_{s}-h_{t})^{2}+(h_{s}-h_{r})^{2}\right),$
(24) $\displaystyle
d^{\left(1\right)}=-3r_{h}\left(y_{s}^{2}+\left(h_{s}-h_{t}\right)^{2}\right).$
(25)

###### Proof:

The proof is provided in Appendix E. ∎

####  III-C2 $S_{s}\geq S_{i}$ case

###### Proposition 5

By by taking the 1st derivative of $\rho$ with respect to $r_{1,h}$ and
setting it equal to 0, $r_{1,h}^{*}$ can approximated by (26), given at the
top of the next page.

$\displaystyle
r_{1,h}^{*}\approx\frac{-\left(h_{s}-h_{t}\right)^{2}+r_{h}^{2}+\left(h_{s}-h_{r}\right)^{2}+\sqrt{\left(\left(h_{s}-h_{t}\right)^{2}-r_{h}^{2}-\left(h_{s}-h_{r}\right)^{2}\right)^{2}+4r_{h}^{2}\left(y_{s}^{2}+\left(h_{s}-h_{t}\right)^{2}\right)}}{2r_{h}}.$
(26)

###### 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 Results & 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.

###  IV-A Results

We consider the parameters of Table II.

TABLE II: Parameter values used in the simulation.

Parameter | Value
---|---
$f$ |  $140$ GHz
$P_{t}$ |  $1$ W
$W$ |  $2$ GHz
$\mathcal{F}_{{\rm{dB}}}$ |  $10$ dB
$d_{x}$, $d_{y}$ | $\lambda/2$
$h_{s}$ |  $12$ m
$D_{t}$ |  $15$ cm
$e_{t}$, $e_{r}$ | $0.7$
$\Gamma$ | $0.9$

####  IV-A1 Validation of Proposition 2

(a) $h_{t}=6$ m, $h_{r}=3$ m.

(b) $h_{t}=3$ m, $h_{r}=6$ m.

(c) $h_{t}=3$ m, $h_{r}=3$ m.

Figure 2: $\rho$ vs. $r_{1,h}$ for $D_{r}=3$ cm, $r_{h}=30$ m, and
$S_{s}=0.012$ $m^{2}$.

(a) $h_{t}=6$ m, $h_{r}=3$ m.
(b) $h_{t}=3$ m, $h_{r}=6$ m.

Figure 3: $\rho$ vs. $r_{1,h}$ for $D_{r}=3$ cm, $r_{h}=80$ m, and
$S_{s}=0.012$ $m^{2}$. Figure 4: Discriminant of (21) for $h_{t}=6$ m,
$h_{r}=3$ m, $D_{r}=3$ cm, and $r_{h}=80$ m. Figure 5: $r_{1,h}^{*}$ vs.
$y_{s}$ for $h_{t}=6$ m, $h_{r}=3$ m, $D_{r}=3$ cm, and $r_{h}=80$ m. Figure
6: $\rho$ vs. $r_{1,h}$ for $h_{t}=6$ m, $h_{r}=3$ m, $D_{r}=3$ cm, $r_{h}=80$
m, $y_{s}=10$ m, and $S_{s}=0.046$ $m^{2}$.

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), $\rho$ vs. $r_{1,h}$
curves for $S_{s}=0.012$ $m^{2}$, $D_{r}=3$ cm, $r_{h}=30$ m, and different
values of $h_{t}$, $h_{r}$, and $y_{s}$. We note that for the considered value
of $D_{t}$ it holds that $\phi_{HPBW}=1.25^{\circ}$, which means that a highly
pencil beam transmission is enacted. In addition, for the examined scanning
range of $r_{1,h}$ the minimum value of $S_{i}$ is equal to $0.09$ $m^{2}$ and
$0.69$ $m^{2}$ for $y_{s}=5$ m and $y_{s}=15$ m, respectively. Hence, it holds
that $S_{i}>>S_{s}$ throughout the considered $r_{1,h}$ range in both cases.

As we observe from Fig. 2, there is a relatively good match between the exact
and closed-form expressions of $\rho$, which validates (13). In addition, we
observe that for $y_{s}=5$ m $\rho$ is maximized when the RIS is closer to the
TX, closer to the RX, and either closer to the TX or the RX for $h_{t}>h_{r}$,
$h_{t}<h_{r}$, and $h_{t}=h_{r}$, respectively. In addition, $\rho$ is
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
$\rho$ and the local minimum near the middle of the TX-RX distance. On the
other hand, for $y_{s}=15$ m we observe that the optimal placement of the RIS
has moved closer to the middle of the TX-RX distance in the $h_{t}>h_{r}$ and
$h_{t}<h_{r}$ cases. This indicates that the higher $y_{s}$ is, the less
pronounced the effect of the height difference between the TX and RX antennas
is on the optimal RIS placement. Consequently, $\rho$ is 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 $\rho$ vs. $r_{1,h}$ curves for a much larger $r_{h}$
distance and 3 values of $y_{s}$. As we observe from Fig. 3, for small values
of $y_{s}$ the variation among the two local optima of $\rho$ and its local
minimum is notable. In particular, a 12.5 dB difference is observed between
the value that maximizes $\rho$ and the one that minimizes it near the middle
of the TX-RX distance. Furthermore, Fig. 3 again reveals that the two local
optima of $\rho$ do not differ substantially in value. As $y_{s}$ increases,
the corresponding difference diminishes. More specifically, as $y_{s}$
increases (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
$y_{s}$.

The trends of Fig. 3 can be validated by observing how the discriminant of the
3rd degree polynomial of (21), which we denote by $\Delta$, varies with
respect to $y_{s}$. Regarding this, in Fig. 4 we depict $\Delta$ with respect
to $y_{s}$. As we observe from Fig. 4, as $y_{s}$ increases $\Delta$ 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 $S_{s}$ on the optimal RIS
placement, in Fig. 5 we compare the exact $r_{1,h}^{*}$ vs. $y_{s}$ curves,
based on $\eqref{received_power_RIS_Proposition_1}$, with the closed-form
analytically obtained one, based on $\eqref{3rd_degree_polynomial_RIS_small}$,
for $r_{h}=80$ m and 3 values of $S_{s}$. Although there is a relatively close
match of the obtained by simulations exact $r_{1,h}^{*}$ values with the
analytical one for $S_{s}=0.012$ $m^{2}$ and $S_{s}=0.026$ $m^{2}$, there is a
substantial discrepancy between them in the $S_{s}=0.046$ $m^{2}$ case for
low-to-moderate $y_{s}$ values. This is justified by the fact that for the
particular $y_{s}$ values it does not hold that the minimum value of $S_{i}$
in in the considered scanning range of $r_{1,h}$ is much larger than $S_{s}$.
For instance, for $y_{s}=10$ m the minimum value of $S_{i}$, which occurs for
$r_{1,h}=0$, is equal to $0.15$ $m^{2}$. Hence, it does not hold that
$S_{i}>>S_{s}$ throughout the $r_{1,h}$ scanning range, which results in (13)
providing inaccurate results. To substantiate the latter, in Fig. 6 we
illustrate the $\rho$ vs. $r_{1,h}$ curve for $S_{s}=0.046$ $m^{2}$ and
$y_{s}=10$ m. As we observe from Fig. 6, there is a substantially discrepancy
between the exact and the closed-form $\rho$ curves for $r_{1,h}=0$. As
aforementioned, in the particular position $S_{i}$ exhibits its smallest
value, equal to 0.15 $m^{2}$, and, hence, it holds that $S_{s}/S_{i}=0.31$,
which is smaller than 1, but not notably smaller. The latter is the
requirement for (13) to hold.

####  IV-A2 Validation of Proposition 3

(a) $h_{t}=6$ m, $h_{r}=3$ m.
(b) $h_{t}=3$ m, $h_{r}=6$ m.

Figure 7: $\rho$ vs. $r_{1,h}$ for $r_{h}=20$ m, $y_{s}=10$ m, and
$S_{s}>S_{i}$. Figure 8:
$\frac{\max\left\\{G_{r,n}\right\\}}{\min\left\\{G_{r,n}\right\\}}$ vs.
$r_{1,h}$ for $h_{t}=6$ m, $h_{r}=3$ m, $r_{h}=20$ m, $y_{s}=10$ m, and
$S_{s}>S_{i}$. Figure 9: $\frac{r_{1}}{\max\left\\{r_{1,n}\right\\}}$ and
$\frac{r_{2}}{\max\left\\{r_{2,n}\right\\}}$ vs. $r_{1,h}$ for $h_{t}=6$ m,
$h_{r}=3$ m, $D_{r}=5$ cm, $r_{h}=20$ m, $y_{s}=10$ m, and $S_{s}>S_{i}$.
Figure 10:
$\max\left\\{\frac{G_{s}\left(\theta_{i,n}\right)}{G_{s}\left(\theta_{i}\right)}\right\\}$
and
$\max\left\\{\frac{G_{s}\left(\theta_{r,n}\right)}{G_{s}\left(\theta_{r}\right)}\right\\}$
vs. $r_{1,h}$ for $h_{t}=6$ m, $h_{r}=3$ m, $D_{r}=5$ cm $r_{h}=20$ m,
$y_{s}=10$ m, and $S_{s}>S_{i}$. Figure 11: $r_{1,h}^{*}$ vs. $y_{s}$ for
$h_{t}=6$ m, $h_{r}=3$ m, $r_{h}=20$ m, $D_{r}=1$ cm, and $S_{s}>S_{i}$.

As far as the validation of Proposition 3 is concerned, in Fig. 7 we depict
the $\rho$ vs. $r_{1,h}$ curves for $r_{h}=20$ m, $y_{s}=10$ m, and different
$D_{r}$ under the assumption that $S_{s}>S_{i}$ throughout the examined
$r_{1,h}$ range. In particular, $S_{i}$ takes its largest value for
$r_{1,h}=40$ m, which is equal to 7.97 $m^{2}$. 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
$\rho$ is maximized.

In addition, from Fig. 7 we further observe that the higher $D_{r}$ is, the
larger the deviation of the exact value of $\rho$, based on
$\eqref{received_power_RIS_Proposition_1}$, with its closed-form counterpart,
based on (14), is for larger $r_{1,h}$ values. This discrepancy is attributed
to the fact that as $D_{r}$ increases the footprint of the main lobe of the RX
antenna on the RIS reduces and, consequently, $G_{r,n}$ 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
$r_{1,h}$ 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 $G_{r,n}$ across the
illuminated RIS region as $D_{r}$ increases, in Fig. 8 we illustrate how the
ratio $\frac{\max\left\\{G_{r,n}\right\\}}{\min\left\\{G_{r,n}\right\\}}$
varies with respect to $r_{1,h}$ for 3 values of $D_{r}$. As we observe, for
$D_{r}=1$ cm $G_{r,n}$ remains almost constant across the illuminated RIS
region, but it starts varying as $D_{r}$ increases. In particular, it varies
substantially for $D_{r}=5$ cm in the $r_{1,h}>10$ m region, which justifies
the increasing discrepancy in the particular region between the exact and
closed-form results as $r_{1,h}$ increases.

On the other hand, from Fig. 9, which depicts the ratios
$\frac{r_{1}}{\max\left\\{r_{1,n}\right\\}}$ and
$\frac{r_{2}}{\max\left\\{r_{2,n}\right\\}}$ with respect to $r_{1,h}$, we see
that throughout the $r_{1,h}$ scanning range there are small variations of the
distances between the TX (RX) antenna center and the $n_{th}$ RU, which
indicates that $r_{1,n}$ and $r_{2,n}$ can be closely approximated by $r_{1}$
and $r_{2}$, respectively, for each $n$. Furthermore, small variations are
also observed for $G_{s}\left(\theta_{i,n}\right)$ and
$G_{s}\left(\theta_{r,n}\right)$ with respect to
$G_{s}\left(\theta_{i}\right)$ and $G_{s}\left(\theta_{r}\right)$,
respectively, across the $r_{1,h}$ scanning range, as Fig. 10 depicts.
Consequently, from Fig. 8, Fig. 9, and Fig. 10, we conclude that the
variations of $G_{r,n}$ over the illuminated RIS region are the reason for the
large discrepancy between the exact numerical and theoretical results in the
$r_{1,h}>10$ m region for $D_{r}=5$ 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 $y_{s}$ for $D_{r}=1$ cm. As we see, there is a relatively close
match of the two curves, which validates (26). In addition, the higher $y_{s}$
is the larger $r_{1,h}^{*}$ becomes.

###  IV-B Design Guidelines

Let us now recapitulate on the main outcomes regarding the $S_{i}>>S_{s}$ and
$S_{i}\leq S_{s}$ cases, 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.

####  IV-B1 $S_{i}>>S_{s}$ case

  * •

For relatively small values of $y_{s}$ compared with $r_{h}$, there are two
positions, one close to the TX and the other close to the RX, that locally
maximize $\rho$, 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 $h_{t}>h_{r}$, whereas it is the opposite for
$h_{t}<h_{r}$. Moreover, the difference in magnitude between the two optima is
relatively small for small values of $r_{h}$, 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 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 $y_{s}$ increases, 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 $y_{s}$
value above which there is only one real value of $r_{1,h}$ that maximizes
$\rho$ that 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 $h_{t}>h_{r}$ or $h_{r}<h_{t}$.

####  IV-B2 $S_{i}\leq S_{s}$ case

  * •

Regardless of whether $h_{t}>h_{r}$ or $h_{r}<h_{t}$, there is only one
optimum of $\rho$ with respect to $r_{1,h}$ that is located closer to the RX.
In addition, a substantially higher $\rho$ is achieved by placing the RIS at
the particular point rather than close to the TX or the middle of the TX-RX
distance.

  * •

$D_{r}$ should be notably smaller than $D_{t}$ so that (14) holds. Under such
a condition, $G_{r,n}$ is approximately constant throughout the illuminated
RIS region and approximately equal to $G_{r}^{max}$. 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.

## Acknowledgements

The authors would like to cordially thank the associate editor and anonymous
reviewers, whose comments and suggestions have led to a substantial
improvement of this work.

## Appendices

## Appendix A

## Proof of Lemma 1

Based on (1), for the total normalized transmit power, which we denote by
$P_{dish}^{tot}$, it holds that

$\displaystyle
P_{dish}^{tot}=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}E^{2}\left(\phi\right)d\phi.$
(27)

As far as $\phi_{0}$ is concerned, which encompasses the main lobe, it can be
computed by finding the points for which $\frac{\pi
D_{t}\sin\left(\phi\right)}{\lambda}=3.83$ since $J_{1}\left(3.83\right)=0$.
Hence, it holds that

$\displaystyle\phi_{0}=2\sin^{-1}\left(\frac{3.83\lambda}{\pi
D_{t}}\right)=2\sin^{-1}\left(\frac{1.22\lambda}{D_{t}}\right).$ (28)

Consequently, the normalized transmit power that is corresponding to the main
lobe, which we denote by $P_{dish}^{main\;lobe}$, is given by

$\displaystyle
P_{dish}^{main\;lobe}=\int_{-\frac{\phi_{0}}{2}}^{\frac{\phi_{0}}{2}}E^{2}\left(\phi\right)d\phi.$
(29)

Let us denote the ratio of the normalized power inside the main lobe over the
total normalized transmit power as a function of $\phi_{HPBW}$ by
$\rho_{main\;lobe}\left(\phi_{HPBW}\right)$. Consequently, it holds that
$\rho_{main\;lobe}\left(\phi_{HPBW}\right)=P_{dish}^{main\;lobe}/P_{dish}^{tot}$
and for $\phi_{HPBW}<\frac{\pi}{12}$ it holds that
$\rho_{main\;lobe}\left(\phi_{HPBW}\right)>\rho_{main\;lobe}\left(\frac{\pi}{12}\right)$.
$\frac{\pi}{12}$ corresponds to the 15∘ minimum HPBW value upper limit for
which the transmission can be considered as pencil beam according to our
assumptions.
$\rho_{main\;lobe}\left(\phi_{HPBW}\right)>\rho_{main\;lobe}\left(\frac{\pi}{12}\right)$
holds due to the fact that as $\phi_{HPBW}$ decreases the main lobe becomes
sharper, which means that more power is concentrated inside the main lobe
compared with the $\phi_{HPBW}=\frac{\pi}{12}$ case. Hence,
$\rho_{main\;lobe}\left(\phi_{HPBW}\right)$ increases.

As far as the value of the term $\frac{D_{t}}{\lambda}$ for which
$\phi_{HPBW}=\frac{\pi}{12}$ is achieved, it holds that

$\displaystyle E^{2}\left(\frac{\pi}{24}\right)=0.5$
$\displaystyle\Rightarrow\left(\frac{2\lambda}{\pi
D_{t}}\frac{J_{1}\left(\frac{\pi
D_{t}\sin\left(\frac{\pi}{24}\right)}{\lambda}\right)}{\sin\left(\frac{\pi}{24}\right)}\right)^{2}=0.5$
$\displaystyle\Rightarrow\frac{D_{t}}{\lambda}=3.94.$ (30)

For $\frac{D_{t}}{\lambda}=3.94$ under which $\phi_{HPBW}=\frac{\pi}{12}$ is
achieved, it holds that

$\displaystyle\rho_{main\;lobe}\left(\frac{\pi}{12}\right)$
$\displaystyle=\frac{\int_{-\sin^{-1}\left(\frac{1.22}{3.94}\right)}^{\sin^{-1}\left(\frac{1.22}{3.94}\right)}\left(\frac{2}{\pi
3.94}\frac{J_{1}\left(\pi
3.94\sin\left(\phi\right)\right)}{\sin\left(\phi\right)}\right)^{2}d\phi}{\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{2}{\pi
3.94}\frac{J_{1}\left(\pi
3.94\sin\left(\phi\right)\right)}{\sin\left(\phi\right)}\right)^{2}d\phi}$
$\displaystyle=0.97.$ (31)

By taking into account that
$\rho_{main\;lobe}\left(\phi_{HPBW}\right)>\rho_{main\;lobe}\left(\frac{\pi}{12}\right)$
for $\phi_{HPBW}<\frac{\pi}{12}=15^{\circ}$, the proof of Lemma 1 is
concluded.

## Appendix B

## Proof of Lemma 2

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

$\displaystyle\frac{\sin\left(\frac{\phi_{0}}{2}\right)}{\alpha}=\frac{\sin\left(\overset{\wedge}{\text{C}}\right)}{r_{1}},$
(32)

where $\overset{\wedge}{\text{C}}$ denotes the angle of corner C and can be
calculated as

$\displaystyle\overset{\wedge}{\text{C}}=\frac{\pi}{2}-\frac{\phi_{0}}{2}-\theta_{i}.$
(33)

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

The eccentricity of the elliptical footprint can be evaluated as

$\displaystyle\epsilon=\frac{\sin\left(\theta_{i}\right)}{\sin\left(\frac{\pi}{2}-\frac{\phi_{0}}{2}\right)},$
(34)

or equivalently as in (8). Finally, $\beta$ can be obtained as in (7). This
concludes the proof.

## Appendix C

## Proof of Proposition 1

The incident electric field on the $n_{th}$ RU of the RIS illuminated area can
be obtained as

$\displaystyle\textbf{E}_{n}=E_{n}e^{-j\frac{2\pi}{\lambda}r_{1,n}}{\mathbf{n}}_{o},\quad
n=1,2,...,M,$ (35)

where $E_{n}$ is the amplitude of the incident wave and ${\mathbf{n}}_{o}$ is
a unitary vector that is perpendicular to the 2D plane that the electric field
lies on [36, Example 11-3]. It holds that

$E_{n}=\sqrt{\frac{2\eta P_{t}G_{t,n}}{4\pi r_{1,n}^{2}}},$ (36)

where $\eta$ is the free-space impedance. As a consequence, the power density
at the $n_{th}$ RU of the RIS can be expressed as

$\displaystyle P_{n}=\frac{E_{n}^{2}}{2\eta}$ (37)

or, with the aid of (36), as

$\displaystyle P_{n}=\frac{P_{t}G_{t,n}}{4\pi r_{1,n}^{2}},$ (38)

where $\eta$ is the free-space impedance. Thus, the incident power at the
$n_{th}$ RU can be evaluated as

$\displaystyle P_{i,n}=P_{n}A_{n},$ (39)

where $A_{n}$ stands for the effective aperture of the $n_{th}$ illuminated RU
and can be obtained as

$\displaystyle
A_{n}=\frac{\lambda^{2}}{4\pi}G_{s}{\left(\theta_{i,n}\right)}.$ (40)

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

$\displaystyle
P_{i,n}=\left(\frac{\lambda}{4\pi}\right)^{2}\frac{P_{t}G_{t,n}G_{s}\left(\theta_{i,n}\right)}{r_{1,n}^{2}}.$
(41)

As a result and due to the energy conservation law, the reflected power
density by the $n_{th}$ RU, which is captured by the RX antenna, can be
expressed as

$\displaystyle
P_{r,n}=\frac{P_{i,n}\Gamma^{2}G_{s}\left(\theta_{r,n}\right)}{4\pi
r_{2,n}^{2}}.$ (42)

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

$\displaystyle
P_{r,n}=\frac{\lambda^{2}}{\left(4\pi\right)^{3}}\frac{P_{t}\Gamma^{2}G_{t,n}G_{s}{\left(\theta_{i,n}\right)}G_{s}{\left(\theta_{r,n}\right)}}{r_{1,n}^{2}r_{2,n}^{2}}.$
(43)

The power captured by the receiver from the $n_{th}$ RU is given by

$\displaystyle P_{R,n}$ $\displaystyle=P_{r,n}\frac{\lambda^{2}}{4\pi}G_{r,n}$
$\displaystyle=\left(\frac{\lambda}{4\pi}\right)^{4}\frac{P_{t}\Gamma^{2}G_{t,n}G_{r,n}G_{s}{\left(\theta_{i,n}\right)}G_{s}{\left(\theta_{r,n}\right)}}{r_{1,n}^{2}r_{2,n}^{2}}.$
(44)

Moreover, the corresponding electric field observed at the receiver from the
$n_{th}$ RU can be evaluated as

$\mathbf{E}_{R,n}=E_{R,n}e^{-j\left(\theta_{n}+\frac{2\pi\left(r_{1,n}+r_{2,n}\right)}{\lambda}\right)}{\mathbf{a}_{o}},$
(45)

where $\mathbf{a}_{o}$ is a unitary vector perpendicular to the 2D plane that
the reflected electric field lies on and $\theta_{n}$ is adjustable the phase
response of the $n_{th}$ RU. Additionally, $E_{R,n}$ can be computed as

$\displaystyle E_{R,n}$ $\displaystyle=\sqrt{2\eta P_{R,n}},$ (46)

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

$\displaystyle E_{R,n}$
$\displaystyle=\sqrt{2\eta\left(\frac{\lambda}{4\pi}\right)^{4}\frac{P_{t}\Gamma^{2}G_{t,n}G_{r,n}G_{s}{\left(\theta_{i,n}\right)}G_{s}{\left(\theta_{r,n}\right)}}{r_{1,n}^{2}r_{2,n}^{2}}}.$
(47)

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

$\displaystyle\mathbf{E}_{R}=\sum_{n=1}^{M}\mathbf{E}_{R,n}.$ (48)

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

$\displaystyle\mathbf{E}_{R}$
$\displaystyle=\left(\frac{\lambda}{4\pi}\right)^{2}\sqrt{2\eta
P_{t}\Gamma}\sum_{n=1}^{M}\sqrt{\frac{G_{t,n}G_{r,n}G_{s}{\left(\theta_{i,n}\right)}G_{s}{\left(\theta_{r,n}\right)}}{r_{1,n}^{2}r_{2,n}^{2}}}$
$\displaystyle\times
e^{-j\left(\theta_{n}+\frac{2\pi\left(r_{1,n}+r_{2,r}\right)}{\lambda}\right)}{\mathbf{a}_{o}},$
(49)

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

$\displaystyle P_{R}=\frac{\left|\mathbf{E}_{R}\right|^{2}}{2\eta},$ (50)

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

$\displaystyle P_{R}$
$\displaystyle=\left(\frac{\lambda}{4\pi}\right)^{4}P_{t}\Gamma^{2}$
$\displaystyle\times\left|\sum_{n=1}^{M}\sqrt{\frac{G_{t,n}G_{r,n}G_{s}{\left(\theta_{i,n}\right)}G_{s}{\left(\theta_{r,n}\right)}}{r_{1,n}^{2}r_{2,n}^{2}}}e^{-j\left(\theta_{n}+\frac{2\pi\left(r_{1,n}+r_{2,n}\right)}{\lambda}\right)}\right|^{2}.$
(51)

By assuming that the optimal phase shift is induced by each RU so as to
maximize $P_{R}$, i.e.,

$\displaystyle\theta_{n}=-\frac{2\pi\left(r_{1,n}+r_{2,n}\right)}{\lambda},$
(52)

the received power can be rewritten as

$\displaystyle P_{R}$
$\displaystyle=\left(\frac{\lambda}{4\pi}\right)^{4}P_{t}\Gamma^{2}\left|\sum_{n=1}^{M}\sqrt{\frac{G_{t,n}G_{r,n}G_{s}{\left(\theta_{i,n}\right)}G_{s}{\left(\theta_{r,n}\right)}}{r_{1,n}^{2}r_{2,n}^{2}}}\right|^{2},$
(53)

which concludes the proof.

## Appendix D

## Proof of Lemma 3

Let us denote the ratio of the normalized power included within a step
function in the interval
$\left[-\frac{\phi_{HPBW}}{2},\frac{\phi_{HPBW}}{2}\right]$ over the
normalized power included within the FNBW by $\kappa\left(\phi_{HPBW}\right)$.
It holds that

$\displaystyle\kappa\left(\phi_{HPBW}\right)=\frac{\phi_{HPBW}}{P_{dish}^{main\;lobe}}=\frac{\phi_{HPBW}}{\int_{-\frac{\phi_{0}}{2}}^{\frac{\phi_{0}}{2}}\left(\frac{2\lambda}{\pi
D_{t}}\frac{J_{1}\left(\frac{\pi
D_{t}\sin\left(\phi\right)}{\lambda}\right)}{\sin\left(\phi\right)}\right)^{2}d\phi}.$
(54)

For the minimum HPBW of value of 15∘ required for the transmit beam to be
considered pencil beam, it holds that
$\kappa\left(\frac{\pi}{12}\right)=0.97$. Moreover,
$\kappa\left(\phi_{HPBW}\right)$ is a monotonically increasing function as
$\phi_{HPBW}\downarrow$ since the smaller the beamwidth is the sharper the
main lobe becomes. Consequently, it can be more accurately approximated by a
step function with magnitude equal to $G_{t}^{max}$ in the interval
$\left[-\frac{\phi_{HPBW}}{2},\frac{\phi_{HPBW}}{2}\right]$.

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
$\mu\left(\phi_{HPBW}\right)$, it holds that

$\displaystyle\mu\left(\phi_{HPBW}\right)=\frac{\phi_{HPBW}}{P_{dish}^{tot}}=\frac{\phi_{HPBW}}{\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{2\lambda}{\pi
D_{t}}\frac{J_{1}\left(\frac{\pi
D_{t}\sin\left(\phi\right)}{\lambda}\right)}{\sin\left(\phi\right)}\right)^{2}d\phi}.$
(55)

For the minimum HPBW of value of 15∘ it holds that
$\mu\left(\frac{\pi}{12}\right)=0.94$. Moreover, for the same reason as in the
$\kappa\left(\phi_{HPBW}\right)$ case $\mu\left(\phi_{HPBW}\right)$ is a
monotonically increasing function as $\phi_{HPBW}\downarrow$.

## Appendix E

## Proof of Proposition 4

The proof begins by plugging (13) into (15) and rewriting $\rho$ as

$\displaystyle\rho=\left(\frac{\lambda}{4\pi}\right)^{4}\frac{16P_{t}\Gamma^{2}\left(S_{s}\right)^{2}G_{t}^{max}G_{r}^{max}}{d_{x}^{2}d_{y}^{2}N_{0}}\frac{F\left(r_{1,h}\right)}{G\left(r_{1,h}\right)},$
(56)

where

$\displaystyle F\left(r_{1,h}\right)$
$\displaystyle{=}\cos\left(\tan^{-1}\left(\frac{\sqrt{r_{{1},{h}}^{2}+\left(h_{s}-h_{t}\right)^{2}}}{y_{s}}\right)\right)$
$\displaystyle\times\cos\left(\tan^{-1}\left(\frac{\sqrt{\left(r_{{1},{h}}-r_{h}\right)^{2}+\left(h_{s}-h_{r}\right)^{2}}}{y_{s}}\right)\right)$
(57)

and

$\displaystyle G\left(r_{1,h}\right)$
$\displaystyle{=}\left(r_{{1},{h}}^{2}+y_{s}^{2}+\left(h_{s}-h_{t}\right)^{2}\right)$
$\displaystyle\times\left(\left(r_{h}-r_{{1,h}}\right)^{2}+y_{s}^{2}+\left(h_{s}-h_{r}\right)^{2}\right).$
(58)

From (56), we observe that the end-to-end SNR depends on $r_{1,h}$ through the
ratio $\frac{F\left(r_{1,h}\right)}{G\left(r_{1,h}\right)}$. Hence, the
optimum $r_{1,h}$ that maximizes $\rho$ can be obtained by evaluating the
roots of the first derivative of
$\frac{F\left(r_{1,h}\right)}{G\left(r_{1,h}\right)}$ with respect to
$r_{1,h}$. The first derivative of
$\frac{F\left(r_{1,h}\right)}{G\left(r_{1,h}\right)}$ 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
$a^{\left(1\right)}r_{1,h}^{3}+b^{\left(1\right)}r_{1,h}^{2}+c^{\left(1\right)}r_{1,h}+d^{\left(1\right)}=0$.

$\displaystyle\frac{d\left(\frac{F\left(r_{1,h}\right)}{G\left(r_{1,h}\right)}\right)}{dr_{1,h}}$
$\displaystyle=-\left(6r_{1,h}^{3}-9r_{h}r_{1,h}^{2}+3\left(2y_{s}^{2}+r_{h}^{2}+\left(h_{s}-h_{t}\right)^{2}+\left(h_{s}-h_{r}\right)^{2}\right)r_{1,h}-3r_{h}\left(y_{s}^{2}+\left(h_{s}-h_{t}\right)^{2}\right)\right)$
$\displaystyle\times\left(\frac{\left(y_{s}^{2}+r_{1,h}^{2}+\left(h_{s}-h_{t}\right)^{2}\right)\left(y_{s}^{2}+\left(r_{h}-r_{1,h}\right)^{2}+\left(h_{s}-h_{r}\right)^{2}\right)}{\left(h_{s}-h_{t}\right)^{2}\left(h_{s}-h_{r}\right)^{2}}\right)^{-\frac{1}{2}}$
$\displaystyle\times\left(\left(y_{s}^{2}+r_{1,h}^{2}+\left(h_{s}-h_{t}\right)^{2}\right)\left(y_{s}^{2}+\left(r_{h}-r_{1,h}\right)^{2}+\left(h_{s}-h_{r}\right)^{2}\right)\right)^{-2}.$
(59)

## Appendix F

## Proof of Proposition 5

The proof begins by plugging (16) into (15) and rewriting $\rho$ as

$\displaystyle\rho$
$\displaystyle\approx\left(\frac{\lambda}{4\pi}\right)^{4}\frac{16P_{t}\Gamma^{2}G_{t}^{max}G_{r}^{max}\pi^{2}\sin^{4}\left(\frac{\phi_{HPBW}}{2}\right)}{d_{x}^{2}d_{y}^{2}N_{0}}$
$\displaystyle\times F\left(r_{1,h}\right)H\left(r_{1,h}\right),$ (60)

where

$\displaystyle
H\left(r_{1,h}\right)=\frac{r_{{1},{h}}^{2}+y_{s}^{2}+\left(h_{s}-h_{t}\right)^{2}}{\left(r_{h}-r_{{1,h}}\right)^{2}+y_{s}^{2}+\left(h_{s}-h_{r}\right)^{2}}\frac{1-\frac{\sin^{2}\left(\theta_{i}\right)}{\cos^{2}\left(\frac{\phi_{HPBW}}{2}\right)}}{\cos^{4}\left(\frac{\phi_{HPBW}}{2}+\theta_{i}\right)}.$
(61)

Hence, the optimum $r_{1,h}$ that maximizes $\rho$ can be obtained by
evaluating the roots of the first derivative of
$F\left(r_{1,h}\right)H\left(r_{1,h}\right)$ with respect to $r_{1,h}$. Before
computing the corresponding derivative, we simplify things by taking into that
for pencil-beam transmissions it holds that $\phi_{HPBW}<<1$. Hence,
considering that $\cos\left(\frac{\phi_{HPBW}}{2}\right)\approx 1$ and
$\cos\left(\frac{\phi_{HPBW}}{2}+\theta_{i}\right)\approx\cos\left(\theta_{i}\right)$,
$H\left(r_{1,h}\right)$ can be approximated as

$\displaystyle
H\left(r_{1,h}\right)\approx\frac{r_{{1},{h}}^{2}+y_{s}^{2}+\left(h_{s}-h_{t}\right)^{2}}{\left(r_{h}-r_{{1,h}}\right)^{2}+y_{s}^{2}+\left(h_{s}-h_{r}\right)^{2}}\frac{1}{\cos^{2}\left(\theta_{i}\right)}.$
(62)

As a result, the first derivative of
$F\left(r_{1,h}\right)H\left(r_{1,h}\right)$ can be approximated by (63) given
at the top of the next page.

$\displaystyle\frac{d\left(F\left(r_{1,h}\right)H\left(r_{1,h}\right)\right)}{dr_{1,h}}\approx-\frac{3\sqrt{(r_{1,h}^{2}+y_{s}^{2}+\left(h_{s}-h_{t}\right)^{2}}\left(r_{h}r_{1,h}^{2}+\left(\left(h_{s}-h_{t}\right)^{2}-r_{h}^{2}-\left(h_{s}-h_{r}\right)^{2}\right)r_{1,h}-r_{h}\left(y_{s}^{2}+\left(h_{s}-h_{t}\right)^{2}\right)\right)}{\left(\left(r_{h}-r_{{1,h}}\right)^{2}+y_{s}^{2}+\left(h_{s}-h_{r}\right)^{2}\right)^{\frac{5}{2}}}.$
(63)

Consequently, $r_{1,h}$ is obtained as one of the two solutions of
$r_{h}r_{1,h}^{2}+\left(\left(h_{s}-h_{t}\right)^{2}-r_{h}^{2}-\left(h_{s}-h_{r}\right)^{2}\right)r_{1,h}-r_{h}\left(y_{s}^{2}+\left(h_{s}-h_{t}\right)^{2}\right)=0$.
As it is shown in Section IV, between its two real roots the one that
maximizes $\rho$ is the one closer to RX, given by (26).

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|  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 University 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 Engineering
(ECE) diploma degree and Ph.D. degree in Wireless Communications from the
Aristotle University 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 candidate at University of Peloponnese in collaboration with
the National Centre for Scientific Research-“Demokritos”. His current research
interests are related 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
scholarship 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 Telecommunications 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 Piraeus. She received the Diploma in Electrical and
Computer Engineering from the National Technical 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).
---|---
|  Symeon Chatzinotas  (S’06–M’09–SM’13) is currently 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 Development projects
for the National Center for Scientific Research Demokritos, the Center of
Research and Technology Hellas and the Center of Communication 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.
---|---