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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)
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dF(r1,h)
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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 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.
