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Coverage analysis of reconﬁgurable intelligent surface assisted THz

wireless systems

Alexandros–Apostolos A. Boulogeorgos, Senior Member, IEEE, and

Angeliki Alexiou, Member, IEEE

This paper presents a connectivity analysis of reconﬁgurable intelligent surface (RIS) assisted terahertz (THz) wireless systems.

Speciﬁcally, a system model that accommodates the particularities of THz band links as well as the characteristics of the RIS is

reported, accompanied by a novel general end-to-end (e2e) channel attenuation formula. Based on this formula, we derive a closed-

form expression that returns the optimal phase shifting of each reﬂection unit (RU) of the RIS. Moreover, we provide a tractable

e2e channel coefﬁcient approximation that is suitable for analyzing the RIS-assisted THz wireless system performance. Building

upon the aforementioned approximation as well as the assumption that the user equipments are located in random positions within

a circular cluster, we present the theoretical framework that quantiﬁes the coverage performance of the system under investigation.

In more detail, we deliver a novel closed-form expression for the coverage probability that reveals that there exists a minimum

transmission power that guarantees 100% coverage probability. Both the derived channel model as well as the coverage probability

are validated through extensive simulations and reveal the importance of taking into account both the THz channel particularities

and the RIS characteristics, when assessing the system’s performance and designing RIS-assisted THz wireless systems.

Index Terms—Channel modeling, Connectivity analysis, Coverage probability, Reconﬁgurable intelligent surfaces, Terahertz

wireless systems.

NOMENCLATURE

3D Three Dimensional

6G Sixth Generation

AF Amplify-and-Forward

AHBF Azimuth Half-Power Beamwidth

AP Access Point

CDF Cumulative Distribution Function

e2e End-to-End

EHPB Elevation Half-Power Beamwidth

HITRAN HIgh Resolution TRansmission AbsorptioN

ITU International Telecommunication Union

LoS Line-of-Sight

PDF Probability Density Function

PL Path-Loss

RIS Reconﬁgurable Intelligent Surface

RU Reﬂection Unit

RX Receiver

SNR Signal-to-Noise-Ratio

THz Terahertz

TX Transmitter

UE User Equipment

I. INTRODUCTION

ACCORDING to the international telecommunication

union (ITU), the global mobile trafﬁc is expected to

continue its exponential growth, reaching 5zettabytes per

month by 2020 [1]. This increase is driven by emerging data

rate hungry applications, like virtual, augmented and extended

reality, virtual presence by means of holographic projection,

autonomous vehicles, and others [2]–[4]. Looking forward

The authors are with the Department of Digital Systems, University

of Piraeus Piraeus 18534 Greece (e-mails: al.boulogeorgos@ieee.org, alex-

iou@unipi.gr).

This work has received funding from the European Commission’s Horizon

2020 research and innovation programme (ARIADNE) under grant agreement

No. 871464.

to the sixth generation (6G) era, two main approaches have

been identiﬁed as candidate technology enabler to support

these unprecedented trafﬁc demands [5], [6]. The ﬁrst one

is to exploit higher-frequency bands, with emphasis to the

terahertz (THz) [7]–[14], while the second one lies to the

use of reconﬁgurable intelligent surfaces (RISs) capable of

alternating their electromagentic properties and thus devising

a beneﬁcial wireless propagation environment [15]–[20].

Scanning the technical literature, we can observe a fast

growing research effort on analyzing, optimizing, designing,

developing and demonstrating wireless THz systems [21]–

[38]. In more detail, in [21] and [22], Jornet et. al used

radiative transfer theory to extract a propagation model for

nano-scale THz communications, while, in [23], Yang et.

al presented a channel model for body-centric THz nano-

scale networks. Moreover, in [25], the authors reported a

simpliﬁed path-loss (PL) model for the 275 −400 GHz band.

This work was extended in [26], in order to include the

impact of secondary reﬂections, and it was used as a basic

propagation model for several works that analyze the wireless

THz system performance, such as [27]–[30], and propose

physical and/or medium access control strategies, like [31]–

[33]. Meanwhile, in [34], the authors introduced a PL model

for nano-sensor THz networks for plant foliage applications,

while, in [35], a propagation model for intra-body nano-scale

communications was proposed. Moreover, in [36], the au-

thors presented a multi-ray THz propagation model. Likewise,

in [37], the authors revealed and quantiﬁed the detrimental

effect of blockage in THz wireless systems, whereas, in [38],

the authors presented a testbed for THz communications in

the 275 to 325 GHz. Additionally, in [39], the impact of

human blockage in low-THz wireless systems was discussed.

Similarly, in [40], the authors performed coverage analysis

in THz wireless systems that experience blockage. Finally,

in [41], the authors evaluated the effect of blockage in the

association process in THz wireless systems.

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All the aforementioned contributions agree that line-of-sight

(LoS) channel attenuation and blockage are the main limiting

factors of THz wireless systems. To break the barriers set

by blockage, recently, some research works proposed the use

of RIS [6], [16]–[19], [42]–[47]. In particular, in [6], [18],

and [19], the authors explained how RIS can be used to

mitigate the impact of blockage and introduced the idea of

reﬂected LoS links. In this direction, in [42], the authors

conducted an asymptotic uplink ergodic capacity study, as-

suming that the transmitter (TX)-RIS and RIS-receiver (RX)

channels follow Rician distribution, whereas, in [16] and [17],

the optimization framework for the maximization of the RX

received power was presented. Similarly, in [43] the joint

maximization of the sum-rate and energy efﬁciency was stud-

ied for a multi-user downlink scenario, in which connectivity

was established by means of reﬂected LoS. Additionally,

in [44], an error analysis was performed for RIS-assisted

non-orthogonal multiple access networks. Moreover, in [45],

di Renzo et. al highlighted the fundamental similarities and

differences between RISs and relays. In the same direction,

in [46], the authors compared the performance of RIS-assisted

systems against decode-and-forward relaying ones in terms

of energy efﬁciency, while, in [47], the authors conducted

a performance comparison between RIS and amplify-and-

forward (AF) relays in terms of average received signal-to-

noise-ratio (SNR), outage probability, diversity order and gain,

symbol error rate and ergodic capacity, which revealed that,

in general, RIS-assisted wireless systems can outperform the

corresponding AF relaying ones. Finally, in [48], the impact of

hardware imperfections in RIS-assisted wireless systems was

quantiﬁed in terms of outage probability and diversity order.

To the best of the authors knowledge, there are only a few

published works that examine the use of RIS in THz wireless

systems [49]–[52]. In [49] and [50], although the directional

nature of the THz links was taken into account, the PL

characteristics of the transmission path were neglected, while,

in [52], the design characteristics of the RIS as well as its

functionality were not taken into account. Finally, in [51], the

impact of molecular absorption loss was ignored, despite of its

paramount importance in the THz band, which was highlighted

in several previous contributions (see e.g., [27], [33], [53], [54]

and reference therein). The main reason behind this is the lack

of tractable channel model for RIS-assisted systems operat-

ing in the THz band that accommodates both the building

blocks of such systems and the particularities of the THz

band. In more detail, this model should take into account

the transceivers’ antenna gains as well as their position in

respect to the RIS position, the transmission frequency, the

characteristics of the reﬂection units (RUS), namely number

and dimensions of reﬂection elements, reﬂection coefﬁcients,

antenna patterns and phase shifts of each one of the RU, as

well as the environmental conditions, while being tractable in

order to become a useful tool for analyzing the performance of

such systems. Motivated by this, this paper focuses on cover-

ing this gap by providing a low-complexity channel model that

takes into account the particularities of the THz propagation

medium as well as the physical characteristic of the RIS.

Building upon this model, we present a comprehensive system

model for RIS-assisted THz wireless systems that support

broadcasting and we conduct coverage analysis that reveals

their limitations. Speciﬁcally, the contribution of this paper is

as follows:

•We provide a system model for RIS-assisted THz wireless

communications and we employ electromagnetic theory

tools to derive a general expression for the end-to-end

(e2e) channel attenuation. This expression takes into

account not only the access point (AP)-RIS and RIS-

user equipment (UE) distances, but also the RIS size, the

radiation pattern and the reﬂection coefﬁcient of the RIS

unit cell, the AP and UE antenna gain, the transmission

frequency, as well as the environmental conditions. Note

that they have been two already published contributions

that provided the e2e pathloss in RIS-assisted wireless

sytsems [55], [56]. However, both [55] and [56] refer to

low frequency band communications; thus, they neglect

the impact of molecular absorption loss.

•Building upon the channel attenuation expression, we

provide a simple closed-form expression that determines

the phase shift that each RIS element should impose in

order to steer the beam radiated from the RIS towards a

desired direction.

•Next, we present a tight channel coefﬁcient approxima-

tion that enables the performance analysis of the RIS-

assisted THz wireless system.

•Based on the channel coefﬁcient approximation, we

present the theoretical framework that returns the cumula-

tive distribution function (CDF) of the e2e channel coef-

ﬁcient of the RIS-assisted THz wireless system. In more

detail, we consider a RIS-assisted downlink scenario in

which a single AP broadcasts to a number of UEs that

are located in random positions within a circular cluster

of radius equals to the one of the RIS to center of the

cluster footprint, and we obtain closed-form expressions

for the e2e channel coefﬁcient CDF.

•Finally, we use the e2e channel statistics to extract

an insightful closed-form expression for the coverage

probability.

The rest of the paper is structured as follows: Section II

focuses on presenting the system and channel model accom-

panied by the latter’s tight approximation, while Section III

presents the statistical analysis of the e2e channel coefﬁcient

as well as the closed-form expression for the coverage prob-

ability. Numerical results veriﬁed by respective simulations

and accompanied by quantitative assessment are reported in

Section IV. Finally, closing remarks and key observations of

this work are summarized in Section V.

Notations: The operator |·|respectively denotes the absolute

value, whereas exp (x)stands for the exponential function.

Additionally, √xreturns the square root of x, while sin (·),

cos (·),sin−1(·),cos−1(·)and sinc (·)respectively denote the

sine, cosine, arc sine, arc cosine, and sinc functions.

II. SY ST EM & CHA NN EL MO DE L

In this section, we present the system model and we extract

general formulas and approximations for the e2e channel

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Microcontroller

Varactor

RIS

AP

re

Nu UEs cluster

d1d

UE 1

UE nu

UE Nu

y

xz

dx

dy

1

n

N

.

.

.

.

.

.

1m

. . . . . . . . . M

φθ

UE 2

x’

y’

r

θnu

Fig. 1. System model.

coefﬁcient. It is organized as follows: Section II-A presents the

system model, while Section II-B reports the channel model.

A. System model

As illustrated in Fig. 1, we consider the downlink scenario

of a RIS-assisted wireless THz system, in which a single AP

serves NuUE, that form a circular cluster with center O0(0,0)

and radius re, through a RIS. Moreover, it is assumed that, due

to blockage, no direct link between the AP and UEs can be es-

tablished. The AP and the nu−th UE, with nu∈ {1,·· · , NU}

are equipped with high directional antennas of gains GAP and

Gnu, respectively, which point at the center of the RIS. The

RIS consists of M×Northogonal unit cells of dimensions dx

and dyand it is assumed that it has knowledge concerning the

position of the center of the cluster and its radius. Moreover,

the AP can use different codebooks in order to adjust its

beamwidth and focusing point in the RIS. Based on the the

used codebook different size of the RIS, i.e., different number

of RUs are employed. This allows the system to adjust the RIS

main lobe footprint in order to equal the cluster radius. Notice

that this functionality require that the cluster size is known to

the AP. A three dimensional (3D) Cartesian system is deﬁned

with its center being at the center of the RIS and the RIS to

be at its x-y plane. Hence, the position of the RU, Um,n can

be obtained as

dm,n =n−1

2dxxo+m−1

2dyyo+ 0 zo,(1)

with n∈1−N

2,N

2and m∈1−M

2,M

2. Also, xo,

yo, and zostand for the unitary vectors at the x,y, and z

direction, respectively. Let, θt

m,n and θr

m,n,nube the elevation

angle from the (m, n)RU, Um,n, to the AP and to the

nu−th UE, respectively, while φt

m,n and φr

m,n,nustand for

the corresponding azimuth angle. Finally, we use lt

m,n and

lm,n,nuto respectively deﬁne the distances from AP to the

Um,n RU and the one from the Um,n RU to the nu−th UE.

B. Channel model

In this section, we assume that the distance between the

center of the RIS and the nu−th UE is known, and we

evaluate the received power at the nu−th, we identify the

optimal phase shift of each Um,n in order to steer the RIS-

generated beam to a desired direction, and we obtain the

channel coefﬁcient as a function of the RIS speciﬁcations and

THz-speciﬁc characteristics.

For a given nu−th UE position, the following theorem

returns the received power at the nu−th UE.

Theorem 1: The received power at the nu-th UE can be

evaluated as

Pr=LnuPAP ,(2)

where Lnucan be expressed as in (3), given at the top of the

following page. In (3),

ζ1n−1

2dx+ζ2m−1

2dy=λφm,n

2π,(4)

and

Gt=GAP GGnu.(5)

Additionally, φm,n and |R|are respectively the controllable

phase shift and the absolute value of the reﬂection coefﬁcient

introduced by the (m, n)RU, while Ur(θ, φ),Ut(θ, φ)and G

are the normalized received, the normalized transmitted power

ratio patterns and the unit cell gain, respectively. Moreover,

d1and dnurepresents the AP to the center of the RIS

distance and the one from the center of the RIS to the nu-

th UE. Meanwhile, θiand φiare respectively the elevation

and the azimuth angles from the center of the RIS to the AP,

while θrand φrrespectively denotes the elevation and the

azimuth angles from the center of the the RIS to the center

of the cluster. Finally, in (3), κ(f)stands for the molecular

absorption coefﬁcient and can be obtained as in (6), given at

the top of the next page[54]1. In (6),

A(µ) = a1(1 −µ) (a2(1 −µ) + a3),(7)

B(µ) = (b1(1 −µ) + b2)2,(8)

C(µ) = c1µ(c2µ+c3),(9)

E(µ) = (e1µ+d2)2,(10)

F(µ) = f1µ(f2µ+f3),(11)

G(µ)=(g1µ+g2)2,(12)

I(µ) = i1µ(i2µ+i3),(13)

J(µ)=(j1µ+j2)2,(14)

K(µ) = k1µ(k2µ+k3),(15)

L(µ)=(l1µ+l2)2,(16)

M(µ) = m1µ(m2µ+m3),(17)

N(µ)=(n1µ+n2)2,(18)

R(µ, f ) = µ

r1

(r2+r3fr4),(19)

with a1= 5.159 ×10−5,a2=−6.65 ×10−5,a3= 0.0159,

b1=−2.09 ×10−4,b2= 0.05,c1= 0.1925,c2= 0.135,

c3= 0.0318,e1= 0.4241,e2= 0.0998,f1= 0.2251,f2=

0.1314,f3= 0.0297,g1= 0.4127,g2= 0.0932,i1= 2.053,

i2= 0.1717,i3= 0.0306,j1= 0.5394,j2= 0.0961,k1=

1In order to compute the molecular absorption coefﬁcient, radiative trans-

fer theory [57] and the high resolution transmission molecular absorption

(HITRAN) database [58] are widely used. However, since, in practice THz

wireless systems are expected to operate in the 100 −500 GHz band, we

employ a simpliﬁed model for this band, which was introduced in [25] and

then extended in [54]. Note that this model, although it is a theoretical one,

is heavily based on the HITRAN database, which contains experimental data.

In other words, it has been veriﬁed by experimental data.

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Lnu=M2N2dxdyλ2|R|2Ur(θi, φi)Ut(θr, φr)Gt

64π3d2

1d2

nu

sinc2Nπ

λ(sin (θi) cos (θi) + sin (θr) cos (φr) + ζ1)dx

sinc2π

λ(sin (θi) cos (θi) + sin (θr) cos (φr) + ζ1)dx

×sinc2Mπ

λ(sin (θi) sin (φi) + sin (θr) sin (φr) + ζ2)dy

sinc2π

λ(sin (θi) sin (φi) + sin (θr) sin (φr) + ζ2)dyexp (−κ(f) (d1+dnu)) (3)

κ(f) = A(µ)

B(µ) + f

100c−q12+C(µ)

E(µ) + f

100c−q22+F(µ)

G(µ) + f

100c−q32

+I(µ)

J(µ) + f

100c−q42+K(µ)

L(µ) + f

100c−q52+M(µ)

N(µ) + f

100c−q62+R(µ, f )(6)

0.177,k2= 0.0832,k3= 0.0213,l1= 0.2615,l2= 0.0668,

m1= 2.146,m2= 0.1206,m3= 0.0277,n1= 0.3789,n2=

0.0871,r1= 0.0157,r2= 2 ×10−4,r3= 0.915 ×10−112,

r4= 9.42,q1= 3.96,q2= 6.11,q3= 10.84,q4= 12.68,

q5= 14.65, and q6= 14.94. Moreover, cis the speed of light,

and µis the volume mixing ratio of the water vapor and can

be obtained as

µ=p1(p2+p3P) exp p4(T−p6)

T+p5−p6,(20)

where p1= 6.1121,p2= 1.0007,p3= 3.46 ×10−8,

p4= 17.502,p5= 240.97 oK, and p6= 273.15 oK.

Furthermore, Tstands for the air temperature, and Pis the

atmospheric pressure.

Proof: Please refer to Appendix A.

Remark 1: To steer the beam at the desired direction θr=θo

and φr=φo, the parameters ζ1and ζ2should be

ζ1=−(sin (θi) cos (φi) + sin (θo) cos (φo)) (21)

and

ζ2=−(sin (θi) sin (φi) + sin (θo) sin (φo)) .(22)

In this case, based on (4), the phase shift of the (m, n)element

can be obtained as in (23), given at the top of the following

page. In this case, according to (3), the maximum path-gain

is

Lmax

nu=M2N2dxdyλ2|R|2Ur(θo, φo)Ut(θo, φo)Gt

64π3d2

1d2

nu

×exp (−κ(f) (d1+dnu)) .(24)

From (24), the channel gain of the received signal at the nu

UE at a speciﬁc distance dnucan be expressed as

hnu=MN pdxdyλ|R|pUr(θi, φi)Ut(θr, φr)Gt

8π3/2d1dnu

×exp −1

2κ(f) (d1+dnu).(25)

The following Lemma returns a tight approximation of the

channel coefﬁcient.

Lemma 1: For realistic scenarios, the channel coefﬁcient can

be tightly approximated as

hnu≈hl

nuX,(26)

where

hl

nu=MN pdxdyλ|R|pUt(θo, φo)Ur(θi, φi)Gtκ(f)

16π3/2d1

×exp −1

2κ(f)d1.(27)

and

X=2

κ(f)dnu−1.(28)

Proof: From (25), the equivalent e2e channel coefﬁcient

can be rewritten as

hnu=hl

nuY,(29)

where

Y=exp −1

2κ(f)dnu

1

2κ(f)dnu

.(30)

Next, by taking into account that, in practice, κ(f)<< 0.1

and by applying the Maclaurin series of the exponential

function [59] in (30), Ycan be tightly approximated as

Y ≈ X.(31)

Finally, by substituting (31) into (29), we obtain (26). This

concludes the proof.

III. PERFORMANCE ANALYSIS

This section is focused on presenting the theoretical frame-

work for the quantiﬁcation of the coverage performance of the

RIS-assisted THz wireless system. In particular, Section III-A

present a closed-form expressions for the CDF of the e2e

channel coefﬁcient, whereas, Section III-B, builds upon this

expression and extract a novel closed-form formula for the

coverage probability.

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φo

m,n =−2π

λn−1

2(sin (θi) cos (θi) + sin (θo) cos (φo)) dx−2π

λm−1

2(sin (θi) sin (θi) + sin (θo) sin (φo)) dy

(23)

A. Statistical analysis of the channel coefﬁcient

We assume that the RIS is placed at height LRfrom the

nu−th UE and that it targets at (φo, θo). In the far-ﬁeld, the

euclidean distance between the center of the RIS and the nu

UE can be approximated as

dnu≈d−rnucos (θnu),(32)

where dis the euclidean distance between the center of the

RIS and the center of the cluster, rnuis the distance of the

nuUE from the center of the cluster and θnuis the physical

direction of the nuUE with respect to the coordination system

deﬁned by the center of the cluster. Note that the UE location

is uniformly distribution in a disc of radius re; hence, the

probability density function of rnucan be obtained as

frnu(r) = 2r

r2

e

,with r∈[0, re].(33)

Moreover, θnucan be modeled as a uniform distribution with

probability density function (PDF) that can be expressed as

fθnu(x) = 1

2π,with x∈[0,2π].(34)

Finally, note that rnuand θnuare independent.

The following theorem returns the PDF and the CDF of the

random variable

Z=rnucos (θnu).(35)

Theorem 2: The PDF and CDF of Xcan be respectively

obtained as

fZ(x) = 2pr2

e−x2

πr2

e

,with x∈[−re, re](36)

and

FZ(x) =

0, x ≤ −re

1

2+1

π

x√r2

e−x2

r2

e+ 2 sin−1x

re,−re< x < re

1, x ≥re

.

(37)

Proof: Please refer to Appendix B.

The following theorem returns the CDF of dnu.

Theorem 3: The CDF of dnucan be expressed as in (38),

given at the top of the next page.

Proof: Please refer to Appendix C.

The following theorem returns the CDF of the e2e channel

coefﬁcient, assuming that the UE is located within the maxi-

mum path-gain region2.

Theorem 4: Assuming that the UE is located within the

maximum path-gain region, the CDF of the e2e channel

coefﬁcient can be obtained as in (39), given at the top of the

next page.

Proof: Please refer to Appendix D.

2As maximum path-gain region, we deﬁne the area in which the following

condition holds: |Lnu/Lmax

nu|< , where stands for the maximum

acceptable path-gain reduction in comparison with Lmax

nu.

B. Coverage probability

As coverage probability, we deﬁne the probability that the

received power at the nu−th UE that is within the maximum

path-gain region, to be above a predetermined threshold Pth,

i.e.,

Pc= Pr (Pr≥Pth |re=rth ),(40)

where rth denotes the radius of the maximum path-gain region.

Note that different applications require different levels of

minimum achievable spectral efﬁciency, which can be deﬁned

as

C= log21 + Pr

No,(41)

where Nostands for the noise power. Thus, for a spectral

efﬁciency requirement, Cth, the received power should be at

least equal to Pth. Hence, (41) can be rewritten as

Cth = log21 + Pth

No,(42)

or equivalently

Pth =2Cth −1No.(43)

The following theorem returns a closed-form expression for

the coverage probability.

Theorem 5: The coverage probability can be obtained as

in (44), given at the top of the following page. In (44),

rth <2

κ(f)−d, (45)

otherwise, Pc= 0.

Proof: Please refer to Appendix E.

Remark 2: From Theorem 5, it becomes evident that in order

to achieve a coverage probability that equals 1, a minimum

transmission power, which can be evaluated as

Pmin

AP =Pth hl

nu−22

κ(f) (d+rth)−1−2

,(46)

should be used.

IV. NUMERICAL RES ULT S & DISCUSSION

In this section, we report numerical results, accompanied

by related discussions, which highlight the effectiveness of

RIS-assisted THz wireless systems, reveals their limitations

as well as the relationship with the system parameters and

performance. In this direction, unless otherwise stated, we

investigate the following insightful scenario. We consider

standard environmental conditions, i.e., relative humidity 50%,

atmospheric pressure 101325 Pa, and temperature 296oK. The

AP transmission antenna gain is 50 dBi, which according

to [30], [60], [61] is a realistic value for THz wireless systems,

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Fdnu(x) =

0, x < d −re

1

2−1

π

(d−x)√r2

e−(d−x)2

r2

e−2 sin−1d−x

re, d −re≤x≤re+d

1, x ≥d+re

(38)

Fhnu(x) =

0, x < hl

nu2

κ(f)(d+re)−1

1

2+1

π

d−2hl

nu

κ(f)(x+hl

nu)!sr2

e−(d−2hl

nu

κ(f)(x+hl

nu))2

r2

e+ 2 sin−1

d−2hl

nu

κ(f)(x+hl

nu)

re

,

hl

nu2

κ(f)(d+re)−1≤x≤hl

nu2

κ(f)(d−re)−1

1, x ≥hl

nu2

κ(f)(d−re)−1

(39)

Pc=

1, Pth <hl

nu22

κ(f)(d+rth)−12PAP

1

2−1

π

d−2hl

nu

κ(f)rPth

PAP +hl

nu

v

u

u

u

tr2

th−

d−2hl

nu

κ(f)rPth

PAP +hl

nu

2

r2

th −2 sin−1

d−2hl

nu

κ(f)rPth

PAP +hl

nu

rth

,

hl

nu22

κ(f)(d+rth)−12PAP ≤Pth ≤hl

nu22

κ(f)(d−rth)−12PAP

0, Pth ≥hl

nu22

κ(f)(d−rth)−12PAP

(44)

while the UE received antenna gains are 20 dBi. The antenna

pattern of the RUs is described by [62], [63]

U(θ, φ) = cos (θ), θ ∈[0,π

2]and φ∈[0,2π]

0,otherwise.(47)

Thus, Gcan be obtained as

G=Z2π

0Zπ

2

0

U(θ, φ) sin (θ) dθdφ, (48)

which by substituting (47) and performing the integration

returns G= 4. Moreover, |R|is set to 0.9, which is in-line

with [64]. The cluster radius can be obtained as

re=dsin 1

2θHP,(49)

where θHP stands for the azimuth half-power beamwidth

(AHPB) of the RIS. Finally, without loss of generality, we

assume that Nu= 1. Of note, in what follows, we use

continuous lines and markers to respectively denote theo-

retical and simulation results. Additionally, we employ the

ﬁnite element method (FEM) in order to verify the channel

model. Notice that this approach has been previously used

in several published contributions (see e.g., [65],[66] and

reference therein), due to its capability to accurate model

electromagnetic propagation mechanisms. Finally, respective

Monte Carlo simulations are employed to verify the coverage

probability theoretical framework.

In Fig. 2, the PL is demonstrated as a function of the

RU to the center of cluster distance for different values of

transmission frequency and M=N, assuming that d1= 1 m

Fig. 2. PL vs dfor different values of M=Nand f.

and that the optimal phase shifts are performed by each RU of

the RIS. In this ﬁgure, continuous lines are used to represent

the PL derived by (24), while dashed lines are employed to

denote the PL approximation, which is evaluated according

to (26). Finally, markers denotes FEM-based simulations.

We observe that the PL approximation perfectly match the

analytical frameworks; thus, the approximation framework

is veriﬁed. Moreover, both the analytical and approximated

results concise with the simulations, which veriﬁes the PL

evaluation framework. As expected, for ﬁxed fand M=N,

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as dincreases, the PL also increases. For instance, for f=

300 GHz and M=N= 100, the PL increases by about

20 dB, as dchanges from 1to 10 m. Moreover, for given

dand M=N, as fincreases, the PL also increases. For

example, for M=N= 10 and d= 10 m, the PL increased

by approximately 10 dB, as fincreases from 100 to 300 GHz.

Finally, we observe that, for ﬁxed fand d, the PL decreases

as the number of RUs increases. For instance, for d= 10 m,

and f= 100 GHz, the PL decreases by approximately 40 dB,

as the M=Nincreases from 10 to 100.

Figure 3 demonstrates the e2e PL as a function of the

nu−th UE direction, for different size of RIS, assuming that

d1=dnu = 1 m, the transmission frequency, f= 100 GHz,

dx=dy= 0.3 mm,|R|= 0.9,GAP = 50 dBi,Gnu =

20 dBi,θi=π

4,φi=π, and no phase shifting is introduced

by any RU. Of note the results presented in this ﬁgure has been

veriﬁed by FEM-based simulations. However, for the sake

of readability no markers have been placed. Notice that this

special case can be considered as RIS performance benchmark

and it represents scenarios in which the direction of the nu−th

UE is unknown to the RIS. Figure 3.a illustrates the e2e PL

as a function of φr, for different number of M=N, and

θr=π

4. From this ﬁgure, we observe that, for given M=N,

the PL is minimized for φr= 0o. This is in-line with (3)

and (4), where, for the special case in which φm,n = 0, it

can be extracted that ζ1=ζ2= 0 and that, for θi6= 2kπ,

φi6= 2kπ, and θr6= 2kπ, with kinteger, the path-gain is

maximized for φr= 2kπ. Moreover, it becomes apparent

that as M=Nincreases, the minimum PL decreases. For

example, for M=N= 10, the minimum e2e PL equals

42.83 dB, while, for M=N= 10, it is 2.83 dB. However,

as the RIS size increases, the azimuth AHPB decreases. For

instance, for M=N= 10, the AHPB is approximately equal

to 74o, which results to a cluster with radius that is equal

to 0.6 m whereas, for M=N= 100, it is about 8o, which

result to a cluster radius of 7 cm. This indicates that as the RIS

size increases, or equivalently as the number of the used RUs

increases, the size of the cluster that can be served, decreases.

Of note, from (49), it becomes evident that as dincreases,

realso increases. For example, for d= 10 m, in the case in

which M=N= 100, the cluster radius becomes equal to

0.7 m, while, in the case in which M=N= 10, it is equal

to 6 m. Similarly, Fig. 3.b depicts the e2e PL as a function

of θr, for different number of M=N, and φr=π

4. Again,

as the RIS size increases, the minimum PL and the elevation

half-power beamwidth (EHPB) decrease. Meanwhile, Fig. 3.c

presents the PL as a function of φrfor θr=π

4, assuming

that θo=π/6,φo=π/3and that the phase shifting of each

element is provided according to (23), whereas, in Fig. 3.d,

the corresponding PL against θr, for φr=π

4is depicted. By

comparing Fig. 3.a with Fig. 3.c and Fig. 3.b with Fig. 3.d,

we observe that by imposing appropriate phase shifting to

each RIS RU, the PL pattern is rotated in order to achieve

the minimum PL at the desired direction.

In Fig. 4, the PL is presented as a function of the nu−th

UE direction, for different transmission frequencies, assuming

that d1=dnu = 1 m,dx=dy= 0.3 mm,|R|= 0.9,

GAP = 50 dBi,Gnu = 20 dBi,θi=π

4,φi=π,

θo=π/6, and φo=π/3. As expected the minimum PL

is observed for φr=π/3. Moreover, it is apparent that for

a ﬁxed φr, as the transmission frequency increases, the PL

also increases. Finally, we observe that as the transmission

frequency increases, the AHPB decreases.

Figure 5 depicts the PL as a function of ffor different

values of M=N, assuming that θi=θr=θo=

φr=φo=π

4,φi=3π

4,d1=dnu= 10 m, and

dx=dy= 0.3 mm. The theoretical results perfectly match

the simulations, verifying the proposed channel model. From

this ﬁgure, it is revealed that there exists two frequency

regions, the ﬁrst one from 370 to 390 GHz and the second one

from 430 to 455 GHz, in which the PL is maximized. This

is due to water molecules resonance. In other words, from

100 to 500 GHz, there exists three transmission windows;

the ﬁrst one from 100 to 365 GHz, the second one from

375 to approximately 430 GHz, and the third one from 460

to 500 GHz. Outside these regions, for ﬁxed Mand N, as

the transmission frequency increases, the PL also increases.

For example, for M=N= 20, as fincreases from 100

to 300 GHz, the PL increases for about 10 dB. Finally, it is

observed that, for a given transmission frequency, as the RIS

size increases, the PL decreases. For example, as M=N

increases from 10 to 100, the PL decreases for about 40 dB.

In Fig. 6, the PL is plotted as a function of the atmospheric

temperature and relative humidity, for different values of f,

assuming M=N= 100,d1= 1 m,dnu= 10 m,dx=dy≈

0.3 mm,θi= 45o,φi= 180o, and θr=θo=φr=φo= 45o.

In more detail, in Fig. 6.a, fis set to 100 GHz, while,

in Fig. 6.b, it equals 200 GHz. Likewise, in Fig. 6.c, fis

300 GHz, whereas, in Fig. 6.d, the transmission frequency is

set to 380 GHz, which is one of the worst scenarios, since

it is one of the resonant frequencies of water molecules.

Furthermore, in Fig. 6.e, the transmission frequency is equal

to 400 GHz, while, in Fig. 6.f, it is set to 450 GHz that is one

of the worst scenarios, due to the fact that in this frequency

the molecular absorption in maximized. Notice that, according

to [67]–[69], the frequencies 100,200,300 and 400 GHz

are within the THz transmission windows. As expected, for

a ﬁxed atmospheric temperature, as the humidity increases,

the water molecules increases; thus, the molecular absorption

and the PL increase. For instance, in the worst case scenario

in which f= 380 GHz, for an atmospheric temperature of

273oK, the PL increases by approximately 2 dB as relative

humidity increases from 10% to 90%. On the other hand,

for f= 100 GHz, for the same atmospheric temperature,

the PL increase does not surpass 0.25 dB as the relative

humidity changes from 10% to 90%. The aforementioned ex-

amples reveals the importance of taking into account both the

transmission frequency and the atmospheric conditions, when

evaluating the link budget of of RIS-assisted THz systems.

Similarly, for a given relative humidity, as the atmospheric

temperature increases, the PL also increases. For example, for

f= 380 GHz and a relative humidity equals 50%, the PL

increases by 2 dB as the atmospheric temperature increase

from 270oKto 290oK. On the contrary, for f= 100 GHz,

the same relative humidity, the PL increases by approximately

0.01 dB as the atmospheric temperature increase from 270oK

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(a) (b)

(c) (d)

Fig. 3. a) PL vs φr, for θr=π

4and φm,n = 0o, for all mand n, b) PL vs θr, for different φr=π

4and φm,n = 0o, for all mand n. c) PL vs φr, for

θr=π

4,θo=π/6, and φo=π/3. d) PL vs θr, for φr=π

4,θo=π/6, and φo=π/3.

Fig. 4. PL vs φr, for θr=π

4,θo=π/6,φo=π/3, and different values

of f.

to 290oK. Moreover, for f= 200 GHz, the same temperature

variation causes a PL increase in the range of 0.05 dB,

while, for f= 400 GHz, it results to 0.5 dB. This indicates

that as the frequency increases the impact of environmental

Fig. 5. PL vs ffor different RIS sizes.

conditions to the PL become more severe. Finally, from this

ﬁgure, it becomes evident that neglecting the absorption loss

would lead to a frequency-dependent PL error in the range of

[0.1,22.7 dB]. This highlights the importance of taking into

account the molecular absorption loss when evaluating the PL

and the performance of RIS-assisted THz systems.

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(a) (b) (c)

(d) (e) (f)

Fig. 6. PL vs temperature and relative humidity, for (a) f= 100 GHz, (b) f= 200 GHz, (c) f= 300 GHz, (d) f= 380 GHz, (e) f= 400 GHz, and

(f) f= 450 GHz.

Fig. 7. PL vs temperature and frequency.

Figure 7 illustrates the PL as a function of the air tempera-

ture and the transmission frequency, assuming M=N= 100,

d1= 1 m,dnu= 10 m,dx=dy≈0.3 mm,θi= 45o,

φi= 180o, and θr=θo=φr=φo= 45o. As expected, for a

given transmission frequency, as the air temperature increases,

the PL also increases. For example, for f= 250 GHz, the PL

increases by about 0.1 dB, as the air temperature increases

from 270 to 320oK. Moreover, from this ﬁgure, it is veriﬁed

that there exist two frequency regions in which the PL is

maximized. In these regions, temperature variations cause a

more severe impact on PL. For instance, increasing the air tem-

perature from 270 to 280oKresults in 0.02 dB PL increase,

if the transmission frequency is equal to 280 GHz, while, the

same temperature increase cause a 0.5 dB PL increase, when

the transmission frequency is set to 383 GHz. This indicates

the importance of taking into account the air temperature and

its variations, when selecting the transmission frequency.

Figure 8 presents the coverage probability as a function

of PAP /Pth, for different values of f,M=Nand d,

assuming that φi=π,φr= 0, and θi=θr=π

4. From

this ﬁgure, we observe that the analytical results coincide with

the simulations, verifying the derived expressions. Moreover,

for given f,M=Nand d, as PAP

Pth increases, the coverage

probability also increases. For example, for f= 200 GHz,

M=N= 10 and d= 5 m,Pcincreases from 12.6% to

60.5%, as PAP /Pth increases from 50 to 55 dB. Meanwhile,

for ﬁxed dand PAP

Pth , although, as the RIS size increase,

rth decreases, the path-gain increases; hence, the Pcalso

increases. For instance, for f= 200 GHz,d= 5 m and

PAP

Pth = 40 dB, as M=Nincreases from 10 to 50,rth

decreases from approximately 6.02 to 2.59 m; however, Pc

increases from 0to 1. Additionally, we observe that for given

RIS size and PAP

Pth , as dincreases, Pcdecreases. Finally, by

comparing Figs. 8a-c, it becomes apparent that for given PAP

Pth ,

M=Nand d, as fincreases, the Pcdecreases. For example,

for M=N= 50,d= 15 m, and PAP

Pth = 25 dB,Pcreduces

from 1to 0as the transmission frequency changes from 100

to 300 GHz.

Figure 9 depicts the coverage probability as a function of the

transmission frequency, for different values of PAP

Pth , assuming

M=N= 100,φi=π,φr= 0,θi=θr=π

4and d= 15 m.

As expected, for a given PAP

Pth , as the transmission frequency

increases, the PL increases; thus, the coverage probability

decreases. Likewise, for a ﬁxed transmission frequency, as

PAP

Pth increases, the received power increases; as a result,

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(a) (b) (c)

Fig. 8. Pcvs PAP /Pth for different RIS sizes and d, assuming a) f= 100 GHz, b) f= 200 GHz, and c) f= 300 GHz.

Fig. 9. Pcvs frequency for different PAP /Pth.

the coverage probability also increases. For example, for a

transmission frequency that is equal to 250 GHz, as PAP

Pth

increases from 40 to 50 dB, the coverage probability increases

from 0to 1. Finally, from this ﬁgure, it becomes evident that as

PAP

Pth increases, the range of frequencies for which the coverage

probability equals 1increases.

Figure 10 presents the coverage probability as a function

of d, for different values of PAP /Pth, RIS size, and a)

f= 100 GHz, b) f= 200 GHz, and c) f= 300 GHz,

assuming φi=π,φr= 0, and θi=θr=π

4. As

expected, for given transmission frequency, RIS size and PAP

Pth ,

as dincreases, the average PL increases; thus, the cover-

age probability decreases. For example, for f= 200 GHz,

M=N= 50 and PAP

Pth = 30 dB, the coverage probability

reduces from 1to 0, as dincreases from 5to 10 m. Similarly,

for f= 100 GHz,M=N= 50 and PAP

Pth = 30 dB,

Pcchanges from 1to 0, as dincreases from 10 to 15 m.

Moreover, we observe that by increasing the RIS size, we

can counterbalance the transmission distance restriction. For

instance, for f= 200 GHz,PAP

Pth = 30 dB, and d= 20 m,

Pcincreases from 0to 1as M=Nchanges from 50

to 100. Likewise, it is observed that for ﬁxed transmission

frequency, RIS size and d, as PAP /Pth increases, the coverage

probability also increases. For example, for f= 200 GHz,

d= 10 m and M=N= 100, as PAP /Pth increases

from 20 to 30 dB, the coverage probability changes from

approximately 40% to 100%. This indicates that another ap-

proach to countermeasure the transmission distance restriction

is to increase the AP transmission power. Finally, within the

same transmission window, for ﬁxed PAP

Pth ,M=N, and

d,Pcincreases as the transmission frequency decreases. For

example, for PAP

Pth = 30 dB,M=N= 100, and d= 40 m,

Pcchanges from 0to 1, as fdecreases from 200 to 100 GHz.

This reveals that another approach to increase the transmission

distance is to decrease the transmission frequency.

Figure 11 depicts the coverage probability as a function

of PAP /Pth and air temperature for different values of f,

assuming M=N= 100,φi=π,φr= 0, and θi=θr=π

4.

As expected, for a given temperature as PAP /Pth increases,

the coverage probability increases. Moreover, for a ﬁxed

PAP /Pth, as the air temperature increases, the PL increases;

hence, the coverage probability decreases. For example, for

f= 380 GHz and PAP /Pth = 60 dB, the coverage probabil-

ity decreases from 1to 0, as the air temperature increases

from 270 to 300oK. Similarly, for f= 100 GHz and

PAP /Pth = 5 dB, as the temperature changes from 270 to

280 oK,Pcdecreases from 1to 0. Finally, from this ﬁgure it

becomes evident that the minimum transmission power that is

required to achieve a coverage probability that is equal to 1,

increases as the air temperature increases.

V. CONCLUSIONS

This contribution presented a theoretical framework for RIS-

assisted THz wireless system coverage performance evalu-

ation. In more detail, we described the system model and

we employed electromagnetic theory tools in order to extract

a generalized formula for the e2e path-gain. This formula

revealed the relationships between the RIS speciﬁcations,

namely size, number of RIS RUs, RU size and reﬂection

coefﬁcient, RU’s radiation patterns, as well as phase shift of

each RU, the transmission parameters, such as transmission

frequency AP to center of RIS and center of RIS to UE

distance, AP transmission and UE reception antenna gains,

azimuth and elevation angles from the AP to the center of the

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(a) (b) (b)

Fig. 10. Pcvs dfor different values of PAP /Pth and RIS size, assuming a) f= 100 GHz, b) f= 200 GHz, and c) f= 300 GHz.

(a) (b) (c)

(d) (e) (f)

Fig. 11. Pcvs Tand PAP /Pth , assuming a) f= 100 GHz, b) f= 200 GHz, c) f= 300 GHz, d) f= 380 GHz, e) f= 400 GHz and f)

f= 450 GHz.

RIS as well as from the center of the RIS to the UE, and THz-

speciﬁc parameters, like the environmental conditions that

affect the molecular absorption. Building upon this expression,

we determined the optimal phase shift of each RU in order to

steer the RIS-generated beam to a desired direction. Next, we

obtained an exact expression for the e2e channel coefﬁcient

as well as a tight approximation that allows us to quantify

the system performance. Based on this expression, we ﬁrst

obtained the statistics of the e2e channel, we deﬁned the

coverage probability of the RIS-assisted THz wireless system

and derived a novel closed-form and insightful expression for

its quantiﬁcation. This expression revealed that there exists

a minimum AP transmission power that can guarantee a

coverage probability of 100%. This minimum AP transmission

power depends not only on the system’s characteristics, but

also on the environmental conditions. Finally, we veriﬁed

the accuracy of the theoretical framework through respective

simulations, which highlighted the dependency between the

transmission power and maximum bandwidth that can be

used in RIS-assisted THz systems. The results highlight the

importance of taking into account the molecular absorption

loss when evaluating the PL and the performance of RIS-

assisted THz wireless systems. Therefore, this work is ex-

pected to contribute on analyzing, simulating, and designing

RIS-assisted THz systems. Moreover, it is expected to play a

key role on devising physical layer and medium access control

algorithms in RIS-assisted THz systems.

The performance analysis of RIS-assisted THz wireless

systems were conducted under the assumptions that (i) the

center of the cluster is perfectly known to the RIS controller,

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and (ii) the cluster radius is perfectly known to the AP. It

would be interesting to relax the aforementioned assumptions

and present new evaluation investigation assuming partial

knowledge of the cluster center and radius. Moreover, the

fading characteristics should be taken into account. Motivated

by this, our future effort will focus on the study of RIS-assisted

THz system performance in fading environments under partial

cluster position and radius knowledge. Finally, inspired by the

fact that in several realistic scenarios the users are mobile and

since the proposed path-loss model can be also applied in this

scenario, we intend to provide the theoretical framework for

quantifying the performance of RIS-assisted THz systems that

operate in dynamic environments in which the UEs are mobile.

APPENDICES

APPENDIX A

PROO F OF T HE OR EM 1

Since lt

m,n >> λ, where λis the wavelength of the

transmission signal, the power of the incident signal into the

Um,n can be obtained as

Pi

m,n = exp −κ(f)lt

m,nGAP PAP

4πlt

m,n2

×Urθt

m,n, φt

m,nSU,(50)

where SUstand for the aperture of the RIS unit cell and can

be expressed as

SU=dxdy.(51)

By substituting (51) into (50), we can express the power of

the incident signal as

Pi

m,n = exp −κ(f)lt

m,nUrθt

m,n, φt

m,n

×GAP PAP

4πlt

m,n2dxdy.(52)

Hence, the electric ﬁeld of the incident signal into Um,n can

be written as

Ei

n,m =s2ZoPi

m,n

dxdy

exp −j2πlt

m,n

λ,(53)

where Zois the air characteristic impedance.

Based on the energy conversation law, the total reﬂected

signal power by the Umn unit cell can be obtained as

Pr

m,n =R2

m,nPi

m,n.(54)

or equivalently

Pr

m,n = exp −κ(f)lt

m,ndxdy

4πlt

m,n2|Rm,n |2

×Urθt

m,n, φt

m,nGAP PAP .(55)

By assuming that lm,n,nu>> λ, we can obtain the power

of the received signal at the nuUE from the Um,n RIS unit

cell as

Pm,n,nu= exp −κ(f)lt

m,n +lm,n,nu

×G Ur(θi, φi)Pr

m,n

4π(lm,n,nu)2Ut(θr, φr)Sr

nu,(56)

where Sr

nuis the aperture of the nu-th UE receive antenna

and can be obtained as

Sr

nu=Gnuλ2

4π.(57)

With the aid of (57), (56) can be rewritten as

Pm,n,nu=dxdyλ2Ur(θi, φi)Ut(θr, φr)G|Rm,n|2GAP Gnu

(4π)3(lm,n,nu)2lt

m,n2

×exp −κ(f)lt

m,n +lm,n,nuPAP .(58)

As a result, the electrical ﬁeld of the received signal at the nu

UE from the Um,n RIS unit cell can be expressed as

Em,n,nu=s2Zo

Pm,n,nu

Sr

nu

exp −j2π

λlt

m,n +lm,n,nu,

(59)

which, based on (58), can be equivalently written as

Em,n,nu=Rm,np2ZodxdyUr(θi, φi)Ut(θr, φr)GGAP PAP

4πlm,n,nult

m,n

×exp −1

2κ(f) + j2π

λlt

m,n +lm,n,nu.(60)

Then, the total electric ﬁeld at the nuUE can be evaluated as

Er

nu=

M

2

X

m=−M

2+1

N

2

X

n=−N

2+1

Em,n,nu,(61)

which, by substituting (60) and taking into account that in

the far-ﬁeld |Rm,n|≈|R|, can be equivalently expressed as

in (62), given at the top of the following page.

The AP position can be obtained as

rt=d1sin (θi) cos (φi)xo+d1sin (θi) sin (φi)yo

+d1cos (θi)zo.(63)

Thus, by employing (1) and (63), the distance between the AP

and the Um,n can be obtained as in (64), given at the top of

the following page. By employing the Taylor expansion in (64)

and keeping only the ﬁrst term, the distance between the AP

and the Um,n can be approximated as

lt

m,n ≈d1−sin (θi)cos (φi)n−1

2dx

−sin (θi)sin (φi)m−1

2dy(65)

Following the same steps, we can prove that

lm,n,nu≈dnu−sin (θr)cos (φr)n−1

2dx

−sin (θr)sin (φr)m−1

2dy.(66)

By substituting (65) and (66) into (62), and taking into

account that in practice dxand dyare at the order of λ/10,

while d1, dnu >> λ, we can tightly approximate the electric

ﬁeld at the nuUE as in (67), given at the top of the following

page. In (67),

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Er

nu=|R|p2ZodxdyUr(θi, φi)Ut(θr, φr)GGAP PAP

4π

M

2

X

m=−M

2+1

N

2

X

n=−N

2+1

exp −1

2κ(f) + j2π

λlt

m,n +lm,n,nu+jφm,n

lm,n,nult

m,n

(62)

lt

m,n = d1sin (θi) cos (φi)−n−1

2dx2

+d1sin (θi) sin (φi)−m−1

2dy2

+d2

1cos2(θi)!1/2

(64)

Er

nu≈|R|p2ZodxdyUr(θi, φi)Ut(θr, φr)GGAP PAP

4πd1dnu

exp −1

2κ(f) (d1+dnu)

×

M

2

X

m=−M

2+1

N

2

X

n=−N

2+1

exp j2π

λd1+dnu−βm,n +λ

2πφm,n (67)

βm,n =d1−sin (θi) cos (θi)n−1

2dx

−sin (θi) sin (θi)m−1

2dy

+dnu−sin (θr)cos (φr)n−1

2dx

−sin (θr)sin (φr)m−1

2dy.(68)

The received signal power at the nuUE can be evaluated as

Pr=|Er

nu|2

2Zo

Sr

nu,(69)

which, with the aid of (57) and (67), can be written as

Pr=dxdyλ2|R|2Ur(θt, φt)Ut(θr, φr)GGAP GnuPAP

64π3d2

1d2

nu

×exp (−κ(f) (d1+dnu)) |γ|2.(70)

In (70), γcan be evaluated as in (71), given at the top of the

following page, or equivalently

γ=γ1γ2,(72)

where γ1and γ2respectively obtained as in (73) and (74),

given at the top of the following page. In (73) and (74), ζ1

and ζ2are deﬁned in (4). By taking into account the sum of

geometric progression theorem, (73) can be rewritten as

γ1=exp −jNπ

λδ1−exp jNπ

λδ1

exp −jπ

λδ1−exp jπ

λδ1,(75)

or equivalently γ1=sin(Nπ

λδ1)

sin(π

λδ1),or

γ1=Nsinc Nπ

λδ1

sinc π

λδ1,(76)

where

δ1= (sin (θi) cos (θi) + sin (θr) cos (φr)) dx+ζ1.(77)

By substituting (77) into (76), we get

γ1=Nsinc Nπ

λ(sin (θi) cos (φi) + sin (θr) cos (φr) + ζ1)dx

sinc π

λ(sin (θi) cos (φi) + sin (θr) cos (φr) + ζ1)dx.

(78)

Similarly, (74) can be expressed as

γ2=Msinc Mπ

λ(sin (θi) sin (φi) + sin (θr) sin (φr) + ζ2)dy

sinc π

λ(sin (θi) sin (φi) + sin (θr) sin (φr) + ζ2)dy.

(79)

Finally, by substituting (78) and (79) into (72) and then to (70),

we obtain (2). This concludes the proof.

APPENDIX B

PROO F OF TH EO RE M 2

Since θnuis uniformly distributed with lower and upper

bounds 0and 2π, respectively, cos (θnu)follows arc sine

distribution with lower and upper bounds −1and 1. Thus,

the PDF of D= cos (θnu)can be expressed as

fD(y) = (1

π√1−y2,−1< y < 1

0,otherwise .(80)

According to [70], since Dand rnuare independent, the PDF

of Zcan be evaluated as

fZ(x) = Zre

|x|

frnu(r)fDz

r1

rdr.(81)

By substituting (33) and (80) into (81), we can rewrite the

PDF of Xas

fZ(x) = 2

πr2

eZre

x

r

√r2−x2dr,(82)

which, after performing the integration, can be expressed

as (36).

The CDF of Xcan be evaluated as

FZ(x) = Zx

−re

fZ(y) dy,(83)

>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <14

γ=

M

2

X

m=−M

2+1

N

2

X

n=−N

2+1

exp j2π

λd1+dnu−βm,n +λ

2πφm,n (71)

γ1=

N

2

X

n=−N

2+1

exp j2π

λsin (θi) cos (θi)n−1

2dx+ sin (θr)cos (φr)n−1

2dx+ζ1 (73)

γ2=

M

2

X

m=−M

2+1

exp j2π

λsin (θi) sin (θi)m−1

2dy+ sin (θr)sin (φr)m−1

2dy+ζ2 (74)

which, with the aid of (36) and after some algebraic manipu-

lations, can be rewritten as

FZ(x) = 2

πZx

re

−1p1−u2du.(84)

By employing [71], (84) can be written as (37). This concludes

the proof.

APPENDIX C

PROO F OF TH EO RE M 3

From (32), we can obtain the CDF of dnuas

Fdnu(x) = Pr (dnu≤x),(85)

or equivalently

Fdnu(x) = Pr (Z ≥ d−x),(86)

or

Fdnu(x) = 1 −FZ(d−x),(87)

which, with the aid of (37), returns (38).

APPENDIX D

PROO F OF TH EO RE M 4

The CDF of the e2e channel coefﬁcient is deﬁned as

Fhnu(x) = Pr (hnu≤x),(88)

which, based on (26), can be rewritten as

Fhnu(x) = Pr X ≤ x

hl

nu.(89)

By employing (28), (89) can be expressed as

Fhnu(x) = Pr dnu≥2hl

nu

κ(f)x+hl

nu!,(90)

or equivalently

Fhnu(x)=1−Pr dnu≤2hl

nu

κ(f)x+hl

nu!,(91)

or

Fhnu(x)=1−Fdnu 2hl

nu

κ(f)x+hl

nu!,(92)

which, with the aid of (38), returns (39). This concludes the

proof.

APPENDIX E

PROO F OF TH EO RE M 5

From (40), with the aid of (2) and (26), we can evaluate the

coverage probability as

Pc= Pr hnu≤rPth

PAP

re=rth!,(93)

or

Pc=Fhnu rPth

PAP

re=rth!,(94)

which, by employing (39) gives (44). This concludes the proof.

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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 national and European communication and

networks projects. During 2017, he joined the information technologies

institute (ITI) at the Centre for Research & Technology Hellas (CERTH),

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 professor at the Department of

ECE 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 60 technical

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

international conferences. Furthermore, he is the holder of 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 span in the area of wireless communications and

networks with emphasis in high frequency communications, optical wireless

communications, as well as communications and digital signal processing

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.

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 Technical

University of Athens in 1994 and the PhD in Elec-

trical 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 Com-

munications 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

Presidents 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, coordi-

nation 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).