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Performance Analysis of Reconﬁgurable

Intelligent Surface-Assisted Wireless

Systems and Comparison with Relaying

ALEXANDROS–APOSTOLOS A. BOULOGEORGOS1, (Senior Member, IEEE ), and

ANGELIKI ALEXIOU1(Member, IEEE )

1Department of Digital Systems, University of Piraeus Piraeus 18534 Greece (e-mails: al.boulogeorgos@ieee.org, alexiou@unipi.gr)

Corresponding author: Alexandros-Apostolos A. Boulogeorgos (e-mail: al.boulogeorgos@ieee.org).

This work has received funding from the European Commission’s Horizon 2020 research and innovation programme under grant

agreement No. 871464 (ARIADNE).

ABSTRACT In this paper, we provide the theoretical framework for the performance comparison of

reconﬁgurable intelligent surfaces (RISs) and amplify-and-forward (AF) relaying wireless systems. In

particular, after statistically characterizing the end-to-end (e2e) wireless channel coefﬁcient of the RIS-

assisted wireless system, in terms of probability density function (PDF) and cumulative density function

(CDF), we extract novel closed-form expressions for the instantaneous and average e2e signal-to-noise ratio

(SNR) for both the RIS-assisted and AF-relaying wireless systems. Building upon these expressions, we

derive the diversity gain of the RIS-assisted wireless system as well as the outage probability (OP) and

symbol error rate (SER) for a large variety of Gray-mapped modulation schemes of both systems under

investigation. Additionally, the diversity order of the RIS-assisted wireless system is presented as well as

the ergodic capacity (EC) of both the RIS-assisted and AF-relaying wireless systems. Likewise, high-SNR

and high-number of metasurfaces (MS) approximations for the SER and EC for the RIS-assisted wireless

system are reported. Finally, for the sake of completeness, the special case in which the RIS is equipped

with only one MS is also investigated. For this case, the instantaneous and average e2e SNR are derived,

as well as the OP, SER and EC. Our analysis is veriﬁed through respective Monte Carlo simulations, which

reveal the accuracy of the presented theoretical framework. Moreover, our results highlight that, in general,

RIS-assisted wireless systems outperform the corresponding AF-relaying ones in terms of average SNR,

OP, SER and EC.

INDEX TERMS Amplify-and-forward, Average signal-to-noise-ratio, Beyond 5G systems, Ergodic

capacity, High-signal-to-noise-ratio approximation, Meta-surfaces, Multipath fading, Outage probability,

Performance analysis, Reconﬁgurable intelligent surfaces, Symbol error rate, Theoretical framework.

NOMENCLATURE

2D Two dimensional

3D Three dimensional

AF Amplify-and-forward

AWGN Additive white Gaussian noise

B5G Beyond ﬁfth generation

BER Bit error rate

BPAM Binary pulse amplitude modulation

BPPM Binary pulse position modulation

BPSK Binary phase shift keying

CDF Cumulative density function

D Destination

e2e End-to-end

EC Ergodic capacity

EM Electromagnetic

KPM Key performance metric

MISO Multi-input single-output

MS Metasurface

NOMA Non-orthogonal multiple access

OP Outage probability

PAM Pulse amplitude modulation

PDF Probability density function

PS Phase shift

PSK Phase shift keying

VOLUME 4, 2016 1

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A.–A. A. Boulogeorgos et al.: Reconﬁgurable Intelligent Surface-Assisted vs Relaying Wireless Systems

QAM Quadrature amplitude modulation

QPSK Quadrature phase shift keying

R Relay

RF Radio frequency

RIS Reconﬁgurable intelligent surface

RV Random variable

S Source

SER Symbol error rate

SINR Signal-to-interference-plus-noise-ratio

SNR Signal-to-noise-ratio

ZMCG Zero-mean complex Gaussian

I. INTRODUCTION

The evolution of the wireless world towards the beyond ﬁfth

generation (B5G) era comes with higher reliability, data-rates

and trafﬁc demands, which is driven by innovative applica-

tions, such as unmanned mobility, three dimensional (3D)

media, augmented and virtual reality [1]–[4]. Technological

advances, such as massive multiple-input multiple-output,

full-duplexing, and millimeter-wave communications, have

been advocated, due to their increased hardware cost, power

consumption [5]–[7], as well as their need to operate in un-

favorable electromagnetic (EM) wave propagation environ-

ment, where they have to deal with a number of medium par-

ticularities [8], [9].

As a remedy, the exploitation of the implicit randomness

of the propagation environment through reconﬁgurable in-

telligent surfaces (RISs), in order to improve the quality of

service and experience, attracts the eyes of both academia and

industry [3], [10]–[12]. Most RIS implementations consist of

two dimensional (2D) metasurface (MS) arrays, which are

controlled by at least one microcontroller, and are capable of

altering the incoming EM ﬁeld in a customized manner [13].

In more detail, each MS can independently conﬁgure the

phase shift (PS) of the EM signal incident upon it; hence,

they are able to collaboratively create a preferable wireless

channel [14]. In other words, RIS can amplify-and-forward

(AF) the incoming signal without employing a power am-

pliﬁer. Due to this functionality, the technological approach

that can be considered equivalent and has the most similarity

to RIS is AF-relaying [5], [15]–[17]. As a consequence, the

question of whether RIS-assisted systems can outperforms

AF-relaying ones and under which conditions arises.

A. RELATED WORK

Scanning the technical literature, a lot of research effort

was put on the design, demonstration, optimization, and

analysis of RIS and RIS-assisted wireless systems (see e.g.,

[5], [12], [14]–[16], [18]–[36] and references therein). For

example, in [18], the authors introduced a RIS that consists

of 102 MSs operates in 2.47 GHz, for indoor applications.

Similarly, in [19], the authors reported a reconﬁgurable MS

with adjustable polarization, scattering and focusing control,

while, in [20], intelligent walls, which were equipped with

frequency-selective MSs, were presented. Likewise, in [21],

a MS capable of rotating a linear polarized EM wave by 90o

was reported, whereas, in [22], an ultra-thin MS based on

phase discontinuities was proposed to manipulate EM waves

in the microwave band. Moreover, in [23], a RIS design

that employed varactor-tuned resonators in order to enable

tunable PS by adjusting the bias voltage applied to the var-

actor, was delivered, while, in [24] and [25], its functionality

was demonstrated. Finally, in [26], RIS elements whose EM

response were controlled by PIN diodes were reported.

From the optimization point of view, in [27], an asymp-

totic uplink ergodic data-rate investigation of RIS-assisted

systems under Rician fading was performed, while, prelim-

inary optimization frameworks for the maximization of the

total received power in RIS-assisted wireless systems were

reported in [14] and [28]. Speciﬁcally, in [14], the values

of the PSs, which were created by the MSs, were opti-

mized in a RIS-assisted single-user multiple-input-single-

output (MISO) wireless system, whereas, in [28], the authors

solved the same problem, in a more realistic scenario, in

which the RIS consisted by a ﬁnite number of discrete PSs.

Moreover, in [29], optimal linear precoder, power alloca-

tion and RIS phase matrix designs that used the large-scale

statistics channel knowledge and aimed at maximizing the

minimum signal-to-noise-plus-interference ratio (SINR) at

the base-station were reported. Likewise, in [5] the problem

of maximizing the weighted sum rate of all users through

jointly optimizing the active precoding matrices at the base-

stations and the PSs at RIS-assisted multi-user wireless net-

works is formulated and solved, while, in [16], the joint

maximization of the sum-rate and the energy efﬁciency was

investigated for a multi-user downlink scenario. Furthermore,

in [30], the optimization problem of simultaneous wireless

information and power transfer in RIS-assisted systems was

studied. Meanwhile, in [32], a downlink multi-user scenario,

in which a multi-antenna base-station, which is capable of

performing digital beamforming, communicates with various

users through a ﬁnite-size RIS was presented and an iterative

algorithm was designed in order to maximize the sum rate.

From the theoretical analysis point of view, in [15]

and [31], the authors provided a symbol error rate (SER)

upper-bound for RIS-assisted wireless systems. It is worth

noting that these upper-bounds are quite tight for RIS uti-

lizations with high-number of MSs, but, for a low-number

of MSs, they are not so accurate. Similarly, in [33], a bit

error rate (BER) analysis was provided for RIS-assisted non-

orthogonal multiple access (NOMA) systems. Again, the

authors employed the central limit theorem in order to model

the distribution of the equivalent base station-user equipment

channel. As a consequence, the results are accurate only for

scenarios in which the RIS consists of a large number of

MSs. In [34], Jung et. al provided an asymptotic analysis of

the uplink sum-rate of a RIS-assisted system, assuming that

the established channels follow Rician distribution. Finally,

in [35], Björnson et. al compared the performance of RIS-

assisted systems against decode-and-forward relaying ones in

terms of energy efﬁciency, assuming deterministic channels,

while, in [36], Renzo et. al revealed the key differences and

2VOLUME 4, 2016

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A.–A. A. Boulogeorgos et al.: Reconﬁgurable Intelligent Surface-Assisted vs Relaying Wireless Systems

similarities between RISs and relays. Also, in [36], simula-

tions were used in order to compare RIS and relays in terms

of data-rate.

On the other hand, there are several published contribu-

tions that investigate the performance of AF-relaying assisted

wireless systems(see e.g., [37]–[50] and reference therein).

In more detail, in [37], the authors reported closed-form

expressions for the probability density function (PDF) and

cumulative density function (CDF) of the end-to-end (e2e)

signal-to-noise-ratio (SNR), assuming that the intermediate

channels are Rayleigh distributed. Similarly, in [38] and

in [39], tight-approximations for the system’s SER were pre-

sented. Moreover,in [40] an asymptotic SER analysis of AF-

relaying systems was conducted, while, in [41] and in [42],

SER lower bounds were presented assuming that differential

and frequency shift keying modulation schemes are respec-

tively used. Furthermore, in [43], closed-form expressions

for the outage probability (OP) of AF cognitive relay net-

works were presented accompanied by a lower bound SER

expressions. Likewise, in [44], the authors reported closed-

form expressions for the OP of switch-and-stay AF relay

networks, whereas, in [45], OP approximations for selection

AF relaying systems were presented. Moreover, in [46]–

[49], the authors provided closed-form expressions for the

OP and bounds for the ergodic capacity (EC) of a dual-

hop variable-gain AF relaying system. Similarly, for the

corresponding ﬁxed-gain AF relaying system, in [50], the

authors derived EC approximations. To sum up, the literature

review revealed that, although a great amount of effort was

put on analyzing the performance of AF relaying systems,

no closed-form tractable expression for the average SNR

and EC was reported. Finally, to the best of the authors

knowledge, no generalized expression for the SER of such

systems was presented.

B. MOTIVATION, NOVELTY AND CONTRIBUTION

Despite of the paramount importance that RIS-assisted wire-

less systems are expected to play in B5G setups, their perfor-

mance has been only assessed in terms of SER lower-bounds.

Likewise, to the best of the authors knowledge, regardless of

their similarities with the AF-relaying systems, no analyti-

cal comparison between RIS-assisted and conventional AF-

relaying wireless systems has been conducted. Motivated by

this, this work focuses on presenting the theoretical frame-

work that quantiﬁes the performance of the RIS-assisted

wireless system. Moreover, an analytic comparison between

the aforementioned wireless systems, in terms of average

SNR, OP, SER, and EC, is conducted. In more detail, the

technical contribution of this paper is as follows:

•Novel analytical expressions for the PDF and CDF

of e2e wireless fading channel coefﬁcient of the RIS-

assisted wireless system, are derived, which take into

account the number of the RIS MSs and assume that the

source (S)-RIS and RIS-destination (D) links experience

Rayleigh fading. Notice that this is the ﬁrst time that

the aforementioned expressions are reported in the lit-

erature. Moreover, closed-form expressions for the PDF

and the CDF of the e2e wireless fading channel for the

special case, in which the RIS is equipped with only one

MS, are also presented.

•Next, the instantaneous and average e2e SNR for the

RIS-assisted wireless system are derived. Building upon

them, the diversity gain of the RIS-assisted wireless sys-

tem is extracted, as well as close-form expressions for

the PDF and CDF of its e2e SNR. Furthermore, analytic

expressions for the SNR statistical characterization, for

the special case in which the RIS is equipped with

only one MS, are provided. From these expressions, the

diversity gain is analytically evaluated for both cases of

single and multiple-MSs.

•To quantify the outage performance of the RIS-assisted

wireless system, we derive low-complexity closed-form

expressions for its OP for both cases, in which the

RIS is equipped with multiple and single MS. These

expressions reveal the relation between the number of

MS, the transmission power and spectral efﬁciency with

the system outage performance; hence, they provide

useful insights and can be used as design tools.

•Closed-form expressions for the SER of a large vari-

ety of Gray-mapped modulation schemes for the RIS-

assisted wireless systems are provided, for both cases

in which the RIS consists of multiple and single MS.

Furthermore, tight low-computational complexity ap-

proximations for the SER in the high-SNR regime

are extracted.

•Building upon the high-SNR SER approximation, we

derive a simple closed-form expression for the RIS-

assisted wireless system diversity order, which high-

lights that the diversity order is a linear function of the

number of MSs.

•Analytical expressions for the EC of the RIS-assisted

wireless system for both cases, in which the RIS is

equipped with multiple and a single MS, are also re-

ported. Likewise, tight high-SNR and high-MS number

approximations for the EC are derived. Moreover, an al-

ternative more elegant EC expression, which is capable

of providing interesting observations, is extracted.

•Finally, in order to compare the RIS-assisted wireless

system with the corresponding AF-relaying one, we

provide the analytical framework for the derivation of

the average e2e SNR, OP, SER, and EC of the AF-

relaying wireless system.Note that although the PDF

and CDF of the e2e SNR of the AF-relaying wireless

system has been initially presented in [37], to the best

of the authors knowledge, this is the ﬁrst time that

closed-form expressions for the average e2e SNR, SER,

and EC are reported in the technical literature. Finally,

comparative results, which shows the superiority of the

RIS-assisted system against the AF-relaying one, are

presented.

It is worth-noting that, for the special case in which RIS

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A.–A. A. Boulogeorgos et al.: Reconﬁgurable Intelligent Surface-Assisted vs Relaying Wireless Systems

is equipped with a single MS, the e2e equivalent channel

follows a double Rayleigh distribution. This distribution has

been extensively examined in the literature [51], [52]. As a

result, the OP expressions that are presented in this paper

for this special case have been previously reported in sev-

eral published works, including [51], [53] under different

frameworks. On the other hand, this is the ﬁrst time that the

generalized SER and EC expressions, which refer the single-

MS RIS system, are presented in the literature.

C. ORGANIZATION AND NOTATIONS

The remainder of this work is organized as follows: Section II

provides the RIS-assisted and AF-relaying wireless system

models as well as the statistical characterization of the RIS-

assisted e2e wireless channel. Next, Section III presents

the performance analysis of both the RIS-assisted and AF-

relaying wireless systems, while, Section IV reports respec-

tive numerical results and discussions. Finally, a summary of

this work accompanied by closing remarks and key observa-

tions are provided in Section V.

Notations

The operators E[·],V[·]and |·|respectively denote the statis-

tical expectation, variance, and the absolute value, whereas

exp (x)and log2(x)respectively stand for the exponential

and the binary logarithmic functions. Additionally, ln (x)

refers to the natural logarithm of x, while √xand lim

x→a(f(x))

respectively return the square root of xand the limit of

the function f(x)as xtends to a. Furthermore, min (A)

returns the minimum value of the set Aand (x)ndenotes the

Pochhammer operator [54, eq. (19)]. Likewise, csc(x)and

acsc(x)respectively give the cosecant and the arc cosecant

of x, while sec(x)returns the secant of x[55, ch. 6].

The upper and lower incomplete Gamma functions [56, eq.

(8.350/2), (8.350/3)] are respectively denoted by Γ (·,·)and

γ(·,·), while the Gamma function is represented by Γ (·)[56,

eq. (8.310)]. The Qand error functions are respectively

represented by Q(·)[57, ch. 2] and erf(·)[58, eq. (7.1.1)],

whereas Kv(·)and Iv(·)are respectively the modiﬁed Bessel

function of the second [58, eq. (9.6.2)] and ﬁrst kind of

order v[58, eq. (9.6.3)]. Moreover, F0(·),E(·), and K(·)

respectively represent the polygamma function of the zero

order [58, eq. (6.4.1)], the elliptic integral function [58,

eq. (17.1.1)], and the complete elliptic integral function of

the ﬁrst kind [58, eq. (17.3.1)]. Furthermore, 2F1(·,·;·;·)

stands for the Gauss hypergeometric function [58, eq.

(4.1.1)], while pFq(a1,··· , ap;b1,··· , bq;x)is the general-

ized hypergeometric function [56, eq. (9.14/1)]. Meanwhile,

U(a, b, x)and Gm,n

p,q x

a1, a2,··· , ap

b1, b2,··· , bqrespectively

represent the conﬂuent hypergeometric function of second

kind [56, ch. 9.2], and the Meijer’s G-function [56, eq.

(9.301)], whereas Hm,n

p,q z

(a1, b1),··· ,(ap, bp)

(c1, d1),··· ,(cp, dp)and

Gm1,m2:m3,n1:n2,n3

p1,p2:p3,q1:q2,q3·,··· ,·

·,··· ,··,··· ,·

·,··· ,··,··· ,·

·,··· ,·

x, y

FIGURE 1. System model of the RIS-assisted wireless system.

is the Fox H-function [59, eq. (8.3.1/1)] and the generalized

Meijer G-function of two variables [60].

II. SYSTEM MODEL

In this section, the system models of the RIS-assisted and

AF-relaying wireless systems are provided. In more detail,

in Section II-A, the RIS-assisted wireless system model is

reported accompanied by the statistical characterization of

its e2e wireless channel, while the system model of the

corresponding AF-relaying setup is delivered in Section II-B.

A. RIS-ASSISTED WIRELESS SYSTEM

As depicted in Fig. 1, for the RIS-assisted wireless system,

we consider a scenario, in which a single-antenna S node

communicates with a single-antenna D node through a RIS,

that consists of NMSs. The baseband equivalent fading

channels between S and the i-th MS of the RIS, hi, as well

as the one between the i−th MS and D, gi, are assumed

to be independent, identical, slowing varying, ﬂat, and their

envelop follow Rayleigh distributions with scale parameters

being equal to 11. For clarity, we highlight that, as usual

practice, the deterministic path-gain is not considered in the

fading coefﬁcients hiand gi.

The baseband equivalent received signal at D can be ex-

pressed as [15]

y=

N

X

i=1

higirix+n, (1)

where ndenotes the additive white Gaussian noise (AWGN)

and can be modeled as a zero-mean complex Gaussian

(ZMCG) process with variance equal No. Additionally, ri

represents the i-th MS response and can be obtained as

ri=|ri|exp (jθi),(2)

with θibeing the PS applied by the i−th reﬂecting MS of the

RIS. In this work, it is assumed that the reﬂected units of the

RIS are equipped with varactor-tuned resonators that are able

to achieve tunable PS by adjusting the bias voltage applied to

the varactor [24]. Additionally, we assume that the phases of

1This assumption was used in several previously published works includ-

ing [15], [16], [61], [62] and references therein. This assumption originates

from the fact that even if the line-of-sight links between S-RIS and RIS-D

are blocked, there still exist extensive scatters.

4VOLUME 4, 2016

10.1109/ACCESS.2020.2995435, IEEE Access

the channels hiand giare perfectly known to the RIS, and

that the RIS choices the optimal phase shifting, i.e.,

θi=−(φhi+φgi),(3)

where φhiand φgiare respectively the phases of hiand gi.

Likewise, without loss of generality, it is assumed that the

reﬂected gain of the i−th MS, |gi|, is equal to 1. Notice that

according to [63], this is a realistic assumption. Hence, (2)

can be simpliﬁed as

ri= exp (−j(φhi+φgi)) .(4)

Additionally, by employing (4), (1) can be re-written as

y=Ax +n, (5)

where Ais the e2e baseband equivalent channel coefﬁcient

and can be obtained as

A=

N

X

i=1 |hi||gi|.(6)

From (6), it is evident that the system experiences a diversity

gain that depends on the number of MSs. Next, we provide

the theoretical framework for the characterization of the e2e

channel coefﬁcient.

Statistical characterization of the e2e channel

The following theorem returns closed-form approximation

for the PDF and CDF of A.

Theorem 1. The PDF and CDF of Acan be respectively

evaluated as

fA(x) = xa

ba+1Γ(a+ 1) exp −x

b(7)

and

FA(x) = γ1 + a, x

b

Γ (a+ 1) ,(8)

where

a=k2

1

k2−1,(9)

and

b=k2

k1

,(10)

with

k1=Nπ

2,(11)

and

k2= 4N1−π2

16 .(12)

Proof: Please refer to Appendix A.

Special case: For the special case in which the RIS consists

of a single MS, i.e. N= 1,Ais the product of two inde-

pendent and identical Rayleigh distributed random variables

(RVs); thus, it follows a double Rayleigh distribution and its

SD

hr

gr

R

FIGURE 2. System model of the AF relay-assisted wireless system.

PDF and CDF can be respectively obtained as [51, eqs. (3),

(4)]

fs

A=xK0(x)(13)

and

Fs

A= 1 −xK1(x),(14)

where K0(x)and K1(x)represent the modiﬁed Bessel func-

tions of the second kind of order 0and 1, respectively.

B. AF-RELAYING WIRELESS SYSTEM

The block diagram of the AF-relaying wireless system is

illustrated in Fig. 2. In this setup, we consider that that S

communicates with D through an AF relay (R) node. All the

involved nodes are equipped with a single radio frequency

(RF) chain that feeds a single-antenna. By assuming that the

transmitted by S data symbol, xconveys through a ﬂat fading

channel hr, the received signal at R can be obtained as

yr=hrx+nr,(15)

where |hr|is modeled as a Rayleigh process with scale

parameter equals 1. Likewise, nris a ZMCG process with

variance Noand stands for the AWGN at R.

According to the AF-relaying protocol, R ampliﬁes the

received signal and re-transmit it to D. Thus, the baseband

equivalent received signal at D can be expressed as

yd=grGyr+nd,(16)

or equivalently

yd=√Ggrhrx+Gnr+nd,(17)

where G,grand ndare independent and respectively stand

for the R ampliﬁcation gain, the R-D channel coefﬁcient, and

the AWGN at D. Of note, |gr|and ndare respectively mod-

eled as a Rayleigh process with scale parameter 1and zero-

mean complex Gaussian process with variance equals No.

For the sake of fairness, we assume that the relay have perfect

knowledge of both the hrand gr[64]–[66].

III. PERFORMANCE ANALYSIS

This section focuses on presenting the theoretical framework

for the performance analysis of both the RIS-assisted and

AF-relaying wireless systems. In particular, Section III-A

is devoted to the extraction of the key performance met-

rics (KPMs) for RIS-assisted wireless systems, whereas the

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10.1109/ACCESS.2020.2995435, IEEE Access

KPMs for the AF relaying wireless system are reported

in Section III-B. The expressions that are presented here

provides insightful remarks and are expected to be used in the

design of RIS-assisted systems as well as their comparison

against corresponding AF-relaying ones.

A. RIS-ASSISTED WIRELESS SYSTEMS

The organization of this section is as follows: Section III-A1

presents closed-form expressions for the instantaneous and

average e2e SNR as well as its statistical characterization.

Based on these expressions, the theoretical framework for

the system outage performance is provided in Section III-A2,

while analytical expressions for the evaluation of the SER

are reported in Section III-A3. Finally, Sections III-A4

and III-A5 respectively deliver the analytical framework for

the evaluation of the diversity order and EC.

1) SNR

According to (5), the instantaneous e2e SNR of the RIS-

assisted wireless system can be obtained as

ρ=A2ρs,(18)

where

ρs=Es

No

,(19)

with Esbeing the S transmitted power.

Theorem 2 returns a closed-form expression for the aver-

age e2e SNR, while Theorem 3 delivers closed-form expres-

sions for its PDF and CDF.

Theorem 2. The average e2e SNR can be obtained as

E[ρ] = Γ(a+ 3)b2

Γ(a+ 1) ρs.(20)

Proof: Please refer to Appendix B.

From (20), it is evident that the diversity gain of the RIS-

assisted wireless system can be evaluated as

GRIS =Γ(a+ 3)b2

Γ(a+ 1) ,(21)

or equivalently,

GRIS = (a+ 1)2b2,(22)

which by employing (9)–(12) can be rewritten as

GRIS =16 −π2

2π2π2

16 −π2N2

.(23)

Interestingly, (23) reveals that the only way to increase the di-

versity gain of the RIS-assisted wireless system is to increase

the number of MSs in the RIS.

Next, we characterize the statistics of the e2e SNR. In this

direction, Theorem 3 provides novel closed-form expressions

for its PDF and CDF.

Theorem 3. The PDF and the CDF of the e2e SNR can be

respectively evaluated as

fρ(x) = 1

2ba+1Γ (a+ 1) ρ

a+1

2

s

xa−1

2exp −1

brx

ρs(24)

and

Fρ(x) =

γa+ 1,1

bqx

ρs

Γ(a+ 1) .(25)

Proof: Please refer to Appendix C.

Special case: For the special case in which N= 1, the

following lemmas return closed-form expressions for the

PDF, CDF and average equivalent e2e SNR.

Lemma 1. For N= 1, the CDF and PDF of the equivalent

e2e SNR can be respectively obtained as

Fs

ρ(x)=1−rx

ρs

K1rx

ρs(26)

and

fs

ρ(x) = 1

4ρs

K0rx

ρs−1

2√ρsxK1rx

ρs

+1

4ρs

K2rx

ρs.(27)

Proof: Please refer to Appendix D.

Lemma 2. For N= 1, the average equivalent e2e SNR can

be obtained as

E[ρs]=4ρs.(28)

Proof: Please refer to Appendix E.

From (28), it becomes evident that the diversity gain of the

single-MS RIS-assisted wireless system is equal to 4.

2) Outage probability

The OP is deﬁned as the probability that the e2e instanta-

neous SNR falls below a predetermined threshold, ρth, i.e.

Po=Pr(ρ≤ρth),(29)

or equivalently

Po=Fρ(ρth),(30)

which, by employing (25), can be written as

Po=

γa+ 1,1

bqρth

ρs

Γ(a+ 1) .(31)

Moreover, by employing (9)–(12), (31) can be rewritten as

Po=

γπ2

16−π2N, 2π

16−π2qρth

ρs

Γπ2

16−π2N.(32)

From (32), we observe that, for a ﬁxed ρth

ρs, as Nincreases,

the OP decreases; thus, the outage performance improves.

Similarly, for a given N, as ρth

ρsincreases, the OP decreases.

6VOLUME 4, 2016

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Special case: For the special case in which N= 1, the OP

can be obtained as

Ps

o=Fs

ρ(ρth),(33)

or, by using (26), as

Ps

o= 1 −rρth

ρs

K1rρth

ρs.(34)

From (34), it is observed that in the special case in which

N= 1, the outage performance of the RIS-assisted wireless

system depend only from ρth

ρs, i.e., the transmitted signal

characteristics, namely spectral efﬁciency and transmission

power. In more detail, as the spectral efﬁciency of the trans-

mission scheme increases, ρtalso increases; thus, ρth

ρsin-

creases and an outage performance degradation is observed.

On the other hand, as the transmission power increases, ρth

ρs

decreases; therefore, the OP also decreases.

3) SER

The following theorem returns a closed-form expression for

the SER of the RIS-assisted wireless system.

Theorem 4. The SER can be analytically evaluated as

in (35), given at the top of the next page.

Proof: Please refer to Appendix F.

In (35), cand dare modulation speciﬁc constants. For

example, the SER of binary phase shift keying (BPSK) can

be obtained for c=d= 1, while for Mpulse-amplitude

modulation (M-PAM) for c=2(M−1)

Mand d=3

M2−1.

Likewise, for c= 1 and d= 0.5, we can obtain the

SER for the case in which binary pulse position modulation

(BPPM) is employed, whereas, for c= 2 and d= 1,

the one of quadrature phase shift keying (QPSK) can be

evaluated. Finally, for c= 2 and d= sin2π

M, the

SER of M-phase shift keying (M-PSK) can be obtained,

while, for c= 4 1−1

√Mand d=3

2

1

M−1, (35) returns

the SER of M-quadrature amplitude modulation (M-QAM)

with M > 4.

The following corollary provides a high-SNR approxima-

tion of the SER.

Corollary 1. In the high-SNR regime, the SER can be ap-

proximated as in (36), given at the top of the next page.

Proof: In the high-SNR regime, i.e., for ρs→ ∞, the

following expressions holds:

lim

y→02F4a+ 1

4,a+ 3

4;1

4,1

2,3

4,a+ 5

4;y= 1,(37)

lim

y→02F4a+ 2

4,a+ 4

4;1

2,3

4,5

4,a+ 6

4;y= 1,(38)

lim

y→02F4a+ 3

4,a+ 5

4;3

4,5

4,3

2,a+ 7

4;y= 1 (39)

and

lim

y→02F4a+ 4

4,a+ 6

4;5

4,3

2,7

4, y= 1,(40)

where

y=1

256b4d2ρ2

s

.(41)

Hence, by substituting (37)-(40), we obtain (36). This con-

cludes the proof.

From (36), we observe that the ﬁrst term of the sum is

the dominant one. This observation leads to the following

remark: the SER is a decreasing function of ρsand a, and

an increasing function of c. This indicates that as the trans-

mission power increases and/or the number of MS increases,

the SER decreases, while as modulation order increases, the

error performance degrades.

Special case: For the special case in which N= 1,

the following lemma returns a closed-form expression for

the SER.

Lemma 3. For N= 1, the SER can be analytically com-

puted as

Ps

e=√π

4

c

2U1

2,0,1

4dρs−1.(42)

Proof: Please refer to Appendix G.

According to (42), in the special case in which N= 1, the

SER increases, as the modulation order increases, while, as

ρsincreases, the SER decreases.

4) Diversity order

Theorem 5 returns the diversity order of the RIS-assisted

wireless system.

Theorem 5. The diversity order of the RIS-assisted wireless

system can be calculated as

D=N

2

π2

16 −π2.(43)

Proof: Please refer to Appendix H.

Based on (43), the diversity order is a linearly increasing

function of N. Note that in the same conclusion was extracted

in [15] and [31].

5) Ergodic capacity

The following theorems return two equivalent and novel

closed-form expressions for the EC.

Theorem 6. The EC of the RIS-assisted wireless system can

be analytically computed as in (44), given at the top of the

next page.

Proof: Please refer to Appendix I.

Theorem 7. The EC of the RIS-assisted wireless system can

be alternatively evaluated as in (45), given at the top of the

next page.

Proof: Please refer to Appendix J.

Notice that (44) returns the EC as a sum of well-deﬁned

special functions that can be directly evaluated in several

software packages, such as Mathematica, Mapple, Matlab,

etc. However, it is quite difﬁcult or even impossible to obtain

VOLUME 4, 2016 7

10.1109/ACCESS.2020.2995435, IEEE Access

Pe=c

2√π(a+ 1)ba+1da+1

2ρ

a+1

2

s

Γa+3

4

Γ (a+ 1) 2F4a+ 1

4,a+ 3

4;1

4,1

2,3

4,a+ 5

4;1

256b4d2ρ2

s

−c

2√π(a+ 2)ba+2da+2

2ρ

a+2

2

s

Γa+4

4

Γ (a+ 1) 2F4a+ 2

4,a+ 4

4;1

2,3

4,5

4,a+ 6

4;1

256b4d2ρ2

s

+c

4√π(a+ 3)ba+3da+3

2ρ

a+3

2

s

Γa+5

4

Γ (a+ 1) 2F4a+ 3

4,a+ 5

4;3

4,5

4,3

2,a+ 7

4;1

256b4d2ρ2

s

−c

12√π(a+ 4)ba+4da+4

2ρ

a+4

2

s

Γa+5

4

Γ (a+ 1) 2F4a+ 4

4,a+ 6

4;5

4,3

2,7

4,a+ 8

4;1

256b4d2ρ2

s(35)

Pe,s ≈c

2√π(a+ 1)ba+1da+1

2

Γ(a+3

4)

Γ(a+1)

ρ−a+1

2

s−c

2√π(a+ 2)ba+2da+2

2

Γa+4

4

Γ (a+ 1)ρ−a+2

2

s

+c

4√π(a+ 3)ba+3da+3

2

Γ(a+5

4)

Γ(a+1)

ρ−a+3

2

s−c

12√π(a+ 4)ba+4da+4

2

Γa+5

4

Γ (a+ 1)ρ−a+4

2

s(36)

C=a2−a

(a−1)2

log2b2ρs+2a2−a

ln(2)(a−1)2

F0(3 + a)

+π

ln(2)(2 + a)ba+2Γ(a+ 1)ρa

2+1

s

csc aπ

21F21 + a

2;3

2,2 + a

2,−1

4b2ρs

+π

(a+ 1)ba+1 ln (2) Γ(a+ 1)ρ

a+1

2

s

sec aπ

21F2a+ 1

2;1

2,a+ 3

2,−1

4b2ρs

+1

ln(2)(a−1)2b2ρs2F31,1; 2,1−a

2,3−a

2,−1

4b2ρs(44)

C= 2 ln (2) b2Γ(a+ 1)ρsH1,4

4,3b2ρs

(0,1),(0,1),(−a−2,2),(−a−3,2)

(0,1),(−a−3,2),(−1,1) (45)

insightful observations from this expression. On the other

hand, a more elegant expression for the EC is presented

in (45), which can be evaluated directly in Mathematica, by

rewritten the Fox H function as a generalized Meijer’s G-

function. Moreover, from (45), it is revealed that the EC is an

increasing function of ρsand N.

The following corollaries present high-SNR and high-N

approximations for the EC.

Corollary 2. In the high SNR regime, the EC can be approx-

imated as in (46), given at the top of the next page.

Proof: For ρs→ ∞,y=−1

4b2ρs→0. Moreover,

lim

y→01F21 + a

2;3

2,2 + a

2,−y= 1,(47)

lim

y→01F2a+ 1

2;1

2,a+ 3

2,−y= 1 (48)

and

lim

y→02F31,1; 2,1−a

2,3−a

2,−y= 1.(49)

Thus,in the high SNR regime (44) can be approximated as

in (46). This concludes the proof.

Corollary 3. In the high SNR and Nregime, the EC can be

approximated as

Cρ,N ≈1

ln(2)(a−1)2b2ρs

+a2−a

(a−1)2

log2b2ρs

+2a2−a

ln(2)(a−1)2

F0(3 + a).(50)

Proof: In the high SNR regime, as N→ ∞,a→ ∞;

hence, since Γ (a+ 1) is an increasing function, as N→ ∞,

8VOLUME 4, 2016

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Cρ≈1

ln(2)(a−1)2b2ρs

+a2−a

(a−1)2

log2b2ρs+2a2−a

ln(2)(a−1)2

F0(3 + a)

+π

ln(2)(2 + a)ba+2Γ(a+ 1)ρa

2+1

s

csc aπ

2+π

(a+ 1)ba+1 ln (2) Γ(a+ 1)ρ

a+1

2

s

sec aπ

2(46)

Γ (a+ 1) → ∞, or equivalently 1

Γ(a+1) →0. This indicates

that the terms

A1=π

ln(2)(2 + a)ba+2Γ(a+ 1)ρa

2+1

s

csc aπ

2(51)

and

A2=π

(a+ 1)ba+1 ln (2) Γ(a+ 1)ρ

a+1

2

s

sec aπ

2(52)

tents to 0. Thus, (46) can be approximated as in (50). This

concludes the proof.

Special case: In the special case in which N= 1, the

following lemma returns a closed-form expression for the

EC.

Lemma 4. For N= 1, the EC can be obtained as

Cs=1

8 ln(2)ρ2

s

G3,1

1,31

4ρ2

s−1

−1,−1,0

−1

4 ln(2)ρs

G3,1

1,31

4ρ2

s−1

2

−1

2,−1

2,−1

2

+1

8 ln(2)ρ2

s

G4,1

2,41

4ρ2

s−1,0

−1,−1,−1,1.(53)

Proof: Please refer to Appendix K.

B. AF-RELAYING WIRELESS SYSTEMS

In this section, we revisit the theoretical framework of the

AF-relaying wireless systems and, after deﬁning their instan-

taneous SNR and presenting its PDF and CDF, we extract

closed-form expressions for their average SNR, OP, SER

and EC.

1) SNR

In the AF-relaying wireless system, based on (17) and by

assuming variable ampliﬁcation, the e2e instantaneous SNR

can be obtained as [67, Eq. (2.144)]

ρr=ρ1ρ2

ρ1+ρ2+ 1,(54)

or approximately

ρr≈ρ1ρ2

ρ1+ρ2

,(55)

where

ρ1=|hr|2ρs(56)

and

ρ2=|gr|2ρR,(57)

with

ρR=Er

No

(58)

and Erdenoting the R transmitted power.

The PDF and the CDF of ρrcan be respectively written

as [37, eqs. (19) and (27)]

fρr(x) = 2

ρsρR

xexp −1

ρs

+1

ρRx

×ρs+ρR

√ρsρR

K12x

√ρsρR+ 2K02x

√ρsρR (59)

and

Fρr(x)=1−2x

√ρsρR

exp −1

ρs

+1

ρRx

×K12x

√ρsρR,(60)

where ρsand ρRare respectively the average SNR of the

ﬁrst and second hop and, since |hr|and |gr|are Rayleigh

distributed processes with scale parameters equal 1, they can

be respectively obtained as

ρs= 2ρs(61)

and

ρR= 2ρR.(62)

The following lemma provides a closed-form expression

for the average e2e SNR of the AF-relaying wireless system.

Lemma 5. The average e2e SNR of the AF-relaying wireless

system can be analytically evaluated as

E[ρr] = −β5+β3γ2−2βγ4

(−β2+γ2)3−π

2

β2γ2−γ4

(−β2+γ2)5/2

−3βγ2

(−β2+γ2)2+3β2γ2−2βγ2

(−β2+γ2)5/2arcsc γ

β,

(63)

for β6=γ, and

E[ρr] = 2

3γ.(64)

for β=γ, where

β=1

ρs

+1

ρR

(65)

and

γ=2

√ρsρR

.(66)

Proof: Please refer to Appendix L.

Notice that β=γcorresponds to ρs=ρR.

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2) Outage probability

Similarly to Section III-A2, the OP of the AF-relaying wire-

less system can be obtained as

PAF

o=Fρr(ρth).(67)

3) SER

The following lemma delivers a closed-form expression for

SER of AF-wireless systems.

Lemma 6. In AF-relaying wireless systems, the SER can be

analytically evaluated as

PAF

e=√γc

2rd

2

β+γ

γ2−(β+d)2E1

2−β+γ

2γ

−√γc

4rd

2

β+γ+d

γ2−(β+d)2K1

2−β+γ

2γ−1.(68)

Proof: Please refer to Appendix M.

4) Ergodic capacity

The following lemma provides a closed-form expression for

the EC of the AF-relaying wireless system.

Lemma 7. In AF-relaying wireless systems, the EC can be

obtained as in (69), given at the top of the next page.

Proof: Please refer to Appendix N.

IV. RESULTS & DISCUSSION

This section is focused on verifying the theoretical frame-

work through respective Monte Carlo simulations and report-

ing the RIS-assisted wireless system performance in compar-

ison with the ones of the corresponding AF-relaying wireless

system, in terms of e2e SNR, OP, SER and EC. Of note,

for the sake of fair comparison, in the following results, we

assume that the total transmission power of both the RIS-

assisted and the AF-relaying wireless systems is the same. In

other words, it is assumed that half of the S transmitted power

in the RIS-assisted system is used for the S-R transmission

and the other half for the R-D one in the AF-relaying wireless

system. Finally, unless otherwise stated, in what follows,

we use continuous lines and markers to respectively denote

theoretical and simulation results.

Figure 3 illustrates the PDF of the equivalent e2e channel

of the RIS-assisted wireless system, for different number

of MSs, N. From this ﬁgure, it becomes evident that the

analytical and simulation results coincide; thus, verifying the

presented theoretical framework. Additionally, it is observed

that, as Nincreases, the equivalent e2e channel values also

increases. This indicates that by increasing N, we can im-

prove the diversity gain of the RIS-assisted wireless system.

Figure 4 depicts the average e2e SNR of both the RIS-

assisted and AF-relaying wireless systems as a function of

ρt, for different values of N. Note that ρtrepresents the total

transmission power to noise ratio. In other words, in the RIS-

assisted wireless system, ρt=ρs, while in the AF-relaying

wireless system ρt=ρs+ρR. From this ﬁgure, for the

RIS-assisted system, we observe that, as theoretically proven

in (20) and (28), for a given N, as ρtincreases, the average

e2e SNR linearly increases. The same applies for the AF-

relaying system. Additionally, we observe that, for a given

ρt, as Nincreases, the average e2e SNR improves. In more

detail, for double values of N, the average e2e SNR increases

by about 6 dB. Finally, it is evident that the RIS-assisted

system outperforms the corresponding AF-relaying one, in

terms of e2e average SNR, for all the values of N. Moreover,

notice that even for N= 1, the RIS-assisted system achieves

about 10 dB higher average e2e SNR compared to the AF-

relaying system. This is due to the fact that the AF-relay,

except from the AWGN at D, experiences an ampliﬁed addi-

tional noise, which is generated in R, while the RIS-assisted

wireless system only experience the AWGN at D.

In Fig. 5, the outage performance of the RIS-assisted wire-

less system is quantiﬁed. In more detail, the OP is plotted as

a function of ρt

ρth . As a benchmark, the OP of the AF-relaying

wireless system is provided. As expected, in the RIS-assisted

system, for a ﬁxed N, as ρt

ρth increases, the OP decreases.

For example, for N= 2, as ρt

ρth changes from 20 to 25 dB,

the OP decreases approximately 10 times. Additionally, for a

given ρt

ρth , as Nincreases, the outage performance improves.

This indicates that for a given OP requirement, we can

improve the RIS-assisted wireless system energy efﬁciency

by increasing N. For instance, ρt

ρth can be reduced by about

30 dB, by employing a RIS that consists of 5MSs instead of

one that consists of 2in order to achieve an OP of 10−5.

Finally, it becomes evident that, the RIS-assisted wireless

system outperforms the corresponding AF-relaying ones, in

terms of OP.

In Fig. 6, the error performance of the RIS-assisted system

as a function of ρt, for different M−QAM schemes and N, is

demonstrated. As a benchmark, the SER of the corresponding

AF-relaying systems is also plotted. As expected, for a given

Nand M, as ρtincreases, the SER performance improves.

For example, for N= 10 and M= 16, as ρtshifts from −4

to 0 dB, the SER decreases by about 100 times. Moreover,

we observe that, for ﬁxed Mand a SER requirement, as N

increases, the ρtgain is signiﬁcantly enhanced. For instance,

for M= 16 and SER requirement set to 10−4, a 52 dB

transmission SNR gain is observed as Nincreases from 1

to 10. Additionally, for given Nand SER requirement, as M

increases, a ρtincrease is required. For example, for N= 50

and a target SER equals 10−5, as Mchanges from 4to 64,ρt

needs to be increased by about 13 dB.Finally, by comparing

the SER of the RIS-assisted with the AF-relaying one, we

observe that, in general, RIS-assisted system outperforms the

AF one. Only for N= 1, for the same modulation scheme,

in the high-SNR regime the AF-relaying wireless system

achieves lower SER. However, this comparison is not fair,

since the RIS-assisted system requires one timeslot to deliver

the message to the ﬁnal D, while the AF-relaying one needs

two. Thus, the spectral efﬁciency of the AF-relaying system

is the half of the RIS-assisted one. In other words, under

the same spectral efﬁciency, we observe that the RIS-assisted

10 VOLUME 4, 2016

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CAF =

√π

ln(2)

γ2

(β−γ)3G0,2:1,2:2,0

2,1:2,2:1,2−2,−3

−3

0,0

0,−1

1

2

0,0

1

β−γ,2γ

β−γ

+√π

ln(2)

βγ

(β−γ)3G0,2:1,2:2,0

2,1:2,2:1,2−2,−3

−3

0,0

0,−1

1

2

−1,−1

1

β−γ,2γ

β−γ,for β6=γ

√π

ln(2) γ2G4,1

3,42γ−2,−1,1

2

0,0,−2,−2+G4,1

3,42γ−2,−1,1

2

−1,1,−2,−2,for β=γ

(69)

FIGURE 3. The PDF of the equivalent e2e channel for different N.

FIGURE 4. Average SNR vs ρt, for different N.

FIGURE 5. Outage probability vs ρt

ρth , for different values of N.

FIGURE 6. SER vs ρt, for different values of Nand M, assuming M−QAM.

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10.1109/ACCESS.2020.2995435, IEEE Access

FIGURE 7. Capacity vs ρt, for different values of N.

system outpeforms the AF-relaying one in terms of SER.

Figure 7 depicts the EC as a function of ρt, for different

values of N. In more detail, continuous lines denote the

analytical results, the dashed ones illustrate the high SNR

approximation, while the dashed-dotted ones the high SNR-

and-Napproximation. Likewise, markers are used for the

Monte Carlo simulation results. For the sake of comparison,

the EC of the AF-relaying wireless system is also plotted.

From this ﬁgure, it becomes evident that the theoretical and

simulation results match; hence, the theoretical framework is

veriﬁed. Interestingly, both the high-SNR and the high-SNR-

Napproximations provide excellent ﬁts even in the medium

and low transmission SNR regimes. This is because of the

diversity gain of the RIS-assisted wireless systems. Likewise,

we observe that, for a given N, as ρtincreases, the EC also

increases. For instance, for N= 2, as ρtincreases from 5

to 10 dB, the EC improves by about 34.2%. Moreover, for a

ﬁxed ρt, as Nincreases, the EC also increases. For example,

for ρt= 10 dB, as Nshifts from 50 to 100, the EC increases

for approximately 12.64%. Furthermore, this ﬁgure reveals

that, independently of ρt, as Ndoubles, the EC increases

by about 2 bits/s/Hz. Finally, it is observed that even with

N= 1, the RIS-assisted wireless system outperforms the

AF-relaying one in terms of EC.

V. CONCLUSIONS & FUTURE WORK

The present contribution investigated the efﬁciency of RIS-

assisted wireless system in terms of average SNR, OP, SER,

diversity order, and EC. In more detail, after statistically

characterizing the e2e wireless channel of the RIS-assisted

system, we provided novel closed-form expressions for the

instantaneous and average SNR, as well as its PDF and CDF.

Moreover, we extracted analytical expressions for the OP as

well as the SER of a number of Gray-mapped modulation

schemes. Likewise, low-complexity tight high-SNR approxi-

mations for the SER are also derived accompanied by an an-

alytical expression for the system’s diversity order. Addition-

ally, closed-form expressions for the EC together with low-

complexity tight high-SNR and high-Napproximations are

extracted. As a benchmark, the corresponding performance

metrics of an AF-relaying wireless system was also assessed

and compared. The theoretical results were compared against

respective Monte Carlo simulations, which validated their

accuracy. Our results revealed that as the number of MSs

increases, the diversity gain and order also increase; hence,

the performance of the RIS-assisted wireless systems im-

proves. Additionally, interesting design observations were

extracted. For example, it was reported that as the number

of MSs, from which a RIS consists, doubles, the average

e2e SNR increases for approximately 6 dB, and the EC by

about 2 bits/s/Hz. Finally, it became evident that, in general,

realistic RIS-assisted wireless systems clearly outperform the

corresponding AF-relaying ones in terms of average SNR,

OP, SER, and EC.

The performance assessment and comparison of RIS and

relay-assisted wireless systems were conducted under the as-

sumptions that (i) the intermediate channels are independent,

ﬂat, and Rayleigh distributed, and (ii) there is no direct link

between S and D. It would be of high interest to relax the

aforementioned assumptions and present a new comparison

study. Motivated by this, our future work includes the study

of RIS-assisted systems performance that operate in compos-

ite fading environments in the presence and absence of direct

link between the S and D.

ACKNOWLEDGMENT

The authors would like to thank the editor and anonymous

reviewers for their constructive comments and criticism.

APPENDICES

APPENDIX A

PROOF OF THEOREM 1

From (6), |hi|and |gi|are Rayleigh distributed RVs. Hence,

their product is a double Rayleigh distributed RV. As a

result, Ais the sum of Nindependent and identical double

Rayleigh processes and, according to [68, ch. 2.2.2], its PDF

can be tightly approximated as the ﬁrst term of a Laguerre

series expansion, i.e., (7), where the parameters aand bare

respectively given by (9) and (10), whereas k1and k2can be

obtained as [68, eq. (2.74)]

k1=E[A],(70)

and

k2= 4V[A].(71)

The expected value of Acan be analytically evaluated as

E[A] = E"N

X

i=1 |hi||gi|#,(72)

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or equivalently

E[A] =

N

X

i=1

E[|hi||gi|].(73)

Since |hi|and |gi|are independent RVs, (73) can be rewrit-

ten as

E[A] =

N

X

i=1

E[|hi|]E[|gi|].(74)

Likewise, |hi|and |gi|follow Rayleigh distribution with

variances 1; thus,

E[|hi|] = E[|gi|] = rπ

2.(75)

By substituting (75) into (74), we get

E[A] = Nπ

2.(76)

By substituting (76) into (70), we obtain (11).

Following a similar procedure, the variance of Acan be

obtained as

V[A] = N1−π2

16 .(77)

By substituting (77) into (71), we get (12).

Next, we express the CDF of Aas

FA(x) = ˆx

0

fA(y) dy,(78)

which, with the aid of (7), can be rewritten as

FA(x) = 1

ba+1Γ (a+ 1) I(x),(79)

where

I(x) = ˆx

0

yaexp −y

bdy,(80)

which, by setting z=y

band employing [56, eq. (8.350/1)],

can be written in closed-form as

I(x) = ba+1γa+ 1,x

b.(81)

By substituting (81) into (79), we get (8). This concludes

the proof.

APPENDIX B

PROOF OF THEOREM 2

With the aid of (18), the average e2e SNR can be analytically

written as

E[ρ] = ρsˆ∞

0

x2fA(x) dx,(82)

which, by substituting (7), can be expressed as

E[ρ] = ρs

ba+1Γ(a+ 1) J,(83)

where

J=ˆ∞

0

xa+2 exp −x

bdx.(84)

By employing [56, eq. (8.310/1)], (84) can be written in

closed-form as

J=ba+3Γ(a+ 3).(85)

By substituting (85) into (83), we obtain (20). This concludes

the proof.

APPENDIX C

PROOF OF THEOREM 3

The CDF of ρcan be expressed as [69]

Fρ(x) = Pr (ρ≤x).(86)

By employing (18), (86) can be re-written as

Fρ(x) = Pr A≤rx

ρs,(87)

or equivalently

Fρ(x) = FArx

ρs,(88)

which, by using (8), returns (25). Next, we obtain the PDF of

ρthrough the derivation of (25), i.e,

fρ(x) = dFρ(x)

dx .(89)

This concludes the proof.

APPENDIX D

PROOF OF LEMMA 1

According to (88), the CDF of ρscan be obtained as

Fs

ρ(x) = Fs

Arx

ρs,(90)

or, by employing (14), as in (26). Moreover, the PDF of ρs,

can be obtained as

fs

ρ(x) = dFs

ρ(x)

dx ,(91)

which, by substituting (26) and performing the derivation,

yields (27). This concludes the proof.

APPENDIX E

PROOF OF LEMMA 2

For the special case in which N= 1, the average e2e SNR

can be evaluated as

E[ρs] = ˆ∞

0

xfs

ρ(x) dx,(92)

which, by employing (13) and after some algebraic manipu-

lations, can be rewritten as

E[ρs] = 1

4ρsN1−1

2√ρsN2+1

4ρsN3,(93)

where

N1=ˆ∞

0

xK0rx

ρsdx,(94)

N2=ˆ∞

0

√xK1rx

ρsdx,(95)

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and

N3=ˆ∞

0

xK2rx

ρsdx.(96)

By setting z=qx

ρsin (94)-(96) and employing [70, Eq.

(5.3)], we get

N1= 8ρ2

s,(97)

N2= 4ρ3/2

s(98)

and

N3= 16ρ2

s.(99)

Finally, by substituting (97)-(99) into (93), we get (28). This

concludes the proof.

APPENDIX F

PROOF OF THEOREM 4

By assuming a two-dimensional modulation, the conditional

to the received SNR, ρ, SER can be obtained as [71]

Pe|ρ(x) = cQ√2dx,(100)

where xis the received SNR. Therefore, the average SER can

be obtained as

Pe=ˆ∞

0

Pe|ρ(x)fρ(x) dx.(101)

By substituting (24) and (100) into (101), the average SER

can be rewritten as

Pe=c1

2ba+1Γ (a+ 1) ρ

a+1

2

sL,(102)

where

L=ˆ∞

0

xa−1

2exp −1

brx

ρsQ√2dxdx.(103)

By employing [72, Eq. (B.112)], (103) can be equivalently

expressed as

L=1

2(L1− L2),(104)

where

L1=ˆ∞

0

xa−1

2exp −1

brx

ρsdx (105)

and

L2=ˆ∞

0

xa−1

2exp −1

brx

ρserf √dxdx.(106)

By setting z=√x, (105) and (106) can be respectively

simpliﬁed as

L1= 2 ˆ∞

0

zaexp −1

b√ρs

zdz (107)

and

L2= 2 ˆ∞

0

zaexp −1

b√ρs

zerf √dzdz,(108)

which, by respectively employing [56, Eq. (8.310/1)]

and [73, eq. (06.25.21.0131.01)], can be analytically evalu-

ated as

L1= 2ba+1ρ

a+1

2

sΓ(a+ 1) (109)

and (110), given at the top of the next page. By substitut-

ing (109) and (110) into (104), we can rewrite Las in (111),

given at the top of the next page. Finally, by substituting (111)

into (102), we get (35). This concludes the proof.

APPENDIX G

PROOF OF LEMMA 3

Based on (101), the average SER can be expressed as

Ps

e=ˆ∞

0

Pe|ρ(x)fs

ρ(x) dx,(112)

which, by applying the integration by parts method and

after some mathematical manipulations, can be equivalently

written as

Ps

e=−ˆ∞

0

Fs

ρ(x)fe(x) dx.(113)

Note that, in (113), fe(x)is deﬁned as

fe(x) = dPe|ρ(x)

dx ,(114)

or

fe(x) = −c

2rd

4πx−1/2exp (−dx).(115)

By substituting (26) and (115), we get

Ps

e=−P1+c

2sd

4πρsP2.(116)

where

P1=ˆ∞

0

fe(x) dx (117)

and

P2=ˆ∞

0

exp (−dx) K1√xρsdx.(118)

Notice that

P1= 1.(119)

Moreover, in (118), by setting z=ρs√x, performing

integration by parts and using [58, Eq. (13.3.4)], it can be

analytically obtained as

P2=π

2rρs

dU1

2,0,1

4dρs.(120)

Hence,by employing (119) and (120), (116) can be ﬁnally

rewritten as in (42). This concludes the proof.

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L2= 2ba+1ρ

a+1

2

sΓ(a+ 1) −2Γ a+3

4

√π(a+ 1)da+1

2

2F4a+ 1

4,a+ 3

4;1

4,1

2,3

4,a+ 5

4;1

256b4d2ρ2

s

+2Γ a+4

4

√π(a+ 1)bda+2

2r1/22F4a+ 2

4,a+ 4

4;1

2,3

4,5

4,a+ 6

4;1

256b4d2ρ2

s

−Γa+5

4

√π(a+ 3)b2da+3

2r2F4a+ 3

4,a+ 5

4;3

4,5

4,3

2,a+ 7

4;1

256b4d2ρ2

s

+Γa+6

4

3√π(a+ 4)b3da+4

2ρ3/2

s

2F4a+ 4

4,a+ 6

4;5

4,3

2,7

4,a+ 8

4;1

256b4d2ρ2

s(110)

L=Γa+3

4

√π(a+ 1)da+1

2

2F4a+ 1

4,a+ 3

4;1

4,1

2,3

4,a+ 5

4;1

256b4d2ρ2

s

−Γa+4

4

√π(a+ 1)bda+2

2r1/22F4a+ 2

4,a+ 4

4;1

2,3

4,5

4,a+ 6

4;1

256b4d2ρ2

s

+Γa+5

4

2√π(a+ 3)b2da+3

2r2F4a+ 3

4,a+ 5

4;3

4,5

4,3

2,a+ 7

4;1

256b4d2ρ2

s

−Γa+6

4

6√π(a+ 4)b3da+4

2ρ3/2

s

2F4a+ 4

4,a+ 6

4;5

4,3

2,7

4,a+ 8

4;1

256b4d2ρ2

s(111)

APPENDIX H

PROOF OF THEOREM 5

According to (36), in the high-SNR regime the SER can be

written as

Pe,s ≈ B1ρ−a+1

2

s+B2ρ−a+2

2

s+B3ρ−a+3

2

s+B4ρ−a+4

2

s,

(121)

where

B1=c

2√π(a+ 1)ba+1da+1

2

Γ(a+3

4)

Γ(a+1)

,(122)

B2=−c

2√π(a+ 2)ba+2da+2

2

Γa+4

4

Γ (a+ 1),(123)

B3=c

4√π(a+ 3)ba+3da+3

2

Γ(a+5

4)

Γ(a+1)

(124)

and

B4=−c

12√π(a+ 4)ba+4da+4

2

Γa+5

4

Γ (a+ 1).(125)

Note that, from (122)-(125), it is evident that the terms B1,

B2,B3, and B4are independent from the SNR. Additionally,

from (121), the terms ρ−a+1

2

s,ρ−a+2

2

s,ρ−a+3

2

s, and ρ−a+4

2

s

contribute with diversity order of a+1

2,a+2

2,a+3

2, and a+4

2,

respectively. Thus, the diversity order can be obtained as

D= min a+ 1

2,a+ 2

2,a+ 3

2,a+ 4

2.(126)

Note that, according to (9), (11) and (12), acan be ex-

pressed as

a=Nπ2

16 −π2−1,(127)

which, since N≥1, is always positive. Thus, (126) can be

simpliﬁed as

D=a+ 1

2,(128)

which, by employing (127), can be ﬁnally written as in (43).

This concludes the proof.

APPENDIX I

PROOF OF THEOREM 6

The EC is deﬁned as

C=E[log2(1 + ρ)] ,(129)

or, by employing (18), can be rewritten as

C=Elog21 + ρsA2,(130)

or equivalently

C=ˆ∞

0

log21 + ρsy2fA(y) dy.(131)

By substituting (7) into (131), the EC can be rewritten as

C=xa

ba+1Γ(a+ 1) ˆ∞

0

exp −y

blog21 + ρsy2dy,

(132)

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or equivalently

C=1

ba+1 ln (2) Γ(a+ 1)K,(133)

where

K=ˆ∞

0

yaexp −y

bln 1 + ρsy2dy.(134)

According to [58, eq. (15.1.1)],

ln(x) = (x−1) 2F1(1,1; 2; 1 −x).(135)

By setting x=ρsy2in (135) and substituting the resulting

expression in (134), we obtain

K=ρsˆ∞

0

ya+2 exp −y

b2F11,1; 2; −ρsy2dy.

(136)

Additionally, by applying integration by parts as well as [74,

eq. (07.23.21.0015.01)], (134) can be expressed in closed-

form as in (137), given at the top of the next page. Likewise,

by substituting (137) into (133), and after some algebraic

manipulations, we extract (138), given at the top of the

next page. Finally, by taking into account that Γ(x+n)

Γ(x)=

(x)n, (138) can be rewritten as in (44). This concludes

the proof.

APPENDIX J

PROOF OF THEOREM 7

According to [56, eq. (8.352/2)], (136) can be equivalently

expressed as

K=ρsˆ∞

0

ya+2Γ1,y

b2F11,1; 2; −ρsy2dy,

(139)

which, based on [75, eq. (5)] and [76, eq. (17)], can be

rewritten as

K= 2ρs

×ˆ∞

0

ya+2G2,0

1,2y

b

1

0,1G1,2

2,2ρsy2

0,0

0,−1dy,

(140)

which, with the aid of [77, ch. 2.3], can be expressed in

closed-form as in (141), given at the top of the next page. By

substituting (141) into (133), we obtain (45). This concludes

the proof.

APPENDIX K

PROOF OF LEMMA 4

According to (130), the EC can be expressed as

Cs=ˆ∞

0

log2(1 + ρsx)fs

ρ(x) dx,(142)

or equivalently

Cs=1

ln(2) ˆ∞

0

ln (1 + ρsx)fs

ρ(x) dx,(143)

which, with the aid of (27), can be rewritten as

Cs=1

4 ln(2)ρsC1−1

2 ln(2)√ρsC2+1

4 ln(2)ρsC3,(144)

where

C1=ˆ∞

0

K0rx

ρsln (1 + ρsx) dx,(145)

C2=ˆ∞

0

x−1/2K1rx

ρsln (1 + ρsx) dx (146)

and

C3=ˆ∞

0

K2rx

ρsln (1 + ρsx) dx.(147)

Moreover, by using [56, eq. (8.352/2)], (145)-(147) can be

equivalently written as

C1=ρsˆ∞

0

xK0rx

ρs2F1(1,1; 2; ρsx) dx,(148)

C2=ρsˆ∞

0

x1/2K1rx

ρs2F1(1,1; 2; ρsx) dx (149)

and

C3=ρsˆ∞

0

xK2rx

ρs2F1(1,1; 2; ρsx) dx.(150)

Additionally, with the aid of [78, eq. (03.04.26.0009.01)]

and [76, eq. (17)], (148)-(150) can be respectively ex-

pressed as

C1=ρs

2ˆ∞

0

xG2,0

0,2x

4ρs

0,0

×G1,2

2,2ρsx

0,0

0,−1dx,(151)

C2=ρs

2ˆ∞

0

x1/2G2,0

0,2x

4ρs

1

2,−1

2

×G1,2

2,2ρsx

0,0

0,−1dx (152)

and

C3=ρs

2ˆ∞

0

xG2,0

0,2x

4ρs

1,−1

×G1,2

2,2ρsx

0,0

0,−1dx.(153)

By employing [77, ch. 2.3], (151)-(153) can be analytically

obtained as

C1=1

2ρs

G3,1

1,31

4ρ2

s−1

−1,−1,0,(154)

C2=1

2√ρs

G3,1

1,31

4ρ2

s−1

2

−1

2,−1

2,−1

2(155)

and

C3=1

2ρs

G4,1

2,41

4ρ2

s−1,0

−1,−1,−1,1.(156)

Finally, by substituting (154)-(156) into (144), we ex-

tract (53). This concludes the proof.

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K= 4aba+3Γ(a)ρsln (b√ρs)+6a2ba+3Γ(a)ρsln (b√ρs)+2a3ba+3Γ(a)ρsln (b√ρs)

+ 2a(a+ 1)(a+ 2)ba+3Γ(a)F0(3 + a)−π

(4 + a)bρa

2+1

s

csc aπ

21F22 + a

2;3

2,3 + a

2,−1

4b2ρs

−π

(a+ 3)ρ

a+1

2

s

sec aπ

21F2a+ 3

2;1

2,a+ 5

2,−1

4b2ρs+aba+1Γ(a)2F31,1; 2,1−a

2,−a

2,−1

4b2ρs(137)

C= 4ab2Γ(a)

Γ(a+ 1)ρslog2(b√ρs)+6a2b2Γ(a)

Γ(a+ 1)ρslog2(b√ρs)+2a3b2Γ(a)

Γ(a+ 1)ρslog2(b√ρs)

+2

ln(2)a(a+ 1)(a+ 2)b2Γ(a)

Γ(a+ 1)F0(3 + a)−π

ln(2)(4 + a)ba+2Γ(a+ 1)ρa

2+1

s

csc aπ

21F22 + a

2;3

2,3 + a

2,−1

4b2ρs

−π

(a+ 3)ba+1 ln (2) Γ(a+ 1)ρ

a+1

2

s

sec aπ

21F2a+ 3

2;1

2,a+ 5

2,−1

4b2ρs

+a

ln(2)

Γ(a)

Γ(a+ 1) 2F31,1; 2,1−a

2,−a

2,−1

4b2ρs(138)

K= 2ba+3ρsH1,4

4,3b2ρs

(0,1),(0,1),(−a−2,2),(−a−3,2)

(0,1),(−a−3,2),(−1,1) (141)

APPENDIX L

PROOF OF LEMMA 5

In the case of AF-relaying wireless system, the average e2e

SNR can be obtained as

E[ρr] = ˆ∞

0

xfρr(x) dx,(157)

which, by substituting (59), can be rewritten as

E[ρr] = βγM1+γ2M2,(158)

where

M1=ˆ∞

0

x2exp (−βx)K1(γx) dx (159)

and

M2=ˆ∞

0

x2exp (−βx)K0(γx) dx.(160)

Next, we deliver closed-form expressions for (159)

and (160). In particular, for β6=γ, by employing [56, eq.

(6.621/3)] and [56, eq. (6.624/1)], (159) and (160) can be

respectively expressed as

M1=−β4

γ(−β2+γ2)3+2γ3−β2γ

(−β2+γ2)3

−3πβγ

2 (−β2+γ2)5/2+3βγ

(−β2+γ2)5/2acsc γ

β(161)

and

M2=−3β

(−β2+γ2)2+π

2

2β2+γ2

(−β2+γ2)5/2

−2β

(−β2+γ2)5/2acsc γ

β.(162)

Finally, by substituting (161) and (162) into (158), we

get (63).

On the contrary, for β=γ, (159) and (160) can be

respectively written as

M1=ˆ∞

0

x2exp (−βx)K1(βx) dx (163)

and

M2=ˆ∞

0

x2exp (−βx)K0(βx) dx.(164)

which by employing [56, eq. (6.621/3)] and after some alge-

braic manipulations, can be obtained as

M1=2

5γ3(165)

and

M2=4

15γ3.(166)

Finally, by substituting (165) and (166) into (158), we ob-

tain (64). This concludes the proof.

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APPENDIX M

PROOF OF LEMMA 6

In the case of AF-relaying wireless system, based on (113),

the SER can be analytically evaluated as

PAF

e=−ˆ∞

0

Fρr(x)fe(x) dx,(167)

which, by using (60) and (115), can be rewritten as

PAF

e=−1−γc

2rd

4πD,(168)

where

D=ˆ∞

0

x1/2exp (−(β+d)x)K1(γx) dx.(169)

By employing [79, p. 185], (169) can be expressed as

D=π

2(D1− D2),(170)

where

D1=ˆ∞

0

x1/2exp (−(β+d)x)I1(γx) dx (171)

and

D2=ˆ∞

0

x1/2exp (−(β+d)x)I−1(γx) dx.(172)

Next, by applying [56, eq. (6.622)] into (171) and (172), we

obtain

D1=r2

π

β+γ+d

√γγ2−(β+d)2K1

2−β+d

2γ(173)

and

D2=−2r2

π

β+d

√γγ2−(β+d)2

×E1

2−β+d

2γ.(174)

Next, by substituting (173) and (174) into (170), we derive

D=−2rπ

2

β+d

√γγ2−(β+d)2E1

2−β+d

2γ

+rπ

2

β+γ+d

√γγ2−(β+d)2K1

2−β+d

2γ.(175)

Finally, by substituting (175) into (168), we obtain (68). This

concludes the proof.

APPENDIX N

PROOF OF LEMMA 7

According to (130), the EC can be expressed as

CAF =1

ln(2) ˆ∞

0

ln (1 + x)fρr(x) dx.(176)

By substituting (59) in (176), we obtain

CAF =γ2

ln(2)F1+βγ

ln(2)F2.(177)

where

F1=ˆ∞

0

xexp (−βx) ln(1 + x)K0(γx) dx (178)

and

F2=ˆ∞

0

xexp (−βx) ln(1 + x)K1(γx) dx.(179)

For β6=γ, note that (178) and (179) can be equivalently

written as

F1=ˆ∞

0

xexp (−(β−γ)x) ln(1 + x)

×exp (−γx)K0(γx) dx (180)

and

F2=ˆ∞

0

xexp (−(β−γ)x) ln(1 + x)

×exp (−γx)K1(γx) dx (181)

By employing [56, eq. (8.352/2)] and [58, eq. (15.1.1)], (180)

and (181) can be respectively written as

F1=ˆ∞

0

x2Γ (1,(β−γ)x)2F1(1,1; 2; −x)

×exp (−γx)K0(γx) dx (182)

and

F2=ˆ∞

0

x2Γ (1,(β−γ)x)2F1(1,1; 2; −x)

×exp (−γx)K1(γx) dx,(183)

which, by using [75, eq. (20)], [76, eq. (17)] and [80] can be

equivalently rewritten as

F1=√πˆ∞

0

x2G2,0

1,2(β−γ)x

1

0,1G1,2

2,2x

0,0

0,−1

×G2,0

1,22γx

1

2

0,0dx (184)

and

F2=√πˆ∞

0

x2G2,0

1,2(β−γ)x

1

0,1G1,2

2,2x

0,0

0,−1

×G2,0

1,22γx

1

2

−1,1dx.(185)

With the aid of [81], (184) and (185) can be analytically

evaluated as (186) and (187), given at the top of the next page.

By substituting (186) and (187) into (177), we obtain (188),

given at the top of the next page.

Next, we examine the case in which β=γ. For β=

γ, (178) and (179) can be respectively expressed as

F1=ˆ∞

0

xln(1 + x) exp (−γx)K0(γx) dx (189)

and

F2=ˆ∞

0

xln(1 + x) exp (−γx)K1(γx) dx,(190)

18 VOLUME 4, 2016

10.1109/ACCESS.2020.2995435, IEEE Access

F1=√π(β−γ)−3G0,2:1,2:2,0

2,1:2,2:1,2−2,−3

−3

0,0

0,−1

1

2

0,0

1

β−γ,2γ

β−γ(186)

F2=√π(β−γ)−3G0,2:1,2:2,0

2,1:2,2:1,2−2,−3

−3

0,0

0,−1

1

2

−1,−1

1

β−γ,2γ

β−γ(187)

CAF =√π

ln(2)

γ2

(β−γ)3G0,2:1,2:2,0

2,1:2,2:1,2−2,−3

−3

0,0

0,−1

1

2

0,0

1

β−γ,2γ

β−γ

+√π

ln(2)

βγ

(β−γ)3G0,2:1,2:2,0

2,1:2,2:1,2−2,−3

−3

0,0

0,−1

1

2

−1,−1

1

β−γ,2γ

β−γ,for β6=γ(188)

which, by following the same steps as in the case of β6=γ,

can be rewritten as

F1=√πˆ∞

0

x2G1,2

2,2x

0,0

0,−1G2,0

1,22γx

1

2

0,0dx

(191)

and

F2=√πˆ∞

0

x2G1,2

2,2x

0,0

0,−1G2,0

1,22γx

1

2

−1,1dx.

(192)

By employing [77, ch. 2.3], (191) and (192) can be respec-

tively expressed as

F1=√πG4,1

3,42γ−2,−1,1

2

0,0,−2,−2(193)

and

F2=√πG4,1

3,42γ−2,−1,1

2

−1,1,−2,−2.(194)

By substituting (193) and (194) into (177), we get (195),

given in the top of next page. Finally, by combining (188)

and (195), we obtain (69). This concludes the proof.

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ALEXANDROS-APOSTOLOS A. BOULO-

GEORGOS (S’11-M’16-SM’19) was born in

Trikala, Greece in 1988. He received the Elec-

trical and Computer Engineering (ECE) (5 year)

diploma degree and Ph.D. degree in Wireless

Communications from the Aristotle University of

Thessaloniki (AUTh) in 2012 and 2016, respec-

tively.

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 telecom-

munication and networks projects. During 2017, he joined the information

technologies institute, while from November 2017, he has joined the De-

partment of Digital Systems, ICT School, 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 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 50 techni-

cal 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 Schol-

arship 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.

ANGELIKI ALEXIOU is a professor at the de-

partment of Digital Systems, ICT School, Uni-

versity of Piraeus. She received the Diploma in

Electrical and Computer Engineering from the Na-

tional Technical University of Athens in 1994 and

the PhD in Electrical Engineering from Imperial

College of Science, Technology and Medicine,

University of London in 2000. Since May 2009

she has been a faculty member at the Department

of Digital Systems, where she conducts research

and teaches undergraduate and postgraduate courses in the area of Broad-

band 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 inter-

face 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).

VOLUME 4, 2016 21