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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. In 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.
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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
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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
xa(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 ith 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 (i),(2)
with θibeing the PS applied by the ith 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.
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A.–A. A. Boulogeorgos et al.: Reconfigurable Intelligent Surface-Assisted vs Relaying Wireless Systems
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 ith 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
k21,(9)
and
b=k2
k1
,(10)
with
k1=Nπ
2,(11)
and
k2= 4N1π2
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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A.–A. A. Boulogeorgos et al.: Reconfigurable Intelligent Surface-Assisted vs Relaying Wireless Systems
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
xa1
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)=1rx
ρs
K1rx
ρs(26)
and
fs
ρ(x) = 1
4ρs
K0rx
ρs1
2ρsxK1rx
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(ρρth),(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
16π2N, 2π
16π2qρth
ρs
Γπ2
16π2N.(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.
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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(M1)
Mand d=3
M21.
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 11
Mand d=3
2
1
M1, (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
y02F4a+ 1
4,a+ 3
4;1
4,1
2,3
4,a+ 5
4;y= 1,(37)
lim
y02F4a+ 2
4,a+ 4
4;1
2,3
4,5
4,a+ 6
4;y= 1,(38)
lim
y02F4a+ 3
4,a+ 5
4;3
4,5
4,3
2,a+ 7
4;y= 1 (39)
and
lim
y02F4a+ 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
4s1.(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
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A.–A. A. Boulogeorgos et al.: Reconfigurable Intelligent Surface-Assisted vs Relaying Wireless Systems
Pe=c
2π(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
2π(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
4π(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
12π(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
2π(a+ 1)ba+1da+1
2
Γ(a+3
4)
Γ(a+1)
ρa+1
2
sc
2π(a+ 2)ba+2da+2
2
Γa+4
4
Γ (a+ 1)ρa+2
2
s
+c
4π(a+ 3)ba+3da+3
2
Γ(a+5
4)
Γ(a+1)
ρa+3
2
sc
12π(a+ 4)ba+4da+4
2
Γa+5
4
Γ (a+ 1)ρa+4
2
s(36)
C=a2a
(a1)2
log2b2ρs+2a2a
ln(2)(a1)2
F0(3 + a)
+π
ln(2)(2 + a)ba+2Γ(a+ 1)ρa
2+1
s
csc
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
21F2a+ 1
2;1
2,a+ 3
2,1
4b2ρs
+1
ln(2)(a1)2b2ρs2F31,1; 2,1a
2,3a
2,1
4b2ρs(44)
C= 2 ln (2) b2Γ(a+ 1)ρsH1,4
4,3b2ρs
(0,1),(0,1),(a2,2),(a3,2)
(0,1),(a3,2),(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
4b2ρs0. Moreover,
lim
y01F21 + a
2;3
2,2 + a
2,y= 1,(47)
lim
y01F2a+ 1
2;1
2,a+ 3
2,y= 1 (48)
and
lim
y02F31,1; 2,1a
2,3a
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)(a1)2b2ρs
+a2a
(a1)2
log2b2ρs
+2a2a
ln(2)(a1)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→ ∞,
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Cρ1
ln(2)(a1)2b2ρs
+a2a
(a1)2
log2b2ρs+2a2a
ln(2)(a1)2
F0(3 + a)
+π
ln(2)(2 + a)ba+2Γ(a+ 1)ρa
2+1
s
csc
2+π
(a+ 1)ba+1 ln (2) Γ(a+ 1)ρ
a+1
2
s
sec
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
2(51)
and
A2=π
(a+ 1)ba+1 ln (2) Γ(a+ 1)ρ
a+1
2
s
sec
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
s1
1,1,0
1
4 ln(2)ρs
G3,1
1,31
4ρ2
s1
2
1
2,1
2,1
2
+1
8 ln(2)ρ2
s
G4,1
2,41
4ρ2
s1,0
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
ρrρ1ρ2
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)=12x
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+β3γ22βγ4
(β2+γ2)3π
2
β2γ2γ4
(β2+γ2)5/2
3βγ2
(β2+γ2)2+3β2γ22βγ2
(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
2β+γ
2γ
γc
4rd
2
β+γ+d
γ2(β+d)2K1
2β+γ
2γ1.(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 105.
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 MQAM 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 104, 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 105, 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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A.–A. A. Boulogeorgos et al.: Reconfigurable Intelligent Surface-Assisted vs Relaying Wireless Systems
CAF =
π
ln(2)
γ2
(βγ)3G0,2:1,2:2,0
2,1:2,2:1,22,3
3
0,0
0,1
1
2
0,0
1
βγ,2γ
βγ
+π
ln(2)
βγ
(βγ)3G0,2:1,2:2,0
2,1:2,2:1,22,3
3
0,0
0,1
1
2
1,1
1
βγ,2γ
βγ,for β6=γ
π
ln(2) γ2G4,1
3,42γ2,1,1
2
0,0,2,2+G4,1
3,42γ2,1,1
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 MQAM.
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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] = N1π2
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 (ρx).(86)
By employing (18), (86) can be re-written as
Fρ(x) = Pr Arx
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
4ρsN11
2ρsN2+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) = cQ2dx,(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
xa1
2exp 1
brx
ρsQ2dxdx.(103)
By employing [72, Eq. (B.112)], (103) can be equivalently
expressed as
L=1
2(L1− L2),(104)
where
L1=ˆ
0
xa1
2exp 1
brx
ρsdx (105)
and
L2=ˆ
0
xa1
2exp 1
brx
ρserf dxdx.(106)
By setting z=x, (105) and (106) can be respectively
simplified as
L1= 2 ˆ
0
zaexp 1
bρs
zdz (107)
and
L2= 2 ˆ
0
zaexp 1
bρs
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
4πx1/2exp (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) K1sdx.(118)
Notice that
P1= 1.(119)
Moreover, in (118), by setting z=ρsx, performing
integration by parts and using [58, Eq. (13.3.4)], it can be
analytically obtained as
P2=π
2rρs
dU1
2,0,1
4s.(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) 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
+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
3π(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
2π(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
6π(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 ≈ B1ρa+1
2
s+B2ρa+2
2
s+B3ρa+3
2
s+B4ρa+4
2
s,
(121)
where
B1=c
2π(a+ 1)ba+1da+1
2
Γ(a+3
4)
Γ(a+1)
,(122)
B2=c
2π(a+ 2)ba+2da+2
2
Γa+4
4
Γ (a+ 1),(123)
B3=c
4π(a+ 3)ba+3da+3
2
Γ(a+5
4)
Γ(a+1)
(124)
and
B4=c
12π(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 ρa+1
2
s,ρa+2
2
s,ρa+3
2
s, and ρa+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 π21,(127)
which, since N1, 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) = (x1) 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
׈
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)ρsC11
2 ln(2)ρsC2+1
4 ln(2)ρsC3,(144)
where
C1=ˆ
0
K0rx
ρsln (1 + ρsx) dx,(145)
C2=ˆ
0
x1/2K1rx
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
s1
1,1,0,(154)
C2=1
2ρs
G3,1
1,31
4ρ2
s1
2
1
2,1
2,1
2(155)
and
C3=1
2ρs
G4,1
2,41
4ρ2
s1,0
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 (bρs)+6a2ba+3Γ(a)ρsln (bρs)+2a3ba+3Γ(a)ρsln (bρs)
+ 2a(a+ 1)(a+ 2)ba+3Γ(a)F0(3 + a)π
(4 + a)a
2+1
s
csc
21F22 + a
2;3
2,3 + a
2,1
4b2ρs
π
(a+ 3)ρ
a+1
2
s
sec
21F2a+ 3
2;1
2,a+ 5
2,1
4b2ρs+aba+1Γ(a)2F31,1; 2,1a
2,a
2,1
4b2ρs(137)
C= 4ab2Γ(a)
Γ(a+ 1)ρslog2(bρs)+6a2b2Γ(a)
Γ(a+ 1)ρslog2(bρs)+2a3b2Γ(a)
Γ(a+ 1)ρslog2(bρs)
+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
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
21F2a+ 3
2;1
2,a+ 5
2,1
4b2ρs
+a
ln(2)
Γ(a)
Γ(a+ 1) 2F31,1; 2,1a
2,a
2,1
4b2ρs(138)
K= 2ba+3ρsH1,4
4,3b2ρs
(0,1),(0,1),(a2,2),(a3,2)
(0,1),(a3,2),(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+2γ3β2γ
(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.
VOLUME 4, 2016 17
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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=1γc
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)I1(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
2β+d
2γ(173)
and
D2=2r2
π
β+d
γγ2(β+d)2
×E1
2β+d
2γ.(174)
Next, by substituting (173) and (174) into (170), we derive
D=2rπ
2
β+d
γγ2(β+d)2E1
2β+d
2γ
+rπ
2
β+γ+d
γγ2(β+d)2K1
2β+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 ((βγ)x) ln(1 + x)
×exp (γx)K0(γx) dx (180)
and
F2=ˆ
0
xexp ((βγ)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,(βγ)x)2F1(1,1; 2; x)
×exp (γx)K0(γx) dx (182)
and
F2=ˆ
0
x2Γ (1,(βγ)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(βγ)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(βγ)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
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A.–A. A. Boulogeorgos et al.: Reconfigurable Intelligent Surface-Assisted vs Relaying Wireless Systems
F1=π(βγ)3G0,2:1,2:2,0
2,1:2,2:1,22,3
3
0,0
0,1
1
2
0,0
1
βγ,2γ
βγ(186)
F2=π(βγ)3G0,2:1,2:2,0
2,1:2,2:1,22,3
3
0,0
0,1
1
2
1,1
1
βγ,2γ
βγ(187)
CAF =π
ln(2)
γ2
(βγ)3G0,2:1,2:2,0
2,1:2,2:1,22,3
3
0,0
0,1
1
2
0,0
1
βγ,2γ
βγ
+π
ln(2)
βγ
(βγ)3G0,2:1,2:2,0
2,1:2,2:1,22,3
3
0,0
0,1
1
2
1,1
1
βγ,2γ
βγ,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
3,42γ2,1,1
2
0,0,2,2(193)
and
F2=πG4,1
3,42γ2,1,1
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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Available: http://functions.wolfram.com/07.34.21.0081.01
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
... Reconfigurable intelligent surface (RIS), as a new type of transmission relay, has ability to correct the wireless channel through a highly controllable software, which paves the way for an intelligent and programmable wireless environment [26][27][28]. Because of its novelty and greater gain, the RISassisted NOMA networks have attracted a large part of research effort [29][30][31]. In [29], the performance of theoretical framework for the RIS and AF relaying were compared and results showed that RIS-assisted wireless systems outperform the corresponding AF-relaying ones. ...
... Because of its novelty and greater gain, the RISassisted NOMA networks have attracted a large part of research effort [29][30][31]. In [29], the performance of theoretical framework for the RIS and AF relaying were compared and results showed that RIS-assisted wireless systems outperform the corresponding AF-relaying ones. Further considered in [30], a RIS-aided NOMA network with stochastic geometry model was discussed and the SIC order is proofed to be altered since RISs were able to change the channel quality of users. ...
... where S ∞ m = M!/ðM − mÞ!m! Remark 10. By substituting (29) into (25), it can be derived that the diversity order of the distant user m is equal to m K½Eðφ n Þ 2 /2Varðφ n Þ, which is also related to the order of the distant user, and Nakagami-m parameter m ′ and Ω Monte Carlo simulation repeated 10 6 iterations Two users'power allocations a n = 0:4 a m = 0:6 ...
Article
Full-text available
This paper considers the application of reconfigurable intelligent surface (RIS) to non-terrestrial non-orthogonal multiple access (NOMA) networks. More specifically, the performance of a pair of non-orthogonal users for RIS assisted non-terrestrial NOMA networks is investigated over large-scale fading and Nakagami- m fading cascaded channel. The exact and asymptotic expressions of outage probability are derived for the nearby user and distant user with the imperfect successive interference cancellation (SIC) and perfect SIC schemes. Based on the approximated results, the diversity orders of these two users are obtained in the high signal-to-noise ratios. The simulation results are used to verify the theoretical derivations and find that: 1) The outage behaviors of RIS assisted non-terrestrial NOMA networks outperforms than that of orthogonal multiple access; 2) By increasing the number of reflecting elements of RIS and Nagakami- m fading factors m ′ and Ω , RIS-assisted non-terrestrial NOMA networks are able to achieve the enhanced outage performance.
... Because they require no transmit power amplifiers such as cooperative communication systems or massive MIMO-equipped networks, IRS-based wireless networks can operate in full-duplex mode without amplifying the interferences [22]. Outstanding results in previous works [25][26][27] showed that IRS-assisted wireless systems for long packet transmission achieve higher performance and consume lower transmit power than cooperative communication schemes using relays. ...
... are the amplitude and phase values of channels h ST l and h T l D , respectively, λ κ ¼ d β κ , β is the path-loss exponent [22,24,26,36], κ ST, TD, SP f g , and j 2 ¼ À1. Consequently, h ST l j j and h T l D j j are independent RVs and follow the Rayleigh distribution with means given by ...
... where r l is the unit-adjusted response of the T l , which can be expressed as r l ¼ e jφ l with φ l being the adjustable phase induced by T l [22,26], To maximize γ D IRS in (2), the phases φ l of the T l are optimally selected as φ l ¼ À φ ST l þ φ T l D À Á by varactortuned resonators. The phases of CSI h ST l and h T l D are perfectly known [22,[24][25][26]28]. ...
... For SU systems, [2] derives a closed form expression for the mean SNR where the user (UE) to RIS and RIS to base station (BS) channels experience Rayleigh fading and the direct channel between UE and BS is absent. [3] derives an exact expression for the optimal uplink (UL) mean SNR for systems where the UE-BS channel is rank-1 LOS and the UE-RIS and UE-BS channels are correlated Rayleigh. ...
... The authors again, leverage the mean SNR expression to provide insight on the impact of correlation and the Rician K-factor on the mean SNR. However, the analysis in [2], [3], [5] assumes either perfect RIS reflection or reflections with constant attenuation. ...
... The PDL model in (2) gives losses which are dependent on the RIS phases. The variables L min ≥ 0, θ ≥ 0 and α ≥ 0 are constants dependent on specific circuit implementations [1]. ...
Preprint
In this paper we focus on phase dependent loss (PDL), an important aspect of reconfigurable intelligent surfaces (RIS) where the signals reflected from the RIS elements are attenuated by varying amounts depending on the phase rotation provided by the element. To evaluate the effects of PDL, we analyse the SNR of a SIMO RIS-aided wireless link. We assume that the channel between the base station (BS) and RIS is a rank-1 LOS channel while the user (UE)-BS and UE-RIS are correlated Rayleigh channels. The RIS design is optimal in the absence of PDL and maximizes the SNR in this scenario. Specifically, we derive a closed form expression for the mean SNR in the presence of PDL. The attenuation function used for PDL was developed from a detailed circuit analysis of RIS elements. Leveraging the derived results, we analytically characterise the impact of PDL on the mean SNR. Numerical results are conducted to validate the derived expressions and verify the analysis.
... To tackle the challenges, a deterministic approximation for signal-to-noise-interference-plus-noise-ratio (SINR) was derived in [6], which eases the phase shift matrix design with statistics CSI. To investigate the efficiency of RIS-aided wireless systems, the authors in [7] studied the performance for Rayleigh channels and demonstrated its superior performance over amplify-and-forward (AF) relaying scheme in terms of average symbol error rate (SER), ergodic capacity (EC), etc.. To further enhance the RIS-aided system performance, the multi-RIS assisted systems were studied in [8], and an effective method was proposed to maximize the weighted sum rate. Moreover, the performance of RIS assisted two-way systems over Rayleigh fading channels was investigated in [9], and the exact expressions for OP and EC were derived in closed-form. ...
Preprint
Full-text available
Reconfigurable intelligent surface (RIS) has recently attracted a spurt of interest due to its innate advantages over Massive MIMO on power consumption. In this paper, we study the outage performance of multi-RIS system with the help of hybrid automatic repeat request (HARQ) to improve the RIS system reliability, where the destination received channels are modeled by Rician fading and the phase shift setting only depends on the line-of-sight (LoS) component. Both the exact and asymptotic outage probabilities under Type-I HARQ and HARQ with chase combining (HARQ-CC) schemes are derived. Particulary, the tractable asymptotic results empower us to derive meaningful insights for HARQ-aided multi-RIS system. On the one hand, we find that both the Type-I and the HARQ-CC schemes can achieve full diversity that is equal to the maximal number of HARQ rounds. On the other hand, the closed-form expression of the optimal phase shift setting with respect to outage probability minimization is obtained. The optimal solution indicates that the reflecting link direction should be consistent with direct link LoS component. Finally, the analytical results are validated by Monte-Carlo simulations.
... The advantages of the RIS compared to relaying technologies have been addressed by some investogators. For instance, in [6] RIS-assisted communication are compared with passive relays. The Authors present endto-end expressions for SNR, outage probability, symbol error rate, and the ergodic capacity of relaying and the RISassisted links in fading channels. ...
Preprint
Full-text available
The fifth generating (5G) of wireless networks will be more adaptive and heterogeneous. Reconfigurable intelligent surface technology enables the 5G to work on multistrand waveforms. However, in such a dynamic network, the identification of specific modulation types is of paramount importance. We present a RIS-assisted digital classification method based on artificial intelligence. We train a convolutional neural network to classify digital modulations. The proposed method operates and learns features directly on the received signal without feature extraction. The features learned by the convolutional neural network are presented and analyzed. Furthermore, the robust features of the received signals at a specific SNR range are studied. The accuracy of the proposed classification method is found to be remarkable, particularly for low levels of SNR.
... The authors have provided a set of simulation results to compare the RIS with a DF relay in terms of EE and SE for various system configurations. Further, [114] examined the outage probability, symbol error rate (SER), and EC of RIS-aided communication system and compared them with a relay counterpart. Moreover, the key difference between a relay and RIS is that relay actively processes the received signal before retransmitting an amplified/decoded signal, whereas RIS reflects the incoming signal passively but with beamforming. ...
Thesis
The demands for high data rate services and applications in wireless communication have been growing exponentially for decades. To meet these demands of users, multi-input multiple-output (MIMO) wireless systems have been introduced. Space modulation technique (SMT) is a lately developed transmission scheme for MIMO systems. All transmission schemes introduced in this thesis are based on the SMT principle. F