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# How Much do Hardware Imperfections Affect the Performance of Reconfigurable Intelligent Surface-Assisted Systems?

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In the present work, we investigate the impact of transceiver hardware imperfection on reconfigurable intelligent surface (RIS)-assisted wireless systems. In this direction, first, we present a general model that accommodates the impact of the transmitter (TX) and receiver (RX) radio frequency impairments. Next, we derive novel closed-form expressions for the instantaneous end-to-end signal-to-noise-plus-distortion-ratio (SNDR). Building upon these expressions, we extract an exact closed-form expression for the system’s outage probability, which allows us not only to quantify RIS-assisted systems’ outage performance but also reveals that the maximum allowed spectral efficiency of the transmission scheme is limited by the levels of the transceiver hardware imperfection. Moreover, in order to characterize the capacity of RIS-assisted systems, we report a new upper-bound for the ergodic capacity, which takes into account the number of the RIS’s reflected units (RUs), the level of TX and RX hardware imperfection, as well as the transmission signal-to-noise-ratio (SNR). Finally, two insightful ergodic capacity ceilings are extracted for the high-SNR and high-RUs regimes. Our results highlight the importance of accurately modeling the transceiver hardware imperfection when designing and assessing RIS-assisted wireless systems.
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How much do hardware imperfections affect the performance of
reconﬁgurable intelligent surface-assisted systems?
Alexandros–Apostolos A. Boulogeorgos, Senior Member, IEEE, and Angeliki Alexiou, Member, IEEE,
In the present work, we investigate the impact of transceiver hardware imperfection on reconﬁgurable intelligent surface (RIS)-
assisted wireless systems. In this direction, ﬁrst, we present a general model that accommodates the impact of the transmitter (TX)
and receiver (RX) radio frequency impairments. Next, we derive novel closed-form expressions for the instantaneous end-to-end
signal-to-noise-plus-distortion-ratio (SNDR). Building upon these expressions, we extract an exact closed-form expression for the
system’s outage probability, which allows us not only to quantify RIS-assisted systems’ outage performance but also reveals that the
maximum allowed spectral efﬁciency of the transmission scheme is limited by the levels of the transceiver hardware imperfection.
Likewise, a diversity analysis is provided. Moreover, in order to characterize the capacity of RIS-assisted systems, we report a new
upper-bound for the ergodic capacity, which takes into account the number of the RIS’s reﬂective units (RUs), the level of TX and
RX hardware imperfection, as well as the transmission signal-to-noise-ratio (SNR). Finally, two insightful ergodic capacity ceilings
are extracted for the high-SNR and high-RUs regimes. Our results highlight the importance of accurately modeling the transceiver
hardware imperfection and reveals that they signiﬁcantly limit the RIS-assisted wireless system performance.
Index Terms—Diversity Order, Ergodic Capacity, Outage Probability, Performance Analysis, Reconﬁgurable Intelligent Surfaces.
I. INTRODUCTION
RECONFIGURABLE intelligent surfaces (RISs) have
been recognized as one of the key enablers of the sixth
generation networks [1], [2]. Most RIS designs consists of
two-dimensional (2D) arrays of reﬂective units (RUs) that are
controlled by at least one micro-controller [3]. Each RU can
independently change the phase shift of the electromagnetic
signal incident upon it [4]. By providing collaboration ca-
pabilities between the RUs through the micro-controller, the
implicit randomness of the propagation environment can be
exploited in order to create preferable wireless channels [5].
Scanning the technical literature, several research work
that studied the performance of RIS-assisted systems [6]–
[8], presented comparisons with their predecessors, i.e. re-
lays [9]–[11], and provided optimum information and/or power
transfer policies [12]–[15], can be identiﬁed. In particular,
in [6], Basar et. al provided an upper bound for the symbol
error rate (SER) of RIS-assisted systems, assuming that the
transmitter (TX)-RIS and RIS-receiver (RX) channels are
Rayleigh distributed. Similarly, in [7], the authors presented
a bit error rate analysis for RIS-assisted systems that employ
non-orthogonal multiple access. Additionally, in [8], Zhang
et. al presented an approximation for the achievable data
rate assuming that both the TX-RIS and RIS-RX channels
are independent and Rician distributed. In [9], Renzo et.
al highlighted the fundamental similarities and differences
between RIS and relays. In the same work, simulation results
were provided in order to compare RIS- with relay-assisted
systems in terms of achievable data rate. In [10], the authors
compared RIS with decode-and-forward relays in terms of
energy efﬁciency, while in [11], RIS-assisted systems were
compared against the corresponding relay ones in terms of
This work was supported from the European Commission’s Horizon 2020
research and innovation programme under grant agreement No. 871464
(ARIADNE).
The authors are with the Department of Digital Systems, University
of Piraeus Piraeus 18534 Greece (e-mails: al.boulogeorgos@ieee.org, alex-
iou@unipi.gr).
outage probability, symbol error rate, diversity gain and order
as well as ergodic capacity, assuming that the transceivers
in both RIS- and relay-assisted systems were equipped with
ideal RF front-ends. Moreover, in [12], the authors presented
an energy efﬁciency maximization strategy for RIS-assisted
wireless systems, whereas, in [13], the authors provided a
policy that enables the maximization of the achievable rate
by jointly optimizing the transceivers beamforming precoders
and the RIS phase shifters. Likewise, in [14], a data rate
maximization strategy for RIS-assisted unmanned aerial ve-
hicle networks was reported. Finally, in [15], a simultaneous
wireless information and power transfer scheme for RIS-
assisted systems was discussed.
All the aforementioned works assumed that the transceivers
were equipped with ideal radio frequency (RF) front-ends.
However, in practice, transceiver suffers from hardware im-
perfections, which cause in-phase and quadrature imbalance,
phase noise and nonlinearities [16]–[23]. Recognizing the fact
that hardware imperfections will be one of the main limitations
of the RIS-assisted systems, Zhou et al. studied the spectral
and energy efﬁciency of RIS-assisted multiple-input single-
output systems in the presence of hardware imperfections and
revealed their detrimental impact on the performance of such
systems [24]. Similarly, in [25], the authors characterized the
asymptotic channel capacity of RIS-assisted systems in which
both the TX and RX experience hardware imperfections.
Additionally, in [26], the capacity degradation due to hardware
imperfections in RIS-assisted systems was bounded. How-
ever, in [26], the detrimental effect of fading was neglected.
Furthermore, [27] and [28], the authors also studied the
capacity performance of RIS-assisted wireless systems in the
presence of hardware imperfections, assuming deterministic
wireless channels. Finally, no outage or diversity analysis were
provided in [26]–[28].
To the best of the authors’ knowledge, there is no paper
in the technical literature that examines the impact of hard-
ware imperfections on the outage performance of RIS-assisted
wireless systems and quantiﬁes its ergodic capacity, under
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the assumption that both the TX-RIS and RIS-RX links are
independent and Rayleigh distributed. Motivated by this, the
contribution of this paper is as follows:
We present a general model to accommodate the impact
of transceiver impairments on RIS-assisted systems. This
model also takes into account the effect of fading as well
as the RIS size.
Next, we extract the instantaneous signal-to-noise-plus-
distortion-ratio (SNDR), and we derive novel closed-
form expressions for the outage probability of RIS-
assisted wireless systems. These expressions are capable
of quantifying the outage performance degradation due to
hardware imperfections and reveal that in order to achieve
an acceptable outage probability, the spectral efﬁciency
of the transmission scheme should be constrained by a
hardware imperfection-dependent limit. As a benchmark,
we revisit the outage probability for the special case in
which both the TX and the RX are equipped with ideal
RF front-ends. Note that the outage probability for ideal
TX and RX was initially presented in [11].
Moreover, we provide a diversity order analysis that
reveals that the diversity order of RIS-assisted wireless
systems depends only from the number of RIS’s RUs.
Additionally, we present a low-complexity closed-form
upper bound for the ergodic capacity of RIS-assisted
wireless systems.
Finally, ergodic capacity ceilings are extracted for the
cases in which the signal-to-noise-ratio (SNR) and/or the
number of RIS’s RUs tend to inﬁnity.
The rest of the paper is organized as follows: Section II
describes the RIS-assisted system model that takes into ac-
count the effect of transceiver hardware imperfections. Next,
Section III provides the theoretical framework that assess
the impact of transceiver hardware imperfections on RIS-
assisted wireless systems in terms of outage probability and
ergodic capacity. Section IV presents respective numerical
results, which verify the analysis, accompanied by insightful
discussions and observations. Finally, closing remarks that
summarize the current contribution, are reported in Section V.
Notations: In what follows, the operators E[·],|·|, and
Pr (A)respectively denote the statistical expectation, the ab-
solute value, and the probability of the event A. Moreover,
limxz(f(x)) returns the limit of f(x)as xtends to z.
Additionally, Γ (·),Γ (·,·)and γ(·,·)respectively stand for the
Gamma [29, eq. (8.310)], upper incomplete Gamma [29, eq.
(3.350/3)] and lower incomplete Gamma [29, eq. (3.350/2)]
functions. Finally, (x)nrepresents the Pochhammer opera-
tor [30, eq. (19)].
II. SYS TEM MODEL
As illustrated in Fig. 1, we consider a scenario in which a
single-antenna TX node communicates with a single-antenna
RX node through a RIS, which consists of NRUs. It is
assumed that, due to blockage, no-direct link between TX
and RX can be established. The baseband equivalent fading
channels between the TX and the i-th RU, hi, and the
one between the ith RU and RX, gi, are assumed to be
independent and identical. Moreover, it is assumed that |hi|
and |gi|are Rayleigh distributed with scale parameter being
equal to 1. Of note, several prior published contributions
employ this assumption [6], [8], [11], [12], which originates
from the fact that even if the line-of-sight links between the
TX and RIS as well as RIS and RX are blocked, there still
exist extensive scatters.
The hardware imperfections at the TX cause a mismatch
between the intended transmitted signal, s, and what is actually
generated. As a result, the actual transmitted signal can be
described as
˜s=s+nt,(1)
where ntrepresents the distortion from the TX hardware
imperfections, and can be modeled as a zero-mean complex
Gaussian process with variance
σ2
t=κ2
tPs.(2)
In (2), κtstands for the TX’s error vector magnitude (EVM),
which is in-general a non-negative design parameter, while
Psrepresents the average transmitted power. Of note, accord-
ing to the third generation partnership project (3GPP) long
term evolution advanced (LTE-A), EVM is in the range of
[0.07,0.175] [31]. Moreover, in high frequency systems, such
as millimeter wave and THz ones, EVM may even reach
0.3[32], [33].
At the RX side, the baseband equivalent received signal can
be obtained as
r=
N
X
i=1
higipi˜s+nr+n, (3)
where nris the distortion from the RX hardware imperfec-
tions, and can be modeled as a zero-mean complex Gaussian
process with variance
σ2
r=κ2
r|A|2Ps,(4)
with
A=
N
X
i=1 |hi||gi|,(5)
being the equivalent TX-RIS-RX channel. In (4), κrrepresents
the RX’s EVM. Note that this model has been validated
by several analytical and experimental prior works, includ-
ing [21], [34]–[40] and references therein.
Likewise, nstands for the white Gaussian noise (AWGN)
and can be modeled as a zero-mean complex Gaussian process
with variance No. Moreover, pirepresents the ith RU
response and can be obtained as
pi=|pi|exp (i),(6)
with φistanding for the phase shift that is applied by the ith
RU of the RIS. Without loss of generality, we assume that
|pi|= 1, which is in line with realistic implementations [41].
In the current contribution, we consider a RIS that uses
varactor-tuned RUs, capable of conﬁguring their phase shift by
adjusting the bias voltage applied to the varactor [42]. Next,
by assuming that the RIS has perfect knowledge of the phase
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ADC
ADC
90oj
Σ
Microcontroller
Varactor
TX RX
hi
gi
RIS
r
Re{ }
Im{ }
90o
Σ
s
Non-linearities In-phase and quadrature
imbalance Phase noise
Fig. 1. System model of the RIS-assisted wireless system. Note that in this ﬁgure ˜srepresents the baseband equivalent of the transmitted signal.
of hi,φhi, and the one gi,φgi, and selects the optimal phase
shifting, i.e.
φ=(φhi+φgi),(7)
we can simplify (6) as
pi= exp (j(φhi+φgi)) ,(8)
Next, by applying (8) into (3) and after some mathematical
manipulations, the equivalent received signal at the RX can be
expressed as [11, eq. (6)]
r=A˜s+nr+n. (9)
Finally, by substituting (1) into (9), the baseband equivalent
received signal can be rewritten as
r=As +w+n, (10)
where
w=Ant+nr,(11)
represents the aggregated distortion caused by the TX and RX
hardware imperfections.
Remark 1: From (11), it becomes evident that for a given
channel realization, the aggregated impact of TX and RX
hardware imperfections can be modeled via a zero-mean
random variable process with variance
σ2
w=|A|2κ2
t+κ2
rPs.(12)
Interestingly, (12) reveals that as the transmission power
increases, the level of distortion due to transceivers hardware
imperfections also increases. Finally, note that (10) reduces to
the conventional model that neglects the impact of hardware
imperfections, for κt=κr= 0. In this case, from (12), σ2
w=
0.
III. PERFORMANCE ANALYSIS
In this section, the theoretical framework that quantiﬁes
the impact of transceivers hardware imperfections on the
performance of RIS-assisted wireless systems is presented.
Speciﬁcally, the structure of this section is as follows:
Section III-A provides the instantaneous end-to-end SNDR,
whereas Section III-B presents a novel closed-form expression
for the outage probability. Likewise, Section III-C returns the
diversity order of the RIS-assisted wireless system. Finally,
Section III-D reports ergodic capacity upper bounds and
ceilings, for the cases in which the SNR and/or the number
of RIS’s RUs tend to inﬁnity.
A. SNDR
From (10) and (12), the instantaneous SNDR can be ob-
tained as
ρ=|A|2Ps
(κ2
t+κ2
r)|A|2Ps+No
,(13)
or equivalently
ρ=|A|2
(κ2
t+κ2
r)|A|2+1
ρs
,(14)
where
ρs=Ps
No
,(15)
denotes the transmission SNR.
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B. Outage probability
The following theorem returns a closed-form expression for
the RIS-assisted wireless system outage probability.
Theorem 1. The outage probability of the RIS-assisted
wireless system can be obtained as in (16), given at the top
of the next page. In (16), ρth is the SNR threshold.
Proof: Please refer to Appendix A.
Notice that the SNR threshold is connected with the spectral
efﬁciency of the transmission scheme, r, through
r= log2(ρth + 1) .(17)
Remark 2. From (16) and (17), we observe that the
outage probability is always 1for r > log21
κ2
t+κ2
r+ 1.
This means that the spectral efﬁciency of the transmission
scheme is limited by the levels of the transceivers hardware
imperfections. Finally, note that this result is independent of
the fading characteristic of the channel.
Remark 3. For the ideal case in which both the TX and
RX are hardware imperfection free, i.e. κt=κr= 0, (16)
reduces to
Pid
o=
γπ2
16π2N, 2π
16π2rρth
ρs
Γπ2
16π2N,(18)
which is the same as [11, eq. (31)].
C. Diversity order
The diversity order can be calculated as
D=lim
ρs→∞ log2(Po)
log2(ρs),(19)
which, with the aid of (16) and by assuming that ρth 1
κ2
t+κ2
r
,
can be rewritten as in (20), given at the top of the next page.
After evaluating the limit in (20), we obtain
D=π2
16 π2
N
2.(21)
Notice that, according to (21), the diversity order only depends
on the number of RIS’s RUs and not from the level of
imperfections.
D. Ergodic capacity
In order to characterize the ergodic capacity of RIS-
assisted wireless systems, the following theorem provides an
upper bound.
Theorem 2. The ergodic capacity, C, of RIS-assisted wire-
less systems can be upper bounded as
Clog2
1 + 16π2
2π2 2
16π22
(κ2
t+κ2
r)16π2
2π2 2
16π22+1
ρs
,
(22)
Proof: Please refer to Appendix B.
The Lemma 1 returns a high-SNR ergodic capacity ceiling,
while Lemma 2 provides a high-Nergodic capacity ceiling.
Lemma 1. (High-SNR and high-Nergodic capacity ceiling)
As the transmission SNR tends to inﬁnity or as the number
of RIS’s RUs tends to inﬁnity, the ergodic capacity is con-
strained by
lim
ρs→∞ C= log21 + 1
κ2
t+κ2
r.(23)
Proof: For brevity, the proof is given to Appendix C.
Remark 4. Lemma 1 reveals that the transceiver hardware
imperfections cause an ergodic capacity saturation; thus, the
performance of high-rate systems is constrained. Moreover, it
becomes apparent that in the high-SNR and high-Nregimes,
the performance of the system is independent from the number
of RUs at the RIS and are fully determined by the level
of imperfections.
IV. RES ULTS & DISCUSSION
This section aims at verifying the theoretical framework
provided in Section III by means of Monte Carlo simulations,
assessing the detrimental impact of transceivers hardware
imperfections on RIS-assisted wireless systems, and present-
ing insightful discussions. Unless otherwise stated, in what
follows, we use continuous lines and markers to respectively
denote theoretical and simulation results. Moreover, we de-
ﬁne κ=κt=κr.
Figure 2 demonstrates the outage probability as a function
of ρsfor different values of Nand ρth, assuming κ= 0.1. Of
note, according to (17), a ρth increase is translated to a spectral
efﬁciency increase. From this ﬁgure, we observe that, for ﬁxed
ρth and N, as ρsincrease, the system’s outage performance
improves. For example, for ρth = 10 dB and N= 100,
the outage probability decreases for about 100 times, as ρs
increases from 33 to 32 dB. On the other hand, in order
to achieve the same outage performance improvement, in a
system with N= 10 for the same ρth, the transmission
SNR should be increased for about 5 dB. This indicates that
RIS-assisted systems with higher Nthat, based on (21), have
higher diversity order, achieve higher diversity gains. In this
sense, another way to boost the RIS-assisted system’s outage
performance, for a given ρth and ρs, is to increase N. For
instance, for ρs= 0 dB and ρth = 10 dB, as Nincreases
from 1to 10, a 3orders decrease occurs on the outage
probability. Finally, this ﬁgures reveals that there exists a trade-
off between RIS-assisted system spectral efﬁciency and power
consumption. In more detail, we observe that, for a ﬁxed N
and a predetermined outage probability requirement, in order
to increase the system spectral efﬁciency, i.e. increase ρth, the
transmission SNR should be also increased; thus, the power
consumption would also increase.
Figure 3 depicts the outage probability as a function of the
transmission SNR, for different values of ρth and κ, assuming
N= 5. As a benchmark, the outage performance for the
ideal case in which both the TX and RX does not experience
the impact of hardware imperfections, i.e. κ= 0, is also
provided. Moreover, according to 3GPP, κ= 0.07 is the
lowest achievable EVM for realistic designs, while κ= 0.2
is a realistic value for devices operating in high-frequency
bands [32], [33]. As expected, we observe that independently
of ρsand κ, an outage performance degradation is observed,
as ρth increases. For instance, for κ= 0.07 and ρs=5 dB,
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Po=
γ π2
16π2N, 2π
16π2
1
1(κ2
t+κ2
r)ρth rρth
ρs!
Γπ2
16π2N, ρth 1
κ2
t+κ2
r
1,otherwise
,(16)
D=lim
ρs→∞
log2
γ π2
16π2N, 2π
16π2
1
1(κ2
t+κ2
r)ρth rρth
ρs!
Γπ2
16π2N
log2(ρs)
(20)
Fig. 2. Outage probability vs ρsfor different values of Nand ρth,
assuming κ= 0.1.
Fig. 3. Outage probability vs ρs, for different values of ρth and κ,
assuming N= 5.
the outage probability increases more than 100 times as the ρth
increases from 0to 10 dB. Likewise, for given ρsand ρth, as κ
increases, the outage performance degrades. For example, for
ρs=ρth = 10 dB, as κincreases from 0to 0.15, the outage
probability decreases for about 10 times. This indicates the
importance of accurately modeling the transceivers’ hardware
imperfections when assessing the performance of RIS-assisted
wireless systems. Moreover, we observe that as ρth increases,
the impact of hardware imperfections becomes more severe.
For instance, for ρs=5 dB and ρth = 0, as κincreases
from 0to 0.2, the outage probability increases from 0.017 to
0.021, which is translated into a 23.5% outage performance
degradation, while, for the same ρsand for ρth = 10 dB,
the same κincrease results to an outage probability increase
from 0.22 to 0.94, i.e. the outage probability increases for
approximately 3times. Similarly, as ρsincreases, the impact
of hardware imperfections on the system’s outage performance
become more detrimental. For example, for ρth = 10 dB
and ρs= 0 dB, the outage probability increases for one
order of magnitude as κincreases from 0to 0.2, whereas,
for the same ρth and ρs= 10 dB, the outage probability
increases for more than 100 times, as κincreases from 0to
0.2. To sum up, this ﬁgure reveals that there exist a relationship
between the transmission SNR, transmission scheme spectral
efﬁciency and level of hardware imperfections, which needs
to be taken into account when assessing and designing RIS-
assisted wireless systems.
Figure 4 illustrates the impact of hardware imperfections on
the outage performance of RIS-assisted systems with different
number of RUs. In more detail, the outage probability is
plotted as a function of the transmission SNR, for different
values of Nand κ, assuming ρth = 10 dB. Again, as a
benchmark, the ideal case in which κ= 0 is also depicted.
From this ﬁgure, it also becomes evident that hardware imper-
fections have a detrimental impact on the RIS-assisted system
performance. In particular, we observe that for a given N,
as κincreases, the transmission SNR should be signiﬁcantly
increase in order for a predetermined outage probability re-
quirement to be satisﬁed. For instance, for N= 100 and an
outage probability requirement of 106, the transmission SNR
should be increased approximately 6 dB, if κincreases from 0
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Fig. 4. Outage probability vs ρs, for different values of Nand κ, assuming
ρth = 10 dB.
Fig. 5. Outage probability vs κtand κr, assuming N= 5,ρs=ρth =
10 dB.
to 0.17. Similarly, for N= 10 and κvariation, approximately
the same transmission SNR increase is required to guarantee
a106outage probability requirement. In other words, we
observe that the impact of hardware imperfections on the
RIS-assisted system outage performance is independent of the
number of the RIS’s RUs.
Figure 5 demonstrates the outage probability as a function
of κtand κr, assuming N= 5, and ρs=ρth = 10 dB. From
this ﬁgure, we observe that, for a given κr, as κtincreases,
the outage probability also increases. Similarly, for a ﬁxed κt,
as κrincreases, the system outage performance degrades. For
example, for κt= 0.1, as κrincreases from 0.1to 0.2, the
outage probability increases from 2.43×105to 1.23×104.
Similarly, for κr= 0.1, as κrincreases from 0.1to 0.2, the
outage probability also changes from 2.43 ×105to 1.23 ×
104. These examples reveal the reciprocal nature of TX and
RX hardware imperfections. Finally, it is evident that when
the ρth <1
κ2
t+κ2
r
is violated, the outage probability becomes
equal to 1.
Fig. 6. Ergodic capacity vs ρs, for different values of κ, assuming N= 10.
Figure 6 illustrates the impact of transceiver hardware
imperfections on the RIS-assisted system ergodic capacity. In
more detail, the ergodic capacity is given as a function of ρs,
for different values of κ, assuming N= 10. In this ﬁgure,
continuous lines are used to denote Monte Carlo simulation
results, while for the ergodic capacity upper bound and ceiling
dashed and dash-dotted lines employed. As a benchmark,
the ideal case in which both the TX and RX are hardware
imperfection free, i.e. κ= 0, is also provided. We observe
that for the ideal case, as ρsincreases, the ergodic capacity
also increases. For example, for a ρsincrease from 0to
10 dB, the ergodic capacity increases from approximately 8
to 11 bits/s/Hz. On the other hand, as described in Lemma
1, in the case of non-ideal transceivers, the ergodic capacity
saturates to its ceiling as ρsincreases. As a consequence,
for a ﬁxed ρs, since the ergodic capacity ceiling is solitary
determined by the level of hardware imperfections, i.e. κtand
κr, both the ergodic capacity and its upper bound as well as the
ceiling increase as κdecreases. For example, for ρs= 5 dB,
as κdecreases from 0.2to 0.1, the ergodic capacity increases
from 3.73 to 5.56 bit/s/Hz, while, for the same κvariation,
the upper bound changes from 3.73 to 5.66 and the ceiling
from 3.75 to 5.67. This indicates the detrimental effect of
hardware imperfections on the system’s ergodic capacity. In
other words, it highlights the importance of accurately mod-
eling the transceiver hardware imperfections, when assessing
the ergodic performance of RIS-assisted systems. Finally, from
this ﬁgure, it becomes evident that, for practical values of
ρs, as κincreases, the upper-bound becomes tighter. As a
consequence, for practical values of ρs, the upper bound
derived in (22) can be used as a tight simpliﬁed approximation.
The accuracy of this approximation increases as the level of
hardware imperfections increases.
Figure 7 depicts the ergodic capacity as a function of κtand
κrfor different values of N, assuming ρs= 20 dB. In more
detail, Fig. 7.a delivers the ergodic capacity for N= 10, while
Fig. 7.b the one for N= 100. As expected, for given κrand
N, as κtincreases, the level of the hardware imperfections at
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(a)
(b)
Fig. 7. Ergodic capacity vs κtand κr, for (a) N= 10, (b) N= 100,
assuming ρs= 20 dB.
the TX side increases; hence, the ergodic capacity decreases.
For example, for κr= 0.1and N= 10, as κtincreases
from 0.1to 0.2, the ergodic capacity decreases from 5.67 to
4.39 bit/s/Hz, which corresponds to approximately 22.6%
ergodic capacity degradation, whereas, for κr= 0.1and
N= 100, the same κtchange causes the same ergodic
capacity degradation. Similarly, for ﬁxed κtand N, as κr
increases, the ergodic capacity decreases. In more detail, we
observe that the reciprocity also holds for the case of ergodic
capacity. This mean that regardless of whether the level of
hardware imperfections changes in the TX or RX, it will cause
the same effect on the RIS-assisted system ergodic capacity.
Finally, by comparing Figs. 7.a and 7.b, we observe that for the
case of hardware imperfection-free transceivers, the ergodic
capacity signiﬁcantly increases as Nincreases. However, in
realistic implementations, the level of hardware imperfections
and not the number of the RIS’s RUs determines the system’s
ergodic capacity performance.
In Fig. 8, the ergodic capacity is provided against N, for
different values of κ, assuming ρs= 20 dB. For the sake of
comparison, the ideal case in which κ= 0 is also plotted.
From this ﬁgure, we observe that in the case in which the
Fig. 8. Ergodic capacity vs N, for different values of κ, assuming ρs=
20 dB.
transceivers experience the impact of hardware imperfections,
as the number of RIS’s RUs increases, the ergodic capacity
saturates and approaches log21
κ2
t+κ2
r. In other words, a
speciﬁc number of RUs exists beyond which no ergodic
capacity gain will be observed as Nincreases.
V. CONCLUSIONS
In this contribution, we considered a generalized hardware
imperfections model, which has been validated in several prior
works, in order to assess their impact on RIS-assisted wireless
systems. In this direction, we extracted simple closed-form
expressions for their outage probability and a novel upper
bound for their ergodic capacity, which takes into account the
level of transceivers hardware imperfections, the number of
RUs at the RIS, as well as the transmission SNR and the
spectral efﬁciency of the transmission scheme. Our results
manifested the detrimental impact of transceiver hardware
imperfections on the outage and ergodic capacity performance
of these systems. In more detail, they revealed that there exists
a speciﬁc spectral efﬁciency limit, which sorely depends on
the level of transceiver hardware imperfections, after with the
outage probability becomes 1. Moreover, the importance of
accurately modeling the level of transceiver hardware imper-
fections when evaluating the performance of such systems
is reported. Likewise, it is highlighted that there exists a
capacity ceiling that is independent of the number of RIS’s
RUs; however, it is determined by the TX and RX EVMs.
This ceiling cannot be crossed by increasing the transmission
SNR or altering the propagation medium characteristics. This
is an RIS-assisted wireless system constraint that is expected
to inﬂuence future designs.
ACKNOWLEDGMENT
The authors would like to thank the editor and anonymous
reviewers for their constructive comments and criticism.
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APPENDICES
APPENDIX A
PROO F OF THE ORE M 1
The outage probability is deﬁned as
Po= Pr (ρρth),(24)
which, by substituting (14), can be rewritten as
Po= Pr |A|2
(κ2
t+κ2
r)|A|2+1
ρsρth!,(25)
or
Po= Pr |A|21κ2
t+κ2
rρthρth
ρs.(26)
For 1κ2
t+κ2
rρth 0, or equivalently
ρth 1
κ2
t+κ2
r
,(27)
the outage probability can be written as
Po= Pr A1
p1(κ2
t+κ2
r)ρth rρth
ρs!,(28)
or
Po=FA 1
p1(κ2
t+κ2
r)ρth rρth
ρs!,(29)
where FA(·)is the cumulative density function (CDF) of A.
Next, by employing [11, eq. (8)], we can obtain the ﬁrst branch
of (16)
For 1κ2
t+κ2
rρth <0,
|A|21κ2
t+κ2
rρthis always no-positive; hence,
Pr |A|21κ2
t+κ2
rρthρth
ρs= 1, or, based on (26),
Po= 1.(30)
APPENDIX B
PROO F OF THE ORE M 2
The ergodic capacity can be deﬁned as
C=E[log2(1 + ρ)] ,(31)
which can be equivalently written as
C=Ehlog21 + a
bi,(32)
where
a=|A|2,(33)
and
b=κ2
t+κ2
r|A|2+1
ρs
.(34)
We note that the function log21 + a
(κ2
t+κ2
r)a+1
ρsis concave
of a, for a0, since its second derivative is
1
ln(2)
1
ρs
2aκ2
t+κ2
r1 + κ2
t+κ2
r+1
ρs+ 2κ2
t+κ2
r
ρs
a(κ2
t+κ2
r) + 1
ρs2a+a(κ2
t+κ2
r) + 1
ρs2<0.
(35)
As a consequence, the Jensen’s inequality holds [43], and (32)
can be upper-bounded as
Clog21 + Eha
bi.(36)
However, based on [44, Eq. (35)],
log21 + Eha
bilog21 + E[a]
E[b].(37)
By combining (36) and (37), we obtain
Clog21 + A
B,(38)
where
A=E[a],(39)
and
B=E[b].(40)
Of note, the same approach has been employed in several prior
works including [44] and references therein.
Next, we provide closed-form expressions for (39) and (40).
Based on [45], (39) can be computed as
A=Z
0
x2fA(x) dx,(41)
where fA(x)is the probability density function (PDF) of A1.
With the aid of [11, eq. (7)], (41) can be rewritten as
A=1
16π2
2π2
16π2Γ 2
16π2I,(42)
with
I=Z
0
x2
16π2+1 exp 2π
16 π2xdx.(43)
By setting t=2π
16π2and then employing [29, eq.
(8.310/1)], (43) can be expressed in closed-form as
I=16 π2
2π2
16π2+2
ΓNπ2
16 π2+ 2.(44)
By substituting (44) into (42), we extract
A=16 π2
2π2Γ 2
16π2+ 2
Γ 2
16π2,(45)
or, by employing [],
A=16 π2
2π2Nπ2
16 π22
.(46)
From (34), (40) can be equivalently expressed as
B=Eκ2
t+κ2
r|A|2+1
ρs,(47)
1Note that the PDF, which was provided in [11, eq. (7)] is an extremely
tight approximation with an error that is lower than 106; thus, it can be
used for the evaluation of the upper bound.
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which, according to [45], can be rewritten as
B=κ2
t+κ2
rE|A|2+1
ρs
,(48)
or, with the aid of (33) and (39),
B=κ2
t+κ2
rA+1
ρs
.(49)
By employing (46), (49) can be evaluated as
B=κ2
t+κ2
r16 π2
2π2Nπ2
16 π22
+1
ρs
.(50)
Finally, by substituting (46) and (50) into (38), we ob-
tain (22). This concludes the proof.
APPENDIX C
PROO F OF LEMMA 1
From (22), the ergodic capacity ceiling can be obtained as
in (51), given at the top of the next page. Due to the fact
that, as ρstends to inﬁnity, 1
ρstends to zero, (51) can be
evaluated as in (23).
Similarly, for Ntends to inﬁnity, the ergodic capacity
ceiling can be expressed as
CN= lim
N→∞
C, (52)
which, by employing (22) can be rewritten as
CN= lim
N→∞ log2 1 + K(N)
(κ2
t+κ2
r)K(N) + 1
ρs!!,(53)
where
K(N) = 16 π2
2π2Nπ2
16 π22
.(54)
As Ntends to inﬁnity, K(N)also tends to inﬁnity. Therefore,
by applying the de L’ Hospital rule in (53) and after some alge-
braic manipulations, we extract (23). This concludes the proof.
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Alexandros-Apostolos A. Boulogeorgos (S’11,
M’16, SM’19) was born in Trikala, Greece in 1988.
He received the Electrical and Computer Engineer-
ing (ECE) diploma degree and Ph.D. degree in
Wireless Communications from the Aristotle Uni-
versity of Thessaloniki (AUTh) in 2012 and 2016,
respectively.
From November 2012, he has been a member of
the wireless communications system group of AUTh,
working as a research assistant/project engineer in
various telecommunication and networks projects.
During 2017, he joined the information technologies institute, while from
November 2017, he has joined the Department of Digital Systems, University
of Piraeus, where he conducts research in the area of wireless communications.
Moreover, from October 2012 until September 2016, he was a teaching
assistant at the department of ECE of AUTh, whereas, from February 2017,
he serves as an adjunct lecturer at the Department of Informatics and
Telecommunications Engineering of the University of Western Macedonia and
as an visiting lecturer at the Department of Computer Science and Biomedical
Informatics of the University of Thessaly.
Dr. Boulogeorgos has authored and co-authored more than 45 technical
papers, which were published in scientiﬁc journals and presented at prestigious
international conferences. Furthermore, he has submitted two (one national
and one European) patents. Likewise, he has been involved as member of
Technical Program Committees in several IEEE and non-IEEE conferences
and served as a reviewer in various IEEE journals and conferences. Dr.
Boulogeorgos was awarded with the “Distinction Scholarship Award” of the
Research Committee of AUTh for the year 2014 and was recognized as an
exemplary reviewer for IEEE Communication Letters for 2016 (top 3% of
reviewers). Moreover, he was named a top peer reviewer (top 1% of reviewers)
in Cross-Field and Computer Science in the Global Peer Review Awards 2019,
which was presented by the Web of Science and Publons. His current research
interests spans in the area of wireless communications and networks with
emphasis in high frequency communications, optical wireless communications
and communications for biomedical applications. He is a Senior Member of
the IEEE and a member of the Technical Chamber of Greece.
Angeliki Alexiou received the Diploma in Electrical
and Computer Engineering from the National Tech-
nical University of Athens in 1994 and the PhD in
Electrical Engineering from Imperial College of Sci-
ence, Technology and Medicine, University of Lon-
don in 2000. Since May 2009 she is faculty member
at the Department of Digital Systems, University of
Piraeus, where she conducts research and teaches
undergraduate and postgraduate courses in the area
of Broadband Communications and Advanced Wire-
less Technologies. Prior to this appointment she was
with Bell Laboratories, Wireless Research, Lucent Technologies, now Alcatel-
Lucent, in Swindon, UK, ﬁrst as a member of technical staff (January 1999-
February 2006) and later as a Technical Manager (March 2006-April 2009).
Prof 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. Prof Alexiou is the
elected Chair of the Working Group on Radio Communication Technologies
of the Wireless World Research Forum. She is a member of the IEEE and
the Technical Chamber of Greece. Her current research interests include
radio interface, 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, and
‘cell-less’ architectures based on softwarization, virtualization and extreme
resources sharing. She is the project coordinator of the H2020 TERRANOVA
project (ict-terranova.eu).
... where κ s and κ d are the S transmitter and UAV receiver error vector magnitudes, while P s stands for the average transmitted power. According to [38], [39], [43], κ s and κ d in highfrequency systems, such as millimeter wave and terahertz (THz), are in the range of [0.07, 0.3]. Finally, for the special case in which both the S and UAV are equipped with ideal transceivers, κ s = κ d = 0 [56]. ...
... For the special case in which disorientation and misalignment are present and k A ̸ = j m A as well as m A ̸ = j k A with j = 1, 2, · · · , the OP can be evaluated as in (43), given at the top of the next page. Moreover, the following lemma returns a high-SNR approximation for the OP. ...
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