Received 12 June 2020; revised 14 July 2020; accepted 29 July 2020. Date of publication 5 August 2020; date of current version 26 August 2020.
Digital Object Identifier 10.1109/OJCOMS.2020.3014331
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)
Department of Digital Systems, University of Piraeus, 18534 Piraeus, Greece
CORRESPONDING AUTHOR: A.–A. A. BOULOGEORGOS (e-mail: al. )
This work was supported from the European Commission’s Horizon 2020 Research and Innovation Programme under Grant 871464 (ARIADNE).
ABSTRACT 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) havebeen 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 capa-
bilities 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.,
relays [9]–[11], and provided optimum information and/or
power transfer policies [12]–[15], can be identified. In par-
ticular, 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) chan-
nels 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
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BOULOGEORGOS AND ALEXIOU: HOW MUCH DO HARDWARE IMPERFECTIONS AFFECT PERFORMANCE RIS-ASSISTED SYSTEMS?
relays in terms of energy efficiency, while in [11], RIS-
assisted systems were compared against the corresponding
relay ones in terms of 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 vehicle 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 imperfections, which cause in-phase and quadra-
ture 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 imper-
fections in RIS-assisted systems was bounded. However,
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 hardware
imperfections on the outage performance of RIS-assisted
wireless systems and quantifies its ergodic capacity, under
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 degra-
dation 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 probabil-
ity 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
account 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 proba-
bility 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
absolute value, and the probability of the event A. Moreover,
limx→z(f (x)) returns the limit of f (x) as x tends to z.
Additionally, �(·), �(·, ·) 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)n represents the Pochhammer
operator [30, eq. (19)].
II. SYSTEM 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 N RUs. 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 i−th RU and RX, gi, are assumed to be indepen-
dent 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 actu-
ally generated. As a result, the actual transmitted signal can
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FIGURE 1. System model of the RIS-assisted wireless system. Note that in this figure s̃ represents the baseband equivalent of the transmitted signal.
be described as
s̃ = s+ nt, (1)
where nt represents the distortion from the TX hardware
imperfections, and can be modeled as a zero-mean complex
Gaussian process with variance
σ
2
t = κ2t Ps. (2)
In (2), κt stands for the TX’s error vector magnitude (EVM),
which is in-general a non-negative design parameter, while
Ps represents 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∑
i=1
higipis̃+ nr + n, (3)
where nr is the distortion from the RX hardware imperfec-
tions, and can be modeled as a zero-mean complex Gaussian
process with variance
σ
2
r = κ2r |A|2Ps, (4)
with
A =
N∑
i=1
|hi||gi|, (5)
being the equivalent TX-RIS-RX channel. In (4), κr rep-
resents the RX’s EVM. Note that this model has been
validated by several analytical and experimental prior works,
including [21], [34]–[40] and references therein.
Likewise, n stands for the white Gaussian noise (AWGN)
and can be modeled as a zero-mean complex Gaussian pro-
cess with variance No. Moreover, pi represents the i−th RU
response and can be obtained as
pi = |pi| exp(jφi), (6)
with φi standing for the phase shift that is applied by the i−th
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 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 = As̃+ nr + n. (9)
Finally, by substituting (1) into (9), the baseband equivalent
received signal can be rewritten as
r = As+ w+ n, (10)
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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
σ
2
w = |A|2
(
κ
2
t + κ2r
)
Ps. (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 hard-
ware imperfections, for κt = κr = 0. In this case, from (12),
σ 2w = 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 expres-
sion 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
obtained as
ρ = |A|
2Ps(
κ2t + κ2r
)|A|2Ps + No
, (13)
or equivalently
ρ = |A|
2
(
κ2t + κ2r
)|A|2 + 1
ρs
, (14)
where
ρs =
Ps
No
, (15)
denotes the transmission SNR.
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
bottom of the 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 out-
age probability is always 1 for r > log2(
1
κ2t +κ2r
+ 1). This
means that the spectral efficiency of the transmission scheme
is limited by the levels of the transceivers hardware imper-
fections. 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
Pido =
γ
(
π2
16−π2 N,
2π
16−π2
√
ρth
ρs
)
�
(
π2
16−π2 N
) , (18)
which is the same as [11, eq. (31)].
Po =
⎧
⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
γ
⎛
⎝ π
2
16−π2 N,
2π
16−π2
1√
1−(κ2t +κ2r
)
ρth
√
ρth
ρs
⎞
⎠
�
(
π2
16−π2 N
) , ρth ≤ 1
κ2t +κ2r
1, otherwise,
(16)
D = − lim
ρs→∞
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝
log2
⎛
⎜⎜⎜
⎝
γ
⎛
⎝ π
2
16−π2 N,
2π
16−π2
1√
1−(κ2t +κ2r
)
ρth
√
ρth
ρs
⎞
⎠
�
(
π2
16−π2 N
)
⎞
⎟⎟⎟
⎠
log2(ρs)
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠
(20)
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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
κ2t +κ2r
, can be rewritten as in (20), given at the bottom of
the previous page. After evaluating the limit in (20), we
obtain
D = π
2
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
wireless systems can be upper bounded as
C ≤ log2
⎛
⎜
⎝1 +
(
16−π2
2π
)2(
Nπ2
16−π2
)
2
(
κ2t + κ2r
)(
16−π2
2π
)2(
Nπ2
16−π2
)
2
+ 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-N ergodic capacity ceiling.
Lemma 1 (High-SNR and High-N Ergodic Capacity
Ceiling): As the transmission SNR tends to infinity or as the
number of RIS’s RUs tends to infinity, the ergodic capacity
is constrained by
lim
ρs→∞
C = log2
(
1 + 1
κ2t + κ2r
)
. (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-N regimes,
the performance of the system is independent from the num-
ber of RUs at the RIS and are fully determined by the level
of imperfections.
IV. RESULTS & DISCUSSION
This section aims at verifying the theoretical framework pro-
vided 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 respec-
tively denote theoretical and simulation results. Moreover,
we define κ = κt = κr.
Figure 2 demonstrates the outage probability as a function
of ρs for different values of N and ρth, assuming κ = 0.1.
FIGURE 2. Outage probability vs ρs for different values of N and 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 ρs increase, the system’s out-
age 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 N that, 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 N increases from 1 to 10, a 3 orders decrease occurs
on the outage probability. Finally, this figures reveals that
there exists a trade-off between RIS-assisted system spec-
tral efficiency and power consumption. In more detail, we
observe that, for a fixed N and a predetermined outage prob-
ability 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 , assum-
ing 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 ρs and κ , an outage performance degradation is observed,
as ρth increases. For instance, for κ = 0.07 and ρs = −5 dB,
the outage probability increases more than 100 times as the
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FIGURE 3. Outage probability vs ρs , for different values of ρth and κ ,
assuming N = 5.
ρth increases from 0 to 10 dB. Likewise, for given ρs and ρth,
as κ increases, the outage performance degrades. For exam-
ple, for ρs = ρth = 10 dB, as κ increases from 0 to 0.15, the
outage probability decreases for about 10 times. This indi-
cates 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
κ increases from 0 to 0.2, the outage probability increases
from 0.017 to 0.021, which is translated into a 23.5% out-
age performance degradation, while, for the same ρs and
for ρth = 10 dB, the same κ increase results to an out-
age probability increase from 0.22 to 0.94, i.e., the outage
probability increases for approximately 3 times. Similarly,
as ρs increases, 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 prob-
ability increases for one order of magnitude as κ increases
from 0 to 0.2, whereas, for the same ρth and ρs = 10 dB,
the outage probability increases for more than 100 times, as
κ increases from 0 to 0.2. To sum up, this figure reveals
that there exist a relationship between the transmission SNR,
transmission scheme spectral efficiency and level of hard-
ware 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 dif-
ferent number of RUs. In more detail, the outage probability
is plotted as a function of the transmission SNR, for dif-
ferent values of N and 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 imperfections have a detrimental impact on the
RIS-assisted system performance. In particular, we observe
that for a given N, as κ increases, the transmission SNR
FIGURE 4. Outage probability vs ρs , for different values of N and κ , assuming
ρth = 10 dB.
FIGURE 5. Outage probability vs κt and κr , assuming N = 5, ρs = ρth = 10 dB.
should be significantly increase in order for a predetermined
outage probability requirement to be satisfied. For instance,
for N = 100 and an outage probability requirement of 10−6,
the transmission SNR should be increased approximately
6 dB, if κ increases from 0 to 0.17. Similarly, for N = 10
and κ variation, approximately the same transmission SNR
increase is required to guarantee a 10−6 outage 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 κt and κr, assuming N = 5, and ρs = ρth = 10 dB. From
this figure, we observe that, for a given κr, as κt increases,
the outage probability also increases. Similarly, for a fixed
κt, as κr increases, the system outage performance degrades.
For example, for κt = 0.1, as κr increases from 0.1 to 0.2, the
outage probability increases from 2.43×10−5 to 1.23×10−4.
Similarly, for κr = 0.1, as κr increases from 0.1 to 0.2,
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FIGURE 6. Ergodic capacity vs ρs , for different values of κ , assuming N = 10.
the outage probability also changes from 2.43 × 10−5 to
1.23 × 10−4. These examples reveal the reciprocal nature
of TX and RX hardware imperfections. Finally, it is evident
that when the ρth <
1
κ2t +κ2r
is violated, the outage probability
becomes equal to 1.
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 sim-
ulation 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 ρs increases, the ergodic
capacity also increases. For example, for a ρs increase from
0 to 10 dB, the ergodic capacity increases from approxi-
mately 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 ρs increases. As a con-
sequence, for a fixed ρs, since the ergodic capacity ceiling
is solitary determined by the level of hardware imperfec-
tions, i.e., κt and κr, both the ergodic capacity and its upper
bound as well as the ceiling increase as κ decreases. For
example, for ρs = 5 dB, as κ decreases from 0.2 to 0.1,
the ergodic capacity increases from 3.73 to 5.56 bit/s/Hz,
while, for the same κ variation, 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 modeling the transceiver hard-
ware imperfections, when assessing the ergodic performance
of RIS-assisted systems. Finally, from this figure, it becomes
evident that, for practical values of ρs, as κ increases, the
upper-bound becomes tighter. As a consequence, for prac-
tical values of ρs, the upper bound derived in (22) can
FIGURE 7. Ergodic capacity vs κt and κr , for (a) N = 10, (b) N = 100, assuming
ρs = 20 dB.
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 κt
and κr for 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 κr and N, as κt increases, the level of the hardware
imperfections at the TX side increases; hence, the ergodic
capacity decreases. For example, for κr = 0.1 and N = 10,
as κt increases from 0.1 to 0.2, the ergodic capacity decreases
from 5.67 to 4.39 bit/s/Hz, which corresponds to approx-
imately 22.6% ergodic capacity degradation, whereas, for
κr = 0.1 and N = 100, the same κt change causes the same
ergodic capacity degradation. Similarly, for fixed κt and 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
VOLUME 1, 2020 1191
BOULOGEORGOS AND ALEXIOU: HOW MUCH DO HARDWARE IMPERFECTIONS AFFECT PERFORMANCE RIS-ASSISTED SYSTEMS?
FIGURE 8. Ergodic capacity vs N , for different values of κ , assuming ρs = 20 dB.
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
N increases. 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 transceivers experience the impact of hardware
imperfections, as the number of RIS’s RUs increases, the
ergodic capacity saturates and approaches log2(
1
κ2t +κ2r
). In
other words, a specific number of RUs exists beyond which
no ergodic capacity gain will be observed as N increases.
V. CONCLUSION
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 transmis-
sion 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 imperfections when evaluating the
performance of such systems is reported. Likewise, it is high-
lighted 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 propaga-
tion medium characteristics. This is an RIS-assisted wireless
system constraint that is expected to influence future designs.
APPENDIX A
PROOF OF THEOREM 1
The outage probability is defined as
Po = Pr(ρ ≤ ρth), (24)
which, by substituting (14), can be rewritten as
Po = Pr
(
|A|2
(
κ2t + κ2r
)|A|2 + 1
ρs
≤ ρth
)
, (25)
or
Po = Pr
(
|A|2
(
1 −
(
κ
2
t + κ2r
)
ρth
)
≤ ρth
ρs
)
. (26)
For 1 − (κ2t + κ2r )ρth ≥ 0, or equivalently
ρth ≤
1
κ2t + κ2r
, (27)
the outage probability can be written as
Po = Pr
⎛
⎝A ≤ 1√
1 − (κ2t + κ2r
)
ρth
√
ρth
ρs
⎞
⎠, (28)
or
Po = FA
⎛
⎝ 1√
1 − (κ2t + κ2r
)
ρth
√
ρth
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 1− (κ2t +κ2r )ρth < 0, |A|2(1− (κ2t +κ2r )ρth) is always
no-positive; hence, Pr(|A|2(1 − (κ2t + κ2r )ρth) ≤ ρthρs ) = 1, or,
based on (26),
Po = 1. (30)
APPENDIX B
PROOF OF THEOREM 2
The ergodic capacity can be defined as
C = E[log2(1 + ρ)
]
, (31)
which can be equivalently written as
C = E
[
log2
(
1 + a
b
)]
, (32)
where
a = |A|2, (33)
and
b =
(
κ
2
t + κ2r
)
|A|2 + 1
ρs
. (34)
1192 VOLUME 1, 2020
We note that the function log2(1 + a(κ2t +κ2r )a+ 1ρs
) is concave
of a, for a ≥ 0, since its second derivative is
− 1
ln(2)
1
ρs
2a
(
κ2t + κ2r
)(
1 + κ2t + κ2r
)+ 1
ρs
+ 2 κ2t +κ2r
ρs(
a
(
κ2t + κ2r
)+ 1
ρs
)2(
a+ a(κ2t + κ2r
)+ 1
ρs
)2 < 0.
(35)
As a consequence, the Jensen’s inequality holds [43],
and (32) can be upper-bounded as
C ≤ log2
(
1 + E
[a
b
])
. (36)
However, based on [44, eq. (35)],
log2
(
1 + E
[a
b
])
≈ log2
(
1 + E[a]
E[b]
)
. (37)
By combining (36) and (37), we obtain
C ≤ log2
(
1 + AB
)
, (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 =
∫ ∞
0
x2fA(x) dx, (41)
where fA(x) is the probability density function (PDF) of A.
1
With the aid of [11, eq. (7)], (41) can be rewritten as
A = 1(
16−π2
2π
) Nπ2
16−π2
�
(
Nπ2
16−π2
)I, (42)
with
I =
∫ ∞
0
x
Nπ2
16−π2 +1 exp
(
− 2π
16 − π2 x
)
dx. (43)
By setting t = 2π
16−π2 and then employing [29, eq.
(8.310/1)], (43) can be expressed in closed-form as
I =
(
16 − π2
2π
) Nπ2
16−π2 +2
�
(
Nπ2
16 − π2 + 2
)
. (44)
1. Note that the PDF, which was provided in [11, eq. (7)] is an extremely
tight approximation with an error that is lower than 10−6; thus, it can be
used for the evaluation of the upper bound.
By substituting (44) into (42), we extract
A =
(
16 − π2
2π
)2 �
(
Nπ2
16−π2 + 2
)
�
(
Nπ2
16−π2
) , (45)
or, by employing ,
A =
(
16 − π2
2π
)2(
Nπ2
16 − π2
)
2
. (46)
From (34), (40) can be equivalently expressed as
B = E
[(
κ
2
t + κ2r
)
|A|2 + 1
ρs
]
, (47)
which, according to [45], can be rewritten as
B =
(
κ
2
t + κ2r
)
E
[
|A|2
]
+ 1
ρs
, (48)
or, with the aid of (33) and (39),
B =
(
κ
2
t + κ2r
)
A + 1
ρs
. (49)
By employing (46), (49) can be evaluated as
B =
(
κ
2
t + κ2r
)(16 − π2
2π
)2(
Nπ2
16 − π2
)
2
+ 1
ρs
. (50)
Finally, by substituting (46) and (50) into (38), we
obtain (22). This concludes the proof.
APPENDIX C
PROOF OF LEMMA 1
From (22), the ergodic capacity ceiling can be obtained as
in (51), given at the bottom of the page. Due to the fact
that, as ρs tends to infinity,
1
ρs
tends to zero, (51) can be
evaluated as in (23).
Similarly, for N tends 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)(
κ2t + κ2r
)
K(N) + 1
ρs
))
, (53)
where
K(N) =
(
16 − π2
2π
)2(
Nπ2
16 − π2
)
2
. (54)
As N tends to infinity, K(N) also tends to infinity. Therefore,
by applying the de L’ Hospital rule in (53) and after some
algebraic manipulations, we extract (23). This concludes
the proof.
lim
ρs→∞
C ≤ lim
ρs→∞
⎛
⎜
⎝log2
⎛
⎜
⎝1 +
(
16−π2
2
)2(
Nπ2
16−π2
)
2
(
κ2t + κ2r
)(
16−π2
2
)2(
Nπ2
16−π2
)
2
+ 1
ρs
⎞
⎟
⎠
⎞
⎟
⎠ (51)
VOLUME 1, 2020 1193
BOULOGEORGOS AND ALEXIOU: HOW MUCH DO HARDWARE IMPERFECTIONS AFFECT PERFORMANCE RIS-ASSISTED SYSTEMS?
ACKNOWLEDGMENT
The authors would like to thank the Editor and anonymous
reviewers for their constructive comments and criticism.
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ALEXANDROS–APOSTOLOS A.
BOULOGEORGOS (Senior Member, IEEE)
was born in Trikala, Greece, in 1988. He
received the Electrical and Computer Engineering
(ECE) Diploma and Ph.D. degrees in wireless
communications from the Aristotle University
of Thessaloniki (AUTh) in 2012 and 2016,
respectively.
From November 2012, he has been a Member
of the Wireless Communications System Group,
AUTh, working as a Research Assistant/Project
Engineer in various telecommunication and networks projects. In 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.
From October 2012 until September 2016, he was a Teaching Assistant
with the Department of ECE, AUTh, whereas, from February 2017,
he serves as an Adjunct Lecturer with the Department of Informatics
and Telecommunications Engineering, University of Western Macedonia
and as an Visiting Lecturer with the Department of Computer Science
and Biomedical Informatics, University of Thessaly. He has authored
and coauthored 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. His current research interests spans in the area of wireless commu-
nications and networks with emphasis in high frequency communications,
optical wireless communications and communications for biomedical
applications.
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). 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.
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. He is a
member of the Technical Chamber of Greece.
ANGELIKI ALEXIOU (Member, IEEE) received the
Diploma degree in electrical and computer engi-
neering from the National Technical University
of Athens in 1994 and the Ph.D. degree 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 with the Department
of Digital Systems, University of Piraeus, 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, currently Alcatel-Lucent, Swindon, U.K., first as a Member
of Technical Staff (January 1999 to February 2006) and later as a Technical
Manager (March 2006 to April 2009). She is the Project Coordinator of
the H2020 TERRANOVA project (ict-terranova.eu). 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, vir-
tualization and extreme resources sharing. He is a co-recipient of Bell Labs
Presidentś 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. He 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.
VOLUME 1, 2020 1195
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/BookAntiqua
/BookAntiqua-Bold
/BookAntiqua-BoldItalic
/BookAntiqua-Italic
/BookmanOldStyle
/BookmanOldStyle-Bold
/BookmanOldStyle-BoldItalic
/BookmanOldStyle-Italic
/BookshelfSymbolSeven
/Century
/CenturyGothic
/CenturyGothic-Bold
/CenturyGothic-BoldItalic
/CenturyGothic-Italic
/CenturySchoolbook
/CenturySchoolbook-Bold
/CenturySchoolbook-BoldItalic
/CenturySchoolbook-Italic
/ComicSansMS
/ComicSansMS-Bold
/CourierNewPS-BoldItalicMT
/CourierNewPS-BoldMT
/CourierNewPS-ItalicMT
/CourierNewPSMT
/EstrangeloEdessa
/FranklinGothic-Medium
/FranklinGothic-MediumItalic
/Garamond
/Garamond-Bold
/Garamond-Italic
/Gautami
/Georgia
/Georgia-Bold
/Georgia-BoldItalic
/Georgia-Italic
/Haettenschweiler
/Helvetica
/Helvetica-Bold
/HelveticaBolditalic-BoldOblique
/Helvetica-BoldOblique
/Impact
/Kartika
/Latha
/LetterGothicMT
/LetterGothicMT-Bold
/LetterGothicMT-BoldOblique
/LetterGothicMT-Oblique
/LucidaConsole
/LucidaSans
/LucidaSans-Demi
/LucidaSans-DemiItalic
/LucidaSans-Italic
/LucidaSansUnicode
/Mangal-Regular
/MicrosoftSansSerif
/MonotypeCorsiva
/MSReferenceSansSerif
/MSReferenceSpecialty
/MVBoli
/PalatinoLinotype-Bold
/PalatinoLinotype-BoldItalic
/PalatinoLinotype-Italic
/PalatinoLinotype-Roman
/Raavi
/Shruti
/Sylfaen
/SymbolMT
/Tahoma
/Tahoma-Bold
/Times-Bold
/Times-BoldItalic
/Times-Italic
/TimesNewRomanMT-ExtraBold
/TimesNewRomanPS-BoldItalicMT
/TimesNewRomanPS-BoldMT
/TimesNewRomanPS-ItalicMT
/TimesNewRomanPSMT
/Times-Roman
/Trebuchet-BoldItalic
/TrebuchetMS
/TrebuchetMS-Bold
/TrebuchetMS-Italic
/Tunga-Regular
/Verdana
/Verdana-Bold
/Verdana-BoldItalic
/Verdana-Italic
/Vrinda
/Webdings
/Wingdings2
/Wingdings3
/Wingdings-Regular
/ZapfChanceryITCbyBT-MediumItal
/ZWAdobeF
]
/NeverEmbed [ true
]
/AntiAliasColorImages false
/CropColorImages true
/ColorImageMinResolution 200
/ColorImageMinResolutionPolicy /OK
/DownsampleColorImages false
/ColorImageDownsampleType /Average
/ColorImageResolution 300
/ColorImageDepth -1
/ColorImageMinDownsampleDepth 1
/ColorImageDownsampleThreshold 1.50000
/EncodeColorImages true
/ColorImageFilter /DCTEncode
/AutoFilterColorImages false
/ColorImageAutoFilterStrategy /JPEG
/ColorACSImageDict <<
/QFactor 0.76
/HSamples [2 1 1 2] /VSamples [2 1 1 2]
>>
/ColorImageDict <<
/QFactor 0.76
/HSamples [2 1 1 2] /VSamples [2 1 1 2]
>>
/JPEG2000ColorACSImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 15
>>
/JPEG2000ColorImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 15
>>
/AntiAliasGrayImages false
/CropGrayImages true
/GrayImageMinResolution 200
/GrayImageMinResolutionPolicy /OK
/DownsampleGrayImages false
/GrayImageDownsampleType /Average
/GrayImageResolution 300
/GrayImageDepth -1
/GrayImageMinDownsampleDepth 2
/GrayImageDownsampleThreshold 1.50000
/EncodeGrayImages true
/GrayImageFilter /DCTEncode
/AutoFilterGrayImages false
/GrayImageAutoFilterStrategy /JPEG
/GrayACSImageDict <<
/QFactor 0.76
/HSamples [2 1 1 2] /VSamples [2 1 1 2]
>>
/GrayImageDict <<
/QFactor 0.76
/HSamples [2 1 1 2] /VSamples [2 1 1 2]
>>
/JPEG2000GrayACSImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 15
>>
/JPEG2000GrayImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 15
>>
/AntiAliasMonoImages false
/CropMonoImages true
/MonoImageMinResolution 400
/MonoImageMinResolutionPolicy /OK
/DownsampleMonoImages false
/MonoImageDownsampleType /Bicubic
/MonoImageResolution 600
/MonoImageDepth -1
/MonoImageDownsampleThreshold 1.50000
/EncodeMonoImages true
/MonoImageFilter /CCITTFaxEncode
/MonoImageDict <<
/K -1
>>
/AllowPSXObjects false
/CheckCompliance [
/None
]
/PDFX1aCheck false
/PDFX3Check false
/PDFXCompliantPDFOnly false
/PDFXNoTrimBoxError true
/PDFXTrimBoxToMediaBoxOffset [
0.00000
0.00000
0.00000
0.00000
]
/PDFXSetBleedBoxToMediaBox true
/PDFXBleedBoxToTrimBoxOffset [
0.00000
0.00000
0.00000
0.00000
]
/PDFXOutputIntentProfile (None)
/PDFXOutputConditionIdentifier ()
/PDFXOutputCondition ()
/PDFXRegistryName ()
/PDFXTrapped /False
/CreateJDFFile false
/Description <<
/CHS
/CHT
/DAN
/DEU
/ESP
/FRA
/ITA (Utilizzare queste impostazioni per creare documenti Adobe PDF adatti per visualizzare e stampare documenti aziendali in modo affidabile. I documenti PDF creati possono essere aperti con Acrobat e Adobe Reader 5.0 e versioni successive.)
/JPN
/KOR
/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken waarmee zakelijke documenten betrouwbaar kunnen worden weergegeven en afgedrukt. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
/NOR
/PTB
/SUO
/SVE
/ENU (Use these settings to create PDFs that match the “Recommended” settings for PDF Specification 4.01)
>>
>> setdistillerparams
<<
/HWResolution [600 600]
/PageSize [612.000 792.000]
>> setpagedevice