Content uploaded by Alexandros-Apostolos A. Boulogeorgos

Author content

All content in this area was uploaded by Alexandros-Apostolos A. Boulogeorgos on Aug 03, 2020

Content may be subject to copyright.

>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <1

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

>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <2

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,

limx→z(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 i−th 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 i−th RU

response and can be obtained as

pi=|pi|exp (jφi),(6)

with φistanding 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

>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <3

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.

>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <4

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

16−π2N, 2π

16−π2rρth

ρs

Γπ2

16−π2N,(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

C≤log2

1 + 16−π2

2π2Nπ 2

16−π22

(κ2

t+κ2

r)16−π2

2π2Nπ 2

16−π22+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,

>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <5

Po=

γ π2

16−π2N, 2π

16−π2

1

√1−(κ2

t+κ2

r)ρth rρth

ρs!

Γπ2

16−π2N, ρth ≤1

κ2

t+κ2

r

1,otherwise

,(16)

D=−lim

ρs→∞

log2

γ π2

16−π2N, 2π

16−π2

1

√1−(κ2

t+κ2

r)ρth rρth

ρs!

Γπ2

16−π2N

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 10−6, the transmission SNR

should be increased approximately 6 dB, if κincreases from 0

>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <6

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

a10−6outage 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 2.43×10−5to 1.23×10−4.

Similarly, for κr= 0.1, as κrincreases from 0.1to 0.2, the

outage probability also changes from 2.43 ×10−5to 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

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

>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <7

(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.

>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <8

APPENDICES

APPENDIX A

PROO F OF THE ORE M 1

The outage probability is deﬁned as

Po= Pr (ρ≤ρth),(24)

which, by substituting (14), can be rewritten as

Po= Pr |A|2

(κ2

t+κ2

r)|A|2+1

ρs≤ρth!,(25)

or

Po= Pr |A|21−κ2

t+κ2

rρth≤ρth

ρs.(26)

For 1−κ2

t+κ2

rρth ≥0, or equivalently

ρth ≤1

κ2

t+κ2

r

,(27)

the outage probability can be written as

Po= Pr A≤1

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 1−κ2

t+κ2

rρth <0,

|A|21−κ2

t+κ2

rρthis always no-positive; hence,

Pr |A|21−κ2

t+κ2

rρth≤ρth

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

bi,(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 a≥0, 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

C≤log21 + 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

C≤log21 + 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

16−π2

2πNπ2

16−π2ΓNπ 2

16−π2I,(42)

with

I=Z∞

0

xNπ2

16−π2+1 exp −2π

16 −π2xdx.(43)

By setting t=2π

16−π2and then employing [29, eq.

(8.310/1)], (43) can be expressed in closed-form as

I=16 −π2

2πNπ2

16−π2+2

ΓNπ2

16 −π2+ 2.(44)

By substituting (44) into (42), we extract

A=16 −π2

2π2ΓNπ 2

16−π2+ 2

ΓNπ 2

16−π2,(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 10−6; thus, it can be

used for the evaluation of the upper bound.

>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <9

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.

REFERENCES

[1] S. Dang, O. Amin, B. Shihada, and M.-S. Alouini, “What should 6G

be?” Nature Electronics, vol. 3, no. 1, pp. 20–29, Jan. 2020.

[2] G. Gui, M. Liu, F. Tang, N. Kato, and F. Adachi, “6G: Opening new

horizons for integration of comfort, security and intelligence,” IEEE

Wireless Commun., pp. 1–7, Mar. 2020.

[3] M. di Renzo, M. Debbah, D.-T. Phan-Huy, A. Zappone, M.-S.

Alouini, C. Yuen, V. Sciancalepore, G. C. Alexandropoulos, J. Hoydis,

H. Gacanin, J. d. Rosny, A. Bounceur, G. Lerosey, and M. Fink, “Smart

radio environments empowered by reconﬁgurable ai meta-surfaces: An

idea whose time has come,” EURASIP Journal on Wireless Communi-

cations and Networking, vol. 2019, no. 1, pp. 1–20, May 2019.

[4] L. Dai, B. Wang, M. Wang, X. Yang, J. Tan, S. Bi, S. Xu, F. Yang,

Z. Chen, M. D. Renzo, C.-B. Chae, and L. Hanzo, “Reconﬁgurable

intelligent surface-based wireless communications: Antenna design, pro-

totyping, and experimental results,” IEEE Access, vol. 8, pp. 45 913–

45 923, Mar. 2020.

[5] C. Pan, H. Ren, K. Wang, W. Xu, M. Elkashlan, A. Nallanathan, and

L. Hanzo, “Intelligent reﬂecting surface for multicell MIMO communi-

cations,” IEEE Trans. Wireless Commun., May 2020.

[6] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M. Alouini, and

R. Zhang, “Wireless communications through reconﬁgurable intelligent

surfaces,” IEEE Access, vol. 7, pp. 116 753–116 773, Aug. 2019.

[7] V. C. Thirumavalavan and T. S. Jayaraman, “BER analysis of recon-

ﬁgurable intelligent surface assisted downlink power domain NOMA

system,” in International Conference on COMmunication Systems &

NETworkS (COMSNETS), Bengaluru, India, Jan. 2020.

[8] H. Zhang, B. Di, L. Song, and Z. Han, “Reconﬁgurable intelligent

surfaces assisted communications with limited phase shifts: How many

phase shifts are enough?” IEEE Trans. Veh. Technol., vol. 69, no. 4, pp.

4498 – 4502, Apr. 2020.

[9] M. di Renzo, K. Ntontin, J. Song, F. H. Danufane, X. Qian, F. Lazarakis,

J. de Rosny, D. T. Phan-Huy, O. Simeone, R. Zhang, M. Debbah,

G. Lerosey, M. Fink, S. Tretyakov, and S. Shamai, “Reconﬁgurable in-

telligent surfaces vs. relaying: Differences, similarities, and performance

comparison,” IEEE Open Journal of the Communications Society, vol. 1,

pp. 798 – 807, Jun. 2020.

[10] E. Bjornson, O. Ozdogan, and E. G. Larsson, “Intelligent reﬂecting

surface versus decode-and-forward: How large surfaces are needed to

beat relaying?” IEEE Wireless Commun. Lett., vol. 9, no. 2, pp. 244–

248, Feb. 2020.

[11] A.-A. A. Boulogeorgos and A. Alexiou, “Performance analysis of recon-

ﬁgurable intelligent surface-assisted wireless systems and comparison

with relaying,” IEEE Access, vol. 8, pp. 94463–94 483, May 2020.

[12] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and

C. Yuen, “Reconﬁgurable intelligent surfaces for energy efﬁciency in

wireless communication,” IEEE Trans. Wireless Commun., vol. 18, no. 8,

pp. 4157–4170, Aug. 2019.

[13] B. Di, H. Zhang, L. Song, Y. Li, Z. Han, and H. V. Poor, “Hybrid

beamforming for reconﬁgurable intelligent surface based multi-user

communications: Achievable rates with limited discrete phase shifts,”

IEEE J. Sel. Areas Commun., pp. 1–1, Jun. 2020.

[14] S. Li, B. Duo, X. Yuan, Y.-C. Liang, and M. D. Renzo, “Reconﬁgurable

intelligent surface assisted UAV communication: Joint trajectory design

and passive beamforming,” IEEE Wireless Commun. Lett., pp. 1–1, Jan.

2020.

[15] C. Pan, H. Ren, K. Wang, M. Elkashlan, A. Nallanathan, J. Wang, and

L. Hanzo, “Intelligent reﬂecting surface enhanced MIMO broadcasting

for simultaneous wireless information and power transfer,” IEEE J. Sel.

Areas Commun., Jun. 2020.

[16] T. Schenk, RF Imperfections in High-Rate Wireless Systems:

Impact and Digital Compensation. The Netherlands: Springer,

2008. [Online]. Available: https://www.ebook.de/de/product/11431698/

tim schenk rf imperfections in high rate wireless systems.html

[17] A.-A. A. Boulogeorgos, V. M. Kapinas, R. Schober, and G. K. Kara-

giannidis, “I/Q-imbalance self-interference coordination,” IEEE Trans.

Wireless Commun., vol. 15, no. 6, pp. 4157 – 4170, Jun. 2016.

[18] A.-A. A. Boulogeorgos, P. C. Sofotasios, B. Selim, S. Muhaidat,

G. K. Karagiannidis, and M. Valkama, “Effects of RF impairments

in communications over cascaded fading channels,” IEEE Trans. Veh.

Technol., vol. 65, no. 11, pp. 8878 – 8894, Nov. 2016.

[19] A.-A. A. Boulogeorgos, “Interference mitigation techniques in modern

wireless communication systems,” Ph.D. dissertation, Aristotle Univer-

sity of Thessaloniki, Thessaloniki, Greece, Sep. 2016.

[20] A.-A. A. Boulogeorgos and G. K. Karagiannidis, “Energy detection in

full-duplex systems with residual RF impairments over fading channels,”

IEEE Wireless Commun. Lett., vol. 7, no. 2, pp. 246–249, Apr. 2018.

[21] X. Yang, M. Matthaiou, J. Yang, C. Wen, F. Gao, and S. Jin, “Hardware-

constrained millimeter-wave systems for 5G: Challenges, opportunities,

and solutions,” IEEE Commun. Mag., vol. 57, no. 1, pp. 44–50, Jan.

2019.

[22] E. Soleimani-Nasab, M. Matthaiou, M. Ardebilipour, and G. K. Kara-

giannidis, “Two-way AF relaying in the presence of co-channel inter-

ference,” IEEE Trans. Commun., vol. 61, no. 8, pp. 3156–3169, Aug.

2013.

[23] L. Anttila, M. Valkama, and M. Renfors, “Circularity-based I/Q im-

balance compensation in wideband direct-conversion receivers,” IEEE

Trans. Veh. Commun., vol. 57, no. 4, pp. 2099–2113, Jul. 2008.

[24] S. Zhou, W. Xu, K. Wang, M. Di Renzo, and M. Alouini, “Spectral and

energy efﬁciency of IRS-assisted MISO communication with hardware

impairments,” IEEE Wireless Commun. Lett., pp. 1–1, Apr. 2020.

[25] Y. Liu, E. Liu, and R. Wang, “Energy efﬁciency analysis of intelligent re-

ﬂecting surface system with hardware impairments,” arXiv:2004.09804,

Apr. 2020.

[26] S. Hu, F. Rusek, and O. Edfors, “Capacity degradation with modeling

hardware impairment in large intelligent surface,” in IEEE Global

Communications Conference (GLOBECOM). Abu Dhabi, United Arab

Emirates: IEEE, Dec. 2018.

[27] J. V. Alegria and F. Rusek, “Achievable rate with correlated hardware

impairments in large intelligent surfaces,” in IEEE 8th International

Workshop on Computational Advances in Multi-Sensor Adaptive Pro-

cessing (CAMSAP). Le gosier, Guadeloupe: IEEE, Dec. 2019.

>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <10

lim

ρs→∞ C≤lim

ρs→∞

log2

1 + 16−π2

22Nπ 2

16−π22

(κ2

t+κ2

r)16−π2

22Nπ 2

16−π22+1

ρs

(51)

[28] Z. Xing, R. Wang, J. Wu, and E. Liu, “Achievable rate analyses and

phase shift optimizations on intelligent reﬂecting surface with hardware

impairments,” arXiv:2005.14411, May 2020.

[29] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and

products, 6th ed. San Diego: Academic Press, 2000.

[30] H. Srivastava and J. Choi, Introduction and Preliminaries. Oxford:

Elsevier, 2011.

[31] H. Holma and A. Toskala, LTE for UMTS: Evolution to LTE-Advanced,

2nd ed. USA: Wiley Publishing, Mar. 2011.

[32] A.-A. A. Boulogeorgos, E. N. Papasotiriou, and A. Alexiou, “Analytical

performance assessment of THz wireless systems,” IEEE Access, vol. 7,

pp. 11 436–11 453, Jan. 2019.

[33] S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther,

A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick,

C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless

sub-THz communication system with high data rate,” Nat. Photonics,

vol. 7, pp. 977 EP–, Oct. 2013.

[34] C. Studer, M. Wenk, and A. Burg, “MIMO transmission with residual

transmit-RF impairments,” in International ITG Workshop on Smart

Antennas (WSA), Bremen, Germany, Feb. 2010, pp. 189–196.

[35] M. Wenk, MIMO-OFDM testbed : challenges, implementations, and

measurement results, ser. Series in microelectronics. Konstanz:

Hartung-Gorre, 2010.

[36] B. E. Priyanto, T. B. Sorensen, O. K. Jensen, T. Larsen, T. Kolding, and

P. Mogensen, “Assessing and modeling the effect of RF impairments on

UTRA LTE uplink performance,” in IEEE 66th Vehicular Technology

Conference, Baltimore, MD, USA, Sep. 2007, pp. 1213–1217.

[37] E. Bj ¨

ornson, M. Matthaiou, and M. Debbah, “Massive MIMO systems

with hardware-constrained base stations,” in IEEE International Confer-

ence on Acoustics, Speech and Signal Processing (ICASSP), Florence,

Italy, May 2014, pp. 3142–3146.

[38] E. Bj ¨

ornson, A. Papadogiannis, M. Matthaiou, and M. Debbah, “On

the impact of transceiver impairments on AF relaying,” in Proc. IEEE

International Conference on Acoustics, Speech and Signal Processing

(ICASSP), Vancouver, BC, Canada, May 2013.

[39] A.-A. A. Boulogeorgos, E. N. Papasotiriou, and A. Alexiou, “Analytical

performance evaluation of thz wireless ﬁber extenders,” in IEEE 30th

Annual International Symposium on Personal, Indoor and Mobile Radio

Communications (PIMRC), Istanbul, Turkey, Sep. 2019, pp. 1–6.

[40] A.-A. A. Boulogeorgos, N. D. Chatzidiamantis, and G. K. Karagiannidis,

“Energy detection spectrum sensing under RF imperfections,” IEEE

Trans. Commun., vol. 64, no. 7, pp. 2754–2766, Jul. 2016.

[41] V. S. Asadchy, M. Albooyeh, S. N. Tcvetkova, A. D´

ıaz-Rubio, Y. Ra'di,

and S. A. Tretyakov, “Perfect control of reﬂection and refraction using

spatially dispersive metasurfaces,” Phys. Rev. B, vol. 94, no. 7, Aug.

2016.

[42] X. Tan, Z. Sun, J. M. Jornet, and D. Pados, “Increasing indoor spec-

trum sharing capacity using smart reﬂect-array,” in IEEE International

Conference on Communications (ICC), Kuala Lumpur, Malaysia, May

2016, pp. 1–6.

[43] S. G. Krantz, Handbook of Complex Variables. Springer Science &

Business Media, Oct. 1999.

[44] E. Bj ¨

ornson, M. Matthaiou, and M. Debbah, “A new look at dual-hop

relaying: Performance limits with hardware impairments,” IEEE Trans.

Commun., vol. 61, no. 11, pp. 4512–4525, Nov. 2013.

[45] J. J. Shynk, Probability, random variables, and random processes:

Theory and signal processing applications. Hoboken, New Jersey:

Wiley, 2013.

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).