Content uploaded by Kezhi Wang

Author content

All content in this area was uploaded by Kezhi Wang on Apr 30, 2020

Content may be subject to copyright.

arXiv:2004.09854v1 [cs.IT] 21 Apr 2020

1

Spectral and Energy Efﬁciency of IRS-Assisted

MISO Communication with Hardware Impairments

Shaoqing Zhou, Wei Xu, Senior Member, IEEE, Kezhi Wang, Member, IEEE,

Marco Di Renzo, Fellow, IEEE, and Mohamed-Slim Alouini, Fellow, IEEE

Abstract

In this letter, we analyze the spectral and energy efﬁciency of an intelligent reﬂecting surface (IRS)-assisted

multiple-input single-output (MISO) downlink system with hardware impairments. An extended error vector mag-

nitude (EEVM) model is utilized to characterize the impact of radio-frequency (RF) impairments at the access

point (AP) and phase noise is considered for the imperfect IRS. We show that the spectral efﬁciency is limited

due to the hardware impairments even when the numbers of AP antennas and IRS elements grow inﬁnitely large,

which is in contrast with the conventional case with ideal hardware. Moreover, the performance degradation at

high SNR is shown to be mainly affected by the AP hardware impairments rather than the phase noise of IRS. We

further obtain the optimal transmit power in closed form for energy efﬁciency maximization. Simulation results are

provided to verify these results.

Index Terms

Intelligent reﬂecting surface, hardware impairments, downlink spectral efﬁciency, energy efﬁciency.

I. INT RO DUC TION

INTELLIGENT reﬂecting surface (IRS) has recently been acknowledged as a promising new tech-

nology to realize spectral-, energy- and cost-efﬁcient wireless communication for the ﬁfth generation

network and beyond [1]. IRS is a planar array consisting of a large number of low-cost reﬂecting elements,

S. Zhou is with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China (e-mail:

sq.zhou@seu.edu.cn).

W. Xu is with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China, and also with

Purple Mountain Laboratories, Nanjing 211111, China (e-mail: wxu@seu.edu.cn).

K. Wang is with the Department of Computer and Information Sciences, Northumbria University, Newcastle upon Tyne NE1 8ST, U.K.

(e-mail: kezhi.wang@northumbria.ac.uk).

M. Di Renzo is with Universit´e Paris-Saclay, CNRS and CentraleSup´elec, Laboratoire des Signaux et Syst`emes, Gif-sur-Yvette, France.

(e-mail: marco.direnzo@centralesupelec.fr).

M.-S. Alouini is with the Division of Computer, Electrical, and Mathematical Science and Engineering, King Abdullah University of

Science and Technology, Thuwal 23955-6900, Saudi Arabia (e-mail: slim.alouini@kaust.edu.sa).

2

which independently induce phase adjustments on impinging signals to conduct reﬂecting beamforming.

Signiﬁcantly different from existing technologies, IRS reconﬁgures wireless communication environment

between transmitter and receiver via programmable and highly controllable intelligent reﬂection. Moreover,

it avoids active radio-frequency (RF) chains and operates passively for short range coverage enhancement

so that it can be densely deployed in a ﬂexible way with affordable hardware cost and energy consumption.

Traditional communication theories may no longer be applied because the IRS-assisted wireless system

consists of both active and passive components, instead of solely active entities [2]. Researches on channel

estimation, IRS beamforming design and system performance analysis are on the way. Two efﬁcient uplink

channel estimation schemes were proposed in [3] for IRS-assisted multi-user systems with various channel

setups. In [4], transmit precoding and passive IRS phase shifts were jointly optimized for simultaneous

wireless information and power transfer systems. Ergodic spectral efﬁciency of an IRS-assisted massive

multiple-input multiple-output system was characterized in [5] under Rician fading channel. In [6], spectral

efﬁciency of an IRS-aided multi-user system was studied under proportional rate constraints and an

iteratively optimizing solution was proposed with closed-form expressions. Secrecy rate was maximized

in [7] for an IRS-assisted multi-antenna system by alternately optimizing transmitting covariance and IRS

phase shifts. IRS was also shown to be effective in enhancing the performance of cell-edge users [8].

In practice, precise phase control is infeasible at IRS due to hardware limitations and imperfect channel

estimation. Corresponding researches are still in their infancy. Discrete phase shifts were considered for

IRS-assisted multi-user communication in [9], where a hybrid beamforming optimization algorithm was

proposed for sum rate maximization. In addition to non-ideal IRS, the impacts of RF impairments at

transmitter on the performance of an IRS system have not been clear. To capture the aggregate impacts

of various RF impairments, a generalized model named extended error vector magnitude (EEVM) was

proposed in [10] for cellular transmitters.

In this letter, we focus on an IRS-assisted multiple-input single-output (MISO) system with hardware

impairments at both access point (AP) and IRS. Theoretical expression of spectral efﬁciency is derived for

this non-ideal case. We discover that the performance is limited even with increasing numbers of elements

at both the AP and IRS. The impact of phase noise at IRS diminishes at high SNR. Meanwhile, we obtain

a closed-form solution to the optimal power design for maximizing energy efﬁciency. The optimal power

increases with RF impairments.

3

II. SY STE M MODE L

A. Signal Model

We consider a MISO downlink system where an IRS consisting of Nreﬂecting elements is deployed

to assist the communication from an M-antenna AP to a single-antenna user. The IRS is triggered

by an attached smart controller connected to the AP. Denote the reﬂection matrix of IRS by Θ=

diag{ζ1ejθ1, ζ2ej θ2,...,ζNejθN}, where ζn∈[0,1] and θn∈[0,2π)for n= 1,2,...,N are respectively the

amplitude reﬂection coefﬁcient and the phase shift introduced by the nth reﬂecting element. In practice,

each reﬂecting element is usually designed to maximize the signal reﬂection. Without loss of generality,

we set ζn= 1 for all n[11]. The direct link between AP and user is blocked by obstacles, such as buildings

or human body, which is common in the communication at high-frequency bands, like millimeter wave.

Thus it would be better to deploy IRS at positions where line-of-sight (LoS) communication is ensured

for both AP-to-IRS and IRS-to-user links.

Considering the ﬂat-fading model, the channel from the AP to IRS and that from the IRS to user are

respectively denoted by H1and hH

2. Both channels are assumed to be LoS, which are represented by

H1=αaN(φa

r, φe

r)aH

M(φa

t, φe

t),hH

2=βaH

N(ϕa

t, ϕe

t),(1)

where αand βare the corresponding strength of path AP-to-IRS and IRS-to-user, φa

r(φe

r) is the azimuth

(elevation) angle of arrival (AoA) at IRS, φa

t(φe

t) and ϕa

t(ϕe

t) are the azimuth (elevation) angles of

departure (AoD) at AP and IRS, respectively, and aX(ϑa, ϑe)is the array response vector. We consider

uniform square planar array (USPA) with √X×√Xantennas. The array response vector can be written as

aX(ϑa, ϑe) = [1,...,ej2πd

λ(xsin ϑasin ϑe+ycos ϑe),...,ej2πd

λ((√X−1) sin ϑasin ϑe+(√X−1) cos ϑe)]T,(2)

where dand λare the antenna spacing and signal wavelength, and 0≤x, y < √Xare the antenna indices

in the planar. Assume that the AP knows the channel state information (CSI) of both H1and hH

2. Channel

estimation methods for communication with IRS can be found in [11][12].

With the errors caused by imperfect RF chains at AP, we adopt the EEVM in [10] to model the transmit

signal, which can be written as

x=χws+nRF ,(3)

where sis the signal satisfying E[|s|2] = Pwith Pbeing the transmit power budget, wis the nor-

4

malized beamforming vector at AP, χ=diag{χ(1), χ(2),...,χ(M)}with χ(m) = η(m)ejψ(m)for

m= 1,2,...,M representing the RF attenuation and phase rotation of the mth RF chain with |η(m)| ≤ 1,

and nRF = [nRF (1), nRF (2), . . . , nRF (M)]Trepresents the additive distortion noise with covariance matrix

CnRF . The mapping of χand nRF to particular type(s) of RF impairments, e.g., phase noise, I/Q imbalance

and nonlinearity, could be found in [10, Ch. 7]. For notational simplicity, assume that ψ(m)is uniformly

distributed as U[−δψ(m), δψ(m)]with δψ(m)∈[0, π),nRF (m)∼ CN(0, σ2(m)), and the impairments of all

RF chains fall into the same level, i.e., η(m) = η,δψ(m)=δψ,σ(m) = σand CnRF =σ2IM.

Furthermore, there always exist some phase errors at IRS in implementation. The received signal with

phase errors can be modeled as

y=hH

2e

ΘH1x+u=hH

2e

ΘH1χws+hH

2e

ΘH1nRF +u, (4)

where e

Θ=diag{ej˜

θ1, ej˜

θ2,...,ej˜

θN}with ˜

θn=θn+ˆ

θnbeing the practical phase shift of the nth reﬂecting

element, ˆ

θnis the phase noise due to the fact, e.g., only discrete phase shifts are possible at IRS, and uis

the additive noise with zero mean and variance σ2

u. Assume that ˆ

θnis uniformly distributed as U[−δˆ

θ, δˆ

θ]

with δˆ

θ∈[0, π). Since the distortion noise is independent of channel noise, the received SNR is given by

SNR =P|hH

2e

ΘH1χw|2

(hH

2e

ΘH1)CnRF (hH

2e

ΘH1)H+σ2

u

.(5)

Then we have the downlink spectral efﬁciency as

R= log2(1 + SNR).(6)

B. Power Consumption Model

Before we discuss the power consumption, it needs to be emphasized that the IRS does not consume

any transmit power due to its nature of passive reﬂection. The total power consumption is modeled as [13]

PT=µP +PC,(7)

where µ=ν−1with νbeing the efﬁciency of transmit power ampliﬁer considering the RF impairments

and PCis the total static hardware power dissipated in all circuit blocks. The establishment of (7) models

well under two assumptions: 1) the transmit ampliﬁer operates within its linear region; and 2) the power

consumption PCdoes not rely on the rate of the communication link. Both assumptions are valid in typical

wireless systems.

5

III. SPECT RAL A ND ENE RGY EFFI CIE NCY ANALYSIS

In this section, we quantitatively analyze the downlink spectral and energy efﬁciency and discover the

impact of hardware impairments at both the AP and IRS. The ideal spectral and energy efﬁciency are

retrieved as a special case of our analysis and it is presented for comparison.

A. Spectral Efﬁciency Analysis

Before analyzing the performance, we need to determine the transmit beamforming of AP and the

reﬂecting beamforming of IRS. Since the hardware impairments are unknown and in order to facilitate

the design in practice, the two parameters, wand Θ, are optimized by treating the hardware as ideal.

Maximum ratio transmission (MRT) is adopted for transmit beamforming, i.e.,

w= (hH

2ΘH1)H/khH

2ΘH1k.(8)

We identify the optimal reﬂecting beamforming of IRS by maximizing the received signal power as

Θopt = arg max

Θ|hH

2ΘH1w|2(a)

= arg max

ΘkhH

2ΘH1k2

= arg max

Θ|aH

N(ϕa

t, ϕe

t)ΘaN(φa

r, φe

r)|2kaH

M(φa

t, φe

t)k2

(b)

= arg max

Θ|X

0≤x,y<√N ,

n=√Nx+y+1

ej2πd

λ(xp+yq)+j θn|2,(9)

where (a)is obtained by substituting win (8), (b)makes use of a mapping from the two-dimensional

index (x, y)to the index n,p= sin φa

rsin φe

r−sin ϕa

tsin ϕe

t, and q= cos φe

r−cos ϕe

t. Observing (9), it is

easy to get the optimal phase shift of the nth reﬂecting element as

θopt

n=−2πd

λ(xp +yq),(10)

where x=⌊(n−1)/√N⌋and y= (n−1) mod √N, and ⌊·⌋ represents rounding down the value and

mod means taking the remainder.

Now considering the design of Θopt in (10) and w(Θopt)in (8), we characterize the impacts of both RF

impairments at AP and phase noise at IRS on the downlink spectral efﬁciency in the following Theorem 1.

Theorem 1:The downlink spectral efﬁciency for the massive IRS-assisted MISO with large Mand N

approaches

Ra.s.

−−→ log21 + P M N2η2|αβ|2sinc2(δψ)sinc2(δˆ

θ)

MN 2|αβ|2sinc2(δˆ

θ)σ2+σ2

u.(11)

6

Special Case 1: For the ideal system without any impairments, we let η= 1,σ= 0 and δψ=δˆ

θ= 0

in (11). The downlink spectral efﬁciency reduces to

Rideal = log21 + P

σ2

u

MN 2|αβ|2.(12)

Special Case 2: For high SNR, (11) in Theorem 1 can be further simpliﬁed as

R→log2P+ 2 log2η+ 2 log2sinc(δψ)−log2σ2.(13)

Remark 1:It is concluded from (11) that the non-ideal spectral efﬁciency increases with ηwhile

decreases with parameters δψ,σ2and δˆ

θ. The impact of the phase rotation at AP in terms of δψis in

general more signiﬁcant than that of the phase noise at IRS in terms of δˆ

θ.

Remark 2:The spectral efﬁciency in (11) increases with the transmit power approximately in a log-

arithmic manner similar to the ideal case in (12) but with a different scale. Contrary to the ideal case,

the performance is ultimately upper bounded for increasing Mand N, which is R(M, N )≤¯

R=

log2(1 + η2P

σ2sinc2(δψ)) for all large Mand N.

Remark 3:An interesting observation from (13) is that the spectral efﬁciency at high SNR is merely

limited by the RF impairments at AP rather than the phase noise at IRS, which can be explained from

the perspective that the IRS reﬂecting beamforming simultaneously affects both the desired signal and the

distortion noise under the considerations of hardware impairments at AP and LoS channel. It encourages us

to use cheap IRS with low-resolution phase shifts without much consideration of performance degradation

for large IRS.

B. Energy Efﬁciency Analysis

The energy efﬁciency is deﬁned as the ratio of the spectral efﬁciency to the power consumption, i.e.,

EE ,BR/PTwhere Bis the channel bandwidth. We are interested in the performance at high SNR,

which can be rewritten as

EE =B(log2P+ 2 log2η+ 2 log2sinc(δψ)−log2σ2)

µP +PC

.(14)

In the following Theorem 2, we give a closed-form expression of the optimal transmit power maximizing

the EE in (14).

7

SNR (dB)

0 2 4 6 8 10 12 14 16 18 20

0

5

10

15

Ideal analysis in (12)

Ideal simulations

Non-ideal analysis in (11)

Non-ideal approximation in (13)

Non-ideal simulations

Reﬂecting Elements N

0 10 20 30 40 50 60 70 80 90 100

Spectral Eﬃciency (bits/s/Hz)

0

2

4

6

8

10

12

Ideal analysis in (12)

Ideal simulations

Non-ideal analysis in (11)

Non-ideal simulations

σ2= 0.3

σ2= 0.05

σ2= 0.05,

δψ=π/3

¯

R

Fig. 1. Downlink spectral efﬁciency versus SNR and N.

Theorem 2:The optimal transmit power to maximize the energy efﬁciency is the unique solution as

P⋆=µP −1

CW(µ−1eCAP−1PC),(15)

where W(x)is the Lambert’s W-function and CAP = 2 ln η+ 2 ln sinc(δψ)−ln σ2.

Note that for an ideal system without hardware impairments, the optimal transmit power can be similarly

derived as

P⋆

ideal =µP −1

CW(µ−1eC−1PC),(16)

where C= ln(MN 2|αβ|2)−ln σ2

u. We emphasize that the IRS has continuous phase in the ideal case,

which increases the static hardware power consumption of IRS.

Remark 4:For P⋆in (15) and EE(P⋆)in (14), the optimal transmit power increases with more severe

RF impairments and the corresponding optimal energy efﬁciency decreases.

IV. SIMUL ATION RE SULTS

In this section, simulation results are presented to validate the results in Section III. Assume that

M= 16,N= 64,η= 0.9,δψ=π

18 ,σ2= 0.1,δˆ

θ=π

8,α= 0.1,β= 0.5and µ= 1.1.

We plot the downlink spectral efﬁciency in Theorem 1, special cases and by simulations in Fig. 1. Both

non-ideal case in (11) and ideal case in (12) increase with the transmit power but by respective scales.

The simpliﬁed expression in (13) appears to be fairly tight at high SNR.

8

Transmit Power P(dB)

-5 0 5 10 15 20 25

Energy Eﬃciency (bits/s/Joule)

0

0.2

0.4

0.6

0.8

1

1.2

Ideal analysis

Non-ideal analysis

P⋆

Highest point

η= 0.9,δψ=π/18,σ2= 0.1

η= 0.8,δψ=π/18,σ2= 0.1

η= 0.8,δψ=π/4,σ2= 0.1

η= 0.8,δψ=π/4,σ2= 0.15

Fig. 2. Energy efﬁciency versus P.

We further assume SNR = P

σ2= 10 dB. Fig. 1 shows the spectral efﬁciency versus the number of IRS

reﬂecting elements. As the number goes larger, the hardware impairments lead to limited growth of spectral

efﬁciency, which is consistent with Remark 2, while the ideal case continues increasing logarithmically

with the squared number of elements.

In Fig. 2, we give the energy efﬁciency with various degrees of RF impairments. The optimal transmit

power derived in Theorem 2 matches the highest point of the curve well. When the RF impairments become

worse, higher optimal transmit power is required while the corresponding energy efﬁciency decreases. Note

that the ideal case may obtain poorer performance than the non-ideal case because of larger static hardware

power consumption of continuous-phase IRS.

V. CONCLUSION

In this letter, we demonstrate the downlink spectral and energy efﬁciency of an IRS-assisted system

with hardware impairments. The non-ideal spectral efﬁciency is upper bounded for large numbers of AP

antennas and IRS elements. Specially, the impact of imperfect IRS diminishes at high SNR. The optimal

transmit power for maximizing the energy efﬁciency increases as the RF impairments become more severe.

APPEN DIX A

PROO F O F THEO REM 1

Applying (8) and (10), we can rewrite the downlink spectral efﬁciency in (6) as

R= log2 1 + P|hH

2e

ΘH1χ(hH

2ΘH1)H|2/khH

2ΘH1k2

khH

2e

ΘH1k2σ2+σ2

u!

9

R(c)

= log2

1 +

P|αβ|2

N

P

n=1

ejˆ

θn

2

|aH

M(φa

t, φe

t)χaM(φa

t, φe

t)|2

M |αβ|2

N

P

n=1

ejˆ

θn

2

kaM(φt)k2σ2+σ2

u!

= log2

1 +

P η2|αβ|2PM

m=1 ejψ(m)

2PN

n=1 ejˆ

θn

2/M

M|αβ|2PN

n=1 ejˆ

θn

2σ2+σ2

u

,

(17)

where (c)is obtained by substituting the equations hH

2ΘH1=αβNaH

M(φa

t, φe

t)and hH

2e

ΘH1=αβ×

PN

n=1 ejˆ

θnaH

M(φa

t, φe

t).

For large M→ ∞, we have

1

M

M

X

m=1

ejψ(m)

2

(d)

−−→

a.s. |E[ejψ(m)]|2(e)

=|E[cos ψ(m)]|2(f)

=sinc2(δψ),(18)

where (d)applies the Strong Law of Large Numbers and the Continuous Mapping Theorem [14] which

indicates that the convergence preserves for continuous matrix functions, (e)uses the symmetry of the odd

function sin ψ(m)for ψ(m)∈[−δψ, δψ],(f)is obtained by substituting the probability density function

of variable ψ(m), i.e., fX(x) = 1

2δψfor x∈[−δψ, δψ], and sinc(x) = sin x

x. Similarly, for large N→ ∞,

we have

1

N

N

X

n=1

ejˆ

θn

2

a.s.

−−→ sinc2(δˆ

θ).(19)

Substituting (18) and (19) into (17) completes the proof.

APPEN DIX B

PROO F O F THEO REM 2

By calculating the partial derivative of EE in (14), we have

∂

∂P EE =BP−1(µP +PC)−µ(ln P+CAP )

(ln 2)(µP +PC)2.(20)

Letting the partial derivative be zero, we have

µP (ln P+CAP −1) = PC,(21)

t=ln P

⇐==⇒µet+CAP −1(t+CAP −1) = eCAP −1PC,

(g)

⇒t=W(µ−1eCAP−1PC)−CAP + 1,(22)

10

where (g)uses the fact the Lambert’s W-function is the inverse function of f(W) = W eW. Now

rearranging (22) yields (15).

The remainder proves that (21) has a unique solution. Deﬁne g(P),µP (ln P+CAP −1). It follows

d

dPg(P) = µ(ln P+CAP)>0. Thus g(P)is monotonically increasing with respect to P, which implies

that equation (21) has at most one solution, which is exactly (15).

REFER ENC E S

[1] S. Dang, O. Amin, B. Shihada, and M.-S. Alouini, “What should 6G be?,” Nat. Electron., vol. 3, no. 1, pp. 20–29,

Jan. 2020.

[2] M. Di Renzo et al., “Smart radio environments empowered by reconﬁgurable AI meta-surfaces: An idea whose time has

come,” EURASIP J. Wireless Commun. Netw., no. 129, pp. 1–20, May 2019.

[3] B. Zheng, C. You, and R. Zhang, “Intelligent reﬂecting surface assisted multi-user OFDMA: Channel estimation and

training design,” [Online]. Available: https://arxiv.org/abs/2003.00648v2.

[4] C. Pan et al., “Intelligent reﬂecting surface aided MIMO broadcasting for simultaneous wireless information and power

transfer,” [Online]. Available: https://arxiv.org/abs/1908.04863v4.

[5] Y. Han, W. Tang, S. Jin, C. Wen, and X. Ma, “Large intelligent surface-assisted wireless communication exploiting

statistical CSI,” IEEE Trans. Veh. Technol., vol. 68, no. 8, pp. 8238–8242, Aug. 2019.

[6] Y. Gao, C. Yong, Z. Xiong, D. Niyato, Y. Xiao, and J. Zhao, “Reconﬁgurable intelligent surface for MISO systems with

proportional rate constraints,” [Online]. Available: https://arxiv.org/abs/2001.10845v1.

[7] H. Shen, W. Xu, S. Gong, Z. He, and C. Zhao, “Secrecy rate maximization for intelligent reﬂecting surface assisted

multi-antenna communications,” IEEE Commun. Lett., vol. 23, no. 9, pp. 1488–1492, Sep. 2019.

[8] C. Pan, H. Ren, K. Wang, W. Xu, M. Elkashlan, A. Nallanathan, and L. Hanzo, “Multicell MIMO communications

relying on intelligent reﬂecting surface,” [Online]. Available: https://arxiv.org/abs/1907.10864

[9] B. Di et al., “Hybrid beamforming for reconﬁgurable intelligent surface based multi-user communications: Achievable

rates with limited discrete phase shifts,” [Online]. Available: https://arxiv.org/abs/1910.14328v1.

[10] T. Schenk, RF Imperfections in High-Rate Wireless Systems: Impact and Digital Compensation. Dordrecht, The

Netherlands: Springer, 2008.

[11] B. Zheng and R. Zhang, “Intelligent reﬂecting surface-enhanced OFDM: Channel estimation and reﬂection optimization,”

IEEE Wireless Commun. Lett., Early Access, Dec. 2019.

[12] B. Ning, Z. Chen, W. Chen, and Y. Du, “Channel estimation and transmission for intelligent reﬂecting surface assisted

THz communications,” [Online]. Available: https://arxiv.org/abs/1911.04719v2.

[13] 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.

[14] J. Xu, W. Xu, D. W. K. Ng, and A. L. Swindlehurst, “Secure communication for spatially sparse millimeter-wave massive

MIMO channels via hybrid precoding,” IEEE Trans. Commun., vol. 68, no. 2, pp. 887–901, Feb. 2020.