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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).
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