Analysis of Massive MIMO With Hardware Impairments and Different Channel Models

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by  Fredrik Athley

Analysis of Massive MIMO With Hardware
Impairments and Different Channel Models
Fredrik Athley1, Giuseppe Durisi2, Ulf Gustavsson1
1Ericsson Research, Ericsson AB, Gothenburg, Sweden
2Dept. of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden
Abstract—Massive Multiple-Input Multiple-Output (MIMO) distributed (IID) Rayleigh, and statistical ray-based channel
is foreseen to be one of the main technology components in models. Simulation results are presented together with simple
next generation cellular communications (5G). In this paper,
analytical expressions where the throughput dependence on
fundamental limits on the performance of downlink massive
the system parameters is transparent. Furthermore, the impact
MIMO systems are investigated by means of simulations and
analyticalanalysis.Signal-to-noise-and-interferenceratio(SINR) of phase and amplitude errors in the precoding weights is
and sum rate for a single-cell scenario multi-user MIMO are analyzed.
analyzedfordifferentarraysizes,channelmodels,andprecoding Previous works: A lower bound on the massive MIMO
schemes. The impact of hardware impairments on performance
5 downlink performance has been derived in [2], under the
isalsoinvestigated.Simpleapproximationsarederivedthatshow
1 assumption that the propagation channel is learnt through
explicitly how the number of antennas, number of served users,
0
transmit power, and magnitude of hardware impairments affect the transmission of orthogonal pilot sequences in the uplink.
2
performance. Hardware impairments in massive MIMO have been the sub-

n
ject of several recent theoretical investigations. Most of these
a
J I. INTRODUCTION works model the hardware impairments as power-dependent
7 Massive Multiple-Input Multiple-Output (MIMO) systems Gaussianadditivenoise,whichsimplifiesthethroughputanal-
1 are foreseen to be one of the main technology components yses [3]. More realistic multiplicative hardware models have
in next generation cellular communication systems (5G). The been adopted in e.g., [4]. Comparisons between the downlink
T] basic idea with massive MIMO is to use a large number throughput corresponding to measured propagation channels
I of antenna elements at the base station (BS)—an order of and ideal channel models have been reported in e.g., [5].
s. magnitude larger than used in current systems and much
c larger than the number of concurrently served user equipment II. SYSTEMMODEL
[
(UE)—in order to achieve high spatial resolution and array
Weconsiderasinglecellwithnoexternalinterference.The
1 gain.Thehighspatialresolutionenableshighsystemcapacity
v by spatial multiplexing of UEs. The array gain enables high downlink signals received by K co-scheduled UEs, each with
0 a single antenna, served by a BS with M antenna elements
energy efficiency. Furthermore, the averaging effects obtained
0 are modeled by the following equation
byusingalargenumberofantennaelementsmakethechannel
2 √
4 behave almost deterministically, which has the potential to y= PHWx+e. (1)
0 simplify radio resource management.
. InorderformassiveMIMOtobeeconomicallyfeasible,the Here,y=[ y ··· y ]T isaK×1vectorcontainingthe
1 1 K
0 costperantennaelementanditsassociatedradioandbaseband signals received by the UEs, P is the BS transmit (Tx) power
5 branches must be significantly less than in current systems. and H =[ hT ··· hT ]T is the K ×M channel matrix,
1 K
1 Therefore, the requirements put on such components must be with h being the 1 × M channel vector from the M BS
k
:
v less stringent than in current systems. Hence, massive MIMO antenna elements to UE k. Furthermore, W = [w1···wK]
i can be seen as a paradigm shift from using a few expensive denotes the M ×K precoding matrix, x the K ×1 vector
X
antennas to many cheap antennas [1]. of transmitted symbols, and e a K ×1 zero-mean complex
r
a Contributions: In this paper, a performance analysis Gaussian vector with covariance matrix N0I modeling the
of downlink massive MIMO is presented. The focus is on receiver noise in the UEs.
the impact of basic antenna and channel model parameters When increasing the number of antenna elements in the
on system performance. Downlink signal-to-interference ratio analysis, the Tx power will be scaled as 1/M in order to
(SINR) and sum rate for the matched filter1 (MF) and zero compensatefortheincreasedarraygain.Asaresult,aconstant
forcing (ZF) precoders are computed for a single-cell sce- signal-to-noise ratio (SNR) operating point is maintained.
nario using line-of-sight (LoS), independent and identically More precisely, the Tx power for a given number of antenna
elementsissetsothatacertaintargetSNR,SNR ,isobtained
This work was partly supported by the Swedish Foundation for Strategic t
ResearchundergrantSM13-0028. in the interference-free (IF) case, i.e.,
1Sometimesalsoreferredtoasconjugatebeamformingormaximumratio
transmission. P =N0SNRt/M. (2)
The BS antenna array is assumed to be a uniform lin- and
seeapraararrtaioyn,(UwLhAer)eiλn itshethheowrizaovnelteanlgpthla.nTehweirthad0ia.t6iλonepleamtteernnt E[|hkwj|2]=E(cid:104)(cid:12)(cid:12)hkhHj /(cid:107)hj(cid:107)(cid:12)(cid:12)2(cid:105)=1. (10)
of a single element is modeled according to [6] with 90◦ Hence,
azimuth half-power beam width. Mutual coupling between
PM
arrayelementsisignored.TheUEisassumedtohaveasingle SINR = . (11)
k P(K−1)+N
isotropic antenna. Polarization is not modeled. 0
Using the power normalization in (2) we obtain
III. PERFECTCHANNELSTATEINFORMATION SINR = SNRt . (12)
k 1+SNR (K−1)/M
In this section, the system throughput is analyzed assuming t
that the BS has perfect channel state information (CSI). The 2) ZF: With equal power allocation to all UEs, the ZF
SINR and sum rate are calculated for MF and ZF precoding precoding vector is
schemes. Some analytical results are first derived for an IID (cid:113)
Rayleighchannelmodelandthencomparedwithotherchannel wk =(H†)k/ (cid:13)(cid:13)(H†)k(cid:13)(cid:13)2F, (13)
models by means of simulations.
where (H†) denotes the k-th column of H†, (cid:107)·(cid:107) the
k F
The SINR for UE k for a given channel realization H is
Frobenius norm, and H† = HH(HHH)−1. In order to
given by
simplify the analytical calculations, the ZF precoding matrix
P|h w |2 is approximated by
γk = P (cid:80)K |hk wk|2+N . (3) W=cH†, (14)
j=1 k j 0
j(cid:54)=k
where c is a normalization constant obtained by solving
The maximum achievable rate for UE k is then given by
E[(cid:107)W(cid:107)2]=K. (15)
R =E[log (1+γ )] (4) F
k 2 k
This normalization makes the average transmitted total power
where the expectation is taken over the channel realizations. toallK UEsequaltoK,butdoesnotguaranteesamepowerin
The sum rate is obtained by summing the rates of all concur- allchannelrealizationsandequalpowerallocationtoallUEs.
rently served UEs according to However,foranIIDRayleighchannelthedifferenceturnsout
(cid:88) to be small when the number of antenna elements is large.
R= R . (5)
k Under the assumption of an IID Rayleigh channel, HHH
k is a K×K central complex Wishart matrix with M degrees
A. IID Rayleigh offreedomandcovariancematrixequaltotheidentitymatrix.
In this section, simple approximations of the rate in (4) are It follows then that [7, p. 26]
derivedundertheassumptionofanIIDRayleighchannel,i.e., E(cid:2)Tr(cid:8)(HHH)−1(cid:9)(cid:3)=K/(M −K) (16)
a channel for which H has IID zero-mean complex Gaussian
elements. The expectation in (4) is difficult to compute ana- and the normalization constant is obtained according to
lytically. However, as will be shown by simulations later in c2E(cid:104)(cid:13)(cid:13)H†(cid:13)(cid:13)2(cid:105)=K ⇒ c=√M −K. (17)
this paper, the following approximation is accurate F
The received signal according to (1) is then given by
R ≈log (1+SINR ), (6)
k 2 k
(cid:112) (cid:112)
y= P(M −K)HH†x+e= P(M −K)x+e. (18)
where
PE[|h w |2] Hence,
SINR = k k . (7)
k P (cid:80)Kj=1 E[|hkwj|2]+N0 SINRk =P(M −K)/N0 =SNRt(1−K/M). (19)
j(cid:54)=k
3) Simpleruleofthumb: TheSINRapproximationsin(11)
Analytical approximations of the SINR for MF and ZF pre-
and (19) can be used to derive a simple rule of thumb on
coders are derived below. These can be used together with
howmanyantennasareneededtoreachacertainperformance
(5)–(6) to estimate the sum rate in the cell.
for a given number of co-scheduled UEs. For example, the
1) MF: The matched filter precoding vector for UE k is
numberofantennasneededtoreachanSINRthatis3dBaway
w =hH/(cid:107)h (cid:107), (8) from the IF SINR, SNR , can be obtained by setting SINR=
k k k t
SNR /2. This leads to the following number of antennas for
t
wherethenormalizationby||h ||assuresthesametransmitted
k MF
power in all channel realizations and equal power allocation
M =(K−1)SNR , (20)
t
to all UEs. The expectations in (7) are then given by
and for ZF
E[|hkwk|2]=E(cid:104)(cid:12)(cid:12)hkhHk /(cid:107)hk(cid:107)(cid:12)(cid:12)2(cid:105)=E[(cid:107)hk(cid:107)2]=M, (9) M =2K. (21)
4) Comparison between analytical and simulation results: Figure 2 shows a comparison of performance with the
Figure1showsacomparisonoftheanalyticalapproximations different channel models when the channel gain has been
in (6)–(7), (11), and (19) with the the exact rate expression in normalizedtoone.WithMF,thebestperformanceisachieved
(4) obtained by Monte Carlo simulations.2 The plot shows with a LoS channel, while the LoS channel gives the worst
the average sum rate vs. number of antenna elements for performancewhenusingaZFprecoder.TheMFbehaviorcan
K = 10 and SNR = 10 dB. Clearly, there is excellent beexplainedbythelowsidelobesattainablebyalargeantenna
t
agreement between the analytical and simulation results, also array, which result in good interference suppression in a LoS
for moderately sized arrays. These results provide empirical channel. The cause of the ZF behavior is that there is a gain
support to the approximations made in deriving the analytical penaltywhentwoUEsareseparatedbylessthanabeamwidth
expressions. in a LoS channel. Placing a null in the direction of one UE
will,inthiscase,givealargegaindroptotheotherUE.Inan
Hz)35 uncorrelated channel, however, it is unlikely that two channel
s/ vectors are almost parallel.
p30
b
e ( Theotherchannelmodelsgiveverysimilarperformancefor
at25
sum r20 MMFF,, sainmaulylatitcioanl bthoethSCthMe ManFdaInTdUZcFhapnrneeclodmeorsd.elIst igsivientaelrmesotisntgidteontniocatel pthear-t
e
ag15 ZF, simulation formance as an IID Rayleigh channel. A possible explanation
Aver100 100 200 300 Z4F0, 0analytical500 to this is that different UEs will obtain different realizations
Number of antenna elements of the channel rays. The only correlation between UEs is
via the correlation between the large-scale parameters, such
Fig.1. Averagesumratevs.numberofantennasforanIIDRayleighchannel as angular spread, delay spread, and shadow fading. User-
usingMFandZFprecoders.Comparisonbetweenanalyticalandsimulation
common clusters are not captured by these models. This is
resultswithK=10andSNRt=10dB.
something that may be important when evaluating multiuser
MIMO performance.
B. Comparison of Channel Models
The IID Rayleigh model is reasonable when there is rich
scatteringaroundtheBSandtheUEs.However,inmanycases Hz)35 MF
the environment around the BS is such that there is spatial s/
correlation among the BS antenna elements. We investigate e (bp30
the impact of spatial correlation on massive MIMO by using at25
m r SCM
two types of correlated channel models: su20 ITU
1) LoS channel model: We assume that there is a single, ge 15 LoS
free-space planar wavefront from the BS to each UE. For a vera10 IID
given azimuth angle of departure (AoD), the model is purely A 0 100 200 300 400 500
Number of antenna elements
deterministic. However, the AoDs to different UEs are drawn
ZF
from a uniform distribution over the interval [-60◦, 60◦]. Hz)35
s/
2) Statisticalray-basedchannelmodels: Twodifferentray- bp30
basedmodelsareinvestigated:the3GPPspatialchannelmodel e (
at25
(SCM) [6] and the ITU urban macro (UMa) model [8] with m r SCM
indoor UEs added as described in [9]. su20 ITU
casTehseinIIDterRmasyloeifghanagnudlaLrosSprcehaadn,nwelitmhotdheelsIIrDeprmesoednetlcboerinnegr verage 1105 LIIoDS
A 0 100 200 300 400 500
spatiallywhiteandtheLoSmodelhavingzeroangularspread. Number of antenna elements
The ray-based models lie in between with a mean angular
Fig.2. Averagesumratevs.numberofantennaelementsfordifferentchannel
spread of 15◦ in the SCM model and 14◦ and 26◦ mean
models,top:MF,bottom:ZF.
angular spread for LoS and non-LoS UEs, respectively, in the
ITU UMa model.
The SCM and ITU channel models also include models for
In order to investigate the impact of path loss difference
pathloss.Inordertoisolatetheimpactofpathlossandspatial
between UEs on performance, Figure 3 shows the average
correlation on massive MIMO performance and to be able to
sum rate relative the corresponding IF case as a function
comparewiththeunitgainIIDRayleighchannel,thepathloss
of the number of antenna elements using the ITU UMa
is first removed from channel matrices generated by the SCM
model with and without path loss. The results show that MF
and ITU models.
relative performance is reduced significantly when using path
loss in the channel model, while the impact on ZF relative
2IntheMonteCarlosimulationsofZFperformance,theprecodingvector
in(13)hasbeenused. performance is weak.
e IF 1 E(cid:104)|hkh˜Hk |2(cid:105)=E(cid:34)| (cid:88)M hkmh∗km(cid:15)m|2(cid:35)=
v
ati0.8 m=1
m rate rel0.6 = E(cid:34) (cid:88)M |hkm|4|(cid:15)m|2+ (cid:88)M (cid:88)M |hkm|2|hkn|2(cid:15)m(cid:15)∗n(cid:35).
erage su0.4 MZMFFF,,, www /ppoaa ptthha tllohos slsosss Since E[|z|m4]==1 2σ4 for a compmle=x1Gnn(cid:54)==am1ussian random variable
Av0.2 ZF, w/o path loss z with variance σ2, we have that E[|him|4] = 2. To compute
0 100 200 300 400 500 E[(cid:15) ], we use that the characteristic function, defined as
Number of antenna elements m
ψ(s)=E[exp(jsX)], of a Gaussian random variable X with
Fig. 3. Average sum rate relative IF vs. number of antennas for different mean µ and variance σ2 is given by ψ(s)=ejµs−σ2s2/2. By
precoderswiththeITUUMachannelwithandwithoutpathloss setting s=1, we get
E[ejX]=ejµ−σ2/2. (26)
IV. IMPERFECTCHANNELSTATEINFORMATION
Since, by assumption, φ is a zero-mean Gaussian random
Perfect CSI at the BS cannot be achieved in reality so the variable with variance σ2, we obtain
results in the previous section should be interpreted as upper φ
boundsonperformance.Inthissection,weanalyzetheimpact E[(cid:15)m]=E[1+am]E[ejφm]=e−σφ2/2. (27)
of imperfect CSI using a simple model of hardware impair-
The expectation in (26) then simplifies to
ments.WemodelimperfectCSIasaphaseandamplitudeerror
appliedtothetruechannel.Morespecifically,thechannelused E[|hkh˜Hk |2]=2M(1+σa2)+M(M −1)e−σφ2 (28)
in the downlink transmission is modeled as
which, for large M, can be approximated by
h˜ =(1+a )ejφmh =(cid:15) h (22)
where h˜ iksmthe perturmbed dowknmlink cmhanknmel coefficient E[|hkh˜Hk |2]≈M2e−σφ2. (29)
km
between user k and antenna m, h is the corresponding true Finally,
km
cehrraonrn,ealmc,oaenfdficpiheanst,eaenrrdor(cid:15),mφm(cid:44),a(r1e+asasumm)eejdφtmo.bTehinedaempepnlidteundte, E[|hkh˜Hj |2]=E(cid:12)(cid:12)(cid:12)(cid:12)(cid:88)M himh∗jm(cid:15)m(cid:12)(cid:12)(cid:12)(cid:12)2=M(1+σa2). (30)
zero-mean Gaussian random variables with variances σ2 and (cid:12) (cid:12)
a m=1
σ2, respectively. In the remainder of the paper, we shall refer
φ The SINR approximation in (7) is then given by
to σ as amplitude error, and measure it in dB, and to σ as
a φ
phaTsheeeardroorp,teadndhamrdewasaurreeimitpianirdmeegnretems.odelisnotamodelofa SINRk = 1e+−σσφ221+SNRSN(KRt−1)/M. (31)
a t
particularcomponent.Rather,itcapturestheaggregatedeffect
Hence, the error-free SINR in (11) is reduced by a factor
of all errors in the system. The polar form of this aggregated
exp(−σ2)/(1+σ2) in the presence of phase and amplitude
impairment model is supported by the fact that two of the φ a
errors. The factor exp(−σ2) reflects that a phase error causes
largest impairments, i.e., the power amplifier distorsion [10] φ
the ideal and perturbed channel vectors not to be parallel. An
and the oscillator phase noise [11] are multiplicative. It has
amplitude error does not destroy the alignment, but yields
been shown in [4] that system performance predictions based
a gain reduction by a factor 1/(1 + σ2) due to the weight
on this model are in good agreement with the ones based on a
normalization in (24) which is needed to assure conservation
more refined hardware models.
of energy. If the errors are small, the error factor in (31) can
1) MF: In the case of errors, the MF precoding vector is
be approximated by Taylor expansions according to
given by the perturbed channel vector
wk =h˜Hk /(cid:107)h˜k(cid:107). (23) e−σφ2 ≈ 1−σφ2 ≈ 1 = 1 (32)
1+σ2 1+σ2 1+σ2+σ2 1+σ2
a a a φ
To simplify calculations, we approximate the norm by its
expectedvalueandinsteadusethefollowingprecodingvector where σ2 = σ2 +σ2 is the total error variance. Hence, for
a φ
(cid:113) small errors, the SINR degradation depends only on the sum
w =h˜H/ E[(cid:107)h˜ (cid:107)2]. (24) of the variances of the phase and amplitude errors. The factor
k k k
in (32) is the same as the gain reduction caused by phase and
As we shall later show, this is an accurate approximation for
amplitude errors in phased arrays [12]. Note that this SINR
large M. Using (22) we obtain
reductiondoesnotdependonthenumberofantennaelements.
M Hence, it remains even in the limit M → ∞. However, with
E[||h˜k||2]=E[(cid:88) |(1+am)ejφmhkm|2]=M(1+σa2), (25) theTxpowernormalizationusedinthispaper,thesystemwill
m=1 asymptotically be noise limited; so the asymptotic SINR loss
and can be compensated for by an increase in Tx power.
2) Simple rule of thumb: A rule of thumb for the required MF ZF
1 1
number of antennas to be 3 dB from the IF SINR is easily e e
at0.9 at0.9
derived also in the case of phase and amplitude errors. Using m r m r
the approximations in (31) and (32), the following expression e su0.8 e su0.8
g g
is obtained era0.7 era0.7
M = 11+−σσ22(K−1)SNRt. (33) Relative av00..56 21000 Relative av00..56 21000
500 500
Hence, the required number of antennas is increased by a 0.40 1 2 3 0.40 1 2 3
factor(1+σ2)/(1−σ2)comparedtotheerror-freecasein(20). Amplitude error (dB) Amplitude error (dB)
MF ZF
3) Simulation results: Figure 4 shows a comparison be- 1 1
tweensimulationresultsusinganIIDRayleighchannelmodel ate0.9 ate0.9
aonfdthtehephaapsperoanxdimaamtipolnitsu(d3e1e)–rr(o3r2s)isw2h0e◦natnhdes1tadnBd,arredspdeecvtiiavteiolyn. ge sum r0.8 ge sum r0.8
The agreement between simulation and analytical results is avera0.7 avera0.7
e 0.6 e 0.6
excellent. SinceanalyticalexpressionsforimperfectCSIhave ativ 20 ativ 20
Rel0.5 100 Rel0.5 100
500 500
Hz)35 0.40 10Phase e2r0ror (deg)30 40 0.40 10Phase e2r0ror (deg)30 40
s/
p30
b
e ( Fig.5. Averagesumraterelativetheerror-freecasevs.amplitude(top)and
at25 phase (bottom) error for MF (left) and ZF (right) precoders. K = 10 and
um r20 SNRt=10dB.
s
ge 15 Simulation limited. Therefore, for large arrays, this loss can be com-
a
er Analytical pensated for by an increase in Tx power. It was shown by
Av100 100 200 300 400 500 simulationsthattheZFprecoderismoresensitivetophaseand
Number of antenna elements
amplitudeerrorsthantheMFprecoder.Somesimpleanalytical
Fig.4. Averagesumratevs.numberofantennasforanIIDRayleighchannel approximations for SINR and sum rate have been derived for
usinganMFprecoderwith1dBamplitudeand20◦phaseerrors.Comparison the IID Rayleigh channel model. These approximations show
betweenanalyticalandsimulationresults.K=10andSNRt=10dB. how different parameters impact system performance.
beenderivedonlyfortheMFprecodersomesimulationresults
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