Cellular Systems with Many Antennas: Large System Analysis under Pilot Contamination

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by  Narayanan Krishnan

Cellular Systems with Many Antennas: Large
System Analysis under Pilot Contamination
Narayanan Krishnan, Roy D. Yates, Narayan B. Mandayam
WINLAB; Rutgers, The State University of New Jersey
E-mail: narayank, ryates, narayan @winlab.rutgers.edu
Abstract—Base stations with a large number of transmit number of users, the dedicated training symbols may take up
antennas have the potential to serve a large number of users thecoherencetimeofthechannel.Asthisisundesirable,only
simultaneouslyathigherrates.Theyalsopromisealowerpower
apartofthecoherencetimeisutilizedtolearnthechannel.As
consumptionduetocoherentcombiningatthereceiver.However,
aresult,thepilotsequencesindifferentcellsoverlapovertime-
the receiver processing in the uplink relies on the channel
3
estimates which are known to suffer from pilot interference. frequencyresourceandasaconsequencethechannelestimate
1
In this work, we perform an uplink large system analysis of is corrupted. This is called pilot interference which is found
0
2 multi-cell multi-antenna system when the receiver employs a to be a limiting factor as we increase the number of antennas
matchedfilteringwithapilotcontaminatedestimate.Wefindthe
[3].
n asymptoticSignaltoInterferenceplusNoiseRatio(SINR)asthe
a number of antennas and number of users per base station grow It is shown in [2] that in the limit of infinite number of
J large while maintaining a fixed ratio. To do this, we make use antennas, the SINR using a matched filter receiver is limited

5 of the similarity of the uplink received signal in a multi-antenna by interference power due to pilot contamination. While the
system to the representation of the received signal in CDMA result assumes a regime with finite number of users, we can

] systems. The asymptotic SINR expression explicitly captures the
also envision a regime where the number of users may be
T
effect of pilot contamination and that of interference averaging.
comparable to the number of antennas such as a system with
I This also explains the SINR performance of receiver processing
s. schemesatdifferentregimessuchasinstanceswhenthenumber 50 antenna base stations serving 50 users simultaneously. In
c of antennas are comparable to number of users as well as when this work, we do a large system analysis of uplink multi-cell,
[ antennas exceed greatly the number of users. Finally, we also multi-antenna system when the receiver employs a matched
proposethattheadaptiveMMSEsymboldetectionscheme,which
1 filter to decode the received signal. We let the number of
doesnotrequiretheexplicitchannelknowledge,canbeemployed
v antennas and the number of users per base station grow large
for cellular systems with large number of antennas.
4
simultaneously while maintaining a fixed users to antennas
5 I. INTRODUCTION ratio and observe the SINR in the following cases,
9
0 Cellularsystemswithlargenumberofbasestationantennas 1. when there is a perfect channel estimate,
1. have been found to be advantageous in mitigating the fading 2. when we have a pilot corrupted channel estimate.
effectsofthechannel[2].Inthedownlink,denselow-powered
0 In order to accomplish that we make use of the similarity of
3 base stations operating with power in the order of milliwatts
the uplink received signal in a MIMO system to that of the
1 have the potential to conserve power as compared to the
received signal in a CDMA system [7]. Further we compare
: present systems. In the uplink, coherent receiver processing
v the results of the asymptotic SINR expression so obtained.
i with a large number of antennas reduces the transmitted
X Also, we propose an adaptive filtering method reminiscent
powers of the users. It is shown in [2] that in an infinite
of CDMA systems where the uplink receiver filter at the
r antennaregime,andinabandwidthof20MHz,atimedivision
a base station converges to the desired MMSE filter. While
duplexing system has the potential to serve 40 single antenna
MMSE filtering requires us to estimate the channel to all the
users with an average throughput of 17 Mbps per user.
users in the system, the adaptive MMSE does not require any
However, any advantages offered by multiple transmitters
prior estimation of the channel to any of the users. Using
at the base station can be utilized only by gaining the channel
independent training symbols among users, the receive filter
knowledge between the base station and all the users. This
will converge to the desired MMSE filter.
requires training data to be sent from the users. Hence, a
part of the channel coherence time is utilized for gaining the
II. SYSTEMMODEL
channel knowledge between the base station and all users.
In a typical system the time-frequency resources are divided We consider a system similar to that in [2] with B base
into blocks of coherence-time coherence-bandwidth product, stations and K users per base station. We assume that all
wheresomeresources(timeorfrequency)areusedforchannel the KB users in the system are allocated the same time-
estimation and the rest is used for transmission in uplink or frequency resource. Also, each base station is equipped with
downlink. However, in [1], it is shown that the number of M antennas. The channel vector representing the small scale
pilot symbols required is proportional to the total number of fading between user k in cell j and the antennas in base
users in the system. Hence, as the system scales with the station l is given by a M × 1 vector h(l). The entries of
jk
h(l) are assumed to be independent zero mean i.i.d Gaussian C. Pilot Interference based Channel Estimate
jk
random variables with unit variance. This corresponds to an
In one of the approaches taken in [2], the base station
idealandfavourablepropagationmediumwithrichscattering. is concerned with only knowing the channel to its own K
Thelargescalefadingcoefficients,whichrepresentsthepower usersandspendsonlyK time-frequencyresourcesforchannel
attenuation due to distance and effects of shadowing between estimationinsteadofKB.Everybasestationsimilarlyspends
base station l and kth user in jth cell is given by βj(lk). This is first K time-frequency resources for channel estimation to
constantacrosstheantennasofthecelll.Accordingly,overall its K users. Intuitively, we can think of K users of the
(cid:113)
channel vector is given by g(l) = β(l)h(l). lth base transmitting pilots in K orthogonal times-frequency
jk jk jk
resource to its base. Consequently, the user indexed one will
A. Uplink Transmission
be transmitting pilot symbol to its base station l in the first
We assume that all user’s transmission are perfectly syn-
time slot in a subcarrier. Since the first user in other cells
chronized. Also, while a user’s transmission is intended to its
alsotransmitsatthesametimeandatthesamesubcarrier,the
base station, other base stations also hear the transmission.
received signal is corrupted by other pilots transmission.
Defining q as the symbol transmitted by user k in cell j,
jk This results in an estimate as given below,
wga(ul)ssaiasntheenMtrie×s,1ρn/oMise avsectthoer wupitlhinzkersoigmnaelantounnitoivsaeriraantcioe hˆ(l) =g(l)+√κ(cid:88)g(l), (2)
r lk lk jk
scaled by the number of antennas, the received signal at base j(cid:54)=l
station l is given by, The variable κ which can take values 0 or 1 is in order to
(cid:114) B K distinguish a perfect estimate and a pilot corrupted estimate
y(l) = ρr (cid:88)(cid:88)g(l)q +w(l). (1) respectively. Since our aim is to analyse the effect of pilot
M jk jk
j=1k=1 interference we assume that there is no uncorrelated gaussian
Since processing power is not an issue at the base station, noisefurthercorruptingtheestimate.Ithasbeenshownin[2],
receivedsignalprocessingcanbeemployed.However,thebase [3] that using this estimate and a matched filter decoder the
stationhastohaveanestimateofthechanneltoallusersprior achievable rates are limited by inter-pilot interference as the
to transmission of uplink information. In a system employing number of antenna grows.
OFDM physical layer with time-frequency resources, we can
III. LARGESYSTEMANALYSIS
divide the resources into blocks spanning the coherence-time
SincetheSINRanalysisisidenticaltoallusersinthesystem
coherence-bandwidth product. Although the channel vector
h(l) to each of the users has to be relearned by the base wefocusonlyonuser1inbasestationindexed1.Formatched
jk filtering we project the received signal y(1) in equation (1)
station at the start of every coherence time, once learnt for a √
onto the normalized channel estimate gˆ1 / M. We also drop
subcarrier it remains the same for of all subcarriers spanning 11
thesuperscript(.)(1)fornotationalconvenience.Consequently,
the coherence bandwidth during that coherence time. Let the
theSINRofthefirstusertoitsbasestationwithpilotcorrupted
number of coherent symbols be given by T and the coherent
c channel estimate is given by,
subcarriersbeN .Therefore,ifwefixthenumberofsymbols
c
uusseerd’sfocrhaensntiemlactaionnbetolebaernTt.TsuhcihsothbasetrTva≤tioTncw, aastontoatledofinN[c2T]. SINR = ρr(cid:12)(cid:12)(cid:12)gˆ1HM1g11(cid:12)(cid:12)(cid:12)2 (3)
WofeuwseorusltdhlainkectaonpboeinstuopuptotrhteadtidteispreenldesvaonnthNereaansdthdeepneunmdibnegr gˆ1HM1gˆ11 +ρr(cid:80)Bj=1(cid:80)Kk=1(cid:12)(cid:12)(cid:12)gˆ1HM1gjk(cid:12)(cid:12)(cid:12)2
c {(j,k)(cid:54)=(1,1)}
onitsvaluethenumberofusersK thatcouldbesupportedcan
Theorem 1. Let SINR obtained with matched filtering be as
becomparabletoM.Therefore,itisworthwhiletoinvestigate
given in equation (3), then as M,K → ∞, with K/M = α,
not only M (cid:29)K scenario but also the case when M and K
SINR converges in probability to
large and comparable.
B. Limitations in gaining Channel Knowledge SINR∗ = ρr1+κ((cid:80)Bjβ=121βj1)/β11 (4)
During each coherence time, users in a cell spend some 1+ρ (cid:20)κ ((cid:80)Bj=2βj21)/β11 +BαE[β](cid:21)
pilotsymboltimesoverthesubcarriersspanningthecoherence r 1+κ((cid:80)Bj=2βj1)/β11
bandwidth for channel estimation at the base station and then
where, the expectation is over the distribution of β - the
the transmission of data ensues until the end of the coherence
random variable representing the realizations of large scale
time.Atbasestationl,thenumberofchannelvectorsh(l) that
jk fading gains β(cid:48) s.
needstobelearntisequaltothenumberofusersinthesystem jk
which is KB where K = N T. In order to accomplish that, Proof: Proof given in Appendix
c
the number of pilots required must at least be KB symbol The expression for the asymptotic SINR captures the ef-
times. However, such a system will not be scalable as there fect of both interference due to pilot contamination and the
existssomelargeB forwhichtheproductKB willoccupyall interference averaging due to α(cid:54)=0. The SINR expression in
thecoherencetime.Thisisclearlyundesirableaspilottraining the limit of infinite antennas but finite number of users per
is taking up significant part of a coherence time. cell are obtained when we put α=0 and this corresponds to
the expression for SINR in [2], [3]. It is seen that the SINR SNR = 20, M = 50
15
expression so obtained is limited by the pilot interference
powers. On the other hand with a perfect channel estimate, 10
gˆ(l) =g(l) implying that κ=0 we get the SINR of matched 5
lk lk
filter as in [7]. We can view Theorem 1 as a generalization of 0
the large system analysis for a matched filter receiver with a →
B) −5
pilot contaminated channel estimate. R (d
N−10
SI
IV. NUMERICALEVALUATION
−15
For the numerical evaluation, we consider hexagonal cells
−20
with users uniformly distributed in each of the cells as shown
−25
in Fig. 3. We consider a set up where 6 closest cells are
interfering with the center cell. We assume received powers −300 0.5 1 1.5
fromalltheusersareunityi.eβ =1.WeconsidertheSINR α →
jk
for the user one in the center cell. Fig. 1 plots the asymptotic
Fig.2. WeobservethattherandomrealizationsoftheSINRsareclustered
SINR of the matched filter with a perfect estimate and that of
aroundtheasymptoticlimit.
a matched filterwith apilot corruptedestimate forthe caseof
equalreceivedpowers.Itisseenthatusingapilotinterference
based estimate for matched filtering causes at least a 10 dB
loss in the SINR. This is dependent on the users contributing
v
to pilot interference based estimate and hence dependent on
B. Fig. 2 shows that the random realizations of the SINR for
differentvaluesofαareclusteredaroundtheasymptoticplots. v v
v
SNR = 20, M = 50
10
perfect estimate
pilot corrupted estimate v v
5
v
0
→
B)
R (d −5
N
SI
Fig.3. Inthefavourablecase,thesumofthereceivedpowersofinterferers
−10
contributing to pilot contamination are very less as compared to that of the
desireduser.User1inthecentercellrepresentssuchascenario.TheSINR
−15 withapilotcorruptedestimateisthencomparabletothatofperfectestimate.
Ontheotherhandforuser2inthecentercell,thepilotinterferersreceived
powers are comparable to that of the desired user and represents the worst
−200 0.5 1 1.5 casescenarios.
α →
Fig.1. TheasymptoticlimitofmatchedfilterSINRwithaperfectestimate
is plotted with that of matched filter with a pilot corrupted estimate. The • α=1with50antennasand50usersperbasestationand
numberofantennasconsideredis50andthesignaltonoiseratiois20dB.
κ = 0, representing the case of matched filter receiver
with a perfect channel estimate,
In order to get more intuition under practical scenarios of • α=1with50antennasand50usersperbasestationand
largescalefadinggains,weconsiderthesevencellmodelwith κ=1 as representing the case of matched filtering with
cell radius is R = 2 km, and assume a COST231 model for a pilot corrupted channel estimate but with comparable
propagation loss between the base station and the users. The number of antennas and users.
noise power is assumed to be −174 dBm and user transmit Again, we only consider evaluating the SINR of the first user
power of 23 dBm. We plot the CDF of the SINR in Fig. 4 for at the first base station as the SINR of all other users in the
the following scenarios, system follow the same statistics. As compared to the case
• α=0withκ=0correspondingtomatchedfilteringwith when α = 0 there is at least 25 dB loss in SINR when the
perfect estimate with number of antennas far exceeding number of antenna becomes equal to number of users. In the
the number of user. case of matched filtering with perfect estimate this can be
• α=0andκ=1correspondingtocasewhenweemploy explainedentirelybythenon-zerointerferencepowerbecause
matched filtering at the receiver with a pilot corrupted α=1. When matched filtering with a pilot corrupted channel
channel estimate similar to [2], estimate is employed, the shift is due to the combination of
bothpilotinterferencepowerandthatinterferenceduetoα= trainingsequencesknownatthebasestationaresentbyevery
1. More specifically consider a scenario when user simultaneously so that each user converges to its desired
MMSE filter directly. The explanation for the implementation
(cid:80)B β (cid:80)B β2
j=2 j1 (cid:28)1⇒ j=2 j1 (cid:28)1 (5) of adaptive MMSE is as follows. Given the received signal in
β11 β11 equation (1), we wish to design an MMSE filter c11 for the
Thiscorrespondstothefactthatsumofreceivedpowersofthe firstuserinthefirstcellsothatE[(cid:12)(cid:12)q11−cH11y(cid:12)(cid:12)2]isminimized.
interferersaremuchlessthatthatofdesireduserpower.Under From [8], the filter which minimizes mean square objective
these conditions the SINR of the received signal in equation is given by c11 = (E[yyH])−1E[q1∗1y]. In order to form
(4) is, the filter c11 directly, the channel knowledge gjk from the
first base to all the users is required.A way to overcome this
ρ β
SINR∗ ≈ r 11 difficulty is by using stochastic gradient descent algorithm,
1+ρ BαE[β]
r so that the filter adaptively converges to MMSE as proposed
and hence the SINR of the matched filter with the corrupt for CDMA systems [8]. With 0 ≤ t ≤ T being the index of
channelestimateisasgoodastheSINRwithaperfectchannel time-frequency resources spent for training data, {ψ [t]} the
11
estimate.ThisisseeninthebestcaseSINRrealizationsinFig. T length training sequence for user indexed one and µ > 0,
4.InFig.3,thesituationofuser1inthecentercellrepresents the progressively decreasing step size, a step in the algorithm
the favourable scenario with the interferers contributing to is given by the equation,
the pilot contamination are far such that the condition (5) is
satisfied. c [t]=c [t−1]−µ(ψ [t]−cH[t−1]y[t])y[t]. (7)
11 11 11 11
Ontheotherhandifalltheinterferersgainsarecomparable
to that of the desired users gains represented by It has been shown that there exist positive values of µ so that
the iteration (7) converges to the MMSE filter. Also, as long
(cid:80)Bj=2βj1 ≈B−1 (6) as training data satisfies E[ψjkψl∗i]=0 for {j,k}=(cid:54) {l,i} the
β filterconvergestotheMMSEofuser1incell1.Consequently,
11
the SINR of the MMSE estimate is given by,
then pilot interference contributes negatively to the SINR in
addition to interference averaging. This is seen in Fig. 4 for  −1
tdheesirweodrsutsecras2erSepINreRsenretsalsizuacthioanss.ceInnaFriiog.w3i,ththdeessiirteudatiuosnero2f SINRmse = Mρrg1H1Mρr (cid:88)(cid:88) gjkgjHk+I g11 (8)
attheedgeofthecentercellandtheinterfererscontributingin {(j,k)(cid:54)=(1,1)}
the pilot estimate as shown in the figure. These observations
InFig.4weplottheSINRoftheMMSEfilterassumingthat
are true for all values of α.
thealgorithmhasconvergedtoc .AscomparedtotheSINR
11
with matched filtering, when α = 1, it is seen that MMSE
Antennas per Base = 50, Users per Cell = 50
1 can provide a gain of at least 10 dB because of interference
cancelling capabilities.
0.8
VI. CONCLUSION
→
0.6 In this work we consider a multi-cell multi-antenna system
)
R with large number of antennas and users. We perform a large
N
I SNR
F(S 0.4 SINRM → ∞ systemanalysisandderivetheasymptoticSINR(equation(4))
when a matched filter with a pilot contaminated estimate is
SINRmmse
employed at the receiver. We verified this expression numeri-
0.2 SINRmf−per
callybyseeingthattherandomrealizationsofSINRconverge
SINRmf−est
to the asymptotic limit as we increase the number of antenna.
0
−20 0 20 40 We also showed that when the number of users is comparable
SINR →
to the number of antennas the performance of the matched
filter with corrupt estimate is limited by the interference. This
Fig.4. ThefigureshowstheCDFofSINRforfivedifferentsystemsituations
is used to compare the effect of interference when the number of antennas is in contrast with the case when number of antennas far
andusersarecomparable.WecanseethatMMSEperformsmuchbetterthan exceed the number of users, where the pilot interference term
matchedfilterwithacorruptestimate.
has the effect of only reducing the 5 percentile SINR. We
also proposed an adaptive MMSE symbol detection scheme
wherethereisnonecessitytoestimatethechannel.Thiscould
V. ADAPTIVEMMSEFILTERING help in mitigating the effect of pilot contamination in the
We propose that adaptive MMSE filtering can be used for uplink. However, since the training symbols are limited due
eachusertodecodeitsuplinkdata.AdaptiveMMSEdoesnot to coherence time of the channel, convergence to the MMSE
require the explicit knowledge of the channel. Instead, pilot filter will be an issue and this is a subject of current research.
p
APPENDIXA Using −→ to denote convergence in probability, from [7,
PROOFOFTHEOREM1 Proposition 3.3] we can show that in the limit of M → ∞,
K →∞, with K/M =α,
In equation (3) numerator term can be simplified as,
gˆ1H1g11 = (g1H1+√κ(cid:80)Bj=2gj1)Hg11 (9) Mρr2 (cid:88)BK(cid:12)(cid:12)g1Hgi(cid:12)(cid:12)2 −p→ρrβ11BαE[β]. (13)
M M i=2
=β2 hH11h11 +√κ(cid:88)B (cid:112)β β hHj1h11 (10) Also, for u∈{1,2...B−1}
11 M j1 11 M BK
j=2 Mρr2 (cid:88)(cid:12)(cid:12)guHK+1gi(cid:12)(cid:12)2,
From (10), we observe that, limM→∞gˆ1H1g11/M = β11. In i=2
enqouisaetiotenrm(3g),ivtehnebfiyrsgˆtHtegˆrm/iMn thiseedqeunaolmtoin,ator which is the = Mρr2 (cid:88) (cid:12)(cid:12)guHK+1gi(cid:12)(cid:12)2+ Mρr2 ||guK+1||4, (14)
11 11 i(cid:54)=1,uK+1
1  √ (cid:88)B H(cid:32) √ (cid:88)B (cid:33) −p→ρrβu+1,1BαE[β]+ρrβu2+1,1,
M g11+ κ gj1 g11+ κ gs1 (11) where, in equation (14) the first term converges due to [7,
j=2 s=2 Proposition 3.3] and the second term converges due to weak
= g1H1g11 +κ(cid:88)B (cid:88)B gjH1gs1 +√κ(cid:88)B g1H1gs1 law of large numbers. Assuming a(cid:54)=b, the other terms are of
the form,
M M M
j=2s=2 s=2
BK
+√κ(cid:88)B gjH1g11 (12) Mρr2 (cid:88)gaHgigbTgi∗
M i=2
j=2 = ρr (cid:88) gHg gTg∗
From (12), we observe that, lim gˆHgˆ /M = β + M2 a i b i
κ(cid:80)Bj=1βj1. Converting the subscMri→pt∞of 1th1e1c1hannel vec11tors +i(cid:54)=ρr1,a|,|bg ||2gTg∗+ ρr ||g ||2gHg (15)
whichisdoubleindexedintosubscriptwhichissingleindexed M2 a b a M2 b a b
byusingthedefinitiong =(cid:112)β h (cid:44)(cid:112)β h =
jk jk jk jk (j−1)K+k In a similar approach to analysis of the interference term in
g , the interference term in equation (3) can be
(j−1)K+k [7,Proposition3.3]wecanshowthatinequation(15)thefirst
simplified as,
term converges to its expectation which is 0 and the second
Mρr2 (cid:88)(cid:88) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32)g11+√κ(cid:88)B gi1(cid:33)Hgjk(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2 aSonfindtcheteh,itereadrcmthesromaflsthcooecnaovbneovrvgeeregsteetroimn0sprcbooybnawvbeeirlagiktey.ilnaAwpfrtoeorfbarleabariglriretaynn,guthimnegbseutrhmse.
j k (cid:12) i=2 (cid:12)
terms we get the expression of SINR as in Theorem 1.
(j,k)(cid:54)=(1,1)
= Mρr2 (cid:88)BK(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32)g1+√κB(cid:88)−1guK+1(cid:33)Hgi(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2 [1] T. L. Marzetta, “HowRMEuFcEhRETrNaiCnEinSg is Required for Multiuser
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= Mρr2 (cid:88)BK(cid:12)(cid:12)(cid:12)g1Hgi+√κB(cid:88)−1guHK+1gi(cid:12)(cid:12)(cid:12) Uvonll.im9,itpepd.N35u9m0b-3e6r0o0f,ANnotve.n2n0a1s”0, IEEE Trans. on Wireless Commun.,
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i=2 u=1
= ρr (cid:88)BK(cid:32)|g g |2+√κB(cid:88)−1gHg gT g∗ tCioonmmanudn.P,rveoclo.d1in0g,pinp.M26u4lt0i--C26el5l1T,DADugS.y2s0t1e1ms”,IEEETrans.Wireless
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= Mρr2 (cid:88) |g1gi|2+κ(cid:88)(cid:12)(cid:12)guHK+1gi(cid:12)(cid:12)2 [7] Dht.tpT:/s/aer,xSiv..oHrga/nalbys,/1“2L0i2n.e2a8r7m5vu1lti[ucsse.IrT]receivers: Effective interference,
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1 i sK+1 i uK+1 i 1 i
[9] D. Tse, P. Viswanath, “Fundamentals of Wireless Communications”,
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 CambridgeUniversityPress.
B−1B−1
+κ(cid:88) (cid:88)guHK+1gigsTK+1gi∗

u=1 s=1
u(cid:54)=s