uplink multicell

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Uplink Linear Receivers for Multi-cell MultiuserMIMO with Pilot Contamination: Large System

AnalysisNarayanan Krishnan, Roy D. Yates, Narayan B. Mandayam

WINLAB; Rutgers, The State University of New JerseyE-mail: narayank, ryates, narayan @winlab.rutgers.edu

Abstract Base stations with a large number of transmitantennas have the potential to serve a large number of usersat high rates. However, the receiver processing in the uplinkrelies on channel estimates which are known to suffer frompilot interference. In this work, making use of the similarityof the uplink received signal in CDMA with that of a multi-cellmulti-antenna system, we perform a large system analysis whenthe receiver employs an MMSE lter with a pilot contaminatedestimate. We assume a Rayleigh fading channel with differentreceived powers from users. We nd the asymptotic Signalto Interference plus Noise Ratio (SINR) as the number of antennas and number of users per base station grow large whilemaintaining a xed ratio. Through the SINR expression weexplore the scenario where the number of users being servedare comparable to the number of antennas at the base station.The SINR explicitly captures the effect of pilot contamination andis found to be the same as that employing a matched lter witha pilot contaminated estimate. We also nd the exact expressionfor the interference suppression obtained using an MMSE lterwhich is an important factor when there are signicant numberof users in the system as compared to the number of antennas. Ina typical set up, in terms of the ve percentile SINR, the MMSElter is shown to provide signicant gains over matched lteringand is within 5 dB of MMSE lter with perfect channel estimate.Simulation results for achievable rates are close to large systemlimits for even a 10 -antenna base station with 3 or more usersper cell.

I. INTRODUCTION

Cellular systems with large number of base station antennashave been found to be advantageous in mitigating the fadingeffects of the channel [ 3] while increasing system capacity. Itis shown in [ 3] that in an innite antenna regime, and in abandwidth of 20 MHz, a time division duplexing system hasthe potential to serve 40 single antenna users with an averagethroughput of 17 Mbps per user. However, any advantagesoffered by multiple antennas at the base station can be utilizedonly by gaining the channel knowledge between the basestation and all the users. This requires training data to besent from the users. In a typical system the time-frequencyresources are divided into Physical Resource Blocks (PRBs)of coherence-time coherence-bandwidth product. For eachuser, it is necessary and sufcient to estimate the channel inevery PRB. Thus, some resources (time slots or equivalentlyfrequency channels) are used for channel estimation and therest are used for transmission in uplink or downlink. However,in [2], it has been shown that the number of pilot symbols

required is proportional to the total number of users in thesystem. Hence, as the system scales with the number of users, the dedicated training symbols may take up a signicantportion of the PRB. As this is undesirable, only a part of thecoherence time is utilized to learn the channel. In this case, thepilot sequences in different cells overlap over time-frequency

resources and, as a consequence, the channel estimates arecorrupted. This pilot interference is found to be a limitingfactor as we increase the number of antennas [ 5].

It is shown in [ 3] that in the limit of large number of antennas, the SINR using a matched lter receiver is limitedby interference power due to pilot contamination. While theresult assumes a regime with nite number of users, we canalso envision a regime where the number of users may becomparable to the number of antennas such as a system with50 antenna base stations serving 50 users simultaneously. Inthis work, we do a large system analysis of uplink multi-cell,multi-antenna system when the receiver employs an MMSElter to decode the received signal. We investigate the SINR

in a regime where the number of users per cell is comparableto the number of antennas at the base station. The MMSE lterwhich is designed to maximize the SINR is evaluated when wehave a pilot corrupted channel estimate. We let the number of antennas and the number of users per base station grow largesimultaneously while maintaining a xed users/antennas ratio and observe the SINR for the above two cases as a functionof . To do so we make use of the similarity of the uplink received signal in a MIMO system to that of the receivedsignal in a CDMA system [ 11] .

Much of the research in large MIMO systems with Rayleighchannel can be borrowed from the considerable literature forCDMA systems. The channel vector with i.i.d entries for the

large MIMO system is analogous to the signature sequence ina CDMA system so that antennas contribute to the processinggain. For example, the uplink analysis of an asymptotic regime[11] with both users and signature sequences tending to innitytranslates directly to results in a large MIMO system whensignatures are replaced by antennas. In both systems it isobserved that the asymptotic analysis is a good approximationfor practical number of antennas (signatures) and users. Whilein a CDMA system we assume that the signature sequencesare known, there are practical limitations in learning a mobileradio multi-antenna channel (antenna signatures) in a multi-cellular system, as shown in [ 3]. In this work we explore this

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limitation when users simultaneously estimate the channel andthe estimates are subject to pilot contamination. We focus ourresults in the regime > 0.1 as opposed to recent works suchas [3], [8], [14], [16] which are found to be approximately inthe regime of 0 < 0.1. Further, we compare the resultsof the asymptotic SINR expression so obtained with that of the performance of the matched lter.

A. Related Work

A similar large system analysis in the context of a Network-MIMO architecture was presented in [ 20]. The authors con-cluded that high spectral efciencies can be realized even with50 antennas in their architecture, paralleling the existing liter-ature results in CDMA systems. In [ 16], the results obtainedsuggest to scale the transmission power by the square rootof the number of base station antennas, as opposed to scalingby the number of antennas. This assumes that the transmissionpower during training and data are same. In general we take theapproach in [ 1] where the transmission power can be differentfor the training and data symbols during a coherence time.Joint channel estimation and multi-user detection was consid-ered in [ 25] for a single cell multi-user MIMO DS-CDMAsystems. For xed number of transmit antennas per user andreceive antennas per base station, the authors employ replicamethod for CDMA large system analysis in order to obtainlower bounds on achievable rate for different feedback basedreceiver strategies. Mathematically, our model can be viewedas an extension of the linear receiver scheme considered intheir model to a multi-cell scenario.

In the MU-MIMO literature the asymptotic SINR is alsocalled the deterministic equivalent of the SINR. Recentwork in [10] nds the deterministic equivalent of SINR with

distributed sets of correlated antennas in the uplink. Authors in[14] , have done a considerable work in providing the determin-istic equivalent for beamforming/maximum ratio combiningand regularized zero forcing/MMSE in the downlink/uplink with a generalized channel model taking into account theeffect of pilot contaminated channel estimate. They nd thenumber of antennas required to match a xed percentage of therate of an innite antenna regime. Also, the number of extraof antennas required for the matched lter to equal the rateobtained out of the MMSE lter is shown, implicitly showingthe interference suppression capability of MMSE lter. Wederive in our work the exact amount by which the MMSElter suppresses the interference for a Rayleigh fading channel

and provide some fresh engineering insights in the regimewith > 0.1. Additionally, we derive those results under astochastic rather than an deterministic received power modelas presented in [14, section 4]. A summary of the results isgiven in section I-B.

There have been signicant attempts to mitigate pilot con-tamination in the recent works in Massive MIMO systemswhich are, however, specic to the case when antennas farexceed the number of users served i.e. < 0.1. I n [8],time shifted pilot schemes were introduced to reduce pilotcontamination. There, simultaneous transmission of pilots wasavoided by scheduling only a subset of base stations to

transmit uplink pilots. Simultaneously, other base stationstransmit in the downlink to their users and it is shown thatthe interference created by these downlink transmission can becancelled with a large number of antennas at the base stationestimating the channel. Pilot contamination is now restrictedto base stations in a group that simultaneously transmit uplink pilots. However, this requires that the number of antennas farexceed the number of users. In their recent work, authors in[18] , [19] show that pilot contamination can be avoided usingsubspace based channel estimation techniques. They show thatthe eigenvalues corresponding to the other-cell interferencesubspace can be separated from the in-cell users in a regimewhere is below a threshold. Their analyses assume anideal power controlled situation with strict user schedulingand antennas far exceeding the number of users. By contrast,we examine the operating regime in which 0.1 < anddetermine the effect of pilot contamination on interference andinterference suppression capability of MMSE receiver. Also,even with power control, pilot contamination is prevalent whenlinear MMSE channel estimation is employed.

B. Contributions of our Work

We develop a large system asymptotic expression for theSINR in a Rayleigh fading environment when using anmatched lter and MMSE lter with a pilot contaminatedchannel estimate for a arbitrary user in the system. TheSINR expression is dependent on the number of users andthe number of antennas only through their ratio . If P signal ,P contam , P inter (c) and 2 represent the signal power, inter-ference power due to pilot contaminated estimate, the lterdependent interference power for a linear lter c and thenoise variance respectively, we show that the expression for

asymptotic SINR for both matched lter and MMSE lter canbe generalized to,

SINR (c) =P signal

2 + P contam + P inter (c ). (1)

If c MF and c MMSE denote the matched lter and the MMSElter with a pilot contaminated estimate, then the followingresult summarizes our contribution:

From the expression for SINR it is derived that the totalinterference is the sum of two terms. The rst termgiven by P contam is due to employing pilot contaminatedchannel estimate and the second is the lter dependent

interference term P inter (c ) due to comparable K andM . Also, the the lter dependent term contributes tointerference only when = 0 .

We show that P inter (cMF ) P inter (c MMSE ) = C() 0,where C() is called the interference suppression term.We nd a closed form expression for C( ) showing theinterference suppression capability of the MMSE lter

when > 0. It is derived that P contam (c

MF ) = P contam (c MMSE ) =P contam . The contribution of the pilot contaminated chan-nel estimate to the pilot interference given by P contam issame for both matched lter and MMSE lter.


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Although the lter cMMSE depends on the pilot contami-

nated channel estimate of all the K users in a cell to itsbase station, the P contam is independent of K or M or for all values of > 0 and therefore is the same as whatwas found at = 0 .

As per the authors knowledge, these contributions have notbeen reported yet in the literature.

We validate the theoretical results with simulations. Weshow that even a system with 50 base station antennas eachserving some number of users sufciently qualies for the termlarge system as the users SINRs are close to the asymptoticlimit. Simulation results for achievable rates are close to theoryfor even a 10-antenna base station with 3 or more users percell. The following summarizes the key contributions throughsimulation:

The theoretical results are derived assuming that the sameset of in-cell orthogonal training signals are repeatedacross the cells. However, we also show through simu-lations that in the case of independently generated butnon-orthogonal training signals(with orthogonal in-cell

training), the resulting SINR performance is close to theasymptotic limit.

In an example seven cell set up, the MMSE lter performsthe best in the absence of pilot contamination. We alsoshow an intermediate regime where the MMSE lter withpilot contamination obtains around 7 dB gain over thematched lter with a pilot contaminated estimate.

In terms of the ve percentile SINR, the MMSE re-ceiver is shown to provide signicant gains over matchedltering. Also in most of the operating points , theperformance of the MMSE receiver with pilot estimateis within 5 dB of the MMSE lter with perfect estimate.

We also show that the achievable rates are within a1 bit/symbol of the MMSE lter with perfect estimatewhen the number of users are comparable to the numberof antennas.

I I . S YSTEM M ODEL

We consider a system similar to that in [ 3] with B non-cooperating base stations and K users per base station. Weassume that all KB users in the system are allocated thesame time-frequency resource by a scheduler. Also, each basestation is equipped with M antennas. The channel vectorrepresenting the small scale fading between user k in cell j and the antennas in base station l is given by a M 1vector h ( l)

jk. The entries of h ( l)

jk are assumed to be independent

zero mean i.i.d Gaussian random variables with variance 1/M corresponding to the scaling of transmit power by the numberof receiver antennas at the base station. This corresponds to anideal and favourable propagation medium with rich scattering.A large scale fading coefcient, which represents the powerattenuation due to distance and effects of shadowing betweenbase station l and kth user in j th cell is given by ( l)jk . Weassume that ( l)jk < 1 as we do not expect the received powerto be greater than what is transmitted. This is constant acrossthe antennas of the cell l. Accordingly, overall channel vectoris given by ( l)jk h ( l)jk .

A. Uplink Transmission

We assume that all users transmission are perfectly syn-chronized. Also, while a users transmission is intended forits own base station, other base stations also hear the trans-mission. Dening q jk as the symbol transmitted by user k incell j , w ( l) as the M 1 noise vector with zero mean circularlysymmetric Gaussian entries such that E [w ( l) w (l)

H ] = 2I , the

received signal at base station l is given by,

y ( l) =B

j =1

K

k=1 ( l)jk h ( l)jk q jk + w ( l) . (2)Here, the Signal to Noise Ratio (SNR) of the received signalper receive dimension is given by 1/ 2M . In order to utilizethe advantages offered by multiple antennas, the base stationhas to have an estimate of the channel to all users prior todetection of uplink signals. In a system employing an OFDMphysical layer with time-frequency resources, we can dividethe resources into physical resource blocks (PRBs) containedin the coherence-time coherence-bandwidth product. Although

the channel vector h( l)jk of each user has to be relearned by the

base station at the start of PRB, once learnt for a subcarrierit remains the same for all subcarriers within that PRB. Letthe number of coherent symbols be given by T c and coherentsubcarriers be N c . We dene a Resource Element(RE) to be asubcarrier at a symbol time. Therefore, if we x the numberof resource elements used for estimation to be T such thatT T cN c , a total of T users channel can be learnt. Thisobservation was noted in [ 3]. We would like to point out thatit is relevant here as the number of users that can be supporteddepends on total available coherent resource elements N cT c .Depending on its value the number of users per cell K thatcould be supported can be comparable to M . We dene theload on the system as = K/M throughout this paper. Forexample, in a single cell set up even in very conservativescenarios of short coherence time with T c = 7 symbols andfrequency selective channel with N c = 14 , around 49 usersper cell can be supported at = 0.5 if half the resourcesare used for channel estimation. This is also illustrated ingure 1. Therefore, it is worthwhile to investigate not onlythe M K scenario but also the case when M and K largeand comparable.

B. Limitations in gaining Channel Knowledge

During each coherence time, users in a cell spend some pilot

symbol times in each PRB for channel estimation at the basestation and then data transmission ensues until the end of theblock. At base station l, the number of channel vectors h ( l)jkthat needs to be learnt is equal to the number of users in thesystem which is KB . In order to accomplish that, the numberof pilots required must at least be KB symbol times in orderfor the pilot sequences to be orthogonal across the users in thesystem. However, such a system will not be scalable as thereexists some large B for which the product KB will occupyall the coherent resource elements. In the example illustratedin gure 1 a 7 cell system with 14 users per cell would end upusing all the coherent resource elements if orthogonal channel


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Fig. 1. The PRB is composed of 98 orthogonal Resource Elements(RE) and12 RE for training. The rst user in the rst base station transmits pilots 11in the rst RE and remain silent for the rest of training duration. Similarly,other users in the rst cell transmit pilots during the RE allocated for itstraining. If orthogonal pilots are allocated for all the users in the system,a typical 7 cell system 14 users per cell would consume all the coherentRE without any time for data transmission. In our model we let the trainingsymbols of other cell users jk for all j and k simultaneously during 1 kleading to pilot contamination problem.

which represents the other cell interference and h 1k = h 1k +h 1k the received signal can be rewritten as,

y =K

k=1 1k h 1k q 1k +K

k =1 1k ,h 1k q 1k + z + w (9)where, h 1k is the result of pilot contamination. The MMSE

lter is then given by the expression,

c = E [yy H |h 1kk]1 E [y q 11 |h 1kk]

= S1 11 h 11 , (10)where,S =

K

k =2

1k h 1k h H 1k + ( 1 + 2 + 2)I , (11)

and,

1 I = E zz H =B

j =2

1M

K

k=1

jk I , (12)

2 I =K

k=1

1k E [h 1k h H 1k ]

=B

j =2

1M

K

k=1

jk 1k (k )

I . (13)

As seen from the expression for the lter in equation ( 10) , thelack of channel knowledge of other-cell users and only a partialchannel knowledge of in-cell users shows up as effectivenoise terms 1 and 2 respectively. In order to obtain theexpression we also use the properties of the MMSE estimate

that E h h H = 0 . In an ideal situation, the channel estimationincurs no error and h 1k = h 1k for all k , then

c * = K

k=1

1k h 1k h H 1k + ( 1 + 2)I

1

11 h 11 (14)This is an optimistic scenario which will serve as a benchmark for the performance of the MMSE lter with pilot contamina-

tion.

III. MMSE F ILTER WITH P ILOT C ONTAMINATEDE STIMATE

After processing the received signal using the linear lterc , let P signal , P noise (c ), P contam (c ), P inter (c ) denote the signalpower, noise power, pilot interference power and interferencepower respectively as a function of the lter c . It follows that,

P signal (c) = 11 c H h 11 h H 11 c (15)P noise (c) = 2c H c (16)

P contam (c) = cH B

j =2

j 1h j 1h H j 1 c (17)

P inter (c) = cH B

j =1

K

k=2

jk h jk h H jk c (18)

The received SINR is then given by the expression,

SINR = P signal (c )

P noise (c) + P contam (c ) + P inter (c) (19)

The motivation to dene P contam (c) as in equation ( 17) isbecause the user 1 of other cells is sending pilots in the sameresource element as the user 1 of the rst base station. Henceuser 1 for j = 2 , . . . B contribute to the interference in adifferent way as compared to the rest of the users in the system.This interference contribution is termed as pilot interferenceand is due to pilot contaminated channel estimate being usedto design linear lters. As per the denition the P contam (c) isdependent on the linear lter c and could be different for h 11and c .

Dene j as the random variable representing the largescale fading gain from an arbitrary user in the j th cell.Therefore, jk can be interpreted as the realization of j forthe user k and let =

Bi =1 i . Next, we state the main

theorem of the paper which gives the expression of SINR fora large system when an MMSE lter with a pilot contaminatedestimate is used to decode the received signal.

Theorem 1. As M, K , with K/M = , the SINR at the output of lter c given in equation (10) converges almost surely to

SINR =

111+ ( Bj =2 j 1 ) / 11

2 + (Bj =2

2j 1 )/ 11

1+ ( Bj =2 j 1 )/ 11+ (E [ ] C( ))

(20)

where, the constants C() , 1 , 2 are given by

C() = E 21

21

1 + 21

1+

21

E 21

Bj =2

2j

1 + 21

1


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+21

E 21

Bj =2

2j

1 + 21

12 , (21)

1 = 2 + E [ ]E 21

21

1 + 21

1

1

, (22)

2 = 21 E 21

1 + 21

1

2 1

. (23)

Proof: Proof given in Appendix B.We will see in a large system that Theorem 1 characterizes

the effect of pilot interference power and interference aver-aging. Specically, in order to put Theorem 1 into properperspective we state two propositions which are the resultsfor SINR of a large system with MMSE lter employing aperfect estimate and a matched lter with pilot contaminatedestimate respectively.

Proposition 2. As M, K , with K/M = , the SINR at the output of lter c * given in equation ( 14) converges almost surely to

SINR * = 11 1 = 11

2 + N j =2 E [ j ] + E 1

1+ 1

1

(24)

where, 1 = 2 + Bj =2

E [ j ] + E 1

1+ 1

1

1 .

Proof: We state the proposition without proof as it isstraightforward to obtain it from the large system analysistechniques used for CDMA systems in [11], [13].

SINR * is the SINR with MMSE ltering with a perfect

channel estimate to its own users. This is best case scenarioas compared to the MMSE with a channel estimate. We donot expect the SINR of MMSE lter with estimate to exceedthis SINR * .

Proposition 3. As M, K , with K/M = , the SINR at the output of lter c = h 11 converges almost surely to

SINR =

111+ ( Bj =2 j 1 )/ 11

2 + (Bj =2

2j 1 ) / 11

1+ ( Bj =2 j 1 ) / 11+ E [ ]

. (25)

Proof: Proof given in Appendix C.It is interesting to see that the expression for SINR converges

to a similar expression to the result in matched ltering SINR .If P signal = 211 /

Bj =1 j 1 , P contam =

Bj =2

2j 1 /

Bj =1 j 1

and

P inter (c ) =E [ ], c = h 11E [ ] C( ), c = c

(26)

then we can dene the expression for the asymptotic SINRwith matched ltering and MMSE ltering with pilot contam-inated channel estimate as,

SINR (c) =P signal

2 + P contam + P inter (c ). (27)

Here, P signal is the effective signal power, P contam is termedas the pilot interference power and is the consequence of employing pilot contaminated channel estimate. In the samelines as in [11], the lter dependent term P inter (c ) is calledthe interference averaging term and is relevant when = 0 .Although, M implying that the channel between the user1 is asymptotically orthogonal to channel between any otherusers in the system, because K

with K/M = , the

contribution to sum interference from all the users is non-zeroand is given by P inter (c ) for a linear lter c . If 1 = 11 (1)

2,

then the following equations shows the relationships betweenthe P signal and P contam to that of denitions in equation (15)and ( 17) :

P signal (h 11 ) 1 =

P signal (c )P signal 21

= P signal (28)

P contam (h 11 ) 1 =

P contam (c )P signal 21

= P contam (29)

It is seen that for both the lters h 11 and c , their respective sig-nal powers given by P signal (h 11 ) and P signal (c ) are just scaledversions of P signal . Similarly, P contam (h 11 ) and P contam (c ) arethe scaled versions of P contam . The following can be concludedfor both matched lter as well as MMSE lter with a pilotcontaminated estimate:

The contribution of the pilot contaminated channel esti-mate to interference is given by P contam is same for bothmatched lter and MMSE lter.

The contribution of pilot interference is independent of and is equal to P contam for all values of .

As trivial as the above comments may seem it is not obviousfor the lter c from the denition of lter dependent pilot

interference power P contam in equation ( 17). This is becausethe matrix S in lter c is dependent upon the channel of all the users in the system through the pilot contaminatedchannel estimate h 1k for all k = 1 , . . . , K . However as wesee in the appendix B, the contribution of the matrix S can besummed up into the constant 1 for a large system. Further,the following relations can also be obtained on the interferenceaveraging term P inter (c ) when c = h 11 and c = c :

P noise (c ) + P inter (c )P signal 21

= 2 + P inter (c ), (30)

P noise (h 11 ) + P inter (h 11 ) 1 =

2 + P inter (h 11 ). (31)

We have shown that P inter (h 11 ) P inter (c ) = C() 0implying that interference suppression of amount C() can bealways achieved. The amount of interference suppression C()is of course depended on through equation (21).

The SINR expression in the limit of innite antennas butnite number of users per cell are obtained when we put = 0in equations ( 20) and (25) or in (27) . This corresponds to asimilar expression as for downlink SINR in [ 5, Equation 16].Since the SNR of the uplink received signal in equation (2)is given by SNR = 1/ 2 , it is seen that the SINR expressionso obtained is limited by the pilot interference powers at high


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SNRs and we obtain [3, Equation 13]. This need not necessar-ily be achieved by transmitting at high power per symbol orequivalently by having 2 0. Had the transmission powerof data symbols be not scaled by the number of antennas, theSNR of the received signal would be linearly increasing in M and when = 0 we again obtain [3, Equation 13].

The deterministic equivalent for SINR in Rayleigh fadingfor MMSE lter and the corresponding expression

C( ) rep-

resenting the interference suppression power are our maincontribution of this paper. The terms E[ ] and C( ) can becomputed ofine for a system with the knowledge of largescale fading distribution. It can also be estimated without theknowledge of the large scale fading distribution from userrealizations over time. Depending on the value of E [ ]C()and the operating point a decision can be made whether touse an MMSE lter or an matched lter. As we will see inthe next section, the interference suppression obtained withan MMSE lter is necessary in increasing the outage SINRand achievable rate of the system, when there are considerablenumber of users as represented by the ratio > 0.1. This is as

opposed to the regime in which antennas far outnumber users.In this operating point 0, and then MMSE lter itself maynot be necessary as pilot signals are the main contributor tointerference. This regime with 0 has been well exploredin recent studies in [ 3], [8], [14] , [16] . As mentioned earliermost of our focus for performance analysis is on the regime > 0.1 although the results are perfectly valid for any 0.Across the users in the system the SINR is a random variableby virtue of different received powers of both the signal andthe interferers contributing to pilot contamination. Also, thepilot interference power is random by virtue of the choice of the interferers contributing to pilot contamination.

IV. P ERFORMANCE A NALYSIS

For the numerical evaluation, we consider hexagonal cellswith users uniformly distributed in each of the cells, as shownin Fig. 2. We consider a scenario where 6 closest cells areinterfering with the center cell. We assume 1k = 1 so thatreceived powers from all the users within a cell are unity. Weconsider a high SNR of 20 dB and the received powers fromall the users in other cells are assumed to take a constantvalue of jk = 0 .001, or 0.01, or 0.1 for j = 1 . Theserepresent the contribution of other cell interference for threedifferent idealized scenarios. The interference from other cellsis strong as jk is close to 1. We consider the SINR for the

user one in the center cell. Figs. 3, 4 plot the asymptotic SINRof the MMSE with a pilot corrupted estimate given by SINRfor the case of different received powers. Although in theorythe effect for small scale fading vanishes only with innitenumber of antennas it is seen in Fig. 4 that even for a 50-antenna base station the actual SINR realizations obtainedthrough simulations are very near to the asymptotic limit.We also plot SINR and SINR * as baseline for performancecomparisons. In Fig. 3, it is seen that SINR * is already affectedby other cell interference due jk = 0 .1 for j = 1 . Hence, the SINR is not expected to perform better than that and there isfurther 4 dB loss due to pilot contamination. However, in the

Fig. 2. In the favourable case, the sum of the received powers of interfererscontributing to pilot contamination are very less as compared to that of thedesired user. User 1 in the center cell represents such a scenario. The SINRwith a pilot corrupted estimate is then comparable to that of perfect estimate.On the other hand for user 2 in the center cell, the pilot interferers receivedpowers are comparable to that of the desired user and represents the worstcase scenarios.

other extreme case when the other cell jk s are close to zero,the channel estimate is already better and the SINR performsclose to SINR * . Useful gains employing an MMSE lter withpilot contaminated channel estimate can be obtained when theother cell jk s are neither close to zero or close to unity. Inthis example when the jk = 0 .01 for j = 1 , around 7 dBgains are possible in comparison with matched lter with pilotestimate when operating at = 0.5 as seen from g. 4. While

there is a loss of 3 dB with respect to the perfect MMSE duenature of channel estimate, the reader is reminded that this is aworse case loss. The curves closes in as we decrease whichrepresents the M K scenario and also when increasesas in that case the limitation is now the averaged interferenceterm.

In g. 5 we plot the achievable sum rate for users in therst cell with each users SNR being 20 dB. We assume largeenough coherence time so that the training time need not betaken into account. This is because our focus is on the sumrate achievable with variation in . However, if necessary thesum rate can be easily adjusted based on training overheadwhen coherence time is a signicant factor. We x the number

of antennas and calculate the sum rate with varying asM log2(1 +

SINR ). The three curves corresponds to thereceived powers of all users from other cells being either jk = 0 .001, or 0.01, or 0.1 for j = 1 assuming unit receivedpower from the in-cell users. Sum rates of over 20 bits/symbolare achieved for users when the other cell received powersare below 10 dB of the in-cell received powers. Also, thesimulation with 50-antenna base station is seen to match thetheoretical rates predicted for this set up. The interferencelimited system has the exibility to serve a large number of users at low SINR or a few number of users at a high SINRdepending on the operating point . The plot suggest a optimal


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0 0.5 1 1.510

5

0

5

10

15

2010 dB othercell rcvd powers

S I N R

( d

B )

Perfect/MMSEPilot/MMSEPilot/Matched

Fig. 3. The plot shows the asymptotic SINR of the rst user in the rstcell when MMSE lter with pilot contaminated estimate is used to decodethe received signal along with the baseline comparison criterion of MMSEwith a perfect estimate and matched lter with a pilot contaminated estimatefor an idealized seven cell set up. It is seen that when the other cell receivedpower is just 10 dB lower than that in cell users the MMSE lter with pilotcontaminated estimate performs close to its corresponding matched lter. Thisis because the limitation is now the other cell interference which MMSE lteris not designed to suppress. Hence, we do not expect the MMSE lter withpilot estimate to be useful when j > 0 .1

0 0.5 1 1.55

0

5

10

15

20

2520 dB othercell rcvd powers

S I N R

( d

B )

Perfect/MMSEPilot/MMSEPilot/MatchedPilot/MMSE (Sim)

Fig. 4. The plot shows the asymptotic SINR of the rst user in the rst cellwhen MMSE lter with pilot contaminated estimate is used to decode alongwith the baseline comparison criterion of MMSE with a perfect estimate andmatched lter with a pilot contaminated estimate for an idealized seven cellset up. In this case when the other cell received powers is 20 dB lower than

that of in-cell received powers signicant gains are obtained as compared toa matched lter with pilot contaminated estimate

operating point for which the sum rate is maximum. Forexample when j = 0 .01 and = 0 .8 gives a sum rate of around 88 bits/symbol. Larger causes the SINR to be lowerso that the term outside the log2 is ineffective to increasesum rate while a lower implies that less users are served andhence lesser sum rate. When the other-cell received powers arelarge, the curve attens and the sum rate is constant for mostof the operating points .

0 0.5 1 1.50

20

40

60

80

100

120

140

160

S u m

R a t e

( b i

t s

/ s y m

)

0.001

0.01

0.1

Fig. 5. The plot shows the sum rate of the users in rst cell for differentoperating points of corresponding to a SNR of 20 dB. The three curvescorrespond to three different received powers from other cell users. Themarkers correspond to simulation with 50 -antenna base stations and matchthe theoretical predictions.

In g. 6, we plot the difference of achievable rate per userbetween MMSE lter with a perfect estimate and MMSE lterwith a pilot estimate for different values of other cell interfer-ence power. We do not take into account the training overheadfor comparison assuming we obtain the perfect estimate withthe same training time. We limit ourselves to j < 0.1 since SINR is already close to SINR otherwise. Also in j > 0.1regime, there is signicant other cell interference which boththe perfect estimate based and the pilot based MMSE lterare not designed to suppress, thereby affecting the achievable

rates. We plot ve different curves corresponding to systemoperating points . As seen earlier the total interference withlter c is given by P contam + P inter (c ) and with c* the in-terference power is given by Bj =2 E [ j ] + E

11+ 1

1.

When is small then the signicant loss of rate with c isdue to P contam as the lter dependent interference power forboth lters in negligible. On the hand, when is large P interdominates the P contam and since both the lter are affected byP inter (c ) the rate difference is small. Although when = 1there is only a 0.4 bits/symbol difference the sum rate will beaffected differently. For example in a 50-antenna base stationwith 50 users at j = 0 .1 this could mean that sum rate withpilot contaminated MMSE lter is 20 bits/symbol lower thanthat perfect MMSE lter. On the other hand when = 0 .2,the sum rate difference is 12 bits/symbol.

A. Effect of Pilot Contamination

Through a couple of typical possible realizations of userpositions, we explain the effect of pilot interference in SINR ,SINR and compare it with that of the SINR with a perfectestimate. For illustration, in Fig. 2 consider only distance basedpathloss in large scale fading although the result holds whenshadowing is also present. This is applicable to both matched


9/13

9

0 0.02 0.04 0.06 0.08 0.10

0.2

0.4

0.6

0.8

1

1.2

1.4

j

R a t e

( b

i t s

/ s y m

)

0.20.40.60.81

Fig. 6. The plot shows the loss of rate due to pilot contamination when anMMSE lter with pilot contaminated estimate is used to decode the receivedsignal as compared to perfect estimate MMSE lter. The different curvescorrespond to different values of from 0 .2 to 1 . It is seen that smallerthe the MMSE lter with pilot contaminated estimate performs worse than

the ideal MMSE lter. However, the sum rate for users per base station isdifferent.

lter and MMSE lter. Consider the rst scenario when

Bj =2 j 1 11

1

Bj =2

2j 1

111 (32)

This corresponds to the fact that sum of received powers of theinterferers are much less that that of desired user power. Underthese conditions the SINR of the received signal in equation(25) is,

SINR 11

2 + (E [ ] I )where, I = 0 if matched lter is employed or I = C() if MMSE lter c is employed. As we will see in the next section,typically scenarios show that the interference suppressionpower C() is almost same as what could have been witha lter c* . Hence the SINR of the lter with the corruptchannel estimate is as good as the SINR with a perfect channelestimate. In Fig. 2, the situation of user 1 in the centercell represents the favourable scenario with the interfererscontributing to the pilot contaminated channel estimate arefar such that the condition (32 ) is satised. On the other handif gains of all the interferers are comparable to that of thedesired users, i.e.,

Bj =2 j 1 11 B 1 (33)

then pilot interference contributes negatively to the SINRin addition to interference averaging. This is represented byrealization of user 2 of the center cell in Fig. 2. Therefore, wecan conclude that, as compared to the linear lter with perfectestimate, the lter with a pilot estimate has higher probabilitythat it is less than a given SINR.

0 0.2 0.4 0.6 0.8 110

5

0

5

10

S I N R

( d

B )

Per./TheoryEst./TheoryPer./SimEst./SimDiff.

j /Sim

Fig. 7. The plot shows the ve percentile SINR of the rst user in the rstcell for a seven cell set up in a non-idealized scenario. The received powersfrom users can be different depending on their positions and shadowing andhence the received SINR is random. The theoretical curves are matched withsimulation. The details are described in section IV-B.

B. Five Percentile SINR

In the earlier section we showed that pilot contaminationhas the effect of reducing the outage SINR. In order to getmore intuition under practical scenarios of large scale fadinggains, we consider the seven cell model with cell radius isR = 1 km, and assume a COST231 model for propagationloss between the base station and the users. The noise poweris assumed to be 174 dBm and user transmit power of 23 dBm. We plot the ve percentile of the SINR in Fig. 7 forthe perfect MMSE ltering given by SINR * and MMSE lter

with pilot contamination given by SINR , for varying values

of . To that extend, we compute the interference terms E[ ]and C() ofine by averaging over a sufcient number of user positions. Also, 1 and 2 can be computed ofine fordifferent values of as they are constant for the system anddependent on the large scale fading characteristics. Notice that SINR is devoid of the small scale fading parameters. Also, forSINR * we compute the terms, 1 and the average interferenceE [ ] E

21

11+ 1

1. It is found that E[ ] C = 38 dB and

E [ ] E 21

11+ 1

1 = 36 dB. This shows that in terms of

the interference suppression the performance of both lters c *

and c are almost same. In order to compare the theoretical

expression we also plot the ve percentile SINR which isgenerated using simulations. These involves computing theSINRs for various small scale fading channel realizationsalong with large scale fading. For the simulation we use50 antenna base stations each serving a different number of users corresponding to different . The channel estimate isbased on same in-cell orthogonal training sequences beingrepeated across the cells. Further, we also compare the vepercentile SINR obtained out of actual channel estimation. Inorder for that we assume different independently generated in-cell orthogonal training sequences which are non-orthogonalacross the cells. We perform an MMSE estimation of the


10/13

10

M = 50 M = 10

R per R pilot R per R pilot0 .1 6 .0 4 .70 .2 5 .0 4 .00 .3 4 .3 3 .4 4 .4 3 .60 .4 3 .8 3 .0 4 .0 3 .20 .5 3 .4 2 .7 3 .6 2 .90 .6 3 .1 2 .5 3 .3 2 .70 .7 2 .8 2 .3 3 .0 2 .40 .8 2 .64 2 .1 2 .7 2 .20 .9 2 .4 1 .9 2 .4 2 .01 .0 2 .2 1 .9 2 .3 2 .0

TABLE ITHE TABLE SHOWS THE ACHIEVABLE RATE (BITS / SYMBOL ) WITH MMSE

FILTERING FOR A 50 A ND 10 -ANTENNA BASE STATION SERVINGDIFFERENT NUMBER OF USERS . R per CORRESPONDS TO RATE WITH A

PERFECT ESTIMATE AND R pilot CORRESPONDS TO THE PILOTCONTAMINATED FILTER .

channel and generate MMSE lter using the actual channelestimate given by equation ( 5).

It is seen through Fig. 7, that SINR in equation ( 20) matchesthe ve percentile SINR obtained through simulation and istypically less by about 0.3 dB of the theoretical expression.This is true for both in-cell pilot sequences being repeatedacross the cells as well as different and independent pilot train-ing sequences across the cells. This also implies that the evena 50 antenna base station is large enough for the theoreticalpredictions to be effective in addition to being independentof the effect of small scale fading in the resulting SINR. Aswe increase the number of antennas the theoretical expressionexactly matches the SINR obtained through simulation. Also,the MMSE lter with pilot contamination performs just 5 dBbelow the MMSE lter with perfect channel estimate andthis gap is unambiguously a result of the pilot contaminatedchannel estimate. Also it is seen that even at = 1 , whichimplies a heavily loaded system the ve percentile SINR is

9 dB which is well within the sensitivity of base stationreceivers.Table I shows the achievable rates per symbol for a user

in the central cell using MMSE based detection with perfectestimate and pilot contaminated estimate. The achievable rateis given by R = E[log(1 + SINR )] which here is calculatedby averaging the instantaneous rate over 2 103 realizationsof user positions for user 1. This is the same for all users inthe central cell. Both theory and simulations based on 50 basestations antennas serving different number of users agree to

the numbers shown in the table. It is seen that the differencebetween them is approximately 1 bit/symbol for small >0.1 and closes in when it increases. However, for 1,the difference between the rates increases as effect of pilotinterference will never let the SINR approach SNR.

Further, Table I, also shows the simulated results for theachievable rate for a 10-antenna base station with 3 to 10users. It is seen that even for a 10 antenna base station, thesimulated results agree closely to the earlier results obtainedfrom theory in Table I. This highlights the usefulness of thelarge system analysis in providing accurate predictions forachievable rates for not necessarily large number of antennas

but also contemporary MIMO systems. However, we wouldlike to point out that with more number of antennas we canserve more users at the same rate given below.

V. C ONCLUSION

In this work we found the expression for SINR for a largesystem when a MMSE lter with a pilot contaminated estimate

is employed to decode the received signal. We validated theexpression through simulations and showed that a 50-antennabase station serving different number of users is sufcientenough to employ our large system results. We characterizedthe effect of pilot contamination in that it has the effectof reducing the ve percentile SINRs as compared to theMMSE with perfect estimate for all values of . We alsofound an explicit expression for the interference suppressionpower due to MMSE lter and compared it with that of matched lter. We showed that ve percentile SINR of theMMSE with a pilot contaminated estimate is within 5 dB of MMSE lter with a perfect estimate. We also found that theresults with actual channel estimation match the theoretical

results. In this work we have assumed that the training timeallocated to users is K symbols. In future work, we wish tostudy the training overhead for different values of trainingtime and users depending on the coherence time. We believethis can be done through a large system analysis of trainingtime versus the number of users per cell. It would alsobe interesting to see if the considerable work done by theauthors in [14], in getting a generalized expression for thedeterministic equivalent of the SINR can be further simpliedinto intuitive expressions for other channel models. This willbe of help in realizing engineering conclusions tailored fordifferent channels models like distributed antennas, correlatedantennas, distributed sets of correlated antennas [ 10] to name a

few. Also, recent work has proposed that pilot contaminationas an artefact of linear channel estimation techniques [ 18] .While they have provided a theoretical understanding in anideal situation, practical solutions applicable to a regime withlarge number of users are still to be found. Algorithms frommulti-user detection for CDMA systems are a useful tool whenwe have enough coherence time [ 21]. We are currently lookinginto such adaptive algorithms that could be implemented forshort coherence time scenarios.

A PPENDIX AR ESULTS FROM L ITERATURE

In this section we briey describe the necessary results fromliterature to derive the asymptotic SINR expressions. Theseresults were used previously in the context of CDMA systemsin [11 ][13] .

Lemma 4. [13, Lemma 1] If S is a deterministic M M matrix with uniformly bounded spectral radius for all M . Let q = 1 M q 1 q 2 . . . q M

T where q i s are i.i.d complex

random variables with zero mean, unit variance and niteeight moment. Let r be a similar vector independent of q .Then,

q H Sq 1

M trace {S}, (34)


11/13


12/13

12

trace {Z} = traceB

j =1

K

k =2

jk S 1H k e j e H j H H k S1 (55)

(a )=

B

j =1

K

k =2

jk e H j HH k S 1

2H k e j (56)

(b)

=

B

j =1

K

k =2 jk e

H j H

H k S

2k I 2

k H k a k a H k HH k S1k

1 + 2k a H k H H k S1k H k a k + 2k H k a k a H k H H k S1k H k a k a H k H H k S1k

(1 + 2k aH k H

H k S1k H k a k )2

H k e j (57)

=B

j =1

K

k =2

jk e H j HH k S2k H k e j 2

K

k=2

jk k e H j H H k S2k H k a k a H k H H k S1k H k e j

1 + k a H k HH k S1k H k a k

+

K

k=2

jk 2k e H j H H k S2k H k a k a H k H H k S1k H k a k a H k H H k S1k H k e j

(1 + k a H k HH k S1k H k a k )2

(58)

trace {Z}M a.s.

B

j =1

E j 2 2 1

2

j 121 + 1

21

+ 1

4

j 221

1 + 1 21

2

= B

j =2

E j 2 + B

j =1

E j 1

2 E 1

2 Bj =1 j 12

1 + 1 21

E 1

2 Bj =1 j 12

1 + 1 21

2

= ( 1 + 2)2 + E 21 2

1 + 21

12 E

21

Bj =2

2j 12

1 + 21

1E

21

Bj =2

2j 12

1 + 21

12 (62)

=

0

+ 1 + 2

( + 1 + 2 + 2)

2 dG()

E

21

Bj =2

2j 12

1 + 2

1 1 E

21

Bj =2

2j 12

1 + 21

12 . (63)

From equations (51), (52) and using the denitionof Stieltjes transform [ 24] we can nd that 1 =limz2 + 2 + 2 m(z), and since m(z) is complex analytic2 = lim z2 + 2 + 2

ddz m(z). The values of 1 and 2

can then also be obtained from solving equation ( 48) and itsderivative. The equations are given by,

1 = 2 + E [ ]

E

21

21

1 + 21

1

1

, (53)

2 = 21 E 21

1 + 21

1

2 1

. (54)

Similarly, to evaluate the expression for interference P inter ,trace {Z}can be expanded as in equation (58 ), where, in step(a) we use trace {qq

H

}= qH q and if

S k =k=1 ,k

k H k a k a H k HH k + ( 1 + 2 +

2)I (59)

then in step (b) use matrix inversion lemma as,

S 1 = S 1k I k H k a k a H k H H k S1k1 + k a H k H

H k S1k H k a k

. (60)

Notice that using Lemma 4, the terms H H k S 1k H ka.s.

1 Iand H H k S2k H k

a.s.

2I and it appears repeatedly in equa-tion ( 58) . Therefore, in the limit of innite number of antennas,and for a given , using Lemma 4 we have

H H 1 ZH 1a.s.

trace {Z}

M I (61)

and trace {Z}/M in turn converges to the expression inequation ( 63) . In equation ( 62) the third term E

21 2

1+ 21 1

2

is equal to 0

( + 1 + 2 + 2 ) 2 dG(). This follows from corol-lary 7.

Using equations ( 49), (50), (63) in expressions(40) , (41), (42) and (44) the SINR given in equation (19)converges almost surely to SINR as in expression (20).


13/13

13

A PPENDIX CPROOF OF P ROPOSITION 3

With 1 = 11 (1)2, using the matched lter given by

c = 11 (1) H 1a 1 , the signal power, noise power, pilot interfer-ence power is given by P signal = 1 a 1H H 1 H 1e 1

2, P noise =

1 11

2a H 1 HH 1 H 1a 1 , P contam =

1 11

Bj =1 j 1 a

H 1 H

H 1 H 1e j

2.

Since, H H H a.s.I as M , we have, P signal a.s. 1 11 ,P noise a.s. 1 11 (1) 2 , P contama.s.

1 11Bj =2

2j 1 . Using

Lemma 4 in the interference term we have,

P inter = 1

11a H 1 H

H 1

B

j =1

K

k =2

jk h k h H k H 1a 1 ,

a.s.

1

11a H 1

1M

traceB

j =1

K

k=2

jk h k h H k I a 1 ,

= 1

11

B

j =1

j 11K

B

j =1

K

k=2

jk h H k h k ,

= 1

11

B

j =1

j 1B

j =1

E [ j ] . (64)

Rearranging the terms in equation (19) we get the expressionfor SINR .

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