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BER Analysis of Multi-Cellular MIMO Systemswith Increasing Number of BS Antennas

José Carlos Marinello, Taufik Abrão

Abstract—In this work, salient characteristics of a wire-less communication system deploying a great number ofantennas in the base station (BS), namely Massive MIMOsystem, are investigated. In particular, we found a simpleand meaningful relationship that corroborates a funda-mental assumption in recent related works: according thenumber of BS antennasN grows, the product of the small-scale fading channel matrix with its conjugate transposetends to a scaled identity, with a variability measureinversely proportional to N . Furthermore, analysis of theMassive MIMO system downlink is carried out from a bit-error-rate (BER) performance viewpoint, including somerealistic adverse effects, such as interference from neigh-boring cells, channel estimation errors due to backgroundthermal noise, and pilot contamination, which was recentlyshown to be the only impairment that remains in theMIMO multicell system with infinite number of BS anten-nas. Our numerical result findings show that, in the sameway as with the sum capacity, the pilot contamination alsolimits the BER performance of a noncooperative multi-cellMIMO system with infinite number of BS antennas.

Index Terms—Massive MIMO systems, BER, MultiuserMIMO, Precoding, MMSE.

I. INTRODUCTION

M ULTIPLE-input-multiple-output (MIMO) tech-niques constitutes one of the key features in

most of the recent telecommunications standards, suchas WiFi, WiMAX, LTE [1], [2], due to the large gainsin spectral/energy efficiency they can offer. In particular,multiuser MIMO (MU-MIMO) systems have attractedsubstantial interest since they can achieve spatial multi-plexing gains even when serving single antenna mobileterminals (MT’s) [3]. Advantages of MU-MIMO alsoincludes a larger robustness to most of propagationlimitations present in single-user MIMO, such as antennacorrelation or channel rank loss. Even when the channelstate information (CSI) of some users are highly corre-lated, multiuser diversity can be extracted by efficient

J. C. Marinello is an Master of Science candidate at the Electri-cal Engineering Department, State University of Londrina,. E-mail:zecarlos.ee@gmail.com

T. Abrão is an Associate Professor at the Electrical Engineer-ing Department, State University of Londrina, PR, Brazil. E-mail:taufik@uel.br

techniques of scheduling the time/frequency resources,yielding to a better exploitation of the additional degreesof freedom (DoF) propitiated by the antenna array at theBS. From an information theoretic point of view, gainsof these systems have been demonstrated in [4].

In order to fully exploit the advantages of MIMOsystems, a certain number of recent technical workshave been concerned with the possibility of increasingthe number of BS antennasN to infinity. While earlypapers have focused on the asymptotic limits of suchsystems for pure academic interest, practical issues ofimplementation are each time more present on the latestpapers, maturing the technology and turning it gradu-ally ready to be considered in next telecommunicationsstandards, such as 5G [5]. It was shown in [6] that ina time division duplex (TDD) noncooperative multi-cellMIMO system, that employs uplink training pilots forCSI acquisition, with infinite number of BS antennas,the effects of uncorrelated thermal noise and fast fadingare averaged out. The only factor that remains limitingperformance in Large MIMO is inter-cell interference,which when associated with the finite time available tosend pilot sequences makes the estimated CSI at oneBS “contaminated” by the CSI of users in adjacentcells, in the so-called pilot contamination effect. Thisphenomenon results from unavoidable re-use of reverse-link pilot sequences by terminals in different cells. As aconsequence of increasing the number of BS antennas toinfinity, the transmit power can be designed arbitrarilysmall, since interference decreases in the same rate ofthe desired signal power, i.e., signal-to-interference-plus-noise ratio (SINR) is independent of transmit power [6].

Alternative strategies to achieve better CSI estimatesexist, such as a) frequency division duplex (FDD) [7],in which pilots for CSI acquisition are transmitted indownlink, and fed back to BS in a feedback channel;and b) network MIMO [8], where CSI and informationdata of different coordinated cells are shared among themin a backhaul link, creating a distributed antenna arraythat serves the users altogether. However, both schemesbecomes unfeasible whenN → ∞, since lengths offorward pilot sequences and capacity of backhaul linksincrease substantially withN , respectively [6].

Operating with a large excess of BS antennas com-

2

pared with the number of terminalsK is a challengingbut desirable condition, since some results from randommatrix theory become noticeable [9], [10]. It is known,for instance, that very tall/wide matrices tend to be verywell conditioned, since their singular values distributionappears to be deterministic, showing a stable behavior(low variances) and a relatively narrow spread [11].This phenomenon is quite appreciable to enhance theachievable rates of such systems. Besides, the mostsimple uplink/downlink techniques, i.e., maximum ratiocombining (MRC) and matched filtering (MF) precod-ing, respectively, becomes optimal [11]. The savings inenergy consumption are also remarkable. In the uplink, itis shown in [12] that the power radiated by the terminalscan be made inversely proportional toN , with perfectCSI, or to

√N , for imperfect CSI, with no reduction

in performance, propitiating the implementation of veryenergy-efficient communication systems.

An interesting investigation about precoding tech-niques to be deployed in the downlink of the nonco-operative multi-cell MU-MIMO systems is performedin [13]. Specifically, authors compared MF precoding,also known as conjugate beamforming, and zero forcing(ZF) beamforming, with respect to net spectral-efficiencyand radiated energy-efficiency in a simplified single-cellscenario where propagation is governed by independentRayleigh fading, and where CSI acquisition and datatransmission are both performed during a short coher-ence interval. It is found that, for high spectral-efficiencyand low energy-efficiency, ZF outperforms MF, whileat low spectral-efficiency and high energy-efficiency theopposite holds. A similar result is found for the uplinkin [12], where for low signal-to-noise ratio (SNR), thesimple MRC receiver outperforms the ZF receiver. It canbe explained since, for reduced power levels, the cross-talk interference introduced by the inferior maximum-ratio combining receiver eventually falls below the noiseenhancement induced by zero forcing and this simplereceiver becomes a better alternative. The analog occursfor downlink.

A more rigorous formula for the achievable SINRin Massive MIMO systems is derived in [14]; besides,authors discuss an efficient technique for temporallydistribute the uplink transmissions of pilot sequences,avoiding simultaneous transmissions from adjacent cellsand reducing interference as well, in conjunction withpower allocation strategy. The achieved gains in the sumrate are about 18 times, remaining, however, limitedto the increasingN . On the other hand, a precodingtechnique that eliminates pilot contamination and leadsto unlimited gains withN → ∞ is proposed in [15].However, these gains come at the expense of sharing

the information data between base stations, what canoverload the backhaul signaling channel for high ratesystems, or high number of users per cell.

In [16], the problem of pilot contamination in non-cooperative TDD multi-cell systems is also investigated,and authors derived a closed form expression for a multi-cell minimum mean squared error (MMSE) precodingtechnique, that depends on the set of training sequencesassigned to the users. The technique is obtained asthe solution of an objective function consisting of themean squared error of signals received at the usersin the same cell, and the mean squared interferencecaused at the users in other cells. Hence, achieved gainsare resultant of both intra-cell interference and inter-cell interference reduction, while not requiring excessivesharing of information in backhaul signaling channels.

Many practical difficulties arise when trying to designa BS equipped with a massive number of antennas [11],such as size limitations, correlation and mutual couplingbetween antennas1, energy consumption of RF circuits,among others. Due to these limitations, some works haveinvestigated the benefits of a high dimensional MIMOsystem with a limited number of BS antennas [18], [19].In [18] it was analyzed to which extent the conclusions ofMassive MIMO systems hold in a more realistic setting,whereN is not extremely large compared toK. Authorsderived expressions of how many antennas per MT areneeded to achieve a given percentage of the ultimateperformance limit with infinitely many antennas, andshowed by simulations that MMSE/regularized zero forc-ing (RZF) can achieve the ultimate performance of thesimple MRC/MF schemes with a significantly reducednumber of antennas. An improved network MIMO ar-chitecture is proposed in [19], that achieves MassiveMIMO spectral efficiencies with an order of magnitudefewer antennas. The proposed technique combines co-operation among base stations located in small clusters,ZF multiuser MIMO precoding with suitable inter-clusterinterference mitigation constraints, uplink pilot signalsallocation and frequency reuse across cells.

When considering that user channels are not inde-pendent, but correlated, it can be modelled as a finite-dimensional channel in which the angular domain isseparated into a finite number of distinct directions [20],or angular bins. This more realistic scenario can be foundwhen BS antennas are not sufficiently well separated, orthe propagation environment does not offer rich enoughscattering. It is shown in [20], for the uplink of anoncooperative multicell MU-MIMO, that the system

1These effects become more adverse when BS antennas are dis-tributed in more than one dimension [11], [17].

3

performance with infinite number of BS antennas buta given finite numberΩ of angular bins is equivalentto the performance of a uncorrelated system withΩBS antennas. Indeed, such systems saturates not onlydue to pilot contamination, but also with the finitedimensionality of the channel. Finally, authors foundlower bounds on the capacity of these systems with linearMRC and ZF detectors performed at the receiver side.

In our work, we analytically demonstrate that a fun-damental assumption made in most of works cited aboveholds, i.e., when the number of BS antennas grows, theinner products between propagation vectors for differentterminals grow at lesser rates than the inner products ofpropagation vectors with themselves. In particular, wefind that the product of the small-scale fading matrix withits conjugate transpose tends to a scaled identity, witha variance inversely proportional toN . Furthermore, asmost of previous works have investigated the noncooper-ative multi-cell TDD MU-MIMO system under the sum-capacity viewpoint, herein we adopt the bit-error-rateperformance as salient figure of merit. Our main resultindicates that, in the same way as it occurs with the sum-capacity performance, the BER performance of MassiveMIMO systems saturates (BER floor effect) with increas-ing number of BS antennasN when considering pilotcontamination.

II. SYSTEM MODEL

In the adopted MIMO system, it is assumed that eachone ofL base stations withN transmit antennas commu-nicates withK users equipped with a single-antenna MT.Note that theL base stations share the same spectrumand the same set ofK pilot signals. We denote the1×Nchannel vector between theℓ-th BS and thek-th user ofj-th cell bygℓkj =

√βℓkjhℓkj, in whichβℓkj is the long-

term fading power coefficient, that comprises path lossand log-normal shadowing, andhℓkj is the short-termfading channel vector, that followshℓkj ∼ CN (0, IN ).Flat fading environment was assumed, where the channelmatrix H is admitted constant over the entire frameand changes independently from frame to frame (blockfading channel assumption). Note thatβℓkj is assumedconstant for allN BS antennas. Since time divisionduplex is assumed, reciprocity holds, and thus channelstate information is acquired by means of uplink trainingsequences. During a channel coherence interval, thesymbol periods are divided to uplink pilot transmissions,processing, and downlink transmission. Using orthogonalpilot sequences, the number of sequences available isequal to its length, and thus, due to mobility of the users,the number of terminals served by each BS is limited.The same set ofK orthogonal pilots of lengthK is

used in all cells; hence, for thek-th user of each cellis assigned the sequenceψk = [ψ1,kψ2,k . . . ψK,k], suchthat |ψi,k| = 1 and |ψH

k′ψk| = δkk′ since the sequencesare orthogonal, where·H is the conjugate transposeoperator, andδkk′ is the Kronecker’s delta, i.e.,δkk′ = 1if k = k′ and 0 otherwise.

In the training phase, assuming synchronization in theuplink pilot transmissions, that is the worst case [6], wehave that theN ×K received signal at theℓ-th BS is:

Yℓ =

L∑

j=1

GTℓj

√ρjΨ+N, (1)

whereρj = diag(ρ1j ρ2j . . . ρKj), beingρkj the uplinktransmit power of thek-th user ofj-th cell, ·T is thetranspose operator,Gℓj = [gT

ℓ1j gTℓ1j . . . g

TℓKj]

T , such

that Gℓj = βℓjHℓj, βℓj = diag(β1

2

ℓ1j β1

2

ℓ2j . . . β1

2

ℓKj),Hℓj = [hT

ℓ1j hTℓ1j . . . h

TℓKj]

T , Ψ = [ψ1ψ2 . . . ψK ], andN is aN ×K additive white gaussian noise (AWGN)matrix with zero mean and unitary variance; hence, wecan define the uplink SNR as SNRUL = ρkjβjkj.

To generate the estimated CSI matrixGℓ of theirserved users, theℓ-th BS correlates its received signalmatrix with the known pilot sequences:

GTℓ =

1

KYℓΨ

H =

L∑

j=1

GTℓj

√ρj +N′, (2)

whereN′ is an equivalent AWGN matrix with zero meanand variance1

K. Note that the channel estimated by the

ℓ-th BS is contaminated by the channel of users that usesthe same pilot sequence in all other cells.

Besides, information transmit symbol vector of theℓ-th cell is denoted byxℓ = [x1ℓ x2ℓ . . . xKℓ]

T , wherexkℓ is the transmit symbol to thek-th user of theℓ-th cell, and takes a value from the squared quadra-ture amplitude modulation (M -QAM) alphabet; so, thecomplex-valued symbol (finite) set is given byS =A+

√−1 · A

, where the real-valued finite setA =

±12a; ±3

2a; . . . ; ±√M−12 a

, with

√M representing

the modulation order (per dimension) of the corre-sponding real-valued ASK scheme. The parametera =√

6/(M − 1) is used in order to normalize the power ofthe complex-valued transmit signals to1. Furthermore,we have assumed that the system is determined, i.e.,N ≥ K. For analysis simplicity, using matrix notation,theK × 1 complex-valued received signal by the usersof the ℓ-th cell is written as:

rℓ =

L∑

j=1

√αjGjℓPjxj + nℓ, (3)

4

where αj = diag(α1j α2j . . . αKj), being αkj thedownlink transmit power devoted by thej-th BS to itsk-th user,Pj denotes the complex valuedN×K precodingmatrix of the j-th BS, nℓ ∼ CN (0, IK) represents theAWGN vector with varianceσ2n = 1

2 per dimension,which is observed at theK mobile terminals of theℓ-th cell. Furthermore, the downlink SNR is defined asSNRDL = αkjβjkj

log2M

.

A. Matched Filter Beamforming

The most simple linear precoding technique is thematched filter beamforming, that simply premultipliesthe transmit symbol vector by the conjugate transposeof the estimated small-scale fading matrix:

PMFℓ=

1√γℓHH

ℓ , (4)

where Hℓ = β−1ℓℓ Gℓ = diag(β

−1

2

ℓ1ℓ β−1

2

ℓ2ℓ . . . β−1

2

ℓKℓ) Gℓ,and γℓ is a normalization factor to adjust the pre-coding matrix with the power constraints, such thatγℓ = trace(HH

ℓ Hℓ)/K for this technique.

B. Zero Forcing Beamforming

The zero forcing beamforming is another prominentlinear precoding technique. It is given by the Moore-Penrose pseudo inverse of the estimated small-scalefading channel matrix:

PZFℓ =1√γℓHH

ℓ

(HℓH

Hℓ

)−1. (5)

As shown in [11], in a single-cell MIMO system,the signal received by thek-th user can be writtenas rk = xk√

γ+ nk, evidencing that this technique

totally supresses the intra-cell interference at the cost,however, of decreasing the received SNR, sinceγℓ =trace(HH

ℓ Hℓ)−1/K for this scheme can assume high

values for ill conditioned small-scale fading channelmatrices. This effect is quite similar to the noise en-hancement of its analogue zero forcing receiver, ordecorrelator.

C. Regularized Zero Forcing Beamforming

Other efficient linear preacoding technique is the reg-ularized zero forcing beamforming [21]:

PRZFℓ =1√γℓHH

ℓ

(HℓH

Hℓ + ζIK

)−1, (6)

in which ζ is a parameter of the technique that balancesinterference supression and SNR decrease. Note thatRZF can be seen as a generalization of MF and ZF, sinceRZF reduces itself to these techniques forζ = ∞ and

ζ = 0, respectively. A suitable value forζ is suggestedin [11] asζ = N

20 .

D. Minimum Mean Squared Error Beamforming

The minimum mean squared error beamforming canbe seen as a special case of RZF, in which the parameterζ is optimized under the objective of minimizing themean squared error between information transmit symbolvector and the received signal estimated by BS in single-cell systems [22].

As a result,ζMMSE =Kσ2

n

2·SNRDL ·log2M

.

III. U LTIMATE L IMITS OF MASSIVE MIMO

Most of the existing theoretical results of MassiveMIMO systems are built upon the assumption [6], [11],[14]:

limN→∞

1

NHHH = IK . (7)

This assumption is justified since as the number ofbase station antennas grows, the inner products betweenpropagation vectors for different terminals grow at lesserrates than the inner products of propagation vectors withthemselves. In this section, we analitically prove that thisassumption is valid, by simply investigating the varianceof this product with increasingN , and then applying thisresult to find the asymptotic SINR of the noncooperativemulti-cell MIMO system with infinite number of BSantennas.

A. Asymptotic Orthogonality of Small-Scale Fading Ma-trices

Defining the matrixQ = 1NHHH , we have that

qi,j = 1N

∑Nn=1 hi,nh

∗j,n. In order to determine the

variance of the elements in matrixQ, the followingstatistics should be evaluated:E[qi,i], E[q2i,i], E[qi,j] andE[|qi,j|2] for i 6= j. Since|hi,j| is a random variable withRayleigh distribution, we have thatE[|hi,j|2] = 2σ2 = 1,thusσ = 1√

2, andE[|hi,j |4] = 8σ4 = 2.

1) E[qi,i]

= E

[1

N

N∑

n=1

hi,nh∗

i,n

]= E

[1

N

N∑

n=1

|hi,n|2]

=1

N

N∑

n=1

E[|hi,n|2

]=

1

N

N∑

n=1

1 = 1.

2) E[q2i,i]

= E

1

N2

N∑

n1=1

hi,n1h∗i,n1

N∑

n2=1

hi,n2h∗i,n2

=1

N2

N∑

n1=1

E

|hi,n1|2

N∑

n2=1

|hi,n2|2

5

=1

N2

N∑

n1=1

E[

|hi,n1|4]

+1

N2

N∑

n1=1

E

|hi,n1|2

N∑

n2=1

n2 6=n1

|hi,n2|2

=1

N2

N∑

n1=1

E[

|hi,n1|4]

+1

N2

N∑

n1=1

E[

|hi,n1|2]

N∑

n2=1

n2 6=n1

E[

|hi,n2|2]

=1

N2

N∑

n1=1

2 +1

N2

N∑

n1=1

1N∑

n2=1

n2 6=n1

1

=1

N22N +

1

N2N(N − 1) =

N + 1

N

3) E[qi,j]i 6=j

= E

[1

N

N∑

n=1

hi,nh∗

j,n

]=

1

N

N∑

n=1

E [hi,n]E[h∗j,n

]= 0.

4) E[|qi,j |2]i 6=j

= E[

qi,jq∗i,j

]

=1

N2E

N∑

n1=1

hi,n1h∗j,n1

N∑

n2=1

h∗i,n2

hj,n2

=1

N2

N∑

n1=1

E

|hi,n1|2|hj,n1

|2 + hi,n1h∗j,n1

N∑

n2=1

n2 6=n1

h∗i,n2

hj,n2

=1

N2

N∑

n1=1

E[

|hi,n1|2]

E[

|hj,n1|2]

+1

N2

N∑

n1=1

E [hi,n1]E

[

h∗j,n1

]

N∑

n2=1

n2 6=n1

E[

h∗i,n2

]

E [hj,n2]

=1

N2

N∑

n1=1

1 =1

N.

Since qi,i is a real random variable, we have thatvar(qi,i) = E[q2i,i] − E[qi,i]

2 = 1N

, where var(·)is the variance operator. On the other hand, asqi,j,for i 6= j, is a complex random variable, we havethat var(qi,j) = E [(qi,j − E[qi,j])(qi,j − E[qi,j])

∗] =E[|qi,j − E[qi,j]|2] = E[|qi,j|2] = 1

N.

B. Ultimate SINR With Infinite Number of BS AntennasFor the MF beamforming, from (2), (3), and (4),

we have that the vector of signal received by users(downlink) at theℓ-th cell (ℓ = 1, . . . , L) is:

rℓ =

L∑

j=1

1√

γj

√

αjGjℓβ−1

ℓℓ

(

L∑

i=1

GHℓi

√

ρi +N′∗

)

xj + nℓ. (8)

It is demonstrated in [14] that the downlink SINR ofthe k-th user of theℓ-th cell in this system converges inthe limit of infinite N to:

SINRDLkℓ =

αkℓβ2ℓkℓ/ν

2kℓ∑L

j=1j 6=ℓ

αkjβ2jkℓ/ν

2kj

, (9)

where the asymptotic power normalization factorsνkj =∑Li=1 ρkiβjki+

1K

. Note that this limit depends mainly onthe large scale fading coefficientsβjki, which are relatedto the spatial distribution of the users on the differentcells. In addition, if one assumes power uniformly dis-tributed between users in the same cell and the samepower normalization factorsνkj as in [6], equation (9)simplifies to:

SINRDLkℓ =

β2ℓkℓ∑Lj=1j 6=ℓ

β2jkℓ. (10)

IV. N UMERICAL RESULTS

We adopt in our simulations a multi-cell scenario withL = 4 hexagonal cells, with radius 1600m, whereK = 4users are uniformly distributed in its interior, except ina circle of 100m radius around the cell centered BS. Asingle realization of our spatial model is shown in Fig.1.

−4000 −3000 −2000 −1000 0 1000 2000 3000−3000

−2000

−1000

0

1000

2000

3000

4000

5000

Users Cell 1Users Cell 2Users Cell 3Users Cell 4Base Stations

Fig. 1. A random realization for the spatial distribution ofmobileusers in a multi-cell MIMO system scenario withL = 4 andK = 4.

Besides, we model the log normal shadowing witha standard deviation of 8dB, and the path loss decayexponent equal to 3.8. We simulated two main scenariosof interest from the BER performance metric: in thefirst, the number of users grows at the same rate as thenumber of BS antennas, i.e.,N = K holds; while in thesecond,K remains fixed whileN grows to infinity. Fur-thermore, in the simulations setup, we have consideredunitary frequency-reuse factor, 4-QAM modulation andSNRDL = 10dB. It is important to note that results inthis section were obtained via Monte-Carlo simulationmethod.

6

A. BER Performance under Perfect CSI

Fig. 2 compares the BER performance for both sce-narios in a single-cell systems with perfect CSI. It can beseen for the first scenario (N = K) that performance ofMF and ZF transmitters stay unchanged with the systemdimension increasing, while RZF and MMSE have theirperformance improved. On the other hand, in the secondscenario (K fixed andN increasing), all the techniques’performances are improved, and one can see that MMSEand ZF fastly achieves very small error rates, while forRZF and MF, in this order, this occurs more slowly,meaning that the achieved diversity gain for the MF-MuT MIMO system is quite modest.

4 5 6 7 8

10−4

10−3

10−2

10−1

100

N

MF−MuT MIMOZF−MuT MIMORZF−MuT MIMOMMSE−MuT MIMO

10 20 30 4010

−4

10−3

10−2

10−1

100

N = K

BE

R

Fig. 2. BER performance of the investigated precoding techniques,for 4-QAM and SNRDL

= 10 dB, with the increasing systemdimension: a)N = K; b) K = 4. Single-cell analysis with perfectCSI.

The same comparison is shown in Fig. 3, but now fora multi-cell configuration, where inter-cell interferenceaccounts, with perfect CSI. One can see that the consid-ered Massive MIMO precoding techniques result in highbit error rates for the first scenario, that remain fixedwith the system dimension increasing, except for ZF,that has its performance getting even worse forN = Kgrowing substantially. For the second scenario, all thetechniques improve their performances with increasingN ; an interesting result is that these performances nearlyigualate themselves forN > 100, and that RZF surpassesMMSE in terms of BER performance, since the lasttechnique does not account inter-cell interference in itsderivation.

B. BER Performance under CSI Estimation Errors

When considering the multicell scenario but now withimperfect CSI, that is obtained by means of uplink

101

102

103

10−4

10−3

10−2

10−1

N

MF−MuT MIMOZF−MuT MIMORZF−MuT MIMOMMSE−MuT MIMO

101

102

103

10−1

100

N = K

BE

R

Fig. 3. BER performance of the investigated precoding techniques,for 4-QAM and SNRDL

= 10 dB, with the increasing systemdimension: a)N = K; b) K = 4. Multi-cell analysis with perfectCSI.

training sequences transmitted with SNRUL = 10dB, butneglecting the pilot contamination effect, Fig. 4 showsthe achieved BER performances. It can be seen that thebehavior of the techniques for both scenarios is almostthe same than for perfect CSI, as shown in Fig. 3. There-fore, we can conclude that, when dealing with inter-cellinterference, effect of uncorrelated noise in the estimatedchannel little degrades the system performance.

101

102

103

10−4

10−3

10−2

10−1

N

MF−MuT MIMOZF−MuT MIMORZF−MuT MIMOMMSE−MuT MIMO

101

102

103

10−1

100

N = K

BE

R

Fig. 4. BER performance of the investigated precoding techniques,for 4-QAM and SNRDL

= 10 dB, with the increasing system dimen-sion: a)N = K; b) K = 4. Multi-cell analysis with SNRUL

= 10

dB.

The main result of this paper can be seen whencomparing Fig. 4 with Fig. 5, that shows the performanceof the two scenarios in the complete system model, in-cluding pilot contamination. While for the first scenario

7

the same behavior of the BER performances is found,but with even higher error levels, one can see for thesecond scenario that, when taking pilot contaminationinto account, the BER performance of the consideredprecoding techniques saturates withN → ∞ (BERfloor effect), in contrast to Fig. 4.b. Indeed, one couldalready expect this result, since errors are induced bynoise, intra-cell interference, as well as by inter-cellinterference, which can be compared with the desiredsignal power by means of the SINR. It was alreadyshown in previous works [6] and [14] that noise andintra-cell interference are averaged out in the MIMOsystem with very large number of BS antennas, beingthe inter-cell interference the only remaining impairment,due to pilot contamination. This phenomenom leads tothe saturation of the SINR, and, as shown herein, BERperformance as well. Besides, the different BER levelsbetween ZF, RZF, MMSE compared to MF can bejustified, as explained in [11] and [18], since the firsttechniques converge to its asymptotic limits with fewerBS antennas than the last one, although these limits areusually close.

101

102

103

10−2

10−1

N

MF−MuT MIMOZF−MuT MIMORZF−MuT MIMOMMSE−MuT MIMO

101

102

103

10−1

100

N = K

BE

R

Fig. 5. BER performance of the investigated precoding techniques,for 4-QAM and SNRDL

= 10 dB, with the increasing system dimen-sion: a)N = K; b) K = 4. Multi-cell analysis with SNRUL

= 10

dB and pilot contamination.

V. CONCLUSION

In this work we have demonstrated that the break-through results of Massive MIMO systems downlinkcan also be found from the bit-error-rate performanceperspective. Two scenarios of great interest have beennumerically analysed: when the number of users growsin the same rate as the number of BS antennas, andwhen the number of BS antennas grows keeping fixed

the number of users covered in each cell. We found thatthe first scenario is unable to operate with acceptableperformance in multi-cellular systems when consideringthe linear precoding techniques investigated herein. Onthe other hand, it was found that the performance ofthe linear precoding techniques evaluated in the secondscenario could be made as good as is wanted, by scalingup the number of BS antennas, if the effect of pilotcontamination has not been taken into account. Thiseffect saturates the achievable SINR in the asymptoticlimit of BS antennasN → ∞, as shown in previousworks, and consequently the BER performance becomessaturated as well. Furthermore, we proved a widely usedassumption that the product of the small-scale fadingchannel matrix with its conjugate transpose tends to ascaled identity matrix asN → ∞, and found a closedform expression for the variance of this product. Thislast result is quite useful, since one can determine withwhich reliability the assumption is true according thenumber of antennas deployed in the BS.

ACKNOWLEDGEMENT

This work was supported in part by the National Coun-cil for Scientific and Technological Development (CNPq)of Brazil under Grants 202340/2011-2, 303426/2009-8and in part by Londrina State University - Paraná StateGovernment (UEL).

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José Carlos Marinello received his B.S. inElectrical Engineering (Summa Cum Laude)from Londrina State University, PR, Brazil,in December 2012. He is currently workingtowards his M.S. and Ph.D. at Londrina StateUniversity, Paraná, Brazil. His research inter-ests include physical layer aspects, speciallyheuristic and convex optimization of 3G and4G MIMO systems.

Taufik Abrão (SM’12) received the B.S.,M.Sc., and Ph.D. degrees in electrical engi-neering from the Polytechnic School of theUniversity of São Paulo, São Paulo, Brazil,in 1992, 1996, and 2001, respectively. SinceMarch 1997, he has been with the Communi-cations Group, Department of Electrical Engi-neering, Londrina State University, Londrina,Brazil, where he is currently an Associate Pro-

fessor of Communications engineering. In 2012, he was an AcademicVisitor with the Communications, Signal Processing and Control Re-search Group, University of Southampton, Southampton, U.K. From2007 to 2008, he was a Post-doctoral Researcher with the Departmentof Signal Theory and Communications, Polytechnic University ofCatalonia (TSC/UPC), Barcelona, Spain. He has participated inseveral projects funded by government agencies and industrial com-panies. He is involved in editorial board activities of six journals inthe communication area and he has served as TCP member in severalsymposium and conferences. He has been served as an Editor for theIEEE COMMUNICATIONS SURVEYS & TUTORIALS since 2013.He is a member of SBrT and a senior member of IEEE. His currentresearch interests include communications and signal processing,specially the multi-user detection and estimation, MC-CDMA andMIMO systems, cooperative communication and relaying, resourceallocation, as well as heuristic and convex optimization aspects of3G and 4G wireless systems. He has co-authored of more than 170research papers published in specialized/international journals andconferences.