Research Article An Improved Multicell MMSE Channel...

Research ArticleAn Improved Multicell MMSE Channel Estimation ina Massive MIMO System

Ke Li1 Xiaoqin Song2 M Omair Ahmad3 and M N S Swamy3

1 College of Information Technology Beijing Union University Beijing 100010 China2 College of Electronics and Information Engineering Nanjing University of Aeronautics and Astronautics Nanjing 210016 China3Department of Electrical and Computer Engineering Concordia University Montreal QC Canada H3G IM8

Correspondence should be addressed to Ke Li likebuueducn

Received 22 January 2014 Accepted 9 April 2014 Published 8 May 2014

Academic Editor Kan Zheng

Copyright copy 2014 Ke Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Massive MIMO is a promising technology to improve both the spectrum efficiency and the energy efficiencyThe key problem thatimpacts the throughput of a massive MIMO system is the pilot contamination due to the nonorthogonality of the pilot sequencesin different cells Conventional channel estimation schemes cannot mitigate this problem effectively and the computationalcomplexity is increasingly becoming larger in views of the large number of antennas employed in a massive MIMO systemFurthermore the channel estimation is always carried out with some ideal assumptions such as the complete knowledge of large-scale fading In this paper a new channel estimation scheme is proposed by utilizing interference cancellation and joint processingHighly interfering users in neighboring cells are identified based on the estimation of large-scale fading and then included inthe joint channel processing this achieves a compromise between the effectiveness and efficiency of the channel estimation at areasonable computational cost and leads to an improvement in the overall system performance Simulation results are provided todemonstrate the effectiveness of the proposed scheme

1 Introduction

Recently there has been a great deal of interests in MIMOsystems using very large antenna arrays namely the massiveMIMO or full dimensionMIMO (FD-MIMO) systems [1ndash4]Such systems can provide a significant increase in reliabilitydata rate and energy efficiency with simple signal processing[1] When the number of base station (BS) antennas growslarge the channel vectors between the users and the BSbecome very large-size random vectors and under ldquofavor-able propagationrdquo conditions Then they become pairwiseorthogonal As a consequence with simple linear processingfor example maximum-ratio combining (MRC) or zeroforcing (ZF) assuming that the BS has perfect channelstate information (CSI) the interference from other userscan be cancelled without consuming more time-frequencyresources This dramatically increases the spectral efficiencyFurthermore by using a very large antenna array at BS thetransmit power can be drastically reduced

Practically however the BS does not have perfect CSIInstead it estimates the channels as usual Conventionallyit is done by using uplink pilots Since both the time-frequency resources allocated for pilot transmission and thechannel coherence time are limited the number of possibleorthogonal pilot sequences is limited too and hence the pilotsequences have to be reused in neighbouring cells of cellularsystems Therefore channel estimates obtained in a givencell get contaminated by the pilots transmitted by the usersin other cells This causes pilot contamination [5] that isthe channel estimate at the base station in one cell becomespolluted by the pilots of the users from other cells

Considering the importance of pilot contamination miti-gation to guarantee the good performance of massiveMIMOmany schemes ranging from optimal pilot design and allo-cation advanced channel estimation to downlink precoding[5ndash11] have been proposed to deal with this problem

In [10] a pilot design criterion and Chu sequence-basedpilots have been proposed by exploiting the orthogonality

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 387436 9 pageshttpdxdoiorg1011552014387436

2 International Journal of Antennas and Propagation

of the channel vectors of massive MIMO systems Because ofthe proposed pilots the channel estimate of most terminalsof a cell is only interfered by partial cells rather than all theother cells where the latter is caused by traditional pilots Asa result pilot contamination is mitigated It is also possibleto coordinate the use of pilots or adaptively allocate the pilotsequences to different terminals in the network [8]

Multicell joint channel estimation algorithms [5 8] oreven blind techniques that circumvent the use of pilotsaltogether [7] may mitigate or eliminate the effects of pilotcontamination The approach in [7] exploits the asymptoticorthogonality of the channel vectors in very large MIMOsystems and shows that the channel to each user can beestimated from the covariance matrix of the received signalsup to a remaining scalar multiplicative ambiguity which canbe resolved by utilizing a short training sequence

The new precoding techniques such as pilot contami-nation precoding [11] that take into account the networkstructure can utilize cooperative transmission over a multi-plicity of cellsmdashoutside of the beamforming operationmdashtonullify at least partially the directed interference that resultsfrom pilot contamination Unlike coordinated beamformingover multiple cells which requires estimates of the actualchannels between the terminals and the service-arrays ofthe contaminating cells the pilot-contamination precodingrequires only the corresponding slow-fading coefficients

By overviewing the abovementioned channel estimationschemes it can be seen that for simplicity it is alwaysassumed that the large-scale fading propagation (ie pathlossand shadow fading) is fixed and known 119886 119901119903119894119900119903119894 which isnot the case in practice Furthermore the computationalcomplexity of conventional channel estimations such as leastsquare (LS) and minimum mean square error (MMSE)estimations grows sharply in a massiveMIMO system as thenumber of antennas is much more than that in the currentmultiple antennas systems This problem is getting worsein the case of joint processing in a multicell scenario Asa countermeasure to these problems an improved channelestimation scheme which aims at mitigating the pilot con-tamination at an affordable expense and in a more realisticsituation is proposed in this paper

To deal with the abovementioned problems a new chan-nel estimation scheme is proposed by utilizing interferencecancellation and joint processing Highly interfering usersin neighboring cells are identified based on the estimationof large-scale fading and then included in the joint channelprocessing this achieves a compromise between the effec-tiveness and efficiency of the channel estimation and leads toan improvement in the overall system performance Metricsof the performance evaluation of the channel estimation aredefined and finally simulation results are provided whichvalidate the proposed scheme

The paper is organized as follows Section 2 gives themul-ticell massive MIMO system model and the pilot design Theconventional channel estimator and the proposed multicelljoint MMSE channel estimation are presented in Section 3Section 4 presents the simulation results and the analysis ofcomputational complexity of the proposed scheme FinallySection 5 concludes this paper

Notations (sdot)lowast (sdot)119879 and (sdot)119867 denote the conjugate transposeand conjugate transpose respectively [sdot]

119894119895is the (119894 119895)th entry

of a matrix 119864sdot is the expectation and sdot denotes theFrobenius norm

2 Multicell Massive MIMO System Model

21 System Model Here we consider a massive MIMOcellular systemwith 119871 hexagonal cells as depicted in Figure 1Each cell comprises one base station with119872 antennas and119870single antenna users (119870 ≪ 119872 in the case of amassiveMIMOsystem) All the cells share the same band of frequenciesIn addition the system employs OFDM and works in time-division duplex (TDD) mode

Let the average power (during transmission) at the basestation be 119901

119891and let the average power at each user be 119901

119903 In

one coherence interval the channel responses of119873119886symbols

are assumed to be static which means that the channelestimate is effective only in this interval Consequently a totalof119873119888symbols are used for pilots from all the terminals to BS

and the remaining 119873119886minus 119873119888symbols are used for a two-way

data transmission Apart from that following the model in[1] the channel responses are assumed to be constant in thefrequency smooth interval which consists of119873

119904consecutive

subcarriers Therefore to estimate the channel response ofone subcarrier every terminal can transmit pilots over 119873

119904

subcarriers In the following channel responses of the 119873119904

subcarriers will be estimated As a result the pilot length isgiven by 120591 = 119873

119888119873119904 and thus up to 120591 terminals can transmit

pilots to BS without intracell interferenceThe propagation factor (channel response) between the

119898th base station antenna of the 119897th cell and the 119896th user ofthe 119895th cell is given by

119892119895119897119896119898

= radic120573119895119897119896ℎ119895119897119896119898

(1)

where 120573119895119897119896 are the large-scale fading coefficients composed

of pathloss and shadow fading and are assumed invariant fordifferent antennas of the same BS and ℎ

119895119897119896119898 are the small-

scale fading coefficients and are assumed to be independentand identically distributed (iid) and circularly symmetricGaussian random variables (CN(0 1))We assume the chan-nel reciprocity for the downlink and the uplink that is thepropagation factor ℎ

119895119897119896119898is the same for both the downlink

and the uplink and block fading that is ℎ119895119897119896119898

remainsconstant for the duration of coherence intervals (typically 4ndash30OFDMsymbols)The additive noise signals at all terminalsare iid (CN(0 1)) random variables Then the channelmatrix from the terminals of the 119895th cell to BS in the 119897th cellis given by

G119895119897= D119895119897H119895119897isin C119870times119872

(2)

where H119895119897isin C119870times119872 are small-scale fading coefficients that

is [H119895119897]119896119898

= ℎ119895119897119896119898

D119895119897isin R119870times119870 is a diagonal matrix where

[D119895119897]119896119896

= radic120573119895119897119896

At the beginning of every coherence interval all the usersin all the cells synchronously transmit pilot sequences of

International Journal of Antennas and Propagation 3

1

1

jth cell

K

K

k

k

gjlkm

3 mM

3 mM

middot middot middot



lth cell

Figure 1 A multicell massive MIMO system model

length of 120591 symbols Let radic120591120595119895119896

(where 120595119867119895119896

120595119895119896

= 1) be thepilot vector transmitted by the 119896th user in the 119895th cell andconsider the base station of the 119897th cell Then the vectorreceived at the119898th antenna is given by

119910119897119898

=

119871

sum

119895=1

119870

sum

119896=1

radic119901119903120591120573119895119897119896ℎ119895119897119896119898

120595119895119896

+ 119908119897119898 (3)

where 119908119897119898

isin C120591times1 is the additive noise LetY119897

= [1199101198971

1199101198972

sdot sdot sdot 119910119897119872

] isin C120591times119872 W119897

=

[1199081198971

1199081198972

sdot sdot sdot 119908119897119872] isin C120591times119872 Ψ

119895 119895 = 1 119871 be the

system matrices and Ψ119895= [1205951198951

1205951198952

sdot sdot sdot 120595119895119870

] isin C120591times119870From (1) the signal received at the 119897th base station is given by

Y119897= radic119901119903120591

119871

sum

119895=1

Ψ119895G119895119897+W119897 (4)

By separating the received pilots of the users in the target celland that in the neighboring cells (4) can be rewritten as

Y119897= radic119901119903120591Ψ119897G119897119897+ radic119901119903120591

119871

sum

119895=1119895 = 119897

Ψ119895G119895119897+W119897

(5)

in which the first item is the desired received pilots and thesecond one is the pilots from neighboring cells that is theinterfering pilot signals

22 PilotDesign It is known that the number of pilot symbolsis limited by the channel coherence interval thus it isassumed that 120591 lt 2119870 In addition the systemmatrixΨ

119897must

satisfy the requirementΘ119897119897= 119868119870 whereΘ

119895119897= Ψ119879

119897Ψlowast

119895isin C119870times119870

is the correlation matrix between the pilots Consequently120591 ge 119870 is required which means that the number of usersper cell 119870 should not be larger than the pilot length 120591 Thusthe pilot length should lie in the interval119870 le 120591 lt 2119870

In [10] a pilot design method has been proposed basedon the Chu sequences which have perfect autocorrelationproperty and very low cross correlations The pilots inside

a cell are orthogonal because of different cyclic shifts Thesepilots are reused among the 119871 cells after being multiplied byphase shifts Then the cross-correlation values of the pilotsare different for different terminals which fit the scenarioswith large shadow fading varianceMoreover it is proved thatthe channel estimate of most terminals is only interfered bypartial cells As a result the pilot contamination is mitigatedWe will now briefly describe how the pilots are generated

The cell-specific basic pilot vector of users in the 119895th cellis denoted asradic120591120595

119895

and the 119899th entry of 120595119895

is given by

[120595119895

]

119899

= 119886119899exp 119894

2120587119902119895119899

120591

119899 = 1 2 120591 (6)

where exp 119894(2120587119902119895119899120591) is the phase shift between the cells 119902

119895

is an integer that is not exactly divisible by 120591 and differs from119895 and 119886

119899is the 119899th entry of the Chu sequence a = [119886

0

1198861 119886

120591minus1] and can be expressed as [12]

119886119899= exp 119894119873120587

120591

119899 (119899 + (120591 mod 2)) 119899 = 0 1 120591 minus 1 (7)

119873 is the unique sequence parameter which should be aninteger that is relatively prime to 120591 The pilot vectors of usersin one cell are the cyclic shifted versions of the cell-specificbasic pilot vector The pilot vector of the 119896th user in the 119895thcell is given by

radic120591120595119895119896

= ⟨radic120591120595119895

⟩

119896minus1

(8)

in which ⟨sdot⟩119896is the left circular shift of a vector with shift

length 119896The circular autocorrelation function of the Chu

sequence a is defined as

119903119895=

120591minus1

sum

119899=0

119886119899119886lowast

(119899+119895) mod 120591 119895 = 0 1 120591 minus 1 (9)

Then the autocorrelation values are 1199030= 120591 119903

119895= 0 when

119895 = 0 It can be easily seen that the autocorrelations of vectors120595119895119896

have the same property

3 Multicell MMSE Channel Estimation

31 Conventional Single-Cell Channel Estimation Conven-tional channel estimation relies on correlating the receivedsignal with the known pilot sequence The typical one isreferred here as the LS estimation For the abovementionedsystemmodel the LS estimator for the desired channel of thetarget cell 119897 is given by [8]

H119897119897= radic119901119903120591(D119897119897Ψ119867

119897Ψ119897D119897119897)

minus1

D119897119897Ψ119867

119897Y119897 (10)

The conventional estimator suffers severely from a lack oforthogonality between the desired pilots and the interferingones In particular when the same pilot sequence is reused inall the 119871 cells the estimator can be written as

H119897119897= H119897119897+

119871

sum

119895=1119895 = 119897

H119895119897+

Ψ119867

119897W119897

120591

(11)


which shows that the interfering channels leak directly intothe desired channel estimate

Another well-known linear channel estimation is theMMSE estimator which is known to have a better perfor-mance For the pilot signal received at the 119897th base station asdescribed in (4) if we treat the pilots of other cells arrivingat the antennas of 119897th BS as pure interference then the single-cellMMSE estimation of the channelH

119897119897of the 119897th cell is given

by

H119897119897= radic119901119903120591D119897119897Ψ119867

119897(I + 119901

119903120591Ψ119897D2119897119897Ψ119867

119897)

minus1

Y119897 (12)

The estimator of (12) performswell in single-cell scenarioHowever it is not capable of eliminating pilot contaminationin multicell cases since the pilot signals from users ofneighboring cells are not utilized in the estimation

32 Multicell MMSE Channel Estimation To mitigate theaforementioned pilot contamination in a multicell scenariowhich is always the case in a real network a straightforwardapproach is to combine the pilot signals of the users inneighboring cells in the estimator as in TD-SCDMA system[13]

By reorganizing the pilot matricesΨ119895and channel matri-

ces G119895119897 (4) can be rewritten as

Y119897= radic119901119903120591 [Ψ1sdot sdot sdot Ψ

119871][

[

[

G1119897

G119871119897

]

]

]

+W119897

= radic119901119903120591ΦG119897+W119897

= radic119901119903120591ΦD119897H119897+W119897

(13)

where Φ isin C120591times119870119871 G119897isin C119870119871times119872 D

119897= [

D1119897sdotsdotsdot 0

d

0 sdotsdotsdot D119871119897

] isin

C119870119871times119870119871 H119897= [

H1119897

H119871119897

]isin C119870119871times119872

Then we have the estimation of the channel response ofall the users in all the cells by simply applying the MMSEcriterion on (13) [5] as

H119895119897= radic119901119903120591D119895119897Ψ119867

119895(I + 119901

119903120591

119871

sum

119894=1

Ψ119894D2119894119897Ψ119867

119894)

minus1

Y119897

119895 = 1 sdot sdot sdot 119871

(14)

With this the pilot contamination can be removed jointlyFurthermore the channel estimation of all the 119871 cells can bedescribed by a unified matrix as

H119897= radic119901119903120591D119897Φ119867

(I + 119901119903120591ΦD2119897Φ119867

)

minus1

Y119897 (15)

However the abovementionedmulticell joint channel estima-tion works more in theory than in practice for the followingreasons

Firstly in the abovementioned channel estimationschemes either single-cell or multicell 120573

119895119897119896 are assumed

to be fixed by considering the fact that the large-scalefading (including pathloss and shadow fading) is known tothe receivers of all BS and that only the small-scale fadingcoefficients ℎ

119895119897119896119898 need to be estimated [5 7] In a real

network scenario though changing slowly the large scalefading must be estimated at the receiver and updated fromtime to time Furthermore the estimation error of 120573

119895119897119896

impacts significantly on the performance of uplink datadecoding and downlink transmission (eg precoding andbeamforming)

Secondly for the large-scale antenna array deployed at BSof a massive MIMO system the number of antenna elementsis in the order of several tens or even hundreds Thus thedimension of the matrices in the channel estimation is aboutone order of magnitude higher than in a normal MIMOsystem On the other hand for the multicell joint channelestimation by (14) the dimension of matrices is even largersince all the users in the neighboring cells are included in thematricesTherefore they are difficult to be applied in practicalsystems due to the huge computational burden

In addition for the joint channel estimation scheme of thetarget cell 119897 by (14) it can be seen that the knowledge of thelarge-scale fading from all the cells to the target cell D

119894119897 119894 =

1 sim 119871 is required in advance Actually it is not necessary bythe method as illustrated below

First the full channel response G119897119897instead of only the

small-scale fading H119897119897is estimated by MMSE criterion that

is

G119897119897= radic119901119903120591Ψ119867

119897(I + 119901

119903120591

119871

sum

119894=1

Ψ119894Ψ119867

119894)

minus1

Y119897 (16)

Then by leftmultiplexing both sides of (16) withDminus1119897119897 we have

the estimation of the small-scale fading as

H119897119897= radic119901119903120591Dminus1119897119897Ψ119867

119897(I + 119901

119903120591

119871

sum

119894=1

Ψ119894Ψ119867

119894)

minus1

Y119897 (17)

Compared with (14) obviously the estimation ofD119894119897(119894 = 119897) is eliminated in (17) With this not only the

unnecessary computational burden is eliminated but also theimpact of estimation error of D

119894119897(119894 = 119897) on the performance

of H119897119897

estimation is mitigated But the computationalcomplexity is still huge due to the large number of users andantennas to be considered in the estimation

33 Improved Multicell MMSE Channel Estimation As weknow the purpose of the joint estimation by including theusers of the neighboring cells in the estimation is to mitigatethe interference of the pilots in the neighboring cells soas to improve the performance of channel estimation ofusers in the target cell It is not necessary to estimate thechannels of the users in the neighboring cells In view ofthis one needs to include only the strong interfering users(SIU) in the neighboring cells into the channel estimatorThen a compromise between estimation performance and


computational cost can be achieved by selecting a propernumber of strong interfering users An improved multicelljoint channel estimation is now proposed based on thisreasoning

The basic ideas of the scheme are as follows Firstly thelarge-scale fading is estimated by a matched filter- (MF-)based fast channel estimation Then the strong interferingusers can be decided by sorting the pilot signal strengthAlso the estimation of large-scale fading can be utilizedin the joint channel estimation thereafter Considering thepoor performance of MF in non-Gaussian scenario signalreconstruction and interference cancellation are employed toimprove the estimation performance of the large-scale fadingOn the basis of this both the users of target cell and thestrong interfering users of the neighboring cells are includedin the channel and the system matrices of the multicell jointMMSE estimation Then a precise estimation of the small-scale fading of the target cell users can be achievedThe blockdiagram of the proposed scheme is illustrated in Figure 2

In view of the outstanding autocorrelation and cross-correlation properties of theChu-based pilot sequence and itspromising capability in the reduction of the nonorthogonalityof intercell users here we employ the pilot design given in[10] Details of the proposed scheme are as follows

(1) MF-Based Estimation of Large-Scale Fading of Target CellTaking the second item of (5) as interference the initialchannel estimation of the 119897th cell is obtained by employingthe MF theorem as in [14]

GMF119897119897

=

1

radic119901119903120591

diag (Ψ119867119897Ψ119897)

minus1

Ψ119867

119897Y119897 (18)

Note thatΨ of all the cells are known 119886 119901119903119894119900119903119894 thereforediag (Ψ119867

119897Ψ119897)

minus1

Ψ119867

119897can be calculated and stored in advance

Since the large-scale fading of one user to all the antennasof one BS is almost the same and that in the case of 119872 ≫

119870 the fast fading is assumed to be iid and CN(0 1) thelarge scale fading of each of the users in the target cell can becalculated as

radic120573119897119897119896=

1

119876

E[GMF119897119897]119896

119896 = 1 119870 (19)

Here 119876 is the normalization factor for fast fading which is12535 forCN(0 1)

Let D119897119897= diag(radic120573

1198971198971radic1205731198971198972

radic120573119897119897119870)

(2) Interference Cancellation-Based Large-Scale Fading Esti-mation of Neighboring Cell Users Suppose that there are 119862

119873

neighboring cells around the target cell 119897 Since the pilotstrength of the target cell is always the largest in the receivedpilots one needs to remove it first so as to improve theprecision of the neighboring cell estimation that is

Y119897= Y119897minus radic119901119903120591Ψ119897GMF119897119897 (20)

By supposingY119897as the pilots of all the users of the 119894th cell

arriving at the antenna of the 119897th BS we have

Y119897= radic119901119903120591Ψ119894G119894119897+W119897 (21)

Then the large-scale fading of the links between the users ofthe 119894th cell and the antenna of the 119897th BS can be estimatedusing (18) and (19)

All the 119862119873neighboring cells can be processed sequen-

tially in the same way as in (18) and (19) by replacing the sys-tem matrixΨ

119894with the pilot sequences of the corresponding

neighboring cell With this the large-scale fading of all theusers in all the 119862

119873neighboring cells are obtained

(3) Selection of Strong Interfering Users Sort all the neigh-boring cell users in an ascending order of its large-scalefading Taking the top 119870

1users in the cell list as the ldquostrong

interfering usersrdquo we define the large-scale fading matrix asDCN119897

= diag(radic1205731198991198971radic1205731198991198972

radic1205731198991198971198701

) where 119899 denotes thecell which the 119896th SIU belongs to The value of 119870

1is decided

by the remaining capability for the processing of users otherthan the target cell users Suppose that the overall capability ofthe BS is to estimate the channel response of119870

119886users and that

the number of active users of the target cell is 1198700(1198700⩽ 119870)

then1198701= 119870119886minus 1198700is the number of SIUs to be processed

It should be noted that since the large-scale fading alwayschanges slowly Steps (1) sim (3) need not be carried out afterevery frame but carried out say after every ten frames so asto save unnecessary computational burden

(4) Generation of the System Matrix for Joint Channel Esti-mation Putting together the estimated large-scale fading ofall the 119870

0users in the target cell and the 119870

1SIUs of the

neighboring cells we have a new large-scale fading matrixgiven by

D119897= [

DCT119897

0

0 DCN119897

] isin R119870119886times119870119886 (22)

where DCT119897

= diag(radic1205731198971198971radic1205731198971198972

radic1205731198971198971198700

) The systemmatrix can then be defined accordingly by combining thepilot sequence of the 119870

0users of the target cell and the 119870

1

strong interfering users of neighboring cells as

Ψ119897= [1205951198971

1205951198971198700

1205951198991

1205951198991198701

] isin C120591times119870119886 (23)

Denoting H119897isin C119870119886times119872 as the small-scale fading of the

1198700users of target cell and the 119870

1SIUs of neighbor cells the

received pilot signal by the target cell is then given by

Y119897= radic119901119903120591Ψ119897

D119897H119897+W119897 (24)

where W119897refers to the noise and received pilot signal of all

the users excluding the abovementioned1198700+ 1198701users

(5) MMSE-Based Joint Channel Estimation By applying theMMSE criterion to (24) we have the estimation of the small-scale fading of target cell users along with the SIUs as

H119897= radic119901119903120591 D119897Ψ

H119897(I + 119901

119903120591Ψ119897D2119897Ψ119867

119897)

minus1

Y119897

(25)

in which H119897isin C119870119886times119872 Taking the first119870

0rows of H

119897 we then

have the final estimation of small scale fading of the targetcell


+minus

Yl MF estimation of

MF estimation of

MF estimation of

target cell users

1st neighbor cellusers

users

Dll calculation

Dll calculation

Dll calculation

Pilot reconstructionof target cell users

sim

Selection of stronginterfering users

Joint MMSEestimation

CNth neighbor cell

Figure 2 Block diagram for an improved joint MMSE channel estimation

4 Experimental Results

41 Assumptions of Simulation In order to evaluate theproposed scheme simulations of a massive MIMO systemwere performed The system consists of 7 cells sharing thesame band of frequencies with 6 cells surrounding one celland the cell edges of the 6 cells overlapping the edges ofthe central cell The parameters of the system are givenin Table 1

The terminals are distributed uniformly in the assignedcells (no handover across the cells) The fast fading factor ofevery uplink terminal-BS path is generated randomly follow-ing iid CN(0 1) and updated every frame The distancesbetween every terminal-BS pair are updated every 10 framesThe shadow fading is modeled with log-normal distributionand is the same between each user and all BSsMore explicitlythe large-scale fading coefficient between each user and BScan be expressed as radic120573

119897119897119896= 10

11990810

(119903119895119897119896100)

V when thepower ismeasured inWatts where 119903

119895119897119896is the distance between

the 119896th terminal in the 119895th cell to BS in the 119897th cell 119908 is thecorresponding shadow fading coefficient in dB and denotedas a Gaussian random variable with zero mean and variance1205902

119904(ieN(0 120590

2

119904))

The simulations are performed in an AWGN noisy envi-ronment with the transmit power for both the pilots and thedata being the same for each terminal and controlled by apredefined Tx power to noise ratio

For the pilots used in the simulation the same settingas in [10] is employed That is the sequence parameter 119873in (7) is set to be unity for all the seven cells and thesame sequence is cyclic shifted for terminals in the samecell And the parameters 119902

119894 119894 = 1 2 7 in (6) are set

to be 0 1 2 9 10 11 12 for the seven cells Furthermoreto mitigate the impact of the correlation property of thepilots onto the performance of channel estimation the pilotsequences with less cross correlation are assigned to thestrong interfering users in the simulation

Table 1 Simulation parameters

Parameter ValueCell radius (119877) 500mPathloss exponent (V) 38Carrier frequency (119891

119888) 2GHz

Shadow fading standard deviation (120590119904) 8 dB

Number of subcarriers for pilots (119873119904) 12

Number of symbols for pilots (119873119888) 3

Pilot length (120591) 36Number of cells (119871) 7Number of users per cell (119870) 24Number of strong interfering users in neighborcells (119870

1)

12

Number of BS antennas (119872) 100Frame length 10ms

42 Metrics Used for the Evaluation of Channel EstimationError To evaluate the proposed channel estimation schemethree performance metrics are defined here one for theevaluation of large-scale fading and the other two for thesmall-scale fading

The first one is a mean square error (MSE) of the large-scale fading estimation for all the paths between all the119870times119871

users in the network and the 119897th BS given by

120596 ≜ E(

100381610038161003816100381610038161003816100381610038161003816

radic120573119897119896minus radic120573119897119896

100381610038161003816100381610038161003816100381610038161003816

2

) 119896 = 1 119870 times 119871 (26)

This metric gives an evaluation of the estimation error oflarge-scale fading


Table 2 Impact of large-scale fading estimation and SIU selection on the performance of small-scale fading estimation

Scheme 120576 (dB) 120588

Case 1 proposed scheme with ideal 120573 minus891 0933Case 2 proposed scheme with estimated 120573 minus753 0911Case 3 proposed scheme with ideal 120573 but randomly selected SIUs minus229 0806

The second one is a normalized channel estimation error(120576) of small-scale fading given by

120576 ≜ 10 log10(

119870

sum

119896=1

119872

sum

119898=1

10038171003817100381710038171003817[H119897119897]119896119898

minus [H119897119897]119896119898

10038171003817100381710038171003817

2

10038171003817100381710038171003817[H119897119897]119896119898

10038171003817100381710038171003817

2) (27)

where H119897119897

and H119897119897

are the desired small-scale channelresponse at the 119897th base station and its estimate respectivelyAlthough the channel response of the top119870

1strongest inter-

fering users in the neighboring cells has also been estimatedby the proposed multicell channel estimation scheme thechannel estimation errors of these users are not of interest tous Therefore here we consider only the channel estimationerror of the desired channel of the 119897th cell The smaller thevalue of 120576 the better the channel estimation

The third metric is the channel correlation coefficient asdefined in (28) which gives an evaluation of the normalizedcorrelation property between the channel estimation and thedesired channel

120588 ≜

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

cov (H119897119897H119897119897)

radiccov (H119897119897H119897119897) cov (H

119897119897H119897119897)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)

where cov(a b) refers to the covariance between a and bGenerally the correlation coefficient approaches unity as theperformance of the channel estimation improves Just asin the definition of the channel estimation error only thechannel of the target cell is considered for channel correlationcoefficient

43 SimulationResults By setting different levels of noise andapplying the proposed channel estimation scheme we obtainMSE of the estimation of large-scale fading as illustrated inFigure 3 It can be seen that the estimation error is marginaleven in the case of rather strong background noise beingpresent

To evaluate the impact of the estimation error of the large-scale fading on the small-scale fading estimation we comparethe normalized channel estimation errors and the channelcorrelation coefficients under three schemes the proposedscheme the proposed scheme with an ideal knowledge ofthe large-scale fading and the proposed scheme with anideal knowledge of the large-scale fading but with randomlyselected SIUs The simulation results are given in Table 2which shows that the estimation error of the large-scalefading has a significant impact on the performance of thesmall-scale fading estimation This is mainly due to a wrongselection of SIUs with poor estimation of the large-scalefading By comparing Case 1 and Case 3 it is obvious that

MSE

SNR (dB)

10minus2

10minus3

10minus4

minus5 0 5 10

All cellsTarget cell

Figure 3 Performance of large-scale fading estimation versuschannel condition

Table 3 Performance comparison of channel estimation schemes

Channel estimation scheme 120576 (dB) 120588

Single-cell MMSE (1198701= 0) minus236 0828

Improved multicell MMSE (1198701= 4) minus533 0879





Full-cell MMSE (1198701= 144) minus886 0933

even with an ideal estimation of the large-scale fading awrong selection of SIUs impacts severely on the estimationof the small-scale fading

To further evaluate the proposed scheme we now com-pare the performance of channel estimation in the caseof conventional schemes and the proposed scheme allbased on the estimated large-scale fading by the proposedinterference cancellation-based algorithm The results arelisted in Table 3 Two conventional schemes are consideredhere One is the single-cell MMSE where no neighboringcell users are included in the channel estimation and theother is the multicell MMSE channel estimation with allthe neighboring cell users included in the joint processing(referred as full-cell MMSE channel estimation) The pro-posed scheme with different number of SIUs is compared


Table 4 Comparison of computational cost (unit MIPS)

Channel estimation scheme Large-scale fading estimation Small-scale fading estimation Total Total savingsImproved multicell MMSE (119870

1= 12) 28 1369 1649 mdash

Full-cell MMSE 28 3973 4253 612Single-cell MMSE 28 1132 1412 minus168

as well so as to find a reasonable number of neighboringusers in the joint estimation From Table 3 it can be seen thatthe proposed scheme performs much better than the single-cell MMSE scheme does and approaches that of the idealmulticell MMSE with a slight degradation when the numberof SIUs considered in the joint processing is not less than 24which is equal to the total number of users in a cellWhen thenumber of SIUs is larger than 48 the estimation performanceis almost the same as that of a full-cell MMSE

44 Computational Complexity The computational com-plexity of the proposed algorithm is now evaluated andcompared with that of the conventional ones Suppose thatone instruction cycle is needed for each real multiply-addoperation in the digital signal processor (DSP)

For the proposed algorithm since diag (Ψ119867119897Ψ119897)

minus1

Ψ119867

119897in

(18) of Step (1) can be computed and stored in advancethere is no real time computation consumed for this partThecomputation in (18) costs 119870 times 120591 times 119872 complex multiply-addoperations In (19) additional119870times119872 complex add operationsare needed In total 350400 real operations are needed periteration For Step (2) the pilot reconstruction of the targetcell and interference deduction in (20) cost119870times120591times119872 complexmultiply-add and 120591 times119872 complex subtract operations Alongwith the MF estimation and large-scale fading computationas in (18) and (19) for each neighboring cell the overallcomputational cost in Step (2) is 2455200 real operations periteration The computational burden of Steps (3) and (4) isnegligible For Step (5) the overall computational cost in (25)consists of 119870

119886times 120591 complex multiply 119870

119886real multiply 119870

119886times 120591

complex multiply 120591 times 119870119886times 120591 complex multiply-add 120591 real

add 120591 times 120591 times 119872 complex multiply-add 120591 times 120591 complex matrixinversion and119870

119886times 120591 times119872 complex multiply-add operations

which amount to a total of 1368648 real operations for oneframe

Assume Steps (1) and (2) are computed every 10 framesand Step (5) is computed each frame then the computationcost for the estimation of the large-scale fading (ie Steps(1) and (2)) and the small-scale fading are 28MIPS and1369MIPS respectively Thus the total computation cost forthe proposed algorithm is 1649MIPS

The computational cost for the full-cellMMSE estimationand the single-cell MMSE estimation may be obtained in asimilar manner and the results are compared with those ofthe proposed algorithm in Table 4 From this table it can beseen that the proposed algorithm saves 612 computationalcost with respect to that of the full-cell MMSE scheme andthe additional computational burden over that of the single-cell MMSE is 168 which is rather small

5 Conclusions

In this paper the pilot contamination problem of a massiveMIMO system is discussed In view of the fact that currentlythe channel estimation schemes for massive MIMO systemsalways assume that the large-scale fading is known 119886 119901119903119894119900119903119894and that the computational complexity of conventional chan-nel estimations grows sharply in massive MIMO systems animproved multicell MMSE joint channel estimation schemewhich aims to mitigate the pilot contamination at an afford-able expense and in a more realistic situation has beenproposed

In the proposed scheme the highly interfering users inneighboring cells have been identified first based on the esti-mation of the large-scale fading and then included in the jointchannel processing Metrics of the performance evaluationof the channel estimation have been defined Finally thesimulation results and the computational complexity analysisare performed that verify the performance advantage of theproposed scheme at a reasonable computational cost over thatof the conventional schemes

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant no 61301103

References

[1] T L Marzetta ldquoNoncooperative cellular wireless with unlim-ited numbers of base station antennasrdquo IEEE Transactions onWireless Communications vol 9 no 11 pp 3590ndash3600 2010

[2] E G Larsson F Tufvesson O Edfors and T L MarzettaldquoMassive MIMO for next generation wireless systemsrdquo IEEECommunications Magazine vol 52 no 2 pp 186ndash195 2014

[3] H Q Ngo E Larsson and T Marzetta ldquoEnergy and spectralefficiency of very large multiuser MIMO systemsrdquo IEEE Trans-actions on Communications vol 61 no 4 pp 1436ndash1449 2013

[4] H Ngo E Larsson and T Marzetta ldquoThe multicell multiusermimo uplink with very large antenna arrays and a finite-dimensional channelrdquo IEEE Transactions on Communicationsvol 61 no 6 pp 2350ndash2361 2013

[5] J Jose A Ashikhmin T L Marzetta and S VishwanathldquoPilot contamination problem in multi-cell TDD systemsrdquo inProceedings of the IEEE International Symposiumon Information


Theory (ISIT rsquo09) pp 2184ndash2188 Seoul Republic of Korea July2009

[6] J Jose A Ashikhmin T L Marzetta and S Vishwanath ldquoPilotcontamination and precoding inmulti-cell TDD systemsrdquo IEEETransactions on Wireless Communications vol 10 no 8 pp2640ndash2651 2011

[7] H Q Ngo and E G Larsson ldquoEVD-based channel estimationsfor multicell multiuser MIMO with very large antenna arraysrdquoin Proceedings of the IEEE International Conference onAcousticsSpeech and Signal Processing (ICASSP rsquo12) pp 3249ndash3252 2012

[8] H Yin D Gesbert M Filippou and Y Liu ldquoA coordinatedapproach to channel estimation in large-scale multiple-antennasystemsrdquo IEEE Journal on Selected Areas in Communicationsvol 31 no 2 pp 264ndash273 2013

[9] H Yin D Gesbert M C Filippou and Y Liu ldquoDecontami-nating pilots in massive MIMO systemsrdquo in Proceedings of theIEEE International Conference on Communications (ICC rsquo2013)pp 3170ndash3175 2013

[10] A Hu T Lv H Gao Y Lu and E Liu ldquoPilot design for large-scalemulti-cell multiuserMIMO systemsrdquo in Proceedings of theIEEE International Conference on Communications (ICC rsquo2013)pp 5381ndash5385 2013

[11] A Ashikhmin and TMarzetta ldquoPilot contamination precodingin multi-cell large scale antenna systemsrdquo in Proceedings ofthe IEEE International Symposium on Information Theory (ISITrsquo2012) pp 1137ndash1141 2012

[12] D C Chu ldquoPolyphase codes with good periodic correlationpropertiesrdquo IEEE Transactions on Information Theory vol 18no 4 pp 531ndash532 1972

[13] X Song K Li and A Hu ldquoTwo-stage channel estimationalgorithm for TD-SCDMA system employing multi-cell jointdetectionrdquo Circuits Systems and Signal Processing vol 26 no6 pp 961ndash968 2007

[14] J G ProakisDigital Communications McGraw-Hill NewYorkNY USA 4th edition 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



of the channel vectors of massive MIMO systems Because ofthe proposed pilots the channel estimate of most terminalsof a cell is only interfered by partial cells rather than all theother cells where the latter is caused by traditional pilots Asa result pilot contamination is mitigated It is also possibleto coordinate the use of pilots or adaptively allocate the pilotsequences to different terminals in the network [8]

Multicell joint channel estimation algorithms [5 8] oreven blind techniques that circumvent the use of pilotsaltogether [7] may mitigate or eliminate the effects of pilotcontamination The approach in [7] exploits the asymptoticorthogonality of the channel vectors in very large MIMOsystems and shows that the channel to each user can beestimated from the covariance matrix of the received signalsup to a remaining scalar multiplicative ambiguity which canbe resolved by utilizing a short training sequence

The new precoding techniques such as pilot contami-nation precoding [11] that take into account the networkstructure can utilize cooperative transmission over a multi-plicity of cellsmdashoutside of the beamforming operationmdashtonullify at least partially the directed interference that resultsfrom pilot contamination Unlike coordinated beamformingover multiple cells which requires estimates of the actualchannels between the terminals and the service-arrays ofthe contaminating cells the pilot-contamination precodingrequires only the corresponding slow-fading coefficients

By overviewing the abovementioned channel estimationschemes it can be seen that for simplicity it is alwaysassumed that the large-scale fading propagation (ie pathlossand shadow fading) is fixed and known 119886 119901119903119894119900119903119894 which isnot the case in practice Furthermore the computationalcomplexity of conventional channel estimations such as leastsquare (LS) and minimum mean square error (MMSE)estimations grows sharply in a massiveMIMO system as thenumber of antennas is much more than that in the currentmultiple antennas systems This problem is getting worsein the case of joint processing in a multicell scenario Asa countermeasure to these problems an improved channelestimation scheme which aims at mitigating the pilot con-tamination at an affordable expense and in a more realisticsituation is proposed in this paper

To deal with the abovementioned problems a new chan-nel estimation scheme is proposed by utilizing interferencecancellation and joint processing Highly interfering usersin neighboring cells are identified based on the estimationof large-scale fading and then included in the joint channelprocessing this achieves a compromise between the effec-tiveness and efficiency of the channel estimation and leads toan improvement in the overall system performance Metricsof the performance evaluation of the channel estimation aredefined and finally simulation results are provided whichvalidate the proposed scheme

The paper is organized as follows Section 2 gives themul-ticell massive MIMO system model and the pilot design Theconventional channel estimator and the proposed multicelljoint MMSE channel estimation are presented in Section 3Section 4 presents the simulation results and the analysis ofcomputational complexity of the proposed scheme FinallySection 5 concludes this paper

Notations (sdot)lowast (sdot)119879 and (sdot)119867 denote the conjugate transposeand conjugate transpose respectively [sdot]

119894119895is the (119894 119895)th entry

of a matrix 119864sdot is the expectation and sdot denotes theFrobenius norm

2 Multicell Massive MIMO System Model

21 System Model Here we consider a massive MIMOcellular systemwith 119871 hexagonal cells as depicted in Figure 1Each cell comprises one base station with119872 antennas and119870single antenna users (119870 ≪ 119872 in the case of amassiveMIMOsystem) All the cells share the same band of frequenciesIn addition the system employs OFDM and works in time-division duplex (TDD) mode

Let the average power (during transmission) at the basestation be 119901

119891and let the average power at each user be 119901

119903 In

one coherence interval the channel responses of119873119886symbols

are assumed to be static which means that the channelestimate is effective only in this interval Consequently a totalof119873119888symbols are used for pilots from all the terminals to BS

and the remaining 119873119886minus 119873119888symbols are used for a two-way

data transmission Apart from that following the model in[1] the channel responses are assumed to be constant in thefrequency smooth interval which consists of119873

119904consecutive

subcarriers Therefore to estimate the channel response ofone subcarrier every terminal can transmit pilots over 119873

119904

subcarriers In the following channel responses of the 119873119904

subcarriers will be estimated As a result the pilot length isgiven by 120591 = 119873

119888119873119904 and thus up to 120591 terminals can transmit

pilots to BS without intracell interferenceThe propagation factor (channel response) between the

119898th base station antenna of the 119897th cell and the 119896th user ofthe 119895th cell is given by

119892119895119897119896119898

= radic120573119895119897119896ℎ119895119897119896119898

(1)

where 120573119895119897119896 are the large-scale fading coefficients composed

of pathloss and shadow fading and are assumed invariant fordifferent antennas of the same BS and ℎ

119895119897119896119898 are the small-

scale fading coefficients and are assumed to be independentand identically distributed (iid) and circularly symmetricGaussian random variables (CN(0 1))We assume the chan-nel reciprocity for the downlink and the uplink that is thepropagation factor ℎ

119895119897119896119898is the same for both the downlink

and the uplink and block fading that is ℎ119895119897119896119898

remainsconstant for the duration of coherence intervals (typically 4ndash30OFDMsymbols)The additive noise signals at all terminalsare iid (CN(0 1)) random variables Then the channelmatrix from the terminals of the 119895th cell to BS in the 119897th cellis given by

G119895119897= D119895119897H119895119897isin C119870times119872

(2)

where H119895119897isin C119870times119872 are small-scale fading coefficients that

is [H119895119897]119896119898

= ℎ119895119897119896119898

D119895119897isin R119870times119870 is a diagonal matrix where

[D119895119897]119896119896

= radic120573119895119897119896

At the beginning of every coherence interval all the usersin all the cells synchronously transmit pilot sequences of


1

1

jth cell

K

K

k

k

gjlkm

3 mM

3 mM




lth cell



(where 120595119867119895119896

120595119895119896


119910119897119898

=

119871

sum

119895=1

119870

sum

119896=1

radic119901119903120591120573119895119897119896ℎ119895119897119896119898

120595119895119896

+ 119908119897119898 (3)

where 119908119897119898


= [1199101198971

1199101198972

sdot sdot sdot 119910119897119872

] isin C120591times119872 W119897

=

[1199081198971

1199081198972


119895 119895 = 1 119871 be the


1205951198952

sdot sdot sdot 120595119895119870


Y119897= radic119901119903120591

119871

sum

119895=1

Ψ119895G119895119897+W119897 (4)


Y119897= radic119901119903120591Ψ119897G119897119897+ radic119901119903120591

119871

sum

119895=1119895 = 119897

Ψ119895G119895119897+W119897

(5)



119897must


119895119897= Ψ119879

119897Ψlowast

119895isin C119870times119870





119895


is given by

[120595119895

]

119899

= 119886119899exp 119894

2120587119902119895119899

120591

119899 = 1 2 120591 (6)


119895



0

1198861 119886


119886119899= exp 119894119873120587

120591

119899 (119899 + (120591 mod 2)) 119899 = 0 1 120591 minus 1 (7)


radic120591120595119895119896

= ⟨radic120591120595119895

⟩

119896minus1

(8)




119903119895=

120591minus1

sum

119899=0

119886119899119886lowast

(119899+119895) mod 120591 119895 = 0 1 120591 minus 1 (9)


119895= 0 when





H119897119897= radic119901119903120591(D119897119897Ψ119867

119897Ψ119897D119897119897)

minus1

D119897119897Ψ119867

119897Y119897 (10)


H119897119897= H119897119897+

119871

sum

119895=1119895 = 119897

H119895119897+

Ψ119867

119897W119897

120591

(11)





by

H119897119897= radic119901119903120591D119897119897Ψ119867

119897(I + 119901

119903120591Ψ119897D2119897119897Ψ119867

119897)

minus1

Y119897 (12)






119871][

[

[

G1119897

G119871119897

]

]

]

+W119897

= radic119901119903120591ΦG119897+W119897

= radic119901119903120591ΦD119897H119897+W119897

(13)


119897= [


d


] isin

C119870119871times119870119871 H119897= [

H1119897

H119871119897

]isin C119870119871times119872


H119895119897= radic119901119903120591D119895119897Ψ119867

119895(I + 119901

119903120591

119871

sum

119894=1

Ψ119894D2119894119897Ψ119867

119894)

minus1

Y119897

119895 = 1 sdot sdot sdot 119871

(14)


H119897= radic119901119903120591D119897Φ119867

(I + 119901119903120591ΦD2119897Φ119867

)

minus1

Y119897 (15)



119895119897119896 are assumed




119895119897119896




119894119897 119894 =




is

G119897119897= radic119901119903120591Ψ119867

119897(I + 119901

119903120591

119871

sum

119894=1

Ψ119894Ψ119867

119894)

minus1

Y119897 (16)



H119897119897= radic119901119903120591Dminus1119897119897Ψ119867

119897(I + 119901

119903120591

119871

sum

119894=1

Ψ119894Ψ119867

119894)

minus1

Y119897 (17)




of H119897119897








GMF119897119897

=

1

radic119901119903120591

diag (Ψ119867119897Ψ119897)

minus1

Ψ119867

119897Y119897 (18)


119897Ψ119897)

minus1

Ψ119867




radic120573119897119897119896=

1

119876

E[GMF119897119897]119896

119896 = 1 119870 (19)


Let D119897119897= diag(radic120573

1198971198971radic1205731198971198972

radic120573119897119897119870)


119873





Y119897= radic119901119903120591Ψ119894G119894119897+W119897 (21)










= diag(radic1205731198991198971radic1205731198991198972

radic1205731198991198971198701


1is decided








1SIUs of the


D119897= [

DCT119897

0

0 DCN119897

] isin R119870119886times119870119886 (22)

where DCT119897

= diag(radic1205731198971198971radic1205731198971198972

radic1205731198971198971198700



1


Ψ119897= [1205951198971

1205951198971198700

1205951198991

1205951198991198701

] isin C120591times119870119886 (23)





Y119897= radic119901119903120591Ψ119897

D119897H119897+W119897 (24)




H119897= radic119901119903120591 D119897Ψ

H119897(I + 119901

119903120591Ψ119897D2119897Ψ119867

119897)

minus1

Y119897

(25)


0rows of H

119897 we then



+minus

Yl MF estimation of

MF estimation of

MF estimation of

target cell users


users

Dll calculation

Dll calculation

Dll calculation


sim



CNth neighbor cell





119897119897119896= 10

11990810

(119903119895119897119896100)




119904(ieN(0 120590

2

119904))



119894 119894 = 1 2 7 in (6) are set




119888) 2GHz





1)

12





120596 ≜ E(

100381610038161003816100381610038161003816100381610038161003816


100381610038161003816100381610038161003816100381610038161003816

2

) 119896 = 1 119870 times 119871 (26)




Scheme 120576 (dB) 120588



120576 ≜ 10 log10(

119870

sum

119896=1

119872

sum

119898=1

10038171003817100381710038171003817[H119897119897]119896119898

minus [H119897119897]119896119898

10038171003817100381710038171003817

2

10038171003817100381710038171003817[H119897119897]119896119898

10038171003817100381710038171003817

2) (27)

where H119897119897

and H119897119897


1strongest inter-



120588 ≜

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

cov (H119897119897H119897119897)

radiccov (H119897119897H119897119897) cov (H

119897119897H119897119897)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)




MSE

SNR (dB)

10minus2

10minus3

10minus4

minus5 0 5 10

















1= 12) 28 1369 1649 mdash





minus1

Ψ119867

119897in




119886times 120591







5 Conclusions





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

1

jth cell

K

K

k

k

gjlkm

3 mM

3 mM




lth cell



(where 120595119867119895119896

120595119895119896


119910119897119898

=

119871

sum

119895=1

119870

sum

119896=1

radic119901119903120591120573119895119897119896ℎ119895119897119896119898

120595119895119896

+ 119908119897119898 (3)

where 119908119897119898


= [1199101198971

1199101198972

sdot sdot sdot 119910119897119872

] isin C120591times119872 W119897

=

[1199081198971

1199081198972


119895 119895 = 1 119871 be the


1205951198952

sdot sdot sdot 120595119895119870


Y119897= radic119901119903120591

119871

sum

119895=1

Ψ119895G119895119897+W119897 (4)


Y119897= radic119901119903120591Ψ119897G119897119897+ radic119901119903120591

119871

sum

119895=1119895 = 119897

Ψ119895G119895119897+W119897

(5)



119897must


119895119897= Ψ119879

119897Ψlowast

119895isin C119870times119870





119895


is given by

[120595119895

]

119899

= 119886119899exp 119894

2120587119902119895119899

120591

119899 = 1 2 120591 (6)


119895



0

1198861 119886


119886119899= exp 119894119873120587

120591

119899 (119899 + (120591 mod 2)) 119899 = 0 1 120591 minus 1 (7)


radic120591120595119895119896

= ⟨radic120591120595119895

⟩

119896minus1

(8)




119903119895=

120591minus1

sum

119899=0

119886119899119886lowast

(119899+119895) mod 120591 119895 = 0 1 120591 minus 1 (9)


119895= 0 when





H119897119897= radic119901119903120591(D119897119897Ψ119867

119897Ψ119897D119897119897)

minus1

D119897119897Ψ119867

119897Y119897 (10)


H119897119897= H119897119897+

119871

sum

119895=1119895 = 119897

H119895119897+

Ψ119867

119897W119897

120591

(11)





by

H119897119897= radic119901119903120591D119897119897Ψ119867

119897(I + 119901

119903120591Ψ119897D2119897119897Ψ119867

119897)

minus1

Y119897 (12)






119871][

[

[

G1119897

G119871119897

]

]

]

+W119897

= radic119901119903120591ΦG119897+W119897

= radic119901119903120591ΦD119897H119897+W119897

(13)


119897= [


d


] isin

C119870119871times119870119871 H119897= [

H1119897

H119871119897

]isin C119870119871times119872


H119895119897= radic119901119903120591D119895119897Ψ119867

119895(I + 119901

119903120591

119871

sum

119894=1

Ψ119894D2119894119897Ψ119867

119894)

minus1

Y119897

119895 = 1 sdot sdot sdot 119871

(14)


H119897= radic119901119903120591D119897Φ119867

(I + 119901119903120591ΦD2119897Φ119867

)

minus1

Y119897 (15)



119895119897119896 are assumed




119895119897119896




119894119897 119894 =




is

G119897119897= radic119901119903120591Ψ119867

119897(I + 119901

119903120591

119871

sum

119894=1

Ψ119894Ψ119867

119894)

minus1

Y119897 (16)



H119897119897= radic119901119903120591Dminus1119897119897Ψ119867

119897(I + 119901

119903120591

119871

sum

119894=1

Ψ119894Ψ119867

119894)

minus1

Y119897 (17)




of H119897119897








GMF119897119897

=

1

radic119901119903120591

diag (Ψ119867119897Ψ119897)

minus1

Ψ119867

119897Y119897 (18)


119897Ψ119897)

minus1

Ψ119867




radic120573119897119897119896=

1

119876

E[GMF119897119897]119896

119896 = 1 119870 (19)


Let D119897119897= diag(radic120573

1198971198971radic1205731198971198972

radic120573119897119897119870)


119873





Y119897= radic119901119903120591Ψ119894G119894119897+W119897 (21)










= diag(radic1205731198991198971radic1205731198991198972

radic1205731198991198971198701


1is decided








1SIUs of the


D119897= [

DCT119897

0

0 DCN119897

] isin R119870119886times119870119886 (22)

where DCT119897

= diag(radic1205731198971198971radic1205731198971198972

radic1205731198971198971198700



1


Ψ119897= [1205951198971

1205951198971198700

1205951198991

1205951198991198701

] isin C120591times119870119886 (23)





Y119897= radic119901119903120591Ψ119897

D119897H119897+W119897 (24)




H119897= radic119901119903120591 D119897Ψ

H119897(I + 119901

119903120591Ψ119897D2119897Ψ119867

119897)

minus1

Y119897

(25)


0rows of H

119897 we then



+minus

Yl MF estimation of

MF estimation of

MF estimation of

target cell users


users

Dll calculation

Dll calculation

Dll calculation


sim



CNth neighbor cell





119897119897119896= 10

11990810

(119903119895119897119896100)




119904(ieN(0 120590

2

119904))



119894 119894 = 1 2 7 in (6) are set




119888) 2GHz





1)

12





120596 ≜ E(

100381610038161003816100381610038161003816100381610038161003816


100381610038161003816100381610038161003816100381610038161003816

2

) 119896 = 1 119870 times 119871 (26)




Scheme 120576 (dB) 120588



120576 ≜ 10 log10(

119870

sum

119896=1

119872

sum

119898=1

10038171003817100381710038171003817[H119897119897]119896119898

minus [H119897119897]119896119898

10038171003817100381710038171003817

2

10038171003817100381710038171003817[H119897119897]119896119898

10038171003817100381710038171003817

2) (27)

where H119897119897

and H119897119897


1strongest inter-



120588 ≜

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

cov (H119897119897H119897119897)

radiccov (H119897119897H119897119897) cov (H

119897119897H119897119897)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)




MSE

SNR (dB)

10minus2

10minus3

10minus4

minus5 0 5 10

















1= 12) 28 1369 1649 mdash





minus1

Ψ119867

119897in




119886times 120591







5 Conclusions





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













by

H119897119897= radic119901119903120591D119897119897Ψ119867

119897(I + 119901

119903120591Ψ119897D2119897119897Ψ119867

119897)

minus1

Y119897 (12)






119871][

[

[

G1119897

G119871119897

]

]

]

+W119897

= radic119901119903120591ΦG119897+W119897

= radic119901119903120591ΦD119897H119897+W119897

(13)


119897= [


d


] isin

C119870119871times119870119871 H119897= [

H1119897

H119871119897

]isin C119870119871times119872


H119895119897= radic119901119903120591D119895119897Ψ119867

119895(I + 119901

119903120591

119871

sum

119894=1

Ψ119894D2119894119897Ψ119867

119894)

minus1

Y119897

119895 = 1 sdot sdot sdot 119871

(14)


H119897= radic119901119903120591D119897Φ119867

(I + 119901119903120591ΦD2119897Φ119867

)

minus1

Y119897 (15)



119895119897119896 are assumed




119895119897119896




119894119897 119894 =




is

G119897119897= radic119901119903120591Ψ119867

119897(I + 119901

119903120591

119871

sum

119894=1

Ψ119894Ψ119867

119894)

minus1

Y119897 (16)



H119897119897= radic119901119903120591Dminus1119897119897Ψ119867

119897(I + 119901

119903120591

119871

sum

119894=1

Ψ119894Ψ119867

119894)

minus1

Y119897 (17)




of H119897119897








GMF119897119897

=

1

radic119901119903120591

diag (Ψ119867119897Ψ119897)

minus1

Ψ119867

119897Y119897 (18)


119897Ψ119897)

minus1

Ψ119867




radic120573119897119897119896=

1

119876

E[GMF119897119897]119896

119896 = 1 119870 (19)


Let D119897119897= diag(radic120573

1198971198971radic1205731198971198972

radic120573119897119897119870)


119873





Y119897= radic119901119903120591Ψ119894G119894119897+W119897 (21)










= diag(radic1205731198991198971radic1205731198991198972

radic1205731198991198971198701


1is decided








1SIUs of the


D119897= [

DCT119897

0

0 DCN119897

] isin R119870119886times119870119886 (22)

where DCT119897

= diag(radic1205731198971198971radic1205731198971198972

radic1205731198971198971198700



1


Ψ119897= [1205951198971

1205951198971198700

1205951198991

1205951198991198701

] isin C120591times119870119886 (23)





Y119897= radic119901119903120591Ψ119897

D119897H119897+W119897 (24)




H119897= radic119901119903120591 D119897Ψ

H119897(I + 119901

119903120591Ψ119897D2119897Ψ119867

119897)

minus1

Y119897

(25)


0rows of H

119897 we then



+minus

Yl MF estimation of

MF estimation of

MF estimation of

target cell users


users

Dll calculation

Dll calculation

Dll calculation


sim



CNth neighbor cell





119897119897119896= 10

11990810

(119903119895119897119896100)




119904(ieN(0 120590

2

119904))



119894 119894 = 1 2 7 in (6) are set




119888) 2GHz





1)

12





120596 ≜ E(

100381610038161003816100381610038161003816100381610038161003816


100381610038161003816100381610038161003816100381610038161003816

2

) 119896 = 1 119870 times 119871 (26)




Scheme 120576 (dB) 120588



120576 ≜ 10 log10(

119870

sum

119896=1

119872

sum

119898=1

10038171003817100381710038171003817[H119897119897]119896119898

minus [H119897119897]119896119898

10038171003817100381710038171003817

2

10038171003817100381710038171003817[H119897119897]119896119898

10038171003817100381710038171003817

2) (27)

where H119897119897

and H119897119897


1strongest inter-



120588 ≜

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

cov (H119897119897H119897119897)

radiccov (H119897119897H119897119897) cov (H

119897119897H119897119897)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)




MSE

SNR (dB)

10minus2

10minus3

10minus4

minus5 0 5 10

















1= 12) 28 1369 1649 mdash





minus1

Ψ119867

119897in




119886times 120591







5 Conclusions





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














GMF119897119897

=

1

radic119901119903120591

diag (Ψ119867119897Ψ119897)

minus1

Ψ119867

119897Y119897 (18)


119897Ψ119897)

minus1

Ψ119867




radic120573119897119897119896=

1

119876

E[GMF119897119897]119896

119896 = 1 119870 (19)


Let D119897119897= diag(radic120573

1198971198971radic1205731198971198972

radic120573119897119897119870)


119873





Y119897= radic119901119903120591Ψ119894G119894119897+W119897 (21)










= diag(radic1205731198991198971radic1205731198991198972

radic1205731198991198971198701


1is decided








1SIUs of the


D119897= [

DCT119897

0

0 DCN119897

] isin R119870119886times119870119886 (22)

where DCT119897

= diag(radic1205731198971198971radic1205731198971198972

radic1205731198971198971198700



1


Ψ119897= [1205951198971

1205951198971198700

1205951198991

1205951198991198701

] isin C120591times119870119886 (23)





Y119897= radic119901119903120591Ψ119897

D119897H119897+W119897 (24)




H119897= radic119901119903120591 D119897Ψ

H119897(I + 119901

119903120591Ψ119897D2119897Ψ119867

119897)

minus1

Y119897

(25)


0rows of H

119897 we then



+minus

Yl MF estimation of

MF estimation of

MF estimation of

target cell users


users

Dll calculation

Dll calculation

Dll calculation


sim



CNth neighbor cell





119897119897119896= 10

11990810

(119903119895119897119896100)




119904(ieN(0 120590

2

119904))



119894 119894 = 1 2 7 in (6) are set




119888) 2GHz





1)

12





120596 ≜ E(

100381610038161003816100381610038161003816100381610038161003816


100381610038161003816100381610038161003816100381610038161003816

2

) 119896 = 1 119870 times 119871 (26)




Scheme 120576 (dB) 120588



120576 ≜ 10 log10(

119870

sum

119896=1

119872

sum

119898=1

10038171003817100381710038171003817[H119897119897]119896119898

minus [H119897119897]119896119898

10038171003817100381710038171003817

2

10038171003817100381710038171003817[H119897119897]119896119898

10038171003817100381710038171003817

2) (27)

where H119897119897

and H119897119897


1strongest inter-



120588 ≜

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

cov (H119897119897H119897119897)

radiccov (H119897119897H119897119897) cov (H

119897119897H119897119897)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)




MSE

SNR (dB)

10minus2

10minus3

10minus4

minus5 0 5 10

















1= 12) 28 1369 1649 mdash





minus1

Ψ119867

119897in




119886times 120591







5 Conclusions





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










+minus

Yl MF estimation of

MF estimation of

MF estimation of

target cell users


users

Dll calculation

Dll calculation

Dll calculation


sim



CNth neighbor cell





119897119897119896= 10

11990810

(119903119895119897119896100)




119904(ieN(0 120590

2

119904))



119894 119894 = 1 2 7 in (6) are set




119888) 2GHz





1)

12





120596 ≜ E(

100381610038161003816100381610038161003816100381610038161003816


100381610038161003816100381610038161003816100381610038161003816

2

) 119896 = 1 119870 times 119871 (26)




Scheme 120576 (dB) 120588



120576 ≜ 10 log10(

119870

sum

119896=1

119872

sum

119898=1

10038171003817100381710038171003817[H119897119897]119896119898

minus [H119897119897]119896119898

10038171003817100381710038171003817

2

10038171003817100381710038171003817[H119897119897]119896119898

10038171003817100381710038171003817

2) (27)

where H119897119897

and H119897119897


1strongest inter-



120588 ≜

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

cov (H119897119897H119897119897)

radiccov (H119897119897H119897119897) cov (H

119897119897H119897119897)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)




MSE

SNR (dB)

10minus2

10minus3

10minus4

minus5 0 5 10

















1= 12) 28 1369 1649 mdash





minus1

Ψ119867

119897in




119886times 120591







5 Conclusions





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Scheme 120576 (dB) 120588



120576 ≜ 10 log10(

119870

sum

119896=1

119872

sum

119898=1

10038171003817100381710038171003817[H119897119897]119896119898

minus [H119897119897]119896119898

10038171003817100381710038171003817

2

10038171003817100381710038171003817[H119897119897]119896119898

10038171003817100381710038171003817

2) (27)

where H119897119897

and H119897119897


1strongest inter-



120588 ≜

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

cov (H119897119897H119897119897)

radiccov (H119897119897H119897119897) cov (H

119897119897H119897119897)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)




MSE

SNR (dB)

10minus2

10minus3

10minus4

minus5 0 5 10

















1= 12) 28 1369 1649 mdash





minus1

Ψ119867

119897in




119886times 120591







5 Conclusions





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1= 12) 28 1369 1649 mdash





minus1

Ψ119867

119897in




119886times 120591







5 Conclusions





Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article An Improved Multicell MMSE Channel...

Documents

Transcript of Research Article An Improved Multicell MMSE Channel...