time-varying channels On channel estimation for spatial modulated systems over Digital Signal Processing

Contents lists available atScienceDirect

Digital

Signal

Processing

www.elsevier.com/locate/dsp

On

channel

estimation

for

spatial

modulated

systems

over

time-varying

channels

Yusuf Acar

a

, Hakan Do˘gan

b,

∗

, Erdal Panayırcı

c

a_{DepartmentofElectronicsEngineering,IstanbulKulturUniversity,Bakirkoy,34156,Istanbul,Turkey} b_{DepartmentofElectricalandElectronicsEngineering,IstanbulUniversity,Avcilar,34320,Istanbul,Turkey} c_{DepartmentofElectricalandElectronicsEngineering,KadirHasUniversity,Cibali,34083,Istanbul,Turkey}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline21November2014

Keywords:

Spatialmodulation Iterativechannelestimation Recursiveleastsquare Curvefitting Time-varyingchannel

SpatialModulation(SM)hasbeenproposedrecentlyformultiple-input multiple-output (MIMO)systems tocopewiththeinterchannelinterferenceand toreducethedetectioncomplexityas comparedtothe conventional MIMOsystems. InSMsystem,the data symbolsare transmitted byarandomlyselected active antenna ofaMIMOtransmitter to thereceiver through awirelesschannel.The information is carriedbothbythedatasymbolfromanysignalconstellationsuchasM-aryphaseshiftkeying(M-PSK) orM-aryquadratureamplitudemodulation(M-QAM),by theindexoftheselectedantenna.Thechannel estimationisacriticalprocessatthereceiverduringthecoherentdetectionofthetransmitted symbol andtheantennaindex,randomlyselected.Recently,thechannelestimationofchannelforSMsystems hasbeeninvestigatedbytherecursiveleastsquare(RLS)algorithmforonlyquasi-staticfadingchannels. Inthispaper,anovelchannelestimationisproposed forSMsystemsinthepresenceofrapidly time-varying channels. The Bayesian mean square error (MSE) bound has been derived as a benchmark and theperformance ofthe proposedapproaches is studiedintermsofMSEand bit-errorrate(BER). Computer simulationresults haveconfirmed thatthe proposed iterativechannelestimation technique hassignificantBER/MSEperformance advantagescomparedwithexistingchannelestimationalgorithm proposedearlierintheliterature.

©2014ElsevierInc.All rights reserved.

1. Introduction

Conventional multiple-input multiple-output (MIMO) systems usealltransmitantennastotransmitmultipledatastreams. There-fore,itsperformancedependsonsomeimportantparameterssuch asthe distance between receiver andtransmitter antennas [1,2], inter-channel interference (ICI) at the receiver and inter-antenna synchronization (IAS) at the transmitter [3,4]. For example, it wasshownthatuncorrectIAScausesperformancedegradationfor MIMOsystems[3,4].

Spatial modulation (SM) is a promising MIMO transmission techniquethathasbeenrecentlyproposed[5–7].Thebasic princi-pleoftheSMistousetheindicesofmultipleantennastoconvey informationinaddition to theconventional two-dimensional sig-nalconstellations such as M-ary phase shiftkeying(M-PSK) and

M-ary quadrature amplitude modulation (M-QAM), where M is

theconstellationsize.OptimalSMdecoderatthereceiversearches jointly for all M-ary constellation points and transmit antennas

*

Correspondingauthor.

E-mailaddress:hdogan@istanbul.edu.tr(H. Do˘gan).

to decide on both the transmitted symbol andthe index of the transmittedantennaoverwhichthesymbolistransmitted. Conse-quently, itisan effectivewayto removethe intercarrier interfer-ence(ICI)completelybetweenthetransmitterantennasofaMIMO link.Furthermore,SM doesnot requireIASoftheMIMOlinkand onlyoneradiofrequencychain(RF)isneededatthetransmitter.

The SM technique is different from the transmit antenna se-lection(TAS) since TAS isa closed-loop mechanismandprovides spatial-multiplexing while the SM is open-loop with transmit-diversity [8]. SM technique adds a third dimension to the two-dimensional signal space which is the spatial dimension and it maps multiple information bits into one symbol and the corre-spondingantennaindex. Therefore,thenumberoftotal transmit-tedinformationbitsdependsontheconstellationdiagramandthe total number oftransmitter antennas [9]. Consequently, the spa-tialmodulationhasa veryflexible mechanismthatprovides high spectralefficiencywithlowcomplexity[10].

The receiver has to detect both the transmitted symbol and theactive antennaindexsincethe desiredinformationcarriedby the modulated signal andthe transmit antenna index, chosen at random. In the literature, the antenna index and symbol detec-tion arerealized by meansofoptimal andnon-optimaldetection

http://dx.doi.org/10.1016/j.dsp.2014.11.004

methods [5,11]. It has been shown in [11] and[9] that SM can achieve better errorperformance thanV-BLAST (Vertical-BellLab LayeredSpace–Time)insomecasesundertheassumptionthat per-fectchannelinformation(CSI)isavailableatthereceiver.However, inpractice,wehardly haveaperfectCSIatthereceiver andthus achannelestimatorisemployedtoprovideunknown channel pa-rameters.Recently, theeffect ofimperfect channel estimationon theSM-MIMO systemshas been investigated[12,13].In [12],the leastsquare (LS)estimationtechniqueisemployedforMIMO sys-temsoperatingoverquasi-staticRayleighflatfading channelsand its mean-square error(MSE) performance isinvestigated. In[13], a joint channel estimationwithdata detectionisproposed while assuming the channel correlation matrix is available at the re-ceiver.

Pilot symbol assisted modulation (PSAM) has been generally employed to achieve coherent detection performance in wireless environments [14]. Based on this approach, in [15] the channel estimation for SM systems has been investigated by means of a pilot-based recursive least square (RLS) method while assuming thewireless channel is quasi-static foraduration of atleastone framelength.The RLSalgorithmisknownto possessfast conver-gence,butalsotoyieldhighchannelestimationerrorsonfast fad-ingchannelsmainlybecauseitsolelydependsonthepilotsymbols anddoesnottake themobilityintoaccount [16]. In communica-tionsystems,pilotsymbols,knowntothereceiver,canbeinserted periodically, usually inthe beginning ofeach frame consistingof several transmitted symbols. However, when the channel varies rapidly, pilot symbol sequence cannot be effective to implement thechannelestimationefficiently.

In this work, the pilots are sent out through only one trans-mitantennaateachtimeinstant.Hence,usingpilot-basedchannel estimation,onlytheCSIoftheactivetransmitantennacanbe ob-tainedatthereceiver.Thisleadstoachallengingtaskforthe chan-nel estimationinSM systems over fasttime-varyingchannels. In

[17],performanceboundsfortraining andsuperimposed training-based channel estimation for time-varying flat-fading channels havebeendiscussed.Itwasshownthattheregularperiodic place-ments (RPPs) perform better athigh SNR andfor slowly varying channels, whereas the superimposed scheme is superior for rel-ativelyfast time-varying channels. However, in MIMOsystems it isrequiredthat the pilotsequences transmitted fromeach trans-mitantennashouldbe orthogonalto eachother toprevent inter-antennainterference.Thisachallengingdesignproblemingeneral. Ontheother hand,thisproblemcan beeasily handledin spatial modulated MIMO systems since the pilot sequences transmitted fromeachtransmitantennaaresurely orthogonal eachother due tothefactthattheyaremutuallydisjointatallthetime.

Channel coefficients in a real mobile environment change

smoothly in time. This smoothnesshelps us to employ well de-signedcurvefittingmethodsinordertofurtherimprovethe chan-nel estimation accuracy [18]. In [19], the rectangular-windowed recursiveleastsquaresalgorithmwhereeachtapofthefrequency selectivefading channelmodeled asa polynomialin timeis pro-posed. On the other hand, all channels could not be observed within theduration of asymbol transmission sinceonly one an-tennaisactive atthegivensignalinginterval.Thisalsomotivates usto usecurve fitting methods to interpolate unknown channel durations[20].Itisclearthattotrackthechannelcoefficientsfor datadurationweneedtoemployadecisiondirectedchannel esti-mationscheme.Differentmethods basedondecisiondirectedare alsoproposed toenhancethe trackingcapabilityoftheRLS algo-rithm in[21] forMIMOsystems.In[21],optimizingthe involved windowsizeandforgettingfactorandtheinitializationofthe au-tocorrelationmatrixofRLSarealsoinvestigated.

To the best of our knowledge, there is not any efficient and computationallyfeasiblechannelestimationalgorithm,inthe

pres-enceofarapidlyvaryingchannel,existsfortheSM-MIMOsystems in the literature. Motivated by the existing correspondences be-tween the RLS, thedecision directed channel estimation andthe polynomial fitting, inthispaperanovel channelestimation tech-nique andaniterativereceiverdesign areproposed basedon the curve fitting and the detected symbols employed in a decision-directed mode that ensure excellent tracking for SM-MIMO sys-tems. Weinsertperiodic pilot blockstocope withthe errors, in-troducedindecision-directedchannelestimationmode,duetothe accumulate overbits. Thedatablock lengthbetweenadjacent pi-lot blockscan be adjusted based on the channel mobilityin our proposedscheme.Thisresultsinminimumoverheadforpilot sym-bols. It is known that the iterative receivers provide significant advantages [22–24] over conventional receivers and shown that theSMreceiveremployingtheproposedchannelestimatorhas su-periorperformance ascomparedto theconventionalRLS channel estimation-based receivers. Moreover, we derived analytically an overall BayesianMSE lowerbound forthe channelestimator pro-posed inthiswork toserve asa benchmark.We performednew computer simulations todetermine howthe MSEperformance of ourchannelestimationalgorithmcanapproachthislowerbound.

Notation: Throughout the paper, the following notations and assumptionsareused.Boldandcapitalletters‘A’denotematrices. Boldandsmallletters‘a’denotevectors.diag

{

a

}

isadiagonal ma-trixwitha onitsmain diagonal. Ex,y

[.]

istheexpectationover x

and y. The notations,

(.)

∗,

(.)

T,

(.)

†,

(.)

+,

(.)

−1 and

.

F denote

conjugate,transpose,Hermitian,pseudoinverse,inverseand Frobe-niusnorm,ofamatrixoravectorrespectively.

2. Systemmodel

AnSM-MIMOsystemwithNt transmitantennasandNrreceive

antennasisconsidered.Ingeneral,thetotalnumberofbitsthatis transmittedbya M-arySM-MIMOsystemis

k

=

log2

(

Nt

)

+

log2

(

M

)

(1) whereM representsthetotalnumberofbitspertransmitted sym-bol. At the nth symbol interval the SM mapper takes a random sequence ofk bitsandmapsthem intoan Nt-dimensionalsignal

vectoras

x

(

n

)

=



x1

(

n

),

x2

(

n

),

· · · ,

xNt

(

n

)



T

.

(2)

Onlyoneofxj(n

)

thatisactiveinx

(

n

)

isnonzero.Then,atthe

nth symbol interval, the output of the SM-MIMO system at the transmittercanbeexpressedas

xj

(

n

)





0

· · ·

xq

(

n

)

  

j.transmitted antenna

· · ·

0



T (3)

where j istheactive antennaindex andxq(n

)

isthe qth symbol

fromtheM-ary constellationdiagram.Theother antennasremain silent over thissymbolduration.The symbol xq(n

)

istransmitted from antenna j over an Nr

×

Nt MIMO channel.The observation

modelatreceivercanbeexpressedas

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

y1

(

n

)

..

.

yr

(

n

)

..

.

yNr

(

n

)

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

h1,1

(

n

)

h1,2

(

n

)

· · · ·

h1,Nt

(

n

)

h2,1

(

n

)

h2,2

(

n

)

· · · ·

h2,Nt

(

n

)

..

.

..

.

. .

.

..

.

..

.

..

.

. .

.

..

.

hNr,1

(

n

)

hNr,2

(

n

)

· · · ·

hNr,Nt

(

n

)

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

×

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

0

..

.

xq

(

n

)

..

.

0

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

+

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

w1

(

n

)

..

.

wr

(

n

)

..

.

wNr

(

n

)

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

(4)

where hr,j(n

)

is the channel coefficient between jth transmitter antennaandrth receiverantenna, wr(n

)

iscomplex-valued, zero-meanwhiteGaussiannoise(AWGN)withvariance

σ

2

w.

In the SM-MIMO system, the time-varying, frequency non-selectivefading channel coefficients, hr,j(n

)

, introduce a random amplitude and phase shiftto the transmitted signal. We assume theyaremodeledaswide-sensestationary(WSS)process narrow-band complex Gaussian random variables with means

μ

hr, j and variances

σ

2

hr, j (Rician channel model), having Jakes’ model with theautocorrelationfunctionofthechannel[25]givenby

Rh

(

m

,

n

)



E



hr,j

(

m

)

h∗r,j

(

n

)



=

σ

2 hr,jJ0



2

π

f_d

(

m

−

n

)

Ts



,

(5)

whereTsstandsforthesymbolperiod, f_distheDopplerfrequency

inHertzand J0

(

·)

is thezeroth orderBesselfunction ofthefirst kind.ThereisadominantcomponentintheRicianfading,distinct fromtheRayleighfadingchannels. TheRicianfactor, R,isdefined as the ratio of the power in the Line-of-sight (LOS) component tothepowerofthenon-line-of-sight(non-LOS)multipath compo-nents.Rayleighfading isa specialcaseoftheRicianfading when

R

=

0.Also R

=∝

describesa channelhaving onlyaLOS compo-nent.

Observation model(4)canbewritteninmatrixformasfollows:

y

(

n

)

=

H

(

n

)

xj

(

n

)

+

w

(

n

),

n

=

1

,

2

,

· · · ,

N

.

(6) 3. Optimaldetection

Antenna index detection is a crucial step of the SM scheme sinceonlyonetransmitantennaisactiveamongthesetoftransmit antennasandboth thedata symbol,transmitted bythisantenna, andtheantennaindexshouldbe decidedatthereceiver.Optimal detectorbased onthemaximumlikelihood (ML)principlecan be statedasfollows,[11]:



jML

(

n

),



qML

(

n

)



=

arg max j,q pY



y

(

n

)

|

xj

(

n

),

H

(

n

)



(7) wherexj(n

)

variesfordifferentq and j asindicated in(3).From

(6), theprobability densityfunction(pdf) ofy

(

n

)

, conditionedon xj(n

)

andH

(

n

)

,canbewrittenas:

pY



y

(

n

)

|

xj

(

n

),

H

(

n

)



=

π

−Nr_exp

−

_y

₍

_n

₎

−

_h j

(

n

)

xq

(

n

)



2_F



(8) wherehj(n

)

= [

h1,j(n

),

h2,j(n

),

· · · ,

hNt,j(n

)

]

T is jth columnvector ofthe matrixH

(

n

)

.Using (8),optimal detectorin(7) can be ex-pressedas



jML

(

n

),



qML

(

n

)



=

arg max j,q



gjq

(

n

)



2_F

−

2



e



y†

(

n

)

gjq

(

n

)



(9) wheregjq(n

)

is: gjq

(

n

)

=

hj

(

n

)

xq

(

n

),

1

≤

j

≤

Nt

,

1

≤

q

≤

M

.

(10)

Ifthe receiver detects both



jML(n

)

and



qML(n

)

correctly, they canbe easilyde-mapped andcombinedto getbackto the trans-mittedbits. However, it is clearthat the receiver needs toknow thefullCSI,H where

H

=



H

(

1

),

H

(

2

),

· · · ,

H

(

n

),

· · ·

H

(

N

)



.

(11)

4. Channelestimation

In theSM-MIMO system, the CSI is neededatthe receiver in orderto detectthemodulatedsignal transmittedfortheselected transmitantennaaswell asthe indexoftheantennaselected.In thisworkweproposeanewiterativechannelestimationtechnique which yields a superior error performance for the SM-based re-ceivers. TheRLS algorithm isemployed only fortheinitialization oftheproposediterativealgorithm.

4.1. Initialization

WenowsummarizetheRLSalgorithmwhichwillbeemployed toinitializeouriterativechannelestimationalgorithmbymeansof thepilotsymbolsdenotedbyx(p)

₍

_n

₎

_{.TheRLSalgorithmworksfor} j

=

1

,

· · · ,

Nt,r

=

1

,

· · · ,

Nr.Thepilotsymbolsarefirsttransmitted

sequentially fromtransmit antennas and the channel coefficients betweenthe selected transmit antennaand the receiveantennas are individuallyestimatedbythe RLSalgorithmwithin eachpilot symbol duration as shown inFig. 1. Let yr(p)

(

n

)

,r

=

1

,

2

,

· · · ,

Nr,

be thereceived signal corresponding tothepilot symbols. Atthe receiver, the received signal samples are processed sequentially andthechannelestimatesareupdatedasthenewsamplesarrive. Thissectionexplainsbrieflyhowtheleastsquaresestimatesofthe channel arecomputedrecursively. Giventhesetofpilot symbols, x(p)

_{= [}

_x(p)

₍

₁

_),

_x(p)

₍

₂

_),

_{· · · ,}

_x(p)

₍

_Np)

_]

T _{where N}

p is thetotal

num-ber pilotsymbolsforeach active antenna, andthecorresponding desired responses y(rp)

(

n

)

= [

y( p) r

(

1

),

y( p) r

(

2

),

· · · ,

y( p) r

(

Np)

]

T, the

outputsofthesetoflinearfiltersaredeterminedaccordingto

Ω

r,j

(

n

)

=

hRLS_r,j

(

n

)

x(p)

(

n

),

n

=

1

,

· · · ,

Np (12)

The channelcoefficientsh

ˆ

RLS

r,j

(

n

)

between jth transmitandrth

receive antennasare estimatedrecursivelyinthe time-domain to minimizethesumofthesquarederrorsas

ε

(

n

)

=

n



i=1

λ

(n−i)



y(rp)

(

i

)

−

hRLSr,j

(

i

)

x(p)

(

i

)



2

,

n

=

1

,

· · · ,

Np (13)

wheretheforgettingorweighting factor,

λ

,0

< λ

≤

1 reducesthe influence of the old data. The basic steps of the RLS parameter estimationalgorithmcanbefoundeasilyintheopenliteratureand thuswillnotbegivenhere,[15,26].

4.2. Theiterativereceiver

Receivers with iterative decision-directed channel estimation (DD-CE) are very attractive since they yield superior error per-formance [27],especially when operating inthe presence of fast time-varying channelsandthey need lesspilot symbols as com-pared to the non-iterative channel estimators [28]. Moreover, it wasalsoshownthatthecomputationalcomplexity ofthereceiver

[29]canbereducedsubstantially.

AsmentionedinSection4.1,apilot-aidedRLSalgorithmis em-ployed to find the initial channel estimates. Unlike multi-stream MIMOschemes, theSM requiresa longer time totransmit pilots asshowninFig. 1andthechannelsfordifferenttransmit anten-nasareindividuallyestimatedbythecorrespondingpilotsymbols. The main problemis that we have onlyone active antenna dur-ingtransmissionsothattheotherchannelscouldnotbeknownat thattime.Therefore,thedatasymbolstransmittedfromrandomly selectedantennasaredetectedfirstbymeansoftheinitialchannel estimatesusingpilotsymbols.AsshowninFig. 2,thechannel co-efficients,associatedwiththedetectedsymbols,arethenupdated. Theunknownchannelcoefficientsthatcannotbeestimatedbythe

Fig. 1. Channel estimation based on pilot and data durations.

Fig. 2. Proposed iterative receiver structure.

detectedsymbolsaredeterminedbyacurvefittingtechnique. Con-sequently,thechannelestimationalgorithmhasalsothecapability oftrackingthetime-varyingchannelatthereceiver.

If jth transmitter antenna is assumed to be active (

τ

=

j)

and transmits the data symbol xq(tk

)

at discrete times tk, k

=

1

,

2

,

· · · ,

K withinanobservationframe oflength N (N



K ), the receivedsignalattherth receiverantennacanbewrittenas:

yr

(

tk

)

=

hr,j

(

tk

)

xq

(

tk

)

+

wr

(

tk

)

(14)

In this case, the DD channel estimates at discrete times

t1

,

t2

,

· · · ,

tK canbedeterminedas:

ˆ

h_rDD_,τ=j

(

tk

)

=

yr

(

tk

)/



xq

(

tk

),

tk

∈ {

1

,

2

,

· · ·

N

}

(15)

where



xq(tk) represents the detected symbol at the tkth symbol

duration. Note that there are approximately K

=

N

/

Nt symbols

detected for eachchannel if weassume that the transmitted an-tennasareselectedwithequalprobabilities.Thedetectedsymbols are then updated iteratively,employing the last updated channel estimatesforthenextiteration,asshowninFig. 2.

4.3. Curvefitting

By means of a polynomial curve fitting at discrete times

t1

,

t2

,

· · · ,

tK,theestimatedchannelcoefficients,h

ˆ

rDD,τ

(

tk),

τ

=

j

be-tween

τ

th transmitand jth receiveantennascanbemodeledasa (L

−

1)th degreepolynomial

ˆ

where ur,τ

(

tk

)

is a random modeling errorassumed to be

zero-meanGaussian withvariance

σ

2

u,conditioned onthe scalar

vari-abletk.Thenwehavethefollowingusuallinearmodel:



hDD_r,τ

=

T

θ

r,τ

+

ur,τ (17) where,



hDD_r,τ

=



hDD_r,τ

(

t1

),

hDDr,τ

(

t2

),

· · · ,

hDDr,τ

(

tK

)



T ur,τ

=



ur,τ

(

t1

),

ur,τ

(

t2

),

· · · ,

ur,τ

(

tK

)



T

θ

r,τ

=



θ

r,τ

(

1

), θ

r,τ

(

2

),

· · · , θ

r,τ

(

L

)



T T

=

⎡

⎢

⎢

⎢

⎢

⎣

1 t1

· · ·

tL₁−1 1 t2

· · ·

tL₂−1

..

.

..

.

. .

.

..

.

1 tK

· · ·

tLK−1

⎤

⎥

⎥

⎥

⎥

⎦

where K isthetotalnumberofsamplesforcurve fittingandthe observationmatrixT hastheform ofaVandermondematrix.The minimumvarianceunbias(MVU)estimatorfor

θ

r,τ is[30]

θ

r,τ

=



TTT



−1TT



hDD_r,τ

.

(18)

Thentheresultingcurvefittingforalldiscretetimesn

=

1

,

2

,

· · · ,

N

is



hCF_r,τ

(

n

)

=

L



i=1



θ

r,τ

(

i

)

tin−1

.

(19)

Thetime-varyingchannelcanbeestimatedusing(19)overthe durationofoneframe.Afterthechannelestimationstep,thedata symbolsaredetectedasshowninFig. 2.Notethatthewrong/poor symbolandantennaindexdetections madebythe SM-MIMO re-ceiver may causesome error accumulation that could affect the mean-square error performance of the channel estimation algo-rithm.However,ourextensivecomputersimulationsaswellasthe work in [12] show that wrongly detected antenna indices rarely occurandthepolynomialfitting,employedinthechannel estima-tionalgorithm, reducesthis effectby means ofthe other correct detectedantennaindexes.Periodicpilotblocksarealsoinsertedin thedecision-directedestimationmodetoreducefurthertheerror propagationeffect.

5. Performancelimitsofthechannelestimationalgorithm Toserve asa benchmark, we now derive an overall Bayesian MSEboundforthechannelestimatorproposedinthispaper[31]. The overall MSE forthe channel impulse response vector can be expressedasthesumofthetruncationMSEandestimationMSE:

MSEall

=

MSEtrun

+

MSEest

.

(20) We assume that all the channel coefficients of the spatial modulatedMIMOsystem,betweentransmitandreceiveantennas,

{

hr,τ

(

n

)

}

, r

=

1

,

2

,

· · · ,

Nr,

τ

=

1

,

2

,

· · · ,

Nt, are fast-time varying,

frequencynon-selectiveRayleigh fading and independentof each other.Consequently,the performance boundsobtainedbelow are independentofthelabelofthetransmitantenna,

τ

,selectedfrom theset ofthetransmit antennasrandomlywithequalprobability. Therefore,we will drop thesubscript

τ

fornotational simplicity. ThetruncationMSE,MSEtrun,canbeevaluatedasfollow:

MSEtrun

=

1 NrN Nr



r=1 E_h r,hoptr



h _r

−

hropt



†



h _r

−

hoptr



(21)

where,h _r

= [

hr(1

),

hr(2

),

· · · ,

hr(N

)

]

T _andhopt

r istheoptimal

poly-nomialin(19)whichistheleast-squares fittedtoh _r andits opti-malcoefficientvector

θ



r

= [

θr(

1

),



θr(

2

),

· · · ,



θr(

L

)

]

T isgivenby[32] hropt

= ˜

T

θ



r and

θ



r

=

˜

TTT

˜



−1

_˜

TTh_r

,

(22)

whereT is

˜

N

×

L matrix.From(22)itfollowsthat

hoptr

= Υ

h r (23)

where

Υ

 ˜

T

( ˜

TTT

˜

)

−1T

˜

T.Thanitcanbeeasilyshowfrom(21)that MSEtrun

=

1 Ntr



IN

− Υ

†



Chr

(

IN

− Υ )



,

(24)

where Chr represents the covariance matrix of the channels be-tweenanytransmitandreceiveantennaswhose(m,n)th element

isgivenby(5).

We now evaluate the Bayesian mean-square estimation error, MSEest in(20)asfollows.Receivedsignal givenin (6)model,can beexpressedatdiscretetimest1

,

t2

,

· · · ,

tK,betweenanytransmit

antennaandtherth receiveantennaas

yr

=

Xqhr

+

wr

,

r

=

1

,

2

,

· · · ,

Nr (25) where yr

=



yr

(

t1

),

yr

(

t2

),

· · · ,

yr

(

tK

)



T

,

hr

=



hr

(

t1

),

hr

(

t2

),

· · · ,

hr

(

tK

)



T

,

Xq

=

diag



xq

(

t1

),

xq

(

t2

),

· · · ,

xq

(

tK

)



,

wr

=



wr

(

t1

),

wr

(

t2

),

· · · ,

wr

(

tK

)



T

.

Notethat hr(tk) isindependent ofhr

(

tk) forr

=

r . Using the

relationhr

=

T

θ

r

+

urin(19),wehave

yr

= Ψ

q

θ

r

+

vr

,

r

=

1

,

2

,

· · · ,

Nr (26)

where

θ

r

= [θ

1

,

θ

2

,

· · · ,

θL

]

T,

Ψ

q



XqT and vr

=

wr

+

ur.Sincewe

assume that the polynomial fitting error vector ur is zero-mean

Gaussian withcovariancematrix

σ

2

uIK, itisclearthat theoverall

additivenoisevectorvrisalsoGaussianhavingthecovariance

ma-trix

σ

2_I

K with

σ

2



σ

w2

+

σ

u2.ThentheBayesianMSEerrorforthe

estimatorof

θ

risdefinedas MSEθr

=

1 LNr Nr



r=1 E_θ r,θr



(θ

r

− 

θ

r

)

†

(θ

r

− 

θ

r

)



,

(27)

where

θ



r

= [

θr(

1

),

θr(



2

),

· · · ,



θr(

L

)

]

T isgivenby(18).Thelefthand

sideof(27)canbelowerboundedas

E_θ r,θr



(θ

r

− 

θ

r

)

†

(θ

r

− 

θ

r

)



=

tr



E_θ r,θr



(θ

r

− 

θ

r

)(θ

r

− 

θ

r

)

†



≥ (

Jθr

+

Cθr

)

− 1 ₍₂₈₎

where Jθr is the Fisher information matrix (FIM) and Cθr is the covariancematrixoftheprior probabilitydistributionof

θ

r andit

canbedeterminedfrom(22)as

Cθr

=



T TT



−1TTChrT



TTT



−1 (29)

whereChr istheautocorrelationmatrixofhr withelementsgiven by(5).Ontheotherhand,forthelinearobservationmodelin(26), theFIMof

θ

r isgivenby[30]

Jθr

=

EXq

{(Ψ

†

q

Ψ

q

)

−1

}

σ

2 (30)

Theexpectationin(30)canbetakeneasilyinthecaseof con-stant envelope data symbols. Otherwise, it can be evaluated by usingatightlowerboundE

{

X

} ≥

1

/

E

{

X

}

resulting

Table 1

Complexityanalysis.

Equation number RLS-CE Iterative-CE

× + × + 12Np 4Np 12Np 4Np (15) – – K – (18) – – L2_{K+L K+O} (L2 )+L2 _L2_{K+L K−2L} (19) – – (L−1)N (L−1)N Total 12Np 4Np K(L2+L+1)+N(L−1)+12Np+O(L2)+L2 K(L2+L)+N(L−1)+4Np−2L Jθr

≥

Es

σ

2



T†T



−1 (31)

where Es



E

{|

xq(tk)

|

2

}

and Jθr achieves the lower bound when constantenvelopesymbolsareemployed.

Consequently,themean-squareestimationerrorMSEest forthe channel vector fromany transmitantenna to therth receive an-tennaforr

=

1

,

2

,

· · · ,

Nrisdefinedas

MSEest

=

1 K Nr Nr



r=1 E_h r, hr



(

hr

− 

hr

)

†

(

hr

− 

hr

)



(32) whereignoringthetruncationerror,h



r

=

T

θ



r andhr

=

T

θ

r.Using (28)in(32)andaftersomealgebrawehave

MSEest

≥

1 K tr



T



Jθr

+

C− 1 θr



₋1 TT



.

(33)

Finally,theoverallBayesianMSElower boundoftheproposed channelestimatorcanbeexpressedfrom(20),(24),(31)and(32)

asfollows: MSEall

≥

1 Ntr



IN

− Υ

†



Chr

(

IN

− Υ )



+

Es K

σ

2tr



T



TTT



−1

+

Cθr



₋1 TT



.

(34) 6. Computationalcomplexity

Computationalcomplexity oftheiterativereceiverproposed in thisworkisdeterminedbytheparameters Np, K ,N andL.

Com-putational load to implement the RLS and the iterative channel estimationalgorithmsaresummarizedinTable 1.

The RLS-based initial channel estimates (RLS-CE) are deter-mined by the associated pilot symbols transmitted through the active antennas. Therefore, the whole algorithm requires 12Np

complex multiplications (CMs) and4Np complex additions (CAs)

perchannel[15].

On theother hand,the computational load to implementthe iterative-CE scheme is based on Eqs. (15), (18) and (19)as well ason theRLS-CE algorithmforinitialization. We need K CMsto calculatethechannelbythedetectedsymbolsin(15).Accordingto

(18),computationofTT_{T requires}1 _L2_{K realmultiplications(RMs)} andL2

₍

_K

₋

₁

₎

_{real additions(RAs) where T is K}

_×

_{L matrix. The} computationalcomplexitytoevaluate

(

TTT

)

−1 is2 O

(

L2

)

[33].The productofTT with



hDDr,τ requiresL K CMsandL

(

K

−

1

)

CAs.Then

toevaluatetheterm

(

TT_T

₎

−1_TT



_hDD

r,τ ,ontherighthandsideof(18)

requiresL2_CMsandL2

₋

_{L CAs.Consequently,tocomputeEq.(18)} overallL2K

+

L K

+

O

(

L2

)

+

L2 multiplicationsandL2K

+

L K

−

2L additions arerequired.According to (19), weneed

(

L

−

1

)

N CMs

and

(

L

−

1

)

N CAs.

1 _{Thetotalrealmultiplications(RMs) andrealadditions(RAs)toevaluatethe} multiplicationofL×K matrixwithK×N matrixareL K N andL(K−1)N

respec-tively.

2 _{ThecomputationalcomplexityofanL×L Vandermondematrix inversionis} O(L2_).

Finally,totalcomplexityofthecurvefittingisequalto K

(

L2

+

L

+

1

)

+

N

(

L

−

1

)

+

12Np

+

O

(

L2

)

+

L2 multiplicationsandK

(

L2

+

L

)

+

N

(

L

−

1

)

+

4Np

−

2L additions forper channel.Notethat

it-erationswill requirerecalculationsof(15),(18),(19).In[19],the channelestimationwithapolynomialtime-varyingchannelmodel was investigated andapolynomial order(L) selectionprobability was given. Similarly, in this work, it is concluded that the de-greeofpolynomial takessmallvalueswithintheacceptablerange ofDoppler frequenciesinpracticetotrackthechannel variations. Therefore,thecomputationalcomplexityoftheiterative-CEisquite feasibleforrealapplications.

7. Simulationresults

Inthissection,performanceofan Nt

×

NrSM-MIMOsystemis

investigatedbasedontheproposedchannelestimationforvarious velocitiesofmobileusersinthepresenceofRicianchannelshaving differentRicianfactors.

Two benchmarksare considered in our computer simulations forcomparison:i)theconventionalRLSchannelestimation,which isdenoted as“RLS-CE” inthesequel;ii)perfectchannelstate in-formation(P-CSI).Mainparameterschosenforthesimulationsare asfollows:

•

RLSparametersareselectedas

μ =

0

.

0005 and

λ

=

1.

•

Four receiver antennas are considered in all cases. SM map-pings are shown for 2

×

4 and 4

×

4 in Fig. 3 and Fig. 4

respectively.

•

Thesymboldurationandthecarrierfrequencyareselectedas 1

μ

snand1.8GHzrespectively.

•

We have the same signal-to-noise ratio (SNR) value at each receiverantenna.

•

TheSNR isdefinedas Es

σ2 where Es isenergypersymboland

σ

2 _{isnoisepower.}

•

InallsimulationsexceptFigs. 7,9 and10,oneiterationis em-ployedfortheproposedreceiver.

•

The channel between transmitterandreceiver ismodeled as rapidlytime-varyingRicianfadingchannel whereDoppler ef-fectistakenintoconsideration.

7.1. Applicationscenario-1:2

×

4 SM-MIMOsystem

Biterrorrate(BER)performance oftheSM-MIMOsystemwith twotransmitantennasandfourreceiverantennasareinvestigated using4-QAMsignaling.Totalnumberofpilotanddatasymbolsare selected as Np

=

12 and N

=

216 respectively.In Fig. 5, the BER

performance of the proposed iterative-CE is compared with the RLS-CE schemeassumingP-CSIand V

=

150 km

/

h ( fd

=

250 Hz)

overRicianfadingchannelhavingR

=

7.Theinitialchannel coeffi-cientsaredeterminedfirstbytheRLS-CEtechniqueusingthepilot symbols[15].The iterative-CEis thenimplemented toobtain the enhancedchannelestimatesasdescribedinFig. 2.

ComputersimulationresultsinFig. 5showthattheBER perfor-manceofthe iterative-CEis betterthan thatofthe RLS-CEwhile

Fig. 3. Spatialmodulationmapping:3-bitstransmissionusing4-QAM,twotransmit antennasandfourreceiverantennas.

Fig. 4. Spatialmodulationmapping:3-bitstransmissionusingBPSK,fourtransmit antennasandfourreceiverantennas.

itachievestheBER performanceofthatoftheP-CSI.Inparticular, itisobservedthat about2 dB gainis achievedatBER

=

10−6_,as comparedwiththeRLS-basedreceiver.

The effectof the Ricianchannel factor, R, on the BER perfor-mance is investigated in Fig. 6. It is shown that as R increases

from0to10,theBERincreasesanddifferenceinBERperformance betweentheRLS-CEandtheiterative-CEincreasesforsmallvalues of R.Inother words,theRLS-CE withR

=

0 andthe iterative-CE withR

=

2 havethesameperformancesatSNR

=

16 dB.Therefore, theiterative-CE hasabout2 dB Rician factorgain atBER

=

10−5_, ascomparedwiththeRLS-CE.

7.2.Applicationscenario-2:4

×

4 SM-MIMOsystem

Inordertoshowthe potentialadvantages ofiterative-CE, V

=

150 km

/

h withRayleighfadingchannelisconsideredinFig. 7.Itis shownthattheRLS-CEexhibitsanerrorfloorathighvelocityand highSNRswhile iterative-CE hassimilar BER performance to the P-CSIcase. InFig. 7,itshownthat additionaliterations mayhelp toimprovetheperformance of theiterative-CE,andthat increas-ingthenumberofiterationsenablethealgorithmtoapproachthe performanceoftheP-CSIcase.AsindicatedinFig. 7thatfour iter-ations wouldbesufficientfortheproposed iterative-CEschemeto converge.

Themainobjectiveofthispaperistoproposearobustand ef-ficientchannelestimation tosupportpermanentaccessibilityand highdata rates forusers employing the SM-MIMO systems in a highly mobileenvironment. Therefore,finally,the effectof veloc-ityontheBERperformanceisalsoinvestigatedandthecomputer simulationresultsarepresentedinFig. 8.Ascanbeseen,mobility

Fig. 5. BERcomparisonoftheRLSestimatorandproposediterativebasedchannel estimatorforV=150 km/s,R=7,Nt=2.

Fig. 6. Behaviorofthe BERwith respecttotheRicianfactorsfor V=150 km/s,

SNR=16 dB,Nt=2.

Fig. 7. BERcomparison of4×4 SM-MIMOsystemforRayleighfadingchannelwith

V=150 km/h,R=7,Nt=4.

has a substantial impacton theperformance ofthe RLS-CE tech-niquewhiletheproposediterative-CEismorerobust.

Inmobilewirelesschannelsthebandwidthefficiencyisat abso-lute premium,thereforehighermodulationssuch as16-QAM and 64-QAMare alsoconsidered for4

×

4 SM-MIMOsysteminFig. 9. The severe amplitude andphase fluctuations caused by wireless

Fig. 8. BERcomparisonofchannelestimatorsfordifferentmobilespeedswith4×4 SM-MIMOsystem,R=7,SNR=20 dB.

Fig. 9. BERcomparisonofchannelestimatorsfor highermodulationswith 4×4 SM-MIMOsystem,V=150 km/h,R=0.

channels significantly degrade the BER performance of M-QAM.

In [34], it was shown that the QAM is very sensitive to chan-nel estimationerrors andthe performance degradation of higher order QAM is more serious than that of lower order QAM for

V

=

150 km

/

h andR

=

0.Thisfigureindicatesthat anerrorfloor occursforhigherordermodulations becausetheiterativechannel estimationdependsondetectedsymbols.However,itisconcluded that proposed algorithm isrobust up to BER

=

10−5 for16-QAM modulation.Moreover,itisshownthattheproposedchannel esti-mation significantly outperforms the RLS channel estimation for higher modulations. We conclude from Fig. 10 that much more iterations areneededfor16-QAMand64-QAMto achieve perfor-manceclosetotheperfectCSIcase.

7.3. Meansquareerror(MSE)performance

The SM-MIMO system with four transmit antennas with a

BPSK,16-QAM and 64-QAM modulations are considered for V

=

150 km

/

h and R

=

0.Totalnumberofpilotanddatasymbolsare selected as Np

=

12 and N

=

216 respectively. The proposed

it-erative channel estimator is compared with previously reported RLSchannel estimator,interms ofaverageMSE forawide range of signal to noise ratio (Es/N0) levels. A MSE lower bound is of particular interest to serve as a bench mark when we compare thechannelestimationalgorithms.TheoverallBayesianMSElower boundforthechannelestimator,obtainedanalyticallyinSection5,

Fig. 10. Iterations 4×4 SM-MIMO system, V=150 km/h, R=0.

Fig. 11. MSEcomparisonofchannelestimatorsfortime-varyingchannelwith4×4 SM-MIMOsystems,V=150 km/h,R=0.

isevaluatedasafunctionofSNRinFig. 11.Itcanbeobservedthat theMSEperformanceoftheiterative-CEchannelestimatorisfairly closetotheBMSElowerbounddependingonthesignalingformat employed.However, theMSEperformance oftheRLS-based chan-nel estimationalgorithm hassubstantially lower than thatof the iterative-CEalgorithmandexperiencesa largeerrorfloorathigher SNRsmainlyduetotheeffectoftherapidlyvaryingchannel.

8. Conclusions

Inthispaper,itwasshownthatproposediterativeCEalgorithm technique employed in SM-MIMO systems has superior BER and MSEperformancesinthepresenceofrapidlyvaryingRicianfading channel overtheconventionalRLSbasedmethods.It was demon-strated by computersimulations thatthe RLS-basedchannel esti-matorhasyieldedirreducibleerrorfloorsathighermobilitiesand the mobilityeffectwas moredevastatingeffectforchannelswith lower Ricianfactor. Basedon theextensive computer simulations aswell ason theanalytical MSE analysis, weconcluded that the proposed decision-feedbackSM-MIMOreceiverstructureinwhich a curve fitting technique isemployed to trackthe channel varia-tions inthecaseofhighmobilityprovidedexcellentperformance withmanageablecomputationalcomplexityfordifferentSM-MIMO systems such as 2

×

4 4-QAM, 4

×

4 BPSK. A comparison with other previously known RLS-CE algorithm was also made and it was demonstratedthattheiterative-CEprovidesperformance that

isclosetothatoftheperfectCSIforrealisticfadingconditionsand thattheBERperformanceismorerobustagainstchannelvariations thanthatoftheRLS-CEtechnique.

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YusufAcar receivedthe B.S.E.degree (with

hon-ors) and M.S.E. degree in Electrical and Electronics EngineeringfromIstanbulUniversity,Istanbul,Turkey, in2008and2011,respectively,andiscurrently work-ingtoward the Ph.D. degree fromthe same univer-sity.Since2009,hehasbeenaResearchAssistantat the Department of Electronics Engineering, Istanbul KültürUniversity.Hewasavisitingscholaratthe Pur-dueUniversity, Fort Wayne, USA, between the June 2012and September2012. Hisgeneralresearchinterestscover commu-nication theory,estimation theory,statistical signal processing, and in-formation theory.His currentresearchactivitiesare focusedonwireless communicationconceptswithspecificattentiontoequalizationand chan-nelestimationforspread-spectrumandmulticarriersystems.

HakanDo˘gan receivedtheB.S.EinElectronics

En-gineeringin2001,andtheM.S.E.andPh.D.degreesin ElectricalElectronics Engineering in2003 and 2007, respectively, all from the Istanbul University. From 2001 to 2007 he was employed as a research and teaching assistant atthe faculty ofthe Department ofElectricalandElectronicsEngineering,Istanbul Uni-versity. In 2007, he joined the same faculty as an Assistant Professor.He has been an adjunct profes-sorintheTurkishAirForceacademysince2008.Hisinterestslieinthe areasofestimationtheory,statisticalsignalprocessing,andtheir applica-tionsinwirelesscommunicationsystems.Hiscurrentresearchareasare focused on wireless communication conceptswith specific attention to equalizationandchannelestimationforspread-spectrumandmulticarrier (orthogonal frequency-division multiplexing)systems. Hewas avisiting scholaratthePurdueUniversity,FortWayne,USAfromMay15,2010to 16August,2010.HecurrentlyservesasaviceheadoftheDepartmentof ElectricalandElectronicsEngineering,IstanbulUniversity.

ErdalPanayırcı receivedtheDiplomaEngineering

degreeinElectricalEngineeringfromIstanbul Techni-calUniversity,Istanbul,TurkeyandthePh.D.degreein ElectricalEngineeringandSystemSciencefrom Michi-gan State University, USA. Until 1998 he has been with the Faculty of Electrical and Electronics Engi-neeringattheIstanbulTechnicalUniversity,wherehe wasaProfessorandHeadoftheTelecommunications Chair.Currently,heisProfessorofElectrical Engineer-ing and Head of the Electronics Engineering Department at KadirHas University, Istanbul, Turkey.Dr. Panayırcı’s recent research interests in-cludecommunicationtheory,synchronization,advancedsignalprocessing techniquesandtheirapplicationstowirelesscommunications,coded mod-ulationandinterferencecancellation witharrayprocessing.Hepublished

extensivelyinleadingscientificjournalsandinternationalconferences.He hasco-authoredthebook“PrinciplesofIntegratedMaritimeSurveillance Systems”(Boston,KluwerAcademicPublishers,2000).

Dr.Panayırcıspenttheacademicyear2008–2009,intheDepartment ofElectrical Engineering,Princeton University, NewJersey,USA. He has beentheprincipalcoordinatorofthe 6thand7thFrameEuropeanproject calledNEWCOM(NetworkofExcellentonWirelessCommunications) and

WIMAGIC Strep project representingKadir Has University. Dr. Panayırcı was anEditor forIEEETransactions onCommunicationsin theareasof SynchronizationandEqualizationsin1995–1999.HeservedasaMember ofIEEEFellowCommitteein2005–2008.Hewasthe TechnicalProgram ChairofICC-2006andPIMRC-2010bothheldinIstanbul,Turkey.Presently heisheadoftheTurkishScientificCommissiononSignalsandSystemsof URSI(InternationalUnionofRadioScience).HeisanIEEELifeFellow.