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Research Article ISSN 22779140

ReceivedonMay2012,PublishedonJuly2012 405

INTERNATIONAL JOURNAL OF ADVANCES IN

COMPUTING AND INFORMATION

TECHNOLOGYAn International online open access peer reviewed journal

BERReductioninOFDMSystemsusingZCTMatrixTransform

RohitGarg1,VijayKumar

2

1Lecturer,SKIET,Kurukshetra,INDIA2Lecturer,MaharishiMarkandeshwarUniversity,Mullana(Ambala),India

[email protected]

doi:10.6088/ijacit.12.14009

ABSTRACT

In recent years, Orthogonal Frequency Division Multiplexing (OFDM) is preferred for transmission over a

dispersivechannel.Itisbecauseofitsseveraladvantagessuchashighspectralefficiency,lowimplementation

complexity,lessvulnerabilitytoechoesandnonlineardistortion.However,ithasfewlimitationssuchaspeak-

to-average-powerratio(PAPR)andbiterrorrate(BER),whichdeterminesthesystemspowerefficiency.HighPAPRisoneofthemajordrawbacksinOFDMsystems,whichcausesasignificantlevelofsignaldistortions,

whenthemodulatedsignalsare amplified throughhigh power amplifiers (HPAs). Inparticularly,high PAPR

increasesthecomplexityofAnaloguetoDigital(A/D)andDigitaltoAnalogue(D/A)convertersandalsoreduces

theefficiencyofHPAs.AhighPAPRandhighnoise(e.g.AWGN)maysignificantlydistortthesignal,resulting

inhighBERondemodulationofthereceivedsignal.Therefore,PAPR&BERreductionisveryessential.Forthispurpose,a Zadoff-Chumatrix Transform(ZCT) basedprecodingandpostcoding techniquesarequantitatively

evaluated in this paper in terms of its parameters q and r.From the analysis, it is inferred that anoptimumselectionofqandrcanleadtoasignificantlevelofreductioninBER.

Keywords:Zadoff-ChumatrixTransform;OFDMconventional;HPA;PeaktoAveragePowerRatioandBER.

1. IntroductionOFDM is a key multicarrier modulation technique [1], which provides high spectral efficiency, low

implementationcomplexity[2],lessvulnerabilitytoechoesandnonlineardistortion[3].Duetotheseadvantages

oftheOFDMsystem,itisvastlyusedinvariouscommunicationsystems.However,ithasapracticallimitationof

highbiterrorrateoccurringduetohighPAPRornoise[4].AlargePAPRincreasesthecomplexityoftheanalog

todigital and digitaltoanalog converter and reduces the efficiency of the radio frequency (RF) power

amplifier [5].There are a numberof techniques dealingwith the problem of BER& PAPR. Some of these

include:constellationshaping,nonlinearcompandingtransforms[6],tonereservation[7]andtoneinjection(TI),

clipping and filtering [8], partial transmit sequence [9] and precoding based techniques. However, these

techniquesdoPAPRreductionattheexpenseofincreaseintransmitsignalpower,biterrorrate[7],datarateloss

and computational complexity. The reduction in BER is very essential. Therefore, two novel reduction

techniques,namely,Zadoff-Chumatrix Transform(ZCT)precodingbasedBER reductiontechniqueandZCT

postcodingbasedBER reduction technique for OFDMsystemswere proposed in this paper. In the proposed

schemes, the reshaping oftheZCT iscarriedoutoneway for precoding and anotherway for postcoding.Forprecoding,thereshapingoftheZCTmatrixisrowwiseandisappliedbeforetheIFFT;however,forpostcoding,

thereshapingoftheZCTmatrixisdonecolumnwiseanditisappliedaftertheIFFT.

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RohitGarg,VijayKumarInternationalJournalofAdvancesinComputingandInformationTechnology

2.SystemModel

2.1OFDMSystemModel

TodesigntheproposedZCTprecodingandpostcodingsystem,considertheblockdiagramofanOFDMsystemshown inFigure 1. In Figure1,anOFDMsignal consistsofN subcarriersthat aremodulatedbyN complex

symbolsselectedfromaparticularQAMconstellation.

Figure1:BlockdiagramofgeneralOFDMsystem.

InFigure1, the basebandmodulated symbolsarepassed through serial toparallelconverter which generates

complexvectorofsizeN.ThiscomplexvectorofsizeNcanbeexpressedas

(1)

XisthenpassedthroughtheIFFTblockofsizeNN IFFTmatrix.TheresultedcomplexbasebandOFDMsignalwithNsubcarrierscanthenbewrittenas

n=0,1,...,N1.

(2)

Afterparallel-to-serialconversion,acyclicprefix(CP)witha lengthofNgsamplesisappendedbeforetheIFFT

outputtoformthetime-domainOFDMsymbol,s=[s0,.. ,sN+Ng1].TheusefulpartofOFDMsymboldoesnot

include theNg prefix samples and has duration ofTu seconds. The samples (s) are then amplified, with the

amplifiercharacteristicsisgivenbyfunctionFandtheoutputofamplifierproducesasetofsamplesdenotedbyy.

Atthereceiverfrontend,thereceivedsignalisappliedtoamatchedfilterandthensampledatarateTs=Tu/N.After dropping the CP samples (Ng), the received sequencez, assuming an additive white Gaussian noise

(AWGN)channel,canbeexpressedas

z=F(Wd)+ (3)

Where,thenoisevectorconsistsofNindependentandnormallydistributedcomplexrandomvariableswithzero

mean.Subsequently, thesequencez isfedtothefastFouriertransform(FFT),whichproducesthefrequency-

domainsequenceras

r=WHz (4)

Where,kthelementofrisgivenby

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k=0,1,2,,N-1

(5)

Finally,theestimatedsymbolsvector canbeobtainedfromr.Itistobenotedthatideally,thedemodulationisperformed based on the assumption of perfect symbol timing, carrier frequency, and phase synchronization.

Howeveritisnotpossibletoachieveinpractice;therefore,theBERisintroducedandwhichcanbedefinedasthe

differenceoftheobtaineddemodulatedsignalrkfromtheinputsignal.

2.2ZADOFF-CHUSEQUENCES

Ingeneral,aZadoffChusequenceisacomplex-valuedsequencewhich,whenappliedtoradiosignals,givesrisetoanelectromagneticsignalofconstantamplitude,wherebycyclicallyshiftedversionsofthesequenceimposed

on a signal result in zero cross-correlation with one another at the receiver. Mathematically, ZadoffChu

sequencescanbedefinedforasequenceoflengthLas:

(6)

Where,k=0,1,2L-1,qisanyintegerandrisanyintegerrelativelyprimetoL.Fromtheaboveequation,q

and rcomesinthenumeratoroftheexponentialpowerfraction.Thismeans,thattheirvariationcanleadtoa

significantchangeintheoutput.Thisisthekeyfocusoftheresearchreportedinthispaper.ThisZCTsequence

canbeappliedintwoforms:pre-codingandpost-coding,asdiscussedbelow.

2.3ZCTPRE-CODINGBASEDOFDMSYSTEM

TheblockdiagramofZCTpre-codingbasedOFDMsystemisshowninFigure2.IntheZCTpre-codingbased

OFDMsystem,thebasebandmodulateddataispassedthroughS/Pconvertorwhichgeneratesacomplexvector

ofsizeNthatcanbewrittenasX=[X0,X1,X2XN-1]T.ZCTpre-codingisthenappliedtothiscomplexvector

whichtransformsthiscomplexvectorintonewvectoroflengthNthatcanbewrittenasY=[Y0,Y1,Y2YN-1]T;

where,RisaZCTbasedrow-wiseprecodingmatrixofsizeL=N*N.Byreordering,

(7)

AndthematrixRwithrowwisereshapingcanbewrittenas

(8)

Where,RisaN*N,ZCTcomplexorthogonalmatrixwithlength L2=N*N.Accordingly,pre-codingXgivesrise

toYasfollows:

Y=RX (9)

Or,

(10)

Where, means themth rowand l

th column ofpre-codermatrix. Therefore, the complexbasebandOFDM

signalwithNsubcarrierswithZCTpre-codingisgivenby

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(11)

Theaboveequationcanbeexpressedas theIFFTofconstellationdataXlpre-multipliedwithZCTmatrix.The

BERofthissignalissameasbefore.

Figure2:BlockdiagramofanOFDMsystemwithZCTpre-codingapproach.

2.4ZCTPOST-CODINGBASEDOFDMSYSTEMTheblockdiagramofZCTpost-codingbasedOFDMsystemisshowninFigure3.IntheZCTpost-codingbased

OFDMsystem,thebasebandmodulateddataispassedthroughS/Pconvertorwhichgeneratesacomplexvector

ofsizeNthatcanbewrittenasX=[X0,X1,X2XN-1]T.Then,IFFTisperformedtothiscomplexvectorwhich

transformsthiscomplexvectorintonewvectoroflength Nthatcanbewrittenasy=IFFT{X}=[y0,y1,y2yN-

1]T.Similarly,re-orderingasinpre-codingtohave:

(12)

And,ZCTmatrixCwithcolumnwisereshapingiswrittenas

(13)

Inotherwords,theN2pointlongZadoff-Chusequencefillsthepre-codingmatrixcolumn-wise.Thematrix Cis

N*N, ZCT complex orthogonal matrix with length L2 =N*N. Accordingly, postcodingy gives rise to w as

follows:

w=C.y (14)

Where,CisaZCTcolumnwisepost-codingmatrixofsizeL2=N*N.Forq=1andr=1,thecomplexbaseband

ZCTpostcodingbasedOFDMsignalwithNsubcarrierscanbewrittenas:

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(15)

Where,wmistheIFFTofconstellationdataXlpremultipliedwithquadraticphaseandIFFTpostcoded,andthenalternatedwith1.TheBERofthisZCTpostcodingbasedOFDMsignalissameasbefore.

Figure3:BlockdiagramofanOFDMsystemwithZCTpost-codingapproach.

3.ImplementationTodotheBERanalysisandtoevaluatetheperformanceofZCTprecodingandpostcodingbasedOFDMsystems,

thestepsundertakenareshowninFigure4andaredetailedbelow.

1. TodotheBERanalysisofZCTprecodingandpostcodingbasedOFDM,firstlybinarydataisgeneratedrandomly.

2. ThegeneratedbinarydataisthenconvertedintosymbolsandisthenmodulatedbyM-QAM(whereM=4,

16,64,256).TheZCTmatrixiscalculatedforthelengthofthedata.

3. TheserialdataisthenconvertedintoparalleldataandIFFTisperformed.TheZCTmatrix(Transform)isappliedbeforeandaftertheIFFT,respectivelyforprecodingandpostcoding.

4. TheparallelsignalisthenconvertedtoserialdataandispassedthroughamultipathchannelwithAWGNnoiseadded.

5. Thereceivedsignalisagainconvertedfromserialtoparalleldata.ZCTinversematrixisappliedbeforeandaftertheFFT.Thefinaldemodulatedsignalisparalleldata,whichisthenconvertedintoserialdata.

The serial data is then demodulated to retrieveback the signal. The BER iscalculated by taking the

differenceofthedemodulateddataandtheinputdata.

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Figure4:Implementationprocedure

4.ResultsandDiscussionsToevaluatetheperformanceofZCTprecodingandpostcodingbasedOFDMsystems,extensivesimulationshave

been performed in Matlab. The simulationsare performed for different values ofq andr. To show theBERanalysisofZCTprecodingandpostcodingbasedOFDMsystems,dataisgeneratedrandomlyandthenmodulated

byM-QAM(whereM=4,16,64,256).TheorderofFFTandIFFTistakenas64.Theresultsobtainedare

showninFigures5to11andTable1,forvariouscombinationsofqand r,andEBN0from1to7.Thekey

conclusionsdrawnfromtheresultsarediscussedbelow:

1. FromFigures5to 11, itcanbeinterpreted that the resultsobtainedforZCTprecodingandpostcodingaredifferent.TheBERismorereducedincaseofZCTpostcodingforthevaluesofEBN0(1to7),andq=1,10andr=11and111andM=4,16,64and256.

2. FromFigures5to11,itcanbeinterpretedthattheBERdecreaseswithincreaseinEBN0,forallthreecasesi.e.nocoding,ZCTprecodingandpostcoding.Itisbecause,withincreaseinEBN0,thenoisewilldecrease

andthustherewillbelessnumberofbitsinerror,whichmeanslowBER.

3. FromFigures5to11,forZCTpostcoding,theBERismorereducedwhenvalueofrisincreasedfrom11to111,foranyvalueofq.ThiscanalsobeseenfromdatainTable1.ItisbecauseinZCTpostcoding,thematrix

isappliedcolumn-wisewhichwillleadtohigherreductioninPAPRandthusmorereductioninBER.

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4. FromFigures5to11,forZCTprecoding,theBERisalsosignificantlyreducedwithincreaseinconstellationsizeforanygivenSNR.

5.ItcanalsobeseenthatZCTpostcodinghasquickandsharprolloffthanZCTpre-coding.Also,thereislessirregularityinZCTpostcodingcomparedtoZCTprecoding.

6. As the constellation size increases, i.e., from M = 4, 16, 64, 256; there is significant reduction in BER;however,notwithoutcoding.Therefore,ZCTprecodingandpostcodinghaspotentialforsignificantreduction

inBER.

7. Itcanbealsoseenthatforsomecases,theBERforZCTpostcodingisnotobserved.ItisbecauseitisveryefficientandhasquicklyreducedtheBERtozero.

From the above discussion, the value of q and r significantly changes the BER for any given EBN0 and

constellationsizeM.Also,forsamevaluesofqandr,thevariationinconstellationsizealsosignificantlychanges

theBER. It isalso found that ZCT postcoding is more effective than ZCT precoding in reducing theBER.

However, ZCT precoding and postcoding and its parameters q and r significantly reduces the PAPR also.

Therefore,theselectionofvalueofq,r,MandZCTprecoding/postcodingdependsuponthelevelofPAPRand

BERreductionrequired.Hereinthispaper,itispreferredtochooseasmallvalueofqandlargevalueofrfor

ZCTpostcodingandbothsmallvaluesofqandrforZCTpre-coding.ItispreferredtochooseZCTpostcoding

becauseofitssharproll-offinreducingBER;however,italsodependsuponthedesiredlevelofreductionin

PAPR.

ConstellationSize(M=4)

Figure5:Thesimulatedbinarydataanditsconversationtosymbolsforconstellationsize(M=4)

Figure6:ThesimulateddatamodulatedusingQAM(M=4)andAWGNnoise(EBN0=1,2and3).

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Figure7:ThesimulateddatamodulatedusingQAM(M=4)andAWGNnoise(EBN0=4,5and6)


Figure8:ThevariationforBERwithEBN0forQAM

(M=4)andwithZCTprecodingandpostcodingandq=1

andr=11.

Figure9:ThevariationforBERwithEBN0for

QAM(M=4)andwithZCTprecodingand

postcodingandq=10andr=111.


Figure10:ThevariationforBERwithEN0forQAM(M=16)andwithZCTprecodingandpostcodingand

q=1andr=11.

Figure11:ThevariationforBERwithEN0forQAM(M=16)andwithZCTprecodingand


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Figure12:ThevariationforBERwithEN0forQAM(M=64)andwithZCTprecodingandpostcodingand

q=1andr=11.

Figure13:ThevariationforBERwithEN0forQAM(M=64)andwithZCTprecodingand



Figure14:ThevariationforBERwithEBN0forQAM

(M=256)andwithZCTprecodingandpostcodingand

q=1andr=11.

Figure15:ThevariationforBERwithEBN0for

QAM(M=256)andwithZCTprecodingand


5.Conclusion

ThispaperpresentedtheresultsobtainedonapplyingaZCTpre-codingandpost-codingforreducingBERinOFDMsystemswithM-QAMmodulation,whereasM=4,16,64,256andtheparametersq=1,10andr=11and

111.ItisobservedfromthegraphsthattheresponseofbothZCTpre-codingandpost-codingisdifferentforrand

q.TheBERisfoundtosignificantlyvarywiththevaluesofM,randqforagivenEBN0.Foranyparticularvalue

ofr,qandM,theBERreductionismoreforZCTpost-codingthanforZCTpre-coding.Fromtheresults,itis

preferredtochooseZCTpostcodingbecauseofitssharproll-offandquickreductioninBERforagivenSNR

andM.

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Table1:VariationofBitErrorRatewithEBN0

Constellation

Size

q r EBN0

1 2 3 4 5 6 7

None

BitsinError 239 135 89 35 23 6 3

BER 0.0478 0.0270 0.0178 0.0070 0.0046 0.0012 0.0006

ZCTPostcoding

BitsinError 66 22 16 0 0 0 0

BER 0.0132 0.0044 0.0032 0 0 0 0

ZCTPrecoding

BitsinError 177 108 46 17 8 0 0

4

1

11

BER 0.0354 0.0216 0.0092 0.0034 0.0016 0 0

None

BitsinError 240 145 102 38 21 5 0

BER 0.0480 0.0290 0.0204 0.0076 0.0042 0.0010 0

ZCTPostcoding

BitsinError 16 0 0 0 0 0 0

BER 0.0032 0 0 0 0 0 0

ZCTPrecoding

BitsinError 157 92 39 4 0 0 0

4

10

111

BER 0.0314 0.0184 0.0078 0.0008 0 0 0

None

BitsinError 1411 1355 1274 1179 1110 1008 945

BER 0.2822 0.2710 0.2548 0.2358 0.2220 0.2016 0.1890

ZCTPostcoding


BER 0 0 0 0 0 0 0

ZCTPrecoding


256

1

11

BER 0.0052 0.0014 0 0 0 0 0

NoneBitsinError 1388 1340 1307 1190 1090 951 943

BER 0.2776 0.2680 0.2614 0.2380 0.2180 0.1902 0.1886

ZCTPostcoding


BER 0 0 0 0 0 0 0

ZCTPrecoding


256

10

111

BER 0.0074 0.0030 0.0014 0 0 0 0

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