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So,inthisway,thedevelopmentofhighadvancedtechniquesofdigitaldatatransmissionandtheactualFPGAstageofintegrationmakepossiblethebuildingallthecircuitsthatcompoundsadigitalcommunicationsystemsinanuniquechip.
Tomakeuseofthismoderntechnologiesofdatatransmission,itisnecessarythedevelopmentefficienttechniquesofdigitalmodulation.TheOFDMisthebiggestutilizedrecently,principally,becauseofitusetheefficientFFTtodothemodulation.
ThisworkdescribeshowimplementanOFDMmodemonFPGAusingVHDLfora31subcarrier(channels)OFDMsystemusing64pointsradix-4FFTtimedecimation,aCORDICimplementationtoperformthebutterflycalculus,andeachchannelmodulationwillusea4-QAMconstellation.Thesystemisdividedinatransmissionsectionandareceptionsection.
Blockdiagram
Detaileddescription
Introduction
Basically,OFDMtechniqueconsistsofdividingtheflowofentrancedataoverorthogonalchannels.Inthisway,andbasedontheorthogonalityprinciple,theinterferencebetweeneachtransmissionchannelisminimal.Anotheradvantagethatcomesfromthisapproachisrelatedtotheassumptionsaboutthenoiseineachchannel.Overalargeanduniquepassbandchannelitisdifficult(impossibleinfact,inmanysituations)toassumethatthemodelnoiseisAWGN(AdditiveNoiseGaussianNoise).Thatisimportantbecauseifthemodelisknow,anditiscorrect,onecanselectthebestwayoffrequencyequalization.But,inlargepassbandchannels(tenthsofMhz)thenoisemodelisunknownandunpredictable.Thesolutionofdividingauniquechannelinsubchannelssimplifiestheassumptionsofthenoiseovereachsubchanneland,ofcourse,onecansupposeitapproximatelyAWGN.
Ineachoneofthesesubchannels,adigitalmodulationismade,eachonewithasubcarrierthatpresentsdifferentfrequency,inasuchformthatachanneldoesnotintervenewithanotherone,thuskeepingtheorthogonallity.
Thewayofimplementationandtheorthogonallitypropertyofasystemofthistypecanvarysufficiently:
sincebandpassfilterseachfrequencyuntilsophisticatedtechniquesastheuseoftheFFTwithguardinterval,whichistheusedonefortheOFDM.
Inliterature,thistechniquehasbeencalledforsomedesignationsasorthogonallymultiplexedQAM(O-QAM),parallelquadratureAM,andsoone.However,OFDMforwirelessandDMTforsystemswired(likeDSL)arethemostcommondesignationsthatonecanfindinliterature.Asalltheabovetechniquesbasicallyusethesameprinciple,itiscommontorefertothembymeansofagenericname:
MCM(toMulti-CarrierModulation).Inthatway,MCMisthetechniqueinthegeneraldirection,andOFDMisrelatedtotheimplementationoftechniqueMCMusingtheFFT.
Implementation
Mappingtheconstellation
Theencoderoftheconstellationmapsthembitsofthechannelinapointa+jbintheconstellationofthemodulator.Decodingreceivesthatpointandtheremapasthemtransmittedbits.
Encoderoftheconstellation
Itisimportanttonoticethatinthatmappingitisjustmadeaconversionofbitsforthefasorthatacts,howeveritisnotmadeanymodulation,asinthecaseofQAM,becausethatasshown,itisdonebyIFFT.
Itisnecessarytospecifyhowtheconstellationwillbetobemapped,toimplementthatblock.However,independentlyoftheformatoftheconstellation,theblockencodercanbemadethroughaconsultationataconversiontable,implementedbyLUTthatexistsinLCsofFPGAs.
Forinstance,fora4-QAMconstellationinsuchawaythataandbarebinarynumbersof3bits,andareconvertedtocomplementtwo.
Attemptthattheentranceoftheencoderabinarynumberofmbits,andthattheexitgeneratestwobinarynumbers,oneinphase,the,andotherinquadrature,b,whosesizeisdefinedbyIFFT.
Decoderoftheconstellation
Inthereceiver,thepointoftheconstellationtransmitteditcanhavechangedduetothenoisesofthetransmissionchannel,mistakeinthetimeofsamplingofthereceiverandseveralothercauses.
Thereforeitisnecessarytodefineathresholdsothatitcanbemadethedecisiononwhichpointintheconstellationthereceivedsignisacting.Thatisthefunctionofthedecoder.
Forthesystemexemplifiedabovethebit0isconvertedfor010bandthebit1for110b.Inthatcase,thedecoderisimplementedinasimpleway,stickstothemostsignificant(thatindicatesthesign)bittodothedecoding,andgeneratingabinarynumberofmbitsagain.
Forsystemsinthattheconstellationdiagramislargerthan4-PSKitwillbenecessarytheimplementationofmoreadvancedmethods,likeaneuralnetwork.
TheHemetiansymmetry
AfterhavingmappedtheNchannelsitisappliedtheHermetiansymmetrysothatthemodulationcanbemadebyanIFFT.Itisgenerated2N+2channelsinsymmetry.
Inthereceiver,totheendoftheprocessingofFFT,theyaresenttothedecoderonlyNchannels,beingeliminatedthechannelsgeneratedduetoHermetiansymmetry.
TheHermetiansymmetryisimplementedinagreementwith
fork=1,2...,N-1,whereN=N+1.
Forinstance,totransmitN=3channels(d1,d2,d3)N=4,2N=8andk=1,2,3.Applyingintheaboveequationitisobtainedtheresultshowninthefollowingtable.
KnowingthatX0=XN=4=0+j0,isobtainedinagreementwiththeTable2theresultshowninthefollowingequation:
Inthatway,itcanstayX0andXNalwaysinzero.WhileitismadeXN+1evenX2n-isametotheconjugatedofXn-1.
BeingitconjugatedofZ=a+jbthesameofZ*=a-jb,thentodotheoperationofhavingconjugatedheshouldmovethesignoftheimaginarypart,inotherwords,todoasigninversion.Inhardware,theinversionofsignofabinarynumberincomplementtwo.
The(I)FFT
ThemodulationOFDMcanbemadethroughanIDFT.ThefastimplementationofIDFT,IFFT,canbeusedreducingthetimeofprocessingandtheusedhardware.Thedemodulation,inthesameway,canbemadebyDFT,orbetter,byFFT,thatisitefficientimplementation.
FFTcalculatesDFTwithagreatreductionintheamountofoperations,leavingseveralexistentredundanciesinthedirectcalculationofDFT.Thatefficiencyisgottenatthecostofanadditionalsteptoreverse-orderthedatainordertobedeterminedthefinalresult.Thatadditionalstep,sinceimplementedefficiently,theywon'
tincreaseinasignificantwaythecomplexitycomputationalofthecalculation.Asresult,FFTisanextremelyefficientalgorithmthatprovidesagoodimplementationinhardware.
ForgreatvaluesofN,thecomputationalefficiencyisgottenbeingbrokenDFTsuccessivelyinsmallercalculations.Thatcanbemadesomuchinthedomainofthetime,asinthedomainofthefrequency,asdiscussedahead.
Decimationintimeandinfrequency
Asmentioned,itispossibletodividetheentranceofFFTsuccessivelygeneratingsmallsequenceinthedomainofthetime,forthatthenamedecimationinthetime,TD.ItisalsopossibletobecomeseparatedthesequenceofexitofFFTsuccessively,insteadofdividingtheentrance.Thatimplementationiscalledofdecimationinthefrequency,FD.ThedecimationcanbeusedsomuchforFFTasforIFFT.
Itispossibletorepeattheprocessuntilreachingthepossiblemaximumlevelofdivision.InthatpointthebasicblockdecimationisgeneratedsoitwillbeusedinthewholeFFT(calledbutterfly).AnexampleofthebutterflyoftheFFTradix-2TDisinthefollowingillustration.
AnexampleofaFFTradix-2DT,forN=8areshownbelow.Itisnoticedthatisnecessaryanalterationintheentranceofdata,becausehe/shehastoseparatetheequalpartoftheoddpart.
Inthesamewaythatthedecimationinthetimegeneratesabutterfly,thedecimationinthefrequencygeneratesacorrespondingbutterfly.
Howeverthealterationintheflowofdatawillhavetobedoneintheexit,andnomoreintheentrance.
ImplementationofButterfly
Thesumusedinthebutterflypossessesthesamealgorithm,somuchforFFTasforIFFT.Toavoidoverflowdangerduetosumincomplementtwo,itismadetheextensionofthesigninthebinarynumber,repeatingthemostsignificantbit,likethistotheifitadds2binarynumbersof10bits,wewillhaveoffirsttodotheextensionofthesign,obtaininglikethis,twonumbersof11bits,todothesum,wheretheresultwillalsobeof11bits.Thatprocedurehastobedoneeverytimethatwilladdortosubtractanumber.
Alreadyforthemultiplication,theexithastobeofthesizeofthesumofthenumberofbitsofthetwomultiplicands.Inthatway,todothemultiplicationoftwonumbersof10bits,wewillhavetoanexitof20bits.Thatprocedurehastobedonetoeachmultiplication.
Ifthereisnotimpediment,itcanbemadearotationforright(divisionsfortwo)inthenumbersandtoreduceitssize,sinceintheendoftheprocedureacorrespondingmultiplierisapplied.
Theorderasitwillbemadethebutterflyisdefinedbythedecimationoftheradix.IfitgoesTD,firstitismademultiplicationandlaterthesum.IfitgoesFD,firstit