OOFDM的原理与仿真

PowerefficientopticalOFDMJ.ArmstrongandA.J.LoweryAnewtechniqueforusingorthogonalfrequencydivisionmultiplexing(OFDM)inopticalsystemsispresented.ClippedOFDMisderivedfromabipolarOFDMwaveformbysettingthenegativevaluestozero.Ithasanopticalefficiency8dBbetterthanDCbiasedOFDM.IfonlytheoddOFDMsubcarriersaremodulated,theclippingnoiseisorthogonaltothewantedsignal.Introduction:Orthogonalfrequencydivisionmultiplexing(OFDM)allowshigh-speeddatatransmissionacrossadispersivechannel,soisusedinmanynewandemerginghigh-speedwiredandwirelesscommunicationsystems.However,OFDMisnotusedincommercialopticalcommunicationsystems.ThisisbecauseOFDMsignalsarebipolar,whileinopticalsystemsthatuseintensitymodulation(IM),onlyunipolarsignalscanbetransmitted.AlargeDCbiasisusuallyaddedtoOFDMsignalsrequiringahighmeanopticalpowerwithalowmodulationdepth.Thisisveryinefficient.WeshowhowopticallypowerefficientOFDMsignalscanbedesigned.Wedescribethenewtechniqueappliedtoabasebandsignal,butitisalsoapplicabletopassbandsignals.OFDMsignals:InOFDM,signalsaretransmittedinparallelonanumberofsubcarriersatdifferentfrequencies.Usuallyquadratureamplitudemodulation(QAM)modulateseachsubcarrier.Thetrans-mitterusesaninversefastFouriertransform(IFFT)togenerateasampledwaveform.LetX(m)bethecomplexnumberrepresentingtheconstellationpointonthemthsubcarrierofagivensymbol.Thenthebasebandtimedomainsamplesforthatsymbolaregivenbyx(k)wherexðkÞ¼1NPN
1m¼0XðmÞexp
j2pkmN
ð1ÞandNisthesizeoftheIFFT.Ingeneral,x(k)andX(m)arecomplex.ForbasebandsystemsthefrequencydomainvectorXisconstrainedtohaveHermitiansymmetry,sothatx(k)isreal.Fig.1ashowssamplesofatypicalOFDMbasebandsymbolandthewaveformx(t)whichcouldbegeneratedfromthem.Fig.1bshowsasymbolwhereonlyoddsubcarriersareused.Inthiscase,forclarity,N¼32,althoughtypicalvaluesrangefrom64to8192.OFDMsignalshaveahighpeak-to-averagepowerratio.ForN64thecentrallimittheoremappliesandthedistributionsofx(k)andx(t)areapproximatelyGaussian.Fig.1OFDMtimedomainsignalaAllsubcarriersmodulatedbOnlyoddsubcarriersmodulated———analoguesignal,x(t)––s–––signalsamples,x(k)OpticalOFDM:InthefewpapersthatdescribetheuseofOFDMinopticalIMsystems[1,2]aunipolarsignalxdc(t)isderivedfromx(t)byaddingaDCbias.InFig.2athebiasistwicethestandarddeviationofx(t).Forafixedbias,therewillbeoccasionalOFDMsymbolswithlargenegativepeakswhichwillbeclipped,addingnoisetothesignal.Becausexdc(t)givestheintensityoftheopticalsignal,theaveragetransmittedopticalpowerisapproximatelyequaltotheDCbias,whichinthisexampleis2,foranRMSelectricalpowerofunity.Thusthissystemisveryinefficientintermsofopticalpower.Fig.2OpticalsignalsaDCbiasedOFDMbClippedOFDM———analoguesignal,xdc(t)––s–––signalsamples,xdc(k)Weproposeusingnobias.Inthenewschemethesignalxc(t)showninFig.2bwouldbetransmitted.Allnegativevaluesareforcedtozero.Wewillshowthatifthesubcarrierfrequenciesusedfordatatransmissionarecorrectlychosen,thedatacanberetrievedfromasignalofthisformwithlittle,orforsomeconfigurations,noin-bandclippingnoise.Analysisofclipping:Clippingisamemorylessnonlinearoperationonx(k).Asx(k)hasaGaussiandistribution,Bussgang’stheoremcanbeapplied[3].ThusxcðkÞ¼KxðkÞþdðkÞð2ÞwhereKisaconstantandd(k)isarandomnoiseprocesswhichisuncorrelatedwithx(k)soE[d*(k)x(k)]¼0.Thusmultiplying(2)byx*(k)andtakingtheexpectationE[.]givesK¼E½xcðkÞx*ðkÞ=E½xðkÞx*ðkÞð3Þx(k)isrealsox*(k)¼x(k).Alsoasx(k)issymmetricallydistributedaboutzeroE[xc(k)x(k)]¼E[x(k)x(k)]=2andthereforeK¼1=2andxc(k)¼x(k)=2þd(k).Thusclippingthesignalatzeroreducestheamplitudeofthewantedsignalbyhalf,butadvantageouslyitreducesthemeanopticalpower,E[xc(k)],bymuchmore.xc(k)iszerowithprobability0.5andhasasemi-normaldistributionotherwise,soE½xcðkÞ¼1=ffiffiffiffiffiffi2ppð10zexpð
z2=2Þdz¼1=ffiffiffiffiffiffi2pp½j1ð4ÞThustheaverageopticalpowerofxc(k)isapproximately1=p(2p)comparedto2forDC-biasedopticalOFDM.TheamplitudeofeachOFDMsubcarrierisalsoreducedbyafactorof0.5.Whenboththeelectricalandopticaleffectsofclippingareconsidered,theelectricalamplitudeofeachsubcarrierisincreasedbyafactorof0.52p(2p)comparedwithDC-biasedOFDMwiththesameaverageopticalpower.Thustheoverallimprovementinelectricalsignal-to-noiseratiois20log10(p(2p))¼7.98dB.Clippingalsoaddsanunwantednoisecomponent,d(k).MostofthenoisepowerisinthezerothsubcarrierandiseasilyseparatedfromtheOFDMsignal,butsomeisatotherfrequencies.Frequencydistributionofclippingnoise:Thedistributionoftheenergyofd(k)acrosssubcarrierfrequenciesdependsonwhichsubcarriersareusedintheunclippedOFDMsignal.Ifonlytheoddsubcarriersofx(k)inasymbolarenonzerothenitcanbeshownthatd(k)hascomponentsatevenfrequenciesonly.ConsiderthepropertiesoftheIFFTandFFTandtherelationshipbetweenx(k)andx(kþN=2)foroddsubcarriersonly.Thecomponentofx(k)duetothemthsubcarrierisgivenbyxðm;kÞ¼ð1=NÞXðmÞexpðj2pkm=NÞð5ÞThus,itissimpletoshowthatformoddx(m,k)¼
x(m,kþN=2).ThusanOFDMsymbolconsistingofonlyoddfrequencieshasthepropertythatx(k)¼
x(kþN=2)asshowninFig.1b.ThevalueofX(m)canbecalculatedfromx(k)usinganFFT,XðmÞ¼PN
1k¼0xðkÞexp
j2pkmN
ð6ÞELECTRONICSLETTERS16thMarch2006Vol.42No.6Authorizedlicenseduselimitedto:SHANGHAIUNIVERSITY.DownloadedonJune1