Chapter Traditional Analog Modulation Techniques Mikael Olofsson Modulation techniques - PDF document

Chapter5TraditionalAnalogModulationTechniquesMikaelOlofsson|2002{2007Modulationtechniquesaremainlyusedtotransmitinformationinagivenfrequencyband.Thereasonforthatmaybethatthechannelisband-limited,orthatweareassignedacertainfrequencybandandfrequenciesoutsidethatbandissupposedtobeusedbyothers.Therefore,weareinterestedinthespectralpropertiesofvariousmodulationtechniques.Themodulationtechniquesdescribedherehavealonghistoryinradioapplications.Theinformationtobetransmittedisnormallyananalogsocalledbasebandsignal.Bythatweunderstandasignalwiththemainpartofitsspectrumaroundzero.Especially,thatmeansthatthemainpartofthespectrumisbelowsomefrequency,calledthebandwidthofthesignal.Wealsoconsidermethodstodemodulatethemodulatedsignals,i.e.toregaintheoriginalsignalfromthemodulatedone.Noiseaddedbythechannelwillnecessarilyaectthedemodulatedsignal.Weseparatetheanalysisofthosedemodulationmethodsintoonepartwhereweassumeanidealchannelthatdoesnotaddanynoise,andanotherpartwhereweassumethatthechanneladdswhiteGaussiannoise.5.1AmplitudeModulationAmplitudemodulation,normallyabbreviatedAM,wastherstmodulationtechnique.Therstradiobroadcastsweredoneusingthistechnique.ThereasonforthatisthatAMsignalscanbedetectedveryeasily.Essentially,allyouneedisanonlinearity.Actually,almostanynonlinearitywillsucetodetectAMsignals.Therehaveevenbeenreportsofpeoplehearingsomenearbyradiostationfromtheirstainlesssteelkitchensink.Andsome67 68Chapter5.TraditionalAnalogModulationTechniques (includingtheauthor)haveexperiencedthatwithaguitaramplier.ThecrystalreceiverisademodulatorforAMthatcanbemanufacturedatalowcost,whichhelpedmakingradiobroadcastspopular.InChapter3,Theorem9,wenotedthataconvolutioninthetimedomaincorresondstoamultiplicationinthefrequencydomain.Infact,theoppositeisalsotrue. Theorem 10(Fouriertransformofamultiplication)LetandbesignalswithFouriertransformsand.ThenwehaveFf=()( Proof: TheproofisalongthesamelineastheproofofTheorem8,butstartingwiththeinversetransformofthesuggestedspectrum.Basedonourdenitions,wehave)( 1)(ftdf 1 1deftdf:Wecanrewritetheexpressionaboveas)( 1 1ftddf:Now,set,andweget)( 1 1dd 1td 1td:Finally,weidentifythelasttwointegralsastheinverseFouriertransformsof)and),andweget)( 2 So,multiplyinginthetimedomaincorrespondstoaconvolutioninthefrequencydomain. 5.1.AmplitudeModulation69 101 antennaBPlterdiodeLPlterear-phone 101(a)(b)(c)Figure5.1:(a)AstandardAMsignalforthemessage)=cos(2twith=2and=1.Thedarklineis.(b)Principleofanenvelopedetector.(c)Thecorrespondingoutputfromanenvelopedetector.StandardAMAnAMsignal,),correspondingtothemessagesignal,),isgivenbytheequation)=))cos(2fwhereisreferredtoasthecarrierfrequencyissomenon-zeroconstant,andwheretheconstantischosensuchthatCholdsforall.InFigure5.1aastandardAMsignalispresentedtogetherwiththemessage,whichinthisparticularexampleisacosinesignal.WementionedthatAMsignalscanbedetectedusinganonlinearity.TherstAMreceiverwasthesocalledcrystalreceiver.Itconsistsofanantenna,aresonancecircuit(bandpasslter),adiodeandasimplelow-passlter.Itextractstheenvelope)from),andisthereforeoftencalledanenvelopedetector.ThediodeinFigure5.1bisthenon-linearitythatmakesthedetectionpossible.Thefewsimplecomponentsmakesitpossibletomanufacturethereceiveratalowcost.Inadditiontothat,itdoesn'tevenneedapowersourceofitsown.Thepoweristakendirectlyfromtheantenna.Theoutputpowerisofcourseverysmall,andonlyonelistenercouldusethesmallearphonethatwasused.InFigure5.1c,theAMsignalispresentedtogetherwiththeoutputofanenvelopedetector.Notethattheoutputisverysimilartotheoriginalmessage.Themechanicalpartsintheearphone,andtheearwillfurtherlow-passltertheoutput,sothelistenerwillhearalmostthesamesignalastheonetransmitted.Modernenvelopedetectorshaveampliersinvariousplacesandmaybeimplementeddigitally,butthebasicconstructionisstillabandpasslter,adiode(orsomeotherrectier)followedbyalowpasslter.AnadvantageofenvelopedetectorsisthattheBPlterthatltersouteverythingexcepttheintended 70Chapter5.TraditionalAnalogModulationTechniques MessageCarrierStandardAMjFfcos(2fgjlowersidebanduppersidebandlowersidebanduppersidebandFigure5.2:SpectrumforstandardAMsignals.frequencybandisnotcritical.Itisenoughifitscenterfrequencyisapproximatelycorrect.Inotherwords,itdoesnotneedtoknowthecarrierfrequencyexactly,orthecarrierphaseforthatmatter.WewishtostudythespectrumofAMsignals.SinceanAMsignalistheproductofamessageandacarrier,thatiseasiestdonebasedonTheorem10.Thus,weneedtondtheFouriertransformofcos(2f).FirstconsiderftdffwheretherstequalityisgivenbythedenitionoftheinverseFouriertransform,andwherethelastequalityisgivenbythedenitionoftheunitimpulse.So,wehaveFfcos(2fff 2=1 )+))NowwearereadytoapplyTheorem10on).Let)bethespectrumof)andlet)bethespectrumof).Thenweget)=FfACcos(2fFfAm)cos(2fAC (f fc)+)]+ M(f fc)+)] 5.1.AmplitudeModulation71 Itisleftasanexercisetoverifythatthelastequalityholds.TheinvolvedspectraaredisplayedinFigure5.2.HereAC )+))isreferredtoasthecar-rier,sincethattermcorrespondstoACcos(2f).Theotherpartofthespectrum, )+)),isreferredtoasthesidebands.Thosesidebandsaretheonlypartsofthespectrumthatdependonthemessage).Thesidebandsarecalledupperandlowersidebandsbasedonwheretheyarecomparedtothecarrierfrequency,accordingtothefollowing.Uppersideband )+))for�fLowersideband )+))forfBecauseofthosetwosidebands,thistypeofAMmodulationisoftencalleddoublesidebandAM,abbreviatedAM-DSB.TherearealsootherversionsofAM,buttheycannotbedetectedusinganenvelopedetector.SuppressedCarrierModulationAlltheinformationaboutthemessage)instandardAMisinthesidebands.Thecarrieritselfdoesnotcarryanyinformation,andinthatrespectthecarriercorrespondstounnecessarypowerdissipation.OneversionofAMthatcannotbedetectedusinganenvelopedetectoriscalledAM-SCorAM-DSB-SC,whereSCshouldbeinterpretedasSuppressedCarrier.Forthistypeofmodulation,theconstantissimplysettozero,i.e.wehave)=Am)cos(2fandthecorrespondingspectrumis)= )+))Sothecarrierisremovedfromthespectrum,asthenamesuggests.InFigure5.3,anAM-SCsignalispresentedtogetherwiththemessageandthecorrespondingenvelopedetectoroutput,aswellastheabsolutevalueofthemessage.Notethattheoutputfromanenvelopedetectorinthiscaseisclosetotheabsolutevalueofthemessage.Thus,anenvelopedetectorcannotbeusedtoreceiveAM-SC.DemodulationofAM-SCcaninsteadbedonebymodulatingoncemore.Let)withspectrum)betheoutputofthatmodulation.Thenwehave,similarilyasabove,)=)cos(2f)=Am)cos(2f)= )(1+cos(4f)) 72Chapter5.TraditionalAnalogModulationTechniques 10 10 10(a)(b)(c)Figure5.3:(a)AnAM-SCsignalforthemessage)=cos(2twith=1.Thethicklineis.(b)Thecorrespondingoutputfromanenvelopedetector.(c)TheAM-SCsignaltogetherwithforcomparison. MessageAM-SCDemodulatedFigure5.4:ModulationofAM-SCanddemodulationbymodulatingagain. 5.1.AmplitudeModulation73 andthecorrespondingspectrumis)= )+))= )+ )++2))So,wehaveregained),butwealsohavecopiesof)centeredaround.TheinvolvedspectraaredisplayedinFigure5.4.If,thebandwidthofthemessage),issmallerthan,whichnormallyisthecase,thenthosecopiesdonotoverlapwiththeoriginalspectrum.Thus,wecanuseasuitablelow-passltertoremovetheunwantedcopies.Thefurtherawaytheunwantedcopiesareinthefrequencydomain,thesimplerthatltercanbe.ItshouldbenotedthatstandardAMcanalsobedemodulatedusingthismethod.SingleSidebandModulationSincethereisaone-to-onerelationbetween)and),thereisalsoaone-to-onerelationbetweenthetwosidebands,atleastifissmallerthan.So,inbothstandardAMandAM-SC,weactuallytransmitourdatatwiceinthefrequencydomain.Noinfor-mationislostifweonlytransmitoneofthesidebands.ThistypeofAMisreferredtoasSSB,whichshouldbeinterpretedasSingleSideBand.ThereareSSBversionsofbothstandardAMandAM-SC,andtheycanbeobtainedbyrstgeneratingstandardAMorAM-SC,andthenusingasuitableband-passltertoremovetheunwantedsideband.SpectraofAM-SSBandAM-SSB-SCaredisplayedinFigure5.5.Obviously,SSBmodula-tiononlyneedshalfthebandwidthcomparedtooriginalAMorAM-SC.SSB-modulatedsignalscanalsobedemodulatedbymodulatingagainusingAM-SC,andwestillgetcopiesnear,thathastoberemovedbyalow-passlter.Howeverthesecopiesnowcontainonlyonesideband.SynchronizationforAMDemodulationDemodulationbyremodulationasdescribedaboveisamethodthatcanbeusedfordetec-tionofallvariantsofAM.However,thatdemandsthatwehaveacorrectcarrieravailableinthedemodulator,withbothcorrectfrequencyandatleastapproximatelycorrectphase.ForstandardAMandforAM-SSB,wherethecarrierisavailableinthesignal,itcaneasilybeextractedfromthereceivedsignalusinganarrowBP-lterwithcenterfrequencyForAM-SC,andforAM-SSB-SC,theabsenceofacarriermakesitimpossibletoextractthecarrierinthatway.OnewayforthereceivertoextractacarriersignalfromanAM-SCsignal)=Am)cos(2f 74Chapter5.TraditionalAnalogModulationTechniques AM-SSBAM-SSB-SCuppersidebanduppersidebandFigure5.5:SpectraforAM-SSBandAM-SSB-SC. NarrowLPlterVCOcos(2f)cos(2cos(2fFigure5.6:Aphase-lockedloopforgenerationofawell-denedcarriersignal.Thesignalisgivenby)=cos(2f)cos(2f)= (cos(4f)+cos()).ThedevicelabelledVCOisavoltagecontrolledoscillator. LPlterLPlterNarrowLPlterVCOphaseshiftAm)cos(2fAm)(cos(4f)+cos())Am)(sin(4f)+sin())Am)cos(Am)sin()sin(2sin(22cos(2f2sin(2fFigure5.7:ACostasloopfordetectionofAM-SC. 5.1.AmplitudeModulation75 istoproducethesquare)=)cos(2f)= (1+cos(4f))Whenwesendinformation,theaverageof)isnon-zero,whichmeansthatascaledversionofcos(4f)canbeextractedfrom),againusinganarrowBP-lter,butwithcenterfrequency2.ExtractingcarriersinthosewayswillproducesignalswithafrequencythatisthecorrectcarrierfrequencyintheAMorAM-SSBcaseandtwicethecarrierfrequencyintheAM-SCcase,buttheamplitudecanvaryfromtimetotimedependingontheactualbehaviourofthechannelorthestatisticsoftheinformation.Also,theextractedsignalmayincludenoiseandpartsofthesideband(s).Acleancarrierwithboththecorrectfrequencyandawelldenedamplitude,withoutanynoiseorresiduesfromthesidebands,canbeobtainedfromtheextractedsignalusingphase-lockedloop(PLL).Thereareseveralvariantsofphase-lockedloopsinuse,andasimplevariantisdisplayedinFigure5.6.ThesignalsgiveninFigure5.6assumethattheinputisalreadyacleansinusoid.Inpractice,theinputisanextractedapproximatecarrierwhich,asnotedabove,ispollutedwithnoiseandresiduesfromthesidebands.Aphase-lockedloopisacontolloopthatproducesasinusoidwithconstantamplitude,thecorrectfrequencyandanapproximatephase.Thevoltage-controlledoscillator(VCO)ischosensuchthatthewantedcarrierfrequencycorrespondstozeroinput.ThefrequencyVCOin)oftheoutputoftheVCOisafunctionoftheinputvoltagein,suchthatthederivativeofthatfunctionispositive,i.e.wehave dVinVCOin0.Thecarrierfrequencyinusemaydierslightlyfromthewantedcarrierfrequency,andthephase-lockedloopfollowsthecarrierfrequency,whichmeansthattheinputtotheVCOmaydierslightlyfromzero.InFigure5.6thatmeansthatthesignalproducedbythePLLhasphaseforwhichcos(0holds,inapointwherecos()haspositivederivative.Inotherwords,wehave forsomeinteger.Wemayassumeanyintegervalueof,sincetheproducedsignalisthesamefordierentvaluesof.So,wecanforinstancesaythatthesignalhasthephase ForAM-SC,wherewehavesquaredthesignal,andwherethefrequencyoftheextractedsignalistwicethecarrierfrequency,thefeedbackisequippedwithafrequencydoublerwhichcanforinstancebeasquarer.Theresultingoutputisthenasinusoidwiththecorrectcarrierfrequencyandphaseapproximately Aspecialtypeofphase-lockedloopthatisespeciallywellsuitedfordetectionofAM-SCsignalsistheCostasloop,giveninFigure5.7.Itextractsthecarrierdirectlyfromthesignal.Again,theVCOischosensuchthatthewantedcarrierfrequencycorrespondstozeroinput.Thus,theloopproducesasinusoidwhosefrequencyisthecarrierfrequencywithphaseforwhichsin(20holds,inapointwheresin()haspositivederivative.Theresultingphaseistherefore0.TheoutputoftheCostasloopisthemessagescaledbycos(),butsincewehave0,wealsohavecos(1. 76Chapter5.TraditionalAnalogModulationTechniques BPlterLPlter)+noise)+2cos(2fFigure5.8:DemodulationofAMsignalsinthepresenceofnoise.ThereasonthattheCostasloopworksisthepresenceofbothsidebands,andtheone-to-onemappingbetweenthetwosidebands.Thetwosidebandspointatthecarrierfrequency,andthephaseinformationinthesidebandsgivesusthecarrierphase.TheCostasloopcanthereforenotbeusedforAM-SSB-SC.ThereissimplynowaytoextractthecarrierfromanAM-SSB-SCsignal,duetothefactthatthecarrierisnotavailableinthesignal,andnothinginthespectrumgivesanyhintaboutthecarrierfrequency.Thatisthepricewehavetopayforsuppressingboththecarrierandoneofthesidebands.Therefore,thereareanumberofmodicationsofAM-SSB-SC,thatmakesitpossibletoextractacarrieranyway.Themostsimplemethodisnottosuppressthecarriercompletely.Thenthe{howeverweak{carriercanbeextractedfromthesignal.Anothermethodistosendashortcarrierburst,andletaPLLlockontothatburst.Aftertheburst,theoscillatorcontinuesproducinganinternalcarrierbasedonthatburst.Ofcourse,theoscillatorwillmostprobablydivergefromtheusedcarriereventually.Thereforethecarrierburstisrepeatedregularly.Athirdpossibilityistokeepasmallpartoftheremovedsideband,andinthereceiverltertheothersidebandsimilarily,andthenextractacarrierusingoneofthemethodsabove.ImpactofNoiseinAMDemodulationWewouldliketoanalyzetheimpactofnoiseondemodulationofAMsignals.Forthisanalysisweneedtomakesomeassumptionsaboutthenoiseandaboutthedemodulation.Therstassumptionisthatthenoiseisdominatedbythermalnoise,andthatitisinde-pendentofthemessage.AsmentionedinSection4.1,suchnoisecanbemodeledaswhiteGaussiannoise.WeassumethatthereceivedsignalislteredbyanidealBPlterthatexactlymatchesthebandwidthoftheAMsignalbeforedemodulation.Weassumethatthedemodulationisdonebyremodulationby2cos(2f)asintheCostasloop.WealsoassumethatthedemodulatedsignalislteredbyanidealLPlterthatexactlymatchesthebandwidthofthemessage.SeeFigure5.8.Weneedtointroducesomenotation.Letdenotethebandwidthofthemessage.Letdenotetheone-sidedpowerspectraldensityoftheassumedwhiteGaussiannoise,andlet)denotethenoiseaftertheBPlter.Also,introducethefollowingnotationfortheinvolvedpowers. 5.1.AmplitudeModulation77 :The(expected)powerofthemessage).mod:The(expected)powerofthereceivedmodulatedsignal).Notethatthismeansthatincludesimpactsofthechannel.:The(expected)powerofthemessageafterdemodulationandLPlter.mod:TheexpectedpoweroftheideallyBP-lterednoise)beforedemodula-tion.:TheexpectedpowerofthedemodulatedandLP-lterednoise.Wedenethesignal-to-noiseratio=Pafterdemodulation.Wewillcomparethissignal-to-noiseratioforDSBandSSBmodulationusingthesamesentpowermodtransmittedoverachannelwiththesameFirstweconsiderAM-SC.Thenwehavethesignal)=Am)cos(2fwithbandwidth2andexpectedpowermodP=2sincethecarriercos(2fhasaveragepower12.Afterdemodulation,weregainAm),whichmeansthatwehave.Forthenoise,wehavemod=2WN.Thedemodulatednoise2cos(2fhasexpectedpower2mod,sincethecarrier2cos(2f)hasaveragepower2.HalfofthatexpectedpowerisinthefrequencyintervalW,whiletheotherhalfisinthefrequencyinterval2W.ThelatterpartisremovedbytheLPlter,leavinguswithmod.Finally,thatgivesusthesignal-to-noiseratio Pn=A2P WNForAM-SSB-SC,oneofthesidebandsfromAM-SCisremoved,whichmeansthatthepowermodisreducedtohalfthatofAM-SC.ToproduceanSSBsignalwiththesamepowerasintheDSBcase,wethereforeneedtoamplifythesignalby 2.So,westartwith)= Am)cos(2fandlteroutoneofthesidebands.ThenwehavethesamesentpowermodP=2.Afterdemodulation,wegetascaledversionofthemessage.Moreprecisely,theoutputis p ),whichhaspowerP=2.Forthenoise,wehavemodWN,sincethebandwidthis.Thedemodulatednoise2cos(2f 78Chapter5.TraditionalAnalogModulationTechniques stillhasexpectedpower2mod,sincethecarrier2cos(2f)hasaveragepower2.HalfofthatexpectedpowerisinthefrequencyintervalW,whiletheotherhalfisinthefrequencyinterval2W.Actually,theotherhalfofthepowerisintheinterval2Wifthelowersidebandisused,orintheintervaliftheuppersidebandisused.Inanycase,thepartofthespectrumthatisnear2isremovedbytheLPlter,leavinguswithmod.Finally,thatgivesusthesignal-to-noiseratio Pn=A2P WNi.e.thesamesignal-to-noiseratioasforDSB.5.2AngleModulationAnglemodulationisthecommonnameforaclassofmodulationtechniques,withthatincommonthatthebandwidthofthemodulatedsignalisnotgivenonlybythebandwidthofthemessage,butalsobyaparametercalledthemodulationindex.Bysettingthismodulationindextoasuitablenumber,wecandecidewhatbandwidthtouse,andthelargerthatindexis,thebetteristheobtainedquality.Thesemethods,andcombinationsofthemareusedinradiobroadcastsintheFM-band(88{108MHz).Theideathatanglemodulationisbasedonistoletafunction))ofthemessage)bethephaseofacarrier,i.e.thesentsignalis)=cos(2f)))whereissomenon-zeroconstant,andwhereagainisreferredtoasthecarrierfre-quency.Wesaythat))isthemomentaryphaseof).Thenthephasedeviation)isthedierencebetweenthemomentaryphaseandtheaverageofthemomentaryphase.Typically,)hasaveragezero,andthefunctionischosensuchthat))alsohasaverage0.Thenwehave)=))andthepeakphasedeviationisdenedasmax=maxThepeakphasedeviationisalsocalledthephasemodulationindex,andisdenotedAnalternativeinterpretationofthevaryingphaseofthesignal),istosaythat)hasvaryingfrequency.Wedenethemomentaryfrequencyasmom)= 2
d dt(2f)))= 2
d dt)) 5.2.AngleModulation79 Note,thatthisfrequencyisafunctionoftime,justasthemomentaryphasealsoisafunctionoftime.Thefrequencydeviation)isthedierencebetweenthemomentaryfrequencyandthecarrierfrequency,i.e.)=momandthepeakfrequencydeviationisdenedasmax=maxThefrequencymodulationindexisdenedasmax whereisthebandwidthofthemessage),orratherthehighestfrequencycomponentin).If)isastationarysine,thenthetwomodulationindicesareequal.SpectrumofAngleModulationThespectraofanglemodulatedsignalsaregenerallyhardtodetermine,duetothefactthatanglemodulationisnon-linear.Forthesimpleexample)=cos(2fsin(2f))for,itiseasilyshownthatisboththephasemodulationindexandthefrequencymodulationindexofthatsignal.Itcanalsobeshownthatwehave)= 1)cos(2nfwhere)istheBesselfunctionoforder.Wewillnotatalltrytoperformthatproof.Thespectrumofthatsignalis)= 1 (f+fc+nf)+nf)]TheBesselfunctionoforderisgivenby)==01) !()! +2forpositiveintegers.Itcanalsobewrittenas 80Chapter5.TraditionalAnalogModulationTechniques 02468101214161820Figure5.9:Besselfunctionsforupto7. =1=5=10Figure5.10:Spectrumofananglemodulatedsignalwithmodulationindex,carrierfrequency,acosinemessagewithfrequency.Thehightsofthearrowsdenotingimpulsesrepresenttheamplitudeofthecorrespondingfrequencies. 5.2.AngleModulation81 )= cos(sin(nd;stillforpositiveintegers.Fornegativeintegers,wehave)=(1)Formally,thebandwidthofthissignalisinnite.However,thecoecients)decreaserapidlytowards0for�,seeFigure5.9.Thus,wecanstatethatthebandwidthofthesignalisapproximately2f.Acommonapproximationofthebandwidthis2(+1)whichisknownasCarson'srule.Usingtheidentitymax=f,wecanexpressthebandwidthas21+ max.InFigure5.10,wehaveplottedthespectraforthreedierentvaluesofthemodulationindex,withasineshapedmessage.Notehowthebandwidthgrowswiththemodulationindex.PhaseModulationPhasemodulationisnormallyabbreviatedPMorPhM.Themessage,),isinthistechniqueuseddirectlytodeterminethemomentaryphase,i.e.wehave))=whereissomeconstant.Themodulatedsignal,),isthusgivenby)=cos(2f))Themomentaryfrequencyforthismodulationismom)= 2
d dt(2f))= 2
d dtthefrequencydeviationis)= 2
d dtandthepeakfrequencydeviationismax max dtFinally,thefrequencymodulationindexisgivenby Wmax dtWenoticethatthepeakfrequencydeviationdependsonmax dt.Hence,thebandwidthof)dependsonthebandwidthof),butalsoontheamplitudeof).InFigure5.11a,aPMsignalispresentedtogetherwiththecorrespondingmessage. 82Chapter5.TraditionalAnalogModulationTechniques FrequencyModulationFrequencymodulationisnormallyabbreviatedFM.AsforPM,themessage,),deter-minesthephaseofthecarrier,butnotdirectly.Instead,thederivativeofthephaseisproportionalto),i.e.thephaseisascaledindeniteintegralof).Moreprecisely,wehave))=dt;whereissomeconstant.Themodulatedsignal,),isthengivenby)=cosfdtandthemomentaryfrequencyisgivenbymom)= 2
d dtfdt Thus,themomentaryfrequencyisdirectlygivenbythemessage.Notethatanyindeniteintegralof)canbeusedasthephase.Anaturalchoiceis))=d;whereisthetimeinstancewhenthecommunicationstarts.Thefrequencydeviationis)= andthepeakfrequencydeviationismax maxFinally,thefrequencymodulationindexisgivenby BmaxWenoticethatthefrequencydeviationdependsonmax.Hence,thebandwidthof)dependsontheamplitudeof),butnotonthebandwidthof).InFigure5.11b,anFMsignalispresentedtogetherwiththecorrespondingmessage. 5.2.AngleModulation83 101 101(a)(b)Figure5.11:(a)APMsignal(thinline)forthemessage)=cos(2t(thickline).(b)AnFMsignal(thinline)forthemessage)=cos(2t(thickline).Themodulationindexisinbothcases10andwehave=1DemodulationofPMandFMRecallthatthesentsignalis)=cos(2f)))Thissignalcanbedemodulatedbydeterminingthederivativeofthesignal, dt)=f dt))sin(2f)))Thisgivesusasignalforwhichtheamplitudedependsonthemessage)inawaysimilartoAM-DSB,butitscarrierhasvaryingphase.Thissignalcanthenbedemodulatedusinganenvelopedetector,whichgivesustheenvelopef dt))TheconstanttermcanberemovedusingaBPorHPlter,leavinguswiththesignal dt)).ForPM,wehave))=am).Thismeansthatweneadtointegratethesignal dt))togetthewantedmessage,i.e.wemustproduce d))dAa))whereisthetimeinstancewhenthecommunicationstarted.ForFM,wehave))=dt: 84Chapter5.TraditionalAnalogModulationTechniques d dtEnvelopedetectorBPlterFigure5.12:DemodulationofPM. d dtEnvelopedetectorBPlterFigure5.13:DemodulationofFM. sgn( dtLPltermomFigure5.14:Detectionofmomentaryfrequencyinanglemodulatedsignalsusingzerocrossings.Thenwehavethesignal dt))= dtdtAaTherefore,demodulationofPMandFMcanbedoneasindicatedinFigures5.12and5.13,respectively.Alternatively,PMandFMcanbedemodulatedbyextractingthemomentaryfrequencyofthemodulatedsignal).Thatcanbedonebydetectingthezerocrossingsof).ThedemodulatorinFigure5.14isbasedonthisapproach.Therstblockoutputsthesignofitsinput,i.e.itsoutputissgn()).Thesgnfunctionisdenedassgn()=1,x�0,0,=0,-1,x0.Inpractice,thisblockisanamplierwithveryhighgain.Thesecondblockproducesthederivativeofitsinput.Theresultisthattheoutputofthesecondblockisapositive(negative)impulsewhen)passeszerowithpositive(negative)derivative.Afterthethirdblock,whichisarectier,allthoseimpulsesarepositive,withmomentaryfrequencythatistwicethemomentaryfrequencyof).AllthatislefttoproducethemessageisanLPlter,whichisthelastblockinFigure5.14.Thisproducesasignalthatisproportionaltothemomentaryfrequencymom)of).ForFMthisisessentiallythemessage),andforPMthisisessentially dt).AllwehavelefttodoistoremovetheDCcomponent 5.2.AngleModulation85 thatoriginatesfromthecarrierfrequency,usingaHPlter,orbyreplacingtheLPlterinFigure5.14byaBPlter.ForPM,wealsohavetointegratetheoutputtogetthemessage.ImpactofNoiseinPMandFMDemodulationWeassumethatthedemodulationiscarriedoutasdescribedabove,andweassumethatthenoiseisdominatedbywhiteGaussiannoisewithone-sidedpowerspectraldensityWeusethesamenotationfortheinvolvedpowersaswedidforAMdemodulation.:The(expected)powerofthemessage).mod:Theaveragepowerofthemodulatedsignal).:The(expected)powerofthemessageafterdemodulationandLPlter.mod:TheexpectedpoweroftheideallyBP-lterednoisebeforedemodulation.:TheexpectedpowerofthedemodulatedandLP-lterednoise.Thesignal)isacosinewithamplitude.Theaveragepowermodofthemodulatedsignal)isthereforegivenbymod2.Thevaryingphase-orfrequencyforthatmatter-isirrelevantinthisrespect.ForbothPMandFM,thedemodulatedsignalisAa),whichgivesus.Thebandwidthof)isapproximately2maxTheexpectedpowermodoftheideallyBP-lterednoisebeforedemodulationisthereforemod=2max2=maxAnglemodulationmethodsarenon-linear.ThatmakestheanalysisofdetectioninthepresenceofnoisealotmorecomplicatedthanforAM.Wesimplyskipthatanalysisandstatethenoisepowerforthetwocases,undertheassumptionthatthesignal-to-noiseratioonthechannelmod=Pmodishigh.WestartwithPM.Thenitcanbeshownthatthenoisepowerisgivenby=2WNThisgivesusthesignal-to-noiseratio Pn=A2a2P WNWewouldliketoexpressthissignal-to-noiseratiousingthephasemodulationindexWeget=max))max 86Chapter5.TraditionalAnalogModulationTechniques fromwhichweget maxWeusethisrelationtorewritethesignal-to-noiserationas Pn=p max WNAswecansee,wegetincreasedsignal-to-noiseratiowithincreasedphasemodulationindex.NowweturntoFM.Itcanbeshownthatthenoisepowerisgivenby Thisgivesusthesignal-to-noiseratio Pn=A2a2P Nowwewouldliketoexpressthesignal-to-noiseratiousingthefrequencymodulationindex.Wehavealreadynotedthatwehavemax Wfromwhichweget  maxWeusethisrelationtorewritethesignal-to-noiseratioas =12 max WNHerewegetincreasedsignal-to-noiseratiowithincreasedfrequencymodulationindex.Pre-emphasizedFMTheresultingnoiseafterdemodulatingPMsignalsisevenlydistributedoverfrequenciesfrom0to,whichresemblesthesituationforAM.ThatisnotthecaseforFM,whereinsteadtheresultingnoiseisdominatedbyhighfrequencies(near).Moreprecisely,thepowerspectraldensityoftheresultingnoiseafterdemodulatingFMasdescribedaboveis,wherefisfrequency.Therefore,ifwecouldcombinePMandFMinsuchawaythatPMisusedforhighfrequenciesinthemessage)andFMisusedforlowfrequencies, 5.2.AngleModulation87 wecouldhopeforreducednoisecomparedtoanyofthetwomethodsbythemselves.Suchacombinationexistsandiscalledpre-emphasizedFM.Thenthemessageisrstlteredusingapre-emphasislterwithfrequencyresponse)=1+jf=fTheoutputofthatlteristhenfrequencymodulated.Inthereceiver,afterordinarydemodulationoftheFMsignal,theresultislteredwithaninverselterof)calledade-emphasislter.Thatlterhasfrequencyresponse)= H1(f)=1 1+jf=fTheresultisthatweregaintheoriginalsignal,andthattheresultingnoiseissmallerthanifwewouldhaveusedordinaryPMorFM.ThetransmissionsonthesocalledFMband(88-108MHz)aredoneusingpre-emphasizedFMwith=2122Hz.