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B1PhaseandFrequencyModulationS(t)=Acos[Θ(t)]A=amplitudeΘ(t)=instantaneousphaseω(t)=dtdtΘ()=instantaneousfrequencyphasemodulationΘ(t)=ω0t+kPMf(t)B2f(t)=basebandsignalkPM=phasedeviationconstantfrequencymodulationωω()()()tkftdtdtFM=+=0ΘkFM=frequencydeviationconstantΘ()()ttkfdFMt=+−∞∫ωττ0B3StAtkftPMPM()cos()=+[]ω0ωωPMPMPMtdtdtkdftdt()()()==+Θ0StAtkfdFMFMt()cos()=+−∞∫ωττ0ωωFMFMFMtdtdtkft()()()==+Θ0B4CarrierBasebandSignalAMPMFMB5Instantaneousfrequencydeviation∆ωωω=−==()()()tkftkdftdtFMPM0RMSFrequencyDeviation∆∆ωωrmsFMPMkfkdfdt=()=()=2221/21/2∆ωrmsisrelatedtothebandwidthoftheFM&PMsignals.B6PowerSAASAAPMFM2222222222====cos()cos()θθUnlikeAMsignalsthepowerinFMandPMsignalsisindependentofthebasebandsignalpower.B7NarrowbandFM(NBFM)LetkFMbesmallsothatkfdFMt()ττ−∞∫1ThenSAtkfdFMFMt=+−∞∫cos()ωττ0B8AtkfdFMtcoscos()ωττ0−∞∫−−∞∫AtkfdFMtsinsin()ωττ0cos()sinAtAkfdtFMtωττω00−−∞∫B9SAtkfdFMFMt=+−∞∫cos()ωττ0cos()sin()AtAkfdtSFMtNBFMtωττω00−=−∞∫carrierDSB-SCsignalB10FourierTransformofNBFMSANBFM()[()()]ωπδωωδωω=−++00+−−−++AkFFFM20000()()()()ωωωωωωωωBasebandsignalF(ω)ωBNBFMsignal2Bωω0–ω0SpectrumisverysimilartoanAMsignalB11WidebandFMSAtkfdFMFMt=+−∞∫cos()ωττ0SAkfdtFMFMt=−∞∫cos()cosττω0−−∞∫AkfdtFMtsin()sinττω0B12letf(t)=AmcosωmtthenfdkAttFMtFMmmmFMm()sinsinττωωβω−∞∫==βωωωFMFMmmmkA===modulationindex∆maxB13∆ωmax=kFMAm=maximumfrequencydeviationJacobiequationscos(sin)()()cos()βωββωFMmFMkFMmktJJkt=+=∞∑02122BesselfunctionsB14sin(sin)()sin[()]βωβωFMmkFMmktJkt=−−=∞∑221211ByusingtheJacobiequationsandthefollowingpropertiesofBesselfunctionsJJJJJnnnnnnnn−=−∞∞=−=−=−∑()()()()()()()βββββ1112B15weobtainStAJntFMnFMmn()()cos[()]=+=−∞∞∑βωω0ItcanbeshownthatJn()~ββ01forn+Thus[Seetextpp.166&167forplots]B16BesselFunctionsoftheFirstKindB17B18StAJntFMnFMnmFMFM()~()cos[()]()βωωββ=−++∑+110BandwidthFMFMmFMmFM~()~max21221ββωωββωωωωββ++==∆∆forB19βωFMFMmmkA=f(t)=AmcosωmtSAtkfdFMFMt=+−∞∫cos()ωττ0∆ω=kFMf(t)=kFMAmcosωmt∆ωrmsFMmkA=2B20kAFMFMmFMm==++≅≅212()ββωωCarson’sruleNote:BFM=2(∆ωmax+ωm)ForcommercialbroadcastFMωm=15kHz∆ωmax=75kHzsothatkFMFM~1805Hzandβ=====222∆∆∆∆ωωωωrmsmaxB21FMDetectorsOnetypeofFMdemodulatorRCHighPassFilterMignalFrequencyselectivenetworkDiodedemodulatoremodulatedutputa)Mignalb)i(t)o(t)(c)fc∆ff∆|Vo/Vi||H(jω)|=|Vo/Vi|1B22H(f)xCyCyDEnvdetFMslope-detectioncircuitandwaveforms.SlopeapproximationH(f)|coxC(t)tyC(t)tyD(t)tB23DetectionofFMSignals(FrequencyDiscriminator)AttkfdFMt()cos()ωττ0+−∞∫CtkfdFMtcos()ωττ0+−∞∫CkfttkfdFMFMt[()]sin()ωωττ00++−∞∫envelope1244344imitertEnvelopedetectorFMf(t)B24SincetheamplitudeofanFMsignalisconstant,anyamplitudevariationsofthereceivedsignalareduetonoise.Thelimiterremovesthenoise.B25FMSignalGeneration(ArmstrongMethod)f(t)NBFMGeneratorFrequencyMultiplierWBFMSAtAkfdtNBFMtFMt()cos()sin=−−∞∫ωττω00|kFMf(t)|1B26Asinω0tLocalOscillatorIntegratorDSBModulatorPhaseShifter90˚cosω0tf(t)BFM–kdFMt(•)τ−∞∫B27requencyMultipliernSNBFMt()cosωl0tSSNBFM()ωSWBFMt()SSSSWBFMNBFMn()()ωω=SSNBFMn()ωω±l0SAtkfdNBFMtFMt()cos()=+−∞∫ωττ0B28SAntnkfdWBFMtFMt()cos()=+−∞∫ωττ0∆ω=nkftkftnkftFMFMFM()()()maxmax1but1B29scannedmaterialtogohereCommercialFMtransmitterB30StereoFML(t)+R(t)+Apcosωpt+[L(t)–R(t)]cosωsctL(t)=leftchannelR(t)=rightchannelωp=19kHzωsc=2ωp=38kHzω19kHzPilottoneChannelL+RChannel15kHz23kHz38kHzSubcarrier53kHz(L–R)cosωsctB31scannedmaterialtogohereStereoFMdecoderB321.(40pts)Anuppersidebandsuppressedcarriersignalisdetectedusinganenvelopedetectorwithcarrierreinsertion.ThetotalpowerinthetransmittedsignalisP,thebasebandbandwidthisB(rad/s),andthecarrierfrequencyisωo(rad/s).(a)(15pts)Ifthebasebandsignalisdenotedbyg(t),deriveanexpressionfortheUSB-SCsignalintermsofPandsketchthepowerspectrum.1stHourExamECE359B33b)(15pts)Ablockdiagramofthereceiverisgivenbelow.Deriveanexpressionfortheinputtotheenvelopedetectorintermsofthesignalandreceiverlocaloscillator.Pleasenotethecharacteristicsofthereceiverfilters.c)(10pts)Deriveanexpressionforthereceiveroutputintermsofthebasebandsignalg(t).Pleasenotethecharacteristicsofthereceiverfilters.RFFilterEnvelopeDetectorBasebandFilterOutput∑LocalOscillator++–BBωωωoωo+B-k1-k2AdcosωotB342.(20pts)Considerthefollowingsignal.g(t)=Ae–|t|/T(a)(10pts)ComputeandsketchtheFouriertransformofg(t).(b)(10pts)Whatisthe3dBbandwidthof|G(ω)|?B353.(40pts)Twobasebandsignalsg1(t)andg2(t),eachwithbandwidthB,aretransmittedonthesamecarrierusingquadraturemultiplexing.s(t)=A[g1(t)cosωot+g2(t)sinωot](a)(15pts)Drawablockdiagramofthecoherentreceivercapableofdetectingbothg1(t)andg2(t).Becertaintospecifyallfilterpassbands.(b)(10pts)Ifthemasterlocaloscillatorofthereceiverhasaphaseerrorθ,whatistheoutputofthechanneldesignedtodetectg1(t)?(c)(15pts)Whatisthemaximumpermissiblephaseerrorifthedistortionpowercanbenolargerthan110thepowerinthedetectedsignal?B36Phase-LockedLoopPhaseComparatorLoopFilterh(t)VoltageControlledOscillatorVc=kc(θi–θf)Vo=Vc*hcos(ωot+θi)sin(ωot+θf)VCOfrequencydeviationisproportionaltoVoB37∆∆ωθωττffoffttkVttd()()()()–=⇒=∞∫θττθf