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1MultipathandDopplerEffectsandModelsTherearetwopartsinthislecture.•InPartI,wewillfirstintroducethemutipathpropagationeffectsandDopplerfrequencyshift/spreadeffects.•InPartII,wewillbrieflyintroducemultipathandDopplerchannelmodels.2Background:OverviewsonWirelessChannelModelingWeneedtoaskthefollowingthreeimportantquestionswhiledesigningawirelesscommunicationlink:1.Fading&PowerLoss:Isthesignaltointerferenceplusnoiseratio(SINR)largeenoughforthereceivertodetectthetransmittedsignal?2.SignalDistortion:Canthesignaldistortionbeignored,predictedorremovedsothatweknowhowtorecoverthetransmittedinformationatthereceiver?3.TimeVariation:Canthereceiveradaptfasterenoughtothevariationsoftheabovetwofeatures(SINR&signaldistortion)?AcompletewirelesschannelmodelshouldprovidequantitativemeasuresofSINR,signaldistortionandtimevariation.ForSINR(referredinquestion1),weneedonlytoconsiderthetimeinvarianttransmissionlossatasinglefrequency(thatistheRFcarrierfrequency).Thefrequencyandtimedependentpropertiesofsignalcanbeaddressedinansweringquestions2and3.Signaldistortion(referredinquestion2)iscausedbythefrequencydependentvariationsofthereceivedsignalstrengthandphase.Theprimarysourceoffrequencydependentvariationsismultipathpropagation.Here,weneedonlytoconsidertimeinvariantsituationsandleavethetimevaryingfeaturestoquestion3.Motionsofreceivers,transmittersorwirelessenvironmentsgenerateDopplereffects.WithDopplereffects,signalfrequenciesshiftandspread.TheseDopplereffectswillcausetimevariationsinthereceivedsignalstrengthandwaveshape.Thiskindoftimevaryingfeaturesisusuallyrandomandcanbemodeledasstochasticprocesses.Inordertoaddressingthesethreeimportantissues,wedividewirelesschannelmodelingintothreeparts:•Transmissionloss-singlefrequency(ornarrowbandsignal)-timeinvariantenvironment(orshortobservationtimeperiod)•Frequencydependentchannelimpulseresponseortransferfunction-multiplefrequencies(orbroadbandsignal)-timeinvariantenvironment(orshortobservationtimeperiod)•Timevaryingchannelimpulseresponseortransferfunction-multiplefrequencies(orbroadbandsignal)3-timevaryingenvironment(orlongobservationtimeperiod)Wehavediscussedtransmissionloss(includingpathloss,shorttermfadingandlongtermfading)ofasinglefrequencyresponseintimeinvariantenvironmentsinthepreviouslecture.Bothfrequencydependentandtimevaryingfeaturesofachannelimpulseresponse(ortransferfunction)willbeintroducedinthislecture.4PartI:MultipathandDopplerEffectsAfterstudyingthisnote,studentswillbeableto1.Understandmultipathchanneleffectsinbothtimeandfrequencydomains2.UnderstandDopplereffectsinbothtimeandfrequencydomains3.UnderstandmultipathandDopplereffectsinbothtimeandfrequencydomainsI.MultipathChannelEffects:TimeInvariantCase(NoDopplereffects)Inwirelesscommunicationenvironments,asignaltransmittedfromthetransmitterreachesthereceiverthroughmanydifferentpathsasillustratedinFigure1.Lets(t)isthetransmittedsignal.Thereceivedsignalcanthenbewrittenasasumofmultipatharrivals:1231()(),....LiiLiytastτττττ==−≤≤≤≤∑(1)Here,Listhetotalnumberofmultipatharrivals,aiandτiaretheamplitudeandarrivaltimeoftheithray,respectively.A.s(t)isatimeharmonic(i.e.,singlefrequencyorsinusoidal)signalConsiderthetransmittedsignal()jtsteω=.Then,thereceivedsignalis()11()()with()nnLLjtjjtnnnnytaeHeHaeωτωτωωω−−=====∑∑(2)Figure1:Multipathpropagation5Here,()Hωisdefinedasthetransferfunctionofthemultipathenvironment.Notethatthereceiversignaly(t)remainsasatimeharmonicsignalwiththesameangularfrequencyωasthetransmittedsignals(t).Thus,nodistortioninwaveshapehasoccurredduringthetransmissionofs(t)throughatimeinvariantmultipathenvironment.However,themagnitudeofthesignalhasbeenmodified.Thenewmagnitudeis()Hωwhichisafunctionofangularfrequencyω.WeusethefollowingmatlabcodetogeneratetheFigure2:========================================================clearall;%amplitudesof7multipatharrivalsa=[0.61540.79190.92180.73820.17630.40570.9355];%arrivaltimesof7multipatharrivalst=[0.91690.41030.89360.05790.35290.81320.0099];i=0;%frequencyindexforomega=0:0.05:100;%angularfreuenciesmultipath_arrival=a.*exp(j*omega*t);i=i+1;abs_H(i)=abs(sum(multipath_arrival));%thei-thtransferfunctionendomega=0:0.05:100;plot(omega,abs_H)ylabel('amplitudeoftransferfunction')xlabel('angularfreuency')title('frequencydependentmultipathfading')Here,weuse7multipatharrivals.Theamplitudesandarrivaltimesofthesesevenmultipatharrivalsarerandomlychosen.FromFigure2,themagnitudeofreceivedsignalfluctuatesasangularfrequencychanges.Forsomefrequencies,the7multipatharrivalsinterferedestructivelyandyieldsmall()Hω.Forotherfrequencies,the7multipatharrivalsinterfereconstructivelyandyieldlarge()Hω.Thisphenomenoniscalledmultipathfading.Figure2showsthatmultipathfadingisfrequencydependent.Otherkindsoffadingwillbediscussedinfuturelectures.6Figure2:Multipathfadingisafunctionoffrequency.Sincetheamplitudesandarrivaltimesofmultipatharrivalsdependonlocationsoftransmitterandreceiver,thereceivedsignalstrengthwillalsodependonthelocationsoftransmitterandreceiver.Forexample,consideratworaymodelwherelineofsight(LOS)andreflectedraysar