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1、6.013(New)Lecture6:Multipath,Arrays,andFrequencyReuseA.SuperpositionofphasorsThislecturefocusesonthesuperpositionofduplicatewavesatreceivers,wherethemultiplicityofwavesmayhaveoriginatedfrommultiplereflectorsintheenvironmentorfrommultipletransmittingantennae
2、lements.Superpositionofwavesiseasiesttounderstandwhenonlyonenarrowbandisconsideredatatime;weapproximatesuchbandshereaspuremonochromaticsinusoids.ThesimplestcaseisillustratedintheFigurebelow,wherethewavesAandBareduplicatesandsuperimposeinphasetoyieldoA+Bwith
3、doubleamplitudeandquadruplepower,andsuperimpose180outofphasetoyieldA+Cwithzeroamplitudeandzeropower.Whenthesetwoequal-amplitudewaveso0.5superimpose90outofphase,weobtainA+Dwithdoublepowerandamplitude2A.A+BABttCA+CDFigureAA+DIm{E}DCfRe{E}0A,BFigureBEquation(1
4、)showshowtwoequal-amplitudesinusoidswithphaseoffsetfcombinetoyieldadouble-amplitudewaveatthesamew,butphase-shiftedbyf/2andmultipliedbytheconstantcos(f/2),whichcanbepositive,negative,orzero.Thisfollowsfromtheidentityin(2).coswt+cos(wt+f)=2cos(wt+f/2)cos(f/2)
5、(1){Sincecosa+cosb=2cos([a+b]/2)cos([a-b]/2)}(2)-1-oForourcaseA+B,f=0andweproduceadouble-amplitudewave.ForA+C,f=180o-0.5andcos(f/2)=0;forA+D,f=90andcos(f/2)=2.AconvenientwaytothinkaboutsuchsuperpositionofwavesisintermsofphasorsEcharacterizedbytheirrealandim
6、aginaryparts,assuggestedgraphicallyinFigureB,wherephasorsrepresentthewavesA,B,C,D,andA+D.ThephysicaljwtsignificanceofthephasorEisdefinedby:E(t)=Re{Ee}.ThesignificanceoftherealandimaginarypartsofEfollowfromjwtE(t)=Re{Ee}=Re{[Re{E}+jIm{E}][coswt+jsinwt]}(3)Th
7、erealpartofEthuscorrespondstotheamplitudeofthecoswtterm,andtheimaginarypartcorrespondsto-sinwt.ThiscorrespondenceisconsistentwiththephasorsplottedinFigureB.WecanalsorepresentthephasorEbyitsequivalent:jfE=
8、E
9、e=
10、E
11、cosf+j
12、E
13、sinf=Re{E}+jIm{E}(4)wherefistheangle
14、inthefigurebetweentherealaxisandthephasor.ThusthephasorjwtEerotatescounter-clockwiseastimeadvances(seethedirectionofthearrowforfin-jqFigureB).AphasorEthathasbeendelayedqradianswouldberepresentedby
15、E
16、e.
