Tse 答案 fundamentals of wireless communications.pdf

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1、Chapter1SolutionstoExercises1Chapter2SolutionstoExercisespExercise2.1.1.Letr(t)=r2+(vt)2+2rvtcos(Á).Then,00<[®(µ;Ã;f)expfj2¼f(1¡r(t)=c)g]Er(f;t;r(t);µ;Ã))=:r(t)Moreover,ifweassumethatr0Àvt,thenwegetthatr(t)¼r0+vtcos(Á).Thus,thedopplershiftisfvcos(Á)=c.2.Let(x;y

2、;z)bethepositionofthemobileinCartesiancoordinates,and(r;Ã;µ)thepositioninpolarcoordinates.Then(x;y;z)=(rsinµcosÃ;rsinµsinÃ;rcosµ)³pp´(r;Ã;µ)=x2+y2+z2;arctan(y=x);arccos(z=x2+y2+z2)xy_¡xy_Ã_=x2+y2zr_¡zr_µ_=¡pr21¡(z=r)2WeseethatÃ_issmallforlargex2+y2.Alsoµ_issmal

3、lforjz=rj<1andrlarge.Ifjr=zj=1thenµ=0orµ=¼andv<=rjµ_jsov=rlargeassuresthatµ_issmall.IfrisnotverylargethenthevariationofµandÃmaynotbenegligiblewithinthetimescaleofinterestevenformoderatespeedsv.Herelargedependsonthetimescaleofinterest.Exercise2.2.®cos[2¼f(t¡r(t)

4、=c)]2®[d¡r(t)]cos[2¼f(t¡r(t)=c)]Er(f;t)=+2d¡r(t)r(t)[2d¡r(t)]®cos[2¼f(t+(r(t)¡2d)=c)]¡2d¡r(t)2TseandViswanath:FundamentalsofWirelessCommunication32®sin[2¼f(t¡d=c)]sin[2¼f(r(t)¡d)=c]2®[d¡r(t)]cos[2¼f(t¡r(t)=c)]=+2d¡r(t)r(t)[2d¡r(t)](2.1)whereweappliedtheidentity

5、µ¶µ¶x+yy¡xcosx¡cosy=2sinsin22Weobservethatthe¯rsttermof(2.1)issimilarinformtoequation(2.13)inthenotes.Thesecondtermof(2.1)goesto0asr(t)!dandisduetothedi®erenceinpropagationlossesinthe2paths.Exercise2.3.Ifthewallisontheotherside,bothcomponentsarriveatthemobilefr

6、omtheleftandexperiencethesameDopplershift.<[®expfj2¼[f(1¡v=c)t¡fr0=c]g]<[®expfj2¼[f(1¡v=c)t¡f(r0+2d)=c]g]Er(f;t)=¡r0+vtr0+2d+vtWehavetheinteractionof2sinusoidalwavesofthesamefrequencyanddi®erentamplitude.Overtime,weobservethecompositionofthese2wavesintoasingles

7、inusoidalsignaloffrequencyf(1¡v=c)andconstantamplitudethatdependsontheattenuations(r0+vt)and(r0+2d+vt)andalsoonthephasedi®erencef2d=c.Overfrequency,weobservethatwhenf2d=cisanintegerbothwavesinterferedestructivelyresultinginasmallreceivedsignal.Whenf2d=c=(2k+1)=

8、2;k2Zthesewavesinterfereconstructivelyresultinginalargerreceivedsignal.Sowhenfisvariedbyc=4dtheamplitudeofthereceivedsignalvariesfromaminimumtoamaximum.Thevariationoverfrequ