数字通信原理课后习题解答

SolutionsManualforDigitalCommunications,5thEdition(Chapter2)1PreparedbyKostasStamatiouJanuary11,20081PROPRIETARYMATERIAL.cTheMcGraw-HillCompanies,Inc.Allrightsreserved.NopartofthisManualmaybedisplayed,reproducedordistributedinanyformorbyanymeans,withoutthepriorwrittenpermissionofthepublisher,orusedbeyondthelimiteddistributiontoteachersandeducatorspermittedbyMcGraw-Hillfortheirindividualcoursepreparation.IfyouareastudentusingthisManual,youareusingitwithoutpermission.2Problem2.1a.ˆx(t)=1πZ∞−∞x(a)t−adaHence:−ˆx(−t)=−1πR∞−∞x(a)−t−ada=−1πR−∞∞x(−b)−t+b(−db)=−1πR∞−∞x(b)−t+bdb=1πR∞−∞x(b)t−bdb=ˆx(t)wherewehavemadethechangeofvariables:b=−aandusedtherelationship:x(b)=x(−b).b.Inexactlythesamewayasinpart(a)weprove:ˆx(t)=ˆx(−t)c.x(t)=cosω0t,soitsFouriertransformis:X(f)=12[δ(f−f0)+δ(f+f0)],f0=2πω0.Exploitingthephase-shiftingproperty(2-1-4)oftheHilberttransform:ˆX(f)=12[−jδ(f−f0)+jδ(f+f0)]=12j[δ(f−f0)−δ(f+f0)]=F−1{sin2πf0t}Hence,ˆx(t)=sinω0t.d.Inasimilarwaytopart(c):x(t)=sinω0t⇒X(f)=12j[δ(f−f0)−δ(f+f0)]⇒ˆX(f)=12[−δ(f−f0)−δ(f+f0)]⇒ˆX(f)=−12[δ(f−f0)+δ(f+f0)]=−F−1{cos2πω0t}⇒ˆx(t)=−cosω0te.Thepositivefrequencycontentofthenewsignalwillbe:(−j)(−j)X(f)=−X(f),f0,whilethenegativefrequencycontentwillbe:j·jX(f)=−X(f),f0.Hence,sinceˆˆX(f)=−X(f),wehave:ˆˆx(t)=−x(t).f.SincethemagnituderesponseoftheHilberttransformerischaracterizedby:|H(f)|=1,wehavethat:ˆX(f)=|H(f)||X(f)|=|X(f)|.Hence:Z∞−∞ˆX(f)2df=Z∞−∞|X(f)|2dfPROPRIETARYMATERIAL.cTheMcGraw-HillCompanies,Inc.Allrightsreserved.NopartofthisManualmaybedisplayed,reproducedordistributedinanyformorbyanymeans,withoutthepriorwrittenpermissionofthepublisher,orusedbeyondthelimiteddistributiontoteachersandeducatorspermittedbyMcGraw-Hillfortheirindividualcoursepreparation.IfyouareastudentusingthisManual,youareusingitwithoutpermission.3andusingParseval’srelationship:Z∞−∞ˆx2(t)dt=Z∞−∞x2(t)dtg.Fromparts(a)and(b)above,wenotethatifx(t)iseven,ˆx(t)isoddandvice-versa.Therefore,x(t)ˆx(t)isalwaysoddandhence:R∞−∞x(t)ˆx(t)dt=0.Problem2.21.UsingrelationsX(f)=12Xl(f−f0)+12Xl(−f−f0)Y(f)=12Yl(f−f0)+12Yl(−f−f0)andParseval’srelation,wehaveZ∞−∞x(t)y(t)dt=Z∞−∞X(f)Y∗(f)dt=Z∞−∞12Xl(f−f0)+12Xl(−f−f0)12Yl(f−f0)+12Yl(−f−f0)∗df=14Z∞−∞Xl(f−f0)Y∗l(f−f0)df+14Z∞−∞Xl(−f−f0)Yl(−f−f0)df=14Z∞−∞Xl(u)Y∗l(u)du+14X∗l(v)Y(v)dv=12ReZ∞−∞Xl(f)Y∗l(f)df=12ReZ∞−∞xl(t)y∗l(t)dtwherewehaveusedthefactthatsinceXl(f−f0)andYl(−f−f0)donotoverlap,Xl(f−f0)Yl(−f−f0)=0andsimilarlyXl(−f−f0)Yl(f−f0)=0.2.Puttingy(t)=x(t)wegetthedesiredresultfromtheresultofpart1.PROPRIETARYMATERIAL.cTheMcGraw-HillCompanies,Inc.Allrightsreserved.NopartofthisManualmaybedisplayed,reproducedordistributedinanyformorbyanymeans,withoutthepriorwrittenpermissionofthepublisher,orusedbeyondthelimiteddistributiontoteachersandeducatorspermittedbyMcGraw-Hillfortheirindividualcoursepreparation.IfyouareastudentusingthisManual,youareusingitwithoutpermission.4Problem2.3Awell-knownresultinestimationtheorybasedontheminimummean-squared-errorcriterionstatesthattheminimumofEeisobtainedwhentheerrorisorthogonaltoeachofthefunctionsintheseriesexpansion.Hence:Z∞−∞s(t)−KXk=1skfk(t)#f∗n(t)dt=0,n=1,2,...,K(1)sincethefunctions{fn(t)}areorthonormal,onlythetermwithk=nwillremaininthesum,so:Z∞−∞s(t)f∗n(t)dt−sn=0,n=1,2,...,Kor:sn=Z∞−∞s(t)f∗n(t)dtn=1,2,...,KThecorrespondingresidualerrorEeis:Emin=R∞−∞hs(t)−PKk=1skfk(t)ihs(t)−PKn=1snfn(t)i∗dt=R∞−∞|s(t)|2dt−R∞−∞PKk=1skfk(t)s∗(t)dt−PKn=1s∗nR∞−∞hs(t)−PKk=1skfk(t)if∗n(t)dt=R∞−∞|s(t)|2dt−R∞−∞PKk=1skfk(t)s∗(t)dt=Es−PKk=1|sk|2wherewehaveexploitedrelationship(1)togofromthesecondtothethirdstepintheabovecalculation.Note:Relationship(1)canalsobeobtainedbysimpledifferentiationoftheresidualerrorwithrespecttothecoefficients{sn}.Sincesnis,ingeneral,complex-valuedsn=an+jbnwehavetodifferentiatewithrespecttobothrealandimaginaryparts:ddanEe=ddanR∞−∞hs(t)−PKk=1skfk(t)ihs(t)−PKn=1snfn(t)i∗dt=0⇒−R∞−∞anfn(t)hs(t)−PKn=1snfn(t)i∗+a∗nf∗n(t)hs(t)−PKn=1snfn(t)idt=0⇒−2anR∞−∞Renf∗n(t)hs(t)−PKn=1snfn(t)iodt=0⇒R∞−∞Renf∗n(t)hs(t)−PKn=1snfn(t)iodt=0,n=1,2,...,KPROPRIETARYMATERIAL.cTheMcGraw-HillCompanies,Inc.Allrightsreserved.NopartofthisManualmaybedisplayed,reproducedordistributedinanyformorbyanymeans,withoutthepriorwrittenpermissionofthepublisher,orusedbeyondthelimiteddistributiontoteachersandeducatorspermittedbyMcGraw-Hillfortheirindividualcoursepreparation.IfyouareastudentusingthisManual,youareusingitwithoutpermission.5wherewehaveexploitedtheidentity:(x+x∗)=2Re{x}.DifferentiationofEewithrespecttobnwillgivethecorrespondingrelationshipfortheimaginarypart;combiningthetwoweget(1).Problem2.4Theprocedureisverysimilartotheoneforthereal-valuedsignalsdescribedinthebook(pages33-37).Theonlydifferenceisthattheprojectionsshouldconformtothecomplex-valuedvectorspace:c12=Z∞−∞s2(t)f∗1(t)dtand,ingeneralforthek-thfunction:cik=Z∞−∞sk(t)f∗i(t)dt,i=1,2,...,k−1Problem2.5Thefirstbasisfunctionis:g4(t)=s4(t)√E4=s4(t)√3=−1/√3,0≤t≤30,o.w.Then,forthesecondbasisfunction:c43=Z∞−∞s3(t)g4(t)dt=−1/√3⇒g′3(t)=s3(t)−c43g4(t)=2/3,0≤t≤2−4/3,2≤t≤30,o.wHence:g3(t)=g′3(t)√E3=1/√6,0≤t≤2−2/√6,2≤