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1、SolutionsManualforDigitalCommunications,5thEdition(Chapter2)1PreparedbyKostasStamatiouJanuary11,20081PROPRIETARYMATERIAL.cTheMcGraw-HillCompanies,Inc.Allrightsreserved.NopartofthisManualmaybedisplayed,reproducedordistributedinanyformorbyanymeans,withoutthepriorwrittenpermissionofthepub
2、lisher,orusedbeyondthelimiteddistributiontoteachersandeducatorspermittedbyMcGraw-Hillfortheirindividualcoursepreparation.IfyouareastudentusingthisManual,youareusingitwithoutpermission.2Problem2.1a.Z1∞x(a)xˆ(t)=daπ−∞t−aHence:R1∞x(a)−xˆ(−t)=−daπ−∞−t−a1R−∞x(−b)=−(−db)π∞−t+b1R∞x(b)=−dbπ−∞−
3、t+bR1∞x(b)=db=ˆx(t)π−∞t−bwherewehavemadethechangeofvariables:b=−aandusedtherelationship:x(b)=x(−b).b.Inexactlythesamewayasinpart(a)weprove:xˆ(t)=ˆx(−t)c.x(t)=cosωt,soitsFouriertransformis:X(f)=1[δ(f−f)+δ(f+f)],f=2πω.020000Exploitingthephase-shiftingproperty(2-1-4)oftheHilberttransform:
4、Xˆ(f)=1[−jδ(f−f)+jδ(f+f)]=1[δ(f−f)−δ(f+f)]=F−1{sin2πft}0000022jHence,ˆx(t)=sinω0t.d.Inasimilarwaytopart(c):11x(t)=sinω0t⇒X(f)=[δ(f−f0)−δ(f+f0)]⇒Xˆ(f)=[−δ(f−f0)−δ(f+f0)]2j21−1⇒Xˆ(f)=−[δ(f−f0)+δ(f+f0)]=−F{cos2πω0t}⇒xˆ(t)=−cosω0t2e.Thepositivefrequencycontentofthenewsignalwillbe:(−j)(−j)X
5、(f)=−X(f),f>0,whileˆˆthenegativefrequencycontentwillbe:j·jX(f)=−X(f),f<0.Hence,sinceX(f)=−X(f),wehave:xˆˆ(t)=−x(t).f.SincethemagnituderesponseoftheHilberttransformerischaracterizedby:
6、H(f)
7、=1,wehavethat:Xˆ(f)=
8、H(f)
9、
10、X(f)
11、=
12、X(f)
13、.Hence:Z∞2Z∞2Xˆ(f)df=
14、X(f)
15、df−∞−∞PROPRIETARYMATERIAL.cTheM
16、cGraw-HillCompanies,Inc.Allrightsreserved.NopartofthisManualmaybedisplayed,reproducedordistributedinanyformorbyanymeans,withoutthepriorwrittenpermissionofthepublisher,orusedbeyondthelimiteddistributiontoteachersandeducatorspermittedbyMcGraw-Hillfortheirindividualcoursepreparation.Ifyou
17、areastudentusingthisManual,youareusingitwithoutpermission.3andusingParseval’srelationship:ZZ∞∞xˆ2(t)dt=x2(t)dt−∞−∞g.Fromparts(a)and(b)above,wenotethatifx(t)iseven,ˆx(t)isoddandvice-versa.Therefore,R∞x(t)ˆx(t)isalwaysoddandhence:x(t)ˆx(t)dt=0.−∞Problem2.21.Usingrelations11X(f)=Xl(f−f0
