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1、DSPImplementationofCommunicationSystemsECPE4654Experiment5CarrierRecoveryUsingaSecondOrderCostasLoopApril1,2002XuanChiIntroductionPhase-lockloops(PLLs)havebeenoneofthebasicbuildingblocksinmoderncommunicationsystems.Theyhavebeenwidelyusedincommunications,multimedia
2、andmanyotherapplications.ThetheoryandmathematicalmodelsusedtodescribePLLscomeintwotypes:linearandnon-linear.Non-lineartheoryisoftencomplicatedanddifficulttodealwithinreal-worlddesigns.TherearemanykindsofPhaseLockLoops;theCostasLoop,whichisnamedbyJ.P.Costas,apionee
3、rinsynchronouscommunications,ischosenforthisexperiment.Thereasonisthattheimplementationisquitesimpleandthestructureisverypowerfulandusefulinmanysituations.ObjectiveTheobjectiveofexperiment5istolearnabouttheCostasLoopandhowitworks.Todothis,wewilldesignandimplementa
4、CostasLooptorecoveracarriermodulateAMandBPSKsignal.TheoryA.HilbertTransform:Theoretically,aHilberttransformimpartsa-p/2phaseshiftoftheinputsignalwithoutmodifyingthemagnitudeoftheinputsignal.AHilberttransformerisafilterthatimplementsaHilberttransform.Thetransformer
5、isdefinedbythefrequencyresponsegivenin(1)andillustratedinFigure1.ì-j...w>0HH(w)=-jsgn(w)=íîj....w<0H(w)=1HjHH(w)0w-jFigure1.HilbertTransformFrequencyResponse.Itisundefinedatw=0,althoughgenerallytreatedasH(w=0)=0.FornotationalHpurposeswewilldenotetheHilberttransfor
6、masH{}andtheoutputoftheHilbert)transformofthesignalx(t),H{x(t)},asx(t).2Whenabasebandmessagesignal,m(t),istransmittedonacarrieroffrequencyw,cthefollowingrelationshipalsoholdsH{m(t)cos(wt+q)}=m(t)cos(wt+q-p/2)ccppj-jThiscanbequicklyverifiedbyutilizingtherelationshi
7、psj=e2and-j=e2.ThusinthefrequencydomaintheoutputofaHiberttransformisgivenby11jj(wcc+q)-+(wq)=-+jM(ww)ejMe()22pp11jj(w+q)-jj-+(wq)=+M(ww)eecc22Mee()22æppöæö11jjçwcc+q-÷-ç÷wq+-=+M(ww)eè22øMe()èø22InordertoimplementaHilberttransform,weneedtoexpressitinthetimedomain.T
8、heimpulseresponseofanidealHilberttransformcanbefoundasfollows.p1jnwhHH(n)=òHe(w)2p-p0p1æöjwwnjn=-ç÷òòjeje2pèø-p02ì2sin(pn/2)ï,0n¹=ípnïî0,0n=ThusanidealH
