通信系统的dsp实施(1)

DSPImplementationofCommunicationSystemsECPE4654Experiment5CarrierRecoveryUsingaSecondOrderCostasLoopApril1,2002XuanChi2IntroductionPhase-lockloops(PLLs)havebeenoneofthebasicbuildingblocksinmoderncommunicationsystems.Theyhavebeenwidelyusedincommunications,multimediaandmanyotherapplications.ThetheoryandmathematicalmodelsusedtodescribePLLscomeintwotypes:linearandnon-linear.Non-lineartheoryisoftencomplicatedanddifficulttodealwithinreal-worlddesigns.TherearemanykindsofPhaseLockLoops;theCostasLoop,whichisnamedbyJ.P.Costas,apioneerinsynchronouscommunications,ischosenforthisexperiment.Thereasonisthattheimplementationisquitesimpleandthestructureisverypowerfulandusefulinmanysituations.ObjectiveTheobjectiveofexperiment5istolearnabouttheCostasLoopandhowitworks.Todothis,wewilldesignandimplementaCostasLooptorecoveracarriermodulateAMandBPSKsignal.TheoryA.HilbertTransform:Theoretically,aHilberttransformimpartsa-π/2phaseshiftoftheinputsignalwithoutmodifyingthemagnitudeoftheinputsignal.AHilberttransformerisafilterthatimplementsaHilberttransform.Thetransformerisdefinedbythefrequencyresponsegivenin(1)andillustratedinFigure1.⎩⎨⎧-=-=0....0...)sgn()()(=wHHFigure1.HilbertTransformFrequencyResponse.Itisundefinedatω=0,althoughgenerallytreatedas0)0(==wHH.FornotationalpurposeswewilldenotetheHilberttransformasH{}andtheoutputoftheHilberttransformofthesignalx(t),H{x(t)},as)(tx).0ω0ωj-jHH(ω)3Whenabasebandmessagesignal,m(t),istransmittedonacarrieroffrequency,cwthefollowingrelationshipalsoholds{})2/cos()()cos()(pqwqw-+=+ΗttmttmccThiscanbequicklyverifiedbyutilizingtherelationships2pjej=and2pjej-=-.ThusinthefrequencydomaintheoutputofaHiberttransformisgivenby()()()()222211()()2211()()2211()()22ccccccjjjjjjjjjMejMeMeeMeeMeMewqwqppwqwqppwqwq⎛⎞⎛⎞+--+-⎜⎟⎜⎟⎝⎠⎝⎠=-+=+=+InordertoimplementaHilberttransform,weneedtoexpressitinthetimedomain.TheimpulseresponseofanidealHilberttransformcanbefoundasfollows.()()()00212122sin/2,00,0jnHHjnjnhnHejejennnnpwppwwpwpppp--=⎛⎞=-⎜⎟⎝⎠⎧≠⎪=⎨⎪=⎩∫∫∫ThusanidealHilberttransform’simpulseresponseisseentobe•infiniteinduration•noncausal(astheoutputdependsonfuturevaluesofn)•antisymmetric(equalinmagnitudeandoppositeinsign)•nonzeroforonlyoddvaluesofnBecauseoftheinfiniteduration,itisnotpossibletodirectlyimplementaHilberttransform.However,itcanbecloselyapproximatedbywindowingtheimpulseresponseandimplementingthefilterusingalinearphaseFIRwithanantisymmetricimpulseresponse.Anyofanumberofdifferentwindowingmethodsareappropriate.Additionally,becausetheimpulseresponseisantisymmetric,thefrequencyresponsemustbe0atω=0forevenandoddorderfilters,and0atω=πforanoddorderfilter,thusitwillalsonotbepossibletoimplementtheHilberttransformasanallpassfilter.However,ifthesignaltobephaseshiftedisknown-tobelimitedtoarangeoffrequencies,thenthefiltercanstillbeappropriatelydesigned.4OnemethodtoimplementaHilbertTransformeris()()220.540.46cos,[0,1],10,[1,1],HnnNnevenhnnNnNnoddppg⎧⎡⎤⎛⎞-∈-⎪⎜⎟⎢⎥=--⎝⎠⎨⎣⎦⎪∈-⎩where2Ng⎢⎥=⎢⎥⎣⎦and⎢⎥⎣⎦indicatesthefloorfunction.AnothermethodfordesigningthetransformeristousetheRemezalgorithmwiththeChebyshevapproximationcriterioncanbeemployed.B.AnalyticSignal(Pre-envelope)Ananalyticsignalisdefinedastheoriginalsignalplusjtimestheoriginalsignal’sHilberttransform.Givenasignalx(t),itsanalyticsignal,x+(t)canbeconstructedasˆ()()()xtxtjxt+=+Inthefrequencydomain,theconstructionofananalyticsignalhastheeffectofeliminatingtheoriginalsignal’snegativefrequencycomponentsanddoublingthepositivefrequencycomponents.Thiscanalsobethoughtofasfoldingthenegativefrequencycomponentsintothepositivefrequencycomponentsaboutω=0.Thisresultcanbeformallyexpressedas()()()()()2020000XXXuX⎧⎪===⎨⎪⎩C.ComplexEnvelopeThecomplexenvelopeisformedbymultiplyingthepre-envelopebythecomplexexponential,tje0w-.Thishastheeffectoftranslatingthepre-envelope-w0inthefrequencydomain.Ifw0ischosenasthecenterfrequencyofthepre-envelope,thentheresultingcomplexenvelopewillbetranslatedtobaseband.Mathematically,thecomplexenvelope,()xt%,ofasignal,x(t),isdefinedas()()0jtxtxtew-+=%Inthefrequencydomainthisis()()()002XXX=+=+%5Thuswithaproperlychosenw0,theformationofthecomplexenvelopecanbeusedtodemodulateasignalmodulatedontoacarrier.D.AMExampleConsiderthereceptionofanAMsignal,x(t),withamessagesignal,m(t),modulatedontoacarrierwithamplitudeAcandfrequencyωc,andaphaseoffsetq1(t).Thissignalcanbeexpressedas()()1()cos()ccxtAmtttwq=+or()()()()()11()()12ccjttjttcxtAmteewqwq+-+⎡⎤=+⎢⎥⎣⎦ItsHilberttransformisequalto()()()()()⎥⎦⎤⎢⎣⎡+=-+--+2/2/1121ˆpqwpqwtjtjccceetmAtxItsanalyticsignalisthenjust()()()1qw++=tjccetmAtxω-ωcωcω-ωcωcωωcωωc6Supposetheanalyticsignalismultipliedby()2qw+-tjce,wherewcisthereceiver'sestimateofthecarrierfrequencyandq2isitsestimateofthecarrierphase.Presumingthecarrierfrequencyiscorrectlyestimated,thecomplexenvelopeisgivenby()()()()()()ttjttjccceetmAtx21~qwqw+-+=()()()()()ttjcetmAtx21~qq-=ThustheAMsignalistranslatedtobaseband.However,inthisexpressionthemessagesignalismultipliedbyacomplexexponential.UsingthesubstitutionsΔq(t)=q1(t)-q2(t)andejΔq(t)=cos(Δq(t))+jsin(Δq(t