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1、(Partial)SolutionstoAssignment4pp.81-82DiscreteFourierSeries(DFS)DiscreteFourierTransform(DFT),k=0,1,...N-1,n=0,1,...N-1DiscreteTimeFourierTransform(DTFT)isperiodicwithperiod=2πFourierSeries(FS)FourierTransform(FT)----------------------------------------------------2.1ConsiderasinusoidalsignalQ2.1
2、Considerasinusoidalsignalthatissampledatafrequency=2kHza).Determineanexpressoinforthesampledsequence,anddetermineitsdiscretetimeFouriertransformb)Determinec)Re-computefromandverifythatyouobtainthesameexpressionasin(a)a).ans:=whereandUsingtheformular:b)ans:wherec).ans:Letbethesamplefunction.TheFour
3、iertransformofisUsingtherelationshiporwhereConsideronlytheregionwhere(orthereforewhereEND-----------------------------2.3Foreachshown,determinewhereisthesampledsequence.Thesamplingfrequenceisgivenforeachcase.(b)Hz(d)Hztheory:therelationshipbetweenDTFTandFTiswhereorb.ans:d.ans:omitted(usingthesamem
4、ethodasabove)----------------------------------------------------2.4Inthesystemshown,letthesequencebeandthesamplingfrequencybekHz.Alsoletthelowpassfilterbeideal,withbandwidth(a).DetermineanexpressionforAlsosketchthefrequencyspectrum(magnitudeonly)withinthefrequencyrange(b)Determinetheoutputsignal(
5、a)ansFromclassnotes,wehavewhereisanZOHinterpolationfunctionandWecanwriteFirstly,tofindwhereItcanbefoundasSecondly,findThiscanbesolvedeitherbyFTorDTFT.WecanwritewhereandUsingtheformula:wehaveUsingtheformula,:wehavefromDTFTofy[n]Notetheaboveexpressionistwopulsesatand-thescalingfactoris:whereTherefor
6、e,where(b)ans:AftertheidealLPF,theFouriertransformofTakeinverseFouriertransformof,theoutputsignalis:NoteboththeandθtermsareintroducedbyZOHfunctionwhereisintroducedbecauseisnon-idealandθrepresentsthedelayof----------------------------------------------------Q2.5.WewanttodigitizeandstoreasignalonaCD
7、,andthenreconstructitatalatertime.LetthesignalandletthesamplingfrequencyHz.(a)Determinethecontinuoustimesignalafterthereconstruction.(a)ans:Assuming(ZOH+idealLPF)isused.Thisproblemcanbesolvedbyusingtheresultsdire
