... RelationshipbetweentheContinuousandDiscreteTimeSpectraBecauseDTsignalsoftenoriginatebysamplingCTsignals,itisimportanttodeveloptherelationshipbetweentheoriginalspectrumoftheCTsignalandthespectrumoftheDTsignalthatresults.First,c1999byCRCPressLLCtheCTFTisappliedtotheCTsamplingmodel,andthepropertieslistedaboveareusedtoproducethefollowingresult:F{sa(t)}=Fs(t)∞n=−∞δ(t−nT)= ... (1.17b)whereCisacounterclockwisecontourofintegrationwhichisaclosedpathcompletelycontainedwithintheregionofconvergenceofS(z).RecallthattheDTFTwasobtainedbytakingtheCTFouriertransformoftheCTsamplingmodelrepresentedbysa(t).Similarly,thebilateralz-transformresultsbytakingthebilateralLaplacetransformofsa(t).Ifthelowerlimitonthesummationof(1.17a)istakentoben=0,then(1.17a )and( 1.17b)becometheone-sidedz-transform,whichistheDTequivalentoftheone-sidedLTforCTsignals.ThehierarchicalrelationshipamongthesevariousconceptsforDTsystemsisdiscussedlaterinthischapter,whereitwillbeshownthatthefamilystructureoftheDTfamilytreeisidenticaltothatoftheCTfamily.ForeveryCTtransformintheCTworldthereisananalogousDTtransformintheDTworld,andviceversa.c1999byCRCPressLLCTABLE1.3 ... (1.19b)whereω=ωTisthenormalizedDTfrequencyaxisexpressedinradians.NotethatS(ejωT)=S(ejω)consistsofaninfinitenumberofreplicasoftheCTspectrumS(jω),positionedatintervalsof(2π/T)ontheωaxis(oratintervalsof2πontheωaxis),asillustratedinFig.1.8.NotethatifS(jω)isbandlimitedwithabandwidthωc,andifTischosensufficientlysmallsothatωs>2ωc,thentheDTspectrumisacopyofS(jω)(scaledby1/T)inthebaseband.Thelimitingcaseofωs=2ωciscalledtheNyquistsamplingfrequency.WheneveraCTsignalissampledatorabovetheNyquistrate,noaliasingdistortionoccurs(i.e.,thebasebandspectrumdoesnotoverlapwiththehigher-orderreplicas)andtheCTsignalcanbeexactlyrecoveredfromitssamplesbyextractingthebasebandspectrumofS(ejω)withanideallow-passfilterthatrecoverstheoriginalCTspectrumbyremovingallspectralreplicasoutsidethebasebandandscalingthebasebandbyafactorofT.1.5...