Chapter4DigitalProcessingofContinuous-TimeSignals§4.1DigitalProcessingofContinuous-TimeSignalsDigitalprocessingofacontinuous-timesignalinvolvesthefollowingbasicsteps: (1)Conversionofthecontinuous-timesignalintoadiscrete-timesignal, (2)Processingofthediscrete-timesignal, (3)Conversionoftheprocesseddiscrete-timesignalbackintoacontinuous-timesignal§4.1DigitalProcessingofContinuous-TimeSignalsConversionofacontinuous-timesignalintodigitalformiscarriedoutbyananalog-to-digital(A/D)converterThereverseoperationofconvertingadigitalsignalintoacontinuous-timesignalisperformedbyadigital-to-analog(D/A)converter§4.1DigitalProcessingofContinuous-TimeSignalsSincetheA/Dconversiontakesafiniteamountoftime,asample-and-hold(S/H)circuitisusedtoensurethattheanalogsignalattheinputoftheA/Dconverterremainsconstantinamplitudeuntiltheconversioniscompletetominimizetheerrorinitsrepresentation§4.1DigitalProcessingofContinuous-TimeSignalsTopreventaliasing,ananaloganti-aliasingfilterisemployedbeforetheS/HcircuitTosmooththeoutputsignaloftheD/Aconverter,whichhasastaircase-likewaveform,ananalogreconstructionfilterisused§4.1DigitalProcessingofContinuous-TimeSignalsSinceboththeanti-aliasingfilterandthereconstructionfilterareanaloglowpassfilters,wereviewfirstthetheorybehindthedesignofsuchfiltersAlso,themostwidelyusedIIRdigitalfilterdesignmethodisbasedontheconversionofananaloglowpassprototypeAnti-aliasingfilterS/HA/DD/AReconstructionfilterDSPCompleteblock-diagram§4.2Samplingof

Continuous-timeSignalsLetga(t)beacontinuous-timesignalthatissampleduniformlyatt=nT,generatingthesequenceg[n]where

g[n]=ga(nT),-<n<

withTbeingthesamplingperiodThereciprocalofTiscalledthesamplingfrequencyFT,i.e.,FT=1/T§4.2.1EffectofSamplingintheFrequencyDomainwetreatthesamplingoperationmathematicallyasamultiplicationofga(t)byaperiodicimpulsetrainp(t):§4.2.1EffectofSamplingintheFrequencyDomainp(t)consistsofatrainofidealimpulseswithaperiodTasshownbelowThemultiplicationoperationyieldsanimpulsetrain:§4.2.1EffectofSamplingintheFrequencyDomaingp(t)isacontinuous-timesignalconsistingofatrainofuniformlyspacedimpulseswiththeimpulseatt=nTweightedbythesampledvaluega(nT)ofga(t)atthatinstantt=nTTheformoftheCTFTofgp(t)isgivenbyTherefore,Gp(jΩ)isaperiodicfunctionofΩconsistingofasumofshiftedandscaledreplicasofGa(jΩ),shiftedbyintegermultiplesofΩTandscaledby1/TEffectofSamplingintheFrequencyDomainKeywords!§4.2.1EffectofSamplingintheFrequencyDomainRecallNow,theCTFTGp(jW)isaperiodicfunctionofWwithaperiodWT=2p/T

∴∴§4.2.1EffectofSamplingintheFrequencyDomainAssumega(t)isaband-limitedsignalwithaCTFTGa(j)asshownbelowThespectrumP(j)ofp(t)havingasamplingperiodT=2/Tisindicatedbelow§4.2.1EffectofSamplingintheFrequencyDomainTwopossiblespectraofGp(j)areshownbelow§4.2.1EffectofSamplingintheFrequencyDomainItisevidentfromthetopfigureonthepreviousslidethatifT>2m,thereisnooverlapbetweentheshiftedreplicasofGa(j)generatingGp(j)

Ontheotherhand,asindicatedbythefigureonthebottom,ifT<2m,thereisanoverlapofthespectraoftheshiftedreplicasofGa(j)generatingGp(j)§4.2.1EffectofSamplingintheFrequencyDomainIfT>2m,ga(t)canberecoveredexactlyfromgp(t)bypassingitthroughanideallowpassfilterHr(j)withagainTandacutofffrequencycgreaterthanmandlessthanT-masshownbelow§4.2.1EffectofSamplingintheFrequencyDomainThespectraofthefilterandpertinentsignalsareshownbelow§4.2.1EffectofSamplingintheFrequencyDomainOntheotherhand,ifT<2m,duetotheoverlapoftheshiftedreplicasofGa(j),thespectrumGp(j)cannotbeseparatedbyfilteringtorecoverGa(j)becauseofthedistortioncausedbyapartofthereplicasimmediatelyoutsidethebasebandfoldedbackoraliasedintothebaseband§4.2.1EffectofSamplingintheFrequencyDomainSamplingtheorem-Letga(t)beaband-limitedsignalwithCTFTGa(j)=0for||>mThenga(t)isuniquelydeterminedbyitssamplesga(nT),-nif

T2m where

T=2/TKeywords!§4.2.1EffectofSamplingintheFrequencyDomainTheconditionT2misoftenreferredtoastheNyquistconditionThefrequencyT/2isusuallyreferredtoasthefoldingfrequency★§4.2.1EffectofSamplingintheFrequencyDomainThehighestfrequencymcontainedinga(t)isusuallycalledtheNyquistfrequencysinceitdeterminestheminimumsamplingfrequencyT=2mthatmustbeusedtofullyrecoverga(t)fromitssampledversionThefrequency2miscalledtheNyquistrate§4.2.1EffectofSamplingintheFrequencyDomainThetermofthepreviousequationfork=0isthebasebandportionofGp(j),andeachoftheremainingtermsarethefrequencytranslatedportionsofGp(j)

ThefrequencyrangeiscalledthebasebandorNyquistbandKeywords!§4.2.1EffectofSamplingintheFrequencyDomainOversampling-ThesamplingfrequencyishigherthantheNyquistrateUndersampling-ThesamplingfrequencyislowerthantheNyquistrateCriticalsampling-ThesamplingfrequencyisequaltotheNyquistrateNote:Apuresinusoidmaynotberecoverablefromitscriticallysampledversion§4.2.1EffectofSamplingintheFrequencyDomainIndigitaltelephony,a3.4kHzsignalbandwidthisacceptablefortelephoneconversationHere,asamplingrateof8kHz,whichisgreaterthantwicethesignalbandwidth,isused§4.2.1EffectofSamplingintheFrequencyDomainInhigh-qualityanalogmusicsignalprocessing,abandwidthof20kHzhasbeendeterminedtopreservethefidelityHence,incompactdisc(CD)musicsystems,asamplingrateof44.1kHz,whichisslightlyhigherthantwicethesignalbandwidth,isused§4.2.1EffectofSamplingintheFrequencyDomainExample4.1(p177)-Considerthethreecontinuous-timesinusoidalsignals:TheircorrespondingCTFTsare:§4.2.1EffectofSamplingintheFrequencyDomainThesethreetransformsareplottedbelow§4.2.1EffectofSamplingintheFrequencyDomainThesecontinuous-timesignalssampledatarateofT=0.1sec,i.e.,withasamplingfrequencyT=20rad/secThesamplingprocessgeneratesthecontinuous-timeimpulsetrains,g1p(t),g2p(t),andg3p(t)TheircorrespondingCTFTsaregivenby

§4.2.1EffectofSamplingintheFrequencyDomainPlotsofthe3CTFTsareshownbelow§4.2.1EffectofSamplingintheFrequencyDomainThesefiguresalsoindicatebydottedlinesthefrequencyresponseofanideallowpassfilterwithacutoffatc=T/2=10andagainT=0.1TheCTFTsofthelowpassfilteroutputarealsoshowninthesethreefiguresInthecaseofg1(t),thesamplingratesatisfiestheNyquistcondition,hencenoaliasing§4.2.1EffectofSamplingintheFrequencyDomainMoreover,thereconstructedoutputispreciselytheoriginalcontinuous-timesignalIntheothertwocases,thesamplingratedoesnotsatisfytheNyquistcondition,resultinginaliasingandthefilteroutputsareallequaltocos(6pt)§4.2.1EffectofSamplingintheFrequencyDomainNote:Inthefigurebelow,theimpulseappearingatΩ=6πinthepositivefrequencypassbandofthefilterresultsfromthealiasingoftheimpulseinG2(jΩ)atΩ=-14πLikewise,theimpulseappearingatΩ=6πinthepositivefrequencypassbandofthefilterresultsfromthealiasingoftheimpulseinG3(jΩ)atΩ=26π§4.2Samplingof

Continuous-timeSignalsNow,thefrequency-domainrepresentationofga(t)isgivenbyitscontinuos-timeFouriertransform(CTFT):

Thefrequency-domainrepresentationofg[n]isgivenbyitsdiscrete-timeFouriertransform(DTFT):§4.2.1EffectofSamplingintheFrequencyDomainWenowderivetherelationbetweentheDTFTofg[n]andtheCTFTofgp(t)TothisendwecomparewithAndmakeuseofg[n]=ga(nT),-∞<n<∞（4.3）（4.6）§4.2.1EffectofSamplingintheFrequencyDomainObservation:WehaveG(ejω)=Gp(jΩ)|Ω=ω/TOr,equivalently,Gp(jΩ)=

G(ejω)|ω=ΩTFromtheaboveobservationand（4.14a）§4.2.1EffectofSamplingintheFrequencyDomainWearriveatthedesiredresultgivenby§4.2.1EffectofSamplingintheFrequencyDomainTherelationderivedonthepreviousslidecanbealternatelyexpressedasFromOrfromItfollowsthatG(ejω)isobtainedfromGp(jΩ)byapplyingthemappingΩ=ω/T★★Keywords!§4.2.1EffectofSamplingintheFrequencyDomainAnotherExample4.2inPage179TheeffectofsamplinginthefrequencydomaincanbeinvestigatedusingMATLAB

Anexponentiallydecayingcontinuous-timesignalissampledattwodifferentrates.Infigure4.7-4.9,wecompareitstwospectrumsampledby2Hzand2/3Hz.§4.2.2RecoveryoftheAnalogSignalTheimpulseresponsehr(t)ofthelowpassreconstructionfilterisobtainedbytakingtheinverseDTFTofHr(j)

^Wenowderivetheexpressionfortheoutput oftheideallowpassreconstructionfilterHr(j)asafunctionofthesamplesg[n]§4.2.2RecoveryoftheAnalogSignalThus,theimpulseresponseisgivenbyTheinputtothelowpassfilteristheimpulsetrain

gp(t):§4.2.2RecoveryoftheAnalogSignalSubstitutinghr(t)=sin(ct)/(Tt/2)intheaboveandassumingforsimplicity

c=T/2=/T,weget^*^^Therefore,theoutputoftheideallowpassfilterisgivenby:whichiscalledPoissonsumformula§4.2.2RecoveryoftheAnalogSignalTheidealbandlimitedinterpolationprocessisillustratedbelowIllustrationofPoissonsumformula§4.2.2RecoveryoftheAnalogSignalItcanbeshownthatwhenΩc=ΩT/2inhr(t)=sin(Ωct)/(ΩTt/2)hr(0)=1andhr(nT)=0forn≠0Asaresult,fromWeobserveForallintegervaluesofrintherange-∞<r<∞§4.2.2RecoveryoftheAnalogSignalTherelation

holdswhetherornottheconditionofthesamplingtheoremissatisfiedHowever,ForallvaluesoftonlyifthesamplingfrequencyΩTsatisfiestheconditionofthesamplingtheorem§4.2.3

ImplicationoftheSamplingProcessConsideragainthethreecontinuous-timesignals:g1(t)=cos(6t),g2(t)=cos(14t),andg3(t)=cos(26t)TheplotoftheCTFTG1p(j)ofthesampledversiong1p(t)ofg1(t)isshownbelow§4.2.3

ImplicationoftheSamplingProcessFromtheplot,itisapparentthatwecanrecoveranyofitsfrequency-translatedversionscos[(20k6)t]outsidethebasebandbypassingg1p(t)throughanidealanalogbandpassfilterwithapassbandcenteredat=(20k6)§4.2.3

ImplicationoftheSamplingProcessForexample,torecoverthesignalcos(34pt),itwillbenecessarytoemployabandpassfilterwithafrequencyresponse whereDisasmallnumber§4.2.3

ImplicationoftheSamplingProcessLikewise,wecanrecoverthealiasedbasebandcomponentcos(6pt)fromthesampledversionofeitherg2p(t)org3p(t)bypassingitthroughanideallowpassfilterwithafrequencyresponse§4.2.3

ImplicationoftheSamplingProcessThereisnoaliasingdistortionunlesstheoriginalcontinuous-timesignalalsocontainsthecomponentcos(6pt)Similarly,fromeitherg2p(t)org3p(t)wecanrecoveranyoneofthefrequency-translatedversions,includingtheparentcontinuous-timesignalg2(t)org3(t)asthecasemaybe,byemployingsuitablefilters§4.3SamplingofBandpassSignalsTheconditionsdevelopedearlierfortheuniquerepresentationofacontinuous-timesignalbythediscrete-timesignalobtainedbyuniformsamplingassumedthatthecontinuous-timesignalisbandlimitedinthefrequencyrangefromDCtosomefrequencyTSuchacontinuous-timesignaliscommonlyreferredtoasalowpasssignal§4.3SamplingofBandpassSignalsThereareapplicationswherethecontinuous-timesignalisbandlimitedtoahigherfrequencyrangeL||

HwithL>0Suchasignalisusuallyreferredtoasthebandpasssignal

Topreventaliasingabandpasssignalcanofcoursebesampledatarategreaterthantwicethehighestfrequency,i.e.byensuring

T

2H§4.3SamplingofBandpassSignalsHowever,duetothebandpassspectrumofthecontinuous-timesignal,thespectrumofthediscrete-timesignalobtainedbysamplingwillhavespectralgapswithnosignalcomponentspresentinthesegapsMoreover,ifHisverylarge,thesamplingratealsohastobeverylargewhichmaynotbepracticalinsomesituations§4.3SamplingofBandpassSignalsAmorepracticalapproachistouseunder-samplingLet=H-LdefinethebandwidthofthebandpasssignalAssumefirstthatthehighestfrequencyHcontainedinthesignalisanintegermultipleofthebandwidth,i.e.,

H

=M()＿＿＿＿＿＿＿＿＿§4.3SamplingofBandpassSignalsWechoosethesamplingfrequencyTtosatisfytheconditionT

=2()=2H/M whichissmallerthan2H,theNyquistrateSubstitutetheaboveexpressionin★§4.3SamplingofBandpassSignalsAsbefore,Gp(j)consistsofasumofGa(j)andreplicasofGp(j)shiftedbyintegermultiplesoftwicethebandwidthDWandscaledby1/TTheamountofshiftforeachvalueofkensuresthattherewillbenooverlapbetweenallshiftedreplicasThisleadstonoaliasing§4.3SamplingofBandpassSignalsFigurebelowillustratetheideabehind00§4.3SamplingofBandpassSignalsAscanbeseen,ga(t)canberecoveredfromgp(t)bypassingitthroughanidealbandpassfilterwithapassbandgivenbyL||

H andagainofTNote:Anyofthereplicasinthelowerfrequencybandscanberetainedbypassing throughbandpassfilterswithpassbandsL-k()||

H

-k(),1

kM-1

providingatranslationtolowerfrequencyranges§4.7Sample-and-HoldCircuitFigure4.36ThebasicS/HCircu