數(shù)字信號處理英文版課件：Chap 4 Digital Processing of CT Signals

Chap4DigitalProcessingofCTSignals

Discrete-TimeSignalProcessingofCTS;SamplingofCTSignals;AnalogLowpassFilterDesign;1Mostsignalsintherealworldarecontinuousintime;4.1IntroductionDTsignalprocessingalgorithmsarebeingusedincreasingly;DigitalprocessingofaCTsignalinvolves3basicsteps:SampleaCTsignalintoaDTsignal;

(analog-to-digital(A/D)

converter)ProcesstheDTsignal(binaryword);ConverttheprocessedDTsignalbackintoCTsignal.

(digital-to-analog(D/A)

converter)2SimplifiedBlockdiagramofaCTsignalprocessedbyDTsystemSincetheA/Dconversionusuallytakesafiniteamountoftime,asample-and-holdcircuitisusedtoensurethattheanalogsignalattheinputoftheA/Dconverterremainsconstantinamplitudeuntiltheconversioniscompletetominimizetheerrorinitsrepresentation;

C/DConverterDiscrete-TimeProcessor

D/CConverterFig.4.2BlockdiagramofCTsignalprocessedbyDTsystem3

OtheradditionalcircuitsTopreventaliasing,ananaloganti-aliasingfilterisemployedbeforetheS/Hcircuit;TosmooththeoutputsignaloftheD/Aconverter,whichisastaircase-likewaveform,ananalogreconstructionfilterisused.SampleandholdDigitalSystemD/A

Anti-aliasingfilterA/DcompensatedreconstructionfilterFig.4.1CompletedblockdiagramrepresentationofaCTsignalprocessedbyDTsystem4NormalizeddigitalangularfrequencyExample

:Normalizeddigitalangularfrequency0:ThesampledDTsignal:CTsignal:5EffectofsamplingintheFrequency-Domain

Supposeacontinuous-timesignal:

ga(t)Samplingsequence:

g

[n]

samplingperiod:

T

;

samplingfrequency:FT=1/T

CTFTGa(j)ofsignal

ga(t)is:6

EffectofsamplingintheFrequency-DomainTheDTFTG(ej)ofasequenceg[n]isgivenby

RelationbetweenG(ej)andGa(j)

Conversionfromimpulsetraintodiscrete-timesequencega(t)gp(t)p(t)g[n]=ga(nT)PeriodsamplinginMathematics7

PeriodsamplinginMathematics8

CTFTGp(j)ofgp(t)

AccordingtothedefinitionofCTFTAccordingtothemodulationtheoremofCTFT9

Effectofsamplinginthefrequency-domain

Gp(j)isaperiodicfunctionoffrequencyconsistingofasumofshiftedandscaledreplicasofGa(j),

shiftedbyintegerof

Tandscaledby1/T.

Basebandsignal:

thetermontheright-handsideofEq.(4.16)for

k=0iscalledbasebandportionofGp(j).

Baseband/Nyquistband:frequencyrange

T/2<T/210Illustrationofthefrequency-domainEffectsoftime-domainsamplingNooverlap11

EffectofsamplingintheFrequency-Domain

ItisevidentfromthefigurethatifT2m,thereisnooverlapbetweentheshiftedreplicasof

Ga(j)generating

Gp(j).

IfT

2m

,ga(t)

canberecoveredexactly

fromgp(t)by

passingifthroughan

ideallowpassfilter

Hr(j)

with

gain

T

anda

cutofffrequency

c

greaterthanm

andlessthanTm

.

12

Hr(j)ga(t)gp(t)p(t)ga(t)

EffectofsamplingintheFrequency-Domain13Illustrationofthefrequency-domaineffectsoftime-domainsamplingOverlap14

EffectofsamplingintheFrequency-Domain

Ontheotherhand,ifT<2m,thereisanoverlap

ofthespectraoftheshiftedreplicasofGa(j)

generating

Gp(j).

IfT

<2m

,duetotheoverlap

oftheshiftedreplicas

of

Ga(j),thespectrumGp(j)

cannotbeseparatedbyfilteringtorecover

Ga(j)

becauseofthedistortioncausedbyapartofreplicasimmediatelyoutsidethebasebandbeing

foldedBackor

aliasedintothebaseband.15SamplingTheoremSupposethat

ga(t)beaband-limitedsignalwiththenga(t)isuniquelydeterminedbyitssamplesg[n]=ga(nT),n=0,±1,±2,···,if

Nyquistconditions:

Foldingfrequency:16SamplingTheorem

NyquistFrequency:

m

Nyquistrate:2mAndthenpassingifthroughanideallowpassfilter

Hr(j)

withagain

Tandacutofffrequency

c

satisfying

Given{g[n]=ga(nT)

},wecanrecoverexactlyga(t)bygeneratinganimpulsetrain,17SeveralSampling

Oversampling:Thesamplingfrequencyishigher

thantheNyquistrate

Undersampling:Thesamplingfrequencyislower

thantheNyquistrateCriticalsampling:ThesamplingfrequencyisequaltotheNyquistrateNote:Apuresinusoidmaynotberecoverablefromitscriticallysampledversion.18ExamplesofsamplingIndigitaltelephony,a3.4kHzsignalbandwidthisadequatefortelephoneconversation;Hence,asamplingrateof8kHz,whichisgreaterthantwicethesignalbandwidth,isused.Inhigh-qualityanalogmusicsignalprocessing,abandwidthof20kHzisusedforfidelity;Hence,inCDmusicsystems,asamplingrateof44.1kHz,whichisslightlyhigherthantwicethesignalbandwidth,isused.19

ExamplesofsamplingExample4.3Consider3CTsinusoidalsignals:ThecorrespondingDTFTsare:TheyaresampledatarateofT=0.1sec,orsamplingfrequencyT=20rad/sec.20TheCTFTofthethreesignals:TheCTFTofthesampledimpulsetrains:

Commentsonexample4.3

Inthecaseofg1(t),thesamplingratesatisfiestheNyquistconditionandthereisnoaliasing;ThereconstructedoutputispreciselytheoriginalCTsignalg1(t);

Intheothertwocases,thesamplingratedoesnotsatisfytheNyquistcondition,resultinginaliasing,andoutputssreallequaltothealiasedsignalg1(t)=cos(6t);23

InthefigureofG2p(j),theimpulseappearingat

=6inthepositivefrequencypassbandofthe

lowpassfilterresultsfromthealiasingoftheimpulseinG2(j)at=14;

InthefigureofG3p(j),theimpulseappearingat

=6inthepositivefrequencypassbandofthe

lowpassfilterresultsfromthealiasingoftheimpulseinG3(j)at=26;

Commentsonexample4.324

RelationbetweenG(ej)andGa(j)Since:therefore:or:25Soor

G(ej)isobtainedfromGp(j)simplybyscalingaccordingtotherelation

RelationbetweenG(ej)andGa(j)26RecoveryoftheAnalogSignalSampleandholdDigitalSystemD/A

Anti-aliasingfilterA/DcompensatedreconstructionfilterFig.4.1DetailedblockdiagramrepresentationofaCTsignalprocessedbyDTsystem

ThelowpassreconstructionfilterHr(j):

TheinputtoHr(j)isimpulsetraingp(t);

27

Theimpulseresponsehr(t)ofthelowpassreconstructionfilterisobtainedbytakingtheinverseCTFTofHr(j):RecoveryoftheAnalogSignal28TheoutputofHr(j)isgivenbyga(t)ga(t)Withassuming:c=T/2=/T.RecoveryoftheAnalogSignal29

4.3SamplingofBandpassSignals

BandpassCTSignalBandwidth:=HL;Assuming:H=M?(),Misaninteger;Samplingrate:T=2()=2H/M;CTFTofthesampledimpulsetrain:30

IllustrationofBandpassSamplingT=2()NoaliasingBandpassfilterRecoverthe

bandpass

signal31

FrequencytranslationFrequencytranslation:Anyofthereplicasinthelowerfrequencybandscanberetainedbypassinggp(t)throughbandpass

filterswithpassbands:providingatranslationoftheoriginalbandpass

signaltolowerfrequencyranges.32

4.4AnalogLowpassFilterDesignFilterSpecification

:

Passband:

Stopband:

Passbandedgefrequency:p;

Stopbandedgefrequency:s.33Frequency-DomainCharacterizationoftheLTIDTSystem

Peakpassband

ripple:

p=20log10(1p)

dB(4.35)Minimumstopband

attentuation:

s=20log10(s)

dB(4.36)RipplesareusuallyspecifiedindBas:

Peakripplevalueinthepassband:p;Peakripplevalueinthestopband

:s.34

Normalizedspecificationsforanalog

lowpassfilter

Themaximumvalueofthemagnitudeinthepassbandisassumedtobeunity(1);

Passbandripple:Themaximumvalueofthemagnitudeinthepassband

Maximumstopbandripple:35TwoadditionalparametersTransitionratio/selectivityparameter:

k<1forlowpassfilterDiscriminationparameter:usually,36ButterworthApproximation

N-thorderbutterworthfilter:alsocalledamaximallyflatmagnitudefilterGainindB:Atdc,i.e.,

=0:At

=c:c3dBcutofffrequency.37Typicalmagnituderesponsewithc=1Twoparameters:the3-dBcutofffrequency

c

andtheorderN

completelycharacterizeaButterworthfilter.candNaredeterminedfrom:p,,s,1/A.38

ObtaincandN

Solvetheequationsof(4.40)39

TransferfunctionofButterworth

lowpassfilterwhereThedenominatorDN(s)isknownastheButterworthpolynomialoforderN.40

ExampleanaloglowpassButterworthFilterExample4.8DeterminethelowestorderofatransferfunctionHa(s)havingamaximallyflatcharacteristicwitha1-dBcutofffrequencyat1kHzandaminimumattenuationof40dBat5kHz.Solution:Obtain:41ObtainA:OrderN:LetNbetheminimuminteger,soN=4.ChebyshevApproximation

Type1ChebyshevApproximation

:

ChebyshevpolynomialoforderN:

orrecurrencerelationofChebyshevpolynomial:43

TypicalType1Chebyshev

lowpassfilter

passbandripple:

stopbandattenuation:

1/Aats44

ObtainNandtransferfunctionPolepl

oftransferfunctionHa(s)

45

Obtaintransferfunction

ChebyshevIfiltertransferfunction:46Type2ChebyshevApproximation

Type2ChebyshevApproximation

:(4.52)47

ExampleofChebyshevIIlowpassFilterExample4.9DeterminetheminimumorderNrequiredtodesignalowpassfilterwithatype1Chebyshevortype2Chebyshev

(specifications:a1-dBcutofffrequencyat1kHzandaminimumattenuationof40dBat5kHz).Solution:Obtain:48ObtainA:OrderN:LetNbetheminimuminteger,soN=3.Note:

ChebyshevorderislowerthanButterworthorder.4.5DesignofothertypeanalogfiltersSpectraltransformationmethodisusedtodesignothertypesoffilters;Stepsfordesignothertypesoffilters;Step1:DevelopthespecificationsofaprototypeanaloglowpassfilterHLP(s)

fromthespecificationsoftheDesiredanalogfilterHD(s)

usingfrequencytransformation;Step2:Designtheprototypeanaloglowpassfilter;Step3:DeterminethetransferfunctionHD(s)

ofthedesiredanalogfilterbyapplyingtheinverseoffrequencytransformationtoHLP(s).50

MarksoftheprototypeanddesiredfiltersToeliminatetheconfusion,Sign:

theprototypeanaloglowpassfilter:HLP(s)

Laplacetransformvariables

thedesiredanalogfilter:HD(s)

Laplacetransformvariable

sTransformbetweenHLP(s)andHD(s)

or51thepassbandedgefrequencyofdesiredanaloghighpassfilterHHP(s).

TransformationtotheHighpassFilterwherethepassbandedgefrequencyofprototypeanaloglowpassfilterHLP(s);

Ontheimaginaryaxis,52Mappingofimaginaryaxisins-domainto-domainsLowpassfilterpassbandHIghpassfilterpassband53

ExampleofHighpassfilterdesignExample4.18DesignananalogButterworthhighpassfilter,withspecifications:Solution:Passbandedgefrequency:4kHz,passbandripple:0.1dB;Stopbandedgefrequency:1kHz,stopbandattenuation:40dB;Forprototy