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High-Speed Multibit Delta-Sigma ADCs with On-Line Digital Error Correction

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1

High-SpeedMultibitDelta-SigmaADCswith

On-LineDigitalErrorCorrection

P´eterKiss,Un-KuMoon,JohnT.Stonick,andG´aborC.Temes

I.INTRODUCTION

Multibitdelta-sigmamodulatorsarewidelyusedinrecentanalog-to-digitalconverter(ADC)implemen-tations[1,Chapter8].Duetotheirimprovedstability,theycanachievethesamehighsignal-to-noiseratio

()withlowerordermodulatorandloweroversamplingratio(OSR),i.e.higherspeed,thantheirsingle-bitcounterparts.Also,theuseofamultibitfeedbackrelaxesslew-rateandsettling-timerequirementsfortheopampsusedintheintegrators[2],[3].However,single-loopmultibitdelta-sigmaADCsimposestringinentrequirementsforthefeedbackmultibitdigital-to-analogconverter(DAC),whichshouldprovidethesamelinearityastheoverallconverter.

SeveraltechniqueshasbeenusedtoaddressthelinearityofthemultibitDAC.Mismatchshapingran-domizesthenonlinearityerror,andhigh-passfilterstheresultinguncorrelatederror(“noise”)signal[1,Sec-tion8.3.3].Althoughthistechniqueisverypopular,itiseffectiveathighOSRsonly,anditrequiresfairlylargechiparea,especiallyforquantizerslargerthan5bits.Calibrationtechniquesweresuccessfullyimple-mented,andimpressivemeasuredperformanceswereobtained[3],[4].However,thepreviouscalibrationtechniquesworkoffline,sotheymaynotfollowtheenviromentalchanges.Recently,abackgroundcali-brationtechniquewaspresented[5],butwhichrequireshighOSR(Note:itrequireshighOSRasitwas

filterismissing,seedetailslater).Ananalogalternative,i.e.analogmismatchcor-presented;

rectedswitched-capacitorDAC,showsapromisingsolutiontotheproblem[6],[7].

Thisreportpresentsanon-linedigitalcorrectionmethodformultibitdelta-sigmaADCs.Itoriginatesfrom[2],[3],butitworksonline,anditiseffectiveatlow-OSR,i.e.high-speed,applicationsalso.Whilethemismatch-shapingtechniquespreadsandshapesthenonlinearityerror(“dirt”),thisdigitalcorrectioncancelsitout(“vacuum-cleaning”versus“sweeping”),soitismoreeffective.Apracticalimplementationusingtheproposedtechniquemayachieveaspurious-freedynamicrange()ofover100dBfor,basedonavailablesimulation[8]results.

II.DESCRIPTION

OFTHE

DIGITALCORRECTION

Asecond-ordermultibitdelta-sigmaADC()ispresentedinFig.1.Theproposedtechniquewillbeexplainedusingthisparticularstructureasanillustrativeexample;however,itcanbeappliedtoanymultibitdelta-sigmaADC.

Thedigitaloutputofisgivenby

(1)

and,inparticular,forthecoefficientvaluesused(

,

,

,and

),eq.(1)becomes

(2)

A.Nonlinear

Thequantizationerrorofthequantizeroccursaftertheintegrators.Therefore,itishigh-passshapedbythenoisetransferfunctionofthemodulator,asshownineq.(1),andaverageddueto

arefilteredbythesame,whichusuallytheoversamplingoperation.Thenonlinearitiesof

DepartmentofElectricalandComputerEngineering,OregonStateUniversity,OR97331-3211,USA,E-mail:kpeter@ece.orst.edu,

July10,2000(reviewedAugust10,2000).

2

-11/2FirstIntegrator-12SecondIntegrator+-bitsDAC+RAM-Fig.1.Second-ordermultibitdelta-sigmaADC(

)withdigitalcorrection.

attenuatesthemtoaninsignificantlevel,comparedwiththeotherdistortionspresentinapracticalcircuit.

,thenonlinearitiesofmaylimitthemodulator’slinearitybelowHowever,atalowOSR,e.g.

thetargeted100dB,soitmustbeconsideredintheanalysis.Apossiblesolutionistoincludeintoa2-0MASHstructure(Fig.2)[9],[10],[11,Section3.C],whosefirst-stagequantizationnoisecancellationlogicburiesthenonlinearityofthefirst-stageADC()underthenoisefloor.Tohandlethisprobleminasingle-loopdelta-sigmamodulator,onemaytrytorandomizethisnonlineardistortioninasimilarwaywiththedynamic-elementmatchingtechniquepresentedin[12]ortheOSRmustbeincreased.

+++-161/16bitsFig.2.2-0MASHwithdigitally-correctedasfirststage.

B.Nonlinear

.DigitalCorrection

ThenonlinearitiesofthefeedbackmultibitDAC()enterintothesamenodeastheinputsignal,andmaycauseharmonicdistortioninthedigitaloutputofthedelta-sigmaADC.Iftheseerrorsaremeasuredbyaseconddelta-sigmaADC(,Fig.1)andaccurateestimatesareobtainedandstoredinaRAM,thenthisinformationcanbeusedfordigitallycorrectingforthenonlinearitiesintroduced

.by

seesthenonlinearitytransferfunctiontowardtheoutput.There-Accordingtoeq.(1),

shouldbefilteredwiththedigitalestimatefore,tocancelitoutfrom,themeasureddigitalestimate

of,beforebeingsubtractedfrom

(3)

Assumingthatthemeasurementofwasperformedattherequiredaccuracy,,thelinearitywillbelimitedonlybythemismatchbetweenand,whichshouldnotbecritical.of

C.Correctionwithout

Althoughitisobvioustofilterwith

usedinpreviouslymentioned[3]and[5],and

beforebeingsubtractedfrom

wasaddeddirectlyto,i.e.

,suchafilterwasnot

intheir

3

case

-

(4)(5)

Because

equals1while.Sinceisanegativefeedbacksignalofaclosed-loop

delta-sigmamodulator,approximatestheinputatlowfrequencies,whentheintegratorsprovidelarge

),theloopgainprovidesonlymodestnegativefeedback,loopgain.However,atlowOSRs(e.g.

willcontainunsuppresseddistortions.Sowill.Inconclusion,isnecessaryforand-low-OSR(i.e.high-speed)delta-sigmaADCswhichusethistypeofdigitalcorrection.

-

III.MEASURING

THE

MULTIBIT

Thedigitalcorrectiontechniquepreviouslydescribedassumedthatthestatictransferfunctionof

wasmeasuredandstoredinadigitalRAM.Acquiringanaccurateestimateoftheintegralnonlinearity(INL)errorscanbedonewithaseconddelta-sigmaADC(,Fig.1).Asecond-ordersingle-.Next,themeasurementmethodwillbebit(i.e.inherentlylinear)delta-sigmaADCwaschosenforpresentedforresistor-stringandunit-elementDACs.

A.Resistor-stringDAC

Aresistor-stringDACispresentedinFig.3(a).ThisspecificDACcanbeusedasatwo-inputandtwo-outputcircuitinthesametime.ItcanfunctionasafeedbackDACinthedelta-sigmaloop,and,simultane-.Thedigitalrampsignalselectstheconsecutivelevelsously,itsnodevoltagescanbemeasuredby

of,andholdsthemforclockperiods,untiltheresultisstored()in

(Fig.1).Notethatisthedecimated(low-passfilteredandtheRAMatalocationaddressed()by

downsampled)digitaloutputof.

istolerantonanalogcircuitimperfections,asfinitegainandcapacitormis-Theperformanceof

match,duetotheinherentrobustnessofsingle-loopdelta-sigmamodulators.Also,thegainerrorinthemeasurementpathdoesnotruintheeffectivenessofthedigitalcorrection,sinceallthevaluesarescaledinthesameway,sothenonlinearityinformationgetspreserved.Finally,theaccuracyofismainlydefinedbytheOSRusedin,thatis,bythenumberofsamplesusedintheaveraging

performsaDCmeasurement,soonlypracticalconsiderationslimitthechoiceoftheOSR.process.

Therefore,thetradeoffbetweendurationandtheaccuracyofthemeasurementmustbetailoredtoreachthedesiredperformance.

B.Unit-elementDAC

Theabovementionedmethodcanbeeasilyextendedforunit-elementDACs.Ifonespareelementisused

[5],thaneachelementcanbeconsecutivelymeasuredby,whiletheotherelementsperformin

2.thefeedback-DACoperationin

Analternativemethodformeasuringaunit-elementDACispresentedinFig.3(b).Sincetheoffsetandgainerrorsofadataconverterdonotcountinitslinearity,onecanalwaysconsiderthatandareaffectedbythesameerrorbutwithoppositesign

(6)(7)

Thescalingfactorbetweenanaloganddigitalsignalswasconsidered1inthisreport.Inageneralcase,idealineq.(5),andwouldbetheoutputofthenonlinearaffectedby.ThankstoGertCauwenberghsofJohnsHopkinsUniversity,Baltimore,MD21218.

wouldstandfortheoutputofan

4

,toDAC,tocalib.+-......resistor-stringDACdigital-to-analogconverterbuiltfromunitelements(a)(b)

Fig.3.DACsusedin

:(a)resistor-stringDAC;(b)unit-elementDAC.

Whilethefirstelementsareselectedbyaparticularvalueof,andtheiranalogoutputisusedinthe

,theremainingelementscanbeselectedby,andtheiranalogoutputismeasuredby.

should“wait”untilthesamedigitaloutputoccurs,thatis,Inordertomeasurethesame,

isvalidonlywhen.Thisleadstoanincreaseofthedurationofthemeasurementcomparedtotheresistor-stringmethod.Notethatthenonuniformstatisticsofactuallyhelpsthisprocess,becausethemoreoftenusedelementswillbemoreaccuratelymeasured,whichaidstheeffectivenessofthedigitalcorrection.

IV.PRACTICALCONSIDERATIONS

A.Nonlinear

AND

SIMULATIONRESULTS

.DigitalCorrection

,Insimulationsweconsideredasecond-order5-bit(32-level)delta-sigmaADCwithanonlinear

arepresentedinFig.4.Alineargradienterrorof0.3%[13,Sec-presentedinFig.1.TheINLsof

tion4.3.1]wasassumed;theoffsetandgainerrorswereremoved.AsshowninFig.5,theuncorrecteddigitaloutput(inFig.1)ishighlydistortedbythisnonlinear.Thetotalharmonicdistortion

dBandtheisonly38.3dBforaninputsinewaveofV(fullscaleis

V).

Ontheotherhand,theproposedmethodcompletelycancelstheseerrorsforperfecterrorcorrection(real

,nonlinear,ideal,perfectmeasurement:,andanalogcircuitsfor

),asshownby.Moreover,perfectmatchingofthefilters:-dBcanstillbeachievedeveninpracticalsituation,asshownby,anddetailedinthefollowing-sections.

B.MashIttoSeeBetter!

powerinthedigitaloutputof,anddemonstrateInordertosuppressthequantization-noise

itslinearity(withoutincreasingexcessivelythenumberofsamplesfortheFFTs),a2-0MASHwasbuiltfor

asitsfirststage,a-simulationpurposes[9],[10],[11,Section3.C].ThisMASHcontains

fromtheglobalbitquantizerasitssecondstage,andadditionalcoefficientsanddigitalfilterstocancel

(Fig.2).Thelinearityofthefirststagestillremainscritical,sothedistortionsintroducedbytheoutput

showupunchangedin.However,thenonlinearitiesofthefirst-stageADC()getsnonlinear

attenuatedoverpracticallimitsinduetothefirst-stagequantizationnoisecancellationprincipleoftheMASH.

arepresentedinFig.6whichcorrespondstothesameconditionsasthosefromFig.5.OneTheFFTsof

canobservethattheonlydifferencebetweenFigs.5and6isthatthelevelofthequantizationnoisepowerismuchlowerfortheMASH.

5

10.5v1a(v1)0−0.5−10x 100eDAC(v1)−5−351015v1202530−10−15−20051015v12025300V1corr−perfect−50SNR(V1corr−perfect)=48.1 dB @ BS2=0FFT @ 32768 samples 64× averaged −100−150000.050.10.150.20.250.30.350.40.450.5SNR(V1uncorr)=39.2 dB @ BS2=0 FFT @ 32768 samples 64× averagedV1uncorr−50−100−150000.050.10.150.20.250.30.350.40.450.5SNR(V1corr−real)=48.1 dB @ BS2=0FFT @ 32768 samples 64× averaged V1corr−real−50−100−150000.050.10.150.20.250.30.350.40.450.5V1corr−NejadSNR(V1corr−Nejad)=48.1 dB @ BS2=0FFT @ 32768 samples 64× averaged −50−100−15000.050.10.150.20.250.3Normalized frequency [f/fS]0.350.40.450.5Fig.5.UncorrectedandcorrectedoutputofforV,,nonlinear,ideal,dBand%;correctionwithperfectestimates:.Fromtoptobottom:—correctionwithperfect-filter:;—withoutcorrection;—correctionwith;-—correctionwith.-60Vmcorr−perfect−50SNR(V1corr−perfect)=84.5 dB @ BS2=0FFT @ 32768 samples 64× averaged −100−150000.050.10.150.20.250.30.350.40.450.5SNR(V1uncorr)=39.8 dB @ BS2=0 FFT @ 32768 samples 64× averagedVmuncorr−50−100−150000.050.10.150.20.250.30.350.40.450.5SNR(V1corr−real)=84.4 dB @ BS2=0FFT @ 32768 samples 64× averaged Vmcorr−real−50−100−150000.050.10.150.20.250.30.350.40.450.5Vmcorr−NejadSNR(V1corr−Nejad)=70.2 dB @ BS2=0FFT @ 32768 samples 64× averaged −50−100−15000.050.10.150.20.250.3Normalized frequency [f/fS]0.350.40.450.5Fig.6.UncorrectedandcorrectedoutputoftheMASHforthesameconditionsasinFig.5.C.Correctionwithout

Notethatdoesnotmeetthetargeted100dBlinearitybecausetheloopgainofthedelta-sigma-,anddBresults.Aquantitativemeasureofthisloopgainmodulatorispoorat

(Tab.IandFig.7).Theharmonicdistortionfromappearscanbegivenbythemagnitudeof

bythemagnitudeof,asdemonstratedinTab.II.Thiscanbeexpressedbyattenuatedin

[dB],(8)

,,representsthefrequencyofthe-thharmonic.Thenumerical3valuesfromTab.IIwerewhere

takenfromFig.6,andtheyobeyeq.(8)withlessthan1.2dBerror.

dB),ifonewantstocorrectwithout,Forexample,assumingan8-bitlinearDAC(

thanthe50dBattenuationnecessaryforthetargeted100dBlinearityrequires4.Worstcase

,sothesecondharmonicoccursatthehighestfrequencyofthesituationwasconsidered,i.e.

Notethatthecorrectquantitativemeasureofthenegativefeedbackisgivenby,whichattenuatesthenonlinearitiesofthequantizer,

andalsoforcestowardsatlowfrequencies.The“filter”assumesoversamplinganditshouldbeusedin(quantization)noisecalculations,butnotfor(distortion)tones.Therefore,asfarastheauthorisaware,thecalculationsgivenin[14,Section5.1.2.1,pp.99–100]areincorrect,i.e.toooptimistic.

ThecoarseestimatefromTab.Iwouldbe.

7

TABLEI

ATTENUATIONPROVIDEDBYTHELOOPGAINFORFIRST-,ANDSECOND-ORDERDSM.

5122561286432168

[dB]4438322620148[dB]88766452402816[dB]76675849403122[dB]1231089277624732

Legend:–first-ordernoiseshaping;–second-ordernoiseshaping(bothapplyfortones);

–noiseshapingandoversampling(appliesforquantizationnoise).

10245010185138

4

251317

236421612513

20100−10−20Attenuation, 20*log10|1−z|, 20*log10|(1−z)|−12OSR = 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1NS1 = −50, −44, −38,−32,−26,−20,−14, −8, −2, 3, 6 [dB]NS2 = −101, −88, −76,−64,−52,−40,−28,−16, −5, 6, 12 [dB]NS1+OS= −85, −76, −67,−58,−49,−40,−31,−22,−13, −4, 5 [dB]NS2+OS= −138,−123,−108,−92,−77,−62,−47,−32,−17, −2, 13 [dB]1 2 4 −30−40−50−6032 −70−80−90128 64 16 8 −1−100−110−120512 −130−1401024 10−3256 NS2NS2+OS NS1 NS1+OS 10Normalized frequency [f/fS]−210−1Fig.7.Attenuationprovidedbytheloopgainforfirst-,andsecond-orderDSM.inband.Inconclusion,themethodsof[3]and[5]canoperateonlyatsuchhighOSRstomeet

dBthetargetedspecifications.Ontheotherhand,themethodproposedinthisreportachieves

regardlessthelowOSRof4,bysimplyusing.

TABLEII

ATTENUATIONPROVIDEDBY

FOR

.

Legend:

,eq.(8)

[dB]

[dB][dB][dB]

123-0.91-40.7-74.1-43.2-30.5-23.5-0.91-71.4-98.1-0.91-71.2-97.6–normalizedfrequency

4-104.2-18.5-123.9-122.7

D.Mismatchbetween

and

Duetotheanalogcircuitimperfectionspresentin,thedigitalfilterdoesnotmatchperfectlytheactualanalog.DetailedcalculationsaregivenintheAppendixofSectionV.Fig.8

8

presentsthefrequencyresponseofboththeanalog5andtheestimateddigital6nonlinearitytransferfunctions,

and,respectively.FiniteopampgainofdBandcapacitormismatchi.e.

%wereassumedinthesimulations.Asmalldifferencecanbeobservedforhighfrequencies.of

Also,theandfromeq.(1)arerepresentedinthefrequencydomain.Thedislocationof

,aswellasthefinitegainatlowfrequenciesoftherealarethepolesandzerosofthereal

shown.

and(Fig.8)leadstoimperfecterrorcorrection.However,Themismatchbetween

eveninthepresenceofpracticalvaluesofdBand%,thecorrectedoutput-hasdB,asshowninFigs.5and6.Notethatinordertoimprovefurtherthelinearityofthe

dB),betteropampsmustbeused(dB)in.Thecorrection(e.g.,capacitormatchingbecomesthelimitingfactoronlyfordB.10NLTF(z) [dB]86420NLTF(z) −− actually seen by eDACNLTFd(z) −− digital implementation980.310−40.410−30.510200−20−40−60−80−100−120−140−1601010.5−4−210−10.10.05STF1(z) [dB]0−0.1−0.2−0.31010.5NTF1(z)022−4−0.05−0.15−0.25ideal circuitsreal circuits 10−3NTF1(z) [dB]ideal circuitsreal circuits 10−310−210−110−210−1ideal circuitsreal circuitsNTF1(z)1.520−0.5−1−2−0.5−1−2−1.5−1−0.50 0.51−1.5−1−0.50 0.511.52Fig.8.Theeffectofanalogcircuitimperfections(dBand%)inasecond-orderDSM.E.Effectsof

Next,theeffectsofstringDAC,

onthedigitalcorrectionwillbeinvestigatedbysimulations.Foraresistor-samplesneedtobeprocessedbyinordertoprovideasufficiently

:analogfilter,affectedbyanalogcircuitimperfectionsasfiniteopampgainandcapacitormismatch.

:digitalfilter,rigidestimateof,calculatedforidealcoefficientvalues,doesnotfollowtheanalogcircuitimperfections.

9

accurateestimatefordB,asshownbyinFig.9.Therefore,4.2secondsare-isclockedwithMHz.Notethatanalognecessaryforacompletebackgroundcalibration,if

dBandcapacitormismatch=0.1%,wereassumedforcircuitimperfections,suchfinitegain

.TheeffectofthegainerrorintheestimationofisshowninFig.9.However,thisdoes

waslimitedbytheanalognotaffecttheeffectivenessofthedigitalcorrection.Thelinearityof-(SectionIV-D),andnotby.Also,thetargeteddBwascircuitimperfectionsinachieved,asshownbyforsamplesinFig.9.-0V1corr−real−500.005SNR(V1corr−real)=68.2 dB @ BS2=4096eDAC(v1)0.40.5FFT @ 32768 samples 64× averaged0−0.005−0.01−0.0200.10.20.3−0.0250.005SNR(V1corr−real)=84.4 dB @ BS2=65536eDAC(v1)FFT @ 32768 samples 64× averaged0−0.005−0.01−0.0200.10.20.30.40.5−0.0250.005SNR(V1corr−real)=84.8 dB @ BS2=262144eDAC(v1)FFT @ 32768 samples 64× averaged0−0.005−0.01−0.0200.10.20.30.40.5−0.0250.005SNR(V1corr−real)=84.9 dB @ BS2=1048576eDAC(v1)FFT @ 32768 samples 64× averaged0−0.005−0.01−0.0200.10.20.30.4Normalized frequency [f/fS]0.5−0.025051015v1202530−0.015051015v1202530−0.015051015v1202530−0.015051015v1202530−0.015−100−1500V1corr−realV1corr−realV1corr−real−50−100−1500−50−100−1500−50−100−15010

whichcouldbedemonstratedsimilarlywithTab.II.However,thesedistortionsdisappearfromtheoutput

ofanidealMASH(i.e.withidealanalogcircuitsandperfectlylinear),andaregreatlyat-ofarealMASH(i.e.withanalogcircuitimperfections),asshowninFig.11.tenuatedintheoutput

Moreover,theresultsshowninFig.9remainstillvalid,thatis,thedigitalcorrectioniseffectiveevenfornonlinearandnonlinear.NotethatthehighsensitivityoftheMASHADConanalogcir-,canbeeffectivelycuitimperfections,whichleadstofirst-stagequantizationnoiseleakage,e.g.inovercomebyadaptivedigitalcompensation[15],[11],[16],[17],[18].

V.APPENDIX.ANALOGIMPERFECTIONS

IN

Theactualisaffectedbythegainandpoleerrorsoftheintegratorsduetocapacitor-ratiomis-matchofthegainstages,andfinitegainoftheopamps.Thetransferfunctionofanidealandarealswitched-capacitornoninvertinganddelayingintegrator[19,Section10.2],takingintoconsiderationthefinitegainoftheopamps,aregivenby,respectively

(10)

(11)

(12)

Inaddition,duetocapacitormismatch,

randomvariablewithatypicalstandarddeviationof

,where

isazero-centeredGaussian-distributed

,eq.(12)becomes7

(13)

Itturnsoutthatfor

fromFig.1

(14)

whereisthegainofADC/DACblock(i.e.betweenand

andcapacitormismatchaffectedbyfiniteopampgain

coefficientsshouldbemodeledasfollows

,usually,),andthecoefficientsare,asdescribedbyeq.(13).Ineq.(14)the

(15)(16)

(17)(18)

Ontheotherhand,

whichiscalculatedforidealgainfactorsgiveninFig.1.

Similarcalculationscanbefoundin[15],[18],[17],[20],[21],[22],[23],etc.

11

Quantized output, y [V]10.50−0.5−1−1−0.50Analog input, u [V]0.51Nonlinearity of the quantizer121086420x 10−3−1−0.50Analog input, u [V]0.510−50SNR(VADC1)=58.5 dBFFT @ 32768 samples 64× averagedVADC1−100−15000.050.10.150.20.250.30.350−50V1−100−15000.050.10.150.20.250.30.35SNR(V1)=48.2 dBFFT @ 32768 samples 64× averaged0−50Vm−100−15000.050.10.150.2Normalized frequency [f/fS]SNR(Vm)=84.5 dB (real MASH)FFT @ 32768 samples 64× averagedSNR(Vm)=102.6 dB (ideal MASH)FFT @ 32768 samples 64× averaged0.250.30.35Fig.11.FFTsduetothenonlinearwhereexceptforforandV,%.,ideal,dBand%for–12

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