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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