Idle-Tones-and-Dithering-in-Sigma-Delta-Modulators

IdleTonesandDitheringinSigma-DeltaModulators{anActualOverview{AndreasBuslehnerandPeterSÄoser¤Abstract|Thispaperdealswithidletonesandhowtoovercomethem.Idletonesareunwanteddiscretespectralpeaksinthesignalbandwhichoc-curduetoperiodicpatternsatthemodulatorout-put.Firstmethodsarediscussedhowidletonescanbedetected.Furthertheoriginofidletonesinsigma-delta(§¢)modulatorsisalluded.Intheendthispaperpresentscommonmethodshowtoreduceoreveneliminatetonesin§¢modulatorsnameddithering.Thepurposeofditheringistodecorre-lateandwhitenthepowerspectraldensityspectrumofthequantizationerror.Furthervariousimple-mentationsofditheringin§¢analog-to-digitalcon-verters(ADCs)areevaluatedandtheprosandconsarediscussed.Keywords|Sigma-deltamodulators,noise-shap-ingmodulators,dithering,idletones.1IntroductionALTHOUGHwhitenoiseapproximationisatbestsuspiciousandatworstsimplywrongin§¢modulatoranalysis,estimationscanbedoneandprin-ciplesof§¢modulatorscanbeexplained.Surpris-inglysimulationsandactualcircuitimplementationsshowgoodagreementswithadditivewhitenoiseanal-ysis.Unfortunately§¢modulatorsalsooftenexhibitunstablebehaviornotpredictablebyusingthewhitenoiseanalysistechnique.Additionallythistechniquefailstopredictnon-harmonictones.Thisfactises-peciallytrueforsingle-bitmodulators.Non-harmonictones,whicharealsosometimesreferredasidlechan-neltonesorpatternnoisearenotcausedbycircuitim-perfections;theyexisteveninidealsigma-deltamod-ulators.Thesetonesorrepetitivepatterns,respec-tively,ofthemodulatoroutputarenotharmonicallyrelatedandoccurasoddchirporpops[1].Idletonesareespeciallyunwantedinaudioapplications,sincehumanhearingcandetecttonesburiedbeneathwhitenoise1.Beforeinvestigatingidletones,itisimportantto¯ndmethodshowtodetectandobserveidletones.Itisinstructivetoinvestigatethefollowingexample[2]:Multiplyingarandomwhitenoisesequence,whichis¤BothauthorsarewiththeDepartmentofElectronics,GrazUniversityofTechnology(TUG),Austria,A-8010Graz,In®eldgasse12.(e-mail:¯[e-mail])1Theearcanperceivetonesthataremorethan20dBbelowtheintegratednoise°oor(reportedin[2]).0500100015002000250030003500400045005000−15−10−5051015SampleNumberAmplitudeFig.1:Multiplicationof32harmonicallyrelatedsinusoids(K=32),eachhavingthesameamplitude,withapseudo-randomnoise.independentandidenticallydistributed(i.i.d.),withthesumofharmonicsinewavesequences.Thesumoftheharmonicallyrelatedsinewavesisgivenbyxharm[n]=PKk=1cos(kn!0);where!0isthefunda-mentalfrequency.Furtheritisassumedthattheran-domwhitenoisesequencehasarectangularproba-bilitydensityfunctionboundedbetween§0:5.TheresultofthismultiplicationinthetimedomainleadstothesequenceplottedinFig.1whereperiodicpat-ternnoisesequencesrepeatedapproximatelyevery512samplescanbedetected.Thereforesomeonewouldas-sumethatpeaksoccuratmultiplesof!0inthepowerspectrum.However,lookingatthecomputedpowerspectrumofthissignal,showninFig.2,itisreveal-ingthattheFFTshowsnooutstandingperiodicityatmultiplesof!0.Thiscanbeexplainedbythefollowingconsideration:Amultiplicationoftwosignalsinthetimedomainresultsinaconvolutionofthesesignalsinthefrequencydomain.Sincetheharmonicallyre-latedsinusoidsaremultipliedwithapseudo-randomnoise,theoveralloutputsignalspectrumalsolookslikeanotherwhite-noisespectrum.Thereforethereisaneedforanothermethodwhichallowstodetectsuchperiodictones.Thesimplestmethodtoovercomethisproblemisusingtheautocorrelationfunction.Thediscreteautocorrelationisgivenforalimiteddiscretesignalsequencex(n)by[3]^©yy[m]=1QQ¡jmj¡1Xn=0x[n]x[n+m]:(1)Theoutputoftheautocorrelationsequenceat0,10−510−410−310−210−1−100−90−80−70−60−50−40−30−20NormalizedfrequencyPowerDensityindBFig.2:PowerspectrumofFig.1basedsimplyononewin-dowed128kHannwindowedFFT.Note,therearenoout-standingperiodicityatmultiplesof!0,whicharemarkedwithcircles.01000200030004000500000.20.40.60.81SampleshiftPower01000200030004000500000.20.40.60.81SampleShiftPower(b)(a)Fig.3:AutocorrelationofFig.1,where(a)iscomputedfrom128ksamplesand(b)iscomputedfrom32ksamples.©yy(0)iswellknownasthemean-squarevalueofthesignalx[n].Theautocorrelationsequenceofastochas-ticnoisesequenceshouldasymptoticallytendtowardzero,whiletheautocorrelationofaperiodicsequenceresultsinperiodicpatternsandpeaksfor©yy(6=0).However,Fig.3(a)showstheautocorrelationofthenoisemodulatedsinusoidalsequencefromFig.1.Itrevealsaperiodicityevery512samplesshifts,whichistheperiodofthefundamentalfrequency!0.Compar-ingFig.3(a)andFig.3(b)weseethatwithincreasingnumberofsamplesthedetectedpeaksoftheperiodicsequencestendstowardzero.Sofollowingimportantconclusioncanbedrawn:Idletonesin§¢modulatorscanonlybereliablyde-tectedwiththeaidoftheautocorrelationfunctionhav-ingamoderatenumberofinputsamples.2Tonesin§¢ModulatorsConsideringa¯rst-order§¢modulatorwithasmallDCinput.Thedigitaloutputofthemodulatorwillbeaperiodicsequenceof1'sand-1's.Thisrepeat-ingoutputpattern(idletone)canoccurinthebase-bandspectrum.In§¢modulatorswithmoreactiveinputsignals,theoutputsequenceistypicallymorecomplex.Nonetheless,theconceptunderlyingtonalbehavioristhatrepeatingpatternsinthequantizer