英文資料：1IntroductionTheuseofthesystem-on-chip(SoC)techniqueallowsforanunprecedenteddegreeofminiaturizationofrubidiumatomicfrequencystandards.Acompactdirectdigitalfrequencysynthesizer(DDFS)playsanimportantroleinourminiaturizedrubidiumatomicfrequencystandardsformodulated5.3125MHzsinewavegeneration.DDFSsareabletogeneratesingle-phaseorquadraturesinusoidswithexcellentfrequencyresolution,goodspectralpurity,veryfastfrequencyswitching1,andphasecontinuityonswitching.Generally,aDDFSconsistsofaphaseaccumulator,aphasetosineconverter(sineROM),andaDAC.Ateachclockpulse,thephaseincrementwordinafrequencyregisterisaddedtothephasevaluepreviouslyheldinthephaseaccumulator.Thephasevalueisgeneratedusingthemodulo2joverflowingpropertyofajbitphaseaccumulator2.Theoutputfrequencyistherateoftheoverflows,wherepisthephaseincrementword,jisthenumberofphaseaccumulatorbits,fclkisthesystemclockfrequency,andfoutistheoutputfrequency.Thepreviousconstraintisrequiredbythesamplingtheorem.ThespectralpurityoftheconventionalDDFSispartlydeterminedbytheresolutionofthevaluesstoredintheROM.ItisdesirabletoincreasetheresolutionoftheROM,butsometimeshigherresolutionmeanslargerROMsize.However,theaccessspeedandthemaximumoutputfrequencydecreaseastheROMsizeincreases.LargerROMstoragealsomeanshigherpowerconsumption,lowerreliability,andgreatlyincreasedcosts.SoitissignificanttoreducethesizeoftheROMundertheconditionofmeetingthehighresolutionrequirementoftherubidiumatomicfrequencystandard.Themostelementarytechniqueofcompressionistostoreonly/2radofsineinformation,andtogeneratetheROMsamplesforthefullrangeof2byexploitingthequarter-wavesymmetryofthesinefunction3.Thequarter-wavememorycanbefurthercompressedbyusingmodifiedSunderlandtechnique,sine-phasedifferencetechnique,Quadlineapproximation(QLA)technique,andquantizationanderrorROM(QE-ROM)technique.Thephaseofaquarterofasinewavecanbedecomposedas=+,where,andare,respectively,themostsignificantbits(MSBs),themiddlebits,andtheleastsignificantbits(LSBs).ThewholecompressionprocessinthispaperisdepictedinFig.1.A10biton-chipcurrentsteeringDACisalsoimplementedtoconvertthedigitalamplitudewordtoanequivalentanalogamplitude.2PhaseaccumulatorA32bitphaseaccumulatorisusedinthisDDFS.Figure2showstheblockdiagramofthephaseaccumulator.Thephaseaccumulatorisbasedona32bitripplecarryadder,withastringoffulladdersthatoperateonthesameclockphase.Theoutputsofthefulladdershavebuilt-inregisters,andthesumbitsfeedbackinternallytoperformaccumulation.Amultiplexer(MUX)isusedtoselectthephaseincrementwordfromthe32bitphaseincrementwordsstoredinregisters.Inordertoguaranteethephasecontinuity,theselectsignalshouldbesynthesizedwiththesystemclocksignaloftheDDFS.3Phasetosineconverter3.1SinefunctionsymmetrytechniqueThefull-periodsinewavecanbereconstructedwithonly/2radofthesineinformationbyexploitingthequarter2wavesymmetryofthesinefunction.Figure3showsthedetailsofthismethod.Sincethemostsignificanttwobitsofthephaseaccumulatorrepresentthequadrantofthesinefunction,themostsignificantbit(MSB)isusedasthesignbitoftheresult,andthesecondmostimportantbit(2ndMSB)isusedtocontrolwhetherthephaseshouldbeincreasingordecreasing4.Thelowj-2bitsofthephaseaccumulatoroutputaresenttoacomplementcontrolledbythe2ndMSBtogenerateaddressesforthequarter-sineROM.Theslopeofthesawtoothisinvertedforthesecondquadrant,asshowninFig.3.Thewave-formattheoutputofthequarter-sineROMisthequantifiedsinewave.Thefull-periodsinewaveisgeneratedattheoutputofthesecondcomplement,whichconsistsofm-1exclusive-orgatesandaninverter.Thus,theROMcapacitydecreasesattheexpenseofadditionallogic.3.2ModifiedSunderlandtechniqueToreducetheROMsize,thisDDFSusesthemodifiedSunderlandtechniquebasedonsimpletrigonometricidentities5.Thephaseaddressofaquarterofasinewavecanbedecomposedas=+,whereistheMSBs,isthemiddlebits,andistheLSBs.Thequarter-wavesinefunctionisgivenbywherejisthenumberofthephaseaccumulatoroutputbits.Equation(2)canbesimplifiedfurthertoTheinformationfromthefirsttermontherightofEq.(3)isstoredintoacoarseROM.ThesecondtermontherightofEq.(3)isstoredintoafineROM.Thismethodworksbyintroducingthe1/2LSBsoffsetsintothephaseandamplitudeofthesineROMsamples.However,significantsavingsinROMsizecanberealizedduetothesmallmagnitudesofandrelativeto.The10bitphasedataofaccumulatoroutputsisdividedintothreepartsinthisdesign.Computersimulationsdeterminetheoptimumpartitioningratio=4,=3,and=36.3.3Sine2phasedifferencetechniqueThesineamplitudeofthefirsttermontherightofEq.(3)isreducedusingthesinephasedifferencetechnique,andtwobitsofthewordlengthcanbesavedbyonlystoringthedifferenceofthesineamplitudeanditsphase.3.4QLAtechniqueTheQLAwaveformisusedtoapproximatethesine-phasedifference,whichiscalculatedusingEq.(4).TheQLAapproximationcanbeexpressedwithfollowingfourexpressions.Thedatafor0(+)/2j-11/8aregeneratedthroughshiftingdownthephase(+)/2j-2by1bit.Thedatafor1/8(+)/2j-11/4aregeneratedthroughshiftingdownthephase(+)/2j-2by2bitandbychangingthefirstandsecondMSBsofthephaseto“10”.Thedatafor0(+)/2j-11/4andthedatafor1/4(+)/2j-11/2aresymmetric.Acomplementisneededtoreconstructthesymmetricwaveform.Twobitsofwordlengthcanbesaved.cr(+)=y(+)-qla(+)(10)maxcr(+)0126maxy(+)(11)3.5QE2ROMtechniqueBasedonthecontinuityofthedatathatneedcompressing,theROMsizeoftheDDFScanbefurtherreducedusingtheQE-ROMtechnique.Inthisdesign,thesecondtermontherightofEq.(3)andcr(+)arebothcompressedusingtheQE-ROMtechnique.TheROMsizecanbereducedto2lm+2anbitsusingtheQE-ROMtechnique,wherel,m,a,andnare,respectively,thelengthoftheaddressofthequantizationROM,thelengthofthedatainthequantizationROM,thelengthoftheaddressoftheoriginaldata,andthelengthofthedataintheerrorROM.TheremaybeseveralgroupsofparametervaluesthatminimizetheROMsize.Tofindthese,thefollowingalgorithmisused.(1)Findthemaximumamplitudeoftheoriginaldataandfigureouthowmanybitsittakestorepresentthisvalue.Themaximumvalueofmcanberepresentedbymaxm.(2)Setm=maxm.(3)Setl=a,whichistheaddresslengthoftheoriginaldata.(4)Calculatethequantizedvalues.(5)CalculatetheerrorsbetweentheoriginaldataandthequantizationROMdata.(6)Determinehowmanybitsittakestorepresenttheseerrors.(7)CalculatethetotalROMsize(2lm+2an).(8)Decreaselby1.Ifl0,thengoto(9)orelserepeattheprocessfrom(4)to(7).(9)Decreasemby1.Ifm0,thengoto(10)orelserepeattheprocessfrom(4)to(7).(10)DeterminetheoptimumvaluesoftheaboveparametersthatminimizethetotalROMsize.cr(+)isstoredintoacoarseROMthatisdividedintoaquantizationROMandanerrorROMusingtheQE-ROMtechnique.TheabovecalculationsindicatethattheminimumsizeofthecoarseROMfora10bitoutputDDFSis480bit(253+273=480).ThesecondtermontherightofEq.(3)isstoredintoafineROM,whichisalsodividedintoaquantizationROMandanerrorROMusingtheQE-ROMtechnique.TheminimumsizeofthefineROMfora10bitoutputDDFSis288bit(251+272=288).SothetotalROMsizefora10bitsineoutputisonly768bit.3.6ArchitectureofthephasetosineconverterFigure4showstheblockdiagramofthephasetosineconverter,whichconsistsofcomplement,multiplexers(MUXs),ROMs,andadders.ThecomplementareusedtorecoverthefullwaveoutputfromthequartersineROMbyinvertingthephaseandamplitudeappropriately.FourcolumnMUXsandthreeaddersarealsorequired.Figure5showstheintermediateresultsduringtheapproximat