Application-Notes-on-Direct-Time-Domain-Noise-Anal

July20061Version1.0ApplicationNotesonDirectTime-DomainNoiseAnalysisusingVirtuosoSpectrePurposeThisdocumentdiscussesthetheoreticalbackgroundondirecttime-domainnoisemodeling,andpresentsapracticalapproachthatgeneratesrandomnoisesignalswithgivenpowerspectraldensityinthetimedomain.Thisenablesthedirecttime-domainnoiseanalysis,ortheso-calledtransientnoiseanalysis,ofalinearornonlinearsystem.ThisdocumentfurtherdescribestheimplementationandusemodeloftransientnoiseanalysisinVirtuosoSpectre,anditsapplicationondirecttime-domainnoiseanalysisoflinearandnonlinearnoisysystems.IntroductionToassesstheimpactofdevicenoiseoncircuitbehavior,smallsignalapproximationismostcommonlyusedincommercialnoiseanalysistools.Itassumesdevicenoisecontributionsinadesignaresmallenoughthatitdoesnotaltertheoperatingpointorperiodicstateofinterest.Asaresult,noiseanalysisbecomesalinearproblem,andnoisecanbeseparatedfromsignalsbymeansofthesuperpositionprinciple.Noisepropertiesofaparticulardesigncanbesolvedinthefrequencydomainbylinearizingthecircuitequationaroundtheoperatingpointorperiodicstateofinterest.Althoughitiswidelyadoptedinvariousdesignpractices,thelimitationofthistraditionalapproachisobvious.Itproducesinaccurateresultswhennoiseislarge,and/orwhenthecircuitishighlynonlinear.Furthermore,thereisnosimplewaytoverifythevalidityofthesmall-signalassumptionthatthesignalandnoiseareindeedindependent.Theoretically,directnoiseanalysisinvolvessolvingstochasticdifferentialequations.Whenrandomforcesareaddedtoacircuit,allcircuitunknownsbecomerandomvariables.Thetaskofcircuitsimulationbecomessolvingtheprobabilitydistributionsofcircuitunknowns.Certainsimplificationhastobemadetomakethisapproachdoable.Forexample,allrandomvariablescanbeassumedGaussianprobabilitydistributionthatallowsanalyticaveraging,orthecircuitequationcanbeformulatedandapproximatedusinglowerorderdistributionmoments.Howeverevenwiththesesimplifications,theresultingnonlinearproblemistoocomplicatedtosolve,andithasnotfoundanycommercializedapplication.Whenultimateaccuracyisneeded,directnoiseanalysisinthetimedomainistheonlyviablesolution.ItenablesthedesignertoexaminehowanonlinearsystemrespondstonoiseincaseswherenoisetoApplicationNotesonDirectTime-DomainNoiseAnalysisusingVirtuosoSpectreJuly20062Version1.0signalratioislarge,andoperatingpoint,orperiodicstateofinterestmaybeaffectedbythenoise.Thisisaverycostlyapproach.Devicenoisemodelsneedtobeevaluatedandrandomnoisesourcesneedtobegeneratedateachtimesstepoftransientanalysis.Timestepisalsoforcedtobeuniformlysmallasnoisebandwidthistypicallylargerthancircuitbandwidth.Inaddition,alargenumberofrepetitivesimulationsneedtobeperformed,orthesimulationhastospanlargenumberofperiodiccyclesinordertodrawmeaningfulstatisticalcharacteristics.ThenecessityofdirectnoiseanalysisintimedomainhasbeenrecognizedbyvariousresearchersandcommercialEDAcompanies.Thekeydifferenceishowtomodelnoisesourcesintimedomainthatcanbeeasilypluggedintotransientanalysis.Variousapproacheshavebeenpublishedsofar.Bolcato,atal[1]modelseachnoisesourceasasumofsinusoidsoverthefrequencyrangeofinterest,withrandomphase,andwithamplitudeequalstothegivennoisepowerspectraldensity.Recentlyfrom[2],itwasshownthatthemagnitudeofnoisesignalinfrequencydomainisalsoarandomnumberwithRayleighprobabilitydistributionaroundgivenpowerspectraldensity,thusrandomnoisespectrumcanbeproperlygeneratedinthefrequencydomain.However,bothnoisemodelingapproachesareCPUintensive--asthetimedomainnoisesignalsareobtainedusinginverseFouriertransformation,andmemoryintensive--asalltimedomainnoisesourceshavetobepre-calculatedandstoredoverthefulldurationoftransientanalysis.Inthefollowingsections,wewillfirstsummarizethetypicaldevicenoisemodels--thermal,shot,flicker,andnoisedatatable.Thenwewilldescribeadifferentwayofprovidingtime-domainwhitenoisesourcesanditsextensiontohandlefrequencydependentnoisemodels.Ourapproachsatisfiesstricterperformanceandcapacityrequirements.Asaresult,CPUtimeusedfornoisesignalgenerationismuchlessthanthatfordeviceevaluation,additionalallocatedmemoryismuchlessthanthatusedfordevicesandmatrixsolver.Finallyandveryimportantly,devicenoisecodedevelopedinexistingSpectreCMIdevicesandVerilog-Amodulesisre-usedfortime-domainnoisemodeling,nomodelcodechangeiseverneeded.DeviceNoiseModelsDevicenoisemodelstypicallyhaveanumberofindependentnoisesourcesconnectedtothedeviceterminalsorinternalnodes.Eachnoisesourceischaracterizedbyitsnoisespectraldensity,whichisafunctionofbiasvoltageand/orfrequency.Mostoftenuseddevicenoisemodelsare:Thermalnoiseofaresistorwithconstantspectraldensity,biasdependentshotnoise,n24kTR---------=n22qv()Iv()⋅=ApplicationNotesonDirectTime-DomainNoiseAnalysisusingVirtuosoSpectreJuly20063Version1.0biasandfrequencydependentflickernoise.Inaddition,noisefilecanbeassociatedtoanindependentsourcetomodelarbitraryfrequencydependentnoiseprofile,obtainedfrommeasurementorbehavioralmodeling.Thethermalnoiseandshotnoisearebothconsideredwhitenoiseprocesses,whichimpliestheirspectraldensityisflatwithfrequencyandthenoiseisnotselfcorrelated.Thefrequencydepen