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1、SystemIdentification6.435SET1–ReviewofLinearSystems–ReviewofStochasticProcesses–DefiningaGeneralFrameworkMuntherA.DahlehLecture16.435,SystemIdentification1Prof.MuntherA.DahlehReviewLTIdiscrete–timesystemstransferfunctionNote(differentnotations)Lecture16.435,SystemIde
2、ntification2Prof.MuntherA.DahlehStability⇔(realrational)polesofareoutsidethediscStrictcausalitysystemhasadelay.StrictStabilityLecture16.435,SystemIdentification3Prof.MuntherA.DahlehFrequencyResponseLecture16.435,SystemIdentification4Prof.MuntherA.DahlehLecture16.435,
3、SystemIdentification5Prof.MuntherA.DahlehSupposethenforstablesystems.Lecture16.435,SystemIdentification6Prof.MuntherA.DahlehPeriodogramsGiven:theFouriertransformisgivenbyDiscrete–FourierTransform(DFT)Lecture16.435,SystemIdentification7Prof.MuntherA.DahlehInverseDFTPe
4、riodogramNiceProperty:Parsaval’sequalityLecture16.435,SystemIdentification8Prof.MuntherA.Dahleh=energycontainedineachfrequencycomponent.Properties:-periodicity-IfisrealExample:(consider)Lecture16.435,SystemIdentification9Prof.MuntherA.DahlehLecture16.435,SystemIdenti
5、fication10Prof.MuntherA.DahlehExample:(Why?)Lecture16.435,SystemIdentification11Prof.MuntherA.DahlehThisrepresentationisvalidoverthewholeinterval[1,N]sinceuisperiodicoverN.Lecture16.435,SystemIdentification12Prof.MuntherA.DahlehPassingthroughafilterClaim:Lecture16.43
6、5,SystemIdentification13Prof.MuntherA.DahlehProof:Notethat:Lecture16.435,SystemIdentification14Prof.MuntherA.DahlehHence:Lecture16.435,SystemIdentification15Prof.MuntherA.DahlehStochasticProcessesDefinition:Astochasticprocessisasequenceofrandomvariableswithajointpdf.
7、Definition:=mean=Correlation/Covariance=CovarianceLecture16.435,SystemIdentification16Prof.MuntherA.DahlehTraditionaldefinitions:isWide-sensestationary(WSS)ifconstant“Thismaybealimitingdefinition.”Wewilldiscussshortly.Lecture16.435,SystemIdentification17Prof.MuntherA
8、.DahlehCommonFrameworkforDeterministicandStochasticSignals•Atypicalsetupnoise(stochastic)experimentoutput(mixed)(deterministic)(loo
