时间平均和系综平均

Chapter1MathematicalMethodsInthischapterwewillstudybasicmathematicalmethodsforcharacterizingnoisepro-cesses.Thetwoimportantanalyticalmethods,stochasticfunctionsandFourieranalysis,areintroducedhere.Thesetwomethodswillbeusedfrequentlythroughoutthistext.1.1TimeAveragevs.EnsembleAverageNoiseisastochasticprocessconsistingofarandomlyvaryingfunctionoftimeandthusisonlystatisticallycharacterized.Onecannotargueasingleeventatacertaintime;onecanonlydiscusstheaveragedquantityofasinglesystemoveracertaintimeintervalortheaveragedquantityofmanyidenticalsystemsatacertaintimeinstance.Theformeriscalledtimeaverageandthelatterensembleaverage.LetusconsiderNidenticalsystemswhichproducenoisywaveformsx(i)(t),asshowninFig.1.1.Figure1.1:Ensembleaveragevs.timeaverage.1Onecandefinethefollowingtime-averagedquantitiesforthei-thmemberoftheen-semble:x(i)(t)=limT!11TZT2¡T2x(i)(t)dt;(mean=first-ordertimeaverage)(1.1)x(i)(t)2=limT!11TZT2¡T2hx(i)(t)i2dt;(meansquare=second-ordertimeaverage)(1.2)Á(i)x(¿)´x(i)(t)x(i)(t+¿)=limT!11TZT2¡T2x(i)(t)x(i)(t+¿)dt:(autocorrelationfunction)(1.3)Onecanalsodefinethefollowingensemble-averagedquantitiesforallmembersoftheensembleatacertaintime:hx(t1)i=limN!11NNXi=1x(i)(t1)=Z1¡1x1p1(x1;t1)dx1;(mean=first-orderensembleaverage)(1.4)hx(t1)2i=limN!11NNXi=1hx(i)(t1)i2=Z1¡1x21p1(x1;t1)dx1;(meansquare=second-orderensembleaverage)(1.5)hx(t1)x(t2)i=limN!11NNXi=1x(i)(t1)x(i)(t2)(1.6)=Z1¡1x1x2p2(x1;x2;t1;t2)dx1dx2:(covariance=second-orderjointmoment)Here,x1=x(t1),x2=x(t2),p1(x1;t1)isthefirst-orderprobabilitydensityfunction(PDF),andp2(x1;x2;t1;t2)isthesecond-orderjointprobabilitydensityfunction.p1(x1;t1)dx1istheprobabilitythatxisfoundintherangebetweenx1andx1+dx1atatimet1andp2(x1;x2;t1;t2)dx1dx2istheprobabilitythatxisfoundintherangebetweenx1andx1+dx1atatimet1andalsointherangebetweenx2andx2+dx2atadifferenttimet2.Anensembleaverageisaconvenienttheoreticalconceptsinceitisdirectlyrelatedtotheprobabilitydensityfunctions.Ontheotherhand,atimeaverageismoredirectlyrelatedtorealexperiments.Onecannotprepareaninfinitenumberofidenticalsystemsinarealsituation.Theoreticalpredictionsbasedonensembleaveragingareequivalenttoex-perimentalmeasurementsoftimeaveragingwhen,andonlywhen,thesystemisaso-called“ergodicensemble.”Onecansaythatensembleaveragingandtimeaveragingareidenti-calforastatistically-stationarysystem,butaredifferentforastatistically-nonstationarysystem.Wewillseethoseconceptsnext.21Chapter1MathematicalMethods1.1Timeaveragevs.ensembleaveragetimeaverage:mean:mean-square:variance:auto-correlation:ensembleaverage:mean:mean-square:variance:2covariance:first-orderprobabilitydensityfunctionsecond-orderprobabilitydensityfunctionIftimeaverage=ensembleaverage“ergodicensemble”1.2Stationaryvs.non-stationaryprocesses•Ifk-thorderprobabilitydensityfunctionisinvariantwithrespecttotheshiftoftimeorigin,stationaryoforderk•Ifastochasticprocessisstationaryofanyorderk=1,2,,strictlystationary