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1、IntroductiontoSpectralAnalysisDonPercival,AppliedPhysicsLab,UniversityofWashington•Q:whatisspectralanalysis?•oneofthemostwidelyusedmethodsfordataanalysisingeophysics,oceanography,atmosphericscience,astronomy,engineering(alltypes),...•methodisusedwithtimeseries•letxt
2、denotevalueoftimeseriesattimet;forexample,x=43◦=temperatureatGHSat8AMon10thdayof200310•fourexamplesoftimeseriesx1,x2,...,x127,x1284(a)15(b)xt−10−6−1512(c)7(d)xt1008−7064128064128tt•Q:howwouldyoudescribethese4series?•spectralanalysisdescribesxt’sbycomparingthemtosine
3、sandcosines1SinesandCosines:I•Q:whatdosinesandcosineshavetodowithtimeseries?•plotsofsin(u)andcos(u)versusuasugoesfrom0to4π20−202π4π02π4πuu2•letu=2πtfort=1,2,...,12812822•plotsofsin(2πt)andcos(2πt)versust12812820−2064128064128tt2•canregardaboveasartificialtimeseries:s
4、t=sin(2πt)etc1282•caninterpretas‘2cyclesovertimespanof128’1282SinesandCosines:II7•nowletu=2πtfort=1,2,...,12812877•plotsofsin(2πt)andcos(2πt)versust12812820−2064128064128tt•nowgetserieswith7cyclesovertimespanof128k•ingeneralplotsofsin(2πt)etcversusthavekcycles128k•q
5、uantityiscalledfrequencyofsineorcosine128(usuallyusevariableftodenotefrequency)ifkissmall,sinetimeseriesissaidtohavelowfrequencyifkislarge,sinetimeseriesissaidtohavehighfrequencyk•amplitude(maximumrangeofvariation)ofst=sin(2πt)is112877•plotsof0.5sin(2πt)and1.5sin(
6、2πt)12812820−2064128064128tt3SinesandCosines:III•let’saddtogethersinesandcosineswithdifferentfrequenciesfirstcolumnusessinesandcosinesofamplitude1secondandthirdcolumnsuserandomamplitudesf=1128f=3128f=5128f=7128f=9128f=11128f=13128f=15128f=17128f=19128xt012801280128t
7、tt•conclusion:bysumminguplotsofsinesandcosineswithdifferentamplitudes,cangetartificialtimeseriesthatresembleactualtimeseries4GoalofSpectralAnalysis•givenatimeseriesxt,figureouthowtoconstructitusingsinesandcosines;i.e.,towritekkxt=aksin(2π128t)+bkcos(2π128t)k•abovecall
8、ed‘Fourierrepresentation’foratimeseries(namedafter19thCenturyFrenchmathematician)•allowsustoreexpresstimeseriesinastandardway•differenttime