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Spectral Analysis and Time Series

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Spectral Analysis and Time Series SSppeecctrtraall AAnnaallyyssiiss aanndd TTiimmee SSeerriieess Andreas Lagg Parrt II:: fundameentntaallss Parrt IIII: FourFourieerr sseerriiess Parrt IIIII: WWavvelettss on timme sseerriieess cllassiification definitionion why wavelet prob. densiity func. method transforms? auto-correlatilation properties fundamentals: FT, STFT and power spectral densiity convolutilution resolutionon prproblobleemmss cross-ccororrrelatiationn correlatiations multiresolution appliicationsons lleakage / wiindowiing analysiis: CWTT pre-processiing iirregular griid DWT samplinging noise removal trend removal EExxeercrcisiseess A.. Lagg ± Spectral Analylysisis BaBassiicc ddeessccrriippttiioonn ooff pphhyyssiiccaall ddaattaa determiinistic: descriibed by expliicciitt mmathemathematiaticalal rrelelatiationon k xt=X cos t t n nono determiinistic: no way to predict an exact value at a future instant of time n d e t e r m A.. Lagg ± Spectral Analylysisis ii n ii s t ii c CCllaassssiiffiiccaattiioonnss ooff ddeetteerrmmiinniissttiicc ddaattaa Determiinistic Perioiodic Nonperiiodicic Siinusoidaldal Complexex Periiodic Allmost perioiodic Transiient A.. Lagg ± Spectral Analylysisis SiSinnuussooiiddaall ddaattaa xt=X sin2 f 0 t T =1/ f 0 time history frequency spectrogrogram A.. Lagg ± Spectral Analylysisis CCoommpplleexx ppeerriiooddiicc ddaattaa xt=xt±nT n=1,2,3,... a xt = 0 a cos 2 n f t b sin 2 n f t 2 ∑ n 1 n 1 ((T = fundamental periiod) time history frequency spectrogrogram A.. Lagg ± Spectral Analylysisis AlAlmmoosstt ppeerriiooddiicc ddaattaa xt=X 1 sin2 t1 X 2 sin3t2 X 3 sin50 t3 no highest common diviisoror --> iinfinfininitelyy long periiodod TT time history frequency spectrogrogram A.. Lagg ± Spectral Analylysisis TrTraannssiieenntt nnoonn--ppeerriiooddiicc ddaattaa alll non-periiodic data other than almost periiodic data Ae−at t≥0 xt= { 0 t0 Ae−at cos bt t≥0 xt= { 0 t0 A c≥t≥0 xt= { 0 ct0 A.. Lagg ± Spectral Analylysisis CCllaassssiiffiiccaattiioonn ooff rraannddoomm ddaattaa Random Data Statiionary Nonstationary Ergodicic Nonnerrggoddiic Speciial ccllassiifications of nonstationarity A.. Lagg ± Spectral Analylysisis ssttaattiioonnaarryy // nnoonn ssttaattiioonnaarryy colllection of samplee funcfunctions = ensemble data can be (hypotheticallly) descriibed by computing ensemblee avavererages (averaging over multiple measurements / sample functions) mean value (first moment): 1 N x t1= lim ∑ xk t1 N ∞ N k=1 autocorrelelationon funcfunctionon ((jjoint moment):: 1 N Rx t1, t1= lim ∑ xk t1 xk t1 N ∞ N k=1 stationary: x t1=x , Rx t1, t1=Rx weaklly stationary: x t1=x , Rx t1, t1=Rx A.. Lagg ± Spectral Analylysisis eerrggooddiicc // nnoonn eerrggooddiicc Ergodic random process: properties of a stationary random process descriibedbed byby ccomomputiputingng avaverages over only one singlele samsample funnction in thee ensemblle mean value of k-th sample function: T 1 x k=lim ∫ xk tdt T ∞ T 0 autocorrelelationon funcfunctionon ((jjoint moment):: T 1 Rx , k=lim ∫ xk t xk tdt T ∞ T 0 ergodic: x k=x , Rx , k=Rx A.. Lagg ± Spectral Analylysisis BaBassiicc ddeessccrriippttiivvee pprrooppeerrttiieess ooff rraannddoomm ddaattaa meean square vallues propabiillityity ddennsisitty ffunctiionn autocoorrrrellaatiioonn functtiioonnss power spectral densityity functiions (from now on: assume random data tto be stationaryionary and ergodic) A.. Lagg ± Spectral Analylysisis Mean square values Mean square values (mean values and variances) T 2 1 2 descriibesbes genergeneralal intensiity of random data: x = lim ∫ x tdt T ∞ T 0 rms 2 rout mean square value:ue: x = x T often convenient: 1 static component descriibed by mean value: x = lim ∫ xtdt T ∞ T 0 T 1 dynamiic component descriibed by variiance: 2 2 x = lim ∫[ xt−x ] dt T ∞ T 0 2 2 2 = x−x standard deviiation: x = x A.. Lagg ± Spectral Analylysisis PrProobbaabbiilliittyy ddeennssiittyy ffuunnccttiioonnss descriibesbes thethe prprobabiobabilliityty thatthat the datadata wiillll assume a value wiithin some defined range at any instant of time k T x Prob[ xxt≤x x] = lim , T x=∑ ti T ∞ T i=1 for smalalll x : Prob[ xxt≤x x] ≈ px x proobabiillityity density fufuncnctiionn Prob[ xxt≤x x] 1 T x px = lim = lim lim x 0 x x 0 x [T ∞ T ] probabiillityity diistrriibubutition funncctionon Px = Prob[xt≤x] x = ∫ pd −∞ A.. Lagg ± Spectral Analylysisis IIllullussttrarattioionn:: pprorobbaabbilitilityy ddeennssitityy ffuunnccttioionn sample time historiieses:: siine wave (a) siine wave + random noise narrow-band random noise wiide-band random noise alll 4 cases: mean valueue x=0 A.. Lagg ± Spectral Analylysisis IIllullussttrarattioionn:: pprorobbaabbilitilityy ddeennssitityy ffuunnccttioionn probabiilliityty densensiityty functitionon A.. Lagg ± Spectral Analylysisis AuAuttooccoorrrreellaattiioonn ffuunnccttiioonnss descriibesbes thethe genergeneral dependence of the data values at one time on the values at another timme. T 1 Rx = lim ∫ xt xtdt T ∞ T 0 2 x = Rx ∞ x = Rx 0 (not for speciial ccasases lliike siine waveses)) A.. Lagg ± Spectral Analylysisis IIllullussttrarattioionn:: AACCFF autocorrelatiation functions (autocorrelogram) A.. Lagg ± Spectral Analylysisis IIllullussttrarattioionnss autocorrelation function of a rectangular pulse xt xt− autocorrelation functionon A.. Lagg ± Spectral Analylysisis PPoowweerr ssppeecctratrall ddeennssitityy fufunnccttiioonnss (also called autospecttral densitity functionsions) descriibebe thethe general frequency composiition of the data in terms of the spectral densiity of its mean square value mean square value in frequency range f , f f : T 1 2 2 f , f = lim xt , f , f dt x ∫ T ∞ T 0 portionion of x(t) inin (f,ff+∆f) 2 definition of power specpectrtral densdensiity funcfunctition: x f , f ≈ G x f f 2 T x f , f 1 1 2 G x f = lim = lim lim ∫ xt , f , f dt f 0 f f 0 f T ∞ T [ 0 ] iimportant property: spectral densiity function is related to the autocorrelelationon funcfunctionon byby aa FFouriier transform: ∞ ∞ −i 2 f G x f = 2 ∫ Rx e d = 4∫ Rx cos2 f d −∞ 0 A.. Lagg ± Spectral Analylysisis IIllullussttrarattioionn:: PPSSDD powwer spectralal densitity ffuncuncttiions siine wave Diirac delta function at f=ff0 siine wave + random noisise narrowbandband noiise ªwhiteºeº noiisse:: spectrum is uniform over broadband alll frequenciies noiise A.. Lagg ± Spectral Analylysisis JJooinintt pprrooppertiertieess ooff rraannddoomm ddatataa untill now: descriibed properties of an iindiviidual random process Jointnt prprobaobabillity density ffuncuncttiionsons jjoiint propertiies in the amplliittudeude doomaiin Crososss-cororrreellaattion functtiions jjoiint propertiies in the tiime domainin jjoiint probabilliity measurement Crososss-spepecttrrall dedensnsiity functtiions jjoiint propertiies in the frequency domaiin A.. Lagg ± Spectral Analylysisis CCrroossss--ccoorrrrelelaattiioonn ffuuncncttiioonn descriibesbes thethe genergeneral dependence of one data set to another T 1 Rxy = lim ∫ xt ytdt T ∞ T 0 siimiillar to autocorrrelatiationon funcfunctitionon Rxy =0 functions are uncorrelatedated cross-ccoorrelatiion measurement typical crosss--ccorrelation plot (cross-correlogram): sharp peaks indicate the exiistence of a corrrelelatiationon between x(t) and y(t) for sspeciific titimmee didissplplacemementsents A.. Lagg ± Spectral Analylysisis AAppppllicicaattiioonnss MMeeaassuurreemmentent ofof ttiimeme ddeellayayss 2 siignals: different offset different S/N time delay 4s often used: ©©discrete© cross correlation coefficiient llag =ll, for l >=0: N−l ∑ xk−xykl−y k=1 Rxy l = N N 2 2 ∑ xk−x ∑ yk−y k=1 k=1 A.. Lagg ± Spectral Analylysisis AAppppllicicaattiioonnss DDeetteectctionion anandd rrececovovereryy ffrromom ssiignals in noissee 3 siignals: noise free repliica of the siignal (e.g. model) 2 noisy siignals cross correlation can be useded toto determiine if theoretietical siignal is present in data A.. Lagg ± Spectral Analylysisis PrPree--pprroocceessssiinngg OOppeerraattiioonnss samplinging cconsiderations trend removal fillteriing methods ssaammpplilinngg cutoff frequency (=Nyquist frequency or folding frequencequency)) 1 f = c 2 h A.. Lagg ± Spectral Analylysisis TrTreenndd rreemmoovvaall often desiirable before performiing a spectral analysiis Leasstt--ssquaquare method: time seriies: ut K desired fit k desired fit u=∑ bk nh n=1,2,... , N (e.g. polynomiial): k=0 N 2 Lsq-Fiit:t: mmiinimiizze Qb=∑ un−u n n=1 N set partialal ∂Q l = 2u −u [−nh ] deriivatives to 0: ∑ n n ∂ bl n=1 K N N kl l K+1 equations: ∑ bk ∑ nh = ∑ un nh k=0 n=1 n=1 A.. Lagg ± Spectral Analylysisis DDiiggiittaall ffiilltteerriinngg A.. Lagg ± Spectral Analylysisis eenndd ooff ppaartrt II ...... A.. Lagg ± Spectral Analylysisis SSppeecctrtraall AAnnaallyyssiiss aanndd TTiimmee SSeerriieess Andreas Lagg Parrt II:: fundameentntaallss Parrt IIII: FourFourieerr sseerriiess Parrt IIIII: WWavvelettss on timme sseerriieess cllassiification definitionion why wavelet prob. densiity func. method transforms? auto-correlatilation properties fundamentals: FT, STFT and power spectral densiity convolutilution resolutionon prproblobleemmss cross-ccororrrelatiationn correlatiations multiresolution appliicationsons lleakage / wiindowiing analysiis: CWTT pre-processiing iirregular griid DWT samplinging noise removal trend removal EExxeercrcisiseess A.. Lagg ± Spectral Analylysisis FoFouurriieerr SeSerriieess aanndd FaFasstt FoFouurriieerr TrTraannssffoorrmmss Standard Fouriier sereriieses procedure: iif a transformed sample record x(t) is periiodic wiith a periiod Tp (fundamentall frequency f1=1/Tp), then x(t)
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