SSppeecctrtraall A Annaallyyssiiss a anndd T Tiimmee S Seerriieess Andreas Lagg

Parrt II:: fundameentntaallss Parrt IIII: FourFourieerr sseerriiess Parrt IIIII: WWavvelettss on timme sseerriieess

cllassiification definitionion why wavelet prob. densiity func. method transforms? auto­correlatilation properties fundamentals: FT, STFT and power spectral densiity convolutilution resolution pron problobleemmss cross­ccororrrelatiationn correlatiations multiresolution appliicationsons lleakage / wiindowiing analysiis: CWTT pre­processiing iirregular griid DWT samplinging noise removal trend removal EExxeercrcisiseess

A.. Lagg – Spectral Analylysisis BaBassiicc ddeessccrriippttiioonn ooff pphhyyssiiccaall ddaattaa

determiinistic: descriibed by expliicciit mt mathemathematiaticalal r relelatiationon

k xt=X cos t  t

n non detero miinistic: 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

xt=X sin2 f 0 t

T =1/ f 0

time history frequency spectrogrogram

A.. Lagg – Spectral Analylysisis CCoommpplleexx ppeerriiooddiicc ddaattaa

xt=xt±nT  n=1,2,3,... a xt = 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

xt=X 1 sin2 t1  X 2 sin3t2  X 3 sin50 t3

no highest common diviisoror ­ ­> i infinfininitelyy long periiod Tod T

time history frequency spectrogrogram

A.. Lagg – Spectral Analylysisis TrTraannssiieenntt nnoonn­­ppeerriiooddiicc ddaattaa alll non­periiodic data other than almost periiodic data

Ae−at t≥0 xt= { 0 t0

Ae−at cos bt t≥0 xt= { 0 t0

A c≥t≥0 xt= { 0 ct0

A.. Lagg – Spectral Analylysisis CCllaassssiiffiiccaattiioonn ooff rraannddoomm ddaattaa

Random Data

Statiionary Nonstationary

Ergodicic Nonnerrggoddiic Speciial c cllassiifications of nonstationarity

A.. Lagg – Spectral Analylysisis ssttaattiioonnaarryy // nnoonn ssttaattiioonnaarryy colllection of sample funce functions = ensemble data can be (hypotheticallly) descriibed by computing ensemble ave avererages (averaging over multiple measurements / sample functions) mean value (first moment): 1 N x t1= lim ∑ xk t1 N ∞ N k=1 autocorrelelation funcon function (on (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 descriibed bybed by c comomputiputing avng averages over only one single samle sample funnction in thee ensemblle mean value of k­th sample function: T 1 x k=lim ∫ xk tdt T ∞ T 0 autocorrelelation funcon function (on (jjoint moment):: T 1 Rx  , k=lim ∫ xk t xk tdt 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 gener generalal intensiity of random data: x = lim ∫ x tdt T ∞ T 0 rms 2 rout mean square value: ue: x = x

T often convenient: 1 static component descriibed by mean value: x = lim ∫ xtdt T ∞ T 0

T 1 dynamiic component descriibed by variiance: 2 2  x = lim ∫[ xt−x ] dt T ∞ T 0 2 2 2 = x−x standard deviiation:  x =  x

A.. Lagg – Spectral Analylysisis PrProobbaabbiilliittyy ddeennssiittyy ffuunnccttiioonnss descriibesbes the pr the probabiobabilliityty that the data w that the data wiillll assume a value wiithin some defined range at any instant of time k T x Prob[ xxt≤x x] = lim , T x=∑ ti T ∞ T i=1 for smalalll : x Prob[ xxt≤x x] ≈ px x proobabiillityity density fufuncnctiionn Prob[ xxt≤x x] 1 T x px = lim = lim lim  x  0  x  x 0  x [T ∞ T ] probabiillityity diistrriibubutition funncctionon Px = Prob[xt≤x] x = ∫ pd  −∞

A.. Lagg – Spectral Analylysisis IIllullussttrarattioionn:: pprorobbaabbilitilityy ddeennssitityy ffuunnccttioionn sample time historiieses:: siine wave (a) siine wave + random noise narrow­band random noise wiide­band 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 the gener the general dependence of the data values at one time on the values at another timme.

T 1 Rx  = lim ∫ xt xtdt T ∞ T 0

2 x =  Rx ∞ x = Rx 0 (not for speciial c casases l liike siine waveses))

A.. Lagg – Spectral Analylysisis IIllullussttrarattioionn:: AACCFF autocorrelatiation functions (autocorrelogram)

A.. Lagg – Spectral Analylysisis IIllullussttrarattioionnss autocorrelation function of a rectangular pulse

xt xt−

autocorrelation functionon

A.. Lagg – Spectral Analylysisis PPoowweerr ssppeecctratral l ddeennssitityy fufunnccttiioonnss (also called autospecttral densitity functionsions) descriibe the generbe the 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 xt , f , f  dt x ∫ T ∞ T 0 portionion of x(t) in in (f,ff+∆f) 2 definition of power specpectrtral dens densiity func functition: x  f , f  ≈ G x  f  f 2 T x  f , f  1 1 2 G x  f  = lim = lim lim ∫ xt , f , f  dt  f  0  f  f 0  f T ∞ T [ 0 ] iimportant property: spectral densiity function is related to the autocorrelelation funcon function byon by a F a Fouriier 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

“white” e” 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

Crososss­cororrreellaattion functtiions jjoiint propertiies in the tiime domainin jjoiint probabilliity measurement Crososss­spepecttrrall dedensnsiity functtiions jjoiint propertiies in the frequency domaiin

A.. Lagg – Spectral Analylysisis CCrroossss­­ccoorrrrelelaattiioonn ffuuncncttiioonn descriibesbes the gener the general dependence of one data set to another T 1 Rxy  = lim ∫ xt ytdt T ∞ T 0 siimiillar to autocorrrelatiation funcon functitionon

Rxy =0 functions are uncorrelatedated

cross­ccoorrelatiion measurement

typical crosss­­ccorrelation plot (cross­correlogram): sharp peaks indicate the exiistence of a corrrelelatiation on between x(t) and y(t) for s speciific ti timme die dissplplacemementsents

A.. Lagg – Spectral Analylysisis AAppppllicicaattiioonnss

MMeeaassuurreemmentent of of 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−xykl−y k=1 Rxy l = N N 2 2 ∑ xk−x ∑ yk−y  k=1 k=1 A.. Lagg – Spectral Analylysisis AAppppllicicaattiioonnss DDeetteectction ion anand d rrececovovereryy ffrromom ssiignals in noissee

3 siignals: noise free repliica of the siignal (e.g. model) 2 noisy siignals

cross correlation can be used to ed to determiine if theoretietical siignal is present in data

A.. Lagg – Spectral Analylysisis PrPree­­pprroocceessssiinngg OOppeerraattiioonnss

sampling cing considerations 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

Leasstt­­ssquaquare method: time seriies: ut K desired fit k desired fit u=∑ bk nh n=1,2,... , N (e.g. polynomiial): k=0 N 2 Lsq­Fiit: mt: miinimiizze Qb=∑ un−u n n=1 N set partialal ∂Q l = 2u −u [−nh ] deriivatives to 0: ∑ n n ∂ bl n=1 K N N kl 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 A Annaallyyssiiss a anndd T Tiimmee S Seerriieess Andreas Lagg

Parrt II:: fundameentntaallss Parrt IIII: FourFourieerr sseerriiess Parrt IIIII: WWavvelettss on timme sseerriieess

cllassiification definitionion why wavelet prob. densiity func. method transforms? auto­correlatilation properties fundamentals: FT, STFT and power spectral densiity convolutilution resolution pron problobleemmss cross­ccororrrelatiationn correlatiations multiresolution appliicationsons lleakage / wiindowiing analysiis: CWTT pre­processiing 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) can be represesenteted bby the Fouurieer r seeriieses:

∞ a0 xt = ∑ aq cos 2q f 1 t  bq sin 2q f 1 t 2 q=1 T 2 where aq = ∫ xtcos2q f 1 t dt q=0,1,2,... T 0 T 2 bq = ∫ xtsin 2q f 1 t dt q=1,2,3,... T 0

A.. Lagg – Spectral Analylysisis FFoouurierier r sseerieriess pprorocceedduurere ­ ­ mmeetthhoodd sample record of finite length, equallly spaced sampled:

xn=xnh n=1,2,... , N

Fouriier seriies passiing through these N data values: N /2 2q t N /2−1 2q t xt = A  A cos  B sin 0 ∑ q T ∑ q T q=1  p  q=1  p 

Fiillll i in parn partiticcular points: t=nh , n=1,2,... , N , T p=Nh , xn=xnh = ... 1 N 1 N A = x = x A = x cos n coefficciientsents Aq and Bq: 0 ∑ n  N /2 ∑ n N n=1 N n=1 2 N 2q n N Aq = ∑ xn cos q=1,2,... , −1 N n=1 N 2 2 N 2q n N Bq = ∑ xn sin q=1,2,... , −1 N n=1 N 2 iinefficiient & sllow => Fast Fouriier Trafos developed

A.. Lagg – Spectral Analylysisis FouFourriieerr TrTrananssfoforrmmss ­­ PrProoppeerrttiieses

LLinineeaaritrityy DFT {xn} ⇔ {X k } DFT

{yn} ⇔ {Y k } DFT

a{xn}b{yn} ⇔ a{X k }b{Y k }

∗ SSyymmmmeettryry {X k } = {X−k }

ℜ{X k } is even ℑ{X k } is odd

DFT CCircircuulalar r ttimimee sshhififtt −i k n0 {x } ⇔ {e X } n−n0 k DFT i k n {e 0 y } ⇔ {Y } n k−k0

A.. Lagg – Spectral Analylysisis UUssiingng FFTFFT ffor Cor Coonnvvololuuttiioonn

∞ r * s ≡ ∫ r st−d  −∞ origiinal data Convollutiion Theorem: FT r * s ⇔ R f S f  response function Fouriier transform of the convolution iis product of the indiviidual Fouriier transforms convolvolveed dat dataa disccrettee ccaassee:: N 2 / (note how tthe response function ion ffor negativive timimes isis wrapped around and stored at the extreme right end of the array) r * s j ≡ ∑ s j−k r k k=−N /21 Convolution Theorem: constraints: N /2 FT duration of r and s are not the same ∑ s j−k r k ⇔ Rn Sn k=−N /21 siignal iss not per not periiodic

A.. Lagg – Spectral Analylysisis TTrereaattmmeenntt ooff eenndd eeffffeeccttss bbyy zzeeroro ppaaddddiningg constraint 1: simplimply expand response function to llength N by padding iit wiith zth zereros constraint 2: extend data at one end wid with a number of zeros equal to the max. positiive / negative duratiion of r (whiichever iis larger)

A.. Lagg – Spectral Analylysisis FFFFTT ffoor r CCoonnvvooluluttioionn

1. zerro­o­pad data 2. zerro­o­pad response function (­> data and response function have N elements) 3. callculate FFT of data and response function 4. multiplyly FFT of data wiith FFFT T of rof responsponse funce functition 5. callculate inverse FFT for this product

DDeeccoonnvvooluluttioionn

­> undo smeariing caused by a r a resesponsponse funce functitionon use steps (1­3), and then: 4. diviide FFT of convolved data wiith FFT of response function 5. callculate inverse FFT for this product

A.. Lagg – Spectral Analylysisis CCoorrrreellaattiioonn // AuAuttooccoorrrreellaattiioonn wwiitthh FFTFFT definition of cororrelelation / autocorrelation see first lecture ∞ Corr g , h = g* h = ∫ gthd  −∞ Corrrellattiion on TheTheororeemm:: FT Corr g , h ⇔ G f  H * f 

Auto­Correllatiion:

FT 2 Corr g , g ⇔ ∣G f ∣ discrete correlation theorem: N−1

Corr g , h j ≡ ∑ g jk hk k=0 FT * ⇔ Gk H k

A.. Lagg – Spectral Analylysisis FoFouurriieerr TrTraannssffoorrmm ­­ pprroobblleemmss ssppeeccttraral l leleaakkaaggee

T =nT p T ≠nT p

A.. Lagg – Spectral Analylysisis reredduucciningg leleaakkaaggee bbyy wwininddoowwiningg (1(1))

Applyiing wiindowiing (apodizziing)ng) func function to data ron to data recorord:d: xt = xtwt (oriiginal data record x wiindowiing funcng function)

xn = xn wn

No Windowndow: Bartleetttt WWiindowndow:: Hammmiing ng WWiindowndow::

∣t−T /2∣ wt=1− wt=0.540.46∗cost T /2

Blaacckkmman n Window:: Hann nn WWindowndow: Gaussiiaan n WWiindow:

wt = 0.420.5∗cost 2 t wt=cos   2 0.08cos2 t 2 wt=exp−0.5a t 

A.. Lagg – Spectral Analylysisis reredduucciningg leleaakkaaggee bbyy wwininddoowwiningg (2(2))

withhoout t windodowiinngg wiith Gaussiian wiindowndowiing

A.. Lagg – Spectral Analylysisis NNoo ccoonnssttaanntt ssaammpplinlingg ffrereqquueennccyy

Fouriier transformation requires constant sampliing (data poidata points at equal distances) ­> not the case for most physiical data

Solutiion: Intnterrpolpolaationon lliinear:

iinear interpolation between yk and yk+1 IDL> idata=iinterpol(data,t,t_reg) quadratic: iimportant:

quadratic interpolation uson usiing yng yk­1, yk and yk+1 IDL> idata=iinterpol(data,t,t_reg,/quadraticc) iinterpolation changes sampliing rate! lleast­square quadratic lleast­square quadratic fit usiing y , y , y and y k­1 k k+1 k+2 ­> careful choice of IDL> idata=iinterpol(data,t,t_reg,/lsq) new (regular) time gre griid spliine necessary! IDL> idata=iinterpol(data,t,t_reg,/spliine) IDL> idata=spliine(t,data,t_reg[,tensiion])

A.. Lagg – Spectral Analylysisis FFoouurriieer r TTrraannssoorrmm oonn irrirreegguulalarr ggrriiddddeedd ddaattaa ­­ IInntteerrppoolalattioionn

oriiginal data: siine wave + noise FT of oriiginal data iirregular sampliing of data (measurement) iinterpolation: liinear, lsq, spliine, quadratic ''re­sampliing' FT of interpolated data

A.. Lagg – Spectral Analylysisis NNoioissee rreemomovvaall

Frequequencncyy tthrhresshold ((lowowpapassss)) make FT of data set high frequenciieses to to 00 transorm back to time domain

A.. Lagg – Spectral Analylysisis NNoioissee rreemomovvaall signall threesholhold d fforor weakk ffrreequencnciies (dB­threessholhold) make FT of data set frequenciies wiith ampliitudes below a given threshold to 0 transorm back to time domainn

A.. Lagg – Spectral Analylysisis OOppttiimmaall FFiilltteerriinngg wwitithh FF FFTT

normal siituation won wiith mth measured data:ed data: ∞ st = ∫ r t−ud  underllyiing, uncorrupted signall u(u(t)) −∞ + reesppoonsee funfunctitioon oof meaassureremementt r( r(t)t) ct = stnt = smeared siignal s(t) + noiise n(t) = smearedd, noisisy siiggnnall c(t) esttiimmaatee trueue ssignagnall u(tt) witth:h: C  f  f  U  f  = R f   f ,t = optiimall fiillterr (Wiiener fiillter)

A.. Lagg – Spectral Analylysisis CCaalclcuulalattioionn ooff ooppttimimaal l ffiltilteerr reconstructed siignal and unc and uncorrrupted supted siignalgnal s should be cllose in least­square sense: ­> miinimiizze e ∞ ∞ 2 2 ∫∣u t−ut∣ dt = ∫∣U  f −U  f ∣ df −∞ −∞ 2 [S f N  f ] f  S f  ⇒ ∂ − = 0 ∂ f ∣ R f  R f ∣ 2 ∣S f ∣ ⇒  f  = 2 2 ∣S f ∣ ∣N  f ∣

additional informatiation: power spectral densiity can often be used to disentangle noise function N(f) from smmeared ed siignal S((f)

A.. Lagg – Spectral Analylysisis UUssiingng FFTFFT ffor Por Poowweerr SpSpeeccttrruum m EsEsttimimaattiionon discrete Fouriierer tr transansforformm of c of c(t) ­> Fouriier coefficiients: N−1 2 i j k / N C k=∑ c j e k=0,... , N−1 j=0

­> periiodogram estimate of power spectrum: Periiodogram 1 2 P0 = P f 0= ∣C0∣ N 2 1 2 2 P f k  = ∣C k∣ ∣C N−k∣ N 2 [ ] 1 2 P f c = P f N /2= ∣C N /2∣ N 2

A.. Lagg – Spectral Analylysisis eenndd ooff FFTT

A.. Lagg – Spectral Analylysisis SSppeecctrtraall A Annaallyyssiiss a anndd T Tiimmee S Seerriieess Andreas Lagg

Parrt II:: fundameentntaallss Parrt IIII: FourFourieerr sseerriiess Parrt IIIII: WWavvelettss on timme sseerriieess

cllassiification definitionion why wavelet prob. densiity func. method transforms? auto­correlatilation properties fundamentals: FT, STFT and power spectral densiity convolutilution resolution pron problobleemmss cross­ccororrrelatiationn correlatiations multiresolution appliicationsons lleakage / wiindowiing analysiis: CWTT pre­processiing iirregular griid DWT samplinging noise removal trend removal EExxeercrcisiseess

A.. Lagg – Spectral Analylysisis IInnttrroodduuccttiioonn ttoo WWaavveelleettss

why wavelelet transansformforms?? fundamentals: FT, short term FT and resolution problems multiresolution analysisiss:: continous wavelet transform multiresolution analysisiss:: discrete wavelet transform

A.. Lagg – Spectral Analylysisis FouFourriieerr:: lloosstt t timimee iinfnfoorrmmaattiioonn

6 Hz, 4 Hz, 2 Hz, 1 Hz 6 Hz + 4 Hz + 2 Hz + 1 Hz

A.. Lagg – Spectral Analylysisis SSoolluuttiioonn:: SShhortort TiTimeme FoFouurriieerr Tra Trannssffoorrmm (STFT)

perform FT on 'wiindowed' function: example: rectangular wiindow move wiindow in smalll steps over data perform FT for every timime step

STFT  f ,t ' = ∫[ xtt−t ']e−i 2 f t dt t

A.. Lagg – Spectral Analylysisis SShhoorrtt TiTimeme FoFouurriieerr Tra Trannssffoorrmm

STFT

STFT­spectrogram showhows both time and frequency information!

A.. Lagg – Spectral Analylysisis SShhoorrtt TiTimeme FoFouurriieerr Tra Trannssffoorrmm:: P Prroobblleemm narrow wiindow function ­> good time resololution wiide wiindow function ­> good frequency resolution

Gauss­functions as wiindows

A.. Lagg – Spectral Analylysisis SSoolluuttiioonn:: WWaavveelleett TrTraannssffoorrmmaattiionon time vs. frequency resolution is intriinsiic problem (Heissenberenberg Uncertainty Priinciiple) approach: analyze the siignal at different frequenciies wiith different resolutions ­> mmululttiiressololututiion on aananallyyssiiss ((MRMRAA))

CCoonnttininuuoouuss WWaavveelelett TTrarannssffoormrm siimiillar to STFT: but: siignal is multipliied wiith a the FT of the wiindowed sed siignals are function (the wavellet) not taken transform is calculated (no negative frequenciies) separately for different segments The wiidth of the wiindow is changed of the time domain as the transform is c comomputed forputed for every siingle spectral component

A.. Lagg – Spectral Analylysisis CCoonnttiinnuuoouuss WWaavveelleett TrTraannssffoorrmm

  1 * t− CWT x = x  , s = ∫ xt dt ∣s∣ t  s   ... translation parameter, s ... scale parameter t ... mother wavelet  = small wave mother wavelet:et: finite length (compactly supported) ­>>''llet'et' osciillllatoratoryy ­>>''wwavave' scale le parameter s repllaccess frefreqqueenccy iy in STTFTT functions for different regions are deriivved fromom this function ­> 'mother''

A.. Lagg – Spectral Analylysisis TheThe ScScaallee

siimiillar to sccalales used in maps: high scale =e = non detailled global viiew (of the siignalgnal)) llow scale = detailled viiew iin practical appliications: llow scales (= hhiigh frgh frequencequenciieses) appear usuallly as short bursts or spikes high scaleses ( (= l low fr frequenciieses) last for entire siignalgnal scaliing dillates (stretches out) or compresses a siignal: s > 1 ­> dillation s < 1 ­> compressiion

A.. Lagg – Spectral Analylysisis CComomppuuttaatitioonn o off tthhee CCWWTT

siignal to be analyzed: x(t), mother wavavelet: Met: Morllet oret or M Mexexiican Hat

start wiith scale s=1 (llowest scale, highest frequency) ­> most compressed wavelet

shift wavelet in time from t0 to t1 iincrease s by smalll value

shift dililated wavelet from t0 to t1 repeat steps for allll scalesles

A.. Lagg – Spectral Analylysisis CCWWT T ­­ ExExaammppllee

siignal x(t) axes of CWT: transllatiation and on and scale (not time and frequency) transllation ­> timme scale ­> 1/frequency

A.. Lagg – Spectral Analylysisis TiTimmee aanndd FrFreeqquueennccyy RReessoolluuttiioonn every box corresponds to a value of the wavelelet transform in the time frequency plane

alll boxes have constant area Δf Δt = const. llow frequenciies: high resololution on iin f, low time resolutiution high frequenciiees: good tis: good time resolutionon

STFT: time and frequency resolution is constant (alll boxes are the same)

A.. Lagg – Spectral Analylysisis WWaavveelleettss:: M Maatthheemmataticicaall ApAppprrooaacchh

WL­transform:   1 * t− CWT x = x  , s = ∫ xt dt ∣s∣ t  s 

Mexiican Hat waveletlet: Morllet wavellet: −t2 t2 2 − 1 2 t i a t 2 t = e 2 −1 t = e e s3  2 

iinveersrse We WL­tr­transfoform: 1 1 t− xt =  , s  d  ds 2 ∫∫ x 2 s c s  s   2 1/2 admmiissiibiilliity ∞ ∣ ∣ FT condiitionn: c={2∫ d } ∞ with   ⇔ t −∞ ∣∣

A.. Lagg – Spectral Analylysisis DDiissccrreettiizzaattiioonn ooff CCWWT:T: WWaavveelleett SeSerriieess

­> sampliing the time – frequency (or scale) plane advantage: sampling high for high frequencies (low scales) s1 sampling high for high frequencies (low scales) N = N scale s and rate N 2 1 1 1 s2

sampling ring rate can be decreased for low f 2 frequenciies (high scales) N 2 = N 1 f 1 scale s2 and rate N2

continuous wavelletet discrete wavvellett

1 t− − j/2 − j  =    t = s s t−k  ,  orthonormal  , s s  s  j , k 0 0 0 j , k  j , k * WL­transformation  x =∫ xt j , k tdt WL­transformation t  j , k reconstruction of xt = c   t ∑∑ x j , k signal j k signal

A.. Lagg – Spectral Analylysisis DDiissccrreettee WWaavveelleett TrTraannssffoorrmm (DWT) discretized continuous wavelet transform is only a sampled versiion of the CWT The discrete wete wavavelelet transform (DWT) has siignignifificcant advant advantages for for iimplementation in computers.

excelllent tutoriial: http://users.rowan.edu/~poliikar/WAVELETS/WTtutoriial.htmll

IDL­Wavelet Tools: IDL> wv_applet

Wavelet expert at MPS: Rajat Thomas

A.. Lagg – Spectral Analylysisis eenndd ooff WWaavveelelettss

A.. Lagg – Spectral Analylysisis ExExeerrcciisseess

Parrt II:: Fouriieer AAnanallyyssiiss (Andreas Lagg)

Instructions: http://www.liinmpi.mpg.de/~llagg

Parrt IIII: WWavvellettss (Rajat Thomas)

Semiinar room Tiime: 15:00

A.. Lagg – Spectral Analylysisis EExxeerrcciissee:: GGalaliilleeoo mmagagnenettiicc ffiieelldd data set from Galiilleo magnetomagnetometereter Tiips: (synthesiized) restore,'glll_data.sav'' fille: glll_data.sav, contains: use IDL­FFT total magnetic field remember basiic plasma radial distance physiics formula for the time in seconds iion cycllotron on wavwave: q B gyro  = , f = gyro m gyro 2 yyoouur r ttaasskkss:: Background: If tthe densitity of ionsions isis high enough tthey will ill excitite ion ion cyclotlotron waves Which ions are present? during gyration ion around the magneticic ffield ield lineslines. This gyration ion frequency Is the time resolutilution of only depends on mass per charge and on the magnitude of the magneticic field.ield. the magnetometer In a lowlow­beta plasma the magneticic ffield ield dominatinates over plasma efffects. sufficiient to detect The magneticic field ield shows only very litlitttle le infinfluencluence from tthe plasma and electrons or protons? can be considered idered as a magneticic dipole. http:///www.scienciencemag.org/cgi/content//full/274/5286/396

A.. Lagg – Spectral Analylysisis LiLiteterraattuurree

Random Data: Analysiis and Measurement Procedures Bendat and Piiersol, W, Wiilleyey Intersscciience, 1971

The Analysiis of Tiime Seriies: An Introduction Chriis Chatfihatfield, Chapman and Halll / CRC, 2004

Tiime Sereriieses Analysiis and Its Appliiccationsons Shumway and Stoffer, Spriinger, 2000

Numeriical Reciipies in C Cambriidge Universiityty P Prress, 1988­1992 http://www.nr.com/

The Wavelet Tutoriial Robi Poliikar, 2001 http://users.rowan.edu/~poliikar/WAVELETS/WTtutoriial.htmll

A.. Lagg – Spectral Analylysisis