ESS 265 Spring Quarter 2005 Time Series Analysis: Some Fundamentals of Spectral Analysis

Transcription

1 ESS 65 Srig Qurtr 5 Tim Sris ysis: Som Fudmts of Sctr ysis Lctur My, 5

2 Fourir Sris y riodic fuctio ttt whr ωt is th riod c xrssd s Fourir sris t c cos t s si t ω ω t must stisfy th coditio T t dt < y rso fuctio stisfyig th ov coditio c xdd s fuctio of si d cos comt To fid th cofficits us th foowig rtioshis which rsut cus si d cos r orthogo. c s T T T T T T t dt tcos ωt dt;,,,... tsi ωt dt;,,...

3 Som Usfu Prortis of Fourir Sris Priodicity Fourir sris r riodic with dfid riod. Th Fourir sris covrgs o th rquird fuctio oy i th giv itrv. Ev d odd fuctios- Th si is odd -t-t whi cosi is v -tt. Fit v fuctios with cosi d odd fuctio with si. Hf-riod sris fuctio t dfid oy o th itrv,t c fit with ithr sis or cosis. Lst squrs roximtio- Wh Fourir sris xsio of cotiuous fuctio t is tructd ftr trms it is st squrs fit to th origi. Th r of th squrd diffrc tw th two fuctios is miimisd d gos to ro s icrss. Gis homo- Th Fourir sris covrgs i th st-squrs ss wh t is discotiuous ut th tructd sris oscits r th discotiuity.

4 Th Fourir Itgr Th Fourir itgr trsform FIT Fω of fuctio ft is dfid s F ω iωt f t dt Th ivrs trsform is f t i ω ω t F dω Th right hd sid is fiit if Shift thorm f t dt < ft Fω ft-t -iwt Fω Drivtiv dfdt -iωfω Covoutio thorm ' ' f t g t t dt ' FωGω Symmtry Ft R F-ωF * ω Prsv f t dt F ω dω

5 Th Fourir Itgr Trsform Cotiud Th coditio f t dt < is vry rstrictiv. It ms tht sim fuctios i ftcost. or si ωt wo t wor. This c fixd y usig th Dirc dt fuctio δt δt, t ; Th sustitutio rorty ms itgrtio trivi y th dfiitio of th FIT to th dt fuctio.this givs δ t f t dt f iωt dω For mochromtic wvs δ t t f t dt f t δ t iωt dt iωt iωt δ t; dt δ ω; δ ω δ ω cosω tdt siω tdt i [ δ ω ω δ ω ω ] [ δ ω ω δ ω ω ]

6 Th -Trsform ssum w hv sris of msurmts i vy tim or sc {},,,, - -trsform is md y crtig oyomi i th comx vri usig Ortios o th trsform hv coutrrts i th tim domi. Imgi mutiyig th -trsform y This w trsform is tht you woud gt if you shiftd th origi tim sris y o uit i tim. I this cs is cd th uit dy ortor. Mutiictio of two -trsforms is cd discrt covoutio. Th discrt covoutio thorm is wht giv th -trsform its owr. Cosidr th roduct of d B ch of diffrt gth M B C

7 Th -Trsform Cotiud St d chg th ordr of summtio This is th -trsform of tim sris of gth M-. c* is th discrt covoutio of d Th discrt covoutio thorm for -trsformtios is giv y th foowig ottio: Divisio y -trsform is dcovoutio. s og s th c d This is rcursiv rocdur cus th rsut of ch qutio is usd i susqut qutios. M M c B B c

8 Th Discrt Fourir Trsform Sustitut -iω t ito th -trsform qutios d ormi y This qutio is comx Fourir sris tht is cotiuous fuctio of frqucy tht hs discrtisd so tht whr ν is th smig frqucy. Th discrt Fourir trsform is giv y This formu trsforms th vus of th tim squcs { } } ito othr squc of umrs with Fourir cofficits { }. Th ivrs qutio to rcovr th origi tim sris from th Fourir cofficits is: msurs tim i uits of t u to mximum of T t. msurs frqucy itrvs of νt u to mximum of v s ν t. t i ω ω ν ω t T,,,..., ; i ω i

9 mitud, Powr d Phs Sctr Th Fourir cofficits dscri th cotriutio of th rticur frqucy ω ν to th origi tim squc. iφ R sig with just o frqucy is si wv. R is th mximum Φ dfis th iiti oit i th cyc. R ottd gist is cd th mitud sctrum R ottd gist is th owr sctrum. Φ is g tht dscris thhs of this frqucy with th tim sris d th corrsodig ot is hs sctrum. Th Shift Thorm mutiictio of th DFT y -iwdt wi dy th squc y o smig itrv. I othr words shiftig th tim squc o sc wi muti th DFT cofficit y -i. Th owr sctrum i ot chgd y th hs is rtrdd y. I drivig th covoutio thorm w omittd trms i th sum ivovig mts of or with suscrits which outsd of th scifid rgs, to - for d to M- for. This is o ogr corrct of riodic fuctios. Prcticy this md to wor y ddig with ros to xtd oth sris to gth M.

10 Diffrtitio d Itgrtio Diffrtitio d itgrtio oy y to cotiuous fuctios of tim so w st t t d w t so th DFT coms t Diffrtitig with rsct to tim givs d dt M this discrt y sttig t t d c so tht & This is th ivrs DFT so d iω must trsforms of ch othr. Diffrtitio with rsct to tim is quivt to mutiictio y frqucy i th frqucy domi. Itgrtio with rsct to tim is quivt to divisio i th frqucy domi. iω & iω t iω t d dt t t & iω i

11 Prsv s Thorm From th dfiitio of th DFT w fid Usig yquist s thorm whr δ is th Krocr dt. Digrssio o th Krocr dt - δ for d for such tht Prsv s thorm gurts th quity of rgy tw th tim d frqucy domis. i i i δ δ i δ,,

12 Th Fst Fourir Trsform Th DFT rquirs sum ovr trms for ch of frqucis. Thus th tot umr of ccutios rquird gos s. This ws mjor imdimt to doig sctr ysis. Th fst Fourir trsform FFT owd this to do much fstr. Suos tht is divisi y. Sit th DFT ito two rts. This sum rquirs us to form th qutity i ccutios d th doig this tims. For frqucis this ms ccutios or rductio of fctor of for rg ccutios. This rocdur c rtd. s og s is owr of th sum c dividd og tims, with tot of 4og ortios. i i i i,,,..., ; i ω

13 isig d Sho s Smig Thorm ω i Th trsformtio ir d r xct. Bfor digitiig w hd cotiuous fuctios of tim. Th riodicity of th DFT givs. ;,,,..., If th dt r r t th comx cojugt d show tht - *. Ths qutios dfi isig which is th mig of highr frqucis ito th rg to ftr digitistio highr frqucis r md to owr frqucis. Fourir cofficits for frqucis ov r dtrmid xcty from th first. ov thy r riodic d tw d th rfct with th sm mitud d hs chg of. Th DFT ows us to trsform r vus i th tim domi ito y umr of comx vus. Th highst migfu cofficit i th DFT is d th corrsodig frqucy is th yquist frqucy ν t. W c rcovr th origi sig from digitid sms rovidd th origi sm cotid o rgy ov th yquist frqucy. To rroduc th origi tim sris t from its sms t w c us Sho's thorm K t t t siυ υ t t i

14 Trig I id uivrs dt woud cotiuous fuctio of tim goig forvr, th th Fourir itgr trsform woud giv th sctrum. I rity dt r imitd d th fiit gth T imits th frqucy scig to Th smig itrv t imits th mximum migfu frqucy to ν. It is frquty usfu to ssum fiit tim sris is sctio of ifiit sris. This is cd widowig d is chivd y mutiyig th tim sris y ox cr cd tr tht is ro outsid of th widow d o isid. Sic mutiictio i th tim domi is th sm s covoutio i th frqucy domi this is quivt to covoutio with th DFT of ox cr which is si d sid os. sig is srd cross rg of frqucis. This is cd sctr g. W c imrov this y usig widow with diffrt sid os. For rsovig s w wt rrow fuctio i th frqucy domi ut tht ms rod fuctio i th tim domi th ucrtity rici. Tim widowig c rduc ois y smoothig th sctrum- ois rductio coms t th xs of rsoutio. υ T

15 Fitrig: Th Ruig vrg Fitrig is covoutio with scod- usuy shortr- tim sris. Bdss fitrs imit rgs of frqucis from th tim sris. Low-ss fitrs imit frqucis ov crti frqucy. High-ss fitrs imit frqucis ov crti frqucy. Th rg of frqucis owd is cd th ss d. Th critic frqucis r cd cut-off frqucis. ruig vrg i which ch mmr of tim sris is rcd y vrg of M ighorig mmrs is fitr. It is covoutio of th origi tim squc with oxcr fuctio. Ituitivy w woud xct th ruig vrg to rmov high frqucis d thrfor ow-ss fitr. Sic covoutio th tim domi is mutiictio i th frqucy domi w hv mutiid th DFT of th tim sris with th Fourir trsform of th oxcr. This is ot id ow-ss fitr cus of th sid os d th ctr t through ot of rgy ov th dsird frqucy. Th mitud sctrum is xcty ro t th frqucis MT whr th ox cr trsform is ro so this fitr is grt if you wt to imitd giv frqucy.

16 Th Fourir Trsform of Box Cr Th Discrt Fourir Trsform of ox cr hs ctr d oscitios i frqucy.

17 Fitrig: Som Exms id ow-ss fitr shoud hv mitud sctrum tht is ro outsid of th cut-off frqucy d o isid it. Gi's homo rvts such fitr from doig good jo s ow-ss fitr. W d to tr th fitr i w trd th widow. Gios uss th Buttrworth fitr F ω ω ω whr ω C is th C cut-off frqucy t which th rgy is hvd. cotros th shrss of th cut-off. Th corrsodig high-ss fitr is. dss fitr c md y shiftig th frqucy og th frqucy xis to ctr it roud ω F ω. ω ω ω otch fitr c md from. To costruct th tim squc scify th hs usuy ro d t th ivrs Fourir trsform. F F h ω F ω [ ] C ω F ω

18 Corrtio Th cross corrtio of two tim sris d is dfid y whr is th g, d M r th gths of th tim sris. Th sum is ovr M ossi roducts. utocorrtio is cross corrtio of tim squc with itsf. Th corrtio cofficit is qu to th cross corrtio ormid to giv o wh th two tim sris r idtic rfcty corrtd. ψ is for rfct corrtio d - for ticorrtio. M c φ ψ