Astronomische Waarneemtechnieken (Astronomical Observing Techniques)

Transcription

1 Astroomische Wreemtechiee (Astroomicl Observig Techiques) 5 th Lecture: 1 October Fourier Series. Fourier Trsorm 3. FT Exmples i 1D D 4. Telescope PSF 5. Importt Theorems Sources: Le boo, Brcewell boo, Wiipedi

2 Je Bptiste Joseph Fourier From Wiipedi: Je Bptiste Joseph Fourier (1 Mrch My 1830) ws Frech mthemtici d physicist best ow or iititig the ivestigtio o Fourier series d their pplictios to problems o het trser d vibrtios. A Fourier series decomposes y periodic uctio or periodic sigl ito the sum o (possibly iiite) set o simple oscilltig uctios, mely sies d cosies (or complex expoetils). Applictio: hrmoic lysis o uctio (x,t) to study sptil or temporl requecies. Fourier Series Fourier lysis = decompositio usig si() d cos() s bsis set. Cosider periodic uctio: x x The Fourier series or (x) is give by: 0 cos si 1 with the two Fourier coeiciets: x b x b 1 1 xcosx dx xsixdx

3 Exmple: Swtooth Fuctio Cosider the swtooth uctio: x x or x x x The the Fourier coeiciets re: b 1 1 x cos xsi x dx 0! x dx 1 (cos() issymmetricroud 0) 1 d hece: 0 x cosx b six si x Exmple: Swtooth Fuctio () x si x

4 Wiipedi: Leohrd Euler ( ) ws pioeerig Swiss mthemtici d physicist. He mde importt discoveries i ields s diverse s iiitesiml clculus d grph theory. He lso itroduced much o the moder mthemticl termiology d ottio. describes the reltioship betwee the trigoometric uctios d the complex expoetil uctio: e i cos i si With tht we c rewrite the Fourier series i i terms o the bsic wves e

5 Deiitio o the Fourier Trsorm The uctios (x) d F(s) re clled Fourier pirs i: F i xs s x e dx For simplicity we use x but it c be geerlized to more dimesios. The Fourier trsorm is reciprocl, i.e., the bc-trsormtio is: ixs x Fse ds Requiremets: (x) is bouded (x) is squre-itegrble x dx (x) hs iite umber o extrems d discotiuities ubouded bouded Note tht my mthemticl uctios (icl. trigoometric uctios) re ot squre itegrble, but essetilly ll physicl qutities re. Properties o the Fourier Trsorm (1) SYMMETRY: The Fourier trsorm is symmetric: I F x P x Q x s P x cosxs i eve 0 0 Q odd dx xsixsdx

6 Properties o the Fourier Trsorm () SIMILARITY: The expsio o uctio (x) cuses cotrctio o its trsorm F(s): 1 s x x F Properties o the Fourier Trsorm (3) LINEARITY: F s Fs TRANSLATION: DERIVATIVE: i s x e Fs x x is Fs ADDITION:

7 Importt 1-D Fourier Pirs

8 Specil 1-D Pirs (1): the Box Fuctio Cosider the box uctio: x 1 0 or - x elsewhere - With the Fourier pirs x si s s d usig the similrity reltio we get: sic s (s) x sic s - Specil 1-D Pirs (): the Dirc Comb Cosider : isx x x e dx FT x 1 Dirc comb series o delt-uctios, spced t itervls o T: x x x x Fourier series 1 ix / T x e (x) Note: the Fourier trsorm o Dirc comb is lso Dirc comb Becuse o its shpe, the Dirc comb is lso clled impulse tri or smplig uctio. (x)(x)

9 Side ote: Smplig (1) Smplig mes redig o the vlue o the sigl t discrete vlues x o the vrible o the x-xis. x x x The itervl betwee two successive redigs is the smplig rte. The criticl smplig is give by the Nyquist-Sho theorem: Cosider uctio bouded support s,. The, smpled distributio o the orm g x x with smplig rte o: 1 x s m x x x s is eough to recostruct (x) or ll x. m s m F, where F(s) hs Side ote: Smplig () Smplig t y rte bove or below the criticl smplig is clled oversmplig or udersmplig, respectively. Oversmplig: Udersmplig: redudt mesuremets, ote lowerig the S/N Alisig: exmple: A mily o siusoids t the criticl requecy, ll hvig the sme smple sequeces o ltertig +1 d 1. They ll re lises o ech other.

10 Side ote: Bessel Fuctios (1) Friedrich Wilhelm Bessel ( ) ws Germ mthemtici, stroomer, d systemtizer were irst deied by the mthemtici Diel Beroulli d the geerlized by Friedrich Bessel. The Bessel uctios re coicl solutios y(x) o Bessel's dieretil equtio: or rbitrry rel or complex umber, the so-clled order o the Bessel uctio. 0 y x x y x x y x These solutios re: 0!! 1 x x J Side ote: Bessel Fuctios () Bessel uctios re lso ow s cylider uctios or cylidricl hrmoics becuse they re oud i the solutio to Lplce's equtio i cylidricl coordites. 0!! 1 x x J

11 Specil -D Pirs (1): the Box Fuctio Cosider the -D box uctio with r = x + y : r 1 or r 1 0 or r 1 r J 1 correspodig FT: (with 1 st order Bessel uctio J 1 ) Exmple: opticl telescope Aperture (pupil): Focl ple: The similrity reltio r J 1 mes tht lrger telescopes produce smller Poit Spred Fuctios (PSFs)! Specil -D Pirs (): the Guss Fuctio Cosider -D Guss uctio with r = x + y : e r e similrity e r e Note: The Guss uctio is preserved uder Fourier trsorm!

12 Importt -D Fourier Pirs

13 Exmple 1: cetrl obscurtio, moolithic mirror (pupil) o support-spiders 39m telescope pupil FT = imge o poit source (log scle) Exmple : cetrl obscurtio, moolithic mirror (pupil) with 6 support-spiders 39m telescope pupil FT = imge o poit source (log scle)

14 Exmple 3: cetrl obscurtio, segmeted mirror (pupil) o support-spiders 39m telescope pupil FT = imge o poit source (log scle) Exmple 4: cetrl obscurtio, segmeted mirror (pupil) with 6 support-spiders 39m telescope pupil FT = imge o poit source (log scle)

15 Covolutio (1) The covolutio o two uctios, ƒg, is the itegrl o the product o the two uctios ter oe is reversed d shited: h x x g x u g x u du

16 g x x F G s s Covolutio () Note: The covolutio o two uctios (distributios) is equivlet to the product o their Fourier trsorms: h x x gx FsGs H s Covolutio (3) Exmple: (x) : str g(x) : telescope trser uctio x g x h x The h(x) is the poit spred uctio (PSF) o the system Exmple: Covolutio o (x) with smooth erel g(x) c be used to smoothe (x) Exmple: The iverse step (decovolutio) compoets, e.g., removig the sphericl berrtio o telescope.

17 Cross-Correltio The cross-correltio (or covrice) is mesure o similrity o two wveorms s uctio o time-lg pplied to oe o them. x x gx u gx udu The dierece betwee cross-correltio d covolutio is: Covolutio reverses the sigl ( - Cross-correltio shits the sigl d multiplies it with other Iterprettio: By how much (x) must g(u) be shited to mtch (u)? The swer is give by the mximum o (x) Covolutio d Cross-Correltio The cross-correltio is mesure o similrity o two wveorms s uctio o oset (e.g., time-lg) betwee them. x x gx u gx udu Exmple: serch log durtio sigl or shorter, ow eture. The covolutio is similr i ture to the cross-correltio but the covolutio irst reverses the sigl to clcultig the overlp. h x x gx u gx udu Exmple: the mesured sigl is the itrisic sigl covolved with the respose uctio Wheres covolutio ivolves reversig sigl, the shitig it d multiplyig by other sigl, correltio oly ivolves shitig it d multiplyig (o reversig).

18 Auto-Correltio The uto-correltio is cross-correltio o uctio with itsel: x x x u x udu + + Wiipedi: The uto-correltio yields the similrity betwee observtios s uctio o the time seprtio betwee them. It is mthemticl tool or idig repetig ptters, such s the presece o periodic sigl which hs bee buried uder oise. Power Spectrum The Power Spectrum S o (x) (or the Power Spectrl Desity, PSD) describes how the power o sigl is distributed with requecy. The power is ote deied s the squred vlue o the sigl: S s F s The power spectrum idictes wht requecies crry most o the eergy. The totl eergy o sigl is: s Applictios: spectrum lyzers, clorimeters o light sources, S ds

19 theorem (or ) sttes tht the sum o the squre o uctio is the sme s the sum o the squre o trsorm: x dx Fds s Iterprettio: The totl eergy cotied i sigl (t), summed over ll times t is equl to the totl eergy F(v) summed over ll requecies v. Wieer-Khichi Theorem The Wieer Khichi (lso Wieer Khitchie) theorem sttes tht the power spectrl desity S o uctio (x) is the Fourier trsorm o its uto-correltio uctio: F F s FT x x s F * s Applictios: E.g. i the lysis o lier time-ivrit systems, whe the iputs d outputs re ot squre itegrble, i.e. their Fourier trsorms do ot exist.

20 Fourier Filterig Exmple Exmple te rom Top let: sigl is I just rdom oise? Top right: power spectrum: high-requecy compoets domite the sigl Bottom let: power spectrum expded i X d Y to emphsize the low-requecy regio. The: use Fourier ilter uctio to delete ll hrmoics higher th 0 Bottom right: recostructed sigl sigl cotis two bds t x=00 d x=300. Covolutio Cross-correltio Auto-correltio Power spectrum theorem Wieer-Khichi theorem h Overview x x gx u gx udu x x gx u gx udu x x x u x udu S F s F s x dx Fds s s FT x x F s F * s