The Fourier Transform

Transcription

1 /9/ Th ourr Transform Jan Baptst Josph ourr Effcnt Data Rprsntaton Data can b rprsntd n many ways. Advantag usng an approprat rprsntaton. Eampls: osy ponts along a ln Color spac rd/grn/blu v.s. Hu/Brghtnss Why do w nd rprsntaton n th frquncy doman? ourr Transform Orgnal Problm Dffcult soluton Soluton of Orgnal Problm Invrs ourr Transform Problm n rquncy Spac Rlatvly asy soluton Soluton n rquncy Spac 3 4

2 /9/ How can w nhanc such an mag? Transforms. Bass unctons.. Mthod for fndng th mag gvn th transformcoffcnts. 3. Mthod for fndng th transform coffcnts gvn th mag. Y Coordnat Grayscal Imag V Coordnats Transformd Imag X Coordnat U Coordnats 6 Chang of Bass Th ourr bass functons u (a,a ) v (a 3,a 4 ) u Bass unctons ar sns and cosns v a 4 a v a 3 u sn() v a a, u u v v a a u u cos() sn(4) whr c, b c T* b c () b() * 7

3 /9/ Th ourr bass functons Th transform coffcnts dtrmn th ampltud and phas: Evry functon quals a sum of sns and cosns 3 sn() A a sn() a sn() -a sn(+φ) + sn(3) B A+B +.8 sn() C A+B+C +.4 sn(7) D A+B+C+D 9 Sum of cosns only symmtrc functons ourr Coffcnts Sum of sns only antsymmtrc functons f() C + C cos() + S sn() C cos() + S sn() +... Trms ar consdrd n pars: C cos() + S sn() R sn(+ θ ) whr R C + S S and θ tan C Usng Compl umbrs: cos(), sn() C cos() + S sn() R θ { { Ampltud+phas 3

4 /9/ Th D Contnuous ourr Transform Th Contnuous ourr Transformfnds () gvn th (cont.) sgnal f(): B π ( ) f() d Th Invrs Contnuous ourr Transform composs a sgnal f() gvn (): f ( ) s a compl wav functon for ach w. π ( ) d 3 Contnuous vs sampld Sgnals Mov from f() ( R) to f( j ) (j Z) by samplng at qualntrvals. f(j) f( +j ) j,,,..., - f() f(j) f( + j ) f( 4 f( + ) +3 ) f() f(3) 4 f() f( + ) 3 3 f( ) f() Th dscrt ourr bass functons ar Th D Dscrt ourr Transform (DT) b π ( ).... ( ) f( ), b( ) f ( ) π,,,..., - or frquncy th ourr coffcnt s: Matlab: fft(f); π π θ / ( R ) π C cos + S sn ( ) R θ Th Invrs Dscrt ourr Transform (IDT) s dfnd as: f ( ) ( ) π Matlab: fft(f);,,,..., - 6 4

5 /9/ Dscrt ourr Transform - Eampl () () () [ +3 -π +4 π +4-3π ] [--]- (3) f() [ 3 4 4] ( ) (f() + f() + f() + f(3)) (+3+4+4) 3 ( ) [ +3 -π/ +4 π +4-3π/ ] [-+] ( ) ( ) [ +3-3π/ +4 3π +4-9π/ ] [--] DT of [ 3 4 4] s [ 3 (-+) - (--) ] 7 Th ourr Transform - Summary () s th ourr transform of f(): { f( ) } ( ) f() s th nvrs ourr transform of (): { ( ) } f( ) f() and () ar a ourr par. f() s a rprsntaton of th sgnal n th Spatal Doman and () s a rprsntaton n th rquncy Doman. 8 Th ourr transform () s a functon ovr th compl numbrs: ( ) R θ Th rquncy Doman ( ) R θ R θ R tlls us how much of frquncy s ndd. θ tlls us th shft of th Sn wav wth frquncy. Altrnatvly: ( ) a b + f() Th sgnal f() Ampltud (spctrum) and Phas ( ) a b + Ral Imag a tlls us how much of cos wth frquncy s ndd. b tlls us how much of sn wth frquncy s ndd. Ral and Imagnary 9

6 /9/ R - s th ampltud of (). θ - s th phas of (). R * () () - s th powr spctrum of (). If f() has a lot of fn dtals, R wll b hgh for hgh. If f() s "smooth, R wll b low for hgh. Eampls 3 sn() + sn(3) +.8 sn() +.4 sn(7) Dmo Th Dlta uncton: f() ( ) g δ( π ( ) δ( ) d f( ) δ R θ lm δ( ) ; δ( ) d ( ) ( ) ) d g Th Constant uncton: f() f ( ) π ( ) d δ( ) R θ ourr ourr Ral Imag Ral Imag 3 4 6

7 /9/ A Bass uncton: f () π Th Cosnuncton: ( ) f ( ) cos π f() π π ( ) d ourr ( ) π d δ( ) R θ f() δ + δ + π π π ( ) ( + ) d [ ( ) ( )] ourr - R θ Ral Imag Ral Imag - 6 Th Sn uncton: ( ) f ( ) sn π δ + δ π π π ( ) ( ) d [ ( ) ( )] f() R θ ourr π/ - -π/ Ral Imag Th Wndow uncton (rct): f() -.. ourr ( ) f < rct ( ) othrws. π. ( π) sn d π R snc ( π)

8 /9/ Proof: -/ / f() rct / () { / othrws Th Gaussan uncton: f ( ) ( ) π π ( ) f ( ) π π π π π [ ] / / d π π [ ] / π / d [ cos( π) sn( π) cos( π) sn( π) ] () f() ourr R sn( π ) SIC ( ) π 9 3 Th Comb uncton: () c δ ( mod ) { c } δ mod C ( ) Proprts of Th ourr Transform Lnarty: [ α f] α [ f] Dstrbutv (addtvty): f() c () ourr R C / () / DC (avrag): Symmtrc: If f() s ral thn, [ f + f ] [ f] [ ] + f ( ) f( ) d * ( ) ( ) thus ( ) ( ) 3 3 8

9 /9/ Dstrbutv: f() { f + g} { f} + { g} g() + f+g Translaton: Transformatons π [ f( )] ( ) Th ourr Spctrum rmans unchangd undr translaton: () + G() π ( ) ( ) ()+G() 33 Scalng: [ f( a ) ] a a 34 Eampl Translaton:.8.6 D Imag ral((u)) mag((u)) (u) 8 6 Chang of Scal: f() f { f( ) } ( ) thn { f( a) } () a a Translatd - - f() () f() () Dffrncs:

10 /9/ Chang of Scal: f() () End f(). (/) f(/) () 37 38