Lecture 11: A Fourier Transform Primer

Transcription

1 PHYS 34 Fall 1 ecture 11: A Fourier Trasform Primer Ro Reifeberger Birck aotechology Ceter Purdue Uiversity ecture 11 1

2 f() I may edeavors, we ecouter sigals that eriodically reeat f(t) T t

3 Such reeatig sigals ca be well uderstood usig trigoometric fuctios Periodic fuctios: si, cos, si, cos,. si, cos Combiatorial roerties: 1 1 si si si cos 1 1 coscos cos( ) cos( ) si si cos( ) cos( ) d d si cos cos si d d Orthogoality: cos d si d cos msi d cos m cos d m m si msi d m m 3

4 Fourier s Theorem (18): AY eriodic fuctio of (or t) that is eriodic i (or T) ca be writte as a sum of sies ad cosies. I what follows, let s focus o eriodic sigals as a fuctio of legth rather tha time. Assume that we kow f(). f ( ) a a cos si b o 1 (1) () where a f ( )cos d b f ( )si d () a o 1 f ( ) d ote: Eq. 1 is called a Fourier Series 4

5 Cosider the derivatio of the b equatio o revious slide f( ) ao a cos si 1 b Fourier s Theorem m Multily l both sides by si : m m si f( ) ao si m m a cos si si si 1 b 1 Itegrate from to : m m si f( ) d ao si d first term m a cos si d 1 mm b si si d 1 secod term third term 5

6 First Term: m m a si da cos o o m Secod Term: 1 1 a a m cos si d cos ( m) cos ( m) ( m) ( m) cos ( m) cos ( m) 1 1 a 1 ( m) ( m) ( m) ( m) cos ( m) cos [( m) ( m)] rewrite ad ead ( m) ( m) cos ( m) cos ( m) si ( m) si ( m) ( m ) cos ( m) 1si ( m) cos ( m) ( m) ( m) 6

7 1 a m cos si d cos ( m) cos ( m) 1 1 a 1 ( m) ( m) ( m) ( m) a cos ( m ) ( m) ( m) ( m) ( m) a 1 1 ( m ) ( m ) ( m ) ( m ) if ( m) is eve or odd Third Term: mm b si si d 1 si ( m) si ( m) b 1 ( m) ( m).. whe m 7

8 m whe m, b si si d b m si 1 4 m si m b d b m m 4 m 8

9 What we have show m m si f( ) d ao si d 1 m a cos si d m b si si d 1 whe m bm whem m bm si f ( d ) qed.. Please, do the same aalysis for the a o ad a terms 9

10 otatio: ote that the argumets of the si ad cos terms i the Fourier sum ca be writte i terms of waveumbers (k) or wavelegths (λ) f( ) ao a cos si 1 b ao a cos si ; 1 k b k k ao a cos si ; 1 b 1

11 EXAMPE: a eriodic fuctio f() f() Reeat distace A f ( ) ao a cos si 1 b The coefficiets ca be evaluated usig iformatio from the lot: 1 1 ao f( ) d Ad A a f( )cos d si A si A A cos d 11

12 b f( )si d cos cos 1 A A A si d A 1 cos To make it simle, ick =/. The. a b a o A for all A A1 cos for odd (1,3,5...) for eve (,4,6...) 1 A si ote that whe =/, A a with 1 are idetically zero. This is related to the symmetry of f() about =. 1

13 What it all meas f() Reeat distace A Remember, we set =/ f ( ) a o 1 a cos b si 1 A A si 1,3,5 13

14 The magitude of the b coefficiets whe =/ ad A= a o =1/.4 b

15 Plottig the sum of the first few terms whe =/ ad A= Harmoics 1,3,5,7, sum f() (micros) 15

16 The Fourier Trasform What haes whe >>? f ( ) a cos si o a 1 b Progressively more terms are required i the Fourier Series before the waveform is adequately aroimated. I the limit, whe the waveform cosists of oly OE ulse (what s the eriodicity?), a cotiuous distributioib ti of siess ad cosies s are required. 16

17 A A Eamle f() A sigle ulse I this case, f() o loger eeds to be eriodic. To recostruct f(), we ow require a cotiuous distributio of sies ad cosies (k is o loger quatized!). The sums become itegrals. Istead of a ad b, f() is ow secified by some fuctio g(k) accordig to 1 ik f ( ) g( k) e dk Oce f() is kow, the fuctio g(k) ca be calculated from 1 ik g ( k ) f ( e ) d ote: This is called a Fourier Trasform, ot a Fourier Series 17

18 Poits to Poder 1 ik gk ( ) f( e ) d 1. Remember, you kow f() AD you must secify a k. I geeral, g(k ) will be a comle umber 3. The Fourier Trasform is really a ideal model because it requires a fuctio f() that goes from - to + like a double it requires a ifiite umber of k ifiity 4. For every k you secify, you eed to erform the itegral to get the value of g(k ) so you eed to erform a ifiite umber of itegrals 5. Sometimes, the itegratio works out so at the ed of the day, you have a aalytical form for g(k ) this is somewhat of a secial case 6. I a Fourier Series, you ca uderstad oe term, but i a Fourier trasform, it s very difficult to comrehed a sigle term because there are a ifiite umber of them, they sa both ositive ad egative values of k AD g(k ) for a secific k ca be a comle umber 18

19 What the Fourier Trasform does A f() a sigle ulse Pick a k ik ' e k i k cos( ' ) si( ' ) 1 cos( k' ) 1 i isi( k' ) i 19

20 Itegratig Grahically A A ia ia A a sigle ulse 1 Re[ g( k ')] f ( )cos( k' ) d i Im[ gk ( ')] f( )si( k' ) d Re[g(k )] Im[g(k )] k k

21 1 ik ' 1 ik ' g ( k ') f ( ) e d Ae d Workig out the math ik ' ik ' A e A e 1 ik ' ' ik ik ' ik ' ik ' ik ' A ik ' e e A ik ' e e e e ik / ik ' / ik ' ik ' A ik ' e e e k' si A ik ' e k' 1 i k' / ote that ow, there is O restrictio o k 1

22 k si A ik gk ( ) e k k k si si A k A k gk ( ) cos i si k k * g( k) g ( k) g( k) A k si k gk ( ) ( A ) Magitude g(k) k / /

23 Check it out! htt://het.colorado.edu/simulatios/sims.h?sim=fourier_makig_waves 3