Introduction to Fourier Transform

Transcription

1 EE354 Signals and Sysms Inroducion o Fourir ransform Yao Wang Polychnic Univrsiy Som slids includd ar xracd from lcur prsnaions prpard y McClllan and Schafr

2 Licns Info for SPFirs Slids his work rlasd undr a Craiv Commons Licns wih h following rms: Ariuion h licnsor prmis ohrs o copy, disriu, display, and prform h work. In rurn, licnss mus giv h original auhors crdi. Non-Commrcial h licnsor prmis ohrs o copy, disriu, display, and prform h work. In rurn, licnss may no us h work for commrcial purposs unlss hy g h licnsor's prmission. Shar Alik h licnsor prmis ohrs o disriu drivaiv works only undr a licns idnical o h on ha govrns h licnsor's work. Full x of h Licns his hiddn pag should kp wih h prsnaion 4/4/28 23, JH McClllan & RW Schafr 2

3 LECURE OBJECIVES Rviw Frquncy Rspons Fourir Sris Dfiniion of Fourir ransform x d Rlaion o Fourir Sris Exampls of Fourir ransform pairs 4/4/28 23, JH McClllan & RW Schafr 3

4 Fourir Sris: Priodic x x x x a k k / 2 / 2 a k k x k d Fourir Synhsis Fundamnal Frq. 2π / 2πf Fourir Analysis 4/4/28 23, JH McClllan & RW Schafr 4

5 Squar Wav Signal x x + a k 2 2 a k k / 4 / 4 / 4 k /4 k d πk / 2 πk / 2 2πk 4/4/28 23, JH McClllan & RW Schafr 5 sinπk / 2 πk

6 Spcrum from Fourir Sris a k sin πk πk / 2 k, ±, ± 3, K k ± 2, ± 4, K 4/4/28 23, JH McClllan & RW Schafr 6

7 Wha if x is no priodic? Sum of Sinusoids? Non-harmonically rlad sinusoids Would no priodic, u would proaly non-zro for all. Fourir ransform givs a sum acually an ingral ha involvs ALL frquncis can rprsn signals ha ar idnically zro for ngaiv.!!!!!!!!! 4/4/28 23, JH McClllan & RW Schafr 7

8 Limiing Bhavior of FS /4/28 23, JH McClllan & RW Schafr 8

9 Limiing Bhavior of Spcrum 2 4 Plo a k 8 4/4/28 23, JH McClllan & RW Schafr 9

10 4/4/28 23, JH McClllan & RW Schafr FS in h LIMI long priod Fourir Synhsis Fourir Analysis π π π d x a x k k k a d x d x a k k / 2 / 2 a π d 2 lim π k 2 lim lim a k

11 x Fourir ransform Dfind For non-priodic signals 2 π d Fourir Synhsis x d Fourir Analysis 4/4/28 23, JH McClllan & RW Schafr

12 Exampl : x a u a d a a + a+ d a + a > a + 4/4/28 23, JH McClllan & RW Schafr 2

13 4/4/28 23, JH McClllan & RW Schafr 3 Frquncy Rspons Frquncy Rspons is h Fourir ransform of h H u h + u h

14 4/4/28 23, JH McClllan & RW Schafr 4 Magniud and Phas Plos a H + a H an a a H H

15 Exampl 2: x < / 2 > / 2 /2 d d /2 /2 /2 /2 /2 /2 /2 sin / 2 / 2 4/4/28 23, JH McClllan & RW Schafr 5

16 x < / 2 > / 2 sin / 2 / 2 No h widr is x, h narrowr in w! 4/4/28 23, JH McClllan & RW Schafr 6

17 4/4/28 23, JH McClllan & RW Schafr 7 Exampl 3: > < x π sin d d x π π 2 2 x π π 2 2

18 4/4/28 23, JH McClllan & RW Schafr 8 > < x π sin No h widr is w, h narrowr in x!

19 Exampl 4: x δ δ d Shifing Propry of h Impuls δ d 4/4/28 23, JH McClllan & RW Schafr 9

20 x δ 4/4/28 23, JH McClllan & RW Schafr 2

21 Exampl 5: 2πδ x 2π 2πδ d x 2πδ x 2πδ 4/4/28 23, JH McClllan & RW Schafr 2

22 x cos πδ + πδ + 4/4/28 23, JH McClllan & RW Schafr 22

23 Wha aou sin? 4/4/28 23, JH McClllan & RW Schafr 23

24 Fourir ransform of a Gnral Priodic Signal If x is priodic wih priod, x k a k k a k x k d k hrfor, sinc 2πδ k k 2π a k δ k 4/4/28 23, JH McClllan & RW Schafr 24

25 Squar Wav Signal x x a k / 2 a k k d + k d k k / 2 k / 2 k /2 4/4/28 23, JH McClllan & RW Schafr 25 πk πk

26 Squar Wav Fourir ransform x x π a k δ k k 4/4/28 23, JH McClllan & RW Schafr 26

27 al of Fourir ransforms cos c c c x πδ πδ + + δ x > < x π sin 2 / 2 / sin 2 / 2 / x > < u x + 2 c x c πδ sin c c c x πδ πδ + + x δ

28 READING ASSIGNMENS his Lcur: Chapr, Scs. - o -4 4/4/28 23, JH McClllan & RW Schafr 28