Fourier Series and Spectrum

Transcription

1 EE54 Signls nd Sysms Fourir Sris nd Spcrum Yo Wng Polychnic Univrsiy Mos of h slids includd r xrcd from lcur prsnions prprd by McCllln nd Schfr

2 Licns Info for SPFirs Slids his wor rlsd undr Criv Commons Licns wih h following rms: Aribuion h licnsor prmis ohrs o copy, disribu, disply, nd prform h wor. In rurn, licnss mus giv h originl uhors crdi. Non-Commrcil h licnsor prmis ohrs o copy, disribu, disply, nd prform h wor. In rurn, licnss my no us h wor for commrcil purposs unlss hy g h licnsor's prmission. Shr Ali h licnsor prmis ohrs o disribu driviv wors only undr licns idnicl o h on h govrns h licnsor's wor. Full x of h Licns his hiddn pg should b p wih h prsnion /5/8, JH McCllln & RW Schfr

3 Wh is Fourir Sris? Any rl, priodic signl wih fundmnl frq. f/ cn b rprsnd s h sum of complx xponnil signls wih frq f SPECRUM: SPECRUM: plo of, Complx Ampliud for -h Hrmonic ANALYSIS: ANALYSIS: Drmin cofficins from x SYNHESIS: SYNHESIS: Gnring x from _ / d x { } N f f x

4 /5/8, JH McCllln & RW Schfr 4 Exmpl: Wri s sum of sin pi nd sin 9pi hn xpnd using complx xponnil sin x x

5 /5/8, JH McCllln & RW Schfr 5 Exmpl sin x x

6 /5/8, JH McCllln & RW Schfr 6 Exmpl In his cs, nlysis us rquirs picing off h cofficins. sin x x

7 Anlysis: x Sp : drmin h fundmnl priod of h signl, Shors inrvl whr signl rps or sisfy x x Sp : using h following formul o compu _: x ω d /5/8, JH McCllln & RW Schfr 7

8 /5/8, JH McCllln & RW Schfr 8 Ex. : SQUARE WAVE...4 x..4 sc. for < < x

9 FS for SQUARE WAVE { } / x d. /.4 /.4 d.4 /.4.4. /5/8, JH McCllln & RW Schfr 9

10 DC Cofficin: / x d x d Ar. d..4.4 /5/8, JH McCllln & RW Schfr

11 Fourir Cofficins is funcion of Complx Ampliud for -h Hrmonic his on dosn dpnd on h priod, ±, ±, l ±, ± 4, l /5/8, JH McCllln & RW Schfr

12 Spcrum from Fourir Sris ω /.4 5 ±, ±, l ±, ± 4, l /5/8, JH McCllln & RW Schfr

13 Ex. : Rcifid Sin Wv { } / x / / / sin / / / / / / d ± / / /5/8, JH McCllln & RW Schfr / d d / / d / Hlf-Wv Rcifid Sin / d

14 /5/8, JH McCllln & RW Schfr 4 ± vn? odd / / 4 / / 4 / / / / / / FS: Rcifid Sin Wv { } 4 ±

15 Spcrum Show plo /5/8, JH McCllln & RW Schfr 5

16 Fourir Sris Synhsis HOW do you APPROXIMAE x? x / d Us FINIE numbr of cofficins N N f x * whn x is rl /5/8, JH McCllln & RW Schfr 6

17 Fourir Sris Synhsis /5/8, JH McCllln & RW Schfr 7

18 Synhsis: s & rd Hrmonics y cos 5 cos 75 /5/8, JH McCllln & RW Schfr 8

19 /5/8, JH McCllln & RW Schfr 9 Synhsis: up o 7h Hrmonic sin5 7 sin5 5 sin5 cos5 y

20 Fourir Synhsis x N sin ω sinω l /5/8, JH McCllln & RW Schfr

21 Gibbs Phnomnon Convrgnc DISCONINUIY of x hr is lwys n ovrshoo 9% for h Squr Wv cs /5/8, JH McCllln & RW Schfr

22 Fourir Sris Dmos Fourir Sris Jv Appl Grg Slbugh Inrciv hp://usrs.c.gch.du/mcclll/5/fsdmo_slbugh/fourir.hml MALAB GUI: fsrisdmo hp://usrs.c.gch.du/mcclll/mlbguis/indx.hml /5/8, JH McCllln & RW Schfr

23 fsrisdmo GUI /5/8, JH McCllln & RW Schfr

24 Fourir Sris Jv Appl /5/8, JH McCllln & RW Schfr 4

25 /5/8, JH McCllln & RW Schfr 5 Alrn Forms of FS Gnrlly, w hv h FS rprsnion If x is rl, _^*_{-} Conug symmry Proof f x * f f x

26 /5/8, JH McCllln & RW Schfr 6 Alrn Forms of FS * f f x A A f A x θ θ θ ; ; cos

27 HISORY Jn Bpis Josph Fourir 87 hsis mmoir On h Propgion of H in Solid Bodis H! Npolonic r hp://www-groups.dcs.s-nd.c.u/~hisory/mhmicins/fourir.hml /5/8, JH McCllln & RW Schfr 7

28 /5/8, JH McCllln & RW Schfr 8

29 READING ASSIGNMENS his Lcur: Fourir Sris in Ch, Scs -4, -5 & -6 Rviw: nir Chp