Revisiting what you have learned in Advanced Mathematical Analysis

Transcription

1 Fourir sris Rvisiing wh you hv lrnd in Advncd Mhmicl Anlysis L f x b priodic funcion of priod nd is ingrbl ovr priod. f x cn b rprsnd by rigonomric sris, f x n cos nx bn sin nx n cos x b sin x cosx b whr h cofficins r compud s f x dx n f xcos nx dx n,,3,! sinx! nd b n f xsin nx dx H. C. So Pg Smsr A,

2 Considr coninuous-im signl x x, for ll H. C. So Pg Smsr A, x priodic wih priod > : : fundmnl priod ω : fundmnl frquncy in rdins/scond x cn lso b rprsnd s complx xponnil funcion x ω ω ω!! ω As x cn b complx-vlud, ll s r gnrlly complx : DC componn ±ω ± : fundmnl firs hrmonic componns ± ω ± : scond hrmonic componns : hird hrmonic componns ω

3 H. C. So Pg 3 Smsr A, Exmpl Considr priodic signl x, which is of h form 3 3 x whr,,, h fundmnl frquncy ω or. x conins only h firs hr hrmonic componns, funcion rl cos6 3 cos cos ] [ 3 ] [ ] [ x

4 s hrmonic componn DC componn nd hrmonic 3rd hrmonic componn H. C. So Pg Smsr A,

5 As in h bov xmpl, for rl priodic signl x, is complx conug quls islf, i... x x Sinc h conug of ω is ω, w hv ω ω ω x con x or x, hus ω ω ω [ ] R{ } Wriing s polr form, i.., x θ A, givs A cos ω θ H. C. So Pg 5 Smsr A,

6 Alrnivly, wri B C, x [ B cos ω C sin ω] o summriz, for rl priodic signl, h following hr forms of Fourir sris rprsnion r quivln: ω summion of complx xponnils A cos ω θ [ B cos ω C sin ω] summion of cosin funcions of non-zro phs summion of cosin nd sin funcions wih zro phs H. C. So Pg 6 Smsr A,

7 Compuing Fourir Sris Cofficins Givn ω x, nω ω nω n ω x Ingring boh sids from o, nω x d d [ d] n ω n ω Knowing h n,, n ω, x nω d d n n,, ±,, for n ±! H. C. So Pg 7 Smsr A,

8 Exmpl Find h Fourir sris cofficins of x sin ω. Soluion: By Eulr s rlion, x sin ω hus,, Exmpl 3 Find h Fourir sris cofficins of Soluion: ω ω, nd x sin ω cos ω cos for n ±, ± 3,! ω ω ω ω ω x. H. C. So Pg 8 Smsr A,

9 hus,,,, for n ±3, ±,! R{ } Im{ } Ampliud { } n Im{ R{ } } Phs H. C. So Pg 9 Smsr A,

10 Exmpl Considr h following priodic squr wv Find is Fourir sris rprsnion. Soluion: h signl is priodic wih priod. Also, i is n vn signl. o, < < x, ohrwis Ovr h spcific priod from H. C. So Pg Smsr A,, h signl is dfind s,

11 I is sy o show h ω x d x [, ω hus w cn do h ingrion ovr ] ω x d ω d d For, d, For, ω ω sin ω d ω H. C. So Pg Smsr A,

12 8 6 H. C. So Pg Smsr A,

13 Convrgnc Problms of Fourir sris h problms: h ingrl ω x ω x x ω d my no convrg, i.. Evn if ll s r fini, h summion b qul o h originl signl x. ω my no Virully ll priodic signls rising in nginring do hv Fourir sris rprsnion, wihou convrgnc problms. d H. C. So Pg 3 Smsr A,

14 Dirichl Condiions of Convrgnc For priodic signl x o hv convrgd Fourir sris, x mus b bsoluly ingrbl ovr ny priod, i.. x d < for ny x d unboundd ims of oscillion x hs fini numbr of mxim & minim ovr ny priod 3 x hs fini numbr of disconinuous ovr ny priod infini numbr of disconinuiis H. C. So Pg Smsr A,

15 Convrgnc Disconinuiis If priodic signl hs no disconinuiis, is Fourir sris rprsnion! convrgs, nd! quls h originl signl vry vlu of. If h signl hs fini numbr of disconinuiis in ch priod, is Fourir sris rprsnion! quls h originl signl vrywhr xcp h disconinuiis! convrgs o h midpoin of x ch disconinuiy H. C. So Pg 5 Smsr A,

16 Exmpl 5 x, < <, ohrwis Expnding x, ω x, whr sin ω nd L i.. x N dno h Fourir sris runcd h N h hrmonics, N N N ω x x N bcoms good pproximion of x if N is lrg nough. H. C. So Pg 6 Smsr A,

17 H. C. So Pg 7 Smsr A,

18 Propris of Fourir sris Noions: h rlion h rprsnd by Assum x x s r Fourir sris cofficins of FS. nd y b FS h sm fundmnl priod or Signl Linriy im shifing Frquncy shifing Mω FS nd x nd. ω FS cofficins Ax By A Bb x ω x M x is y hv H. C. So Pg 8 Smsr A,

19 Signl FS cofficins Conugion x * im rvrsl x im scling x α priod chngd o α Priodic convoluion x y d Muliplicion Diffrniion Ingrion b x y dx d l l b ω x d ω l H. C. So Pg 9 Smsr A,

20 Exmpl 6 Driv h Fourir sris rprsnion of h following signl. Priod Soluion: Dfin g ovr priod, L ρ g < < g FS ρ, for, ρ [ d d] sin / d d, H. C. So Pg Smsr A,

21 Alrnivly, w cn us propris of Fourir sris o solv i. Rclling h signl x in Exmpl : g cn b xprssd s By h propry of im-shif, x FS g x, wih & ω x sin ω, / / FS Subrcing h DC offs of from, ρ, sin,,, H. C. So Pg Smsr A,

22 H. C. So Pg Smsr A, No h whn w chos h inrvl [,: d d / 8 / 8 [ ]

23 [ ] " for ingr sin / / / / [ / / ] sin / On h ohr hnd, if w choos [-, d d H. C. So Pg 3 Smsr A,

24 H. C. So Pg Smsr A, [ ] [ ] / / sin / 8 / 8 You cn choos inrvls [3,7, [,8, c. M us of you cn g h sm nswr.

25 Fourir sris of vn & odd signls Rl vn signls By h propry of im rvrsl, w hv x is vn x x x is rl hrfor, w hv For rl vn signl { } r rl: } r vn: { x FS x FS x FS. for ll < <. x cos ω ω ω H. C. So Pg 5 Smsr A,

26 Rl odd signls Similr o h bov drivion, w hv. For rl odd signl { } r purly imginry: { } r odd: 3 ; L C C is rl nd x, whr C C ω ω C C sin ω H. C. So Pg 6 Smsr A,

27 Prsvl s horm Avrg powr ovr priod x d Sum of squrd mgniuds of ll hrmonics Rcll ω x Avrg powr of h h hrmonic ω d h ol vrg powr ovr priod quls h sum of h vrg powrs in ll of h hrmonic componns. H. C. So Pg 7 Smsr A,

28 Proof: / / x d / mω m m n m / m mn m / * n / n mn ω nω d * d Exmpl 7 Suppos w wn o us whr hrmonics, i.. g N o pproxim g of Exmpl 6 g N dnos h Fourir sris runcd h N h N so h N N ρ N ω g. Wh is h minimum vlu of g N conins ls 9% powr of g? H. C. So Pg 8 Smsr A,

29 H. C. So Pg 9 Smsr A, Soluion: Avrg powr of g d d g ry N, w hv.6 / sin ρ ρ ρ ry N, sin ρ ρ ρ odd is, ρ ry 3 N, w hv / 3 sin > ρ ρ ρ Hnc h minimum vlu of N is 3.

30 Discr-im Fourir Sris! Fourir Sris is frquncy nlysis ool for coninuous-im priodic signls whil Discr-im Fourir Sris DFS is usd for nlyzing discr-im priodic signls! In fc, DFS cn b drivd from h Fourir Sris I cn show you if you r inrsd on his! Similr o coninuous-im cs, discr-im signl x [n] is priodic wih fundmnl priod N if x [ n] x[ n N] whr N is h smlls ingr for which h quion holds. h fundmnl frquncy is dfind s ω / N.! Howvr, noic h smpling coninuous-im priodic signl dos no ncssrily giv discr-im priodic squnc. H. C. So Pg 3 Smsr A,

31 Exmpl 8 Considr coninuous-im priodic signl cos.5 Smpling x wih smpling priod of s givs [n] { x[ n]} {! cos.5,cos,cos.5,cos,!} x is no priodic H. C. So Pg 3 Smsr A, x. Considr nohr coninuous-im priodic signl y cos.5. Smpling y wih smpling priod of s yilds { y[ n]} {! cos.5,cos,cos.5,cos,!} {!,,,,,!} y [n] is priodic nd h fundmnl priod is N W cn us DFS o nlyz y [n] bu no x [n]. In fc, w cn us discr-im Fourir rnsform DF, which will b discussd lr, o nlyz x [n].

32 Fourir Sris & LI Sysms h oupu of coninuous-im priodic signl x o LI sysm wih impuls rspons h is givn by y x h x h d whr y is lso priodic wih h sm fundmnl frquncy s. h is, if x ω ω x y b only phss nd mgniuds r chngd H. C. So Pg 3 Smsr A,

33 H. C. So Pg 33 Smsr A, Proof: d x h d h x h x y Considr h h componn of x, i.., ω. h sysm oupu du o his componn is: ω ω ω d h d h y whr w cn s h ω d h b Combining ll hrmonic componns, w hv b y y ω

34 H. C. So Pg 3 Smsr A, Exmpl 9 Givn LI sysm wih impuls rspons u h nd n inpu signl.5cos x. Find h sysm oupu y. Soluion: cos cos d d d d d u d x h y

35 H. C. So Pg 35 Smsr A, sin cos sin cos sin cos R sin cos R R * y

36 Exprss y s funcion of cos y only, w g cosn cos sinn sin cos cos n sin Compring x.5cos nd y, w s h hy r of h sm frquncy bu hir phss nd mgniuds r diffrn. Nvrhlss, w noic h compuing h convoluion is dious vn for simpl signl nd sysm. W will solv his problm gin wih h us of Fourir rnsform, which is shown o b sir. H. C. So Pg 36 Smsr A,