Fourier Techniques Chapters 2 & 3, Part I

Transcription

1 Fourir chiqus Chaprs & 3, Par I Dr. Yu Q. Shi Dp o Elcrical & Compur Egirig Nw Jrsy Isiu o chology shi@i.du usd or h cours: <Modr Digial ad Aalog Commuicaio Sysms>, 4 h Ediio, Lahi ad Dog, Oord Dr. Shi Digial Commuicaios Fourir chiqu Rviw: Compl Epoial Fourir Sris Compl Epoial Fourir Sris Cosidr a sigal, saisis Dirichl codiios: a has oly a ii umbr o maima & miima. b h umbr o discoiuiis mus b ii. c h discoiuiis mus b boudd. Dr. Shi Digial Commuicaios

2 Compl Epoial Fourir Sris h, d Dr. Shi Digial Commuicaios 3 Compl Epoial Fourir Sris his compl poial Fourir sris rprs acly, cp a a poi o ump discoiuiy whr i covrgs o h arihmic ma o h l- ad righ-had limis Dirichl codiios ar suici codiios, o cssary codiios. Ousid h irval, ohig is guarad All o h priodic ucios i pracic oby Dirichl codiios Dr. Shi Digial Commuicaios 4

3 Suici Codiio ad Ncssary Codiio Codiio ad Sam Suici codiio s.c.: I s.c. is saisid, i is guarad ha h sam holds. i I s.c. is o saisid, i is o clar i h sam holds. Ncssary codiio.c.: I h sam holds, i is cssary ha.c. holds. oly i I.c. holds, i is o clar i h sam holds. Som codiio is cssary ad suici i ad oly i i. Dr. Shi Digial Commuicaios 5 Compl Epoial Fourir Sris h abov compl poial Fourir sris slid 3 ca b viwd as a compl orhoormal sris pasio φ, ±, ±,... h compl poial Fourir sris ar uiqu: I o ids a F.S. or h hr is o ohr F.S. pasio or Dr. Shi Digial Commuicaios 6 3

4 Diiio A s o M sigals Orhogoaliy y, y,..., ym is said o b orhogoal, i b c i yi y d i Dr. Shi Digial Commuicaios 7 Eampl: Siusoidal ucio sik ɷ A orhogoal sigal O ampl o M orhogoal sigals πk Si yk b < < b ohrwis k,,..., M I h s, M dir rqucis ar: k :,,..., b b b M b No: Igraio o Si i π, igraio o SiSi i π. Dr. Shi Digial Commuicaios 8 4

5 5 Dr. Shi Digial Commuicaios 9 Compl Epoial Fourir Sris Coicis: im avrag Di h im avrag as d lim d d υ υ υ I o priodic υ I priodic υ Dr. Shi Digial Commuicaios Compl Epoial Fourir Sris Coicis: im avrag h im avrag o, or dc valu o d d d si cos si cos

6 Compl Epoial Fourir Sris Coicis: Symmry I is ral, h * Accordig o diiio Also ca b s rom abov Magiud: Phas agl : Ev symmry Odd symmry Dr. Shi Digial Commuicaios Compl Epoial Fourir sris Coicis: Symmry I is ral ad v, i.., - cos Q si 443 A odd ucio o akig igraio i a v irval wih rspc o ais. is ral is v w.r.. Dr. Shi Digial Commuicaios 6

7 Compl Epoial Fourir sris Coicis: Symmry I is ral ad odd, i.., -- si Imagiary, odd ucio o Q cos 443 odd A odd ucio o akig igraio i a v irval wih rspc o ais. Dr. Shi Digial Commuicaios 3 Fourir chiqu Rviw: rigoomric Form o Fourir Sris Compac orm o rigoomric FS also rrrd o as Polar orm FS cos is ral is Fourir Coici Dr. Shi Digial Commuicaios 4 7

8 8 Dr. Shi Digial Commuicaios 5 rigoomric Form o Fourir Sris coiu,,, si,,, cos si cos A d d B d A B A A L L Quadraur orm o FS Dr. Shi Digial Commuicaios 6 rigoomric Form o Fourir Sris I ihr h rigoomric or h compl poial orm o h F.S. : h avrag o or dc compo o : h udamal compo : h scod harmoic compo : h hird harmoic compo 3 M Maig

9 9 Dr. Shi Digial Commuicaios 7 rigoomric Form o Fourir Sris drivaio rom compl poial orm [ ][ ] [ ][ ] si cos si cos si cos si cos si.cos ormula Eulr Q Dr. Shi Digial Commuicaios 8 I is ral, is v, is odd, w.r.. [ ] [ ] [ ] cos si si cos cos si cos cos si si si cos cos si cos cos si si si cos cos

10 Dr. Shi Digial Commuicaios 9 A ormula i drivaio cos Dr. Shi Digial Commuicaios Diiio o Sic Fucio Also, calld irpolaig ucio. Ev ucio o. Equal o as & si si or π as is a igr. No ha sic du o L Hopial s rul. sic hibis siusoidal oscillaios wih ampliud dcrasig coiuously as /. c si si

11 Figur o sic Dr. Shi Digial Commuicaios Eampl: Rcagl ucio ad is FS Gral orm: τ < ohrwis < ohrwis τ Mos gral orm: A,, A τ Dr. Shi Digial Commuicaios τ

12 Figur o rc Dr. Shi Digial Commuicaios 3 / d / / / co s si d / / si d co s / si π S i π si c π π π Dr. Shi Digial Commuicaios 4

13 Rcagl widow ucio ad is FS Wh,. Wh,. hs mak ss or rc, do hy? Dr. Shi Digial Commuicaios 5 Spcrum FS o A Rcagular Puls rai h puls rai is a sigal o gra irs i digial commuicaios A priodic squial o rcagular pulss Dr. Shi Digial Commuicaios 6 3

14 Spcrum o h Rcagl Puls rai Is compl Fourir sris coici / A A π d π A A π / si π / V π / whr V is h Fourir rasorm o a priod o w [which is a rcagl]. Dr. Shi Digial Commuicaios 7 { A,, A π, odd v c δ W Li spcrum o priodic ucios Dr. Shi Digial Commuicaios 8 4

15 Magiuds o FS coicis, C, o h priodic sigal w Dr. Shi Digial Commuicaios 9 O horm: Li Spcra or A Priodic Sigal I w is a priodic sigal wih priod, h is spcrum Fourir rasorm is: W δ whr, ad ar compl / poial FS coicis, ad δ is a ui impuls ucio. Dr. Shi Digial Commuicaios 3 5

16 Aohr horm: FS Coicis o A Priodic Sigal I w is a priodic sigal wih priod,, ad whr v w w, < v, lswhr h V, V Fv { }, ad /. Dr. Shi Digial Commuicaios 3 Plas rad yoursl, pag 5, Eampl.8, Figur., pag 5, Eampl.9, Figur.4 Dr. Shi Digial Commuicaios 3 6

17 Diiio Comm: Fourir rasorm π π F Compar h pair o ormula wih ha o FS h slid No h similariy d d IF Dr. Shi Digial Commuicaios 33 Rcall: Compl Epoial Fourir Sris d Dr. Shi Digial Commuicaios 34 7

18 Fourir rasorm Dirichl s Codiios. Sigl-valud wih a ii umbr o maima ad miima. A ii umbr o discoiuiis i ay ii im irval 3. Absoluly igrabl, i.., d < hy ar suici codiios or a sigal o hav a F. Dr. Shi Digial Commuicaios 35 Fourir rasorm Loosly spakig, rom Figur. pag 9, o ca s Wh li spcra, dsr, Wh li spcra coiuous, Implis: F o ui rcagl ucio sic ucio * Bgiig o Ch3 i coais a good dscripio o his. Dr. Shi Digial Commuicaios 36 8

19 Ampliud & Phas Spcra F Spcrum θ Compl i gral, v is ral. : Ampliud Spcrum : Phas Spcrum θ Dr. Shi Digial Commuicaios 37 Ampliud & Phas Spcra: Symmry I is ral, is v i.., θ is odd i.., θ θ us similar o h symmry o FS Coicis. Dr. Shi Digial Commuicaios 38 9

20 Ampliud & Phas Spcra: Symmry Q R I m cosπ d siπ d Ev w.r.. Odd w.r.. [ R ] [ I ] Im θ arcg R 443 odd Dr. Shi odddigial Commuicaios 39 m Ev w.r.. Odd w.r.. Symmry Propris I is ral ad v, [ -, v o ] h is ral & v [ - v o ] Q R cos π d, v o v o Im si π d odd o Dr. Shi Digial Commuicaios 4

21 Symmry Propris I is ral ad odd, [ --, odd o ] h is imagiary, odd o. Q R cos π d odd o Im si π d v o, odd o Dr. Shi Digial Commuicaios 4 A τ o Eampl τ τ τ τ Cosidr mporarily Dr. Shi Digial Commuicaios 4

22 τ τ F { } A c o s π d s i π d τ τ τ s i π π A π τ s i A π s i π τ Aτ Aτ s i c π τ π τ Dr. Shi Digial Commuicaios 43 Ar usig h shi propry o F w ca hav: i.., im-dlay propry o F a w slids lar Aτ si c π τ π Dr. Shi Digial Commuicaios 44

23 Diiio Covoluio A spcial yp o igraio. : is a paramr as ar as h igraio is cocrd. λ λ λ dλ λ dλ Dr. Shi Digial Commuicaios 45 Covoluio For a liar sysm: h y y h * h igrad is ormd rom ad by hr opraios:. im rvrsal o λ or λ. im shiig o obai λ or λ 3. Muliplicaio o λ ad λ or ad λ λ Dr. Shi Digial Commuicaios 46 3

24 Covoluio Graphical aalysis o drmi Dir sags Dir igraio limis No a asy ob Fourir rasorm chiqu Mak i asir i h rasormd domai Also calld rqucy domai Dr. Shi Digial Commuicaios 47 Covoluio horm whr Dr. Shi Digial Commuicaios 48 4

25 Liariy o F Suprposiio horm a a a a whr a,a ar cosas Proo: From diiio o F Dr. Shi Digial Commuicaios 49 im-dlay horm π whr Proo: From diiio o F Dr. Shi Digial Commuicaios 5 5

26 Scal-Chag horm a a a Proo: i a > i a < Usig variabl subsiu ad d. o F Dr. Shi Digial Commuicaios 5 Scal-Chag horm Maig: a, i a >, h diagram graph o a is shrikd by a im. Si ad Si Dr. Shi Digial Commuicaios 5 6

27 Dualiy horm I h Proo: From diiio F{ } π d IF π d Dr. Shi Digial Commuicaios 53 Frqucy raslaio horm π Proo: From diiio o IF Dr. Shi Digial Commuicaios 54 7

28 8 Dr. Shi Digial Commuicaios 55 Modulaio horm Proo: Usig Frqucy raslaio horm cos π [ ] cos π π π Dr. Shi Digial Commuicaios 56 Diriaio horm d d π

29 Igraio horm λ dλ δ π Dr. Shi Digial Commuicaios 57 Sigulariy Fucios A impora subclass o apriodic sigals. wo will b iroducd: δ : h ui impuls ucio. u : h ui sp ucio. Dr. Shi Digial Commuicaios 58 9

30 h Ui Impuls Fucio h dla ucio Diiio : δ d is a coiuous ucio a his propry is calld h siig propry Diiio : δ δ d < < Dr. Shi Digial Commuicaios 59 h Ui Impuls Fucio h dla ucio Diiio 3: lim δ ε Dr. Shi Digial Commuicaios 6 3

31 Ay sigal ucio havig a ui ara ad zro widh i h limi as som paramr approachs zro is a suiabl rprsaio or δ. Eampls: riagl ucio. im domai Frq. domai F δ F δ δ Dr. Shi Digial Commuicaios 6 Aohr way o say h abov wo rlaios: δ δ π d π d O mor impora ormula: φ δ φ δ Dr. Shi Digial Commuicaios 6 3

32 Spcrum o a Siusoid Cosidr Acosπ F: A π π π d o o A π o π o d Dr. Shi Digial Commuicaios 63 Q δ π d A [ δ δ ] Dr. Shi Digial Commuicaios 64 3

33 For y A si π A Y Q [ δ δ ] π π π Y d A Dr. Shi Digial Commuicaios 65 F o cos ad si y Dr. Shi Digial Commuicaios 66 33

34 34 Dr. Shi Digial Commuicaios 67 Covoluio o a ucio wih a ui impuls ucio h origial ucio Similarly, i rqucy domai:.,. i ad F δ δ δ Q * * δ δ Dr. Shi Digial Commuicaios 68 Eampl

35 35 Dr. Shi Digial Commuicaios 69 [ ] [ ] Z Z Z Z Z Z Z Z Z δ δ δ δ δ δ Q Dr. Shi Digial Commuicaios 7 Ui Sp Fucio u,, < λ δ udid d u > λ d du δ

36 36 Dr. Shi Digial Commuicaios 7 h ui rcagular puls ca b did as: u u Dr. Shi Digial Commuicaios 7 Muliplicaio horm

37 Eampls Eampl Sic, A Aτ si c τ τ Aτ si c τ A τ A τ dualiy Dr. Shi Digial Commuicaios 73 Eampl :. Aδ A Eampls [ or, δ ] liariy Dr. Shi Digial Commuicaios 74 37

38 . 3. Aδ A Aδ A π [ or, δ ] im dlay liariy Dr. Shi Digial Commuicaios A Aδ π rqucy raslaio Proo:. F{ δ } δ π d π. δ δ, δ : v ucio Dr. Shi Digial Commuicaios 76 38

39 Summary Fourir sris: Compl poial sris rigoomric sris: compac ad quadraur Propris Fourir rasorm Diiio Propris Spcial ucios: sic, rc, dla, sp Ohrs: Orhogaaliy Dirichl codiios Som chiqus i mahmaical maipulaio abls 3. ad abl 3. Dr. Shi Digial Commuicaios 77 39