Part B: Transform Methods. Professor E. Ambikairajah UNSW, Australia

Transcription

1 Par B: rasform Mhods Profssor E. Ambikairaah UNSW, Ausralia

2 Chapr : Fourir Rprsaio of Sigal. Fourir Sris. Fourir rasform.3 Ivrs Fourir rasform.4 Propris.4. Frqucy Shif.4. im Shif.4.3 Scalig.4.4 Diffriaio i im.4.5 Diffriaio i Frqucy.5 Problm Sh B Profssor E. Ambikairaah UNSW, Ausralia

3 rasform Mhods h sudy of sigals ad sysms usig siusoidal rprsaio is rmd Fourir Aalysis. h Fourir Sris FS applis o coiuous-im priodic sigals If h sigal is coiuous im ad opriodic, h rprsaio is rmd h Fourir rasform F Profssor E. Ambikairaah UNSW, Ausralia

4 . Fourir sris Ay priodic sigal x [assumig fii powr] wih a priod of scods ca b rprsd as a summaio of si wavs ad cosi wavs. his rprsaio is kow as h rigoomric Fourir sris: i.. x A A cos B si h fudamal frqucy of x is radias pr sc or f Hz. f Profssor E. Ambikairaah ad UNSW, Ausralia

5 I gral, h sigal is also coiuous a scod harmoic a Hz, a hird harmoic a Hz c. h Fourir cofficis A ad B ca b calculad dircly from h sigal usig h followig wo quaios: A B,,,,,...,... 3 x cos d x si d Profssor E. Ambikairaah UNSW, Ausralia

6 Exampl : Cosidr h squar wav x. h priod of h squar wav is scods. Calcula h Fourir cofficis. - x Profssor E. Ambikairaah UNSW, Ausralia

7 cos cos cos cos d x d x d x d x A For his paricular wavform all h A cofficis ar zro ad hc hr will b o cosi wav rms i h summaio of quaio x. h B cofficis ar cos si si si si d d d x d x B Profssor E. Ambikairaah UNSW, Ausralia

8 For v valus of h B cofficis ar also zro. x A A cos B cos si si x si si3 si fudamal d harmoic 4 h harmoic 5 h harmoic As mor harmoics ar addd h summaio gs closr ad closr covrgs o x. Profssor E. Ambikairaah UNSW, Ausralia

9 Qusio 4 Cosidr h squar wav x show i Fig. 3. By calculaig h Fourir sris 4 cofficis, fid h valu of k if h 5h harmoic compo of x is qual o 5 A B 5 k k x - - -k x cos d x cos d x si d si d 4 k cos 5 k 5 5 k x cos d x cos d si d x si d Figur 3 k cos Profssor E. Ambikairaah UNSW, Ausralia [6 marks]

10 By makig us of h gral rlaios: W ca rwri cos si x much mor cocisly as x α α x. d [ ] A [ ] whr ad A cos B si A B α, >, α Profssor E. Ambikairaah A I gral h cofficis α will hrfor hav a complx valu. UNSW, Ausralia

11 Exampl : τ x τ τ Evalua h complx Fourir sris. Profssor E. Ambikairaah UNSW, Ausralia

12 α α x d τ si τ τ α τ Alhough h Fourir cofficis α ar complx, i his paricular xampl hy ar ral. α 3 d his is oly spcifid a paricular frqucis ad ach frqucy has a ampliud. Profssor E. Ambikairaah UNSW, Ausralia

13 x α α α α α α cos x α α si τ τ τ α W hav α α - Whr Profssor E. Ambikairaah UNSW, Ausralia

14 . Fourir rasform F [8] Alhough h Fourir sris is a powrful cocp, i suffrs from o disadvaag, i.. i is oly applicabl o priodic sigals. Mos sigals of pracical imporac ar o priodic. h Fourir rasform pair is giv by X x x d X d Profssor E. Ambikairaah UNSW, Ausralia

15 Exampl: Evalua h Fourir rasform of a rcagular puls show blow: τ τ x A τ τ τ τ A d x d x X Profssor E. Ambikairaah UNSW, Ausralia

16 si X Aτ Sic fucio τ τ τ Aτ X τ 4 τ No: h Fourir rasform is a coiuous fucio of frqucy ad X rprss h disribuio of ampliud wih frqucy Profssor E. Ambikairaah UNSW, Ausralia

17 Exampl: Fid h Fourir rasform of x δ. δ X X δ d Profssor E. Ambikairaah UNSW, Ausralia

18 Exampl: Fid h F of x -a u, a > a d u X a a X Profssor E. Ambikairaah UNSW, Ausralia

19 Exampl : Fid h ivrs Fourir rasform of h rcagular spcrum show blow: X - c c X > Profssor E. Ambikairaah UNSW, Ausralia c c

20 .3 Ivrs F: c c c c d x d X x ; x c c c c si si x c c c si c / c / c x c c c X A ad Psi rofssor E. Ambikairaah UNSW, Ausralia

21 Exampl : Fid h Fourir rasform of x δ-a F{ δ a} δ a a h ampliud spcrum of h impuls fucio is F{δ -a} -a d Sic F{δ-a} -a F - { -a } δ a Profssor E. Ambikairaah UNSW, Ausralia

22 From h ivrs fourir rasform, if a a d δ d a δ F - { -a } δ a Irchagig variabls, w also hav d δ a d δ Profssor E. Ambikairaah UNSW, Ausralia a

23 Exampl: F F { } d { } δ h ampliud spcrum of F{ } d δ δ- δ - is show blow: Profssor E. Ambikairaah UNSW, Ausralia

24 Exampl : F { } { } { } cos F F F δ δ h ampliud spcrum of cos is show blow - { cos } F Profssor E. Ambikairaah UNSW, Ausralia

25 Summary : x A F Aτ X τ τ τ τ A rcagular im-domai puls is rasformd o a sic fucio i frqucy. A c / x c IF A X - c c Profssor E. Ambikairaah A sic fucio i im is rasformd o a rcagular fucio i frqucy. UNSW, Ausralia

26 Exampl : Usig h quaio for h F, valua h frqucy-domai rprsaio for h followig sigals. Skch h magiud spcra. a x -3 u- b x - Profssor E. Ambikairaah c x - u UNSW, Ausralia

27 a x -3 u d X d d x X 3 3 a 9 X X X -.5 Profssor E. Ambikairaah UNSW, Ausralia

28 b x - d d d d d x X X X X X x - X Profssor E. Ambikairaah UNSW, Ausralia

29 c x - u d d X 4 4 a 4 4 X X X.5 X 4-4 Profssor E. Ambikairaah UNSW, Ausralia

30 Exampl : Drmi h im-domai sigals corrspodig o h followig Fs. X - u X - 3 X as show i blow: - X X X - X 4-4 Profssor E. Ambikairaah UNSW, Ausralia

31 i X - u x d d x ii X - d d x Profssor E. Ambikairaah UNSW, Ausralia

32 iii si d d x Profssor E. Ambikairaah UNSW, Ausralia

33 .4 Propris [8].4. Frqucy-shif propry k F x X k Usig h frqucy-shif propry o drmi h F of h complx siusoidal puls. y ohrwis Profssor E. Ambikairaah UNSW, Ausralia

34 W xprss y as h produc of a complx siusoid ad a rcagular puls x Usig h prvious xampl, w obai F x X ad usig h frqucy-shif propry W obai F x X si ohrwis y F si Profssor E. Ambikairaah UNSW, Ausralia

35 .4. im -shif propry of F x F X Exampl: Usig F of h rcagular puls x show i Figur a blow, drmi h F of h im-shifd rcagular puls, y, show i Figur b. - Fig a x y Fig b Profssor E. Ambikairaah UNSW, Ausralia

36 Firs w o y x-, so h im-shif propry implis ha hus w hav Y - X X si, τ Y si Profssor E. Ambikairaah UNSW, Ausralia

37 .4.3 Scalig propry If y xa, h a X a Y Exampl: x - L x b h rcagular puls > x Us h F of x ad h scalig propry o fid h F of h scal rcagular puls y - > y Profssor E. Ambikairaah UNSW, Ausralia

38 Wh, No ha y y X si x, hc applicaio of h scalig propry givs x Y X X si a a si a Profssor E. Ambikairaah UNSW, Ausralia

39 .4.4 Diffriaio i im d d Exampl: x d d F x X a u a u X F X a a Profssor E. Ambikairaah UNSW, Ausralia

40 Ls vrify his rsul by diffriaig ad Fidig h F dircly: d d a u F a a a a a u u a a δ δ a Profssor E. Ambikairaah UNSW, Ausralia

41 .4.5 Diffriaio i Frqucy F d x d X Diffriaio i Frqucy corrspods o muliplicaio i im by Profssor E. Ambikairaah UNSW, Ausralia

42 Qusio 5 Usig h F pair [x ad X] giv blow ad h F propriss aachd, valua h frqucy domai rprsaio Y for h followig sigal y. F x X ohrwis y F x * x X. X Y x * x X. X 4 si x - si y - Profssor E. Ambikairaah UNSW, Ausralia [6 marks]

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