Module 2: z-transform and Discrete Systems

Transcription

1 Module : -Trasform ad Discrete Systems Prof. Eliathamy Amikairajah ajah Dr. Tharmarajah Thiruvara School of Electrical Egieerig & Telecommuicatios The Uiversity of New South Wales Australia

2 The -Trasform Similar i meaig ad operatio to the Laplace Trasform i the cotiuous time domai. is a complex variale which plays a role similar il to s i the Laplace Trasform. ELEC97

3 Differece Equatio Frequecy Respose -Trasform ad Trasfer Fuctio System Structure Iverse - trasform ad Impulse Respose

4 Defiitio of -Trasform The trasform of a sequece x is either writte as Zx or, more commoly, X Its full mathematical defiitio is: X x where re jθ θ digital frequecy I practice, we will oly cosider real time i.e. causal sigals i.e. x[] if < X x ELEC97

5 The -trasform is a power series with a ifiite umer of terms ad so may ot coverge for all values of. The regio where the -trasform coverges is kow as the regio of covergece ROC ad i this regio the values of X are fiite. Example: Fid the trasform ad ROC of the step fuctio. x[] X <... This is a geometric series with a commo ratio of -. The series coverges if - < or equivaletly if > X... 5

6 So the Regio Of Covergece ROC is see to e ouded y the circle, the radius of the pole of X Example: Fid the trasform ad droc of the sigal x[] a X a a a for a < or equivaletly, > a. 6

7 Example: Fid the trasform ad ROC of the sigal x[] {,, -7,, 3} 3 7 ] [ x X ROC: etire plae except. 7

8 Properties of -Trasform The trasform X of a sigal x has a umer of importat properties Liearity: ax[ ] y[ ] ax Y Z k Shift: x[ k] X Time reversal: [ ] x X or X Z Multiplicatio y expoetial sequece: a x[ ] X a Differetiatio i the -domai: x [ ] X 8 d d

9 Covolutio: Z{x[]*y[]} X.Y Proof: Proof: [ ] [ ] [ ] [ ] } * { k h k x Z h x Z k [ ] [ ] [ ] [ ] k Lt k h k k h k x k [ ] [ ] [ ] [ ] m h k x k m Let k h k x k m k [ ] [ ] [ ] [ ] m h k x m h k x m k m k [ ] [ ] [ ] [ ] m h k x m h k x m k k m k [ ] [ ] H X H X m h k x k m H X Exercise: Prove all the other properties

10 Examples: Fid the -trasform of the followig sigals Examples: Fid the trasform of the followig sigals a x[] u[-]. [] [ ] u u ; ROC: < [ ] [ ] y[] -α u[--] α < Y α α α < < Y α α <

11 Tale of the trasforms of useful sequeces Basic -Trasforms Sequece -Trasform. Uit sample δ δ -k -k. Uit step u /- 3. Expoetial a u /-a siθ. Siusoidal siθ o u cosθ 5. Uit ramp u cosθ o u cosθ cosθ -Trasform Properties Sequece -Trasform Note: N 3 N... x X x-k -k X a x X/a x- X - x*y X.Y x - dx/d

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13 Example: Determie the -trasform of the followig differece equatio y a x x- c x- Gettig the trasform of each term gives: Y ax - X c - X Y a - c - X, The trasfer fuctio H Y X a c 3 ELEC97

14 Exercise: Dt Determie the trasfer fuctio for the followig differece equatio y a x a x- y- y- ELEC97

15 The iverse -trasform The iverse -trasform allows us to recover the discrete time sequece x - [X] where X is the -trasform of x Usig partial fractio Power series method Evaluatig cotour itegral 5 ELEC97

16 Example Example Fid x for the followig X X g f f ti Nt 3 3 X X 5 3 form : so proper fractio, Not a X X, ] [ ; ; trasforms, - iverse takig ad fractio expasio partial Perfromig x X X 6

17 Example Example Fid x for the followig X g X B A ] [ Ζ Ζ x 7 [ ], >

18 Example Determie the trasfer fuctio for the followig structure: X a Y Z - a Z - Z - a Z - 8 ELEC97

19 Pole-ero descriptio of discrete-time time systems The eros of a -trasform H are the values of for which H. The poles of a -trasform are the values of forwhichh. a a a H... a... M M L L a p p p M L 9 ELEC97

20 The uit circle re jθ where r magitude is real ad θ is the digital frequecy; -π θ π re jθ r e r jθ If r the is a circle of uit radius referred to as uit circle. Im -plae r Re ELEC97

21 Pole-ero diagram A plot of the positio of the poles ad the eros i the complex plae is kow as a pole-ero diagram Poles are show with a x o the diagram Zeroes are show as a o o the diagram Multiple poles, or eroes, at a poit are idicated d with a umer eside the x or o o the diagram ELEC97

22 Example: Poles ad eros Cosider differece equatio y x - x- -. y- A H,.. eros fid Firstly B H 5., at is ero Therefore A., B poles fid The f ELEC97.,. at is pole Therefore B

23 Pole-Zero Diagram for y x -x- -.y-.5 ary part Imagi Real part 3

24 Example: Poles ad eros Cosider differece equatio yx -.8 y- -.9 y B A H, A eros fid Firstly B 9. 8.,, B poles fid The at is ero doule Therefore , at is pole Therefore ± ELEC j ± ± ±

25 Pole-Zero Diagram for y x -.8y- -.9y-.5 Imagi ary part Real 5 part

26 Example: Oe Real Pole iside Uit Circle H, h[ ] α α α α < [ ]; alpha.8; a [ -alpha]; suplot,,,plae,a; suplot,,,imp,a; 6 ELEC97

27 -plae oe real pole iside uit circle.5.5 impulse respose expoetial decay, as θ Imagi ary Part h Real Part sample o ELEC97

28 Example: Complex Cojugate Poles Cosider H with H ero at origi ad re complex cojugate poles Impulse respose iverse trasform h[ ] jθ re jθ r si θ siθ θ This is the impulse respose of the secod order system with complex poles. The impulse respose will decay away to ero if r <. 8 ELEC97

29 iverse -trasform of H jθ jθ θ re re H jθ jθ re re jθ jθ re re jθ jθ jθ jθ re re re re r jθ jθ re re h [ ] si θ siθ θ jθ jθ e e jθ jθ j si θ re re jθ jθ jθ jθ h[ ] e re u[ ] e re u [ ] j si θ 9

30 r ] [ si ] [ u e e j r h j j θ θ θ ] [ ] si[ si ] [ u r h θ θ ] [ ] si[ ] [ si u r h θ θ ] [ ] si[ si ] [ u h θ θ 3

31 Example: Complex Cojugate Poles iside Uit Circle H r r < ad jθ θ j re re r cos θ θ π plae,a to plot pole-ero diagram, where is the umerator cofficiets ad a is deomiator coefficiets. imp,a,n to plot impulse respose with N poits r.7; theta pi/; [ ];a [ *r*costheta r^]; plae, a; imp,a; ELEC97

32 Case I.5 Poles iside uit circle.5 Expoetial decay siewave.5.5 Imagi ary part -.5 θ -θ Decay Evelope Real part 3 ELEC97

33 Example: Complex Cojugate Poles o the Uit Circle r, θπ/ Case II.5.5 cosie wave, o decay Im magiary Par rt h Real Part sample o

34 Example: Complex Cojugate Poles outside tid the Uit UitCircle r>, >θ θπ/ Case III.5.5 Cosie wave icreasig! Imagiary Part θ h Real Part sample o

35 Example: Complex Cojugate Poles r<, θ two real poles iside uit circle Imagi ary Part Real Part 35 expoetial decay, as θ h sample o

36 Example: Complex Cojugate Poles r<, θπ/ Im magiary Part Real Part h sample o 36

37 Example: Complex Cojugate Poles r<, θ3π/ 3 Imagia ry Part h expoetial decay siewave, e icreased frequecy Real Part sample o 37

38 Example: Complex Cojugate Poles r<, θπ.5 expoetial decay siewave, icreased frequecy highest frequecy fs/.5 Imag giary Part Real Part h sample o 38

39 Example: Complex Cojugate Poles r.5 poles ad eros at the origi.5 expoetial decay r ary Part.5.5 Imagi h Real Part sample o 39

40 System Staility A ecessary ad sufficiet coditio that a system oth stale ad causal must have all its poles of the trasfer fuctio, H, iside the uit circle of the -plae, i.e., r <. If the poles are o the uit circle it is called margially stale. The eros may lie iside, o, or outside the uit circle ELEC97

41 Exercise The trasfer fuctio of a discrete-time system has poles at.5,. ±j. ad eros at - ad. Sketch the pole-ero diagram for the system. Derive the system trasfer fuctio H from the pole-ero diagram. 3 Develop the differece equatio. Draw a lock diagram structure of the discrete system. ELEC97

42 Exercise From the pole-ero diagram, write the trasfer fuctio.5.8 Imag giary part Real part

43 Aswer H.6 3 ELEC97

44 Exercise From the pole-ero diagram, write the trasfer fuctio Imag giary part -.8 θπ/ Real part

45 Aswer H ELEC97

46 Staility of a Geeral Secod Order System Geeral secod order system H, are coefficiets Two eros at origi p,p H Poles at p, p p, p ± For staility p <, p < > < For complex poles < or < ± 6 ELEC97

47 For staility,, < < < p p p p < < <

48 Real Poles For real poles ± > If the two poles are real the they must have a value etwee - ad for the system to e stale y < ± < < < ± < < ad < < < < ± ad ad < > < < ad > <

49 Staility Triagle The staility coditios defie a regio i the coefficet plae, which is the form of a triagle paraola Complex Cojugate Poles - - Real Poles

50 Secod Order Resoat System k H k r cosθ r > r cosθ ad r cos θ θ... resoat digital frequecy θ 5 Re r θ p re p re jθ jθ p p Im ELEC97

51 Digital Oscillators Digital oscillator ca e descried y simple differece equatio Usig tale of - trasforms we get P p Acos θ cosθ cos θ Y Let P X > y cos θ y y x cos θ x 5 ELEC97

52 Oscillator Realisatio Oscillator has o iput so x y cosθ y- y- y - y- - y- yacosθ y- - y-

53 Oscillator Iitial Coditios To otai y Acosθ the followig iitial coditios must e used. y Acos A y- Acos-θ Acosθ The frequecy may e tued y chagig g coeffeciet, for oscillator - cosθ 53 ELEC97

54 Oscillator Desig Example Desig a discrete system that will oscillate at a aalogue frequecy of 5H ad amplitude A5 assumig a samplig frequecy of 8kH ππ f θ a fs cosθ Coditios for oscillatio y y 5 y ELEC97

55 y Acos 5 3 fa 5; fs 8; - theta *pi*fa/fs; - N 5; -3 A 5; -*costheta; - ; y []; -5 y A*costheta; t Sample Numer y A; for 3 : N y -*y--*y-; *y ; ed; stemy; hold o,ploty,'g'; xlael'sample Numer'; ylael'y Acosθ ELEC97

56 Sie ad Cosie Oscillators I modulatio schemes, oth - sie ad cosie oscillators are eeded. A structure that delivers sie ad cosie waves simultaeously is - show here. cosθ cosθ siθ -siθ ycosθ xsiθ 56 ELEC97

57 Cosider the followig trigoometric equatios: Cosθ cosθ cosθ siθ siθ Let Therefore Replace y - y cosθ ad x siθ y cosθ y y siθ x y cosθ y- x- siθ. Similarly si θ siθ cosθ siθ cosθ Therefore Replace y - x siθ y x cosθ x siθ y- x- cos θ. Usig equatios ad figure previous page ca e otaied

58 Example: A siusoidal oscillator is show i Figure elow: a Write the differece equatio for the Figure. Assumig x Asiθ, ad y- ; y-. Show, y aalysig the differece equatio, that the applicatio of a impulse at serves the purpose of egiig the siusoidal oscillatio ad prove that the oscillatio is self sustaiig thereafter. By settig the iput ero, ad uder certai iitial coditios, siusoidal oscillatio ca e otaied usig the structure show i Figure. Fid these iitial ii coditios. dii x y - Z - -cosθ θ resoat - Z - frequecy of the digital oscillator

59 a y - y- y- x y y cosθ y- y- A siθ δ cosθ y- y- A siθ δ Therefore y A siθ y cosθ y y- A siθ δ Therefore y cosθ Asiθ Asiθ y cosθ y y A siθ δ y cosθ Asiθ A siθ A cosθ { cosθ A siθ - siθ } A [cos θ -] siθ A [ 3siθ - si 3 θ ] y A si3θ ad so forth. ELEC97

60 y cosθ y- y- x y cosθ y- y- For oscillatio y- ad y- - A siθ {sie term is required} Therefore y - A siθ A siθ Iitial coditios: y- ; y- -A siθ x - Z - Z y y- - Z - y-

61 Relatioship etwee the -trasform ad the Laplace trasform Let e st T samplig period e σ jω T T σ T jω T σ T jθθ Thus e e σt e e e ad θ { ωt θ} As ω varies from to, s-plae is mapped to - plae The etire jω axis i the s-plae is mapped oto the uit circle. The left had s-plae is mapped to the iside of the uit circle ad the right had s- plae maps to the outside of the uit circle 6 ELEC97

62 s-plae is mapped to the -plae jω s-plae σ θπ/ θπ θ -π Im -plae θ Re Uit circle I terms of frequecy respose, the jω axis is the most importat i the s-plae. I this case σ ad the frequecy respose poits i the s-plae are related to poits o the - plae uit circle y σt jθ e. e { σ o} } e jθ

63 Frequecy Respose Estimatio The frequecy respose of a system ca e readily otaied from its -trasform For example, if we set e jθ, that is to evaluate the -trasform aroud the uit circle, we otai the Fourier Trasform of a discrete-time system Hθ. jθ Δ H H e H θ jθ e π θ π 63 ELEC97

64 Aalogue Domai Discrete-Time Domai xt ---- tt x Xs Laplace Trasform X -Trasform s jω e jθ Δ Xjω Xω ωaalogue frequecy jω s-plae θπ jθ Δ Xe Xθ θdigital frequecy Im Stale Stale regio σ regio -plae Re Uit circle ELEC97

65 Relatioship etwee Xθ ad X There is a very simple relatioship etwee the trasform of a sequece x, X, ad the frequecy trasform of the sequece, Xθ Cosider the dfii defiig equatios for the two trasforms: X x X θ x e j θ Clearly, Xθ is got from X with replaced y e jθ 65 ELEC97

66 Complex Nature of Xθ Xθ is, i geeral, a complex umer ad as with ihall complex umers it ca e expressed i polar, rather tha Cartesia form X θ a i X θ e j φ θ X θ a Magitude of X θ φ θ ta Phase agle of X θ a 66 ELEC97

67 Displayig Frequecy Trasform Hece, the frequecy trasform of a sigal is ormally plotted o two separate graphs The first is the magitude of Xθ versus θ - Frequecy vs Magitude Respose The secod is the phase agle of Xθ versus θ - Frequecy vs Phase Respose 67 ELEC97

68 Example: Determie Xθ where xδ a impulse. X replace ye jθ Xθ agitude M.5 Impulse Frequecy Repose Xθ Phas se Agle Frequecy Theta Frequecy Theta -ππ -f s / π f s / θ f A impulse sequece cotais ALL frequecies with a amplitude of p q q p ad a phase of o

69 a Example: If H < a <, Fid H θ. H θ H θ H jθ e ae jθ a cosθ ja siθ a cosθ a cos θ a Xθ asiθ /a /-a -π -f s / π f s / f θ 69

70 Example: Cosider xδ-k, a impulse delayed y k samples Determie Xθ. X -k replace y e jθ Xθ e - jkθ Xθcoskθ - jsi kθ This has a Xθ for all frequecies ut a phase agle φθ -kθ Delayed Impulse Frequecy Repose Magitude Frequecy Theta Agle Phase Frequecy Theta

71 Example: Cosider the differece equatio: y x x- x- H - - Replacig with e jθ yields Hθ e -jθ e -jθ e -jθ [e jθ e -jθ ] e -jθ [Cosθ] Magitude of Ηθ: Ηθ Cosθ Phase agle of Hθ: φθ -θθ Magitude Mag gitude db le radias Phase Ag 5 Frequecy Respose of y[]x[]x[-]x[-] Frequecy Theta Frequecy Theta Frequecy Theta Use freq commad i Matla l for plottig frequecy resposes

72 Exercise The frequecy respose of a ideal lowpass filter i the fudametal iterval -π θ π is give y H θ θ θ c θ θ θ π a Fid the impulse respose of the ideal lowpass filter Sketch the impulse respose for θ c π/6 c 7 ELEC97

73 d e H h a j θ θ π θ Xθ d e j c θ π θ θ π / e e d e j j j c c c ] [ θ π θ θ θ θ π -π θ c -θ c j e e ] [si ] [ θ π c ] [si θ π ; π θ c h /8 si 8 si π π π h 8-8 π π

74 Z-trasform Summary Defiitio ad properties Iverse Z-Trasform Pole ero diagram Staility of secod order system Oscillators Estimatig frequecy respose from pole ero diagram 7 ELEC97

75 Challegig questio If a impulse respose of a system is h[] -α u[--] show that it is a ocausal system 75