UNIT V: Z-TRANSFORMS AND DIFFERENCE EQUATIONS. Dr. V. Valliammal Department of Applied Mathematics Sri Venkateswara College of Engineering

Transcription

1 UNIT V: -TRANSFORMS AND DIFFERENCE EQUATIONS D. V. Vllimml Deptmet of Applied Mthemtics Si Vektesw College of Egieeig

2 TOPICS:. -Tsfoms Elemet popeties.. Ivese -Tsfom usig ptil fctios d esidues. Covolutio theoem..fomtio of diffeece equtios 5.Solutios of diffeece equtios usig -Tsfom

3 Itoductio The -Tsfom pls impott ole i the commuictio egieeig. I commuictio egieeig thee e two bsic tpes of sigls e ecouteed. The e cotiuous time sigl d discete time sigls. The cotiuous time sigls e defied b the idepedet vible time d e deoted b fuctio ft..

4 O the othe hd, discete time sigls e defied ol t discete set of vlues of the idepedet vible d e deoted b sequece {x}. Fo the cotiuous time sigl, Lplce tsfom d Fouie tsfom pl impott ole. - Tsfom pls impott ole i discete time sigl lsis.

5 Defiitio Let {x} be sequece defied fo, ±, ±, ±,. The the two sided -tsfom of the sequece x is defied s whee is complex vible i geel. Defiitio {x} X If {x} is csul sequece, i.e., x fo <, the the -tsfom educes to oe-sided -tsfom d is defied s {x} X x Note: The ifiite seies will be coveget ol fo ceti vlues of depedig o the sequece x. x x

6 Defiitio The ivese -tsfom of {x} X is defied s Defiitio - {x} {x} The uit smple sequece δ is defied s the sequece with vlues Defiitio 5 δ fo fo Tht uit step sequece u hs vlues u fo fo <

7 Defiitio 6 If ft is fuctio defied fo discete vlues of t whee t T,,,, T beig the smplig peiod, the -tsfom of ft is defied s Now we follow the ottios i [ft F d ii {x} X [ft] ft iii We shll mostl del with oe sided - tsfom which will be hee fte efeed to s - tsfom. ft

8 Theoem The -tsfom is lie i.e., i [ftbgt] [ft] b[gt] ii [{x} b{}] {x} b{} Poof i [ft bgt] [ft bgt] ft b [ft] b[gt] gt ii [{x} b{}] F bg [x b] x b X by {x} b{}

9 Theoem Fequec shiftig i ii Poof i ii F [ ft] X [ x] ft ft] [ ft F x x] [ x X

10 Theoem i [ft] d d [ft] d d F ii Poof i [x] [ft] d d ft [x] X Diffeetitig w..t d d d d [ft] d d F df d ft [ft] ft ft [ft] d F d

11 ii X {x} x, diffeetitig w..t d d X x x {x {x} d d X

12 Theoem i ii [ft T] [F f] k f T f T f[k T] [ft kt] F f T k Poof i [ft T ft T m m f[ T] fmt fmt m m fmt m fmt m Extedig this esult, we get m m f Put [F f] m

13 ii kt]} {f[ T] [ft m k Put kt] f[ k m k m fmt k m m k fmt k m m m m k fmt fmt k k T] f[k... ft ft f F

14 Theoem 5 Shiftig theoem If F the T T [ft] [e ft] F[e ] Poof [e T ft] e T ft fte T F[e T ] [F] e T

15 Theoem 6 Iitil vlue theoem If [ft] F the Poof f lim F F [ft] ft f T f T f T... ft ft f... Tkig limit s lim F f

16 Theoem 7 Fil vlue theoem If [ft] F the Poof lim ft t lim F [ f t T f t] [ f T T f T ] [ f t T ] [ f t] [ f T T f T] F f F [ f T T f T]

17 Tkig limit s lim F f lim [ft T ft] [ft T ft] lim [ft f ft ft... f[ T] ft] lim F f lim f[ T] f f f f f f lim ft t lim F

18 Covolutio of Sequeces The covolutio of two sequeces {x} d {} is defied s x * w w k xk k if the sequeces e o-csul k xk k if the sequeces e csul The covolutio of two fuctios ft d gt is defied b ft gt k fktg[ kt]

19 Theoem Covolutio theoem i ii if {x} Poof i Let X d {} Y, the {x } X Y if {ft} F d { gt} G, the {ft gt} F G {x} X {} Y X Y x x k k x k k

20 B defiitio [x ] Fom equtio d [x ] k [x ] X Y xk k [x] [] Note: [XY] x [X] [Y]

21 ii If F d G e oe sided -tsfom of ft d gt Q m FG fmt gt m ft gt m k [fmtgt [ft gt] [ft gt] k [ft gt] F G m fktg{ kt} fktg{ kt} ]

22 -Tsfoms of some bsic fuctios Result { δ} Poof δ fo fo {δ} δ

23 Result Whee u is uit step sequece [] Note : {k} k{} k if >

24 Result Poof if } { } { > } {, if / } { <

25 Result Poof if u} { } { > u} { if / <, < if } { >

26 Result 5 Poof {} d {} {.} {} d d b Theoem {x} {x} d d d {}

27 Result 6 Poof } { } { d d } { {x} d d {x} Theoem b } { d d } {

28 Result 7 Poof { } b Theoem { { {x} } {. } d d d d d d {x} {} - }

29 Result 8 { } { } Poof :

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31 Result Fid the tsfom of the sequeces f d g Poof {f} { } { } { } {}

32 {g} { } { } { } {}

33 Result : Fid the tsfom Poof { } θ θ ii d i si } cos { cos si } si { cos cos } cos {,.. cos si cos si cos si cos ] si cos ][ si cos [ ] si cos [ si cos } si cos { si cos } si cos { } { } { : } {, } { d get we P I d P R Equtig i i i i i i i i i e put e e besult e e get we e put tht kow We i i i i i i i θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ

34 Result Fid / Poof log L log x x x x... log log

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38 Result

39 Result Result 5

40 Result 6 Result 7

41 Ivese -tsfoms The ivese -tsfom of X defied s - [X] x Whe X [x]. X c be expded i seies of scedig powes of -, b biomil expoetil, logithmic theoem, the coefficiet of - i the expsio gives - [X]. - [X] c be foud out b oe of the followig methods.

42 Methods to fid ivese -tsfom: - [X] c be foud out b oe of the followig methods. imethod-i Usig Covolutio theoem iimethod-ii UsigCuchs esidue theoem iiimethod-iii Usig Ptil Fctios method

43 Model I : Usig covolutio theoem. Usig covolutio theoem, fid the ivese tsfom of Solutio:. ]... [

44 . Usig covolutio theoem, fid the ivese tsfom of Solutio:.

45 . Usig covolutio theoem, fid the ivese tsfom of Solutio: b b b. b b b b b b b

46 b b b... b b b b b b /b b /b b b b b

47 . Usig covolutio theoem, fid the ivese tsfom of Solutio: / / / / / / / /

48

49 5. Usig covolutio theoem, fid the ivese tsfom of Solutio:.

50 ... / / / /

51 6. Usig covolutio theoem, fid the ivese tsfom of Solutio:

52 Equtio becomes [... ]

53 6. Usig covolutio theoem, fid the ivese tsfom of Solutio:....

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55 7. Usig covolutio theoem, fid the ivese tsfom of Solutio:

56 .

57 Model II : Usig Cuch s esidue theoem B usig the theo of complex vibles, it c be show tht the ivese -tsfom is give b x πi c X. d Whee c is the closed cotou which cotis ll the isolted sigulities of X d cotiig the oigi of the -ple i the egio of covegece.

58 B Cuch s Residue theoem. x Sum of the esidue of X - t the isolted sigulities. Whee. Residue fo simple pole is lim [ X.. Residue o ode t the pole is ] d lim! d X

59 . Fid usig esidue method. Solutio: Let { f } f sum of the esidues of. t its poles. i.e. f sum of the esidues of t its poles. Poles of f. e, is the simple pole d is the pole of ode.

60 lim Re s lim lim! Re d d s lim d d }. { lim }. { Re Re s s f

61 . Fid the ivese tsfom of b esidue method. Solutio: Let { } sum of the esidues of f f. t its poles. i.e. f sum of the esidues of t its poles. Poles of f. e is the pole of ode.

62 lim! Re d d s } { lim! d d } { lim! d d ]. lim [! ] [ ] [ Re s f

63 Model III : Usig Ptil Fctios Method Whe X is tiol fuctio i which the deomito is fctoisble, X is esolved ito ptil fctios d the -[X] is deived s the sum of the ivese -tsfoms of the ptil fctios.. Fid Solutio: A B A B Put, we get A A

64 , B get we Put B

65 . Fid usig ptil fctio method. Solutio: Let f f Put A B C A B C, we get B B Put, we get C C Coeff. of, A C A A

66 . Fid b the method of ptil fctios. Solutio: Let f f A B C A B C Put, we get A Coeff. of, A B B 8A A B

67 C B of Coeff,. C C / / / f f } { f si cos π π

68 . Fid the ivese -tsfom of Solutio: Let f f Put A B C D A B C D Coeff. of, we get B, B A C Coeff. of, A B C D A C D A C D

69 A C C A - f f } { f cos π

70 lim Res lim 6 8 Re Re s s f

71 Applictios of -tsfom i Solvig Fiite Diffeece Equtios -tsfom c be pplied i solvig diffeece equtio. Usig the eltios. whee Y [ ] ] [ X m x i m ] [ ] [ Y ii Y iii ] [ ] [ Y iv

72 Applictios of -tsfom i Solvig Fiite Diffeece Equtios. Solve u 6u 9u with u u usig -tsfom. Solutio: Give u 6u 9u Tkig tsfom o both sides, we get u ] 6 [ u ] 9 [ u ] [ { u u u} 6{ u u} 9u { u } 6{ u } 9u

73 9 6 u u u u C B A C B A 5, A get we Put 5 A 5, C get we Put 5 C

74 B A of Coeff,. B 5 5 B 5 / 5 / 5 / u u } { u u u e i

75 . Solve u u u with u, u usig - tsfom. Solutio: Give u u u Tkig tsfom o both sides, we get ] [ ] [ ] [ u u u } { } { u u u u u u } { } { u u u u u 6] 7 [ u

76 7 7 u 7 7 C B A 7 7 C B A 7 8, A get we Put A 7 7, C get we Put C B A of Coeff,. B B u u } { u u u e i..

77 . Solve with, usig - tsfom. Solutio: Give Tkig tsfom o both sides, we get ] [ ] [ ] [ } { } { } { } { ] [

78 A B C A B C Put, we get 9 6 C6 Put, we get C 8 A A 8 Coeff. of, A B C B B B

79 /8 5/ / } { e i

80 . Usig -tsfom solve - - give tht, -. Solutio: Chgig ito i the give equtio, it becomes, Tkig tsfom o both sides, we get ] [ ] [ ] [ } { } { } { } { 7 7

81 7 7 B A 7 B A 5 7, A get we Put 5 A 5 7, B get we Put 5 5B B } { e i..

82 5. Usig -tsfom method solve give tht. Solutio: Give Tkig tsfom o both sides, we get ] [ ] [ } { } {

83 C B A C B A, A get we Put A B A of Coeff,. B B C B of Coeff,. C C } { si cos.. π π e i

84 6. Fom the diffeece equtio whose solutio is Solutio: Give A B A B A B [A B] [A B] A B [A B] [A B] A B Elimitig A d B fom equtios, d, we hve

85 [ 8 8 ] [ ] [ ] i. e.

86 7. Deive the diffeece equtio fom A B- Solutio: Give A B- A- B [A B]- -[A B]- -A- -B [A B]- 9[A B]- 9A- 9B Elimitig A d B fom equtios, d, we hve

87 9 9 [ 7 7 ] [9 9] [ ] i. e. 6 9

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