UNIT V: -TRANSFORMS AND DIFFERENCE EQUATIONS D. V. Vllimml Deptmet of Applied Mthemtics Si Vektesw College of Egieeig
TOPICS:. -Tsfoms Elemet popeties.. Ivese -Tsfom usig ptil fctios d esidues. Covolutio theoem..fomtio of diffeece equtios 5.Solutios of diffeece equtios usig -Tsfom
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..
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.
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
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 <
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
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{}
Theoem Fequec shiftig i ii Poof i ii F [ ft] X [ x] ft ft] [ ft F x x] [ x X
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
ii X {x} x, diffeetitig w..t d d X x x {x {x} d d X
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
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
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
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
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]
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
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]
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
B defiitio [x ] Fom equtio d [x ] k [x ] X Y xk k [x] [] Note: [XY] x [X] [Y]
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} ]
-Tsfoms of some bsic fuctios Result { δ} Poof δ fo fo {δ} δ
Result Whee u is uit step sequece [] Note : {k} k{} k if >
Result Poof if } { } { > } {, if / } { <
Result Poof if u} { } { > u} { if / <, < if } { >
Result 5 Poof {} d {} {.} {} d d b Theoem {x} {x} d d d {}
Result 6 Poof } { } { d d } { {x} d d {x} Theoem b } { d d } {
Result 7 Poof { } b Theoem { { {x} } {. } d d d d d d {x} {} - }
Result 8 { } { } Poof :
Result Fid the tsfom of the sequeces f d g Poof {f} { } { } { } {}
{g} { } { } { } {}
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
Result Fid / Poof log L log x x x x... log log
Result
Result Result 5
Result 6 Result 7
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.
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
Model I : Usig covolutio theoem. Usig covolutio theoem, fid the ivese tsfom of Solutio:. ]... [
. Usig covolutio theoem, fid the ivese tsfom of Solutio:.
. Usig covolutio theoem, fid the ivese tsfom of Solutio: b b b. b b b b b b b
b b b... b b b b b b /b b /b b b b b
. Usig covolutio theoem, fid the ivese tsfom of Solutio: / / / / / / / /
... 7 7 7
5. Usig covolutio theoem, fid the ivese tsfom of Solutio:.
... / / / /
6. Usig covolutio theoem, fid the ivese tsfom of Solutio:
Equtio becomes [... ]
6. Usig covolutio theoem, fid the ivese tsfom of Solutio:....
9 9 6 9 6 9 8 9 8. 9
7. Usig covolutio theoem, fid the ivese tsfom of Solutio:.......
.
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.
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
. 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.
lim Re s lim lim! Re d d s lim d d }. { lim }. { Re Re s s f
. 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.
lim! Re d d s } { lim! d d } { lim! d d ]. lim [! ] [ ] [ Re s f
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
, B get we Put B
. 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
. 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
C B of Coeff,. C C / / / f f } { f si cos π π
. 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
A C ----- - C A - f f } { f cos π
lim Res lim 6 8 Re Re s s f
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
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
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
B A of Coeff,. B 5 5 B 5 / 5 / 5 / u 5 5 5 u 5 5 5 } { u u u e i. 5 5. 5..
. 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
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..
. Solve with, usig - tsfom. Solutio: Give Tkig tsfom o both sides, we get ] [ ] [ ] [ } { } { } { } { ] [
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 8 8 8 8 8 8 8 5
/8 5/ /8 8 5 8 } { 8 5 8 8 5 8 5 8.. e i
. Usig -tsfom solve - - give tht, -. Solutio: Chgig ito i the give equtio, it becomes, Tkig tsfom o both sides, we get ] [ ] [ ] [ } { } { } { } { 7 7
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..
5. Usig -tsfom method solve give tht. Solutio: Give Tkig tsfom o both sides, we get ] [ ] [ } { } {
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
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
[ 8 8 ] [ ] [ ] 6 8 8 8 8 i. e.
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
9 9 [ 7 7 ] [9 9] [ ] 5 7 8 7 8 i. e. 6 9