DIGITAL SIGNAL PROCESSING LECTURE 5

Transcription

1 DIGITAL SIGNAL PROCESSING LECTURE 5 Fll K8-5 th Semester Thir Muhmmd tmuhmmd_7@yhoo.com Cotet d Figures re from Discrete-Time Sigl Processig, e by Oppeheim, Shfer, d Buck, 999- Pretice Hll Ic.

2 The -Trsform Couterprt of the Lplce trsform for discrete-time sigls Geerlitio of the Fourier Trsform Fourier Trsform does ot eist for ll sigls Defiitio: X Compre to DTFT defiitio: X ( ) [ jω jω ( e ) [ e is comple vrible tht c be represeted s r e jω Substitutig e jω will reduce the -trsform to DTFT

3 The -trsform d the DTFT The -trsform is fuctio of the comple vrible Coveiet to describe o the comple -ple If we plot e jω for ω to π we get the uit circle Im ( ) jω X e Uit Circle r ω Re π π

4 Covergece of the -Trsform ( jω ) jω DTFT does ot lwys coverge [ DTFT does ot lwys coverge X e Emple: [ u[ for > does ot hve DTFT Comple vrible c be writte s r e jω so the - trsform ( jω ) jω [( ) ( X re re [ r ) covert to the DTFT of [ multiplied with epoetil sequece r e jω For certi choices of r the sum mybe mde fiite [ r - < e

5 Regio of Covergece (ROC) ROC: The set of vlues of for which the -trsform coverges The regio of covergece is mde of circles Im Re Emple: -trsform coverges for vlues of.5<r< ROC is show o the left I this emple the ROC icludes the uit circle, so DTFT eists

6 Regio of Covergece (ROC) Emple: Does't coverge for y r. DTFT eists. It hs fiite eergy. DTFT coverges i me squre sese. [ cos ( ω ) o Emple: Does't coverge for y r. [ It does t hve eve fiite eergy. But we defie useful DTFT with impulse fuctio. siω π c

7 Emple : Right-Sided Epoetil Sequece [ u [ X ( ) u [ ( ) For Covergece we require < Hece the ROC is defied s Im o Re < > Iside the ROC series coverges to ( ) ( ) X Regio outside the circle of rdius is the ROC Right-sided sequece ROCs eted outside circle

8 Emple : Left-Sided Epoetil Sequece [ [ [ [ u ( ) [ ( ) ( ) u X ROC < < < : < < < ( ) X

9 Emple : Two-Sided Epoetil Sequece ROC : < [ [ [ u u < + + ROC : > > Im ( ) + X oo + +

10 Emple 4: Fiite Legth Sequece [ ( u [ u [ N ) [ N otherwise N N N X ( ) N N N ( ) ( ) ROC : N < <

11 Some commo -trsform pirs SEQUENCE TRANSFORM ROC δ [ ALL u [ u [ > < δ [ m m All ecept ( if m > ) ( if m < ) or <

12 Some commo -trsform pirs u u [ [ u u [ [ [ ω u[ ( ) ( ) [ cosω [ cosω ROC : > ROC : < ROC : > ROC : < cos ROC : > +

13 Some commo -trsform pirs [ si ω u [ [ siω [ cosω + ROC : > [ cos r cosω ω u ROC : r cosω + r [ r [ [ > r [ r ω u[ [ r siω [ r cosω si ROC : > + r r N otherwise N N ROC : >

14 Properties of The ROC of -Trsform The ROC is rig or disk cetered t the origi DTFT eists if d oly if the ROC icludes the uit circle The ROC cot coti y poles The ROC for fiite-legth sequece is the etire -ple ecept possibly d The ROC for right-hded sequece eteds outwrd from the outermost pole possibly icludig The ROC for left-hded sequece eteds iwrd from the iermost pole possibly icludig The ROC of two-sided sequece is rig bouded by poles The ROC must be coected regio A -trsform does ot uiquely determie sequece without specifyig the ROC

15 Stbility, Cuslity, d the ROC Cosider system with impulse respose h[ The -trsform H() d the pole-ero plot show below Without y other iformtio h[ is ot uiquely determied d > or <½ or ½< < If system stble ROC must iclude uit-circle: ½< < If system is cusl must be right sided: >

16 -Trsform Properties: Lierity Nottio Lierity [ X( ) ROC R [ + b [ X ( ) + bx ( ) ROC R R Note tht the ROC of combied sequece my be lrger th either ROC This would hppe if some pole/ero ccelltio occurs Emple: R [ u[ - u[ - N Both sequeces re right-sided Both sequeces hve pole Both hve ROC defied s > I the combied sequece the pole t ccels with ero t The combied ROC is the etire ple ecept

17 -Trsform Properties: Time Shiftig Here o is iteger [ o X ( ) ROC R o If positive the sequece is shifted right If egtive the sequece is shifted left The ROC c chge The ew term my dd or remove poles t or Emple ( ) X > [ u [ - -

18 -Trsform Properties: Multiplictio by Epoetil ROC is scled by o o [ X( / o ) ROC o R All pole/ero loctios re scled If o is positive rel umber: -ple shriks or epds If o is comple umber with uit mgitude it rottes u[ ROC : > - - Emple: We kow the -trsform pir jω o jωo [ r cos( ω ) u[ ( re ) u[ + ( re ) u[ Let s fid the -trsform of o / jωo re / ( ) + > r X jω re o

19 -Trsform Properties: Differetitio dx ( [ ) ROC R d Emple: We wt the iverse -trsform of X( ) log ( + ) > Let s differetite to obti rtiol epressio dx d ( ) dx( ) + d + Mkig use of -trsform properties d ROC [ ( ) u[ [ ( ) u[

20 -Trsform Properties: Cojugtio * * [ X ( ) ROC R * X ( ) [ X ( ) [ [ ( ) [ ( ) [ { [ } X

21 -Trsform Properties: Time Reversl ROC is iverted [ X( / ) ROC Emple: [ u[ R Time reversed versio of ( ) < X

22 -Trsform Properties: Covolutio [ [ X( ) X( ) ROC : R R Covolutio i time domi is multiplictio i - domi Emple: Let s clculte the covolutio of [ u[ d [ u[ ( ) ROC : ( ) ROC : X > Multiplictios of -trsforms is Y ( ) X ( ) X ( ) X > ( )( ) ROC: if < ROC is > if > ROC is > Prtil frctiol epsio of Y() + y ( u u ) Y ( ) ssume ROC : [ [ [ >