from definition we note that for sequences which are zero for n < 0, X[z] involves only negative powers of z.
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1 We ote that for the past four examples we have expressed the -trasform both as a ratio of polyomials i ad as a ratio of polyomials i -. The questio is how does oe kow which oe to use? [] X ] from defiitio we ote that for sequeces which are ero for <, X[] ivolves oly egative powers of. Similarly for sequece which are ero for >, X[] ivolves oly the positive powers of. Properties of the Regio of Covergece for the Z-trasform Uder this sectio we will try to explore the properties of the ROC for the Z-trasforms. Property : The ROC of X[] cosists of a rig i the -plae cetered about the origi. Property 2: The ROC does ot cotai ay poles. Property 3: If ] is of fiite duratio the the ROC is the etire -plae except possibly at ad/or. X [] 2 ] for ot equal to ero or ifiity each term i the sum will be fiite ad cosequetly X[] will coverge. Property 4: If ] is a right-sided sequece ad if the circle r is i the ROC the all fiite values of for which > r will also be i the ROC. Property 5: If ] is a left-sided sequece ad if the circle r is i the ROC the all fiite values of for which < < r will also be i the ROC. Lecture otes prepared by Erha A. Ice
2 Property 6: If ] is a two-sided sequece ad if the circle r is i the ROC the the regio of covergece will cosist of a rig i the -plae that icludes the circle r. Property 7: If the -trasform X[] of ] is ratioal, the its ROC is bouded by the poles or exteds to ifiity. Property 8: If the -trasform X[] of ] is ratioal, ad if ] is right sided, the the ROC is the regio i the -plae outside the outermost pole. Furthermore if ] is causal (equal to ero for < ) the the ROC also icludes. Property 9: If the -trasform X[] of ] is ratioal, ad if ] is left sided, the the ROC is the regio i the -plae iside the iermost oero pole. Furthermore if ] is ati-causal (equal to ero for >) the the ROC also icludes the. Examples: Assume that ] δ[]. Fid the -trasform ad the ROC. Z [] δ[ ] δ the ROC cosists of the etire -plae icludig ad. ow let us cosider a delayed impulse ] δ[-]. δ [ ] Z δ[ ] this -trasform is well defied everywhere except at. ROC cosists of the whole plae except the. (it does iclude ). Lecture otes prepared by Erha A. Ice
3 similarly if we cosider ] δ[+] the the -trasform is δ Z [ ] δ[ + ] + The ROC icludes the whole plae (also ) but ot the poit. Justificatio of Property 4. For a right sided sequece the -trasform will be i egative powers of. Assume a right sided sequece is ero prior to some value of say as show i the plot below: ] If the circle r is i the ROC the ]r - is absolutely summable hece FT exists. ow if we cosider r with r > r positive as show below: we ote that r decays faster tha r for r r Sice ] is right sided it ca ot cause the sequece values to become ubouded for egative values of (because ] - for < ). Cosequetly ] r is absolutely summable ad hece r is i the ROC. Lecture otes prepared by Erha A. Ice
4 Cosider the sigal a ] where, a > otherwise The the Z-trasform is: X [] a ( a ) ( a ) a ( a ) a multiplied by Sice ] os of fiite legth it follows from Property 3 that the ROC icludes the etire -plae except possibly the ad or. I fact sice ] is ero for < (causal) the the ROC will exted to ifiity. However sice ] is oero for some positive values of, the ROC will ot iclude the origi: ] ] i. e ubouded We ote that There is a pole of order (-) at. The roots of the umerator polyomial are at: The root at k cacels the pole at a. k ( 2πk / ) j ae, k,,2,..., The remaiig eros are : k ( 2πk / ) j ae, k,2,..., For < a < ad 6 the pole-ero plot is as follows. Im[] (-)st order pole a Re[] Lecture otes prepared by Erha A. Ice
5 Lecture otes prepared by Erha A. Ice
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