INTRO TO DIG SIG PRO
INTRO TO DIG SIG PRO E C E 467
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This 7 page Class Notes was uploaded by Eloy Ferry on Saturday September 26, 2015. The Class Notes belongs to E C E 467 at Clemson University taught by John Gowdy in Fall. Since its upload, it has received 62 views. For similar materials see /class/214305/e-c-e-467-clemson-university in ELECTRICAL AND COMPUTER ENGINEERING at Clemson University.
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Date Created: 09/26/15
ECE 467667 Introduction to Digital Signal Processing LECTURE 8 Continue the previous example Response to 5n 1 82 I Ta This 1 0 1 2 n minus Response to a8n a a2 Iaquot lt This 0 1 2 n 1 g m equals 1 0 n This is 6n 1 with ztransform z with R00 of 21 lt oo ECE T467 667 Introduction to Digital Signal Processing Examples using properties 15 to derive additional important ztransforms Zielwonum g ewum Zunu from Property 3 eimo Z z 1 ZZ Z z eJmo ejmo quot391 quot quot1 elwo ROC 21gt1e1 1 Now consider eiwon e jwon 39 3 summon un 3 un 1 Z e w un 3 e39i un using propertyi 1 z z from preVIous example 2Jzequot z equot 22 equotj 22 e190 I L A z ziei equot 2jz eij X2 e jwi 2le2 ziej 0 e jcoo 1J z2jvsinooO 4 zsinco0 12 gt1 Zjlzz choscoOHJ i ZZZZCOSUJO1 ESE 2 ECE l 467667 Introduction to Digital Signal Processing More on ROC of ztransforms Fact If the ztranstorm converges it must converqe uniformly Therefore since 3 xn Slxnr J a necessary and sufficient condition for the convergence of is xnr quot for which his is satisfied are collectively called the ROC of Xz lt mw21xn r n lt oo where z reim Values of r n Note that if a point 21 newquot is inside the ROC of Xz then all other points 2k new same r value are also in the ROCgt Boundaries of anyROC must be circles centered at origin in the zplane Recall for LSl systems the quotiffquot test for BIBO stability is Zlhnl lt oo This is identical to the test for convergence of Hz if r1 gtLS system is stable iff the ROC of Hz includes the unit circleri in 2 plane Fact A ztransform such as Hz is anemic throughout its ROC gt ROC cannot contain any poles Examgles R00 and stability Consider Hz 39 Unit 5 possible ROC39s circle Case 1 121 lt21g Q39jus 1 System not stable sinceunit circle not in ROC ECE T467 667 Introduction to Digital Signal Processing Case 2 13ltzlt23 13 1 ix System not stable 5 v Case 3 23ltzlt43 System is stable C l i 9 since unit circle is inside the ROC 2 43 63 lt25 Case 4 43ltzlt53 9 System not stable l 43 53 ECE T467667 Introduction to Digital Signal Processing Case 5 zgt53 System not stable 3 Recall for LSI systems Yz 2 Xz Hz To find yn need Inverse ztransform Recall forward z transform Fz fnzquot f 1zf0j gz 2m Ellis 39139467 b bquot mtroctucuon to Digital tugmar I39Tocessmg Consider ldz 9 Contour integral aiongC Z C Circle centered at origin C K K On C 2 rejm k dz rejmi w 1 27139 1 I 27 27C dz re wjdw 139 Idea 14 j2n z 0 re1 0 o c Now consider zquotdzn 7t 1 C 2n n 2n n1 New reimjdoa j We do 0 0 zirnHZTejmemm jrn1 ejwn1 2min i 1 0 1 0 ej2nn11 O n 1 gt Fzdz 2njf15 or f Fzdz C C circle in ROC of Fz centered at the origin ECE T467667 Introduction to Digital signal Processing To find fk k 1z 1 MPY Fz by 2 2 Perform contour integration 1 k 1 0k1 1k1 kk1 L SfFzzquot 1 dz o o f 1zk fOzk1 f1zquot 2 o o fkz 1 o o 27tj fk C gt fk Fzzk dz any k 1 2n C circle in ROC of Fz centered at 20 Inverse z Transform
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