INTRO TO DIG SIG PRO
INTRO TO DIG SIG PRO E C E 467
Popular in Course
Popular in ELECTRICAL AND COMPUTER ENGINEERING
E C E 211
verified elite notetaker
This 8 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 82 views. For similar materials see /class/214305/e-c-e-467-clemson-university in ELECTRICAL AND COMPUTER ENGINEERING at Clemson University.
Reviews for INTRO TO DIG SIG PRO
Report this Material
What is Karma?
Karma is the currency of StudySoup.
You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!
Date Created: 09/26/15
ECE 467667 Introduction to Digital Signal Processing LECTURE10 Finding residue at a pole of order gt 1 Examgle Fz has pole of order 4 atfzpo b b b b Thean az p 1 2 3 4 quot 2 ltzpor z pogt3 2 por for O lt 2 p0 lt r1 due to the existence of Laurent Series Define ltIgtz z p04 Fz DZ Zanz pon4 quot39b1zquotpo3 b z po2 b32quotpob4 n0 1912 n 4X2 p0 gt63 3b1z po2 2b2 2 quot p0 ha I Zan n0 2 Zan n0 n4n3z p0n2 32b1zp02b2 13quot2 i an n 4n 3n 2z p0n1 3 2b1 n0 lt1quotz i 2b d b Im Szggoq Z 3 1 an 1 290 In general the residuebl of Fz at a pole of order M at z p0 is 1 dM M r m F M 1iz391gtrgodz T JZ p 2 V z ECE T467 667 Introduction to Digital Signal Processing Examgle Fz l 2 12z 2 Res of Fz at 22 limz 2Fz 1 1 z2 2 12 Res of Fz at 21 um umW nm391 1 z ndz 2 2 H1 2 2 2412 2 Now use above method toevaluate inverse z transform Given Fz with ROCz gt 2 212Z 2 2 rightsided signal 1 Zn 1 12 27 W2 2 12z 2 and enclose orlgln 1 11 1 z 512712 Resgm at poles mSIde C 2 gt 2 For n 21 only poles inside C are at 21 22 Therefore for n 21 01 Zn 1 at 21 Res at 22 z 1gt2ltz 2 pow2 fn Res ECE T467 667 Introduction to Digital Signal Processing At 211 n 1 n2 n 1 mdz quotmz 2n 1z z 1 z gt1 dz 2 2 2 1 Z 22 lt 1ltn 1gtlt11 n 39quot 1 A1222 n 1 lim 2 2 2 1 z gt2 z Therefore for n 21 fn 2 quot n For nlt1 recall Fz Lf 222 t 1gf0ig 1 2 g 22 J Since ROC lslzl gt 2 all For large 2 as z gt oo fn must be 0 for nlt0 these terms 9 0 otherwise these terms would blow up a z gt co gt imam f0 umvo 2 2 1 z2 Overall result fn 2 1 njum 1 ECE T467 667 Introduction to Digital Signal Processing Notethat f12 1O f221 2O f32231 f423 44 etc in general if the ROC of Fz has the form lzl gt a Then fn must be 0 for all nltO and f0 lim Fz Zv Also If the ROC of Fz has the form al lt z lt oo then fn 0 for a finite no of negative n values and lim Fz co z wo Examgie 4 Let F1z Fz 24 Z 21222 New R00 2 lt z lt co Recall 24Fz lt gt fn 4 gt f1n fn 4 gt f1 l f3 1 4 z and llm F1z 7 gt oo Z o v Z I ECE T467 667 Introduction to Digital Signal Processing Examgle z Fz R0012 lt la 2 a n I fnj Z z 1 1 2 dzquot 2174 C z a 2njCz a 144 For n 2 0 no poles inside C gt fn O for n 2 O For nlt0 poles QLQ present inside C at 20 2 fn 22 O for nlt0 f 1 Re 3 at 20 zz a 1 1 llm z aOZ a a 1 f 2Res 2 at 20 2 z a d 1 1 1 Im Im z gtOdz Z a z gt0 2 432 a2 Simiiarly f 3 1 a39nd 4 I a x a 5 39 ECE T 467 667 Introduction to Digital Signal Processing General method for nding fn when Fz has a ROC of the form lzl lt la Start with fn Fzzquotquot dz n zltr1 Let p gtzl and dz1 2 dp Z p P Let 2 m on C with rltr1 2 p 19 5 on the new contour C in pplane r As angle of 2 on C goes from 0 a 2n counterclockwise angle of p on 0 goes from 27 gt O clockwise n1 1 1 1 1 3111 33 quot F quot H 27 91110 1 DZ 3 lt r1 Reversing directions of integral cancels the sign 1 1 fn F Ep 1dp E Useful when Fz has 3 ROC J 1 inside a circle or between two circles 391 of finite radius l gt fn ZRes Fp quot1 at poles inside 039 pl gtrl where ROC of 1 Fz is z lt M ECE T467 667 Introduction to Digital Signal Processing Apply to previous example Fz E E a ROC l2 lt Ia p39quot 1 at poles inside 0 kv J I No poles inside 6 for nlt1 1 pole at inside 039 a For nlt1 1 1 1 fn Ilm quotquot1 t plp a1 app a a 1 p ajp 1 n1 n a im1 a ns1 f T m E p a a a D We already showed that xn O for n 2 O by working in the z plane for the previous Fz and the same ROC ECE T 467 667 Introduction to Digital Signal Processing Overall result 39 fn a u n 1 B for n gt 1 Go back to general form of Fz Fz f2z2 f 1z fO igltgl all fn must be 0 for n 21 Since ROC includes 20 f0 mm 0
Are you sure you want to buy this material for
You're already Subscribed!
Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'