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 34 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 9 How to evaluate the contour integral in the formula for the inverse z transform Start with CauchvGoursat Theorem If Fz is analytic at all points on and interior to a closed contour C then Fzmz o C Nzlt Not A ratl nal in F m ol m I z e o 2 32 p yno Iasm is analytic except at its poles Existence of Laurent Series Consider Hazy Eli if 20 is a pole of order M of Fz then a positive value Dz r1 exists such that Fz can be represented by the Laurent Series in a region 0 lt iz zo lt r1 Fz ianu zo b1 b2 b n0 Zmzo zmzoy zzo39 39 ECE T467667 Introduction to Digital Signal Processing Q 39z1z 2 Exam le Fz Z 32Z 4 For 2 quotnearquot 23 Fz i a z 3quot b1 b2 n0 n 39 Z z 32 For 2 quotnearquot 24 Fz Z a z 4y n0 zeplane x Now consider 2 m 0 C circle centered at z 20 dcde Let cj L dz gt 2n circle centered at 00 39ECE 391 467 667 Introduction to Digital Signal Processing 39 39 Also fa20quot 4m dm0 iii1 c c 39 gt Fzdz 271 b1 C W 0f FZ at its pole at 2 20 small circle centered at z 20 within region where Laurent Series is valid Now consider an Fz with 3 poles Also consider integrating along a closed contour C not necessarily a circle which encloses all three poles XPi X92 Modified contour C which does n9t enclose any poles ECE T467 667 Introduction to Digital Signal Processing From CauchyGoursat Theorem Fzdz 0 since there are no poles C inside 0 gram j H c 4 i0 W C Fzdz Fzdz ch2dz chzdz C C1 C3 C2 V Small circles centered at poles 2njRes of Hz at z p1 Fi39es of Fz at z p2 Res of Fz at 2 p3 Generalize For a rational function Fz Fzdz 21112 Res of Fz at poles inside C CK gm closed contour How to find residues Case 1 Simple pole at z po Laurent series for Fz near p0 b1 2 po Fz fate p0quot n0 b1 residue lim 2 poFz Z Po L where Laurent Series is valid 4 ECE T467 667 Introduction to Digital Signal Processing Example Fz235 2 3 2 4 2 1 Res at 21 lim 2 1 Fz 54 2 91 Res at 23 Iim3z 3 Fz 2 Res at 24 lim 2 4 Fz 3 2 94 Examgle 1 F z z3z 4 Res at 23 Iim z 3 Fz lim 1 1 2 93 z gt3 z 4 Res at 24 lim 2 4 Fz lim 1 1 z gt4 294 Z 3 ECE T467 667 Introduction to Digital Signal Processing Fz poles at z 12 2 2amp2 Resatzz1 limz l 25 E 1 gtO 2 2 H z 2 639 Use I hopital s rule d Z 15Z 1 lim1dzd Iim1 Z 1 2 1 2 2 Z gt 2 Z V 2 dz 2 1 Res at 22 lim 2 3 75le 9 2 2 z z2 0 d m agkz ma1 quotm z 25 52 1 23 z2 d 2Z2 292 22 1 3 Now consider integrating the same Fz Fz along several different closed contours z z C1 39 KR Fzwz 2ch 3 1 Res t1 Res at2 1O7cj ECE T467667 Introduction to Digital Signal Processing 4Fzdz 27tj2 41 C2 39 Res at 15 1 3 g cj39Fzdz 27113 67 V Ca 1 a U Res at 2 C4 39 4Fzdz 2710 0 C4 3 392 J no poles inside C4
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