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# INTRO TO DIG SIG PRO E C E 467

Clemson

GPA 3.84

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This 6 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 47 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 7 N 1 Continue with example XZ E Z 9 M Z l z 1 I 02 For 21 use I hopital39s rule I N z gt l D Z 2 91 391 N l N 1 or evaluate as 21 2 quot 21 N n0 Z1 39 n0 More on z transform of finitelength signals Consider a signal xn which extends from M lt n s N Xz x MzM x M 1zM 1 x 1z x0lg ZN Ii xn 0 for am nlt0 Iim Xz co zzoo is not in ROC 2 If xn O for w ngt0 ling Xz 00 z390 is not in R00 2 Four cases for finite length signals I l l gt ROC zgt0 n O W n quot 39 xn g I gt lZllt nO n xI 39 ECE T467667 introduction to Digital Signal Processing l l L gt ROC OltZltoo J n0 V I gt ROC all 39z W A xn 0 only for n0 ie xnk 8n Properties of zTransforms 1 Linearigy 3 a1x1n azx2 a1X1z a2X2z ROC ze SR1 H 32 intersection of individual ROC s 2Translation 3 Xn no Z X nokm Let m n no 2 xmz quot quot z39 Z xmz m m m X 2 Same ROC as for Xz possibly excluding 20 nogt0 Z nolt0 Example Let x1nun1 2 2 1 3X1n2 3unl 223 2 ROC zgt1 excluding zoq duet Overall ROC 1ltzltoo ECE T467 667 Introduction to Digital Signal Processing Example Let x2n x1n 1 where x1 n is defined above Elxglniizxiz 23 z 1 ROC 1ltzltoo same ROC as for X1z Now consider an LSI system for which the inputoutput relation is expressed as a linear difference equation yin k aiyn 1 Write as M N Zbkxnk2aiyniyn k0 i1 Take ztransform of both sides using Property 1 linearity Property 2 translation bkzukxh YZLZ ajz j 1 HZ System function for discretetime system 1 Note The frequency response of this system is He5 M4226 when the ROC of Hz includes the unit circle39This is true for all stable systems ECE T467667 Introduction to Digital Signal Processing 3 MPY by an exponential 3 aquotxn 2 xnaquotz39n z Zn WET 39 n Xz ROC Rb ltE R z zlt a aRx lt lzl lt a Rx n i Examgle 3 a un 2 If Z 2H2 54 Z a a ROC zigt1aa 5319 If Xz has a pole or zero at z 20 roei030 then 3 laquotxnl has a pole or Z I O I I zero at 20 ie at Z azo aroe wquot If 39a39 IS real and posmve poles and zeros a move along radial lines agt1 Z0 alt1 DD ECE T467667 Introduction to Digital Signal Processing 4 MPY by a quotrgmgquot A x1n nxn 3 X100 znxltngtzn Consider le A V II lm 1 1 M 3 EL 3 LJ gm z zxnx ngtz ltquot same ROC as for Xz Example W nun 3 mom 5 z49 quotlb 11f 23 1 ROC zgt1 same as ROC of un 5 Convolution in time dorhain ltgt MPY in zdomain Let yn xr391 hn Yz znlynz39 k xkhn kz39n n ECE T467667 Introduction to Digital Signal Processing Let mnk vlt2xkrltmgtzltkgt zhltmzmzxltkzk Tzgtxltzgt k ROC of Yz at least 2 e 91x 091 can be more if a zero cancels a pole R00 is 14gt 1 1121 gt a If1a21thisbecomeszgta zgtmax1a If lt 1 this is 21 gt1 Examgle hn aquotun Hz xn 8n 1 a6n Xz z Yz xz Hz 2 zaz az ROC zgta ROC z lt oo ROC 2 lt oo which is more than 91x n 91 a lt 2 lt oo

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