M5. LTI Systems Described by Linear Constant Coefficient Difference Equations

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1 5. LTI Systes Descied y Lie Costt Coefficiet Diffeece Equtios Redig teil: p.34-4, /22/2 I. Discete-Tie Sigls d Systes

2 Up til ow we itoduced the Fouie d -tsfos d thei popeties with oly ief peview of thei use i the lysis of LTI systes I the followig We will develop i oe detil the epesettio d lysis of LTI systes usig the Fouie d - tsfos 3/22/2 I. Discete-Tie Sigls d Systes 2

3 Tsfos d Thei Popeties Z-tsfo ROC s popeties. Syste fuctio H - h - Output espose YHX Ivese -tsfo: Ispectio, Ptil fctio expsio, Powe seies expsio Popeties: lieity, tie shiftig, tie evesl, diffeetitio, covolutio. X e X x jω x e jω Fouie tsfo x Eigefuctios: e jω Fequecy espose He j ω - h e -jω Output espose Ye j ω He j ω Xe j ω Popeties: syety, lieity, tie shiftig, tie evesl, diffeetitio, psevl s theoe, covolutio, odultio. 3/22/2 I. Discete-Tie Sigls d Systes 3 X e j ω 2 π π π X e x e j ω e The Fouie tsfo coespods the - tsfo o the uit cicle i the -ple jw jw d ω

4 3/22/2 I. Discete-Tie Sigls d Systes 4 Relted to Rtiol Fuctios Fist tie of etio is i the 2 ROC discussio A tiol fuctio X is tio of two polyoils i : XP/Q, Zeos: Pc ; poles: Qd Secod tie of etio is i 3 ivese -ts Ay tiol fuctio X c lwys e expessed s su of siple tes, ech of which is tulted Ivese -tsfo Exple 3.9, pge5 Wht id of systes hs the -tsfo s tiol fuctio? d A B X d c X u d A B x δ, > X

5 LTI Systes Descied y LCCDE Lie costt coefficiet diffeece equtios LCCDE is used to descie suclss of LTI systes, which iput d output stisfy th-ode diffeece equtio s y It gives ette udestdig of how to ipleet the LTI systes, such s x x y Z - - Z - Z - 3/22/2 I. Discete-Tie Sigls d Systes 5 - Z -

6 Exples of LCCDE Systes Exple 2.4 Recusive Repesettio of ccuulto y x y y x y y x Exple 2.5 the ovig-vege syste: IR is h/ 2 u-u- 2 -, the y / 2 Σ 2 x- Also thee is h/ 2 δ- δ- 2 -*u, the y-y- / 2 x-x The diffeece equtio epesettios of the LTI systes is ot uique!!! 3/22/2 I. Discete-Tie Sigls d Systes 6

7 3/22/2 I. Discete-Tie Sigls d Systes 7 Hoogeeous Solutios Fo give iput x p, ssue y p is the coespodig output, so tht thee is the the se equtio with the se iput is stisfied y y output of the fo y y p y h whee y h is y solutio to the hoogeeous equtio y h is clled the hoogeeous solutio p p x y h y

8 Hoogeeous Solutios cotiue The hoogeeous solutio y h hs the fo y h Though Σ -, coefficiets c e deteied, while coefficiets A still eed to e deteied, i.e., set of uxiliy coditios is equied fo the uique specifictio of y fo give x. If syste is chcteied y LCCDE d is futhe specified to e lie, tie-ivit, d cusl, the the solutio is uique. I this cse, the uxiliy coditios e efeed to s iitil-est coditios IRC IRC es if the iput x is eo fo less th soe, the y is lwys eo fo less th 3/22/2 I. Discete-Tie Sigls d Systes 8 A

9 3/22/2 I. Discete-Tie Sigls d Systes 9 Suy fo LCCDE Desciptio The output fo give iput is ot uiquely specified, uxiliy coditios e equied If the uxiliy ifotio is i the fo of sequetil vlues of the output, lte vlues c e otied y egig the LCCDE s ecusive eltio uig fowd i, such s Pio vlues c e otied y egig the LCCDE s ecusive eltio uig cwd i, lie If the syste is iitilly t est, the the syste will e lie, tie-ivit, d cusl x y y x y y

10 Exple 2.6 Recusive Coputtio pge LCCDE desciptio Iput xkδ yy-x Auxiliy coditio y-c Recusive coputtio fo >- d <- Cuslity? Lieity? Tie-ivice? Why. If the uxiliy is the iitil-est coditio with y-c, how out the esult? see pge 39 C you get the -tsfo of the LCCDE? 3/22/2 I. Discete-Tie Sigls d Systes

11 3/22/2 I. Discete-Tie Sigls d Systes Syste Fuctios of LCCDE Systes ovelp X d Y of ROCs ivice tie d lieity d c X Y H X Y x y c c

12 Stility,Cuslity of LCCDE Systes The syste is cusl h is ight-sided sequece The ROC of H is outside the outeost pole The syste is stle 3/22/2 I. Discete-Tie Sigls d Systes 2 h is solutely sule The ROC of H icludes the uit cicle h < fo The LTI syste descied y LCCDE is oth cusl d stle, iff the ROC of the coespodig syste fuctio is outside the outeost pole d icludes the uit cicle Exple 5.3 deteie the ROC of y-5/2y-y-2x

13 Ivese Syste of LCCDE Syste The ivese syste is defied to e the syste with syste fuctio H i such tht if it is cscded with H, the ovell effective syste fuctio is uity, i.e., GHH i The tie-doi equivlece is This iplies tht gh*h i δ H i / H Theefoe, the ivese of LCCDE syste is d H i c 3/22/2 I. Discete-Tie Sigls d Systes 3

14 Ivese Syste of LCCDE cotiue I ode to e ivese syste sesile, the ROC of H i d H ust ovelp Exple5.5 H - -.5/-.9 - with ROC >.9 If H is cusl, its ivese syste will e cusl iff the ROC of H i is >x c If H is stle, its ivese syste will e stle iff x c < A stle, cusl LTI syste hs stle d cusl ivese iff oth eos d poles of H e iside the uit cicle iiu-phse syste 3/22/2 I. Discete-Tie Sigls d Systes 4

15 3/22/2 I. Discete-Tie Sigls d Systes 5 FIR, IIR fo Rtiol Syste Fuctios LCCDE tiol syste fuctios -tsfo d its ivese Ipulse esposes Fo tiol fuctio If the syste is cusl, thee is If thee is t lest oe oeo pole of H is ot cceled y eo, the the syste is IIR syste; othewise, the syste is FIR syste d A B H u d A B h δ

16 Execise Five Pole 3.23 o pge32 of the textoo Pole 5.4 o pge33 of the textoo Pole 5.28,, c-i o pge32 of the textoo 3/22/2 I. Discete-Tie Sigls d Systes 6

UNIT V: Z-TRANSFORMS AND DIFFERENCE EQUATIONS. Dr. V. Valliammal Department of Applied Mathematics Sri Venkateswara College of Engineering

UNIT V: Z-TRANSFORMS AND DIFFERENCE EQUATIONS. Dr. V. Valliammal Department of Applied Mathematics Sri Venkateswara College of Engineering UNIT V: -TRANSFORMS AND DIFFERENCE EQUATIONS D. V. Vllimml Deptmet of Applied Mthemtics Si Vektesw College of Egieeig TOPICS:. -Tsfoms Elemet popeties.. Ivese -Tsfom usig ptil fctios d esidues. Covolutio