Frequency-domain Characteristics of Discrete-time LTI Systems

Transcription

1 requecy-domi Chrcteristics of Discrete-time LTI Systems Prof. Siripog Potisuk LTI System descriptio Previous bsis fuctio: uit smple or DT impulse The iput sequece is represeted s lier combitio of shifted DT impulses. The respose is give by covolutio sum of the iput d the reflected d shifted impulse respose ow, use eigefuctios of ll LTI systems s bsis fuctio

2 Eigefuctios of LTI Systems A eigefuctio of system is iput sigl tht, whe pplied to system, results i the output beig the scled versio of itself. The sclig fctor is kow s the system s eigevlue. Comple epoetils re eigefuctios of LTI systems, i.e., the respose of LTI system to comple epoetil iput is the sme comple epoetil with oly chge i mplitude. s = s + jw = re jw

3 If the iput is lier combitio of comple epoetils, the the output is lso lier combitio of the sme set of comple epoetils both CT d DT. = re jw Z-trsform of DT Sigls The bilterl -trsform of geerl DT sigl is defied s where is comple vrible, i.e., 3

4 Covergece Issue If the ourier trsform of -trsform coverges coverges, the the The -trsform of sequece hs ssocited with it regio of covergece ROC o the comple -ple defied s rge of vlues of for which coverges. E. or r =, the -trsform reduces to the DTT o the uit circle cotour i the comple -ple. Regio of Covergece 4

5 5 E. id the -trsform of the followig right-sided sequece u u 0 This form to fid iverse ZTusig PE E. id the -trsform of the followig left-sided sequece

6 Properties of the ROCs of the -trsform Properties of the ROCs of the -trsform 6

7 Properties of the ROCs of the -trsform The Uilterl Z-trsform - The uilterl -trsform of cusl DT sigl is defied s 0 - Equivlet to the bilterl -trsform of u - Sice u is lwys right-sided sequece, ROC of is lwys the eterior of circle. - Useful for solvig differece equtios with iitil coditios 7

8 E. id the -trsform of the followig sequece = {, -3, 7, 4, 0, 0,..} , 0 3 The ROC is the etire comple - ple ecept the origi. E. id the -trsform of d 3 d with ROC cosistig of the etire - ple. E. 3 id the -trsform of d - d with ROC cosistig of the etire - ple ecept 0. E. 4 id the -trsform of d + d with ROC cosistig of the etire - ple ecept i.e., there is pole t ifiity., 8

9 E. 5 id the -trsform of u with ROC cosistig of the eterior of the uit circle, i.e., u is u 0 cusl or right - sided sequece., E. 6 id the -trsform of Rewritig s sum of left-sided d right-sided sequeces d fidig the correspodig -trsforms, where otice from the ROC tht the -trsform does t eist for b > 9

10 Properties of -Trsform 3 Lierity : by by Properties of -Trsform 4 Z- scle Property: 5 Iitil Vlue : 0 lim 6 il Vlue : Applicbl e oly if lim the ROC of icludes the uit circle, i.e., ll the poles re iside the uit circle 7 Covolutio : h 0

11 Rtiol -Trsform or most prcticl sigls, the -trsform c be epressed s rtio of two polyomils where, d of p, p of b0 D p p p,, the umertor polyomil M,, p 0 re the eroes of,i.e., the roots re the poles of,i.e., the roots the deomitor polyomil. M Rtiol -Trsform It is customry to ormlie the deomitor polyomil to mke its ledig coefficiets oe, i.e., b0 D p p p b M 0 b M b Also, it is cusl sigl, the will be proper rtiol polyomil with M, i.e., # of eroes # of poles. M M

12 where where, for fied r, IDTT DTT A cotour itegrl Iverse -Trsform Sythetic Divisio Method Perform log divisio of the umertor polyomil by the deomitor polyomil to produce the quotiet polyomil 0 q q q r r r r q r q r 0, 0, 0, 0 Write s ormlied rtiol polyomil i by multiplyig the umertor d deomitor by M M r b b b 0 Idetify coefficiets i the power series defiitio of where

13 3 E. id the iverse -trsform of Equtig coefficiets, },0,0,0,0,,0,,0,3,0, { Remrks: This method does t produce closed-form epressio for E. id the iverse -trsform of 3 } 5, 3, {0,,3,3,

14 4 Prtil rctio Method Produce closed-form epressio for Write s sum of terms, ech of which c be iverse -trsformed by usig -trsform tble Suitble for rtiol whose deomitor polyomil hs distict rel poles multiplicity of oe or simple rel poles Prtil rctio Epsio, p p p b i p R p R p i i i i i i,, Give with distict poles i fctored form s epress / i terms of prtil frctio epsio: The iverse -trsform of c be writte s u R p i i i

15 5 E. id the iverse -trsform of u Trsfer uctios Let be oero iput to LTI discrete-time system, d y be the resultig output ssumig ero iitil coditio. The trsfer fuctio, deoted by, is defied: C be determied by tkig the Z-trsform of the goverig LCCDE d pplyig the dely property The system s impulse respose: k M k k k k b k y y Z 0 0 M M b b b Y 0 0 } { } { Y Z y Z Z h

16 BIBO Stbility BIBO = Bouded-iput-bouded-output A lier time-ivrit LTI discrete-time system with trsfer fuctio is BIBO stble if d oly if the poles of stisfy, i p i Tht is, the poles of stble system, whether simple or multiple, must ll lie strictly withi the uit circle i the comple -ple Mrgilly stble simple st order poles o the uit circle 6

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18 E. Cosider d order discrete-time LTI system with y.y 0.3y 0 6 Determie the trsfer fuctio of the system d commet o the stbility of the system. b Determie the ero-stte respose due to uit-step iput d the DC gi of the system. requecy Respose or discrete-time LTI system, the frequecy respose is defied s jw jw Y e e jw e 8

19 9 I terms of trsfer fuctio, w w w, j e j e The frequecy respose is just the trsfer fuctio evluted log the uit circle i the comple -ple. Re Im w e jw periodic i w with period,, s s e T j e f j j e f e e T s j s f j w or geerted by differece eq. with rel coefficiets, Odd fuctio } Re{ } Im{ t Eve fuctio 0, A s

20 E. Cosider d order discrete-time system with 0.64 Plot the mgitude d phse resposes of the system. Determie lso the DC d the high-frequecy gi. 0