8. INVERSE Z-TRANSFORM
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1 8. INVERSE Z-TRANSFORM The proce by whch Z-trnform of tme ere, nmely X(), returned to the tme domn clled the nvere Z-trnform. The nvere Z-trnform defned by: Computer tudy Z X M-fle trn.m ued to fnd nvere Z-trnform. Emple 8.» ym X() 5 6» =/(^+5+6);» trn() U U n = (-)^-(-)^ Four prctcn technque cn be ued to mplement n nvere trnform. They re:. Long dvon. Prtl frcton. Redue theorem. Dfference equton 8. Invere Z-trnform v long dvon For cul equence, the -trnform X() cn be epended nto power ere n. For rtonl X(), convenent wy to determne the power ere n epnon by long dvon. n n bn b n... b b X() c c c... n n... n where c,c,c... re power ere coeffcent. Emple 8. X() X[]=[-]+.[-]+ [-]+-[-5]...
2 Computer tudy The nvere of rtonl -trnform cn lo be redıly clculted ung MATLAB. The functon mp cn be utled for th purpoe. Three veron of th functon re follow: [h,t]=mp(num,den) [h,t]=mp(num,den, L) [h,t]=mp(num,den, L, FT) Where the nput dt cont of the vector num nd den contnng the coeffcent of the numertor nd the denomntor polynoml of the -trnform gven n the decendng power of, the output mpule repone vector h, nd the tme nde vector t. The frt form, the length L of h determned utomtclly by the computer wth t=:l-, where n the remnng two form t uppled by the uer through the nput dt L. In the lt form, the mplng ntervl. The defult vlue of FT. The followng two FT h, t mp num, den fle to nd plot power. emple how pplcton Emple 8. X ( ).» num=[ ]; Power ere coeffcen for X ( )» den=[ -. +];.» L=8;» [,]=mp(num,den,l).5 = = » tem(,, fll, ) 5 6 7
3 X ) ( Emple 8.» num=[ -];» den= [ ];» L=;» [,t]=mp(num,den,l) = » tem(,, fll, ) Power ere coeffcent for X ) ( The Invere Z-Trnform Ung Prtl Frcton We now derve the epreon for the nvere -trnform nd outlne the two method for t computton. j Recll tht, for re, the -trnform G() gven by the equton merely the Fourer trnform of the modfed equence gr. Accordngly, by the nvere Fourer trnform, we hve: g r G(re j )e j d (8.) j By mng the chnge of vrble re, the bove equton cn be converted nto contour ntegrl gven by : g redue of G() t the pole nde C (8.) Note tht theequton mentoned bove need to be equted t ll vlue of whch cn be qute complcted n mot ce. A rtonl G() cn be epreed : G() P() D() 5
4 where P() nd D() re the polynoml n. If the degree M of the numertor polynoml P() greter thn or equl to the degree N of the denomntor polynoml D(), we cn dvde P() by D() nd re-epre G() : M N P ( ) G( ) D( ) where the degree of the polynoml P () le tht tht of D(). The rtonl functon P() clled proper frcton. D() The epreon of Eq (8.) cn be computed n number of wy. Conder the followng ce: Ce : G() proper frcton wth mple pole. Let the pole of G() be t p, =,,,,N, where p re dtnct. A prtl-frcton epnon of G() then of the form : N G () (8.) p where the contnt l n the bove epreon, clled the redue, re gven by: G() ( p ) (8.5) Ech term of the um on the rght-hnd de of Eq.(8.) h n ROC gven by p, therefore, the nvere trnform g[] of G() gven by g N p Note tht the bove pproch wth lght modfcton cn lo be ued to determne the nvere -trnform of noncul equence wth rtonl -trnform. Emple 8.5 Let the -trnform of cul equence g[] be gven by : p U (.) G() (.)(.6) G()..6.. (.). 75 (.)(.6).8 (.) (.6) (.)(.6) g.75(.).75(.6) U.75 Emple 8.6 Ung MATLAB determne the prtl frcton epnon of X(): ().5.5 X 6
5 num=[ ];den=[ ];» [r,p,]=redue(num,den) r = p = X().5.5 = -8» r=[ ];» p=[.977 X-.989 () ];» [num,den]=redue(r,p,) num = den = Multplyng the numertor nd the denomntor by. ().5.5 G Ce. G() h multple pole, for emple, f the pole t of multplcty r nd the remnng N-r pole re mple nd t p,,,,...n r, then the generl prtl-frcton epnon of G() te the form G () MN Nr p r r ( ) where the contnt r (no longer clled the redue for ) re computed ung the formul: r d r G() r ( ) r, =,,,..,r (r )! d() Emple 8.7 X() ; X() ( ) ( ) ( ) X() ( ) ; ( ) d d X() X() ( ) X() ; d d 7
6 Whch reult n the followng tme-ere u Emple 8.8 G() (.5)( ) G() (.5).5 G() ; ( ) ; d G() ( ) d G().5 ( ) Conder the followng three ce: ) g[] (.5) U U U ) U U g[] (.5) U U U ) g[] (.5) U U U Emple () ( ) (.5) X X().5 ( ) where /(.5) n eponentl, /( ) tep functon, nd /( ) rmp functon. Wht dered, however, the prtl frcton epnon of X()/, where: where whch reult n X() (.5)X().5 ( ) 5 ( ).5.5 d ( ) X() d ( ) X() 5 ( / ) 9 ( ( / ) 5 5 )( / ) (( / )) 8
7 5(.5) u Emple 8. 8 Solve ung Mtlb: H() (). (. ).5 H» num=[8]; den=[8 - -];» [r,p,]=redue(num,den) r =...6 p = =[]» [num,den]=redue(r,p,) num =... den = Ung the numertor nd the denomntor coeffcent we hve: X () It cn be een tht the coeffcent wll be me n the equton of the queton f we multply ech coeffcent by 8. Emple 8.. Fnd the nvere Z-trnform of ( ) X() (.5)(.5) X() (.5) (.5) (.5) X() 8 9
8 (.5)X().5 (.5)( ) (.5).5 (.5) X().5 ( ) (.5).5 7 X() d (.5) X() d.5 d d ( ) (.5) (.5) (.5) (.5) (.5) (.5) (.5) U» num=[ ]; den=poly([ ]) den = » [r,p,]=redue(num,den) r = p = = [] 7 Ce. X() h comple pole Emple 8.. The econd-order X() (.5) /( co( ) / ) h non 6 repeted comple root. The prtl epnon of X() defned by: where =. j.5 nd X () ( ) ( ) X() ( )X() ( ) ( ) (.5) ( )
9 Alo note tht ( )X() (.5) ( ) Z u Z ( ) nd u Where =.-j.5=.5ep(-j/6). Therefore, X().5ep( j/ 6) whch correpond to tme-ere, for.55ep( j /).55. co( / 6 /).5ep( j / 6) ep( j / 6).55ep( j /) (ep( j / 6 j /) ep( j / 6 /)) ep( j / 6) whch een to be cul phe-hfted cone wve wth n eponentlly decendng envelope. Alo oberve tht.co( /), whch cn be verfed ung the ntl vlue theorem. 8. Dfference Equton Long dvon cn be ntenve nd tedou computtonl proce. If computer-bed gnl proceng dered, the ue of dfference equton generlly more effcent. X(), where Aume tht the Z-trnform of tme ere X () M b N Recll tht the Z n nd Z n. Therefore, t follow tht: b... M (M ) M M b... b (N ) b N N N
10 The repone cn be multed by mplementng the dfference equton. Emple 8. Conder cul 5 X() ( )(.5) from emple. X() ( ) (.5 ) Whch produce tme-ere (5(.5) )u Then, for, cn be multed ung.5.5 5
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