A METHOD FOR NUMERICAL EVALUATING OF INVERSE Z-TRANSFORM UDC 519.6(045)

Transcription

1 FACTA UNIVERSITATIS Srs: Mcacs Automatc Cotrol ad Rootcs Vol 4 N o 6 4 pp A METHOD FOR NUMERICAL EVALUATING OF INVERSE Z-TRANSFORM UDC Prdrag M Raovć Momr S Staovć Slađaa D Marovć 3 Dpartmt of Matmatcs Faculty of Mcacal Egrg Dpartmt of Matmatcs Faculty of Occupatoal Safty 3 Dpartmt of Matmatcs Faculty of Elctrcal Egrg Uvrsty of Nš Sra ad Motgro Astract W wll dscuss t prolm of fdg t st approxmatos t spac of ral sucs W troduc ortogoal sucs usg Z-Trasform ad apply t approxmatg of vrs Z-Trasform W wll llustrat t y som xampls Ky words: Approxmato Ortogoal Fuctos Z-Trasform INTRODUCTION T Z-Trasform s usd to ta dscrt tm doma sgals to a complxvaral frucy doma It plays a smlar rol to t o t Laplac Trasform dos t cotuous tm doma L t Laplac t Z-Trasform ops up w ways of solvg prolms ad dsgg dscrt doma applcatos Z-Trasform ad vrs Z-Trasform av applcatos umrous sccs: tory of proalty dffrc uatos sgal procssg fltr dsg ad so o Lt { } No a uow suc t spac l wos Z-Trasform s a ow fucto H Z H T suc s t vrs Z-Trasform of H Z - H dfd y - H d N π Γ wr Γ s a cotour t complx pla cotag all pols of H Rcvd August 3 AMS Suct Classfcato: J7 44D5

2 34 PM RAJKOVIĆ MS STANKOVIĆ SD MARINKOVIĆ T fdg of t vrs Z-Trasform s closd wt a lot of trouls W wll try to rcostruct ts uow suc umrcally Trfor w wll rmd o som proprts of t Z-Trasform ad t spac l T rgo of covrgc of t Z-Trasform of s t rag of valus of for wc H s ft T Z-Trasform dfto wll covrg asolutly w t srs of ral umrs covrgs T rato tst for covrgc stats lm + + < > lm R Trfor t srs covrgs outsd t crcl wt t ctr at org ad radus R W df a composto ad a scalar product l y f g { f g } N f g f g f g l T suar of orm of a suc f t spac l s f <ff> A suc { } N s ortogoal wt rspct to r product f m δ m m N wr δ m s Krocr dlta W suppos tat ts suc s ormald y tal valu N Lt S t lar ovr t frst mmrs of ortogoal suc S a a R Our purpos s to fd approxmato wt t proprty m f f S 3 wc w call t st approxmato of S If Z ff ad Z g G t Kowg tat F G f g m m π R d δ + f + m g

3 A Mtod for Numrcal Evaluatg of Ivrs Z-Trasform 35 w coclud tat t s vald π R g f Z d G F Sc Zf g s a olomorpc fucto t pot t scalar product ca rprstd t followg way < π Rs d G F G F g f 4 Hr w wll rmd of t asc facts of -calculus s for xampl ] So - umrs -factorals ad -omals ar dfd y ]! ]! ]! ] ] ] ]! ] ORTHOGONAL SEUENCES W wll start wt t suc E { } N of t sucs dfd y N T suc E s fudamtal l Tat s wy w ca xprss y wr From t otr sd lt us dot y Z Z-Trasform of From Z w av Z Sc lm ad Z w coclud tat must a ratoal fucto wt t moc polyomals of t sam dgr at umrator ad domator Bcaus of ortogoalty w av

4 36 PM RAJKOVIĆ MS STANKOVIĆ SD MARINKOVIĆ Hc wr Now w ca xpad wr By som valuatg w av + + ad At last t coffct s T orm of ca xprssd y Rs < Sc w av Rs ad fally 4

5 A Mtod for Numrcal Evaluatg of Ivrs Z-Trasform 37 3 APPLICATIONS Now w ca xpad ay suc from l t srs From w av c wr c 3 If w dot y H Z t follows Hc w ca rwrt c t form T fucto dfd y H c H 3 c s t st approxmato of t spac l wt rror Accordg to 4 w ca valuat xactly c 33 < H H Rs 34 Exampl 3 Lt + H 3 4 I t Fgur 3 t xact suc s MR Stoc 3] s sow as t cotuous l t ad t approxmato s draw y larg pots Applyg our mtod w fd t approxmato of wos rlatv rror s gv Tal 3

6 38 PM RAJKOVIĆ MS STANKOVIĆ SD MARINKOVIĆ t - Fg 3 Tal 3 approx approx rl rror rl rror T rlatv rror s valuatd y t cogto of t xact suc But vry mportat tg s tat w ca stmat t rror for uow accordg to formula 33 ad 34 Espcally for ts xampl t fucto HH as t pols sd t ut crcl t pots 3 ad 4 Accordg to 34 t suar orm of s Now applyg 33 w ca stmat t suar orms of rrors So for 34 t s * - ad for 56 w gt 34545* - I t formula 33 t orm of rror of approxmato t coffcts c ad t orms wc dpd o t paramtr ta part So y t sutal coc of w ca xrt a fluc to t s of t rror of approxmato

7 A Mtod for Numrcal Evaluatg of Ivrs Z-Trasform 39 REFERENCES R Koo R F Swarttouw Asy-scm of yprgomtrc ortogoal polyomals ad ts - aalogu Rport of Dlft Uvrsty of Tcology vol No A Šm Ortoxpotal polyomals of dscrt argumt Automatc Cotrol Systms ad Computg mtods Pragu A MR Stoc Dgtal sstm upravlaa Nauca ga Bograd I Sra 989 JEDAN METOD ZA NUMERIČKO IZRAČUNAVANJE INVERZNE Z-TRANSFORMACIJE Prdrag M Raovć Momr S Staovć Slađaa D Marovć U radu proučavamo prolm alaža aol aprosmac u prostoru ral ova Stoga orstć Z-trasformu uvodmo ortogoal ov Ov ov upotrlavamo u aprosmrau vr Z-trasformac Mtod lustrumo odgovaraućm prmrma