Z Transforms. Lesson 20 6DT. BME 333 Biomedical Signals and Systems - J.Schesser

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1 Z rasforms Lesso 6D BME 333 Biomedical Sigals ad Systems

2 Z rasforms A Defiitio I a sese similar to the L excet it is associated with discrete time fuctios. Let s assume we hae a cotiuous time fuctio, f t, ad let s create a discrete time fuctio from it, f by samlig it at a rate. We would the hae: f f t t aig the L of f we would hae: st st Fs f f t t e dt f t t e dt f e s BME 333 Biomedical Sigals ad Systems

3 Z rasforms A Defiitio Cotiued Relacig e s with we hae the defiitio of the - trasform: F f Note that this is a -sided summatio. Sice may sigals are ero for t<, we ca also defie a -sided -trasform as: F f Note that both ersios F ca be writte i closed form whe the series coerges BME 333 Biomedical Sigals ad Systems

4 Z-trasform -sided Calculatios - Examles First, recall that the geometric series Examle, let: Ar coerges to f e a for ; otherwise F e a e a ea Usig the result of a geometric series: F ea e a A -r roided that r. ad will coerge whe ea or e a BME 333 Biomedical Sigals ad Systems 3

5 Examle, let: a f e for ; otherwise Examles Cotiued a a F e e let m- we hae: am m a m a e e e where m a a e e Usig the result of a geometric series: a F e a a e e a ad will coerge whe e or e We see that we hae to roide a regio of coergece, i additio, to the fuctio F to comletely defie the -trasform of a f. BME 333 Biomedical Sigals ad Systems a 4

6 -sided -trasforms Useful sice most fuctios we use are ero for t<. F f BME 333 Biomedical Sigals ad Systems 5

7 -sided -trasforms Some examles see. 396 ables 6D. & 6D. for more: Uit samlig fuctio for, otherwise; F u for, otherwise F coerges for u coerges for u coerges for 3 BME 333 Biomedical Sigals ad Systems 6

8 u Proof 3 4 { 3 4 } 3 { 3 } 3 { } { } { } { } { } { } { } coerges for { 3 4 } { 3 } BME 333 Biomedical Sigals ad Systems 7

9 -sided -trasforms Some examles see. 396 ables 6D. & 6D. for more: e a u coerges for ea a e a u a a a coerges for a a BME 333 Biomedical Sigals ad Systems 8

10 Proerties of -trasforms Z{f f } F F Z{af } af Z{f } F Z{f m } m F Z{f } df d If f f f, the F F F BME 333 Biomedical Sigals ad Systems 9

11 Proof of Delay Proerty Gie: Z{f } F Z{ f } F Z{f } Let - m f f f f m m f m m Sice f m f m m m F m BME 333 Biomedical Sigals ad Systems

12 Proof of f Proerty { f } f F f df d d f - f - f d df d df d - f f f { f } BME 333 Biomedical Sigals ad Systems

13 Proerties of -trasforms If Z{f } df ;the d Z{ f } Z{g } dg d where g f but G df ; d dg d Ad { df d d F d } Z{ f } dg d { df d d F d } { df d d F d } Z{ M f } dg ;where G Z{ M f } d BME 333 Biomedical Sigals ad Systems

14 Iertig the -trasform Seeral Methods: aylor series exasio: Sice F f φy f f y f f y f We ca defie a ew fuctio y where y f y f ad ote that this is just a aylor series exasio ad therefore, calculate the coefficiets of this serieswhich are the alues of f by f d! dy Log diisio of the closed form of F should also yield the same results. - y BME 333 Biomedical Sigals ad Systems 3

15 Iertig the -trasform Cotiued Residue Method which relates the, F, to the L, Fs: It ca be show that Fs F residues of at the oles of Fs s -e he residues are calculated for a ole s aof order as : d! ds s a Fs s -e sa BME 333 Biomedical Sigals ad Systems 4

16 BME 333 Biomedical Sigals ad Systems 5 Iertig the -trasform Cotiued d d M M K M M K F F for u f F F Partial Fractio Exasio of sice most discrete fuctios we deal with are of the form ad, therefore, due to, will hae a i its umerator; e.g.,z{ } i F f u u F f u F ad will hae terms of the form which ca be ierted. i i A

17 Aother Iersio Examle F j j F j j A A A* A* j j j j A A j j j j j j j j j 4 j * j A d d j j 3 3 j j j j j j j 3 3 j j j j 4 4 j 4 j 3 j j A * Liewise for B j j BME 333 Biomedical Sigals ad Systems 6

18 Aother Iersio Examle Cotiued j j j 4 j 4 F e e j j j j e e his loos lie the term is related to e e j j df Ze u Zf e d j { } ; Recall { } ; j d df e d d j { u } Ze j j j j e e j j j j e e e e d j j df e e Z e u j d d e Liewise j e e j { } BME 333 Biomedical Sigals ad Systems 7

19 Aother Iersio Examle Cotiued Cotiuig F e j 4 e j e j e j e j 4 e j e j e j f e j 4 e j e j 4 e j u e j 4 e j 4 u { e j 4 j 4 e }u cos 4u BME 333 Biomedical Sigals ad Systems 8

20 BME 333 Biomedical Sigals ad Systems 9 Differece Equatios It ca be show that for a fuctio which equals ero for t<, ay iitial coditios will ot affect the followig formulatio: as :, the resose due to the uit samlig fuctio, Ad we hae the system fuctio which is Or yield: trasform to this system will he alyig the - gie by this equatio : we hae a system If X Y H H a a a b b b X Y X b b b Y a a a x b x b x b y a m y a m y a m m m m m m m m m m

21 BME 333 Biomedical Sigals ad Systems Examle we hae recedig examle, From a Ad 3 3 otherwise. he,, for where 3 Assume we hae u f V Y H V Y V Y V y y y

22 Examle But H which is ot i ay table, - but if: h Z { H } - he, h Z { H } - Z { } h for m m herefore, hm for m Alteratiely, usig Z{ f } F - H ; but Z { } u Ad Z { } Z { } Z { F } f u BME 333 Biomedical Sigals ad Systems

23 Aother Examle r o - r g i - o r g i o + r g i + Let's calculate the odal oltage of this iteratie etwor : By Kirchoff's Curret Law at ode, we hae : o o o i r r / g rg rg i - V o V o o ad let rg H ;, sice c, the or ad / herefore, H H K K K K / H where / h u at By the way this loos lie our first exercise whe f t e for t ; otherwise - the ole art at ad f t e for t ; otherwise - the / ole art. So the left had side of h ca be associated with a -sided Z ad the right had side of h with a -sided Z. BME 333 Biomedical Sigals ad Systems

24 Oe More Examle t R d t dt V V C RC RC t How do we sole this o a digit comuter? d t t t dt RC We use a iteratie techique but first we must aroximate the deriatie. First we defie a discrete ersio of t ow as the forward Euler Algorithm RC RC RC where u RC V K, K V ad t - + u K K BME 333 Biomedical Sigals ad Systems 3

25 Forward Euler Algorithm Cotiued Estimate of the outut oltage s samlig time C..6.4 C C -5 - C BME 333 Biomedical Sigals ad Systems 4

26 BME 333 Biomedical Sigals ad Systems 5 Cotiued ad discrete ersio of First we defie a We use a iteratie techique but first we must aroximate the deriatie. digit comuter? o a How do we sole this t t t t RC dt t d R C t t the bacward Euler Algorithm ow as dt t d, u V K K K K V RC where V V V u RC RC RC RC RC - +

27 Bacward Euler Algorithm Cotiued Estimate of the outut oltage s samlig time C..6.4 C C.6 C BME 333 Biomedical Sigals ad Systems

28 ime Domai to Frequecy Domai rasformatios ime Domai ime Sigal ye Periodic & Cotiuous x t No-Periodic & Cotiuous x t Discrete x Discrete x Quadratic Cotet Fiite Fiite Fiite Fiite a rasformatio ye Fourier Series FS / / x t e j t dt Cotiuous ime Fourier rasform CF X j x t e jt dt Discrete ime Fourier rasform DF X e j ˆ x e j ˆ Discrete Fourier rasform DF L j N X x e Frequecy Domai Discrete Sectrum a Cotiuous Sectral Desity X j Cotiuous Sectral Desity j ˆ X e Discrete Sectral Desity X No-Periodic, Cotiuous & Zero for t < x t Not Necessarily Fiite Lalace rasform X st s x t e dt Comlex Sectrum X s No-Periodic, Discrete & Zero for < x Not Necessarily Fiite -sided Z rasform X x BME 333 Biomedical Sigals ad Systems Comlex Sectrum X 7

29 Problems:.6,.7 Homewor.6 Cosider the LI system ad fid the outut y for x u : y y x, 3.7 Fid x. X BME 333 Biomedical Sigals ad Systems 8

30 Problems: -9, -5.9 Fid the - trasform for x u 3 Homewor u.5 Fid the differece equatio for H 3 6 Fid the outut for a iut of u,a uit ste. BME 333 Biomedical Sigals ad Systems 9

Z - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.

Z - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals. Z - Trasform The -trasform is a very importat tool i describig ad aalyig digital systems. It offers the techiques for digital filter desig ad frequecy aalysis of digital sigals. Defiitio of -trasform: