Digital signal processing: Lecture 5. z-transformation - I. Produced by Qiangfu Zhao (Since 1995), All rights reserved

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1 Digital sigal processig: Lecture 5 -trasformatio - I Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/

2 Review of last lecture Fourier trasform & iverse Fourier trasform: Time domai & Frequecy domai represetatios Uderstad the true face latet factor of some physical pheomeo. Key poits: Defiitio of Fourier trasformatio. Defiitio of iverse Fourier trasformatio. Coditio for a sigal to have FT. Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/

3 Review of last lecture The samplig theorem If the highest frequecy compoet cotaied i a aalog sigal is f, the samplig frequecy the Nyquist rate should be f at least. Frequecy respose of a LTI system: The Fourier trasformatio of the impulse respose Physical meaig: frequecy compoets to keep ~ ad those to remove ~. Theoretic foudatio: Covolutio theorem. Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/3

4 Topics of this lecture Chapter 6 of the tetbook The -trasformatio. Covergece regio of - trasform. Relatio betwee - trasform ad Fourier trasform. Relatio betwee - trasform ad differece equatio. Trasfer fuctio system fuctio. - 変換 - 変換の収束領域 - 変換とフーリエ変換 - 変換と差分方程式 伝達関数 or システム関数 Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/4

5 -trasformatio- 変換 Laplace trasformatio is importat for aalyig aalog sigal/systems. -trasformatio is importat for aalyig discrete sigal/systems. For a give sigal, its -trasformatio is defied by Z[ ] 6.3 where is a comple variable. Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/5

6 Oe-sided -trasform ad two-sided -trasform The -trasform defied above is called oesided -trasform 片側 - 変換. Oe-sided -trasform is meaigful for causal sigals 因果的信号. For o-causal sigals, the summatio should start from mius ifiity, ad the -trasform so defied is called two-sided -trasform. We study oly oe-sided -trasform here. Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/6

7 Covergece regio 収束領域 Fourier trasform eists coverges if ad oly if the sigal is absolutely summable. -trasform eists for some value of. The regio i which the -trasform eists is called the covergece regio. -trasformatio is a more powerful tool for aalyig sigals see Fig. 6. i p. 89. Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/7

8 Eamples Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/8 α α α α α α δ δ δ δ α δ > > for, ] [ for, ] [ for ay, ] [.,ad, trasform of Fid the - Eample 6. Z u u u u Z Z u u

9 Relatio betwee -trasform ad Fourier trasform Fourier trasform is the - trasform o the uit circle. The values of the -trasform o the uit circle equals to those of the Fourier trasform of the same sigal. -plae - j jω e jω e or -j 6.6 Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/9

10 Remarks Table 6. -trasforms of some typical sigals T It is importat to specify the covergece regio i the results. If we use the right table, we ca fid the -trasform of more complicated sigals. r is the ramp sigal Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/

11 Properties of -trasform Liearity : - trasform is a liear trasform a, b if ad the Z[ Z[ Z[ a ] ] b ] a b Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/

12 Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/ Properties of -trasform see later will aalysis, as for system The secod equatio is especially useful ] [ ] [ ] [ the ] [ ad for if Traslatio : m Z m Z Z m m m <

13 Eample 6.4 p. 94 relatio betwee -trasform ad differece equatio Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/3 6. we have trasform, - From liearity ad traslatio property of Give : a a b b b Y b b b Y a Y a Y b b b y a y a y

14 Remarks The purpose of this eample is to show how to fid the -trasform of the system output whe the system is give as a differece equatio. The liearity ad the traslatio properties of - trasform are used here. It is iterestig to see aother way to relate the iput ad the output of a system. Actually, -trasform is oe way for solvig the costat coefficiet differece equatios. Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/4

15 Properties of -trasform 3 Sigal multiplied by a epoetial sigal If is the -trasformatio of, the Z[a ]a - Eample 6.5 siω Z[siω u ] cosω siω Z[ α siω u ] α cosω α α Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/5

16 Properties of -trasform 4 Differetiatio of If is the -trasformatio of, the Z[ ] Eample 6.6 Z d d d d [ r ] Z[ u ] [ ] Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/6

17 Properties of -trasform 5 Covolutio theorem agai Covolutio theorem for -trasformatio y Y * See Eample 6.7 ad Eample 6.8. Matlab6.3 revised syms a ; ha^; Htrash,, ; tras,, HH* ysymsumh*,, Ytrasy,, simplifyy simplifyh Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/7

18 Properties of -trasform 6 Iitial value Fial value lim lim lim See eample 6.9. Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/8

19 The System fuctio of a LTI system or trasfer fuctio 伝達関数 - plae : * time domai : impulse respose : : system output : system iput H Y h y H h Y y Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/9 HY/ is called the trasfer fuctio or system fuctio of the system p.. H is the -trasform of h.

20 HOMEWORK Fid the -trasforms of the followig discrete sigals: Suppose that the differece equatio is give by T とする y - a y- Fid the trasfer fuctio of the correspodig digital filter. hit: see Eample Cofirm Eample 6.8 usig the program with revisios give i p. 96 of the tetbook. Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/

21 Qui ad Self-evaluatio Quies: For a give sigal, what is the oe-sided -trasformatio of?..8 T.6 What is the relatio betwee - trasform ad Fourier trasform? T5.4. T What is the trasfer fuctio 伝達関数 of a digital filter? T4 T3 Name: Studet ID:.

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

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