1 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 The z-transform  The sampling process  The definition and the properties

2 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 The z-transform Classification of signals Sampling of continuous signals The z-transform: definition The z-transform: properties Inverse z-transform Application to systems Comments on stability

3 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Convolution Analogous to Laplace convolution theorem

4 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 k 1 Apply z-transform

5 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Discrete Cosine

6 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Another approach

7 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Dirac function

8 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Sampled step function t u(t) 1 NB: Equivalent to Exp(-  k) as   0

9 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 t t TT Delayed pulse train

10 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Complete z-transform Example:exponential function

11 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Addition and substraction

12 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Multiplication by a constant

13 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP Linearity Application ++

14 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Right shifting theorem t

15 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Right shifting theorem Application Unit step function which is delayed by one sampling period

16 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Left shifting theorem t

17 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Complex translation or damping f(t) is multiplied in continuous domain by Exp(-  t) And then sampled at the rate  T Laplace transform

18 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Application Find the z-transform of sampled at  T knowing that

19 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 t S,fS,f Sum of a function

20 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Difference equation

21 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 kt Example step function

22 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 kt u(t) -u(t-  T) V(t)=u(t)-U(t-  T)

23 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Initial-value theorem If f(t) has a z-transform F(z) and if lim F(z) as z  exists

24 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Final-value theorem

25 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Application:example Initial value Final value Expanding F(z)

26 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 The z-transform Classification of signals Sampling of continuous signals The z-transform: definition The z-transform: properties Inverse z-transform Application to systems Comments on stability

27 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 ? -Reference to tables -Practical identification -Analytic methods -Decomposition -Numerical inversion Inverse

28 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Discrete exponential g(k) Practical identification Sum of a function

29 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 x x x x x x o Re z Im z Laurent seriesCauchy theorem Analytic method Enclosing all singularities of F(z,  )

30 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Partial fraction expansion With Laplace transform With z-transform no such an expansion, one looks for terms like: The function F(z)/z is developed by partial-fraction expansion

31 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 The power series method The coefficients of the series expansion represent the values of f(t) (usually a series of numerical values)

32 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 The z-transform Classification of signals Sampling of continuous signals The z-transform: definition The z-transform: properties Inverse z-transform Application to systems Comments on stability

33 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Continuous Systems in series with an ideal sampler at each input

34 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 In general Continuous Systems in series with an ideal sampler at first input

35 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 given by and by Continuous Systems in series with an ideal sampler at second input

36 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Discrete and continuous Systems in series with an ideal sampler

37 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Continuous and discrete Systems in series with an ideal sampler

38 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Discrete Systems in series with an ideal sampler

39 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Continuous Systems in parallel with an ideal sampler +

40 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Discrete Systems in parallel with an ideal sampler +

41 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 The z-transform Classification of signals Sampling of continuous signals The z-transform: definition The z-transform: properties Inverse z-transform Application to systems Comments on stability

42 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Continuous Systems in series with zero-order hold Transfer function via impulse response t t

43 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Laplace transform

44 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Global transfer function Equal to G(s) with an integrator Z-transform of G(s)

45 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Consequences on the behaviour Z-transform There are n poles of G(z,  ), they depend on n the poles of the transfer function of the continuous system