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G. Baribaud/AB-BDI

Digital Signal Processing-2003. The z-transform. The sampling process The definition and the properties. 6 March 2003. DISP-2003. G. Baribaud/AB-BDI. Digital Signal Processing-2003. The z-transform. • Classification of signals • Sampling of continuous signals

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G. Baribaud/AB-BDI

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  1. Digital Signal Processing-2003 The z-transform • The sampling process • The definition and the properties 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  2. Digital Signal Processing-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 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  3. Digital Signal Processing-2003 Convolution Analogous to Laplace convolution theorem 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  4. 1 k Digital Signal Processing-2003 Apply z-transform 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  5. Digital Signal Processing-2003 Discrete Cosine 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  6. Digital Signal Processing-2003 Another approach 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  7. Dirac function Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  8. u(t) 1 t Digital Signal Processing-2003 Sampled step function NB: Equivalent to Exp(-k) as  0 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  9. Digital Signal Processing-2003 T Delayed pulse train t t 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  10. Digital Signal Processing-2003 Complete z-transform Example:exponential function 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  11. Digital Signal Processing-2003 Addition and substraction 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  12. Digital Signal Processing-2003 Multiplication by a constant 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

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

  14. t Digital Signal Processing-2003 Right shifting theorem 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  15. Right shifting theorem Application Unit step function which is delayed by one sampling period Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  16. t Digital Signal Processing-2003 Left shifting theorem 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  17. Digital Signal Processing-2003 Complex translation or damping f(t) is multiplied in continuous domain by Exp(-t) And then sampled at the rate T Laplace transform 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  18. Find the z-transform of sampled at T knowing that Digital Signal Processing-2003 Application 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  19. S,f t Digital Signal Processing-2003 Sum of a function 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  20. Digital Signal Processing-2003 Difference equation 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  21. Example step function kt kt kt Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  22. u(t) kt -u(t-T) V(t)=u(t)-U(t-T) Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  23. Initial-value theorem If f(t) has a z-transform F(z) and if lim F(z) as z exists Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  24. Digital Signal Processing-2003 Final-value theorem 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  25. Initial value Final value Digital Signal Processing-2003 Application:example Expanding F(z) 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  26. Digital Signal Processing-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 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  27. Digital Signal Processing-2003 Inverse ? -Reference to tables -Practical identification -Analytic methods -Decomposition -Numerical inversion 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  28. Digital Signal Processing-2003 Practical identification Discrete exponential g(k) Sum of a function 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  29. Digital Signal Processing-2003 Analytic method Laurent series Cauchy theorem Im z x x Re z o x x x x Enclosing all singularities of F(z,) 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  30. Digital Signal Processing-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 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  31. Digital Signal Processing-2003 The power series method The coefficients of the series expansion represent the values of f(t) (usually a series of numerical values) 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  32. Digital Signal Processing-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 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  33. Digital Signal Processing-2003 Continuous Systems in series with an ideal sampler at each input 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  34. Digital Signal Processing-2003 Continuous Systems in series with an ideal sampler at first input In general 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  35. and by given by Digital Signal Processing-2003 Continuous Systems in series with an ideal sampler at second input 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  36. Digital Signal Processing-2003 Discrete and continuous Systems in series with an ideal sampler 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  37. Digital Signal Processing-2003 Continuous and discrete Systems in series with an ideal sampler 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  38. Digital Signal Processing-2003 Discrete Systems in series with an ideal sampler 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  39. + Digital Signal Processing-2003 Continuous Systems in parallel with an ideal sampler 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  40. + Digital Signal Processing-2003 Discrete Systems in parallel with an ideal sampler 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  41. Digital Signal Processing-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 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  42. t t Digital Signal Processing-2003 Continuous Systems in series with zero-order hold Transfer function via impulse response 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  43. Digital Signal Processing-2003 Laplace transform 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  44. Equal to G(s) with an integrator Digital Signal Processing-2003 Global transfer function Z-transform of G(s) 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  45. Z-transform Digital Signal Processing-2003 Consequences on the behaviour There are n poles of G(z,), they depend on n the poles of the transfer function of the continuous system 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

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