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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 Classification of signals • Continuous (or analogue) signals • Sampled signals • Discrete (or digital or time) signals 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  4. y(t) Y(s) u(t)U(s) G(s) Continuous signals (Single or multiple) u(t) Discontinuity y(t) t Digital Signal Processing-2003 Continuous (or analogue) signals 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  5. x(t) t x[kT] Amplitude modulated t x[kT] Pulsewidth modulated t Digital Signal Processing-2003 Sampled signal: modulation techniques Original signal 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  6. Digital Signal Processing-2003 Sampled signal x’(t) T t kT (K+1)T • Noted x’(t) • Samples exist only at sampling times • The relative height represents the value (information) • The sampling Tperiod is constant • Able to drive a physical system 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  7. Digital Signal Processing-2003 Physical sample Information Area=h h Dirac (or unit) pulse As  (Distribution) t  NB: The energy of a sample pulse is finite (able to drive a physical system) 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  8. x*(t) T t kT (K+1)T Digital Signal Processing-2003 Discrete (or digital or time) signals • Noted x*(t) • Values defined only at sampling times • The relative height represents the numerical value • The sampling Tperiod is constant • Usable for arithmetic operations • Unable to drive a physical system 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  9. Digital Signal Processing-2003 •Application Samples or steps u(t) y(t) Computer Driver (DAC) G(s) Measure (ADC) Sampling Continuous signal Discrete signal 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  10. • Continuous system y(t) x(t) t t G(s) y(t) x(t) Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  11. • Sampled system x’(t) y(t) t t G(s) y(t) x’(t) Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  12. • Sampled system with hold circuit y(t) u(t) x’(t) t t t u(t) Hold G(s) y(t) x’(t) Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  13. • Discrete system x*(t) y*(t) t t D(z) y*(t) x*(t) D(z) defined by difference equations or by transfer function Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  14. 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

  15. x[k] x[k,] x[k+1] x[k-1]  t t=(k+1)T t=(k-1)T t=kT t=(k+)T x(kT)=x[k,]=x[k]=x k Digital Signal Processing-2003 Sampling time and delay Several notations 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  16. Sampling times t (k-1)T (k+1)T kT Periodic function n=integer………-2,-1,0,1,2,3,4,……….. Digital Signal Processing-2003 Sampling function 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  17. Decomposition in Fourier series n=-1 n=0 n=1 0 Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

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

  19. f(t) Ideal sampler t t Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  20. Sampling process T Continuous Function f(t) Series of samples Sampler (t) Reconstruction ? Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  21. Digital Signal Processing-2003 Analysis in frequency domain • F()= Fourier transform of f(t) •  ()= Fourier transform of (t) • F’()= Fourier transform of f’(t) Convolution in the frequency domain 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  22. Digital Signal Processing-2003 Analysis of F’() 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  23. Digital Signal Processing-2003 Aliasing 0 Primary components Fundamental components Complementary components Complementary components The spectra are overlapping (Folding) Folding frequency 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  24. Window In the frequency domain In the time domain Digital Signal Processing-2003 Reconstruction f’(t) f °(t) f(t) Sampler Filter 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

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

  26. Reconstruction f(t) Interpolation functions t nT (n+1)T (n+2)T (n+3)T Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  27. 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

  28. t Digital Signal Processing-2003 Definition For sampled or discrete signals Apply Laplace transform of f’(t) Factors like Exp(-sT) are involved Unlike the majority of transfer functions of continuous systems It will not lead to rational functions 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  29. Laplace z k t Digital Signal Processing-2003 Definition 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  30. Digital Signal Processing-2003 Representation of a delay 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  31. The operation of taking the z-transform of a continuous-data function, f(t), involves the following three steps: 1- f(t) is sampled by an ideal sampler to get f’(t) 2- Take the Laplace transform of f’(t) 3- Replace by z in F’(s) to get Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  32. A few z-transforms f(t) F(s) F(z)

  33. 1 Digital Signal Processing-2003 S-plane Mapping Primary strip Im[z] z-plane The left half of the primary strip is mapped inside the unit circle Re[z] 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  34. Digital Signal Processing-2003 S-plane Mapping Primary strip The right half of the primary strip is mapped outside the unit circle Imz Z-plane Rez 1 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  35. Digital Signal Processing-2003 S-plane Complementary strip The left half of the complementary strip is also mapped inside the unit circle Imz Z-plane Rez 1 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  36. s-plane properties of F’(s) Complementary strip Complementary strip Primary strip Complementary strip Complementary strip Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  37. Complementary strip X Complementary strip X Primary strip X Complementary strip X Complementary strip X X Poles of F’(s) in primary strip Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  38. Complementary strip X Folded back poles Complementary strip X Primary strip X Complementary strip X Complementary strip X X Poles of F’(s) in complementary strips Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  39. Digital Signal Processing-2003 The constant damping loci s-plane z-plane 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  40. Digital Signal Processing-2003 The constant frequency loci s-plane z-plane 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

  41. Mapping between the s-plane and the z-plane Conclusion: All points in the left half of the s-plane are mapped into the Region inside the unit circle in the z-plane. The points in the right half of the s-plane are mapped into the Region outside the unit circle in the z-plane Digital Signal Processing-2003 13 March 2003 DISP-2003 G. Baribaud/AB-BDI

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