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G. Baribaud/AB-BDI

Digital Signal Processing-2003. The Laplace transform. The linear system concept The definition and the properties. 6 March 2003. DISP-2003. G. Baribaud/AB-BDI. Digital Signal Processing-2003. The Laplace transform. • Continuous linear monovariable systems

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G. Baribaud/AB-BDI

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  1. Digital Signal Processing-2003 The Laplace transform • The linear system concept • The definition and the properties 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  2. Digital Signal Processing-2003 The Laplace transform • Continuous linear monovariable systems • Continuous linear multivariable systems • Impulse response • Convolution • Laplace transform: definition and properties • Inverse Laplace transform • Comments on stability 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  3. Digital Signal Processing-2003 Applications • Operational calculus • Solution for solving problems in engineering, physics • Applied by Heaviside by the end of 19 century • Bromwich, Carson and van der Pol • Applied complex variable early 20 century • A useful tool for system analysis • However we will limit to essentials for our purpose 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  4. f(t) t Digital Signal Processing-2003 Definition of function f(t) • f(t)=0 for t<0 • defined for t>=0 • possibly with discontinuities • f(t)<Mexp(t)[exponential order] • s: real or complex Definition of Laplace transform F(s) 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  5. f(t) t f(t) t Digital Signal Processing-2003 Examples: Dirac 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  6. f(t) t f(t) t Digital Signal Processing-2003 Examples: Heaviside 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  7. f(t) t Digital Signal Processing-2003 Example : Ramp 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  8. f(t) F(s)=L[f(t)] Digital Signal Processing-2003 Some useful Laplace transforms 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  9. Digital Signal Processing-2003 •Linearity 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  10. Example Digital Signal Processing-2003 • Translation [in s] a) if F(s)=L[f(t)] 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  11. f(t) g(t) t Example Digital Signal Processing-2003 • Translation [time] b) if g(t) = f(t-a) for t>a = 0 for t<a a 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  12. Example Digital Signal Processing-2003 •Change of time scale 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  13. f(t) t Digital Signal Processing-2003 • Derivative [first order] 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  14. •If discontinuity in a Digital Signal Processing-2003 • Derivatives of higher order 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  15. Digital Signal Processing-2003 • Derivatives examples 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  16. Equation describing the circuit Laplace transform Digital Signal Processing-2003 Application: RC circuit R v(t) e(t) C 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  17. Digital Signal Processing-2003 • Integral 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  18. More general Digital Signal Processing-2003 Multiplication by t Leibnitz’s rule 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  19. Digital Signal Processing-2003 Division by t 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  20. Digital Signal Processing-2003 • Periodic function 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  21. Digital Signal Processing-2003 Sine and cosine are periodic functions 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  22. Exponential order Digital Signal Processing-2003 •Limit behaviour: Initial value 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  23. Digital Signal Processing-2003 •Limit behaviour: Initial value 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  24. Impulse function Impulse response Digital Signal Processing-2003 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  25. e0 Digital Signal Processing-2003 Step function and zero initial conditions 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  26. Digital Signal Processing-2003 Step function and initial conditions v(0)  0 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  27. Digital Signal Processing-2003 Ramp function 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  28. Digital Signal Processing-2003 Harmonic analysis 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  29. Digital Signal Processing-2003 Forced Transient 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  30. y(t) Convolution x(t) Linear system h(t) Heuristic Digital Signal Processing-2003 Laplace transform  6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  31. Digital Signal Processing-2003 H(j) = Transfer function Argument Module 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  32. Resolvent Matrix State Transition Matrix Digital Signal Processing-2003 Multivariable systems 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  33. Characteristic polynomial Digital Signal Processing-2003 x(t) y(t) Linear system Described by constant coefficient linear differential equations Assuming all derivatives are 0 for Laplace transform 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  34. Transient (free) Forced Digital Signal Processing-2003 x(t) y(t) Linear system Described by constant coefficient linear differential equations Assuming all derivatives are  0 for Laplace transform 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  35. Digital Signal Processing-2003 The Laplace transform • Continuous linear monovariable systems • Continuous linear multivariable systems • Impulse response • Convolution • Laplace transform: definition and properties • Inverse Laplace transform • Comments on stability 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  36. Digital Signal Processing-2003 Inverse Laplace transform Knowing F(s)=L[f(t)]  What about f(t) from F(s) ? Lerch theorem: If f(t) is piece-wise continuous and of exponential order then the inverse transform is unique (This is the case for our functions) 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  37. a) Analytic method On Bromwich contour b) Use of tables method Digital Signal Processing-2003 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  38. Digital Signal Processing-2003 c) Fraction expansion with different roots 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  39. Digital Signal Processing-2003 Example 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  40. Digital Signal Processing-2003 c) Fraction expansion with multiple roots Example 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  41. Digital Signal Processing-2003 d) Series Example 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  42. Q(s) has n distinct zeroes Digital Signal Processing-2003 e) Heaviside development 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  43. Digital Signal Processing-2003 The Laplace transform • Continuous linear monovariable systems • Continuous linear multivariable systems • Impulse response • Convolution • Laplace transform: definition and properties • Inverse Laplace transform • Comments on stability 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  44. Digital Signal Processing-2003  Consequences : stability  All s real and negative : tends towards zero  All s real and at least one  positive : tends towards infinity If s complex then conjugates :   +j and -j : oscillatory behaviour : see above 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  45. Digital Signal Processing-2003 s plane x x x x The zeroes of the characteristic Polynomial must lie on the left side of the s plane 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

  46. Digital Signal Processing-2003 Thank you for your attention 6 March 2003 DISP-2003 G. Baribaud/AB-BDI

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