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The unit step response of an LTI system

The unit step response of an LTI system. Linear constant-coefficient difference equations. +. delay. When n 1,. Causality. Linear constant-coefficient difference equations. +. delay. Determine A by initial condition:. When n = 0 ,. A = 1.

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The unit step response of an LTI system

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  1. The unit step response of an LTI system

  2. Linear constant-coefficient difference equations + delay When n 1, Causality

  3. Linear constant-coefficient difference equations + delay Determine A by initial condition: When n = 0, A = 1

  4. Linear constant-coefficient difference equations + delay Two ways: (1) Repeat the procedure (2)

  5. Linear constant-coefficient difference equations + When t>0, Causality Determine A by initial condition:

  6. Linear constant-coefficient difference equations + Determine A by initial condition: A = 1

  7. Linear constant-coefficient difference equations +

  8. Fourier series representation of continuous-time periodical signal Periodic signal for all t k is an integer Fourier series form a complete and orthogonal bases Complete: no other basis is needed. Kronecker Delta Orthogonal: Orthogonal:

  9. Fourier series representation of continuous-time periodical signal Periodic signal for all t k is an integer

  10. Fourier series representation of continuous-time periodical signal Periodic signal for all t k is an integer e.g.

  11. Fourier series representation of continuous-time periodical signal 0

  12. The response of system to complex exponentials Band limited channel Bandwidth Bandwidth

  13. Fourier series representation of discrete-time periodical signal Periodic signal for all t

  14. Example #1

  15. Properties of discrete-time Fourier series (1) Linearity

  16. (2) Time shifting (3) Time reversal

  17. (4) Time scaling (5) multiplication

  18. (6) Conjugation and conjugate symmetry Real signal Even Real & Even

  19. (7) Parseval’s relation

  20. (8) Time difference (9) Running sum

  21. Example N = 4 [1, 2, 2, 1] [1, 1, 1, 1]

  22. Fourier series and LTI system Output periodic? Periodic signal System response doesn’t have to be periodic.

  23. Filtering • Frequency-shaping filters • Frequency-selective filters (1) Frequency-shaping filters

  24. (1) Frequency-shaping filters

  25. (2) Frequency-selective filters Low-pass high-pass band-pass

  26. Discrete-time

  27. Example: averaging

  28. Continuous-time Fourier transform Aperiodic signal Periodic signal k is an integer

  29. Continuous-time Fourier transform Aperiodic signal Periodic signal k is an integer

  30. Examples

  31. Properties of continuous-time Fourier transform (1) Linearity

  32. Properties of continuous-time Fourier transform (2) Time shifting (3) Time reversal

  33. Properties of continuous-time Fourier transform (4) Time scaling

  34. Properties of continuous-time Fourier transform (5) Conjugation and conjugate summary Real

  35. Example even Even and real

  36. Differential

  37. Integral

  38. Example

  39. Example

  40. Example

  41. Example

  42. Parseval’s relation

  43. Parseval’srelation for continuous-time Fourier series Parseval’srelation for continuous-time Fourier transfer

  44. Example 1.0 -1.0 0.5 -0.5

  45. Example 1.0 -1.0 0.5 -0.5

  46. Example, P. 4.14 (1) real (2) (3) Solution:

  47. Example, P. 4.14 (1) real (2) (3) Solution:

  48. Example, P. 4.14 Solution:

  49. Example, P. 4.14 (3) Solution:

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