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Frequency Response Prof. Seungchul Lee Industrial AI Lab. Most slides from Signals and Systems (MIT 6.003) by Prof. Denny Freeman Time Response Previously, we have determined the time response of linear systems to arbitrary inputs and


  1. Frequency Response Prof. Seungchul Lee Industrial AI Lab. Most slides from Signals and Systems (MIT 6.003) by Prof. Denny Freeman

  2. Time Response • Previously, we have determined the time response of linear systems to arbitrary inputs and initial conditions • We have also studied the character of certain standard systems to certain simple inputs 2

  3. Frequency Response • Only focus on steady-state solution • Transient solution is not our interest any more • Input sine waves of different frequencies and look at the output in steady state • If 𝐻(𝑡) is linear and stable, a sinusoidal input will generate in steady state a scaled and shifted sinusoidal output of the same frequency 3

  4. Response to a Sinusoidal Input • When the input 𝑦 𝑢 = 𝑓 𝑘𝜕𝑢 to an LTI system • Output is also a sinusoid – same frequency – possibly different amplitude, and – possibly different phase angle 4

  5. Response to a Sinusoidal Input • When the input 𝑦 𝑢 = 𝑓 𝑘𝜕𝑢 to an LTI system • Output is also a sinusoid – same frequency – possibly different amplitude, and – possibly different phase angle 5

  6. Fourier Transform • Definition: Fourier transform • 𝐼 𝑘𝜕 𝑓 𝑘𝜕𝑢 rotates with the same angular velocity 𝜕 6

  7. Response to a Sinusoidal Input: MATLAB 7

  8. Response to a Sinusoidal Input: MATLAB transient 8

  9. Frequency Response to a Sinusoidal Input • Two primary quantities of interest that have implications for system performance are: – The scaling = magnitude of 𝐼(𝑘𝜕) – The phase shift = angle of 𝐼(𝑘𝜕) 9

  10. Frequency Response to a Sinusoidal Input: MATLAB • Given input 𝑓 𝑘𝜕𝑢 • 𝑧 = 𝐵𝑓 𝑘(𝜕𝑢+𝜚) 10

  11. From Laplace Transform to Fourier Transform 11

  12. Eigenfunctions and Eigenvalues • Eigenfunctions – If the output signal is a scalar multiple of the input signal, we refer to the signal as an eigenfunction and the multiplier as the eigenvalue 12

  13. Eigenfunctions and Eigenvalues • Fact: Complex exponentials are eigenfunctions of LTI systems. • If 𝑦 𝑢 = 𝑓 𝑡𝑢 and ℎ(𝑢) is the impulse response then • The eigenvalue associated with eigenfunction 𝑓 𝑡𝑢 is 𝐼(𝑡) 13

  14. Rational Transfer Functions • Eigenvalues are particularly easy to evaluate for systems represented by linear differential equations with constant coefficients. • Then the transfer function is a ratio of polynomials in 𝑡 • Example 14

  15. Vector Diagrams • The value of 𝐼(𝑡) at a point 𝑡 = 𝑡 0 can be determined graphically using vectorial analysis. • Factor the numerator and denominator of the system function to make poles and zeros explicit. • Each factor in the numerator/denominator corresponds to a vector from a zero/pole to 𝑡 0 , the point of interest in the s-plane 15

  16. Vector Diagrams • The value of 𝐼(𝑡) at a point 𝑡 = 𝑡 0 can be determined by combining the contributors of the vectors associated with each of the poles and zeros – The magnitude is determined by the product of the magnitudes – The angle is determined by the sum of the angles 16

  17. Frequency Response • Given the system described by • Find the response to the input 𝑦 𝑢 = 𝑓 2𝑘𝑢 17

  18. Vector Diagrams for Frequency Response • The magnitude and phase of the response of an LTI system to 𝑓 𝑘𝜕𝑢 is the magnitude and phase of 𝐼 𝑡 at s = 𝑘𝜕 18

  19. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 19

  20. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 20

  21. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 21

  22. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 22

  23. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 23

  24. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 24

  25. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 25

  26. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 26

  27. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 27

  28. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 28

  29. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 29

  30. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 30

  31. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 31

  32. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 32

  33. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 33

  34. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 34

  35. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 35

  36. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 36

  37. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 37

  38. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 38

  39. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 39

  40. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 40

  41. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 41

  42. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 42

  43. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 43

  44. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 44

  45. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 45

  46. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 46

  47. Vector Diagrams at 𝒕 = 𝒌𝝏 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 47

  48. System Design in S-plane From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 48

  49. Frequency Response (Frequency Sweep): MATLAB 49

  50. Frequency Response and Bode Plots 50

  51. ȁ 𝑰(𝒕) 𝒕←𝒌𝝏 Frequency Response: From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 51

  52. Poles and Zeros • Frequency response • Thinking about systems as collections of poles and zeros is an important design concept. – Simple: just a few numbers characterize entire system – Powerful: complete information about frequency response 52

  53. Bode Plots: Magnitude 53

  54. Asymptotic Behavior: Isolated Zero • The magnitude response is simple at low and high frequencies From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 54

  55. Asymptotic Behavior: Isolated Zero • Two asymptotes provide a good approximation on log-log axes From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 55

  56. Asymptotic Behavior: Isolated Pole • The magnitude response is simple at low and high frequencies From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 56

  57. Asymptotic Behavior: Isolated Pole • Two asymptotes provide a good approximation on log-log axes From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 57

  58. Check Yourself • Compare log-log plots of the frequency-response magnitudes of the following system functions From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 58

  59. Asymptotic Behavior of More Complicated Systems • Constructing 𝐼(𝑡 0 ) From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 59

  60. Asymptotic Behavior of More Complicated Systems • The magnitude of a product is the product of the magnitudes • The log of the magnitude is a sum of logs From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 60

  61. Bode Plot: Adding Instead of Multiplying From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 61

  62. Bode Plot: Adding Instead of Multiplying From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 62

  63. Bode Plot: Adding Instead of Multiplying From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 63

  64. Bode Plot: Adding Instead of Multiplying From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 64

  65. Bode Plots: Angle 65

  66. Asymptotic Behavior: Isolated Zero • The angle response is simple at low and high frequencies From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 66

  67. Asymptotic Behavior: Isolated Zero • Three straight lines provide a good approximation versus log 𝜕 From Signals and Systems (MIT 6.003) by Prof. Denny Freeman 67

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