1. Ch # 11 Fourier Series, Integrals, and Transform 1

2. Fourier Analysis Signals with different frequencies Signal DFT...... White Light Color lights 2

3. Fourier Theory Any functions or signals = Fourier Fourier SeriesFourier Transform Periodic function Aperiodic function Continuous Discrete Continuous Discrete 3

4. Fourier Series and Transform Effectively represent a signal (function) In the form of a linear combination of cosine and sine basis function Fourier series Representing a periodic signal (function) : cosine, sine Fourier Transform Representing a non-periodic signal (function) : x, x 2, e x, cosh x, ln x 4

5. Periodic Function (signals) A function f(x) is called periodic function if f(x) is defined for all real x and if there is some positive number p, called a period of f(x)  Repeat itself every a period (p), 5

6. Continued Periodic Functions: Trigonometric Series 6

7. Trigonometric Series: Fourier Series Function f(x) of period in terms of the simeple funciotn; 1, cosx, sinx,…, cosnx,sinnx,… A linear combination of sine and cosine functions at different frequencies : Fourier Coefficients of the series 7

8. Trigonometric Series: Fourier Series The coefficients of Fourier Series ค่าขนาดของสัญญาณที่เป็นองค์ประกอบ ในรูปของ cosine and sine function ของ a periodic signal ( s(t) ) ใดๆ a n : magnitude of cosine at n th frequency (cos nx) b n : magnitude of sine at n th frequency (sin nx) 8

9. Continued 9 Fourier series of f(x)

10. Fourier Series: Fourier Coefficient Estimation Coeffients = sum of a projection of a periodic function (f(x)) onto Fourier basis function (cosine, sine) for the period 10

11. Fourier Series: Derivation of the Euler formulas (6) : a 0 11

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13. Fourier Series: Derivation of the Euler formulas (6) : a n x cos mx 13 Theorem 3 Term 2:9(a) n=m, Term 3: 9(c)

14. Fourier Series: Derivation of the Euler formulas (6) : b n, x sin mx 14 Theorem 3 term2 : 9(c) term3: 9(b), n=m

15. Fourier Series: Derivation of the Euler formulas (6) : a n 15

16. Fourier Series: Fourier Coefficient Estimation If m = n 16

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18. Continued 18

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22. Convergence and Sum of Fourier Series 22

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24. 24 convergence At discontinuous point x 0 =1 f(x 0 )=average of the left and right-hand limits

25. Functions of Any Period p = 2L 25

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27. Complex Fourier Series 27

28. Complex Fourier Series 28

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30. Complex Fourier Series 30

31. Fourier Transform 31

32. Fourier Transform Non-periodic function คาดว่าจะมีคาบเกิดขึ้นในช่วง มาจาก Complex Fourier Integral 32

33. Fourier transform & its Inverse 33

34. Forward Fourier Transform Project input f(x) onto complex exponential basis function 34

35. Forward Fourier Transform Fourier Coefficients Representing the magnitude of the complex exponential basis function (cosine & sine) at frequency w 35

36. How to Interpret the Weights Real part Imaginary part How much of a cosine of frequencys you need. How much of a sine of frequencys you need. Magnitude Phase How much of a sinusoid of frequencys you need. What phase that sinusoid needs to be. The weights are complex numbers: 36

37. Existence of Fourier Transform f(x) Absolutely integrable on the x-axis Piecewise continuous on every finite interval 37

38. Spectrum Light a superposition of colors (frequencies) Signal a superposition of sinusoidal oscillations (sine & cosine) of all possible frequencies 38

39. Spectrum density Represent total energy of the physical system Hence an integral of from a to b gives the contribution of the frequencies w between a and b to the total energy 39

40. Page 520 40

41. Fourier Transform Properties Linearity Fourier Transform of Derivative Convolution 41

42. Linearity 42

43. Fourier Transform of Derivative 43 Integrating by parts

44. 44 Application of the Operational Formula (9) Find the Fourier transform of from table II formula 9 Solution: we use theorem 3: Fourier Transform of Derivative and formula 9, table III

45. Convolution 45

46. Convolution Theorem 46

47. 47 Proof:

48. Discrete Fourier Transform (DFT) Forward Fourier Transform Inverse Fourier Transform 48 A function f(x) is given only in terms of values at finitely many points, concerns amounts of equally spaced data: telecommunication, time series analysis, etc.

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50. 50 หา a w, w=0,1,…3 w=0  (1,1,1,1)w=1  (1,0,-1,0) w=2  (1,-1,1,-1)w=3  (1,0,-1,0) x x x x

51. 51 หา b w, w=0,1,…3 w=0  (0,0,0,0)w=1  (0,1,0,-1) w=2  (0,0,0,0)w=3  (0,-1,0,1) x x x x

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53. 4-point, DFT + 53 x x x x 412 w=0, (1,1,1,1) w=1, (1,0,-1,0) w=2, (1,-1,1,-1) w=3, (1,0,-1,0) x x x x 040- 4 w=0, (0,0,0,0) w=1, (0,1,0,-1) w=3, (0,-1,0,1) w=2, (0,0,0,0)

54. Fourier Applications Analysis of Linear System (PDE) Power Spectrum Analysis Filtering Noise Convolution / Correlation Low Pass / Band Pass / High Pass Filters Signal Sampling Signal Enhancement Signal Registration 54

55. Power Spectrum Analysis 55

56. Supraventricular tachycardia (SVT) Ventricular tachycardia (VT) 56

57. Filtering Noise DFT f(x,y)F(u,v) H(u,v) Inverse DFT F(u,v)H(u,v) g(x,y) 57

58. Image Enhancement 58

59. Image Enhancement 59

60. Image Enhancement Wiener filter 60

61. Image Registration Using Fourier Time-Shifting property 61

62. Page 529a Continued 62

63. Page 529b Continued 63

64. Page 529c 64

65. Page 530a Continued 65

66. Page 530b Continued 66

67. Page 530c 67

68. Page 531a Continued 68

69. Page 531b Continued 69

70. Page 531c 70

71. Fourier Transform Comp344 Tutorial Kai Zhang

72. Outline Fourier Transform (FT) Properties Fourier Transform of regular signals Exercises

73. Popular forms of FT

74. Period and Frequency Imagine a rod spinning around the center, taking 2 seconds for one round Period T = 2 (second) Ordinary frequency f = 0.5 (Hz or times/second) Angular frequency ω = 1 (degree/second) Remember

75. FT Properties Property 1: time domain shifting (or delay) Proof

76. FT Properties Property 2: frequency domain shifting Proof

77. FT Properties Property 3: scaling Proof

78. FT Properties Property 4: time domain differentiation Proof Question: what about

79. FT Properties Property 5: Symmetry Proof:

80. Some common FT-pairs Impulse function

81. Some common FT-pairs Complex exponential function Using the frequency domain shifting property, and symmetry property

82. Some common FT-pairs Sine function By using the Euler formula and delay property

83. Some common FT-pairs Rectangular function

84. Some common FT-pairs Gaussian function

85. More Example Let f(t)-F(w) be a FT-pair. Now compute the FT of g(t) = f(t)cos(t). using Euler’s Formula and FT frequency shifting property

86. Exercise Compute the FT of the following signals u(t)cos( ω 0 t) u(t)sin( ω 0 t)e -at e -|a|t u(t)e -at u(t)te -at Here u(t) is the heavyside step function