1. Chapter 4 The Fourier Transform EE 207 Dr. Adil Balghonaim

3. Let x p (t) be a periodical wave, then expanding the periodical function Rewriting x p (t) and X n

6. Fourier Transform Pairs

9. Finding the Fourier Transform

11. Example Find the Fourier Transform for the following function

14. Example

17. It was shown previously

18. The Fourier Transform for the following function

22. Example Find the Fourier Transform for the delta function x(t) =  (t)

23. 1-Linearity Proof Properties of the Fourier Transform

26. LetThen Proof Change of variable 2-Time-Scaling (compressing or expanding)

27. Let

28. Now Let Change of variable Since

30. Proof 3-Time-Shifting

31. Example Find the Fourier Transform of the pulse function Solution From previous Example

32. 4-Time Transformation Proof

36. 5-Duality ازدواجية

37. Step 1 from Known transform from the F.T Table Step 2

38. Multiplication in FrequencyConvolution in Time Proof 6- The convolution Theorem

39. Now substitute x 2 (t- ) ( as the inverse Fourier Transform) in the convolution integral

40. Exchanging the order of integration, we have

41. Proof Similar to the convolution theorem, left as an exercise The multiplication Theorem Applying the multiplication Theorem

42. Find the Fourier Transform of following Solution Since

43. System Analysis with Fourier Transform

44. Proof 6- Frequency Shifting

45. Example Find the Fourier Transform for

46. Find the Fourier Transform of the function

47. Since and Therefore Method 1

48. Method 2

49. 7-Differentiation

50. Using integration by parts

51. Since x(t) is absolutely integrable

52. Example Find the Fourier Transform of the unit step function u(t) 7- Integration

54. Proof

64. Find the Transfer Function for the following RC circuit we can find h(t) by solving differential equation as follows Method 1

65. We will find h(t) using Fourier Transform Method rather than solving differential equation as follows Method 2

66. From Table 4-2

67. Method 3 In this method we are going to transform the circuit to the Fourier domain. However we first see the FT on Basic elements

68. Method 3

69. Fourier Transform

72. Find y(t) if the input x(t) is Method 1 ( convolution method) Using the time domain ( convolution method, Chapter 3) Example

73. Using partial fraction expansion (will be shown later) From Table 5-2 Method 2 Fourier Transform Sine Y(  ) is not on the Fourier Transform Table 5-2

74. Example Find y(t) Method 1 ( convolution method)

75. Method 2 Fourier Transform