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

3. Let xp(t) be a periodical wave, then expanding the periodical function
Rewriting xp(t) and Xn

6. Fourier Transform Pairs

7. Fourier Transform Pairs

8. 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) = d(t)

23. Properties of the Fourier Transform
1-Linearity
Proof

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

27. Let

28. Now Let
Change of variable
Since

30. 3-Time-Shifting
Proof

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

38. 6- The convolution Theorem
Multiplication in Frequency
Convolution in Time
Proof

39. Now substitute x2(t-l) ( as the inverse Fourier Transform) in the convolution integral

40. Exchanging the order of integration , we have

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

42. Find the Fourier Transform of following
Solution
Since

43. System Analysis with Fourier Transform

44. 6- Frequency Shifting
Proof

45. Example Find the Fourier Transform for

46. Find the Fourier Transform of the function

47. Method 1
Since
and
Therefore

48. Method 2

49. 7-Differentiation

50. Using integration by parts

51. Since x(t) is absolutely integrable

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

54. Proof

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

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

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. Method 3
Fourier Transform

70. Fourier Transform

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

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

74. Example

75. Method Phasor method
Voltage Division

76. Method 2 Fourier Transform method

79. Let x(t) be a periodical signal
were
Fourier Transform

82. Constant with respect to Fourier Transform

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

84. Method 2 Fourier Transform