1. Chapter 5:
Fourier Transform

2. FOURIER TRANSFORM: Definition of the Fourier transforms
Relationship between Laplace Transforms and Fourier Transforms
Fourier transforms in the limit
Properties of the Fourier Transforms
Circuit applications using Fourier Transforms
Parseval’s theorem
Energy calculation in magnitude spectrum

3. Definition of Fourier Transforms

4. Inverse Fourier Transforms:

5. Example 1:
Obtain the Fourier Transform for the
function below:

6. Solution:
Given function is:

7. Fourier Transforms:

8. FOURIER TRANSFORM: Definition of the Fourier transforms
Relationship between Laplace Transforms and Fourier Transforms
Fourier transforms in the limit
Properties of the Fourier Transforms
Circuit applications using Fourier Transforms
Parseval’s theorem
Energy calculation in magnitude spectrum

9. Relationship between Fourier Transforms and Laplace Transforms
There are 3 rules apply to the use of Laplace transforms to find Fourier Transforms of such functions.

10. Rule 1: If f(t)=0 for t<=0-
Replace s=jω

11. Example:

12. Replace s=jω

13. Rule 2: Inverse negative function

14. Example:
Negative

15. Fourier Transforms

16. Rule 3: Add the positive and negative function

17. Thus,

18. Example 1:

19. Fourier transforms:

20. Example 2:
Obtain the Fourier Transforms for the function below:

21. Solution:

22. Example 3:

23. Solution:

24. Example 4:

25. Solution:

27. FOURIER TRANSFORM: Definition of the Fourier transforms
Relationship between Laplace Transforms and Fourier Transforms
Fourier transforms in the limit
Properties of the Fourier Transforms
Circuit applications using Fourier Transforms
Parseval’s theorem
Energy calculation in magnitude spectrum

28. Fourier Transforms in the limit
Fourier transform for signum function (sgn(t))

31. assume ε→0,

32. Fourier Transforms for step function:

33. Fourier Transforms for cosine function

35. Thus,

36. FOURIER TRANSFORM: Definition of the Fourier transforms
Relationship between Laplace Transforms and Fourier Transforms
Fourier transforms in the limit
Properties of the Fourier Transforms
Circuit applications using Fourier Transforms
Parseval’s theorem
Energy calculation in magnitude spectrum

37. Properties of Fourier Transforms
Multiplication by a constant

38. Addition and subtraction

39. Differentiation

40. Integration

41. Scaling

42. Time shift

43. Frequency shift

44. Modulation

45. Convolution in time domain

46. Convolution in frequency domain:

47. Example 1:
Determine the inverse Fourier Transforms for the function below:

48. Solution:
LAPLACE
TRANSFORMS

49. A and B value: