1. FOURIER TRANSFORMS

2. JELMAAN FOURIER: 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. CIRCUIT APPLICATION USING FOURIER TRANSFORMS
Circuit element in frequency domain:

4. Example 1:
Obtain vo(t) if vi(t)=2e-3tu(t)

5. Solution:
Fourier Transforms for vi

6. Transfer function:

7. Thus,

8. From partial fraction:
Inverse Fourier Transforms:

9. Example 2:
Determine vo(t) if vi(t)=2sgn(t)=-2+4u(t)

10. Solution:

13. JELMAAN FOURIER: 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

14. PARSEVAL’S THEOREM
Energy absorbed by a function f(t)

15. Parseval’s theorem stated that energy also can be calculate using,

16. Parseval’s theorem also can be written as:

17. PARSEVAL’S THEOREM DEMONSTRATION
If a function,

18. Integral left-hand side:

19. Integral right-hand side:

20. JELMAAN FOURIER: 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

21. ENERGY CALCULATION IN MAGNITUDE SPECTRUM
Magnitude of the Fourier Transforms squared is an energy density (J/Hz)

22. Energy in the frequency band from ω1 and ω2:

23. Example 1: The current in a 40Ω resistor is:
What is the percentage of the total energy dissipated in the resistor can be associated with the frequency band 0 ≤ ω ≤ 2√3 rad/s?

24. Solution:
Total energy dissipated in the resistor:

25. Check the answer with parseval’s theorem:
Fourier Transform of the current:

26. Magnitude of the current:

28. Energy associated with the frequency band:

29. Percentage of the total energy associated:

30. Example 2:
Calculate the percentage of output energy to input energy for the filter below:

31. Energy at the input filter:

32. Fourier Transforms for the output voltage:

33. Thus,

34. Energy at the output filter:

35. Thus the percentage: