1. 4.A Fourier Series & Fourier Transform Many slides are from Taiwen Yu

2. Fourier Series: background

3. The Mathematic Formulation Any function that satisfies where T is a constant and is called the period of the function.

4. f(x) x

5. x

6. Example: Find its period. Fact: smallest T

7. Example: Find its period.

8. f(t) = cos2  t + ½ cos4  t Can we get the individual components just from the values of f(t)?

9. A periodic function/signal Decompose a periodic input signal into primitive periodic components. A periodic signal T2T3T t f(t)f(t)

10. What could be the components? (Base/fundamental) period: T fundamental frequency: f 0

11. Synthesis DC Part Even Part Odd Part

12. Orthogonal Functions Call a set of functions {  k } orthogonal on an interval a < t < b if it satisfies

13. Fourier Series: Complex Form

14. Euler’s formula

15. Complex Form of the Fourier Series

17. An FS example: from cos and sin functions

18. Example 1. Find the Fourier series of the following periodic function.

19. Therefore, the corresponding Fourier series is In writing the Fourier series we may not be able to consider infinite number of terms for practical reasons. The question therefore, is – how many terms to consider?

20. When we consider 4 terms as shown in the previous slide, the function looks like the following. 1.5 1 0.5 0 1 1.5 f  () 

21. When we consider 6 terms, the function looks like the following. 1.5 1 0.5 0 1 1.5 f  () 

22. When we consider 8 terms, the function looks like the following. 1.5 1 0.5 0 1 1.5 f  () 

23. When we consider 12 terms, the function looks like the following. 1.5 1 0.5 0 1 1.5 f  () 

24. The red curve was drawn with 12 terms and the blue curve was drawn with 4 terms. 1.5 1 0.5 0 1 1.5 

25. The red curve was drawn with 12 terms and the blue curve was drawn with 4 terms. 0246810 1.5 1 0.5 0 1 1.5 

26. The red curve was drawn with 20 terms and the blue curve was drawn with 4 terms. 0246810 1.5 1 0.5 0 1 1.5 

27. If we add all the terms

28. Fourier transform

29. Fourier coefficients for rect with T = 2

30. Fourier coefficients for rect with T = 4

31. Fourier coefficients for rect with T = 16

32. Fourier transform of rect fn = sinc fn