The Fourier Series

1. The Fourier Series BEE2113 – Signals & Systems R. M. Taufika R. Ismail FKEE, UMP

2. Introduction A Fourier series is an expansion of a periodicfunctionf (t) in terms of an infinite sum of cosines and sines

3. In other words, any periodicfunction can be resolved as a summation of constant value and cosine and sine functions: dc ac

4. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombinedto obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical.

5. f(t) Periodic Function = t + + + + + …

6. where

7. Notice that [*The right sign of the area below graph must be obeyed. +ve sign for area above x-axis & −ve sign for area below x-axis]

8. Symmetry Considerations • Symmetry functions: (i) even symmetry (ii) odd symmetry

9. Even symmetry • Any function f (t) is even if its plot is symmetrical about the vertical axis, i.e.

10. Even symmetry (cont.) • The examples of even functions are:

11. Even symmetry (cont.) • The integral of an even function from −A to +A is twice the integral from 0 to +A −A +A

12. Odd symmetry • Any function f (t) is odd if its plot is antisymmetrical about the vertical axis, i.e.

13. Odd symmetry (cont.) • The examples of odd functions are:

14. Odd symmetry (cont.) • The integral of an odd function from −A to +A is zero −A +A

15. Even and odd functions The product properties of even and odd functions are: • (even)× (even) = (even) • (odd)× (odd) = (even) • (even)× (odd) = (odd) • (odd)× (even) = (odd)

16. Symmetry consideration From the properties of even and odd functions, we can show that: • for even periodic function; • for odd periodic function;

17. How?? [Even function] (even) ×(odd) (even) ×(even) | | (odd) | | (even)

18. How?? [Odd function] (odd) (odd)×(odd) | | (even) (odd) ×(even) | | (odd)

19. The amplitude-phase form is known as the sine-cosine form • We can also express the Fourier series in the cosine form only, that is This form is called as the amplitude-phase form

20. From this form, we can plot the amplitude spectrum, vs. n and the phase spectrum, vs. n. • It can be shown that the combination of cosine and sine function can be expressed as a cosine function only: • Comparing both sides of eqn:

22. Hence where Or in phasor/complex form:

23. Example 1 Determine the Fourier series of the following waveform. Obtain the amplitude and phase spectra.

24. Solution First, determine the period & describe the one period of the function: T = 2 We find that

25. Then, obtain the coefficients a0, an and bn: Or

26. Notice that n is integer which leads , since Therefore, .

27. Notice that or Therefore,

28. Finally,

29. In amplitude-phase form,

30. Amplitude spectrum: Phase spectrum:

31. Some helpful identities For n integers,

32. Notes: • The sum of the Fourier series terms can evolve (progress) into the original waveform • From Example 1, we obtain • It can be demonstrated that the sum will lead to the square wave:

33. (a) (b) (c) (d)

34. (e) (f)

36. Example 2 Given Sketch the graph of f (t) such that Then compute the Fourier series expansion of f (t). Plot the amplitude and phase spectra until the forth harmonic.

37. Solution The function is described by the following graph: T = 2 We find that

38. Then we compute the coefficients: since f(t) is an odd function.

39. Hence,

40. Amplitude spectrum: Phase spectrum: 0.64 0.32 0.21 0.16

41. Example 3 Given (i) Sketch the graph of v (t) such that (ii) Compute the trigonometric Fourier series of v (t). (iii) Express the Fourier series of v (t) in the amplitude-phase form. Then plot the amplitude and phase spectra until the forth harmonic.

42. Solution (i) The function is described by the following graph: v (t) 2 t 2 4 6 8 10 12 0 T = 4 We find that

43. (ii) Then we compute the coefficients: Or

45. since

46. Hence,

47. (iii) Where

48. We obtained that

49. Amplitude spectrum: Phase spectrum: 0.75 0.5 π/2 π 3π/2 2π 0.32 0.22 0.16 −57.5° −78.0° −90° −90° 0 π/2 π 3π/2 2π

50. Example 4 Given Sketch the graph of f (t) such that Then compute the Fourier series expansion of f (t).