 Download Presentation Fourier Series

# Fourier Series

Download Presentation ## Fourier Series

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1. Fourier Series 主講者：虞台文

2. Content • Periodic Functions • Fourier Series • Complex Form of the Fourier Series • Impulse Train • Analysis of Periodic Waveforms • Half-Range Expansion • Least Mean-Square Error Approximation

3. Fourier Series Periodic Functions

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

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

6. Example: Find its period. must be a rational number

7. Example: Is this function a periodic one? not a rational number

8. Fourier Series Fourier Series

9. A periodic sequence f(t) t T 2T 3T Introduction • Decompose a periodic input signal into primitive periodic components.

10. Synthesis T is a period of all the above signals Even Part Odd Part DC Part Let 0=2/T.

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

12. Orthogonal set of Sinusoidal Functions Define 0=2/T. We now prove this one

13. Proof m  n 0 0

14. Proof m = n 0

15. Define 0=2/T. Orthogonal set of Sinusoidal Functions an orthogonal set.

16. Decomposition

17. Proof Use the following facts:

18. f(t) 1 -6 -5 -4 -3 -2 -  2 3 4 5 Example (Square Wave)

19. f(t) 1 -6 -5 -4 -3 -2 -  2 3 4 5 Example (Square Wave)

20. f(t) 1 -6 -5 -4 -3 -2 -  2 3 4 5 Example (Square Wave)

21. Harmonics T is a period of all the above signals Even Part Odd Part DC Part

22. Define , called the fundamental angular frequency. Define , called the n-th harmonicof the periodic function. Harmonics

23. Harmonics

24. harmonic amplitude phase angle Amplitudes and Phase Angles

25. Fourier Series Complex Form of the Fourier Series

26. Complex Exponentials

27. Complex Form of the Fourier Series

28. Complex Form of the Fourier Series

29. Complex Form of the Fourier Series

30. Complex Form of the Fourier Series If f(t) is real,

31. amplitude spectrum |cn|  n phase spectrum  Complex Frequency Spectra

32. f(t) A t Example

33. A/5 0 -120 -80 -40 40 80 120 -150 -100 -50 50 100 150 Example

34. A/10 0 -120 -80 -40 40 80 120 -300 -200 -100 100 200 300 Example

35. f(t) A t 0 Example

36. Fourier Series Impulse Train

37. t 0 Dirac Delta Function and Also called unit impulse function.

38. Property (t): Test Function

39. t T 2T 0 T 2T 3T 3T Impulse Train

40. Fourier Series of the Impulse Train

41. Complex FormFourier Series of the Impulse Train

42. Fourier Series Analysis of Periodic Waveforms

43. Waveform Symmetry • Even Functions • Odd Functions

44. Decomposition • Any function f(t) can be expressed as the sum of an even function fe(t) and an odd function fo(t). Even Part Odd Part

45. Example Even Part Odd Part

46. T/2 T/2 T Half-Wave Symmetry and

47. T/2 T/2 T T/2 T/2 T Quarter-Wave Symmetry Even Quarter-Wave Symmetry Odd Quarter-Wave Symmetry

48. A T T A/2 T T A/2 Hidden Symmetry • The following is a asymmetry periodic function: • Adding a constant to get symmetry property.

49. Fourier Coefficients of Symmetrical Waveforms • The use of symmetry properties simplifies the calculation of Fourier coefficients. • Even Functions • Odd Functions • Half-Wave • Even Quarter-Wave • Odd Quarter-Wave • Hidden

50. Fourier Coefficients of Even Functions