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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