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

10. T is a period of all the above signals
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

14. Proof
m = n

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

16. Decomposition

17. Proof
Use the following facts:

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

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

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

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

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

23. Harmonics

24. Amplitudes and Phase Angles
harmonic amplitude
phase angle

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

32. Example t
f(t) A

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

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

35. Example t
f(t) A

36. Fourier Series
Impulse Train

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

38. Property
(t): Test Function

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

40. Fourier Series of the Impulse Train

41. Complex Form Fourier Series of the Impulse Train

42. Analysis of Periodic Waveforms
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. Half-Wave Symmetry
and T
T/2
T/2

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

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

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

51. Fourier Coefficients of Even Functions

52. Fourier Coefficients for Half-Wave Symmetry
and T
T/2
T/2
The Fourier series contains only odd harmonics.

53. Fourier Coefficients for Half-Wave Symmetry
and

54. Fourier Coefficients for Even Quarter-Wave Symmetry

55. Fourier Coefficients for Odd Quarter-Wave Symmetry

56. Example
Even Quarter-Wave Symmetry T
T/2
T/2 1 1 T
T/4
T/4

57. Example
Even Quarter-Wave Symmetry T
T/2
T/2 1 1 T
T/4
T/4

58. Example
Odd Quarter-Wave Symmetry T
T/2
T/2 1 1 T
T/4
T/4

59. Example
Odd Quarter-Wave Symmetry T
T/2
T/2 1 1 T
T/4
T/4

60. Half-Range Expansions
Fourier Series
Half-Range Expansions

61. Non-Periodic Function Representation
A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

62. Without Considering Symmetry T
A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

63. Expansion Into Even Symmetry
T=2
A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

64. Expansion Into Odd Symmetry
T=2
A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

65. Expansion Into Half-Wave Symmetry
T=2
A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

66. Expansion Into Even Quarter-Wave Symmetry
T/2=2
T=4
A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

67. Expansion Into Odd Quarter-Wave Symmetry
T/2=2
T=4
A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

68. Least Mean-Square Error Approximation
Fourier Series
Least Mean-Square Error Approximation

69. Approximation a function
Use
to represent f(t) on interval T/2 < t < T/2.
Define
Mean-Square Error

70. Approximation a function
Show that using Sk(t) to represent f(t) has least mean-square property.
Proven by setting Ek/ai = 0 and Ek/bi = 0.

71. Approximation a function

72. Mean-Square Error

73. Mean-Square Error

74. Mean-Square Error