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Fourier Series 主講者:虞台文. Content Periodic Functions Fourier Series Complex Form of the Fourier Series Impulse Train Analysis of Periodic Waveforms Half-Range.

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Presentation on theme: "Fourier Series 主講者:虞台文. Content Periodic Functions Fourier Series Complex Form of the Fourier Series Impulse Train Analysis of Periodic Waveforms Half-Range."— Presentation transcript:

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

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

10 Synthesis DC Part Even Part Odd Part T is a period of all the above signals 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 0 m  n 0

14 Proof 0 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)f(t) 1

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

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

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

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 amplitudephase angle

25 Fourier Series Complex Form of the Fourier Series

26 Complex Exponentials

27 Complex Form of the Fourier Series

28

29

30 If f(t) is real,

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

32 Example t f(t)f(t) A

33 40  80  120  -40   -80  A/5 5050 10  0 15  0 -5    0

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

35 Example t f(t)f(t) A 0

36 Fourier Series Impulse Train

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

38 Property  (t): Test Function

39 Impulse Train 0 t T2T2T3T3T TT 2T2T 3T3T

40 Fourier Series of the Impulse Train

41 Complex Form Fourier 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 f e (t) and an odd function f o (t). Even Part Odd Part

45 Example Even Part Odd Part

46 Half-Wave Symmetry and TT/2  T/2

47 Quarter-Wave Symmetry Even Quarter-Wave Symmetry TT/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 T TT A/2  A/2 T 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

52 Fourier Coefficients for Half-Wave Symmetry and TT/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 TT/2  T/2

55 Fourier Coefficients for Odd Quarter-Wave Symmetry T T/2  T/2

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

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

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

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

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

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

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

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

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

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

68 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 S k (t) to represent f(t) has least mean-square property. Proven by setting  E k /  a i = 0 and  E k /  b i = 0.

71 Approximation a function

72 Mean-Square Error

73

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