Download presentation

Published byAlexandra Kent Modified over 4 years ago

1
**For more ppt’s, visit www.studentsaarthi.com**

Fourier Series For more ppt’s, visit

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 50 100 150 -50 -100**

-120 -80 A/5 50 100 150 -50 -100 -150

34
**Example A/10 40 80 120 -40 -120 -80 100 200 300 -100 -200**

-120 -80 A/10 100 200 300 -100 -200 -300

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

Similar presentations

Presentation is loading. Please wait....

OK

Fourier Series & Transforms

Fourier Series & Transforms

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on credit policy in banks Ppt on hong kong airport Ppt on product advertising campaign Ppt on boilers operations with integers Chapter 16 dilutive securities and earnings per share ppt online Ppt on web page designing Ppt on area of trapezium Ppt on economic development in india Ppt on networking related topics about work Ppt online shopping project report