Continous time fourier series notes coep

Continuous time fourier series notes signals and systems

1. Fourier Transform Fourier Transform Analysis: Fourier analysis for Continuous time signals and systems, Continuous time Fourier series and its convergence, Continuous time Fourier Transform, its properties, frequency response 1

2. Continuous time Fourier Series It is convenient to choose a set of orthogonal waveforms as basic signals due to It is mathematically convenient to represent an arbitrary signal as a weighted sum of orthogonal waveform because many of the calculations involving signals are simplified by using such representation. It is possible to visualize the signal as a vector in an orthogonal coordinate system with orthogonal waveforms being coordinates. Representation in terms of orthogonal basis function provides a convenient means of solving for the response of linear systems to arbitrary inputs. 2

3. Continuous time Fourier Series For periodic signals, the set of harmonically related complex exponentials is the convenient choice as orthogonal basis function . Fourier Series The representation of a non sinusoidal periodic signals in terms of complex exponential or equivalently in terms of sine and cosine waveforms . Fourier series is mathematical tool that allows the representation of any periodic signal as the sum of harmonically related sinusoids . 3

4. Fourier Series 4 Any periodic signal can be expressed by Fourier series if If it is discontinuous and there are a finite number of discontinuities in the period T ; It has a finite average value over the period T ; i.e. absolutely integrable It has a finite number of positive and negative maxima in the period T. When these Dirichlet conditions are satisfied, the Fourier series exist.

5. Fourier Series 5 The Fourier series of two types Trigonometric Fourier series and Exponential Fourier series.

6. Trigonometric Fourier Series 6 Trigonometric Fourier series is expressed as Where a , a n and b n are the trigonometric Fourier series coefficients. The coefficient may be obtained from x(t) as  

7. Trigonometric Fourier Series 7 The integrations can be carried out from –T/2 to T/2 or over any other full period. The series with coefficients obtained form the above integrals converges uniformly to the function at all the points of continuity and converges to mean values at the points of discontinuity .  

8. Polar Form Representation of Fourier Series 8 Case I: Let and where and are related to and as We know  

9. Polar Form Representation of Fourier Series 9 The coefficient is the amplitude and is the phase of the n th harmonic . The and contains all the needed information to construct the Fourier series. The plot of as a function of is known as the amplitude spectrum . The plot of as a function of is known as the phase spectrum .  

10. Polar Form Representation of Fourier Series 10 Case II: Let and where and are related to and as We know  

11. Evaluation of Fourier Series Coefficients 11 We know that The average value of a sinusoid over m and n for period T is zero.  

12. Evaluation of Fourier Series Coefficients 12 Also  

13. Evaluation of Fourier Series Coefficients 13 Prove that +… Integrating the above equation over the period 0 to T  

14. Evaluation of Fourier Series Coefficients 14  

15. Evaluation of Fourier Series Coefficients 15 The coefficient a is simply the average value of x(t) over a period.

16. Evaluation of Fourier Series Coefficients 16 Prove that +… Multiplying both side by and Integrating the above equation over the period 0 to T  

17. Evaluation of Fourier Series Coefficients 17  

18. Evaluation of Fourier Series Coefficients 18 Prove that +… Multiplying both side by and Integrating the above equation over the period 0 to T.  

19. Evaluation of Fourier Series Coefficients 19  

20. Symmetry Conditions (Even Symmetry) 20 If x(t) is even signal then the Fourier series coefficients a , a n , and b n are defined as  

21. Symmetry Conditions (Odd Symmetry) 21 If x(t) is odd signal then the Fourier series coefficients a , a n , and b n are defined as  

22. Symmetry Conditions (Half wave Symmetry) 22 This symmetry is defined if the function is shifted one half period and inverted and look identical to original then it is half wave symmetric For is even For is odd  

23. Symmetry Conditions (Half wave Symmetry) 23 Example

24. Exponential Fourier Series 24 The exponential Fourier series is expressed as Where, X n are the exponential Fourier series coefficient of the signal x(t). The integration can be carried out form –T/2 to T/2 or over any other full period. The Fourier series coefficient are known as a frequency domain representation of x(t) because each Fourier coefficient is associated with a complex sinusoid of a different frequency.  

25. Relationship between Trigonometric & Exponential Fourier Series 25 We can express the sine and cosine terms in the trigonometric series by its equivalent exponential. Let  

26. Relationship between Trigonometric & Exponential Fourier Series 26 Substituting the values of a n and b n in X n  

27. Relationship between Trigonometric & Exponential Fourier Series 27 Substituting the n = 0 in  

28. Relationship between Trigonometric & Exponential Fourier Series 28 Consider We know that Therefore, a n is an even Fourier series coefficient  

29. Relationship between Trigonometric & Exponential Fourier Series 29 Consider We know that Therefore, b n is an odd Fourier series coefficient  

30. Relationship between Trigonometric & Exponential Fourier Series 30 Both representations are identical . The reasons for offering the exponential form are that it is more compact and the expression for deriving the exponential coefficients is also more compact. Also, the LTI system response to exponential signals is simpler than that of sinusoids.  

31. Relationship between Trigonometric & Exponential Fourier Series 31 Disadvantage Exponential form can not be visualized as easily as sinusoids . For spontaneous and qualitative understanding the sinusoids are better.

32. Relationship between Trigonometric & Exponential Fourier Series 32 To overcome the difficulty with the exponential form we can always convert the exponential form into trigonometric form using a = X a n = 2 Re{ X n } = X n + X -n b n = 2 Im { X n } = j ( X n – X -n )

33. Line Spectrum 33 It is a plot that shows each of harmonic amplitudes in the signal. The line deceases rapidly for the rapidly converging signals . The line deceases slowly for the signals with discontinuities (triangular, saw tooth, square), since their series have strong high harmonics. The harmonic contents and line spectrum of a signal never change and independent of method of analysis.

34. Line Spectrum 34 The magnitude spectrum is The magnitude spectrum is even function.  

35. Line Spectrum 35 The phase spectrum is The phase spectrum is odd function.  

36. Concept of Negative Frequency 36 The existence of the spectrum at negative frequency is confusing because by definition the frequency is positive quantity. Then what is the negative frequency?

37. Concept of Negative Frequency 37 The negative frequencies are present in the exponential Fourier series because the mathematical model of a signal that requires use of negative frequencies. The exponential spectra are a graphical representation of coefficients as a function of . Existence of spectrum at is an indication that an exponential component exists in the series.  

38. Concept of Negative Frequency 38 The function of frequency (negative frequency) represents the counterpart of signal of (positive frequency), and when these two exponential are combined, they provide a real function (sinusoid) of frequency .  

39. Properties of CTFS 39 Suppose that x(t) is periodic signal with period T and fundamental frequency ω = 2 π /T . If the Fourier series coefficients of x(t) are denoted by X n x(t)↔ X n

40. Properties of CTFS: Linearity 40 Let x(t) and y(t) be the two periodic signal with period T and if x(t)↔ X n ; y(t)↔ Y n Then z(t) = a x(t) + b y(t) ↔ Z n = a X n + b Y n

41. Properties of CTFS: Linearity 41 Let x(t) and y(t) be the two periodic signal with period T and if x(t)↔ X n ; y(t)↔ Y n Then z(t) = a x(t) + b y(t) ↔ Z n = a X n + b Y n  

42. Properties of CTFS: Time Shifting 42 Let x(t) be a periodic signal with period T and if x(t)↔ X n Then  

43. Properties of CTFS: Time Shifting 43 Let x(t) be a periodic signal with period T and if x(t)↔ X n Then    

44. Properties of CTFS: Time Shifting 44 Let x(t) be a periodic signal with period T and if x(t)↔ X n Then When a signal is shifted in time the magnitudes of its Fourier series coefficients remain unaltered.  

45. Properties of CTFS: Frequency Shifting 45 Let x(t) be a periodic signal with period T and if x(t)↔ X n Then  

46. Properties of CTFS: Frequency Shifting 46 Let x(t) be a periodic signal with period T and if x(t)↔ X n Then The frequency shift is corresponding to multiplication in time domain by a complex sinusoid.  

47. Properties of CTFS: Time Reversal 47 Let x(t) be a periodic signal with period T and if x(t)↔ X n Then The time reversal applied to a continuous time signal results in a time reversal of the corresponding sequence of Fourier series coefficients.  

48. Properties of CTFS: Time Reversal 48 if x(t) even then its Fourier series coefficients are also even. x(-t) = x(t), then X -n = X n Similarly, if if x(t) odd then its Fourier series coefficients are also odd. x(-t) = -x(t), then X -n = - X n

49. Properties of CTFS: Time Scaling 49 Let x(t) be a periodic signal with period T and if x(t) ↔ X n Then The time scaling operation changes the period of the signal. Therefore, if x(t) is a periodic signal with period T and fundamental frequency ω = 2 π /T then y(t) = x(at) is a periodic with period T/a and fundamental frequency a ω .  

50. Properties of CTFS: Time Scaling 50 Let x(t) be a periodic signal with period T and x(t)↔ X n Then Let  

51. Properties of CTFS: Time Scaling 51 Let x(t) be a periodic signal with period T and x(t)↔ X n Then The Fourier series coefficients have not changed, the Fourier series representation has changed because of the change in fundamental frequency.  

52. Properties of CTFS: Periodic Convolution 52 Let x(t) and y(t) be a periodic signals with period T and if x(t) ↔ X n ; y(t) ↔ Y n Then For the periodic signals with same period, a special form of convolution is known as periodic convolution and defined by and denoted by  

53. Properties of CTFS: Periodic Convolution 53 Let x(t) and y(t) be a periodic signals with period T and  

54. Properties of CTFS: Multiplication 54 Let x(t) and y(t) be a periodic signals with period T and if x(t) ↔ X n ; y(t) ↔ Y n Then Z n is interpreted as discrete time convolution of X n and Y n .  

55. Properties of CTFS: Multiplication 55 Let n = k+m ; therefore m = n – k; limits will be same as m  

56. Properties of CTFS: Differentiation 56 Let x(t) be a periodic signals with period T and if x(t)↔ X n ; Then The Fourier series representation of periodic signal x(t) is Differentiating both sides  

57. Properties of CTFS: Integration 57 Let x(t) be a periodic signals with period T and X = 0 x(t)↔ X n ; Then The Fourier series representation of periodic signal x(t) is  

58. Properties of CTFS: Integration 58 integrating both sides  

59. Conjugation & conjugate Symmetry 59 Let x(t) be a periodic signals with period T and if x(t)↔ X n ; Then y(t) = x*(t) ↔ Y n = X* -n The complex conjugates of a period signal x(t) results in complex conjugation and time reversal on the corresponding Fourier series coefficients.

60. Conjugation & conjugate Symmetry 60 Case I: If x(t) is real i.e. x*(t) = x(t) Then X * -n = X n X -n = X * n The Fourier series coefficients will be conjugate symmetric. Also, if x(t) is real and even X -n = X * n = Xn So, if x(t) is real and even, Fourier series coefficients are also.  

61. Conjugation & conjugate Symmetry 61 Case II: If x(t) is real and odd i.e., x*(t) = - x(t) Then X -n = X * n = - X n The Fourier series coefficients are purely imaginary and odd.

62. Conjugation & conjugate Symmetry 62 Case III: Even and odd decomposition of real signals If x(t) is real and x(t)↔ X n ; Then x e (t) ↔ Re{ X n } The even part of the signal is x e (t) = ½ [x(t) + x(-t)] Using linearity property, we get x e (t) ↔ ½ [ X n + X -n ] x e (t) ↔ ½ [ X n + X * -n ] x e (t) ↔ ½ 2 Re{ X n } x e (t) ↔ Re{ X n }

63. Conjugation & conjugate Symmetry 63 Case III: Even and odd decomposition of real signals If x(t) is real and x(t)↔ X n ; Then x (t) ↔ j Im { X n } The odd part of the signal is x (t) = ½ [x(t) - x(-t)] Using lineraity property we get x (t) ↔ ½ [ X n - X -n ] x (t) ↔ ½ [ X n - X * n ] x (t) ↔ ½ 2j Im { X n } x (t) ↔ j Im { X n }

64. Parseval’s Theorem for Power Signals 64 Let x(t) be a periodic signals with period T and if x(t)↔ X n ; Then The theorem states that the total average power in a periodic signal equals to the sum of average powers in all its harmonic components.  

65. Parseval’s Theorem for Power Signals 65 Let x(t) be a periodic signals the energy of x(t) is Since all the periods are same therefore for n periods the energy is The average signal power of a periodic signal is      

66. Parseval’s Theorem for Power Signals 66 .  

67. Parseval’s Theorem for Power Signals 67 Find the average power of signal , when is given as   -2 -1 0 1 2 n→ 4 4 2  

68. Systems with periodic inputs 68 The response of an LTI system to a sinusoidal input is known as the frequency response of the system. This is obtained by convolution of the impulse response and a complex sinusoidal input signal   ,  

69. Systems with periodic inputs 69 Thus, the output of the system is a complex sinusoid of the same frequency because the input is multiplied by the complex number H( ω ). The polar form of frequency response is Where |H( ω )|is known as the magnitude response and angle of H( ω ) is phase response. Therefore, the output is The system modifies the amplitude of the input by |H( ω )| and phase by angle of H( ω )  

70. Systems with periodic inputs 70 If the input to an LTI system is given by Then the output of the system is given by Where Y n is the Fourier series coefficients of the output of an LTI system and is given by        

71. Numerical 1 71 Find the trigonometric Fourier series for the waveforms x(t) 10 t 2 π  

72. Numerical 1 72 Find the trigonometric Fourier series for the waveforms  

73. Numerical 1 73  

74. Numerical 2 74 Find the trigonometric Fourier series for the square wave   x(t) A t 2 π π - π - A

75. Numerical 2 75 Find the trigonometric Fourier series for the square wave   x(t) A t 2 π π - π - A

76. Numerical 2 76 Find the trigonometric Fourier series for the square wave

77. Numerical 3 77 Find the trigonometric Fourier series for the triangular wave and plot the line spectrum. x(t) 1 t -1 1 -2 -3 2 3

78. Numerical 3 78 Find the trigonometric Fourier series for the triangular wave

79. Numerical 3 79 Find the trigonometric Fourier series for the triangular wave

80. Numerical 4 80 Find the trigonometric Fourier series for the waveforms x(t) 1 t 1 2 -1 -2 T  

81. Numerical 4 81 Find the trigonometric Fourier series for the waveforms  

82. Numerical 4 82 Find the trigonometric Fourier series for the waveforms

83. Numerical 5 83 Find the trigonometric Fourier series for the waveforms

84. Numerical 4 84 Find the trigonometric Fourier series for the waveforms

85. Numerical 5 85 Find the exponential Fourier series for the waveforms. Using the coefficients of this exponential Fourier series obtain coefficients of trigonometric Fourier series.   x(t) 10 t 2 π

86. Numerical 5 86 As therefore fundamental frequency The average value  

87. Numerical 5 87 The exponential Fourier series is The trigonometric Fourier series coefficients are The trigonometric Fourier series is  

88. Numerical 5 88 Find the exponential Fourier series for the half wave rectified sine wave with period with amplitude and sketch the line spectrum. As therefore fundamental frequency The given waveform for one period may be written as  

89. Numerical 6 89 For the continuous time periodic signal Determine the fundamental frequency and the Fourier series coefficients such that  

90. Numerical 7 90 A continuous time periodic signal is real-valued and has a fundamental period . The non-zero Fourier series coefficients for are Express in the form