1. Fourier Series

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

3. Synthesis DC Part Even Part Odd Part T is a period of all the above signals Let  0 =2  /T.

4. Decomposition

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

11. Example (Square Wave)

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

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

14. Harmonics

15. Amplitudes and Phase Angles harmonic amplitudephase angle

16. Complex Form of Fourier Series

17. Complex Exponentials

18. Complex Form of the Fourier Series

21. If f(t) is real,

22. Example t f(t)f(t) A

23. 40  80  120  -40  0 -120  -80  A/5 5050 10  0 15  0 -5  0 -10  0 -15  0

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

25. Example t f(t)f(t) A 0

26. Waveform Symmetry Even Functions Odd Functions

27. 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

28. Example Even Part Odd Part

29. Half-Wave Symmetry and TT/2  T/2

30. Quarter-Wave Symmetry Even Quarter-Wave Symmetry TT/2  T/2 Odd Quarter-Wave Symmetry T T/2  T/2

31. 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

32. 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

33. Fourier Coefficients of Even Functions

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

36. Fourier Coefficients for Half-Wave Symmetry and

37. Fourier Coefficients for Even Quarter-Wave Symmetry TT/2  T/2

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

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

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

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

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

43. Dirichlet Conditions A periodic signal x(t), has a Fourier series if it satisfies the following conditions: absolutely integrable 1.x(t) is absolutely integrable over any period, namely finite number of maxima and minima 2.x(t) has only a finite number of maxima and minima over any period finite number of discontinuities 3.x(t) has only a finite number of discontinuities over any period

44. We have seen that periodic signals can be represented with the Fourier series aperiodic signals Can aperiodic signals be analyzed in terms of frequency components? Yes, and the Fourier transform provides the tool for this analysis spectra of aperiodic signals The major difference w.r.t. the line spectra of periodic signals is that the spectra of aperiodic signals are defined for all real values of the frequency variable not just for a discrete set of values Fourier Transform

45. Fourier transform Given a signal x(t), its Fourier transform is defined as A signal x(t) is said to have a Fourier transform in the ordinary sense if the above integral converges The Fourier Transform in the General Case

46. The integral does converge if well-behaved 1.the signal x(t) is “well-behaved” absolutely integrable 2.and x(t) is absolutely integrable, namely, well behaved Note: well behaved means that the signal has a finite number of discontinuities, maxima, and minima within any finite time interval The Fourier Transform in the General Case – Cont’d

47. Consider Since in general is a complex function, by using Euler’s formula Rectangular Form of the Fourier Transform

48. can be expressed in a polar form as where Polar Form of the Fourier Transform

49. If x(t) is real-valued, it is Moreover whence Fourier Transform of Real-Valued Signals Hermitian symmetry

50. Consider the even signal It is Example: Fourier Transform of the Rectangular Pulse

51. Example: Fourier Transform of the Rectangular Pulse – Cont’d

52. amplitude spectrum phase spectrum

53. bandlimited B bandwidth of the signal A signal x(t) is said to be bandlimited if its Fourier transform is zero for all where B is some positive number, called the bandwidth of the signal It turns out that any bandlimited signal must have an infinite duration in time, i.e., bandlimited signals cannot be time limited Bandlimited Signals

54. infinite bandwidthinfinite spectrum If a signal x(t) is not bandlimited, it is said to have infinite bandwidth or an infinite spectrum Time-limited signals cannot be bandlimited and thus all time-limited signals have infinite bandwidth However, for any well-behaved signal x(t) it can be proven that whence it can be assumed that Bandlimited Signals – Cont’d B being a convenient large number

55. inverse Fourier transform Given a signal x(t) with Fourier transform, x(t) can be recomputed from by applying the inverse Fourier transform given by Transform pair Transform pair Inverse Fourier Transform

56. Properties of the Fourier Transform Linearity: Linearity: Left or Right Shift in Time: Left or Right Shift in Time: Time Scaling: Time Scaling:

57. Properties of the Fourier Transform Time Reversal: Time Reversal: Multiplication by a Power of t: Multiplication by a Power of t: Multiplication by a Complex Exponential: Multiplication by a Complex Exponential:

58. Properties of the Fourier Transform Multiplication by a Sinusoid (Modulation): Multiplication by a Sinusoid (Modulation): Differentiation in the Time Domain: Differentiation in the Time Domain:

59. Properties of the Fourier Transform Integration in the Time Domain: Integration in the Time Domain: Convolution in the Time Domain: Convolution in the Time Domain: Multiplication in the Time Domain: Multiplication in the Time Domain:

60. Properties of the Fourier Transform Parseval’s Theorem: Parseval’s Theorem: Duality: Duality: if

61. Properties of the Fourier Transform - Summary

62. Example: Linearity

63. Example: Time Shift

64. Example: Time Scaling time compression frequency expansion time expansion frequency compression

65. Example: Multiplication in Time

66. Example: Multiplication by a Sinusoid sinusoidal burst

67. Example: Multiplication by a Sinusoid – Cont’d

68. Example: Multiplication in Time

69. Example: Multiplication by a Sinusoid sinusoidal burst

70. Example: Multiplication by a Sinusoid – Cont’d

71. Example: Integration in the Time Domain

72. Example: Integration in the Time Domain – Cont’d The Fourier transform of x(t) can be easily found to be Now, by using the integration property, it is

73. Example: Integration in the Time Domain – Cont’d

74. Fourier transform of Applying the duality property Generalized Fourier Transform generalized Fourier transform generalized Fourier transform of the constant signal

75. Generalized Fourier Transform of Sinusoidal Signals

76. Fourier Transform of Periodic Signals Let x(t) be a periodic signal with period T; as such, it can be represented with its Fourier transform Since, it is

77. Since using the integration property, it is Fourier Transform of the Unit-Step Function

78. Common Fourier Transform Pairs

79. Laplace Transform

80. Why use Laplace Transforms? Find solution to differential equation using algebra Relationship to Fourier Transform allows easy way to characterize systems No need for convolution of input and differential equation solution Useful with multiple processes in system

81. How to use Laplace Find differential equations that describe system Obtain Laplace transform Perform algebra to solve for output or variable of interest Apply inverse transform to find solution

82. How to use Laplace Find differential equations that describe system Obtain Laplace transform Perform algebra to solve for output or variable of interest Apply inverse transform to find solution

83. What are Laplace transforms? t is real, s is complex! Inverse requires complex analysis to solve Note “transform”: f(t)  F(s), where t is integrated and s is variable Conversely F(s)  f(t), t is variable and s is integrated Assumes f(t) = 0 for all t < 0

84. Evaluating F(s) = L{f(t)} Hard Way – do the integral let

85. Evaluating F(s)=L{f(t)}- Hard Way remember let Substituting, we get: It only gets worse…

86. Table of selected Laplace Transforms

87. More transforms

88. Note on step functions in Laplace Unit step function definition: Used in conjunction with f(t)  f(t)u(t) because of Laplace integral limits:

89. Properties of Laplace Transforms Linearity Scaling in time Time shift “frequency” or s-plane shift Multiplication by t n Integration Differentiation

90. Properties of Laplace Transforms Linearity Scaling in time Time shift “frequency” or s-plane shift Multiplication by t n Integration Differentiation

91. Properties: Linearity Example :Proof :

92. Example :Proof : let Properties: Scaling in Time

93. Properties: Time Shift Example :Proof : let

94. Properties: S-plane (frequency) shift Example :Proof :

95. Properties: Multiplication by t n Example :Proof :

96. The “D” Operator 1.Differentiation shorthand 2.Integration shorthand if then if

97. Properties: Integrals Example : Proof : let If t=0, g(t)=0 forso slower than

98. Properties: Derivatives (this is the big one) Example :Proof : let

99. The Inverse Laplace Transform

100. Inverse Laplace Transforms Background: To find the inverse Laplace transform we use transform pairs along with partial fraction expansion: F(s) can be written as; Where P(s) & Q(s) are polynomials in the Laplace variable, s. We assume the order of Q(s) P(s), in order to be in proper form. If F(s) is not in proper form we use long division and divide Q(s) into P(s) until we get a remaining ratio of polynomials that are in proper form.

101. Inverse Laplace Transforms Background: There are three cases to consider in doing the partial fraction expansion of F(s). Case 1: F(s) has all non repeated simple roots. Case 2: F(s) has complex poles: Case 3: F(s) has repeated poles. (expanded)

102. Inverse Laplace Transforms Case 1: Illustration: Given: Find A 1, A 2, A 3 from Heavyside

103. Inverse Laplace Transforms Case 3: Repeated roots. When we have repeated roots we find the coefficients of the terms as follows:

104. Inverse Laplace Transforms Case 3: Repeated roots.Example ???

105. Inverse Laplace Transforms Complex Roots: An Example. For the given F(s) find f(t)

106. Inverse Laplace Transforms Complex Roots: An Example. (continued) We then have; Recalling the form of the inverse for complex roots;