1. UNIT II Analysis of Continuous Time signal

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

3. 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.

4. Decomposition

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

11. Example (Square Wave)

12. 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

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 amplitude
phase angle

16. Complex Form of Fourier Series

17. Complex Exponentials

18. Complex Form of the Fourier Series

19. Complex Form of the Fourier Series

20. Complex Form of the Fourier Series

21. Complex Form of the Fourier Series
If f(t) is real,

22. Example t
f(t) A

23. 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

24. 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

25. Example t
f(t) A

26. Waveform Symmetry
Even Functions
Odd Functions

27. 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

28. Example
Even Part
Odd Part

29. Half-Wave Symmetry
and T
T/2
T/2

30. Quarter-Wave Symmetry
Even Quarter-Wave Symmetry T
T/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/2
A/2 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

34. Fourier Coefficients of Even Functions

35. Fourier Coefficients for Half-Wave Symmetry
and T
T/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

38. Fourier Coefficients for Odd Quarter-Wave Symmetry

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

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

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

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

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

44. Fourier Transform
We have seen that periodic signals can be represented with the Fourier series
Can aperiodic signals be analyzed in terms of frequency components?
Yes, and the Fourier transform provides the tool for this analysis
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

45. The Fourier Transform in the General Case
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

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

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

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

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

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

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

52. Example: Fourier Transform of the Rectangular Pulse – Cont’d
amplitude spectrum
phase spectrum

53. Bandlimited Signals
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

54. Bandlimited Signals – Cont’d
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
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

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

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

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

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

60. Properties of the Fourier Transform
Parseval’s Theorem:
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. Generalized Fourier Transform
Fourier transform of
Applying the duality property
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. Fourier Transform of the Unit-Step Function
Since
using the integration property, it is

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
let
let

85. Evaluating F(s)=L{f(t)}- Hard Way
remember
let
Substituting, we get:
let
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 tn
Integration
Differentiation

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

91. Properties: Linearity
Example :
Proof :

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

93. Properties: Time Shift
Example :
Proof :
let

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

95. Properties: Multiplication by tn
Example :
Proof :

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

97. Properties: Integrals
Proof :
Example :
let
If t=0, g(t)=0
for so
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:
(expanded)
Case 3: F(s) has repeated poles.
(expanded)

102. Inverse Laplace Transforms
Case 1: Illustration:
Given:
Find A1, A2, A3 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;