1. Continuous-Time Fourier Transform
主講者：虞台文

2. Content Introduction Fourier Integral Fourier Transform
Properties of Fourier Transform
Convolution
Parseval’s Theorem

3. Continuous-Time Fourier Transform
Introduction

4. The Topic Time Discrete Fourier Periodic Series Transform Continuous
Aperiodic

5. Review of Fourier Series
Deal with continuous-time periodic signals.
Discrete frequency spectra.
A Periodic Signal T 2T 3T t
f(t)

6. Two Forms for Fourier Series
Sinusoidal
Form
Complex
Form:

7. How to Deal with Aperiodic Signal?
f(t)
If T, what happens?

8. Continuous-Time Fourier Transform
Fourier Integral

9. Fourier Integral
Let

10. Fourier Integral
F(j)
Synthesis
Analysis

11. Fourier Series vs. Fourier Integral
Period Function
Discrete Spectra
Fourier
Integral:
Non-Period
Function
Continuous Spectra

12. Continuous-Time Fourier Transform

13. Fourier Transform Pair
Inverse Fourier Transform:
Synthesis
Fourier Transform:
Analysis

14. Existence of the Fourier Transform
Sufficient Condition:
f(t) is absolutely integrable, i.e.,

15. Continuous Spectra
FR(j)
FI(j)
|F(j)|
()
Magnitude
Phase

16. Example 1 -1 t
f(t)

17. Example 1 -1 t
f(t)

18. Example t
f(t)
et

19. Example t
f(t)
et

20. Continuous-Time Fourier Transform
Properties of
Fourier Transform

21. Notation

22. Linearity
Proved by yourselves

23. Time Scaling
Proved by yourselves

24. Time Reversal
Pf)

25. Time Shifting
Pf)

26. Frequency Shifting (Modulation)
Pf)

27. Symmetry Property
Pf)
Interchange symbols  and t

28. Fourier Transform for Real Functions
If f(t) is a real function, and F(j) = FR(j) + jFI(j)
F(j) = F*(j)

29. Fourier Transform for Real Functions
If f(t) is a real function, and F(j) = FR(j) + jFI(j)
F(j) = F*(j)
FR(j) = FR(j)
FI(j) = FI(j)
FR(j) is even, and FI(j) is odd.
Magnitude spectrum |F(j)| is even, and phase spectrum () is odd.

30. Fourier Transform for Real Functions
If f(t) is real and even
If f(t) is real and odd
F(j) is real
F(j) is pure imaginary  
Pf)
Pf)
Even
Odd
Real
Real

31. Example:
Sol)

32. Example:
d/2
d/2 1 t
wd(t)
f(t)=wd(t)cos0t

33. Example:
d/2
d/2 1 t
wd(t)
f(t)=wd(t)cos0t

34. d/2
d/2 1 t
wd(t)
Example:
Sol)

35. Fourier Transform of f’(t)
Pf)

36. Fourier Transform of f (n)(t)
Proved by yourselves

37. Fourier Transform of f (n)(t)
Proved by yourselves

38. Fourier Transform of Integral
Let

39. The Derivative of Fourier Transform
Pf)

40. Continuous-Time Fourier Transform
Convolution

41. Basic Concept fi(t) fo(t)=L[fi(t)] Linear System
fi(t)=a1 fi1(t) + a2 fi2(t)
fo(t)=L[a1 fi1(t) + a2 fi2(t)]
A linear system satisfies
fo(t) = a1L[fi1(t)] + a2L[fi2(t)]
= a1fo1(t) + a2fo2(t)

42. Basic Concept fo(t) fi(t) Time Invariant System fi(t +t0) fo(t + t0)

43. Basic Concept fo(t) fi(t) Causal System A causal system satisfies
fi(t) = 0 for t < t0
fo(t) = 0 for t < t0

44. Basic Concept fo(t) fi(t) Causal System
Which of the following systems are causal?
Basic Concept
Causal
System
fi(t)
fo(t) t
fi(t)
fo(t) t0 t0 t
fi(t)
fo(t)
fi(t) t
fo(t) t0

45. Unit Impulse Response (t) h(t)=L[(t)] LTI System f(t) L[f(t)]=?
Facts:
Convolution

46. Unit Impulse Response LTI System (t) h(t)=L[(t)] f(t) L[f(t)]=?
Facts:
Convolution

47. Unit Impulse Response
Impulse Response
LTI System
h(t)
f(t)
f(t)*h(t)

48. Convolution Definition
The convolution of two functions f1(t) and f2(t) is defined as:

49. Properties of Convolution

50. Properties of Convolution
Impulse Response
LTI System
h(t)
f(t)
f(t)*h(t)
Impulse Response
LTI System
f(t)
h(t)
h(t)*f(t)

51. Properties of Convolution
Prove by yourselves

52. Properties of Convolution
The following two systems are identical
Properties of Convolution
h1(t)
h2(t)
h3(t)
h2(t)
h3(t)
h1(t)

53. Properties of Convolution
f(t)

54. Properties of Convolution
f(t)

55. Properties of ConvolutionT
(tT)
f(t)
f(t T) t
f (t) t
f (t) T

56. Properties of Convolution
System function (tT) serves as an ideal delay or a copier.
Properties of Convolution T
(tT)
f(t)
f(t T) t
f (t) t
f (t) T

57. Properties of Convolution

58. Properties of Convolution
Time Domain
Frequency Domain
convolution
multiplication
Properties of Convolution

59. Properties of Convolution
Time Domain
Frequency Domain
convolution
multiplication
Properties of Convolution
Impulse Response
LTI System
h(t)
f(t)
f(t)*h(t)
Impulse Response
LTI System
H(j)
F(j)
F(j)H(j)

60. Properties of Convolution
Time Domain
Frequency Domain
convolution
multiplication
Properties of Convolution
F(j)H1(j)
F(j)H1(j)H2(j)H3(j)
H1(j)
H2(j)
H3(j)
F(j)
F(j)H1(j)H2(j)

61. Properties of Convolution
H(j) p
p 1
Fo(j) 
Fi(j) 
An Ideal Low-Pass Filter

62. Properties of Convolution
H(j) p
p 1
Fo(j) 
Fi(j) 
An Ideal High-Pass Filter

63. Properties of Convolution
Prove by yourselves

64. Properties of Convolution
Time Domain
Frequency Domain
multiplication
convolution
Properties of Convolution
Prove by yourselves

65. Continuous-Time Fourier Transform
Parseval’s Theorem

66. Properties of Convolution
=0

67. Properties of Convolution
If f1(t) and f2(t) are real functions,
f2(t) real

68. Parseval’s Theorem: Energy Preserving