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Continuous-Time Fourier Transform. Content Introduction Fourier Integral Fourier Transform Properties of Fourier Transform Convolution Parseval s Theorem.

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Presentation on theme: "Continuous-Time Fourier Transform. Content Introduction Fourier Integral Fourier Transform Properties of Fourier Transform Convolution Parseval s Theorem."— Presentation transcript:

1 Continuous-Time Fourier Transform

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

3 Continuous-Time Fourier Transform Introduction

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

5 Review of Fourier Series Deal with continuous-time periodic signals. Discrete frequency spectra. A Periodic Signal T2T3T t f(t)f(t)

6 Two Forms for Fourier Series Sinusoidal Form Complex Form:

7 How to Deal with Aperiodic Signal? A Periodic Signal T t f(t)f(t) If T, what happens?

8 Continuous-Time Fourier Transform Fourier Integral

9 Let

10 Fourier Integral F ( j ) Synthesis Analysis

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

12 Continuous-Time Fourier Transform Fourier Transform

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

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

15 Continuous Spectra F R (j ) F I (j ) |F(j )| ( ) Magnitude Phase

16 Example 1 1 t f(t)f(t)

17 Example 1 1 t f(t)f(t)

18 Example t f(t)f(t) e t

19 Example t f(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 ) = F R (j ) + jF I (j ) F( j ) = F*(j )

29 Fourier Transform for Real Functions If f(t) is a real function, and F(j ) = F R (j ) + jF I (j ) F( j ) = F*(j ) F R (j ) is even, and F I (j ) is odd. F R ( j ) = F R ( j ) F I ( j ) = F I ( j ) Magnitude spectrum |F(j )| is even, and phase spectrum ( ) is odd.

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

31 Example: Sol)

32 Example: d/2 1 t wd(t)wd(t) t f(t)=w d (t)cos 0 t

33 Example: d/2 1 t wd(t)wd(t) t f(t)=w d (t)cos 0 t

34 Example: Sol) d/2 1 t wd(t)wd(t)

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 Linear System fi(t)fi(t) f o (t)=L[f i (t)] f i (t)=a 1 f i1 (t) + a 2 f i2 (t)f o (t)=L[a 1 f i1 (t) + a 2 f i2 (t)] A linear system satisfies f o (t) = a 1 L[f i1 (t)] + a 2 L[f i2 (t)] = a 1 f o1 (t) + a 2 f o2 (t)

42 Basic Concept Time Invariant System Time Invariant System fi(t)fi(t) fo(t)fo(t) f i (t +t 0 ) f o (t + t 0 ) f i (t t 0 ) f o (t t 0 ) t fi(t)fi(t) t fo(t)fo(t) t fi(t+t0)fi(t+t0) t fo(t+t0)fo(t+t0) t f i (t t 0 ) t f o (t t 0 )

43 Basic Concept Causal System Causal System fi(t)fi(t) fo(t)fo(t) A causal system satisfies f i (t) = 0 for t < t 0 f o (t) = 0 for t < t 0

44 Basic Concept Causal System Causal System fi(t)fi(t) fo(t)fo(t) t fi(t)fi(t) t fo(t)fo(t) t0t0 t0t0 t0t0 t fi(t)fi(t) t fo(t)fo(t) t0t0 fi(t)fi(t) tt fo(t)fo(t) t0t0 t0t0 Which of the following systems are causal?

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

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

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

48 Convolution Definition The convolution of two functions f 1 (t) and f 2 (t) is defined as:

49 Properties of Convolution

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

51 Properties of Convolution Prove by yourselves

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

53 Properties of Convolution (t) f(t)f(t)f(t)f(t)

54 Properties of Convolution (t) f(t)f(t)f(t)f(t)

55 Properties of Convolution f(t)f(t) f(t T) 0T (t T) t f (t)f (t) 0 t f (t)f (t) 0T

56 Properties of Convolution f(t)f(t) f(t T) 0T (t T) t f (t)f (t) 0 t f (t)f (t) 0T System function (t T) serves as an ideal delay or a copier.

57 Properties of Convolution

58 Time DomainFrequency Domain convolutionmultiplication

59 Properties of Convolution Time DomainFrequency Domain convolutionmultiplication Impulse Response LTI System h(t) Impulse Response LTI System h(t) f(t)f(t) f(t)*h(t) Impulse Response LTI System H(j ) Impulse Response LTI System H(j ) F(j ) F(j )H(j )

60 Properties of Convolution Time DomainFrequency Domain convolutionmultiplication F(j ) F(j )H 1 (j ) H 2 (j ) H 1 (j ) H 3 (j ) F(j )H 1 (j )H 2 (j ) F(j )H 1 (j )H 2 (j )H 3 (j )

61 Properties of Convolution An Ideal Low-Pass Filter 0 F i (j ) 0 F o (j ) 0 H(j ) p p 1

62 Properties of Convolution An Ideal High-Pass Filter 0 F i (j ) 0 H(j ) p p 1 0 F o (j )

63 Properties of Convolution Prove by yourselves

64 Properties of Convolution Prove by yourselves Time DomainFrequency Domain multiplicationconvolution

65 Continuous-Time Fourier Transform Parseval s Theorem

66 Properties of Convolution =0

67 Properties of Convolution If f 1 (t) and f 2 (t) are real functions, f 2 (t) real

68 Parseval s Theorem: Energy Preserving


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