The Continuous-time Fourier

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1 The Continuous-time Fourier Transform Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it

2 Outline Representation of Aperiodic signals: The continuous-time Fourier Transform The Fourier transform for periodic signals Properties of the continuous-time Fourier transform The convolution/multiplication property System characterization by linear constant- coefficient 2

3 4.1 The continuous-time Fourier transform Extend the Fourier series representation to Aperiodic signals Almost all of the signals, including all signals with finite energy can be represented by a linear combination of complex exponentials The representation of in terms of a linear combination takes a form of an integral (rather than a sum) Fourier transform: the resulting spectrum of coefficients in the representation Inverse Fourier transform: use these coefficients to represent the signal as a linear combination of complex exponentials 3

4 4.1 The continuous-time Fourier transform Development of Fourier transform representation of an aperiodic signal Consider a periodic square wave 4

5 4.1 The continuous-time Fourier transform Alternative interpreting 5

6 4.1 The continuous-time Fourier transform As T -> infinite The original periodic signals becomes a rectangular pulse The Fourier series coefficients become more and more closely spaced samples approached by a continuous envelop function For aperiodic signal We think of an aperiodic signal as the periodic signal with arbitrary large period 6

7 4.1 The continuous-time Fourier transform An example: 7

8 4.1 The continuous-time Fourier transform The Fourier series representation: Therefore, define the envelop as We have 8

9 4.1 The continuous-time Fourier transform Combining them together, we have As, we have Fourier Transform Inverse Fourier Transform 9

10 4.1 The continuous-time Fourier transform We call X(jw) as spectrum of x(t) It provides us with the information needed for describing x(t) as a linear combination of the exponential signals with different frequencies Some conclusions: For a periodic signal, the Fourier coefficients can be expressed in terms of equally spaced samples of the Fourier transform of one period Let be a finite-duration signal in one period 10

11 4.1 The continuous-time Fourier transform Convergence of Fourier transform: following the Dirichlet conditions X(t) is absolutely integrable X(t) have a finite number of maxima and minima within any finite interval X(t) has a finite number of discontinuities within any finite interval and each of these discontinuities is finite 11

12 4.1 The continuous-time Fourier transform Example 1: Consider the signal Determine the Fourier transform Solution: 12

13 4.1 The continuous-time Fourier transform Example 2: Consider the signal Determine the Fourier transform Solution: 13

14 4.1 The continuous-time Fourier transform Example 3: Consider the signal Determine the Fourier transform Solution: 14

15 4.1 The continuous-time Fourier transform Example 4: Consider the signal Determine the Fourier transform Solution: 15

16 4.1 The continuous-time Fourier transform 16

17 4.1 The continuous-time Fourier transform Example 5: Consider the signal x(t) whose Fourier transform is Determine x(t) Solution: 17

18 4.1 The continuous-time Fourier transform Sinc function: 18

19 4.1 The continuous-time Fourier transform Some property for sinc function: As W increases, X(jw) becomes broader Main peak of x(t) becomes higher The width of the first lobe ( ) becomes narrower 19

20 4.2 The Fourier transform for periodic signal The Fourier transform of periodic signal Consists of a train of impulses in the frequency domain So we have 20

21 4.2 The Fourier transform for periodic signal Example 1: Consider periodic square wave Determine its Fourier transform Solution: 21

22 4.2 The Fourier transform for periodic signal 22

23 4.2 The Fourier transform for periodic signal Example 2: Consider periodic signals Determine its Fourier transform Solution: 23

24 4.2 The Fourier transform for periodic signal 24

25 4.2 The Fourier transform for periodic signal Example 2: Consider the impulse train Determine its Fourier transform Solution: 25

26 4.3 Properties Linearity: Time shifting: Proof: 26

27 4.2 The Fourier transform for periodic signal Example 1: Consider the signal Determine its Fourier transform Solution: 27

28 4.3 Properties Conjugation: Proof: For real signal: 28

29 4.3 Properties For a real and even signal, is real and even Proof: 29

30 4.3 Properties By decompose the signal as even and odd parts 30

31 4.2 The Fourier transform for periodic signal Example 2: Consider the signal Determine its Fourier transform Solution: 31

32 4.3 Properties Differentiation and integration: Proof: 32

33 4.2 The Fourier transform for periodic signal Example 3: Consider the signal Determine its Fourier transform Solution: 33

34 4.2 The Fourier transform for periodic signal Example 4: Consider the signal Determine its Fourier transform Solution: 34

35 4.3 Properties Time and frequency scaling: Proof: 35

36 4.3 Properties Duality: 36

37 4.3 Properties 37

38 4.3 Properties Example 5: Consider the signal Determine its Fourier transform Solution: 38

39 4.3 Properties Parseval s s relation: Proof: 39

40 4.3 Properties Example 6: Evaluate the following time-domain expressions: 40

41 4.3 Properties Solution: 1 2 5/8 E X ( j ) d 2b 1 1 D g (0) G ( j ) d D j X( j ) d

42 4.4 The convolution property Linear convolution in time domain product in frequency domain Proof: 42

43 4.4 The convolution property In Using Fourier analysis to study LTI system, we require The unit impulse function of the LTI system has Fourier transform Using transform techniques to examine unstable LTI system Use the Laplace transform (in Chapter 9) 43

44 4.4 The convolution property Example 1: Check the time-shifting with unit impulse transform as follows by using Fourier transform Solution: 44

45 4.4 The convolution property Example 2: Determine the Fourier transform of a differentiator Solution: 45

46 4.4 The convolution property Example 3: Determine the Fourier transform of a integrator Solution: the unit impulse response is a unit step function 46

47 4.4 The convolution property Example 4: Accomplish a frequency-selective filter using LTI system Solution: 47

48 4.4 The convolution property Example 5: Consider a system with input and unit impulse function given as follows. Determine the output by using Fourier transform. 48

49 4.4 The convolution property Solution: for a!= b 49

50 4.4 The convolution property Solution: for a = b 50

51 4.4 The convolution property Example 6: Consider a system with input and unit impulse function given as follows. Determine the output by using Fourier transform. 51

52 4.4 The convolution property Solution: 52

53 4.5 The multiplication property The multiplication in the time domain corresponds to the convolution in the frequency domain Amplitude modulation: multiplication of one signal by another Using one signal to scale or modulate the amplitude of the other 53

54 4.5 The multiplication property Example 1: Let s(t) be a signal with spectrum given as Fig. (a). Also, consider the signal of Then 54

55 4.5 The multiplication property 55

56 4.5 The multiplication property Example 2: Determine the Fourier transform of the signal Solution: 56

57 4.5 The multiplication property Consider a system given as follows: 57

58 4.5 The multiplication property Consider a system given as follows: 58

59 4.6 Summary of the properties X(t), y(t) Linearity Time shifting Frequency shifting Conjugation Time reversal e Time Scaling (Period ) Convolution 59

60 4.6 Summary of the properties Multiplication Differentiation in time Integration Differentiation in frequency Conjugate Symmetry for Real Signals real 60

61 4.6 Summary of the properties Symmetry for Real and Even Signals Real and Odd Signals Even-odd Decomposition of Real Signal [x(t) real] Parseval s Relation for Periodic Signals X(t) real and even X(t) real and odd real and even imaginary and odd 61

62 4.6 Summary of the properties 62

63 4.6 Summary of the properties 63

64 4.7 System characterization by linear constant-coefficient t i t differential equations Determine the frequency response of the LTI system of For a LTI system, we have 64

65 4.7 System characterization by linear constant-coefficient t i t differential equations Taking Fourier transform on both sides, we have Apply linearity Apply differentiation property, we have 65

66 4.7 System characterization by linear constant-coefficient t i t differential equations Or equivalently, We thus have 66

67 4.7 System characterization by linear constant-coefficient t i t differential equations Example 1: consider a LTI system (a>0) The frequency response is 67

68 4.7 System characterization by linear constant-coefficient t i t differential equations Example 2: consider a LTI system (a>0) The frequency response is 68

69 4.7 System characterization by linear constant-coefficient t i t differential equations Example 3: consider a LTI system (a>0) The frequency response is 69

70 4.7 System characterization by linear constant-coefficient t i t differential equations Then we have 70

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