elec361: signals and systems topic 4: continuous-time ...amer/teach/elec361/slides/topic4-ft.pdf ·...

o Introduction to Fourier Transformo Fourier transform of CT aperiodic signalso CT Fourier transform exampleso Convergence of the CT Fourier Transformo Convergence exampleso CT Fourier transform of periodic signalso Properties of CT Fourier Transform o Summaryo Appendix:

Transition from CT Fourier Series to CT Fourier Transform

ELEC361: Signals And Systems

Topic 4: Continuous-TimeFourier Transform (CTFT)

Dr. Aishy AmerConcordia UniversityElectrical and Computer Engineering

Figures and examples in these course slides are taken from the following sources:

•A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997

•M.J. Roberts, Signals and Systems, McGraw Hill, 2004

•J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003

2

Fourier Series versus Fourier Transform

Fourier Series (FS): a representation of periodicsignals as a linear combination of complex exponentialsFourier Transform (FT): apply to signals that are not periodic Aperiodic signals can be viewed as a periodic signal with an infinite periodThe CT Fourier Series is a good analysis tool for systems with periodic excitation but the CT Fourier Series cannot represent an aperiodic signal for all timeThe CT Fourier transform can represent an aperiodicsignal for all time

3

Fourier Series versus Fourier Transform: Types of signals

4

Fourier Series versus Fourier Transform: Types of signals

5


Four distinct Fourier representations:Each applicable to a different class of

signalsDetermined by the periodicity properties of the signal and whether the signal is discrete or continuous in time

A Fourier representation is unique, i.e., no two same signals in time domain give the same function in frequency domain

6


In FS representation of periodic signals:

As the period increases T↑, ω0↓the harmonically related components become closer in frequency

As the period becomes infinitethe frequency components form a continuum and the FS sum becomes an integral

FT for aperiodic signals:

TeatxdtetxT

akX tjk

kk

Ttjk

k /2;)(;)(1][ 00

00 πωωω ==== ∑∫∞

−∞=

−

7

Fourier Series versus Fourier TransformPeriodic in TimeDiscrete in Frequency

Aperiodic in TimeContinuous in Frequency

Continuous in Time

Aperiodic in Frequency

Discrete in Time

Periodic in Frequency

∑

∫

∞

−∞=

−

=

⇒⊗

=

⇒⊗

k

tjkk

Ttjk

k

eatx

dtetxT

a

0

0

)(

P-CTDT :SeriesFourier Inverse CT

)(1

DTP-CT :SeriesFourier CT

T

0

T

ω

ω

∫

∑

=

⇒+⊗

=

+⇒⊗∞

−∞=

−

π

ωω

π

ωω

π

ωπ 2

2

2

)(21][

DT PCT :TransformFourier DT Inverse

][)(

PCTDT :TransformFourier DT

deeXnx

enxeX

njj

n

njj

∑

∑

−

=

−

=

−

=

⇒⊗

=

⇒⊗

1

0

NN

1

0

NN

0

0

][1][

P-DTP-DT SeriesFourier DT Inverse

][][

P-DTP-DT SeriesFourier DT

N

k

knj

N

n

knj

ekXN

nx

enxkX

ω

ω

∫

∫

∞

∞−

∞

∞−

−

=

⇒⊗

=

⇒⊗

ωωπ

ω

ω

ω

dejXtx

dtetxjX

tj

tj

)(21)(

CTCT :TransformFourier CT Inverse

)()(

CTCT :TransformFourier CT

8

Outline

Introduction to Fourier TransformFourier transform of CT aperiodic signalsFourier transform examplesConvergence of the CT Fourier TransformConvergence examplesFourier transform of periodic signalsProperties of CT Fourier Transform SummaryAppendix

Transition: CT Fourier Series to CT Fourier Transform

9

Fourier Transform of CT aperiodic signals

Consider the CT aperiodic signal given below:

10


We have: Define:

This means that:

11

As , approaches approaches zero, and the right-hand

side changes to an integralThe pair of equations:

are referred to as a Fourier Transform pair


12

CT Fourier transform for aperiodic signals

Spectrum) Magnitude (Symmetric )( )( signals, realFor

)()(

:) transform(forward analysisFourier

)(21)(

:) transform(inverse SynthesisFourier

ωω

ω

ωωπ

ω

ω

jXjX

dtetxjX

dejXtx

tj

tj

−=

=

=

−∞

∞−

∞

∞−

∫

∫

13

CT Fourier transform for aperiodic signals

14

CT Fourier Transform of aperiodic signal

The CT Fourier Transform expresses a finite-amplitude, real-valued, aperiodic signal x(t)

x(t) can also, in general, be time-limited as a summation (an integral) of an infinite continuum of weighted, infinitesimal-amplitude, complex sinusoids, each of which is unlimited in time Time limited means “having non-zero values only for a finite time”

15

CTFT: Pulse (rectangular) Function Time Domain

16

CTFT: Pulse (rectangular) Function Spectrum

17

CTFT: Exponential Decay Time Domain

18

CTFT: Exponential Decay Spectrum

19

Outline



20

CT Fourier Transform:Example

21


22


23


24

Outline



25

Convergence of CT Fourier Transform

Dirichlet’s sufficient conditions for the convergence of Fourier transform are similar to the conditions for the FS:

1. x(t) must be absolutely integrable

2. x(t) must have a finite number of maxima and minima within any finite interval

3. x(t) must have a finite number of discontinuities, all of finite size, within any finite interval

26

Convergence of CT Fourier Transform:Example 5.4

27

Convergence of CT Fourier Transform: Example 5.4

28


29


The Fourier transform for this example is real at all frequenciesThe time signal and its Fourier transform are

30


31


32


Reducing the width of x(t)will have an opposite effect on X(jω)Using the inverse Fourier transform, we get a time signal which is equal to x(t) at all points except discontinuities (t=T1 and t=-T1), where the inverse Fourier transform is equal to the average of the values of x(t) on both sides of the discontinuity

33


34


This example shows the reveres effect in the time and frequency domains in terms of the width of the time signal and the corresponding Fourier transform

35


36

Convergence of the CT-FT: Generalization

37

Convergence of the CTFT: Generalization

38


which is equal to A, independent of the value of σSo, in the limit as σ approaches zero, the CT Fourier Transform has an area of A and is zero unless f = 0This exactly defines an impulse of strength, A. Therefore

39


40


41

Outline



42

Fourier transform for periodic signals

Consider a signal x(t) whose Fourier transform is given by

Using the inverse Fourier transform, we will have:

This implies that the time signal corresponding to the following Fourier transform

43

Fourier transform for periodic signals

Note that is the Fourier series representation of periodic signals

Fourier transform of a periodic signal is a train of impulses with the area of the impulse at the frequency kω0 equal to the kth coefficient of the Fourier series representation ak times 2π

44

Fourier Transform of Periodic Signal: Example 6.1

45


This example shows the reveres effect in the time and frequency domains in terms of the width of the time signal and the corresponding Fourier transform

46


47

Fourier transform for periodic signals: Example 6.2

48


49


The periodic impulse train and its Fourier transform are very useful in the analysis of sampling systems

50

Fourier Transform of Periodic Signals: Example

51

Outline

Introduction to Fourier TransformFourier transform of CT aperiodic signalsFourier transform examplesConvergence of the CT Fourier TransformConvergence examplesFourier transform of periodic signalsProperties of CT Fourier TransformSummaryAppendix


52

Properties of the CT Fourier Transform

The properties are useful in determining the Fourier transform or inverse Fourier transformThey help to represent a given signal in term of operations (e.g., convolution, differentiation, shift) on another signal for which the Fourier transform is knownOperations on {x(t)} Operations on {X(jω)}Help find analytical solutions to Fourier transform problems of complex signals Example:

⇔

tionmultiplicaanddelaytuatyFT t →−= })5()({

53


The properties of the CT Fourier transform are very similar to those of the CT Fourier seriesConsider two signals x(t) and y(t) with Fourier transforms X(jω) and Y(jω), respectively (or X(f) and Y(f))The following properties can easily been shown using

54


55


56


57


58


59


The time and frequency scaling properties indicate that if a signal is expanded in one domain it is compressed in the other domain. This is also called the “uncertainty principle” of Fourier analysis

60


61

Properties of the CT Fourier: Time Shifting & Scaling: Example

62

Properties of the CT Fourier: Time Shifting & Scaling: Example

63


64


65


66


67

Properties of the CT Fourier Transform: Example 6.4

68

Properties of the CT Fourier Transform:Differentiation Property Example

69

Properties of the CT Fourier Transform: Examples 6.5 & 6.6

70

Properties of the CT Fourier Transform:Differentiation Property Example

71

Properties of the CT Fourier Transform: Differentiation Property: Example

72


73


MultiplicationConvolution Duality Proof

74


75

Properties of the CT Fourier Transform: Convolution Property Example

76


77

Properties of the CT Fourier Transform: Differential Equations

78

Properties of the CT Fourier Transform: Differential Equation Example 6.8

79

Properties of the CT Fourier Transform:Modulation Property: Example

80


81


82

Properties of the CT Fourier Transform:Integration Property

83

Properties of the CT Fourier Transform:Integration Property: Example

84


85

Properties of the CT Fourier Transform:Area Property: Example

86


87


88


89


90

Properties of the CT Fourier Transform: Example 6.4

91

Properties of the CT Fourier Transform:Duality Property: Example

92

Outline



93

CTFT: Summary

94

Summary of CTFT Properties

95

Summary of CTFT Properties

96

CTFT: Summary of Pairs

97

CTFT: Summary of Pairs

98

Outline



99


100


Below are plots of the magnitude of X[k] for 50% and 10% duty cyclesAs the period increases the sinc function widens and its magnitude fallsAs the period approaches infinity, the CT Fourier Series harmonic function becomes an infinitely-wide sinc function with zero amplitude (since X(k) is divided by To)

101


This infinity-and-zero problem can be solved by normalizing the CT Fourier Series harmonic functionDefine a new “modified” CT Fourier Series harmonic function

102


elec361: signals and systems topic 4: continuous-time ...amer/teach/elec361/slides/topic4-ft.pdf ·...

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