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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
Fourier Series versus Fourier Transform
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
Fourier Series versus Fourier Transform
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
Fourier Transform of CT aperiodic signals
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
Fourier Transform of CT aperiodic signals
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
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
20
CT Fourier Transform:Example
21
CT Fourier Transform:Example
22
CT Fourier Transform:Example
23
CT Fourier Transform:Example
24
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
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
Convergence of CT Fourier Transform:Example 5.5
29
Convergence of CT Fourier Transform:Example 5.5
The Fourier transform for this example is real at all frequenciesThe time signal and its Fourier transform are
30
Convergence of CT Fourier Transform: Example 5.6
31
Convergence of CT Fourier Transform: Example 5.7
32
Convergence of CT Fourier Transform: Example 5.7
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
Convergence of CT Fourier Transform: Example 5.8
34
Convergence of CT Fourier Transform: Example 5.8
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
Convergence of CT Fourier Transform: Example 5.8
36
Convergence of the CT-FT: Generalization
37
Convergence of the CTFT: Generalization
38
Convergence of the CTFT: Generalization
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
Convergence of the CTFT: Generalization
40
Convergence of the CTFT: Generalization
41
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
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
Fourier Transform of Periodic Signal: Example 6.1
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
Fourier Transform of Periodic Signal: Example 6.1
47
Fourier transform for periodic signals: Example 6.2
48
Fourier transform for periodic signals: Example 6.2
49
Fourier transform for periodic signals: Example 6.2
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
Transition: CT Fourier Series to CT Fourier Transform
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
Properties of the CT Fourier Transform
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
Properties of the CT Fourier Transform
55
Properties of the CT Fourier Transform
56
Properties of the CT Fourier Transform
57
Properties of the CT Fourier Transform
58
Properties of the CT Fourier Transform
59
Properties of the CT Fourier Transform
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
Properties of the CT Fourier Transform
61
Properties of the CT Fourier: Time Shifting & Scaling: Example
62
Properties of the CT Fourier: Time Shifting & Scaling: Example
63
Properties of the CT Fourier Transform
64
Properties of the CT Fourier Transform
65
Properties of the CT Fourier Transform
66
Properties of the CT Fourier Transform
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
Properties of the CT Fourier Transform
73
Properties of the CT Fourier Transform
MultiplicationConvolution Duality Proof
74
Properties of the CT Fourier Transform
75
Properties of the CT Fourier Transform: Convolution Property Example
76
Properties of the CT Fourier Transform
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
Properties of the CT Fourier Transform
81
Properties of the CT Fourier Transform
82
Properties of the CT Fourier Transform:Integration Property
83
Properties of the CT Fourier Transform:Integration Property: Example
84
Properties of the CT Fourier Transform
85
Properties of the CT Fourier Transform:Area Property: Example
86
Properties of the CT Fourier Transform:Area Property: Example
87
Properties of the CT Fourier Transform:Area Property: Example
88
Properties of the CT Fourier Transform
89
Properties of the CT Fourier Transform
90
Properties of the CT Fourier Transform: Example 6.4
91
Properties of the CT Fourier Transform:Duality Property: Example
92
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
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
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
99
Transition: CT Fourier Series to CT Fourier Transform
100
Transition: CT Fourier Series to CT Fourier Transform
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
Transition: CT Fourier Series to CT Fourier Transform
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
Transition: CT Fourier Series to CT Fourier Transform