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Spring 2007
Signals and Systems
Chapter SS-5The Discrete-Time Fourier Transform
Feng-Li Lian
NTU-EE
Feb07 Jun07
Figures and images used in these lecture notes are adopted fromSignals & Systems by Alan V. Oppenheim and Alan S. Willsky, 1997
NTUEE-SS5-DTFT-2
Feng-Li Lian 2007Outline
Representation of Aperiodic Signals:the Discrete-Time Fourier Transform
The Fourier Transform for Periodic Signals
Properties of Discrete-Time Fourier TransformThe Convolution Property
The Multiplication Property
Duality
Systems Characterized by Linear Constant-Coefficient Difference Equations
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NTUEE-SS5-DTFT-3
Feng-Li Lian 2007Representation of Aperiodic Signals: DT Fourier Transform
DT Fourier Transform of an Aperiodic Signal:
NTUEE-SS5-DTFT-4
Feng-Li Lian 2007Representation of Aperiodic Signals: DT Fourier Transform
DT Fourier Transform of an Aperiodic Signal:
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NTUEE-SS5-DTFT-5
Feng-Li Lian 2007Representation of Aperiodic Signals: DT Fourier Transform
DT Fourier Transform of an Aperiodic Signal:
NTUEE-SS5-DTFT-6
Feng-Li Lian 2007Representation of Aperiodic Signals: DT Fourier Transform
Periodicity of DT Fourier Transform:
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NTUEE-SS5-DTFT-7
Feng-Li Lian 2007Representation of Aperiodic Signals: DT Fourier Transform
High-Frequency & Low-Frequency Signals:
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Feng-Li Lian 2007Representation of Aperiodic Signals: DT Fourier Transform
Example 5.1:
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NTUEE-SS5-DTFT-9
Feng-Li Lian 2007Representation of Aperiodic Signals: DT Fourier Transform
Example 5.2:
NTUEE-SS5-DTFT-10
Feng-Li Lian 2007Representation of Aperiodic Signals: DT Fourier Transform
Example 5.3:
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NTUEE-SS5-DTFT-11
Feng-Li Lian 2007Representation of Aperiodic Signals: DT Fourier Transform
Convergence of DT Fourier Transform:
The analysis equation will converge:
Either if x[n] is absolutely summable, that is,
Or if x[n] has finite energy, that is,
NTUEE-SS5-DTFT-12
Feng-Li Lian 2007Representation of Aperiodic Signals: DT Fourier Transform
Example 5.4:
Approximation
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NTUEE-SS5-DTFT-13
Feng-Li Lian 2007Representation of Aperiodic Signals: DT Fourier Transform
Approximation of an Aperiodic Signal:
NTUEE-SS5-DTFT-14
Feng-Li Lian 2007Outline
Representation of Aperiodic Signals:the Discrete-Time Fourier Transform
The Fourier Transform for Periodic Signals
Properties of Discrete-Time Fourier TransformThe Convolution Property
The Multiplication Property
Duality
Systems Characterized by Linear Constant-Coefficient Difference Equations
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NTUEE-SS5-DTFT-15
Feng-Li Lian 2007Fourier Transform for Periodic Signals
Fourier Transform from Fourier Series:
NTUEE-SS5-DTFT-16
Feng-Li Lian 2007Fourier Transform for Periodic Signals
Fourier Transform from Fourier Series:
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NTUEE-SS5-DTFT-17
Feng-Li Lian 2007Fourier Transform for Periodic Signals
Fourier Transform from Fourier Series:
NTUEE-SS5-DTFT-18
Feng-Li Lian 2007Fourier Transform for Periodic Signals
Example 5.5:
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NTUEE-SS5-DTFT-19
Feng-Li Lian 2007Fourier Transform for Periodic Signals
Example 5.6:
NTUEE-SS5-DTFT-20
Feng-Li Lian 2007Outline
Representation of Aperiodic Signals:the Discrete-Time Fourier Transform
The Fourier Transform for Periodic Signals
Properties of Discrete-Time Fourier TransformThe Convolution Property
The Multiplication Property
Duality
Systems Characterized by Linear Constant-Coefficient Difference Equations
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NTUEE-SS5-DTFT-21
Feng-Li Lian 2007Outline
Time Expansion5.3.7
Differencing in Time5.3.5
Multiplication5.5
Convolution5.4
Time Reversal5.3.6
Conjugation5.3.4
Frequency Shifting5.3.3
Time Shifting5.3.3
Symmetry for Real and Even Signals5.3.4
Conjugate Symmetry for Real Signals5.3.4
Differentiation in Frequency5.3.8
Accumulation5.3.5
Even-Odd Decomposition for Real Signals5.3.4
Symmetry for Real and Odd Signals5.3.4
Parsevals Relation for Aperiodic Signals5.3.9
Linearity5.3.2
PropertySection
NTUEE-SS5-DTFT-22
Feng-Li Lian 2007Properties of DT Fourier Transform
Fourier Transform Pair:
Synthesis equation:
Analysis equation:
Notations:
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NTUEE-SS5-DTFT-23
Feng-Li Lian 2007Properties of DT Fourier Transform
Periodicity of DT Fourier Transform:
Linearity:
Time & Frequency Shifting:
NTUEE-SS5-DTFT-24
Feng-Li Lian 2007Properties of DT Fourier Transform
Example 5.7:
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NTUEE-SS5-DTFT-25
Feng-Li Lian 2007Properties of DT Fourier Transform
Conjugation & Conjugate Symmetry:
NTUEE-SS5-DTFT-26
Feng-Li Lian 2007Properties of DT Fourier Transform
Conjugation & Conjugate Symmetry:
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NTUEE-SS5-DTFT-27
Feng-Li Lian 2007Properties of DT Fourier Transform
Differencing & Accumulation:
NTUEE-SS5-DTFT-28
Feng-Li Lian 2007Properties of DT Fourier Transform
Differentiation in Frequency:
Time Reversal:
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NTUEE-SS5-DTFT-29
Feng-Li Lian 2007Properties of DT Fourier Transform
Time Expansion:
NTUEE-SS5-DTFT-30
Feng-Li Lian 2007Properties of DT Fourier Transform
Time Expansion:
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NTUEE-SS5-DTFT-31
Feng-Li Lian 2007Properties of DT Fourier Transform
Time Expansion:
NTUEE-SS5-DTFT-32
Feng-Li Lian 2007Properties of DT Fourier Transform
Example 5.9:
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NTUEE-SS5-DTFT-33
Feng-Li Lian 2007Properties of DT Fourier Transform
Parsevals relation:
NTUEE-SS5-DTFT-34
Feng-Li Lian 2007Properties of DT Fourier Transform
Example 5.10:
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NTUEE-SS5-DTFT-35
Feng-Li Lian 2007Outline
Representation of Aperiodic Signals:the Discrete-Time Fourier Transform
The Fourier Transform for Periodic Signals
Properties of Discrete-Time Fourier Transform
The Convolution Property
The Multiplication Property
Duality
Systems Characterized by Linear Constant-
Coefficient Difference Equations
NTUEE-SS5-DTFT-36
Feng-Li Lian 2007Convolution Property & Multiplication Property
Convolution Property:
Multiplication Property:
X
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NTUEE-SS5-DTFT-37
Feng-Li Lian 2007Convolution Property
Example 5.11:
LTI System
NTUEE-SS5-DTFT-38
Feng-Li Lian 2007Convolution Property
Example 5.12:
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NTUEE-SS5-DTFT-39
Feng-Li Lian 2007Convolution Property
Example 5.13:
Filter
LTI System
NTUEE-SS5-DTFT-40
Feng-Li Lian 2007Convolution Property
Example 5.13:
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NTUEE-SS5-DTFT-41
Feng-Li Lian 2007Convolution Property
Example 5.14:
NTUEE-SS5-DTFT-42
Feng-Li Lian 2007Convolution Property
Example 5.14:
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NTUEE-SS5-DTFT-43
Feng-Li Lian 2007Convolution Property & Multiplication Property
Convolution Property:
Multiplication Property:
NTUEE-SS5-DTFT-44
Feng-Li Lian 2007Multiplication Property
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NTUEE-SS5-DTFT-45
Feng-Li Lian 2007Multiplication Property
NTUEE-SS5-DTFT-46
Feng-Li Lian 2007Multiplication Property
Example 5.15:
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NTUEE-SS5-DTFT-47
Feng-Li Lian 2007Multiplication Property
Example 5.15:
NTUEE-SS5-DTFT-48
Feng-Li Lian 2007
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Feng-Li Lian 2007
NTUEE-SS5-DTFT-50
Feng-Li Lian 2007
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NTUEE-SS5-DTFT-51
Feng-Li Lian 2007Outline
Representation of Aperiodic Signals:the Discrete-Time Fourier Transform
The Fourier Transform for Periodic Signals
Properties of Discrete-Time Fourier Transform
The Convolution Property
The Multiplication Property
Duality
Systems Characterized by Linear Constant-
Coefficient Difference Equations
NTUEE-SS5-DTFT-52
Feng-Li Lian 2007Duality
DT Fourier Series Pair of Periodic Signals:
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NTUEE-SS5-DTFT-53
Feng-Li Lian 2007Duality
Duality in DT Fourier Series:
NTUEE-SS5-DTFT-54
Feng-Li Lian 2007Duality
Duality between DT-FT & CT-FS:
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NTUEE-SS5-DTFT-55
Feng-Li Lian 2007Duality
NTUEE-SS5-DTFT-56
Feng-Li Lian 2007Outline
Representation of Aperiodic Signals:the Discrete-Time Fourier Transform
The Fourier Transform for Periodic Signals
Properties of Discrete-Time Fourier TransformThe Convolution Property
The Multiplication Property
Duality
Systems Characterized by Linear Constant-Coefficient Difference Equations
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NTUEE-SS5-DTFT-57
Feng-Li Lian 2007Systems Characterized by Linear Constant-Coefficient Difference Equations
A useful class of DT LTI systems:
LTI System
NTUEE-SS5-DTFT-58
Feng-Li Lian 2007Systems Characterized by Linear Constant-Coefficient Difference Equations
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NTUEE-SS5-DTFT-59
Feng-Li Lian 2007Systems Characterized by Linear Constant-Coefficient Difference Equations
Examples 5.18 & 5.19:LTI
System
NTUEE-SS5-DTFT-60
Feng-Li Lian 2007Systems Characterized by Linear Constant-Coefficient Difference Equations
Example 5.20:
LTI System
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NTUEE-SS5-DTFT-61
Feng-Li Lian 2007
Periodic Aperiodic
FSCT
DT(Chap 3)
FTDT
CT (Chap 4)
(Chap 5)
Bounded/Convergent
LT
zT DT
Time-Frequency
(Chap 9)
(Chap 10)
Unbounded/Non-convergent
CT
CT-DT
Communication
Control
(Chap 6)
(Chap 7)
(Chap 8)
(Chap 11)
Signals & Systems LTI & Convolution(Chap 1) (Chap 2)
Flowchart