signals and systems -...
TRANSCRIPT
Spring 2013
信號與系統Signals and Systems
Chapter SS-5The Discrete-Time Fourier Transform
Feng-Li LianNTU-EE
Feb13 – Jun13
Figures and images used in these lecture notes are adopted from“Signals & Systems” by Alan V. Oppenheim and Alan S. Willsky, 1997
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-2Outline
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
ss4-2
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-3Representation of Aperiodic Signals: DT Fourier Transform
DT Fourier Transform of an Aperiodic Signal:ss4-9
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-4Representation of Aperiodic Signals: DT Fourier Transform
DT Fourier Transform of an Aperiodic Signal:ss4-10
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS3-FS-30Fourier Series Representation of CT Periodic Signals
Example 3.5:
Feng-Li Lian © 2011Feng-Li Lian © 2013NTUEE-SS3-FS-33Fourier Series Representation of CT Periodic Signals
Example 3.5:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-5Representation of Aperiodic Signals: DT Fourier Transform
DT Fourier Transform of an Aperiodic Signal:ss4-11
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-6Representation of Aperiodic Signals: DT Fourier Transform
Periodicity of DT Fourier Transform:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-7Representation of Aperiodic Signals: DT Fourier Transform
High-Frequency & Low-Frequency Signals:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-8Representation of Aperiodic Signals: DT Fourier Transform
Example 5.1:
ss4-14
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-9Representation of Aperiodic Signals: DT Fourier Transform
Example 5.2:
ss4-16
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-10Representation of Aperiodic Signals: DT Fourier Transform
Example 5.3:
ss4-18
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-11Representation of Aperiodic Signals: DT Fourier Transform
Example 5.3:
ss3-71
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS3-FS-12Fourier Series Representation of DT Periodic Signals
Example 3.12:
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Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-13Representation 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,
ss4-13
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-14Representation of Aperiodic Signals: DT Fourier Transform
Example 5.4:
Approximation
ss4-17
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-15Representation of Aperiodic Signals: DT Fourier Transform
Approximation of an Aperiodic Signal:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-16Outline
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
ss4-21
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-17Fourier Transform for Periodic Signals
Fourier Transform from Fourier Series:ss4-22
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-18Fourier Transform for Periodic Signals
Fourier Transform from Fourier Series:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-19Fourier Transform for Periodic Signals
Fourier Transform from Fourier Series:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-20Fourier Transform for Periodic Signals
Example 5.5:ss4-24
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-21Fourier Transform for Periodic Signals
Example 5.6:
ss4-25
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-22Outline
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
ss4-26
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-23Outline
Section Property
5.3.2 Linearity
5.3.3 Time Shifting
5.3.3 Frequency Shifting
5.3.4 Conjugation
5.3.6 Time Reversal
5.3.7 Time Expansion
5.4 Convolution
5.5 Multiplication
5.3.5 Differencing in Time
5.3.5 Accumulation
5.3.8 Differentiation in Frequency
5.3.4 Conjugate Symmetry for Real Signals
5.3.4 Symmetry for Real and Even Signals
5.3.4 Symmetry for Real and Odd Signals
5.3.4 Even-Odd Decomposition for Real Signals
5.3.9 Parseval’s Relation for Aperiodic Signals
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-24Outline
Property CTFS DTFS CTFT DTFT LT zT
Linearity 3.5.1 4.3.1 5.3.2 9.5.1 10.5.1
Time Shifting 3.5.2 4.3.2 5.3.3 9.5.2 10.5.2
Frequency Shifting (in s, z) 4.3.6 5.3.3 9.5.3 10.5.3
Conjugation 3.5.6 4.3.3 5.3.4 9.5.5 10.5.6
Time Reversal 3.5.3 4.3.5 5.3.6 10.5.4
Time & Frequency Scaling 3.5.4 4.3.5 5.3.7 9.5.4 10.5.5
(Periodic) Convolution 4.4 5.4 9.5.6 10.5.7
Multiplication 3.5.5 3.7.2 4.5 5.5
Differentiation/First Difference 3.7.2 4.3.4, 4.3.6
5.3.5, 5.3.8
9.5.7, 9.5.8
10.5.7, 10.5.8
Integration/Running Sum (Accumulation) 4.3.4 5.3.5 9.5.9 10.5.7
Conjugate Symmetry for Real Signals 3.5.6 4.3.3 5.3.4
Symmetry for Real and Even Signals 3.5.6 4.3.3 5.3.4
Symmetry for Real and Odd Signals 3.5.6 4.3.3 5.3.4
Even-Odd Decomposition for Real Signals 4.3.3 5.3.4
Parseval’s Relation for (A)Periodic Signals 3.5.7 3.7.3 4.3.7 5.3.9
Initial- and Final-Value Theorems 9.5.10 10.5.9
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-25Properties of DT Fourier Transform
Fourier Transform Pair:• Synthesis equation:
• Analysis equation:
• Notations:
ss4-29
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-26Properties of DT Fourier Transform
Periodicity of DT Fourier Transform:
Linearity:
Time & Frequency Shifting:
ss4-30
ss4-31
ss5-6
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-27Properties of DT Fourier Transform
Example 5.7:ss4-19
ss5-14
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-28Properties of DT Fourier Transform
Conjugation & Conjugate Symmetry:
ss4-34
ss4-35
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-29Properties of DT Fourier Transform
Conjugation & Conjugate Symmetry:
ss4-34
ss4-36
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-30Properties of DT Fourier Transform
Conjugation & Conjugate Symmetry:
ss4-37
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-31Properties of DT Fourier Transform
Differencing & Accumulation:
ss4-40
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-32Properties of DT Fourier Transform
Differentiation in Frequency:
Time Reversal:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-33Properties of DT Fourier Transform
Time Expansion:ss4-45
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-34Properties of DT Fourier Transform
Time Expansion:
ss3-71
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-35Properties of DT Fourier Transform
Time Expansion:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-36Properties of DT Fourier Transform
Time Expansion:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-37Properties of DT Fourier Transform
Example 5.9:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-38Properties of DT Fourier Transform
Parseval’s relation:ss4-50
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-39Properties of DT Fourier Transform
Example 5.10:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-40Outline
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
ss4-51
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-41Convolution Property & Multiplication Property
Convolution Property:
Multiplication Property:
X
ss4-52
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-42Convolution Property
Example 5.11:
LTI System
ss4-57
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-43Convolution Property
Example 5.12:ss4-59
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-44Convolution Property
Example 5.13:
FilterLTI System
ss4-63
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-45Convolution Property
Example 5.13:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-46Convolution Property
Example 5.14:ss4-74
ss5-27
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-47Convolution Property
Example 5.14:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-27Properties of DT Fourier Transform
Example 5.7:ss4-19
ss5-14
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-48Convolution Property & Multiplication Property
Convolution Property:
Multiplication Property:ss4-67
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-49Multiplication Property
ss4-68
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-50Multiplication Property
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-51Multiplication Property
Example 5.15:ss4-71
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-52Multiplication Property
Example 5.15:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-53
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-54
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-55
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-56Outline
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
ss4-47
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-57Duality
DT Fourier Series Pair of Periodic Signals:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-58Duality
Duality in DT Fourier Series:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-59Duality
Duality between DT-FT & CT-FS:
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-60Duality
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-61Outline
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
ss4-75
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-62Systems Characterized by Linear Constant-Coefficient Difference Equations
A useful class of DT LTI systems:
LTI System
ss4-76
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-63Systems Characterized by Linear Constant-Coefficient Difference Equations
ss4-79
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-64Systems Characterized by Linear Constant-Coefficient Difference Equations
Examples 5.18 & 5.19: LTISystem
ss4-81
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-65Systems Characterized by Linear Constant-Coefficient Difference Equations
Example 5.20:
LTI System ss4-82
ss5-45
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-66Chapter 5: The Discrete-Time Fourier Transform
Representation of Aperiodic Signals: the DT FT The FT for Periodic SignalsProperties of the DT FT
• Linearity Time Shifting Frequency Shifting• Conjugation Time Reversal Time Expansion• Convolution Multiplication• Differencing in Time Accumulation Differentiation in Frequency• Conjugate Symmetry for Real Signals• Symmetry for Real and Even Signals & for Real and Odd Signals• Even-Odd Decomposition for Real Signals• Parseval’s Relation for Aperiodic Signals
The Convolution Property The Multiplication PropertyDualitySystems Characterized by Linear Constant-
Coefficient Difference Equations
Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-67
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