Signals Analog and Digital Analog and Digital Data & Signals Periodic & Aperiodic Signals.
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![Page 1: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/1.jpg)
Spring 2014
信號與系統Signals and Systems
Chapter SS-5The Discrete-Time Fourier Transform
Feng-Li LianNTU-EE
Feb14 – Jun14
Figures and images used in these lecture notes are adopted from“Signals & Systems” by Alan V. Oppenheim and Alan S. Willsky, 1997
![Page 2: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/2.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-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
![Page 3: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/3.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-3Representation of Aperiodic Signals: DT Fourier Transform
DT Fourier Transform of an Aperiodic Signal: ss4-9
![Page 4: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/4.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-4Representation of Aperiodic Signals: DT Fourier Transform
DT Fourier Transform of an Aperiodic Signal: ss4-10
![Page 5: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/5.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-5Representation of Aperiodic Signals: DT Fourier Transform
DT Fourier Transform of an Aperiodic Signal: ss4-11
![Page 6: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/6.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-6Representation of Aperiodic Signals: DT Fourier Transform
Periodicity of DT Fourier Transform:
![Page 7: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/7.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-7Representation of Aperiodic Signals: DT Fourier Transform
High-Frequency & Low-Frequency Signals:
![Page 8: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/8.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-8Representation of Aperiodic Signals: DT Fourier Transform
High-Frequency & Low-Frequency Signals:
![Page 9: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/9.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-9Representation of Aperiodic Signals: DT Fourier Transform
Example 5.1:
ss4-14
![Page 10: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/10.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-10Representation of Aperiodic Signals: DT Fourier Transform
Example 5.2:ss4-16
![Page 11: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/11.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-11Representation of Aperiodic Signals: DT Fourier Transform
Example 5.3:
ss4-18
![Page 12: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/12.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-12Representation of Aperiodic Signals: DT Fourier Transform
Example 5.3:
ss3-71
![Page 13: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/13.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS3-FS-13Fourier Series Representation of DT Periodic Signals
Example 3.12:
-80 -60 -40 -20 0 20 40 60 80-0.05
0
0.05
0.1
0.15
0.2
0.25
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-80 -60 -40 -20 0 20 40 60 80-0.04
-0.02
0
0.02
0.04
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0.08
0.1
0.12
0.14
-80 -60 -40 -20 0 20 40 60 80-0.1
0
0.1
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-80 -60 -40 -20 0 20 40 60 80-0.2
0
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1.2
-60 -40 -20 0 20 40 60-0.5
0
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-60 -40 -20 0 20 40 60-0.5
0
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-60 -40 -20 0 20 40 60-0.5
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1.5
2
![Page 14: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/14.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-14Representation 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
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-15Representation of Aperiodic Signals: DT Fourier Transform
Example 5.4:
Approximation
ss4-17
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-16Representation of Aperiodic Signals: DT Fourier Transform
Approximation of an Aperiodic Signal:
![Page 17: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/17.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-17Outline
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
![Page 18: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/18.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-18Fourier Transform for Periodic Signals
Fourier Transform from Fourier Series: ss4-22
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-19Fourier Transform for Periodic Signals
Fourier Transform from Fourier Series:
![Page 20: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/20.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-20Fourier Transform for Periodic Signals
Fourier Transform from Fourier Series:
![Page 21: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/21.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-21Fourier Transform for Periodic Signals
Example 5.5: ss4-24
![Page 22: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/22.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-22Fourier Transform for Periodic Signals
Example 5.6:
ss4-25
![Page 23: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/23.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-23Outline
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
![Page 24: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/24.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-24Outline
Section Property5.3.2 Linearity5.3.3 Time Shifting5.3.3 Frequency Shifting5.3.4 Conjugation5.3.6 Time Reversal5.3.7 Time Expansion5.4 Convolution5.5 Multiplication
5.3.5 Differencing in Time5.3.5 Accumulation5.3.8 Differentiation in Frequency5.3.4 Conjugate Symmetry for Real Signals5.3.4 Symmetry for Real and Even Signals5.3.4 Symmetry for Real and Odd Signals5.3.4 Even-Odd Decomposition for Real Signals5.3.9 Parseval’s Relation for Aperiodic Signals
![Page 25: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/25.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-25Outline
Property CTFS DTFS CTFT DTFT LT zTLinearity 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 10.5.2Frequency 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 10.5.6Time 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 5.5Differentiation/First Difference 3.7.2 4.3.4,
4.3.65.3.5, 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 10.5.7Conjugate Symmetry for Real Signals 3.5.6 4.3.3 5.3.4Symmetry for Real and Even Signals 3.5.6 4.3.3 5.3.4Symmetry for Real and Odd Signals 3.5.6 4.3.3 5.3.4
Even-Odd Decomposition for Real Signals 4.3.3 5.3.4Parseval’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
![Page 26: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/26.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-26Properties of DT Fourier Transform
Fourier Transform Pair:• Synthesis equation:
• Analysis equation:
• Notations:
ss4-29
![Page 27: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/27.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-27Properties of DT Fourier Transform
Periodicity of DT Fourier Transform:
Linearity:
Time & Frequency Shifting:
ss4-30
ss4-31
ss5-6
![Page 28: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/28.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-28Properties of DT Fourier Transform
Example 5.7: ss4-19
ss5-14
![Page 29: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/29.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-29Properties of DT Fourier Transform
Conjugation & Conjugate Symmetry:
ss4-34
ss4-35
![Page 30: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/30.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-30Properties of DT Fourier Transform
Conjugation & Conjugate Symmetry:
ss4-34
ss4-36
![Page 31: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/31.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-31Properties of DT Fourier Transform
Conjugation & Conjugate Symmetry:
ss4-37
![Page 32: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/32.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-32Properties of DT Fourier Transform
Differencing & Accumulation:
ss4-40
![Page 33: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/33.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-33Properties of DT Fourier Transform
Differentiation in Frequency:
Time Reversal:
![Page 34: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/34.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-34Properties of DT Fourier Transform
Time Expansion: ss4-45
![Page 35: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/35.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-35Properties of DT Fourier Transform
Time Expansion:
ss3-71
![Page 36: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/36.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-36Properties of DT Fourier Transform
Time Expansion:
![Page 37: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/37.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-37Properties of DT Fourier Transform
Time Expansion:
![Page 38: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/38.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-38Properties of DT Fourier Transform
Example 5.9:
![Page 39: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/39.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-39Properties of DT Fourier Transform
Parseval’s relation: ss4-50
![Page 40: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/40.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-40Properties of DT Fourier Transform
Example 5.10:
![Page 41: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/41.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-41Outline
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
![Page 42: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/42.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-42Convolution Property & Multiplication Property
Convolution Property:
Multiplication Property:
X
ss4-52
![Page 43: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/43.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-43Convolution Property
Example 5.11:
LTI System
ss4-57
![Page 44: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/44.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-44Convolution Property
Example 5.12: ss4-59
![Page 45: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/45.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-45Convolution Property
Example 5.13:
FilterLTI System
ss4-63
![Page 46: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/46.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-46Convolution Property
Example 5.13:
![Page 47: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/47.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-47Convolution Property
Example 5.14: ss4-74
ss5-27
![Page 48: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/48.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-48Convolution Property
Example 5.14:
![Page 49: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/49.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-49Convolution Property & Multiplication Property
Convolution Property:
Multiplication Property:ss4-67
![Page 50: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/50.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-50Multiplication Property
ss4-68
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-51Multiplication Property
![Page 52: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/52.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-52Multiplication Property
Example 5.15: ss4-71
![Page 53: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/53.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-53Multiplication Property
Example 5.15:
![Page 54: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/54.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-54
![Page 55: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/55.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-55
![Page 56: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/56.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-56
![Page 57: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/57.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-57Outline
Representation of Aperiodic Signals: the Discrete-Time Fourier Transform
The Fourier Transform for Periodic SignalsProperties of Discrete-Time Fourier Transform The Convolution Property The Multiplication PropertyDualitySystems Characterized by
Linear Constant-Coefficient Difference Equations
ss4-47
![Page 58: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/58.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-58Duality
DT Fourier Series Pair of Periodic Signals:
![Page 59: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/59.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-59Duality
Duality in DT Fourier Series:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-60Duality
Duality between DT-FT & CT-FS:
![Page 61: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/61.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-61Duality
![Page 62: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/62.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-62Outline
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
![Page 63: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/63.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-63Systems Characterized by Linear Constant-Coefficient Difference Equations
A useful class of DT LTI systems:
LTI System
ss4-76
![Page 64: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/64.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-64Systems Characterized by Linear Constant-Coefficient Difference Equations
ss4-79
![Page 65: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/65.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-65Systems Characterized by Linear Constant-Coefficient Difference Equations
Examples 5.18 & 5.19: LTISystem
ss4-81
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-66Systems Characterized by Linear Constant-Coefficient Difference Equations
Example 5.20:
LTI System ss4-82
ss5-45
![Page 67: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/67.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-67Chapter 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
![Page 68: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/68.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-68
Periodic Aperiodic
FS CTDT
(Chap 3)FT
DT
CT (Chap 4)
(Chap 5)
Bounded/Convergent
LTzT 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
![Page 69: Signals and Systems - cc.ee.ntu.edu.twcc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss05_DTFT.pdf · Spring 2014 信號與系統 Signals and Systems Chapter SS-5 The Discrete-Time](https://reader031.fdocuments.net/reader031/viewer/2022022509/5ad3b08c7f8b9aff738e5765/html5/thumbnails/69.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS5-DTFT-69Question for Chapter 5
In discrete time, what kind of signals are low frequency or high frequency?
• Chap 1, pages 47-49• Chap 5, pages 7-9
Time expansion and double the width in time domain• Chap 5, pages 33-34• Chap 3, pages 70-71• Chap 5, page 11
Frequency shaping filter on low-freq and high-freq signals• Chap 3, pages 92-95