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![Page 1: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/1.jpg)
Spring 2014
信號與系統Signals and Systems
Chapter SS-4The Continuous-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 - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/2.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-2Outline
Representation of Aperiodic Signals: the Continuous-Time Fourier Transform
The Fourier Transform for Periodic Signals
Properties of Continuous-Time Fourier Transform
The Convolution Property
The Multiplication Property
Systems Characterized by Linear Constant-Coefficient Differential Equations
![Page 3: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/3.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-3Fourier Series Representation of CT Periodic Signals
Example 3.5:
![Page 4: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/4.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-4Fourier Series Representation of CT Periodic Signals
Example 3.5:
![Page 5: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/5.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-5Fourier Series Representation of CT Periodic Signals
Example 3.5:
![Page 6: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/6.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-6Representation of Aperiodic Signals: CT Fourier Transform
CT Fourier Transform of an Aperiodic Signal: Page 193, Ex 3.5
![Page 7: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/7.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-7Representation of Aperiodic Signals: CT Fourier Transform
![Page 8: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/8.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-8Representation of Aperiodic Signals: CT Fourier Transform
![Page 9: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/9.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-9Representation of Aperiodic Signals: CT Fourier Transform
![Page 10: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/10.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-10Representation of Aperiodic Signals: CT Fourier Transform
![Page 11: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/11.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-11Representation of Aperiodic Signals: CT Fourier Transform
![Page 12: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/12.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-12Representation of Aperiodic Signals: CT Fourier Transform
Sufficient conditions for the convergence of FT
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-13Representation of Aperiodic Signals: CT Fourier Transform
Sufficient conditions for the convergence of FT
• Dirichlet conditions:
1.x(t) be absolutely integrable; that is,
2.x(t) have a finite number of maxima and minimawithin any finite interval
3.x(t) have a finite number of discontinuitieswithin any finite intervalFurthermore, each of these discontinuities must be finite
![Page 14: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/14.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-14Representation of Aperiodic Signals: CT Fourier Transform
Example 4.1:
![Page 15: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/15.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-15Representation of Aperiodic Signals: CT Fourier Transform
Example 4.1:
![Page 16: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/16.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-16Representation of Aperiodic Signals: CT Fourier Transform
Example 4.2:
![Page 17: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/17.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-17Representation of Aperiodic Signals: CT Fourier Transform
Example 4.3:
![Page 18: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/18.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-18Representation of Aperiodic Signals: CT Fourier Transform
Example 4.4:
![Page 19: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/19.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-19Representation of Aperiodic Signals: CT Fourier Transform
Example 4.5:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-20Representation of Aperiodic Signals: CT Fourier Transform
sinc functions:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-21Outline
Representation of Aperiodic Signals: the Continuous-Time Fourier Transform
The Fourier Transform for Periodic Signals
Properties of Continuous-Time Fourier Transform
The Convolution Property
The Multiplication Property
Systems Characterized by Linear Constant-Coefficient Differential Equations
![Page 22: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/22.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-22Fourier Transform for Periodic Signals
Fourier Transform from Fourier Series:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-23Fourier Transform for Periodic Signals
Example 4.6:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-24Fourier Transform for Periodic Signals
Example 4.7:
![Page 25: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/25.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-25Fourier Transform for Periodic Signals
Example 4.8:
![Page 26: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/26.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-26Outline
Representation of Aperiodic Signals: the Continuous-Time Fourier Transform
The Fourier Transform for Periodic Signals
Properties of Continuous-Time Fourier Transform
The Convolution Property
The Multiplication Property
Systems Characterized by Linear Constant-Coefficient Differential Equations
![Page 27: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/27.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-27Outline
Section Property4.3.1 Linearity4.3.2 Time Shifting4.3.6 Frequency Shifting4.3.3 Conjugation4.3.5 Time Reversal4.3.5 Time and Frequency Scaling4.4 Convolution4.5 Multiplication
4.3.4 Differentiation in Time4.3.4 Integration4.3.6 Differentiation in Frequency4.3.3 Conjugate Symmetry for Real Signals4.3.3 Symmetry for Real and Even Signals4.3.3 Symmetry for Real and Odd Signals4.3.3 Even-Odd Decomposition for Real Signals4.3.7 Parseval’s Relation for Aperiodic Signals
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-28Outline
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
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-29Properties of CT Fourier Transform
Fourier Transform Pair:
• Synthesis equation:
• Analysis equation:
• Notations:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-30Properties of CT Fourier Transform
Linearity:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-31Properties of CT Fourier Transform
Time Shifting:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-32Properties of CT Fourier Transform
Time Shift => Phase Shift:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-33Properties of CT Fourier Transform
Example 4.9: Ex 4.4
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-34Properties of CT Fourier Transform
Conjugation & Conjugate Symmetry:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-35Properties of CT Fourier Transform
Conjugation & Conjugate Symmetry:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-36Properties of CT Fourier Transform
Conjugation & Conjugate Symmetry:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-37Properties of CT Fourier Transform
Conjugation & Conjugate Symmetry:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-38Properties of CT Fourier Transform
Example 4.10:
Ex 4.1
Ex 4.2
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-39Properties of CT Fourier Transform
Example 4.10:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-40Properties of CT Fourier Transform
Differentiation & Integration:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-41Properties of CT Fourier Transform
FT of u(t) and 1(t):
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-42Properties of CT Fourier Transform
= +
Ex 4.3
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-43Properties of CT Fourier Transform
Example 4.11:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-44Properties of CT Fourier Transform
Example 4.12: Ex 4.4
![Page 45: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/45.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-45Properties of CT Fourier Transform
Time & Frequency Scaling:
![Page 46: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/46.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-46Properties of CT Fourier Transform
Time & Frequency Scaling:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-47Properties of CT Fourier Transform
Duality:
Example 4.4
Example 4.5
![Page 48: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/48.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-48Properties of CT Fourier Transform
Duality:
![Page 49: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/49.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-49Properties of CT Fourier Transform
Duality:
![Page 50: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/50.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-50Properties of CT Fourier Transform
Parseval’s relation:
![Page 51: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/51.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-51Outline
Representation of Aperiodic Signals: the Continuous-Time Fourier Transform
The Fourier Transform for Periodic Signals
Properties of Continuous-Time Fourier Transform
The Convolution Property
The Multiplication Property
Systems Characterized by Linear Constant-Coefficient Differential Equations
![Page 52: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/52.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-52Convolution Property & Multiplication Property
Convolution Property:
Multiplication Property:
X
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-53Convolution Property
From Superposition (or Linearity):
LinearSystem
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-54Convolution Property
From Superposition (or Linearity):
![Page 55: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/55.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-55Convolution Property
From Convolution Integral:
![Page 56: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/56.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-56Convolution Property
Equivalent LTI Systems:
![Page 57: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/57.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-57Convolution Property
Example 4.15: Time Shift
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-58Convolution Property
Examples 4.16 & 17: Differentiator & Integrator
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-59Convolution Property
Example 4.18: Ideal Lowpass Filter
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-60Convolution Property
Filter Design:
FilterLTI System
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Feng-Li Lian © 2014Feng-Li Lian © 2014Signals & Systems: CT & DT Systems
Interconnections of Systems:• Audio System:
NTUEE-SS1-SS-61
Bass Treble EqualizerMicrophone
or Tape Speaker
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-62Convolution Property
Filter Design:
FilterLTI System
RC circuit
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-63Convolution Property
Example 4.19:
FilterLTI System
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-64Convolution Property
Example 4.19:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-65Convolution Property
Example 4.20:
FilterLTI System
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-66Outline
Representation of Aperiodic Signals: the Continuous-Time Fourier Transform
The Fourier Transform for Periodic Signals
Properties of Continuous-Time Fourier Transform
The Convolution Property
The Multiplication Property
Systems Characterized by Linear Constant-Coefficient Differential Equations
![Page 67: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/67.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-67Multiplication Property
Convolution & Multiplication:
Multiplication of One Signal by Another:• Scale or modulate the amplitude of the other signal• Modulation
X
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-68Multiplication Property
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-69Multiplication Property
Example 4.21:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-70Multiplication Property
Example 4.22:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-71Multiplication Property
Example 4.23:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-72Multiplication Property
Bandpass Filter Using Amplitude Modulation:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-73Multiplication Property
Bandpass Filter Using Amplitude Modulation:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-74Multiplication Property
Bandpass Filter Using Amplitude Modulation:
• On Page 349-350, Problem 4.46
X
X
X
X
+
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-75
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-76
![Page 77: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/77.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-77
![Page 78: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/78.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-78Outline
Representation of Aperiodic Signals: the Continuous-Time Fourier Transform
The Fourier Transform for Periodic Signals
Properties of Continuous-Time Fourier Transform
The Convolution Property
The Multiplication Property
Systems Characterized by Linear Constant-Coefficient Differential Equations
![Page 79: Signals and Systems - 國立臺灣大學cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/102-2_ss04_CTFT… · Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform](https://reader033.vdocuments.net/reader033/viewer/2022042601/5f8f987ef4db641c8b07a54d/html5/thumbnails/79.jpg)
Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-79Systems Characterized by Linear Constant-Coefficient Differential Equations
A useful class of CT LTI systems:
LTI System
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-80Systems Characterized by Linear Constant-Coefficient Differential Equations
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-81Systems Characterized by Linear Constant-Coefficient Differential Equations
Examples 4.24 & 4.25:
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-82Systems Characterized by Linear Constant-Coefficient Differential Equations
Example 4.26:LTI System
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-83X(jw) of Aperiodic Signals and a_k of Periodic Signals
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-84X(jw) of Aperiodic Signals and a_k of Periodic Signals
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-85X(jw) of Aperiodic Signals and a_k of Periodic Signals
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-86Chapter 4: The Continuous-Time Fourier Transform
Representation of Aperiodic Signals: the CT FT The FT for Periodic SignalsProperties of the CT FT
• Linearity Time Shifting Frequency Shifting• Conjugation Time Reversal Time and Frequency Scaling• Convolution Multiplication• Differentiation in Time Integration 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 Property
Systems Characterized by Linear Constant-Coefficient Differential Equations
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-87Questions for Chapter 4
Why to study FT• In order to analyze or represent aperiodic signals
How to develop FT• From FS and let T -> infinity
Do periodic signals have FT• Yes, their FT is function of isolated impulses
Why to know the properties of FT• Avoid using the fundamental formulas of FT to compute the FT
What the duality of FT and why• FT and IFT have almost identical integration formulas
Why to know the convolution property• To analyze system response and/or design proper circuits
• To simplify computation
Why to know the multiplication property• For signal modulation with different-frequency carriers• To simplify computation
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Feng-Li Lian © 2014Feng-Li Lian © 2014NTUEE-SS4-CTFT-88
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