Signals and Systems - 台大電機系計算機中心cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/103-2... · 2015-03-27 · Signals and Systems Chapter SS-5 The Discrete-Time

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 © 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:




Periodicity of DT Fourier Transform:


High-Frequency & Low-Frequency Signals:


Example 5.1:

ss4-14


Example 5.2:

ss4-16


Example 5.3:

ss4-18


Example 5.3:

ss3-71

Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS3-FS-12Fourier 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

0.3

-80 -60 -40 -20 0 20 40 60 80-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

-80 -60 -40 -20 0 20 40 60 80-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-80 -60 -40 -20 0 20 40 60 80-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-60 -40 -20 0 20 40 60-0.5

0

0.5

1

1.5

2

-60 -40 -20 0 20 40 60-0.5

0

0.5

1

1.5

2

-60 -40 -20 0 20 40 60-0.5

0

0.5

1

1.5

2


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


Example 5.4:

Approximation

ss4-17


Approximation of an Aperiodic Signal:








Duality



ss4-21

Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-17Fourier Transform for Periodic Signals

Fourier Transform from Fourier Series:ss4-22


Fourier Transform from Fourier Series:


Fourier Transform from Fourier Series:


Example 5.5:ss4-24


Example 5.6:

ss4-25








Duality



ss4-26


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


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


Periodicity of DT Fourier Transform:

Linearity:

Time & Frequency Shifting:

ss4-30

ss4-31

ss5-6


Example 5.7:ss4-19

ss5-14


Conjugation & Conjugate Symmetry:

ss4-34

ss4-35



ss4-34

ss4-36



ss4-37


Differencing & Accumulation:

ss4-40


Differentiation in Frequency:

Time Reversal:


Time Expansion:ss4-45


Time Expansion:

ss3-71


Time Expansion:


Time Expansion:


Example 5.9:


Parseval’s relation:ss4-50


Example 5.10:








Duality



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


Example 5.12:ss4-59


Example 5.13:

FilterLTI System

ss4-63


Example 5.13:


Example 5.14:ss4-74

ss5-27


Example 5.14:


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



Example 5.15:ss4-71


Example 5.15:

Feng-Li Lian © 2013Feng-Li Lian © 2013NTUEE-SS5-DTFT-53




Representation of Aperiodic Signals: the Discrete-Time Fourier Transform





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:


Duality in DT Fourier Series:


Duality between DT-FT & CT-FS:









Duality



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


ss4-79


Examples 5.18 & 5.19: LTISystem

ss4-81


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


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

Documents

Signals and Systems - 台大電機系計算機中心cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/103-2... · 2015-03-27 · Signals and Systems Chapter SS-5 The Discrete-Time