Signals and Systems -...

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

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

DT Fourier Transform of an Aperiodic Signal: ss4-9

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

Example 3.12:

-80 -60 -40 -20 0 20 40 60 80-0.05

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

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

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

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

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

Fourier Transform from Fourier Series: ss4-22

Fourier Transform from Fourier Series:

Example 5.5: ss4-24

Example 5.6:

ss4-25

Duality

ss4-26

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

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

Fourier Transform Pair:• Synthesis equation:

• Analysis equation:

• Notations:

ss4-29

Periodicity of DT Fourier Transform:

Linearity:

Time & Frequency Shifting:

ss4-30

ss4-31

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:

Example 5.9:

Parseval’s relation: ss4-50

Example 5.10:

Duality

ss4-51

Convolution Property:

Multiplication Property:

ss4-52

Example 5.11:

LTI System

ss4-57

Example 5.12: ss4-59

Example 5.13:

FilterLTI System

ss4-63

Example 5.13:

ss5-27

Example 5.14:

Convolution Property:

Multiplication Property:ss4-67

ss4-68

Example 5.15:

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

DT Fourier Series Pair of Periodic Signals:

Duality in DT Fourier Series:

Duality between DT-FT & CT-FS:

Duality

ss4-75

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

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

FS CTDT

(Chap 3)FT

CT (Chap 4)

(Chap 5)

Bounded/Convergent

LTzT DT

Time-Frequency

(Chap 9)

(Chap 10)

Unbounded/Non-convergent

Communication

Control

(Chap 6)

(Chap 7)

(Chap 8)

(Chap 11)

Signals & Systems LTI & Convolution(Chap 1) (Chap 2)

Flowchart

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

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