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Spring 2007

Signals and Systems

Chapter SS-5The Discrete-Time Fourier Transform

Feng-Li Lian

NTU-EE

Feb07 Jun07

Figures and images used in these lecture notes are adopted fromSignals & Systems by Alan V. Oppenheim and Alan S. Willsky, 1997

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Feng-Li Lian 2007Outline

Representation of Aperiodic Signals:the Discrete-Time Fourier Transform

The Fourier Transform for Periodic Signals

Properties of Discrete-Time Fourier TransformThe Convolution Property

The Multiplication Property

Duality

Systems Characterized by Linear Constant-Coefficient Difference Equations

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Feng-Li Lian 2007Representation of Aperiodic Signals: DT Fourier Transform

DT Fourier Transform of an Aperiodic Signal:

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Periodicity of DT Fourier Transform:

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High-Frequency & Low-Frequency Signals:

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Example 5.1:

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Example 5.2:

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Example 5.3:

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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,

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Example 5.4:

Approximation

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Approximation of an Aperiodic Signal:

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Duality


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Feng-Li Lian 2007Fourier Transform for Periodic Signals

Fourier Transform from Fourier Series:

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Example 5.5:

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Example 5.6:

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Duality


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Time Expansion5.3.7

Differencing in Time5.3.5

Multiplication5.5

Convolution5.4

Time Reversal5.3.6

Conjugation5.3.4

Frequency Shifting5.3.3

Time Shifting5.3.3

Symmetry for Real and Even Signals5.3.4

Conjugate Symmetry for Real Signals5.3.4

Differentiation in Frequency5.3.8

Accumulation5.3.5

Even-Odd Decomposition for Real Signals5.3.4

Symmetry for Real and Odd Signals5.3.4

Parsevals Relation for Aperiodic Signals5.3.9

Linearity5.3.2

PropertySection

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Feng-Li Lian 2007Properties of DT Fourier Transform

Fourier Transform Pair:

Synthesis equation:

Analysis equation:

Notations:

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Periodicity of DT Fourier Transform:

Linearity:

Time & Frequency Shifting:

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Example 5.7:

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Conjugation & Conjugate Symmetry:

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Conjugation & Conjugate Symmetry:

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Differencing & Accumulation:

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Differentiation in Frequency:

Time Reversal:

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Time Expansion:

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Time Expansion:

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Time Expansion:

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Example 5.9:

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Parsevals relation:

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Example 5.10:

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Properties of Discrete-Time Fourier Transform

The Convolution Property


Duality

Systems Characterized by Linear Constant-

Coefficient Difference Equations

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Feng-Li Lian 2007Convolution Property & Multiplication Property

Convolution Property:

Multiplication Property:

X

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Feng-Li Lian 2007Convolution Property

Example 5.11:

LTI System

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Example 5.12:

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Example 5.13:

Filter

LTI System

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Example 5.13:

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Example 5.14:

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Example 5.14:

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Feng-Li Lian 2007Convolution Property & Multiplication Property

Convolution Property:

Multiplication Property:

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Feng-Li Lian 2007Multiplication Property

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Example 5.15:

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Example 5.15:

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Feng-Li Lian 2007

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Properties of Discrete-Time Fourier Transform

The Convolution Property


Duality

Systems Characterized by Linear Constant-

Coefficient Difference Equations

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Feng-Li Lian 2007Duality

DT Fourier Series Pair of Periodic Signals:

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Duality in DT Fourier Series:

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Duality between DT-FT & CT-FS:

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Duality


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Feng-Li Lian 2007Systems Characterized by Linear Constant-Coefficient Difference Equations

A useful class of DT LTI systems:

LTI System

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Examples 5.18 & 5.19:LTI

System

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Example 5.20:

LTI System

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Feng-Li Lian 2007

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

mam's file