Discrete-Time Fourier Methods M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1

Discrete-Time Fourier Methods

M. J. Roberts - All Rights Reserved. Edited

by Dr. Robert Akl

1

Discrete-Time Fourier Series Concept

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2

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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Discrete-Time Fourier Series Concept

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

4

Discrete-Time Fourier Series Concept

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

5

Discrete-Time Fourier Series Concept

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

6

Discrete-Time Fourier Series Concept

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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Discrete-Time Fourier Series Concept

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Discrete-Time Fourier Series Concept

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Discrete-Time Fourier Series Concept

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Discrete-Time Fourier Series Concept

The Discrete-Time Fourier Series

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11

The Discrete Fourier Transform

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The Discrete Fourier Transform

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The Discrete Fourier Transform

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14

DFT Example

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The DFT Harmonic Function

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The DFT Harmonic Function

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The DFT Harmonic Function

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The DFT Harmonic Function

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The DFT Harmonic Function

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20

The DFT Harmonic Function

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21

The DFT Harmonic Function

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The DFT Harmonic Function

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23

The DFT Harmonic Function

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The DFT Harmonic Function

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Convergence of the DFT

• The DFT converges exactly with a finite number of terms. It does not have a “Gibbs phenomenon” in the same sense that the CTFS does

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The Discrete Fourier Transform

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DFT Properties

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DFT Properties

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DFT Properties

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The Dirichlet Function

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DFT Pairs

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The Fast Fourier Transform

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The Fast Fourier Transform

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Generalizing the DFT for Aperiodic Signals

Pulse Train

This periodic rectangular-wave signal is analogous to thecontinuous-time periodic rectangular-wave signal used toillustrate the transition from the CTFS to the CTFT.

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DFT of Pulse Train

As the period of the rectangular wave increases, the period of the DFT increases

Generalizing the DFT for Aperiodic Signals

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NormalizedDFT of

Pulse Train

Generalizing the DFT for Aperiodic Signals

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The normalized DFT approaches this limit as theperiod approaches infinity.

Generalizing the DFT for Aperiodic Signals

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Definition of the Discrete-Time Fourier

Transform (DTFT)

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The Discrete-Time Fourier Transform

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DTFT Pairs

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The Generalized DTFT

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The Generalized DTFT

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The Generalized DTFT

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Forward DTFT Example

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Forward DTFT Example

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Forward DTFT Example

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More DTFT Pairs

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DTFT Properties

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DTFT Properties

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DTFT Properties

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DTFT Properties

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DTFT Properties

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DTFT Properties

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DTFT Properties

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DTFT Properties

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DTFT Properties

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Transform Method Comparisons

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Transform Method Comparisons

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Transform Method Comparisons

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Transform Method Comparisons

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Transform Method Comparisons

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Transform Method Comparisons

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Transform Method Comparisons

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Transform Method Comparisons

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Transform Method Comparisons

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Transform Method Comparisons

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67

The Four Fourier Methods

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Relations Among Fourier Methods

Multiplication-Convolution Duality

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Relations Among Fourier Methods

Parseval’s Theorem

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Relations Among Fourier Methods

Time and Frequency Shifting

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Relations Among Fourier Methods

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