Discrete-Time Fourier Methods

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Presentation transcript:

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

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

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

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

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

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

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

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

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

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

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

The Discrete Fourier Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The Discrete Fourier Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The Discrete Fourier Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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

The DFT Harmonic Function M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The DFT Harmonic Function M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The DFT Harmonic Function M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The DFT Harmonic Function M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The DFT Harmonic Function M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The DFT Harmonic Function M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The DFT Harmonic Function M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The DFT Harmonic Function M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The DFT Harmonic Function M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The DFT Harmonic Function M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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

The Discrete Fourier Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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

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

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

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

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

The Fast Fourier Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The Fast Fourier Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Generalizing the DFT for Aperiodic Signals Pulse Train This periodic rectangular-wave signal is analogous to the continuous-time periodic rectangular-wave signal used to illustrate the transition from the CTFS to the CTFT. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Generalizing the DFT for Aperiodic Signals DFT of Pulse Train As the period of the rectangular wave increases, the period of the DFT increases M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Generalizing the DFT for Aperiodic Signals Normalized DFT of Pulse Train M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Generalizing the DFT for Aperiodic Signals The normalized DFT approaches this limit as the period approaches infinity. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Definition of the Discrete-Time Fourier Transform (DTFT) M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The Four Fourier Methods M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Multiplication-Convolution Duality Relations Among Fourier Methods Multiplication-Convolution Duality M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Relations Among Fourier Methods Parseval’s Theorem M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Relations Among Fourier Methods Time and Frequency Shifting M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Relations Among Fourier Methods M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl