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Discrete-Time Fourier Methods
M. J. Roberts - All Rights Reserved. Edited
by Dr. Robert Akl
1
Discrete-Time Fourier Series Concept
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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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
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
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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
The Discrete-Time Fourier Series
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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The Discrete Fourier Transform
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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The Discrete Fourier Transform
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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The Discrete Fourier Transform
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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DFT Example
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The DFT Harmonic Function
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The DFT Harmonic Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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The DFT Harmonic Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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The DFT Harmonic Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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The DFT Harmonic Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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The DFT Harmonic Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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The DFT Harmonic Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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The DFT Harmonic Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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The DFT Harmonic Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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The DFT Harmonic Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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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
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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The Discrete Fourier Transform
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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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.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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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
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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NormalizedDFT of
Pulse Train
Generalizing the DFT for Aperiodic Signals
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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The normalized DFT approaches this limit as theperiod approaches infinity.
Generalizing the DFT for Aperiodic Signals
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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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
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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DTFT Properties
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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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
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Transform Method Comparisons
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Transform Method Comparisons
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Transform Method Comparisons
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Transform Method Comparisons
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Transform Method Comparisons
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Transform Method Comparisons
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Transform Method Comparisons
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Transform Method Comparisons
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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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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