1. Discrete-Time Fourier Methods
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2. Discrete-Time Fourier Series Concept
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3. 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

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

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

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

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

11. The Discrete-Time Fourier Series
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12. The Discrete Fourier Transform
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13. The Discrete Fourier Transform
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14. The Discrete Fourier Transform
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15. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
DFT Example
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16. The DFT Harmonic Function
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17. The DFT Harmonic Function
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18. The DFT Harmonic Function
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19. The DFT Harmonic Function
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20. The DFT Harmonic Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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

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

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

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

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

26. 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
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27. The Discrete Fourier Transform
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28. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
DFT Properties
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29. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
DFT Properties
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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

31. The Dirichlet Function
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32. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
DFT Pairs
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33. The Fast Fourier Transform
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34. The Fast Fourier Transform
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35. 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.
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36. Generalizing the DFT for Aperiodic Signals
DFT of
Pulse Train
As the period of the rectangular wave increases, the period of the DFT increases
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37. Generalizing the DFT for Aperiodic Signals
Normalized
DFT of
Pulse Train
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38. Generalizing the DFT for Aperiodic Signals
The normalized DFT approaches this limit as the
period approaches infinity.
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39. Definition of the Discrete-Time Fourier Transform (DTFT)
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40. The Discrete-Time Fourier Transform
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41. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
DTFT Pairs
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42. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
The Generalized DTFT
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43. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
The Generalized DTFT
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44. 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

45. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Forward DTFT Example
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46. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Forward DTFT Example
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47. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Forward DTFT Example
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48. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
More DTFT Pairs
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49. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
DTFT Properties
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50. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
DTFT Properties
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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

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

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

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

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

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

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

58. Transform Method Comparisons
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59. Transform Method Comparisons
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60. Transform Method Comparisons
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61. Transform Method Comparisons
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62. Transform Method Comparisons
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

63. Transform Method Comparisons
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64. Transform Method Comparisons
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65. Transform Method Comparisons
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66. Transform Method Comparisons
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67. Transform Method Comparisons
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68. The Four Fourier Methods
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69. Multiplication-Convolution Duality
Relations Among Fourier Methods
Multiplication-Convolution Duality
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70. Relations Among Fourier Methods
Parseval’s Theorem
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71. Relations Among Fourier Methods
Time and Frequency Shifting
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72. Relations Among Fourier Methods
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