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The Discrete Fourier Transform

The Discrete Fourier Transform. 主講人：虞台文. Content. Introduction Representation of Periodic Sequences DFS (Discrete Fourier Series) Properties of DFS The Fourier Transform of Periodic Signals Sampling of Fourier Transform Representation of Finite-Duration Sequences

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The Discrete Fourier Transform

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  1. The Discrete Fourier Transform 主講人：虞台文

  2. Content • Introduction • Representation of Periodic Sequences • DFS (Discrete Fourier Series) • Properties of DFS • The Fourier Transform of Periodic Signals • Sampling of Fourier Transform • Representation of Finite-Duration Sequences • DFT (Discrete Fourier Transform) • Properties of the DFT • Linear Convolution Using the DFT

  3. The Discrete Fourier Transform Introduction

  4. Periodicity Periodic Aperiodic Continuous Time Discrete Infinite Finite Duration Signal Processing Methods Fourier Series Continuous-Time Fourier Transform DFS Discrete-Time Fourier Transform and z-Transform

  5. Periodicity Periodic Aperiodic Fourier Series Continuous-Time Fourier Transform Continuous Time DFS Discrete-Time Fourier Transform and z-Transform Discrete Infinite Finite Duration Frequency-Domain Properties Continuous & Aperiodic Discrete & Aperiodic Continuous & Periodic (2) ?

  6. Periodicity Periodic Aperiodic Fourier Series Continuous-Time Fourier Transform Continuous & Aperiodic Discrete & Aperiodic Continuous Time DFT Discrete-Time Fourier Transform and z-Transform Continuous & Periodic (2) ? Discrete Infinite Finite Duration Frequency-Domain Properties Relation? Relation? Relation? Relation?

  7. Periodicity Periodic Aperiodic Fourier Series Continuous-Time Fourier Transform Continuous & Aperiodic Discrete & Aperiodic Continuous Time DFT Discrete-Time Fourier Transform and z-Transform Continuous & Periodic (2) Discrete Infinite Finite Duration Frequency-Domain Properties Discrete & Periodic

  8. The Discrete Fourier Transform Representation of Periodic Sequences --- DFS

  9. Periodic Sequences • Notation: a sequence with period N where r is any integer.

  10. Harmonics Facts: Each has Periodic N. N distinct harmonics e0(n), e1(n),…, eN1(n).

  11. Synthesis and Analysis Notation Both have Period N Synthesis Analysis

  12. Example A periodic impulse train with period N.

  13. n 0 1 2 3 4 5 6 7 8 9 Example

  14. n 0 1 2 3 4 5 6 7 8 9 Example

  15. n 0 1 2 3 4 5 6 7 8 9 Example

  16. n N 0 N n 0 DFS vs. FT

  17. n 0 1 2 3 4 5 6 7 8 9 n 0 1 2 3 4 5 6 7 8 9 Example

  18. n 0 1 2 3 4 5 6 7 8 9 n 0 1 2 3 4 5 6 7 8 9 Example

  19. n 0 1 2 3 4 5 6 7 8 9 n 0 1 2 3 4 5 6 7 8 9 Example

  20. The Discrete Fourier Transform Properties of DFS

  21. Linearity

  22. Shift of a Sequence Change Phase (delay)

  23. Shift of Fourier Coefficient Modulation

  24. Duality

  25. Periodic Convolution Both have Period N

  26. Periodic Convolution Both have Period N

  27. The Discrete Fourier Transform The Fourier Transform of Periodic Signals

  28. Periodicity Periodic Aperiodic Fourier Series Continuous-Time Fourier Transform Continuous & Aperiodic Discrete & Aperiodic Continuous Time DFT Discrete-Time Fourier Transform and z-Transform DFS Continuous & Periodic (2) Discrete Infinite Finite Duration Fourier Transforms of Periodic Signals Sampling Sampling

  29. n N 0 N n 0 Fourier Transforms of Periodic Signals

  30. The Discrete Fourier Transform Sampling of Fourier Transform

  31. n 0 N’1 z-plane z-plane > = < N N’ Equal Space Sampling of Fourier Transform

  32. Equal Space Sampling of Fourier Transform

  33. z-plane Equal Space Sampling of Fourier Transform

  34. N’=9 n 0 8 N=12 n 0 8 12 N=7 n 0 8 12 Example

  35. N’=9 n 0 8 N=12 n 0 8 12 N=7 n 0 8 12 Example Time-Domain Aliasing

  36. Time-Domain Aliasing vs. Frequency-Domain Aliasing • To avoid frequency-domain aliasing • Signal isbandlimited • Sampling ratein time-domain ishigh enough • To avoid time-domain aliasing • Sequence isfinite • Sampling interval(2/N) in frequency-domain issmall enough

  37. DFT vs. DFS • Use DFS to represent a finite-length sequence is call the DFT (Discrete Fourier Transform). • So, we represent the finite-duration sequence by a periodic sequence. One period of which is the finite-duration sequence that we wish to represent.

  38. The Discrete Fourier Transform Representation of Finite-Duration Sequences --- DFT

  39. Definition of DFT Synthesis Analysis

  40. Example

  41. Example

  42. The Discrete Fourier Transform Properties of the DFT

  43. Duration N1 n 0 N11 Duration N2 n 0 N21 Linearity

  44. n 0 N n 0 N n 0 N Circular Shift of a Sequence

  45. Circular Shift of a Sequence

  46. Duality

  47. Example Choose N=10 Re[x1(n)]= Re[X(n)] Re[X(k)] Im[x1(n)]= Im[X(n)] Im[X(k)] X1(k) = 10x((k))10

  48. Linear Convolution (Review)

  49. Circular Convolution both of length N

  50. n0 0 N 0 N 0 0 n0=2, N=5 Example

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