Insight into Discrete Fourier Transform: Theory, Applications, and Properties

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