1. Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete Fourier Transform (DFT) The DTFT provides the frequency-domain ( ) representation for absolutely summable sequences. The z-transform provides a generalized frequency- domain ( ) representation for arbitrary sequences. Defined for infinite-length sequences. Functions of continuous variable ( or ). They are not numerically computable transform.

2. Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Content The Family of Fourier Transform The Discrete Fourier Series (DFS) The Discrete Fourier Transform (DFT) The Properties of DFT The Sampling Theorem in Frequency Domain Approximating to FT (FS) with DFT (DFS) Summary

3. Copyright © 2005. Shi Ping CUC The Family of Fourier Transform Introduction Fourier analysis is named after Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist. A signal can be either continuous or discrete, and it can be either periodic or aperiodic. The combination of these two features generates the four categories of Fourier Transform.

4. Copyright © 2005. Shi Ping CUC The Family of Fourier Transform Aperiodic-Continuous － Fourier Transform

5. Copyright © 2005. Shi Ping CUC The Family of Fourier Transform Periodic-Continuous － Fourier Series

6. Copyright © 2005. Shi Ping CUC The Family of Fourier Transform Aperiodic-Discrete － DTFT

7. Copyright © 2005. Shi Ping CUC The Family of Fourier Transform Periodic-Discrete － DFS (DFT)

8. Copyright © 2005. Shi Ping CUC The Family of Fourier Transform Summary Time functionFrequency function Continuous and AperiodicAperiodic and Continuous Continuous and Periodic( )Aperiodic and Discrete( ) Discrete ( ) and AperiodicPeriodic( ) and Continuous Discrete ( ) and Periodic ( ) Periodic( ) and Discrete( ) return

9. Copyright © 2005. Shi Ping CUC The Discrete Fourier Series (DFS) Definition Periodic time functions can be synthesized as a linear combination of complex exponentials whose frequencies are multiples (or harmonics) of the fundamental frequency Periodic continuous-time function Periodic discrete-time function fundamental frequency

10. Copyright © 2005. Shi Ping CUC The Discrete Fourier Series (DFS)

11. Copyright © 2005. Shi Ping CUC The Discrete Fourier Series (DFS) Because: The is a periodic sequence with fundamental period equal to N

12. Copyright © 2005. Shi Ping CUC The Discrete Fourier Series (DFS)

13. Copyright © 2005. Shi Ping CUC The Discrete Fourier Series (DFS) Relation to the z-transform The DFS represents N evenly spaced samples of the z-transform around the unit circle.

14. Copyright © 2005. Shi Ping CUC The Discrete Fourier Series (DFS) Relation to the DTFT The DFS is obtained by evenly sampling the DTFT at intervals. It is called frequency resolution and represents the sampling interval in the frequency domain.

15. Copyright © 2005. Shi Ping CUC The Discrete Fourier Series (DFS) jIm[z] Re[z] N=8 frequency resolution

16. Copyright © 2005. Shi Ping CUC The properties of DFS The Discrete Fourier Series (DFS) Linearity Shift of a sequence Modulation

17. Copyright © 2005. Shi Ping CUC The Discrete Fourier Series (DFS) Periodic convolution

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19. Copyright © 2005. Shi Ping CUC Introduction The Discrete Fourier Transform (DFT) The DFS provided us a mechanism for numerically computing the discrete-time Fourier transform. But most of the signals in practice are not periodic. They are likely to be of finite length. Theoretically, we can take care of this problem by defining a periodic signal whose primary shape is that of the finite length signal and then using the DFS on this periodic signal. Practically, we define a new transform called the Discrete Fourier Transform, which is the primary period of the DFS. This DFT is the ultimate numerically computable Fourier transform for arbitrary finite length sequences.

20. Copyright © 2005. Shi Ping CUC Finite-length sequence & periodic sequence The Discrete Fourier Transform (DFT) Finite-length sequence that has N samples periodic sequence with the period of N Window operation Periodic extension

21. Copyright © 2005. Shi Ping CUC The definition of DFT The Discrete Fourier Transform (DFT) return

22. Copyright © 2005. Shi Ping CUC The Properties of DFT Linearity N3-point DFT, N3=max(N1,N2) Circular shift of a sequence Circular shift in the frequency domain

23. Copyright © 2005. Shi Ping CUC The Properties of DFT The sum of a sequence The first sample of sequence

24. Copyright © 2005. Shi Ping CUC The Properties of DFT Circular convolution N N N N Multiplication

25. Copyright © 2005. Shi Ping CUC The Properties of DFT Circular correlation Linear correlation Circular correlation

26. Copyright © 2005. Shi Ping CUC The Properties of DFT

27. Copyright © 2005. Shi Ping CUC The Properties of DFT Parseval’s theorem

28. Copyright © 2005. Shi Ping CUC The Properties of DFT Conjugate symmetry properties of DFT and Let be a N-point sequence It can be proved that

29. Copyright © 2005. Shi Ping CUC The Properties of DFT Circular conjugate symmetric component Circular conjugate antisymmetric component

30. Copyright © 2005. Shi Ping CUC The Properties of DFT

31. Copyright © 2005. Shi Ping CUC The Properties of DFT and

32. Copyright © 2005. Shi Ping CUC The Properties of DFT

33. Copyright © 2005. Shi Ping CUC The Properties of DFT Circular even sequences Circular odd sequences

34. Copyright © 2005. Shi Ping CUC The Properties of DFT Conjugate sequences

35. Copyright © 2005. Shi Ping CUC The Properties of DFT Complex-value sequences

36. Copyright © 2005. Shi Ping CUC The Properties of DFT

37. Copyright © 2005. Shi Ping CUC The Properties of DFT Real-value sequences Imaginary-value sequences

38. Copyright © 2005. Shi Ping CUC The Properties of DFT Summary example

39. Copyright © 2005. Shi Ping CUC The Properties of DFT Linear convolution & circular convolution Linear convolution N 1 point sequence, 0≤n≤ N 1 -1 N 2 point sequence, 0≤n≤ N 2 -1 L point sequence, L= N 1 +N 2 -1

40. Copyright © 2005. Shi Ping CUC The Properties of DFT Circular convolution We have to make both and L-point sequences by padding an appropriate number of zeros in order to make L point circular convolution.

41. Copyright © 2005. Shi Ping CUC The Properties of DFT L

42. Copyright © 2005. Shi Ping CUC The Properties of DFT L return

43. Copyright © 2005. Shi Ping CUC The Sampling Theorem in Frequency Domain Sampling in frequency domain

44. Copyright © 2005. Shi Ping CUC The Sampling Theorem in Frequency Domain Frequency Sampling Theorem For M point finite duration sequence, if the frequency sampling number N satisfy: then

45. Copyright © 2005. Shi Ping CUC The Sampling Theorem in Frequency Domain Interpolation formula of

46. Copyright © 2005. Shi Ping CUC The Sampling Theorem in Frequency Domain Interpolation function

47. Copyright © 2005. Shi Ping CUC The Sampling Theorem in Frequency Domain Interpolation function Interpolation formula of return

48. Copyright © 2005. Shi Ping CUC Approximating to FT (FS) with DFT (DFS) Approximating to FT of continuous-time aperiodic signal with DFT CTFT

49. Copyright © 2005. Shi Ping CUC Sampling in time domain Approximating to FT (FS) with DFT (DFS)

50. Copyright © 2005. Shi Ping CUC Truncation in time domain Approximating to FT (FS) with DFT (DFS)

51. Copyright © 2005. Shi Ping CUC Sampling in frequency domain Approximating to FT (FS) with DFT (DFS)

52. Copyright © 2005. Shi Ping CUC demo Approximating to FT (FS) with DFT (DFS)

53. Copyright © 2005. Shi Ping CUC Approximating to FS of continuous-time periodic signal with DFS Approximating to FT (FS) with DFT (DFS)

54. Copyright © 2005. Shi Ping CUC Sampling in time domain Approximating to FT (FS) with DFT (DFS)

55. Copyright © 2005. Shi Ping CUC Truncating in frequency domain Approximating to FT (FS) with DFT (DFS)

56. Copyright © 2005. Shi Ping CUC Some problems Aliasing Otherwise, the aliasing will occur in frequency domain Sampling in time domain: Sampling in frequency domain: Period in time domain Frequency resolution and is contradictory Approximating to FT (FS) with DFT (DFS)

57. Copyright © 2005. Shi Ping CUC Spectrum leakage Spectrum extension (leakage) Spectrum aliasing Approximating to FT (FS) with DFT (DFS) demo

58. Copyright © 2005. Shi Ping CUC Fence effect Frequency resolution demo Approximating to FT (FS) with DFT (DFS)

59. Copyright © 2005. Shi Ping CUC Comments return demo  Zero-padding is an operation in which more zeros are appended to the original sequence. It can provides closely spaced samples of the DFT of the original sequence.  The zero-padding gives us a high-density spectrum and provides a better displayed version for plotting. But it does not give us a high-resolution spectrum because no new information is added.  To get a high-resolution spectrum, one has to obtain more data from the experiment or observation. example Approximating to FT (FS) with DFT (DFS)

60. Copyright © 2005. Shi Ping CUC Summary return The frequency representations of x(n) Time sequence z-transform of x(n) Complex frequency domain DTFT of x(n) Frequency domain DFT of x(n) Discrete frequency domain ZT DTFT DFT interpolation

61. Copyright © 2005. Shi Ping CUC Illustration of the four Fourier transforms Discrete Fourier Series Signals that are discrete and periodic DTFT Signals that are discrete and aperiodic Fourier Series Signals that are continuous and periodic Fourier Transform Signals that are continuous and aperiodic

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70. Copyright © 2005. Shi Ping CUC return N-point real-value sequences

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83. Copyright © 2005. Shi Ping CUC return Suppose Determine Solution