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# Chapter 8 The Discrete Fourier Transform

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1. Chapter 8 The Discrete Fourier Transform Biomedical Signal processing Zhongguo Liu Biomedical Engineering School of Control Science and Engineering, Shandong University 1 Zhongguo Liu_Biomedical Engineering_Shandong Univ.

2. Chapter 8 The Discrete Fourier Transform • 8.0 Introduction • 8.1 Representation of Periodic Sequence: the Discrete Fourier Series • 8.2 Properties of the Discrete Fourier Series • 8.3 The Fourier Transform of Periodic Signal • 8.4 Sampling the Fourier Transform • 8.5 Fourier Representation of Finite-Duration Sequence: the Discrete Fourier Transform • 8.6 Properties of the Discrete Fourier Transform • 8.7 Linear Convolution using the Discrete Fourier Transform

3. Filter Design Techniques 8.0 Introduction

4. 8.0 Introduction • Discrete Fourier Transform (DFT)for finite duration sequence • DFT is a sequence rather than a function of a continuous variable • DFT corresponds to sample, equally spaced in frequency, of the Fourier transform of the signal.

5. 8.0 Introduction • The relationship between periodic sequence and finite-length sequences： • The Fourier series representation of the periodic sequence corresponds to the DFT of the finite-length sequence.

6. Given a periodic sequence with period N so that 8.1 Representation of Periodic Sequence: the Discrete Fourier Series • The Fourier series representation can be written as • Fourier series representation of continuous-time periodic signals require infinite many complex exponentials • Not that for discrete-time periodic signals we have

7. 8.1 Representation of Periodic Sequence: the Discrete Fourier Series • Due to the periodicity of the complex exponential we only need N exponentials for discrete time Fourier series • No need

8. To obtain the Fourier series coefficients we multiply both sides by for 0nN-1 and then sum both the sides , we obtain Discrete Fourier Series Pair • A periodic sequence in terms of Fourier series coefficients

9. Discrete Fourier Series Pair Problem 8.51, HW

10. The Fourier series coefficients of is 8.1 Representation of Periodic Sequence: the Discrete Fourier Series • a periodic sequence with period N,

11. The sequence is periodic with period N 8.1 Representation of Periodic Sequence: the Discrete Fourier Series

12. Synthesis equation: Discrete Fourier Series (DFS) • Let • Analysis equation:

13. N points …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 Ex. 8.1 DFS of a impulse train • Consider the periodic impulse train n

14. N points k …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 Ex. 8.1 DFS of a impulse train

15. N points 1 k …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 N points 1 n …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1

16. - N 1 ~ N points [ ] [ ] ~ å = kn X k x n W N N = n 0 …… …… …… -N N 1 2 -2 -1 0 Example 8.2 Duality in the Discrete Fourier Series • The discrete Fourier series coefficients is the periodic impulse train

17. N points N …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 N points 1 …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 k n

18. N points 1 k …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 N points N N points …… …… 1 …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 N points 1 n …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1

19. Example 8.3 The Discrete Fourier Series of a Periodic Rectangular Pulse Train • Periodic sequence with period N=10 1

20. magnitude phase

21. magnitude phase

22. 8.2 Properties of the Discrete Fourier Series • Linearity: two periodic sequence, both with period N

23. 8.2 Properties of the Discrete Fourier Series • Shift of a sequence Problem 8.52, HW

24. N k …… 2 N-1 0 1 1 1 n 1 …… 2 N-1 0 1 k …… N-1 2 0 1 n …… 2 N-1 0 1 8.2 Properties of the Discrete Fourier Series • Duality

25. 8.2.4 Symmetry Problem 8.53, HW

26. and are two periodic sequences, each with period N and with discrete Fourier series and 8.2.5 Periodic Convolution

27. The sum is over the finite interval • The value of in the interval repeat periodically for m outside of that interval 8.2.5 Periodic Convolution

28. Example 8.4 Periodic Convolution

29. 8.2.5 Periodic Convolution

30. The Fourier series coefficients of is 8.1 Representation of Periodic Sequence: the Discrete Fourier Series • a periodic sequence with period N, Review

31. Synthesis equation: Discrete Fourier Series (DFS) • Let • Analysis equation:

32. 8.2 Properties of the Discrete Fourier Series • Shift of a sequence

33. N k …… 2 N-1 0 1 1 1 n 1 …… 2 N-1 0 1 k …… N-1 2 0 1 n …… 2 N-1 0 1 8.2 Properties of the Discrete Fourier Series • Duality

34. 8.2.5 Periodic Convolution

35. Example 8.4 Periodic Convolution

36. 8.3 The Fourier Transform of Periodic Signal • Periodic sequences are neither absolutely summable nor square summable, hence they don’t have a strict Fourier Transform

37. 8.3 The Fourier Transform of Periodic Signal • We can represent Periodic sequences as sums of complex exponentials: DFS • We can combine DFS and Fourier transform • Fourier transform of periodic sequences • Periodic impulse train with values proportional to DFS coefficients

38. 8.3 The Fourier Transform of Periodic Signal This is periodic with 2 since DFS is periodic • The inverse transform can be written as

39. N points 1 • The DFS was calculated previously to be …… …… …… N points -N N 1 2 -2 -1 0 1 …… …… …… n 2 N-1 N -N -2 -1 0 1 • Therefore the Fourier transform is Ex. 8.5 Fourier Transform of a periodic impulse train • Consider the periodic impulse train

40. 1 …… …… -N N 1 2 -2 -1 0 Relation between Finite-length and Periodic Signals • Consider finite length signal x[n] spanning from 0 to N-1 • Convolve with periodic impulse train • The Fourier transform of the periodic sequence is

41. Relation between Finite-length and Periodic Signals • This implies that • DFS coefficients of a periodic signal can be thought as equally spaced samples of the Fourier transform of one period

42. If is periodic with period N, the DFS are • If for and otherwise Relation between Finite-length and Periodic Signals then

43. Ex. 8.5 Relation between FS coefficients and FT • Consider the sequence • The Fourier transform

44. Ex. 8.5 Relation between FS coefficients and FT • The Fourier transform • The DFS coefficients • Consider the sequence

45. The Fourier transform • The DFS coefficients Ex. 8.5 Relation between FS coefficients and FT • Consider the sequence

46. Consider an aperiodic sequence with Fourier transform ,and assume that a sequence is obtained by sampling at frequency • is Fourier series coefficients of periodic sequence 8.4 Sampling the Fourier Transform

47. Sampling the Fourier Transform

48. N points 1 …… …… …… -N N 1 2 -2 -1 0 Sampling the Fourier Transform

49. Sampling the Fourier Transform

50. Sampling the Fourier Transform • Samples of the DTFT of an aperiodic sequence • can be thought of as DFS coefficients • of a periodic sequence • obtained through summing periodic replicas of original sequence • If the original sequence is of finite length, • and we take sufficient number of samples of its DTFT, • then the original sequence can be recovered by