Understanding Fourier Analysis: Series, Transform, and Application

1. Chapter 7: Fourier Analysis Fourier Analysis = Series + Transform ◎ Fourier Series -- A periodic (T) function f(x) can be written as the sum of sines and cosines of varying amplitudes and frequencies

2. ○ Some function is formed by a finite number of sinuous functions

3. Some function requires an infinite number of sinuous functions to compose

4. Spectrum The spectrum of a periodic function is discrete, consisting of components at dc, , and its multiples, e.g., For non-periodic functions, i.e., The spectrum of the function is continuous

5. ○ In complex form: Compare with

6. Euler’s formula: 7-6

7. 7-7

8. 7-8

9. Continuous case

10. Discrete case: ◎ Fourier Transform • Vector-Matrix form

12. Let 7-12

13. 。Example: f = {1,2,3,4}. Then, N = 4,

15. 7-15

16. Let ○ Inverse DFT

17. 。Example:

18. ◎ Properties ○ Linearity: Show:

19. Application: Noise removal f’ = f + n, n: additive noise It may be easier to identify than n.

20. ○ Scaling: Show: Assignment : Show

21. ○ Periodicity:

22. 7-22

23. ○ Shifting:

24. 。 Example:

25. Compared with: 7-25

27. ◎ Convolution theorem: Convolution: ◎ Correlation theorem Correlation: * : conjugate 7-27

28. ◎ Fast Fourier Transform (FFT) -- Successive doubling method Let Assume Let N = 2M. 7-28

29. 7-29

30. Let --------- (B) Consider 7-30

31. 7-31

32. ---- (C) 7-32

33. ○ Analysis : The Fourier sequence F(u), u = 0 , … , N-1 of f(x), x = 0 , … , N-1can be formed from sequences u = 0 , …… , M-1 Recursively divide F(u) and F(u+M), eventually each contains one element F(u), i.e., u = 0, and F(u) = f(x). 7-33

34. 7-34

35. ○ Example: Input { f(0), f(1), ……, f(7) } Computing needs { f(0), f(2), f(4), f(6) } Computing needs { f(1), f(3), f(5), f(7) } { f(0), f(4)},{ f(2), f(6) } { f(0), f(2), f(4), f(6) } even odd { f(1), f(5)}, {f(3), f(7) } { f(1), f(3), f(5), f(7) } 7-35

36. { f(2)},{ f(6)} { f(0), f(4)} { f(0)},{ f(4)} { f(2), f(6)} even odd even odd { f(3)},{ f(7)} { f(1), f(5)} { f(1)},{ f(5)} { f(3), f(7)} Reorder the input sequence into {f(0), f(4), f(2), f(6), f(1), f(5), f(3), f(7)} ＊Bit-Reversal Rule 7-36

37. ○ FFT Algorithm

38. 7-38

39. 。 Time complexity : the length of the input sequence FT: FFT: Times of speed increasing: N FT FFT Ratio 4 16 8 2.0 8 84 24 2.67 16 256 64 4.0 32 1024 160 6.4 64 4096 384 10.67 128 16384 896 18.3 256 65536 2048 32.0 512 262144 4608 56.9 1024 1048576 10240 102.4

40. ○ Inverse FFT ← Given ← compute i. Input into FFT. The output is ii. Taking the complex conjugate and multiplying by N , yields the f(x)

41. ◎ 2D Fourier Transform ○ FT: IFT:

42. ◎ Properties ○ Filtering: every F(u,v) is obtained by multiplying every f(x,y) by a fixed value and adding up the results. DFT can be considered as a linear filtering ○ DC coefficient:

43. ○ Separability:

44. F(u,v) = F*(-u,-v) ○ Conjugate Symmetry:

45. ○ Shifting

46. ○ Rotation Polor coordinates:

47. ○ Display: effect of log operation

49. ◎ Image Transform

50. ◎ Filtering in Frequency Domain ○ Low pass filtering I FT m IFT