1. Presenter: Hong Wen-Chih 2015/8/11

2. Outline Introduction Definition of fractional fourier transform Linear canonical transform Implementation of FRFT/LCT The Direct Computation DFT-like Method Chirp Convolution Method Discrete fractional fourier transform Conclusion and future work 2015/8/12

3. Introduction Definition of fourier transform: Definition of inverse fourier transform: 2015/8/13

4. Introduction In time-frequency representation Fourier transform: rotation π/2+2k π Inverse fourier transform: rotation -π/2+2k π Parity operator: rotation –π+2k π Identity operator: rotation 2k π And what if angle is not multiple of π/2 ? 2015/8/14

5. Introduction Time-frequency plane and a set of coordinates rotated by angle α relative to the original coordinates. 2015/8/15

6. Fractional Fourier Transform Generalization of FT use to represent FRFT The properties of FRFT: Zero rotation: Consistency with Fourier transform: Additivity of rotations: 2π rotation: Note: do four times FT will equal to do nothing 2015/8/16

7. Fractional Fourier Transform Definition: Note: when α is multiple of π, FRFTs degenerate into parity and identity operator 2015/8/17

8. Linear Canonical Transform Generalization of FRFT Definition: when b≠0 when b=0 a constraint: must be satisfied. 2015/8/18

9. Linear Canonical Transform Additivity property: where Reversibility property: where 2015/8/19

10. Linear Canonical Transform Special cases of LCT: {a, b, c, d} = {0, 1,  1, 0}: {a, b, c, d} = {0,  1, 1, 0}: {a, b, c, d} = {cos , sin ,  sin , cos  }: {a, b, c, d} = {1, z/2 , 0, 1} : LCT becomes the 1-D Fresnel transform {a, b, c, d} = {1, 0, , 1} : LCT becomes the chirp multiplication operation {a, b, c, d} = { , 0, 0,   1 }: LCT becomes the scaling operation. 2015/8/110

11. Implementation of FRFT/LCT Conventional Fourier transform Clear physical meaning fast algorithm (FFT) Complexity : (N/2)  log 2 N LCT and FRFT The Direct Computation DFT-like Method Chirp Convolution Method 2015/8/111

12. Implementation of FRFT/LCT The Direct Computation directly sample input and output 2015/8/112

13. Implementation of FRFT/LCT The Direct Computation Easy to design No constraint expect for Drawbacks lose many of the important properties not be unitary no additivity Not be reversible lack of closed form properties applications are very limited 2015/8/113

14. Implementation of FRFT/LCT Chirp Convolution Method Sample input and output as and 2015/8/114

15. Implementation of FRFT/LCT Chirp Convolution Method implement by 2 chirp multiplications 1 chirp convolution complexity 2P (required for 2 chirp multiplications) + P  log 2 P (required for 2 DFTs)  P  log 2 P (P = 2M+1 = the number of sampling points) Note: 1 chirp convolution needs to 2DFTs 2015/8/115

16. Implementation of FRFT/LCT DFT-like Method constraint on the product of  t and  u (chirp multi.) (FT) (scaling) (chirp multi.) 2015/8/116

17. Implementation of FRFT/LCT DFT-like Method Chirp multiplication: Scaling: Fourier transform: Chirp multiplication: 2015/8/117

18. Implementation of FRFT/LCT DFT-like Method For 3 rd step Sample the input t and output u as p  t and q  u 2015/8/118

19. Implementation of FRFT/LCT DFT-like Method Complexity 2 M-points multiplication operations 1 DFT 2P (two multiplication operations) + (P/2)  log 2 P (one DFT)  (P/2)  log 2 P 2015/8/119

20. Implementation of FRFT/LCT Compare Complexity Chirp convolution method: P  log 2 P (2-DFT) DFT-like Method: (P/2)  log 2 P (1-DFT) DFT: (P/2)  log 2 P (1-DFT) trade-off: chirp. Method: sampling interval is FREE to choice DFT-like method: some constraint for the sampling intervals 2015/8/120

21. Discrete fractional fourier transform Direct form of DFRFT Improved sampling type DFRFT Linear combination type DFRFT Eigenvectors decomposition type DFRFT Group theory type DFRFT Impulse train type DFRFT Closed form DFRFT 2015/8/121

22. Discrete fractional fourier transform Direct form of DFRFT simplest way sampling the continuous FRFT and computing it directly 2015/8/122

23. Discrete fractional fourier transform Improved sampling type DFRFT By Ozaktas, Arikan Sample the continuous FRFT properly Similar to the continuous case Fast algorithm Kernel will not be orthogonal and additive Many constraints 2015/8/123

24. Discrete fractional fourier transform Linear combination type DFRFT By Santhanam, McClellan Four bases: DFT IDFT Identity Time reverse 2015/8/124

25. Discrete fractional fourier transform Linear combination type DFRFT transform matrix is orthogonal additivity property reversibility property very similar to the conventional DFT or the identity operation lose the important characteristic of ‘fractionalization’ 2015/8/125

26. Discrete fractional fourier transform Linear combination type DFRFT DFRFT of the rectangle window function for various angles : (top left) α= 0:01, (top right) α = 0:05, (middle left) α = 0:2, (middle right) α = 0:4, (bottom left) α =π/4, (bottom right) α =π/2. 2015/8/126

27. (a) = 0.01 (b) = 0.05 (c) = 0.2 (d) = 0.4 (e) = π/4 (f) = π/2 2015/8/127

28. Discrete fractional fourier transform Eigenvectors decomposition type DFRFT DFT : F=F r – j F i Search eigenvectors set for N-points DFT 2015/8/128

29. Discrete fractional fourier transform Eigenvectors decomposition type DFRFT Good in removing chirp noise By Pei, Tseng, Yeh, Shyu cf. : DRHT can be 2015/8/129

30. Discrete fractional fourier transform Eigenvectors decomposition type DFRFT DFRFT of the rectangle window function for various angles : (top left) α= 0:01, (top right) α = 0:05, (middle left) α = 0:2, (middle right) α = 0:4, (bottom left) α =π/4, (bottom right) α =π/2 2015/8/130

31. Discrete fractional fourier transform Group theory type DFRFT By Richman, Parks Multiplication of DFT and the periodic chirps Rotation property on the Wigner distribution Additivity and reversible property Some specified angles Number of points N is prime 2015/8/131

32. Discrete fractional fourier transform Impulse train type DFRFT By Arikan, Kutay, Ozaktas, Akdemir special case of the continuous FRFT f(t) is a periodic, equal spaced impulse train N =  2, tanα = L/M many properties of the FRFT exists many constraints not be defined for all values of  2015/8/132

33. Discrete fractional fourier transform Closed form DFRFT By Pei, Ding further improvement of the sampling type of DFRFT Two types digital implementing of the continuous FRFT practical applications about digital signal processing 2015/8/133

34. Discrete fractional fourier transform Type I Closed form DFRFT Sample input f(t) and output F a (u) Then Matrix form: 2015/8/134

35. Discrete fractional fourier transform Type I Closed form DFRFT Constraint: 2015/8/135

36. Discrete fractional fourier transform Type I Closed form DFRFT and choose S = sgn(sin  ) =  1 2015/8/136

37. Discrete fractional fourier transform Type I Closed form DFRFT when   2D  +(0,  ), D is integer (i.e., sin  > 0) when   2D  +( , 0), D is integer (i.e., sin  < 0) 2015/8/137

38. Discrete fractional fourier transform Type I Closed form DFRFT Some properties 1 2 and 3 Conjugation property: if y(n) is real 4 No additivity property 5 When is small, and also become very small 6 Complexity 2015/8/138

39. Discrete fractional fourier transform Type II Closed form DFRFT Derive from transform matrix of the DLCT of type 1 Type I has too many parameters Simplify the type I Set p = (d/b)  u 2, q = (a/b)  t 2 2015/8/139

40. Discrete fractional fourier transform Type II Closed form DFRFT from  t  u = 2  |b|/(2M+1), we find a, d : any real value No constraint for p, q, and p, q can be any real value. 3 parameters p, q, b without any constraint, Free dimension of 3 (in fact near to 2) 2015/8/140

41. Discrete fractional fourier transform Type II Closed form DFRFT p=0: DLCT becomes a CHIRP multiplication operation followed by a DFT q=0: DLCT becomes a DFT followed by a chirp multiplication p=q: F (p,p,s) (m,n) will be a symmetry matrix (i.e., F (p,p,s) (m,n) = F (p,p,s) (n,m)) 2015/8/141

42. Discrete fractional fourier transform Type II Closed form DFRFT 2P+(P/2)  log 2 P No additive property Convertible 2015/8/142

43. Discrete fractional fourier transform The relations between the DLCT of type 2 and its special cases DFRFT of type 2 p = q, s =  1 DFRFT of type 1 p = cot  u 2, q = cot  t 2, s = sgn(sin  ) DLCT of type 1 p = d/b  u 2, q = a/b  t 2, s = sgn(b) DFT, IDFT p = q = 0, s = 1 for DFT, s =  1 for DFT 2015/8/143

44. Discrete fractional fourier transform Comparison of Closed Form DFRFT and DLCT with Other Types of DFRFT DirectlyImprovedLinearEigenfxs.GroupImpulseProposed Reversible   *  Closed form    Similarity      Complexity P2P2 P  log 2 P+ 2P P 2 /2 P  log 2 P+ 2P +2P FFT  2 FFT1 FFT  2 FFT 1 FFT ConstraintsLessMiddleUnableLessMuch Less All orders    PropertiesLessMiddle LessMany Adv./Cvt.NoConvt.Additive Convt. DSP   2015/8/144

45. Conclusions and future work Generalization of the Fourier transform Applications of the conventional FT can also be the applications of FRFT and LCT More flexible Useful tools for signal processing 2015/8/145

46. References [1] V. Namias, ‘The fractional order Fourier transform and its application to quantum mechanics’, J. Inst. Maths Applies. vol. 25, p. 241-265, 1980. [2] L. B. Almeida, ‘The fractional Fourier transform and time-frequency representations’. IEEE Trans. Signal Processing, vol. 42, no. 11, p. 3084-3091, Nov. 1994. [3] J. J. Ding, Research of Fractional Fourier Transform and Linear Canonical Transform, Ph. D thesis, National Taiwan Univ., Taipei, Taiwan, R.O.C, 1997 [4] H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, 1st Ed., John Wiley & Sons, New York, 2000. 2015/8/146

47. References [5] S. C. Pei, C. C. Tseng, M. H. Yeh, and J. J. Shyu,’ Discrete fractional Hartley and Fourier transform’, IEEE Trans Circ Syst II, vol. 45, no. 6, p. 665–675, Jun. 1998. [6] H. M. Ozaktas, O. Arikan, ‘Digital computation of the fractional Fourier transform’, IEEE Trans. On Signal Proc., vol. 44, no. 9, p.2141-2150, Sep. 1996. [7] B. Santhanam and J. H. McClellan, “The DRFT—A rotation in time frequency space,” in Proc. ICASSP, May 1995, pp. 921–924. [8] J. H. McClellan and T. W. Parks, “Eigenvalue and eigenvector decomposition of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust., vol. AU-20, pp. 66–74, Mar. 1972. 2015/8/147