Presenter Hong WenChih Fractional fourier transform 20201125 1

Presenter: Hong Wen-Chih Fractional fourier transform 2020/11/25 1

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 2020/11/25 2

Introduction Definition of fourier transform: Definition of inverse fourier transform: 2020/11/25 3

Introduction In time-frequency representation Fourier transform: rotation π/2+2 k π Inverse fourier transform: rotation -π/2+2 k π Parity operator: rotation –π+2 k π Identity operator: rotation 2 k π And what if angle is not multiple of π/2 ? 2020/11/25 4

. Introduction Time-frequency plane and a set of coordinates rotated by angle α relative to the original coordinates 2020/11/25 5

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 2020/11/25 6

Fractional Fourier Transform Definition: Note: when α is multiple of π, FRFTs degenerate into parity and identity operator 2020/11/25 7

Linear Canonical Transform Generalization of FRFT Definition: when b≠ 0 a constraint: 2020/11/25 when b=0 must be satisfied. 8

Linear Canonical Transform Additivity property: where Reversibility property: where 2020/11/25 9

Linear Canonical Transform Special cases of LCT: {a, b, c, d} = {0, 1, 0}: {a, b, c, d} = {0, 1, 1, 0}: {a, b, c, d} = {cos , 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. 2020/11/25 10

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 2020/11/25 11

Implementation of FRFT/LCT The Direct Computation directly sample input and output 2020/11/25 12

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 2020/11/25 13

Implementation of FRFT/LCT Chirp Convolution Method Sample input and output as 2020/11/25 and 14

Implementation of FRFT/LCT Chirp Convolution Method implement by 2 chirp multiplications 1 chirp convolution complexity 2 P (required for 2 chirp multiplications) + P log 2 P (required for 2 DFTs) P log 2 P (P = 2 M+1 = the number of sampling points) Note: 1 chirp convolution needs to 2 DFTs 2020/11/25 15

Implementation of FRFT/LCT DFT-like Method constraint on the product of t and u (chirp multi. ) (FT) (scaling) (chirp multi. ) 2020/11/25 16

Implementation of FRFT/LCT DFT-like Method Chirp multiplication: Scaling: Fourier transform: Chirp multiplication: 2020/11/25 17

Implementation of FRFT/LCT DFT-like Method For 3 rd step Sample the input t and output u as p t and q u 2020/11/25 18

Implementation of FRFT/LCT DFT-like Method Complexity 2 M-points multiplication operations 1 DFT 2 P (two multiplication operations) + (P/2) log 2 P (one DFT) (P/2) log 2 P 2020/11/25 19

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 2020/11/25 20

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 2020/11/25 21

Discrete fractional fourier transform Direct form of DFRFT simplest way sampling the continuous FRFT and computing it directly 2020/11/25 22

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 2020/11/25 23

Discrete fractional fourier transform Linear combination type DFRFT By Santhanam, Mc. Clellan Four bases: DFT Identity Time reverse 2020/11/25 24

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’ 2020/11/25 25

Discrete fractional fourier transform Linear combination type 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. 2020/11/25 26

(a) = 0. 01 (b) = 0. 05 (c) = 0. 2 (d) = 0. 4 (e) = π/4 (f) = π/2 2020/11/25 27

Discrete fractional fourier transform Eigenvectors decomposition type DFRFT DFT : F=Fr – j Fi Search eigenvectors set for N-points DFT 2020/11/25 28

Discrete fractional fourier transform Eigenvectors decomposition type DFRFT Good in removing chirp noise By Pei, Tseng, Yeh, Shyu cf. : DRHT can be 2020/11/25 29

Discrete fractional fourier transform Eigenvectors decomposition type 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 2020/11/25 30

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 2020/11/25 31

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 2020/11/25 32

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 2020/11/25 33

Discrete fractional fourier transform Type I Closed form DFRFT Sample input f(t) and output Fa(u) Then Matrix form: 2020/11/25 34

Discrete fractional fourier transform Type I Closed form DFRFT Constraint: 2020/11/25 35

Discrete fractional fourier transform Type I Closed form DFRFT and choose S = sgn(sin ) = 1 2020/11/25 36

Discrete fractional fourier transform Type I Closed form DFRFT when 2 D +(0, ), D is integer (i. e. , sin > 0) when 2 D +( , 0), D is integer (i. e. , sin < 0) 2020/11/25 37

Discrete fractional fourier transform Type I Closed form DFRFT Some properties 1 2 3 4 5 6 2020/11/25 and Conjugation property: No additivity property When is small, and Complexity if y(n) is real also become very small 38

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 2020/11/25 39

Discrete fractional fourier transform Type II Closed form DFRFT from t u = 2 |b|/(2 M+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) 2020/11/25 40

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)) 2020/11/25 41

Discrete fractional fourier transform Type II Closed form DFRFT 2 P+(P/2) log 2 P No additive property Convertible 2020/11/25 42

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 2020/11/25 43

Discrete fractional fourier transform Comparison of Closed Form DFRFT and DLCT with Other Types of DFRFT Directly Improved Linear Eigenfxs. Group Impulse Proposed Reversible * Closed form Similarity Complexity P 2 P log 2 P+ P 2/2 P log 2 P+ 2 P 2 P +2 P 2 P FFT 2 FFT 1 FFT Constraints Less Middle Unable Less Much Less All orders Properties Less Middle Less Many Adv. /Cvt. No Convt. Additive Convt. DSP 2020/11/25 44

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 2020/11/25 45

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, 1 st Ed. , John Wiley & Sons, New York, 2000. 2020/11/25 46

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. Mc. Clellan, “The DRFT—A rotation in time frequency space, ” in Proc. ICASSP, May 1995, pp. 921– 924. [8] J. H. Mc. Clellan 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. 2020/11/25 47