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Fractional fourier transform. Presenter: Hong Wen-Chih. 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

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fractional fourier transform

Fractional fourier transform

Presenter:

Hong Wen-Chih

outline
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
introduction
Introduction
  • Definition of fourier transform:
  • Definition of inverse fourier transform:
introduction4
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 ?

introduction5
Introduction

.

Time-frequency plane and a set of coordinates

rotated by angle α relative to the original coordinates

fractional fourier transform6
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

fractional fourier transform7
Fractional Fourier Transform
  • Definition:
  • Note: when α is multiple of π, FRFTs degenerate into parity and identity operator
linear canonical transform
Linear Canonical Transform
  • Generalization of FRFT
  • Definition:

when b≠0

when b=0

  • a constraint: must be satisfied.
linear canonical transform9
Linear Canonical Transform
  • Additivity property:

where

  • Reversibility property:

where

linear canonical transform10
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.
implementation of frft lct
Implementation of FRFT/LCT
  • Conventional Fourier transform
    • Clear physical meaning
    • fast algorithm (FFT)
    • Complexity : (N/2)log2N
  • LCT and FRFT
    • The Direct Computation
    • DFT-like Method
    • Chirp Convolution Method
implementation of frft lct12
Implementation of FRFT/LCT
  • The Direct Computation
  • directly sample input and output
implementation of frft lct13
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
implementation of frft lct14
Implementation of FRFT/LCT
  • Chirp Convolution Method
    • Sample input and output as and
implementation of frft lct15
Implementation of FRFT/LCT
  • Chirp Convolution Method
  • implement by
    • 2 chirp multiplications
    • 1 chirp convolution
  • complexity
    • 2P (required for 2 chirp multiplications) + Plog2P(required for 2 DFTs) Plog2P(P = 2M+1 = the number of sampling points)
  • Note: 1 chirp convolution needs to 2DFTs
implementation of frft lct16
Implementation of FRFT/LCT
  • DFT-like Method
    • constraint on the product of t and u

(chirp multi.) (FT) (scaling) (chirp multi.)

implementation of frft lct17
Implementation of FRFT/LCT
  • DFT-like Method
  • Chirp multiplication:
  • Scaling:
  • Fourier transform:
  • Chirp multiplication:
implementation of frft lct18
Implementation of FRFT/LCT
  • DFT-like Method
    • For 3rd step
      • Sample the input t and output u as pt and qu
implementation of frft lct19
Implementation of FRFT/LCT
  • DFT-like Method
  • Complexity
    • 2 M-points multiplication operations
    • 1 DFT
    • 2P (two multiplication operations) + (P/2)log2P(one DFT) (P/2)log2P
implementation of frft lct20
Implementation of FRFT/LCT
  • Compare
    • Complexity
      • Chirp convolution method:Plog2P (2-DFT)
      • DFT-like Method: (P/2)log2P (1-DFT)
      • DFT: (P/2)log2P (1-DFT)
    • trade-off:
      • chirp. Method: sampling interval is FREE to choice
      • DFT-like method:some constraint for the sampling intervals
discrete fractional fourier transform
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
discrete fractional fourier transform22
Discrete fractional fourier transform
  • Direct form of DFRFT
    • simplest way
    • sampling the continuous FRFT and computing it directly
discrete fractional fourier transform23
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
discrete fractional fourier transform24
Discrete fractional fourier transform
  • Linear combination type DFRFT
  • By Santhanam, McClellan
    • Four bases:
      • DFT
      • IDFT
      • Identity
      • Time reverse
discrete fractional fourier transform25
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’
discrete fractional fourier transform26
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.
slide27

(a) = 0.01

  • (b) = 0.05
  • (c) = 0.2
  • (d) = 0.4
  • (e) = π/4
  • (f) = π/2
discrete fractional fourier transform28
Discrete fractional fourier transform
  • Eigenvectors decomposition type DFRFT
    • DFT : F=Fr – j Fi
    • Search eigenvectors set for N-points DFT
discrete fractional fourier transform29
Discrete fractional fourier transform
  • Eigenvectors decomposition type DFRFT
    • Good in removing chirp noise
    • By Pei, Tseng, Yeh, Shyu
    • cf. : DRHT can be
discrete fractional fourier transform30
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
discrete fractional fourier transform31
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
discrete fractional fourier transform32
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 
discrete fractional fourier transform33
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
discrete fractional fourier transform34
Discrete fractional fourier transform
  • Type I Closed form DFRFT
    • Sample input f(t) and output Fa(u)
    • Then
    • Matrix form:
discrete fractional fourier transform35
Discrete fractional fourier transform
  • Type I Closed form DFRFT
    • Constraint:
discrete fractional fourier transform36
Discrete fractional fourier transform
  • Type I Closed form DFRFT
    • and
    • choose S = sgn(sin) = 1
discrete fractional fourier transform37
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)

discrete fractional fourier transform38
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
discrete fractional fourier transform39
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)u2, q = (a/b)t2
discrete fractional fourier transform40
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)
discrete fractional fourier transform41
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))
discrete fractional fourier transform42
Discrete fractional fourier transform
  • Type II Closed form DFRFT
    • 2P+(P/2)log2P
    • No additive property
    • Convertible
discrete fractional fourier transform43
Discrete fractional fourier transform
  • The relations between the DLCT of type 2 and its special cases
discrete fractional fourier transform44
Discrete fractional fourier transform
  • Comparison of Closed Form DFRFT and DLCT with Other Types of DFRFT
conclusions and future work
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
references
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.

references47
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. AudioElectroacoust., vol. AU-20, pp. 66–74, Mar. 1972.