Fractional fourier transform

1
Fractional fourier transform

• Presenter
• Hong Wen-Chih

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

3
Introduction

• Definition of fourier transform
• Definition of inverse fourier transform

4
Introduction

• In time-frequency representation
• Fourier transform rotation p/22k p
• Inverse fourier transform rotation -p/22k p
• Parity operator rotation p2k p
• Identity operator rotation 2k p
• And what if angle is not multiple of p/2 ?

5
Introduction
.
Time-frequency plane and a set of coordinates
rotated by angle a relative to the original
coordinates
6
Fractional Fourier Transform

• Generalization of FT
• use to represent FRFT
• The properties of FRFT
• Zero rotation
• Consistency with Fourier transform
• Additivity of rotations
• 2p rotation
• Note do four times FT will equal to do nothing

7
Fractional Fourier Transform

• Definition
• Note when a is multiple of p, FRFTs degenerate
  into parity and identity operator
• 
  

8
Linear Canonical Transform

• Generalization of FRFT
• Definition
• 
  when b?0
• 
  when b0
• a constraint must be
  satisfied.

9
Linear Canonical Transform

• Additivity property
• where
• Reversibility property
• where

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.

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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

12
Implementation of FRFT/LCT

• The Direct Computation
• directly sample input and output

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

14
Implementation of FRFT/LCT

• Chirp Convolution Method
• Sample input and output as and

15
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
  2M1 the number of sampling points)
• Note 1 chirp convolution needs to 2DFTs

16
Implementation of FRFT/LCT

• DFT-like Method
• constraint on the product of ?t and ?u
• (chirp multi.) (FT)
  (scaling) (chirp multi.)

17
Implementation of FRFT/LCT

• DFT-like Method
• Chirp multiplication
• Scaling
• Fourier transform
• Chirp multiplication

18
Implementation of FRFT/LCT

• DFT-like Method
• For 3rd step
• Sample the input t and output u as p?t and q?u

19
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

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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

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

22
Discrete fractional fourier
transform

• Direct form of DFRFT
• simplest way
• sampling the continuous FRFT and computing it
  directly

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

24
Discrete fractional fourier transform

• Linear combination type DFRFT
• By Santhanam, McClellan
• Four bases
• DFT
• IDFT
• Identity
• Time reverse

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

26
Discrete fractional fourier transform

• Linear combination type DFRFT
• DFRFT of the rectangle window function for
  various angles
• (top left) a 001,
• (top right) a 005,
• (middle left) a 02,
• (middle right) a 04,
• (bottom left) a p/4,
• (bottom right) a p/2.

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• (a) 0.01
• (b) 0.05
• (c) 0.2
• (d) 0.4
• (e) p/4
• (f) p/2

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Discrete fractional fourier transform

• Eigenvectors decomposition type DFRFT
• DFT FFr j Fi
• Search eigenvectors set for N-points DFT

29
Discrete fractional fourier transform

• Eigenvectors decomposition type DFRFT
• Good in removing chirp noise
• By Pei, Tseng, Yeh, Shyu
• cf. DRHT can be

30
Discrete fractional fourier transform

• Eigenvectors decomposition type DFRFT
• DFRFT of the rectangle window function for
  various angles
• (top left) a 001,
• (top right) a 005,
• (middle left) a 02,
• (middle right) a 04,
• (bottom left) a p/4,
• (bottom right) a p/2

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

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 , tana L/M
• many properties of the FRFT exists
• many constraints
• not be defined for all values of ?

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

34
Discrete fractional fourier transform

• Type I Closed form DFRFT
• Sample input f(t) and output Fa(u)
• Then
• Matrix form

35
Discrete fractional fourier transform

• Type I Closed form DFRFT
• Constraint

36
Discrete fractional fourier transform

• Type I Closed form DFRFT
• and
• choose S sgn(sin?) ?1

37
Discrete fractional fourier transform

• Type I Closed form DFRFT
• when ? ? 2D?(0, ?), D is
  integer (i.e., sin? gt 0)
• when ? ? 2D?(??, 0), D is
  integer (i.e., sin? lt 0)

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

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)??u2, q (a/b)??t2

40
Discrete fractional fourier transform

• Type II Closed form DFRFT
• from ?t??u 2?b/(2M1), 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)

41
Discrete fractional fourier transform

• Type II Closed form DFRFT
• p0 DLCT becomes a CHIRP multiplication
  operation followed by a DFT
• q0 DLCT becomes a DFT followed by a chirp
  multiplication
• pq 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))

42
Discrete fractional fourier transform

• Type II Closed form DFRFT
• 2P(P/2)?log2P
• No additive property
• Convertible

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???u2, q cot???t2, s sgn(sin?)
DLCT of type 1 p d/b??u2, q a/b??t2, s sgn(b)
DFT, IDFT p q 0, s 1 for DFT, s ?1 for DFT
44
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 P2 P?log2P2P P2/2 P?log2P2P P?log2P2P 2P
FFT ? 2 FFT 1 FFT ? 2 FFT 2 FFT 1 FFT
Constraints Less Middle Unable Less Much Much Less
All orders ? ? ? ? ? ? ?
Properties Less Middle Middle Less Many Many Many
Adv./Cvt. No Convt. Additive Additive Additive Additive Convt.
DSP ? ? ? ? ? ? ??
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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

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.

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. 665675, 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 DRFTA
  rotation in time frequency space, in Proc.
  ICASSP, May 1995, pp. 921924.
• 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. 6674, Mar. 1972.