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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

- Definition of fourier transform:
- Definition of inverse fourier transform:

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 ?

Introduction

.

Time-frequency plane and a set of coordinates

rotated by angle α relative to the original coordinates

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 Transform

- Definition:
- Note: when α is multiple of π, FRFTs degenerate into parity and identity operator

Linear Canonical Transform

- Generalization of FRFT
- Definition:

when b≠0

when b=0

- a constraint: must be satisfied.

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

- 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/LCT

- The Direct Computation
- directly sample input and output

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/LCT

- Chirp Convolution Method
- Sample input and output as and

Implementation of FRFT/LCT

- Chirp Convolution Method
- implement by
- 2 chirp multiplications
- 1 chirp convolution
- complexity
- 2P (required for 2 chirp multiplications) + Plog2P(required for 2 DFTs) Plog2P(P = 2M+1 = the number of sampling points)
- Note: 1 chirp convolution needs to 2DFTs

Implementation of FRFT/LCT

- DFT-like Method
- constraint on the product of t and u

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

Implementation of FRFT/LCT

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

Implementation of FRFT/LCT

- DFT-like Method
- For 3rd step
- Sample the input t and output u as pt and qu

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/LCT

- Compare
- Complexity
- Chirp convolution method:Plog2P (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

- 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 transform

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

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 transform

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

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 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.

- (b) = 0.05
- (c) = 0.2
- (d) = 0.4
- (e) = π/4
- (f) = π/2

Discrete fractional fourier transform

- Eigenvectors decomposition type DFRFT
- DFT : F=Fr – j Fi
- Search eigenvectors set for N-points DFT

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 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 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 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 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 transform

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

Discrete fractional fourier transform

- Type I Closed form DFRFT
- Constraint:

Discrete fractional fourier transform

- Type I Closed form DFRFT
- and
- choose S = sgn(sin) = 1

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 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 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 transform

- Type II Closed form DFRFT
- from tu = 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 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 transform

- Type II Closed form DFRFT
- 2P+(P/2)log2P
- No additive property
- Convertible

Discrete fractional fourier transform

- The relations between the DLCT of type 2 and its special cases

Discrete fractional fourier transform

- Comparison of Closed Form DFRFT and DLCT with Other Types of DFRFT

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

[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.

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.

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