1

Introduction of Fractional Fourier Transform (FRFT)

Speaker: Chia-Hao Tsai Research Advisor: Jian-Jiun Ding

Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

10/23/2009

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Outlines

• Introduction of Fractional Fourier Transform (FRFT)

• Introduction of Linear Canonical Transform (LCT)

• Introduction of Two-Dimensional Affine Generalized Fractional Fourier Transform (2-D AGFFT)

• Relations between Wigner Distribution Function (WDF), Gabor Transform (GT), and FRFT

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Outlines (cont.)

• Implementation Algorithm of FRFT /LCT

• Closed Form Discrete FRFT/LCT• Advantages of FRFT/LCT contrast

with FT• Optics Analysis and Optical

Implementation of the FRFT/LCT• Conclusion and future works• References

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Introduction of Fractional Fourier Transform (FRFT)• The FRFT: a rotation in time-frequency plane. • : a operator• Properties of - =I: zero rotation - : FT operator - : time-reverse operator - : inverse FT operator - =I: 2π rotation - : additivity (I: identity operator)

R

0R

R

/2R

R R R

2R

3 /2R

R

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Introduction of FRFT (cont.)

• Definition of FRFT:

( : Kernel of FRFT)

2 2cot cot csc

2 21 cot

( ) , if isn't a multiple of 2

( ), if is multiple of 2

( ), if is multiple of 2

u tj j jutjx t dte e e

x t

x t

( ) ( ( ))Fx u x tOX R

( ) ( , )x t t u dtK

( , )t uK

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Introduction of FRFT (cont.)

• When : identity• When : FT• When isn’t equal to a multiple of , the

FRFT is equivalent to doing times of the FT. -when doing the FT 0.4 times. -when doing the FT 0.5 times. -when doing the FT 2/3 times.

0 0.5

0.5 / (0.5 )

0.2 0.25

/ 3

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Introduction of FRFT (cont.)

• An Example for the FRFT of a rectangle (Blue line: real part, green line: imaginary part)

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Introduction of Linear Canonical Transform (LCT)

• The LCT is more general than the FRFT. The FRFT has one free parameter (), but the LCT has four parameters (a, b, c, d) to adjust the signal.

• The LCT can use some specific value to change into the FRFT.

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Introduction of LCT (cont.)

• Definition of LCT:

with the constraint ad – bc = 1

, , ,, , , ( ) a b c da b c d Fu O x tF

2 2

2

1

2 2

2

1( ) , 0

2

( ), 0

j d j aj utu u

b b b

jcd u

x t dt be e ej b

d x du be

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Introduction of LCT (cont.)

• Additivity property:

where

• Reversibility property:

2 2 2 2 1 1 1 1, , , , , , , , ,a b c d a b c d e f g hF F FO O x t O x t

2 2 1 1

2 2 1 1

a b a be f

c d c dg h

, , , , , , ( )d b c a a b c dF FO O x t x t

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Introduction of LCT (cont.)

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Fractional / Canonical Convolution and Correlation

• FT for convolution:

• FT for correlation :

( ) ( ) ( )y t IFT FT x t FT h t

( ) ( ) ( )y t IFT FT x t FT h t

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Fractional / Canonical Convolution and Correlation (cont.)

• Fractional convolution (FRCV):

• Fractional correlation (FRCR) : type1:

type2:

( ) ( ) ( )F F Fy t x t h tO O O

3 1 21 2 3, . ( ( ), ( )) ( ( )) ( ( ))P P PP P Pcorr F F Fx t h t x t h tO O O O

( ) ( )x t h t transformed domain

( ) ( ) ( )u u uY X H

3 1 21 2 3, . ( ( ), ( )) ( ( )) ( ( ))P P PP P Pcorr F F Fx t h t x t h tO O O O

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Fractional / Canonical Convolution and Correlation (cont.)

• Canonical convolution (CCV):

• Canonical correlation (CCR) :type1:

type2:

( , , , ) ( , , , ) ( , , , )( ) ( ) ( )d b c a a b c d a b c dF F Fy t x t h tO O O

( , , , ),( , , , ),( , , , ) ( , , , ) ( , , , ) ( , , , )( ( ), ( )) ( ( )) ( ( ))a b c d e f g h m n s v m n s v a b c d e f g hcorr corr corr corrx t h t x t h tO O O O

( , , , )( ) ( )a b c dx t h t transformed domain

( , , , ) ( , , , ) ( , , , )( ) ( ) ( )a b c d a b c d a b c du u uY X H

( , , , ),( , , , ),( , , , ) ( , , , ) ( , , , ) ( , , , )( ( ), ( )) ( ( )) ( ( ))a b c d e f g h m n s v m n s v a b c d e f g hcorr corr corr corrx t h t x t h tO O O O

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Introduction of Two-Dimensional Affine Generalized Fractional Fourier Transform (2-D AGFFT)• The 2-D AGFFT that it can be regarded as

generalization of 2-D FT, 2-D FRFT, and 2-D LCT.• Definition of 2-D AGFFT:

where , , ,

, and

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Introduction of 2-D AGFFT (cont.)

where , • Reversibility property:

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Introduction of 2-D AGFFT (cont.)

• When

the 2-D AGFFT can become the 2-D unseparable FRFT which was introduced by Sahin et al.

2 1

2 1

cos cos cos sin,

cos sin cos cos

A

1 2

1 2 1 2

1 2

1 2 1 2

sin cos sin sin

cos cos,

sin sin sin cos

cos cos

B

2 1

2 1

sin cos sin sin,

sin sin sin cos

C

1 1

1 12 2

1 1

1 12 2

cos cos cos sin

cos cos,

cos sin cos cos

cos cos

D

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Introduction of 2-D AGFFT (cont.)

2-D AGFFT

Sahin’ s 2-D unseparable

FRFT

Geometrictwisting

2-D separablecanonical

2-D separable Fresnel

2-D separableFRFT

2-D inverseFourier

2-D forward Fourier

Identity operation

α = β = π/2 α = β = -π/2

a12=a21=b12=b21=c12=c21=d12=d21=0

a11=a22=d11=d22=1c11=c22=0

α = β = 0

(a11,b11,c11,d11)=(cosα,sinα,-sinα,cosα)(a22,b22,c22,d22)=(cosβ,sinβ,-sinβ,cosβ)

B=C=0

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2-D Affine Generalized Fractional Convolution/ Correlation

• 2-D Affine Generalized Fractional Convolution (2-D AGFCV):

-applications of 2-D AGFCV: filter design, generalized Hilbert transform, and mask

• 2-D Affine Generalized Fractional Correlation (2-D AGFCR):

-applications of 2-D AGFCR: 2-D pattern recognition

, , , ( , , , ) ( , , , ), , ,F F Fz x y f x y g x yO O O TT T T-CD -B A A B C D A B C D

, , , *( , , , ) ( , , , ), , ,'' ' '

F F Fz x y f x y x ygO O O E F G H A B C D CA B D

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Relation Between FRFT and WignerDistribution Function (WDF)

• Definition of WDF:

• The property of the WDF: -high clarity -with cross-term problem

*1( , )

2 2 2jw

ft t

t w f dfW e

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Relation Between FRFT and WDF (cont.)

• Why does the WDF have a cross-term problem?• Ans: autocorrelation-term

of the WDF is existed

• If , its WDF will become

*(( ) / 2) (( ) / 2)f t tf

( ) ( ) ( )f t s t r t

( , ) ( , ) ( , ).f s rt w t w t wW W W

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Relation Between FRFT and WDF (cont.)

• Clockwise-Rotation Relation:

• The FRFT with parameter is equivalent to the clockwise-rotation operation with angle for WDF.

( , ) ( cos sin , sin cos )fF u v u v u vW W

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Relation Between FRFT and Gabor Transforms (GT)

• Definition of GT:

• The property of the GT: -with clarity problem -avoid cross-term problem -cost less computation time

2/2 ( /2)1

( , ) ( )2

t jw tf t w f dG e e

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exp 0.0001, when 4.29192

xx

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Relation Between FRFT and GT (cont.)

• Why can the GT avoid the cross-term problem?• Ans: the GT no have the autocorrelation-term

• If , its GT will become

*(( ) / 2) (( ) / 2)f t tf

( ) ( ) ( )f t s t r t

( , ) ( , ) ( , ).f s rt w t w t wG G G

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Relation Between FRFT and GT (cont.)

• Clockwise-Rotation Relation:

• The FRFT with parameter is equivalent to the clockwise-rotation operation with angle for GT.

( , ) ( cos sin , sin cos )fF u v u v u vG G

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Relations Between FRFT, WDF, and GT

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Relations Between FRFT, WDF, and GT(cont.)

• Definition of GWT:

• Clockwise-Rotation Relation:

• Ex1: if then• Ex2: if then

( , ) ( ( , ), ( , ))f f ft w p t w t wC G W

( , ) ,p x y xy ( , ) ( , ) ( , ).f f ft w t w t wC G W

( , ) ,p x y x y ( , ) ( , ) ( , ).f f ft w t w t wC G W

( , ) ( cos sin , sin cos )fF u v u v u vC C

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Implementation Algorithm of FRFT/LCT

• Two methods to implement FRFT/LCT: - Chirp Convolution Method - DFT-Like Method

• To implement the FT, we need to use multiplications.

22 logN N

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Implementation Algorithm of FRFT/LCT (cont.)

Chirp Convolution Method:• For LCT, we sample t-axis and u-axis as and , then the continuous LCT becomes

tp uq

2 22 2

2 2( , , , )1

( ) ( )2

u tMj d j j a

p qq pu tb b ba b c d u tp M

q f pe e eFj b

22 22 21 1

2 2 21

( ).2

Mj d j j ap qq pu tu tb b b t

p Mf pe e e

j b

chirp multiplication

chirp convolution10/23/2009

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Implementation Algorithm of FRFT/LCT (cont.)

• To implement the LCT, we need to use 2 chirp multiplications and 1 chirp convolution.

• To implement 1 chirp convolution, we need to use requires 2 DFTs.

• Complexity: 2P (2 chirp multiplications) + (2 DFTs) (P = 2M+1 = the number of sampling points)

2logP P

2logP P

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Implementation Algorithm of FRFT/LCT (cont.)

• This is 2 times of complexity of FT.

• To implement the LCT directly, we need to use multiplications.

• So, we use Chirp Convolution Method to implement the LCT that it can improve the efficiency of the LCT.

• For FRFT, its complexity is the same as LCT.

2P

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Implementation Algorithm of FRFT/LCT (cont.)

DFT-Like Method:

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

Step.1: chirp multi. Step.2: scaling Step.3: FT Step.4: chirp multi.

1 0 0 1 1 0 1 0, 0

1 1 0 0 1

a b bb

c d d b b a b

21( ) exp( 2 ) ( )t ja b f tf t

2

22 1( ) ( ) ( )abj tt b bt b f btf f e

3 2

1( ) ( )

2jutu t dtfeF

j

24 3( ) exp( 2 ) ( )u jd b uuF F

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Implementation Algorithm of FRFT/LCT (cont.)

• We can implement the LCT:

where

• To implement the LCT, we need to use 2 M-points multiplication operations and 1 DFT.

(P = 2M+1 = the number of sampling points)

22222

2( )

2( , , , )( )2

abj p t t

Mj d p qj f pbq eub Pa b c d u t

p M

bs e eF

j b

2t u P

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Implementation Algorithm of FRFT/LCT (cont.)

• Complexity: 2P (2 multiplication operations) + (1 DFT)

• This is only half of the complexity of Chirp Convolution Method.

2( 2) logP P

2( 2) logP P

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Implementation Algorithm of FRFT/LCT (cont.)

• When using Chirp Convolution Method, the sampling interval is free to choose, but it needs to use 2 DFTs.

• When using the DFT-like Method, although it has some constraint for the sampling intervals, but we only need 1 DFT to implement the LCT.

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Closed Form Discrete FRFT/LCT

• DFRFT/DLCT of type 1: applied to digital implementing of the continuous

FRFT

• DFRFT/DLCT of type 2: applied to the practical applications about digital

signal processing

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Closed Form Discrete FRFT/LCT (cont.)

• Definition of DFRFT of type 1:

when 2D+(0, ), D is integer (sin > 0)

when 2D+(, 0), D is integer (sin < 0)

with the constraint and (2N+1, 2M+1 are the number of points in the time,

frequency domain)

22 2 2 2cot cot2 2 1 2

sin cos( ) ( )

2 1

j n m jNm u j n tM

n N

jm y nY

Me e e

22 2 2 2cot cot2 2 1 2

sin cos( ) ( )

2 1

j n m jNm u j n tM

n N

jm y nY

Me e e

M N 2 sin / (2 1)u t M

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Closed Form Discrete FRFT/LCT (cont.)

• when and D are integer, because we can’t find proper choice for u and t that can’t use as above.

, when = 2D , when = (2D+1)

• The DFRFT of type 1 is efficient to calculate and implement.

D

( ) ( )m y mY ( ) ( )m y mY

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Closed Form Discrete FRFT/LCT (cont.)

• Complexity: 2P (2 chirp multiplications)+ (1 FFT)

• But it doesn’t match the continuous FRFT, and lacks many of the characteristics of the continuous FRFT.

2( 2) logP P

2( 2) logP P

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Closed Form Discrete FRFT/LCT (cont.)

• Definition of DLCT of type 1: (b>0) (b<0)

with the constraint

2 2 2 22

2 2 1 2( , , , )1

( ) ( )2 1

Nd n m aj j jm u n tb M ba b c d

n Nm y ne e eY

M

2 2 2 22

2 2 1 2( , , , )1

( ) ( )2 1

Nd n m aj j jm u n tb M ba b c d

n Nm y ne e eY

M

2 / (2 1)u t b M

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Closed Form Discrete FRFT/LCT (cont.)

• When b = 0, we can’t also find proper choice for u and t that can’t use as above.

(b=0, d is integer)

(b=0, d isn’t integer) where R= (2M+1)(2N+1) and

2 2

2( ,0, , )( ) ( )jcd m u

a c d m d y d meY

2 2 2 sgn( ) 2

2 2 1 2 1( ,0, , )1

( ) ( )N Mc a k m n k

j j jm ua M Na c d

n N k Mm y ne e eY

R

(2 1) / (2 1)u N a t M

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Closed Form Discrete FRFT/LCT (cont.)

• Definition of DLCT of type 2: From DLCT of type 1, we let p = (d/b)u2, q =

(a/b)t2

where M N (2N+1, 2M+1 are the number of points in the time,

frequency domain), s is prime to M

22 2

2 2 1 2( , , )1

( ) ( )2 1

Nj s n m jp j q nm

Mp q sn N

m y ne e eYM

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Closed Form Discrete FRFT/LCT (cont.)

• By setting p = q and s = 1, we can define the DFRFT from the DLCT and get the formula of DFRFT of type 2.

• Definition of DFRFT of type 2:

where M N • Complexity of DFRFT/DLCT of type 2 is the same as

complexity of DFRFT/DLCT of type 1.

2 22

2 2 1 2( )1

( ) ( )2 1

Nj n m jp j pm n

Mpn N

m y ne e eYM

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Advantages of FRFT/LCT contrast with FT

• The FRFT/LCT are more general and flexible than the FT.

• The FRFT/LCT can be applied to partial differential equations (order n > 2). If we choice appropriate parameter , then the equation can be reduced order to n-1.

• The FT only deal with the stationary signals, we can use the FRFT/LCT to deal with time-varying signals.

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Advantages of FRFT/LCT contrast with FT (cont.)

• Using the FRFT/LCT to design the filters, it can reduce the NMSE. Besides, using the FRFT/LCT, many noises can be filtered out that the FT can’t remove in optical system, microwave system, radar system, and acoustics.

• In encryption, because the FRFT/LCT have more parameter than the FT, it’s safer in using the FRFT/LCT than in using the FT.

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Advantages of FRFT/LCT contrast with FT (cont.)

• In signal synthesis, using the transformed domain of the FRFT/LCT to analyze some signal is easier than using the time domain or frequency domain to analyze signals.

• In multiplexing, we can use multiplexing in fractional domain for super-resolution and encryption.

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Applications of FRFT/LCT

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Use FRFT/LCT to Represent Optical Components

• Propagation through the cylinder lens with focus length f:

• Propagation through the free space (Fresnel Transform) with length z:

1 0

2 1

a b

c d f

1 2

0 1

a b z

c d

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Implementation FRFT/LCT by Optical Systems

• Case1:1 0 1 1 0

, 0( 1) / 1 0 1 ( 1) / 1

a b bb

c d d b a b

f1Input Output

d0

f2

Cylinder lens Cylinder lens

Free space

The implementing of LCT (b≠0) with 2 cylinder lenses and 1 free space.10/23/2009

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Implementation FRFT/LCT by Optical Systems (cont.)

01 2

2 2 2, , , for LCT

(1 ) (1 )

b b bf fd

a d

01 2

2 cot( / 2) 2 sin, , for FRFTf f d

{a, b, c, d} = {cosα, sinα, -sinα, cosα}

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Implementation FRFT/LCT by Optical Systems (cont.)

• Case2:1 ( 1) / 1 0 1 ( 1) /

, c 00 1 1 0 1

a b a c d c

c d c

Cylinder lensFree spaceFree space

Input Outputf0

d1 d2

The implementing of LCT (c ≠ 0) with 1 cylinder lenses and 2 free space.10/23/2009

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Implementation FRFT/LCT by Optical Systems (cont.)

1 0 2

2 ( 1) 2 2 ( 1), , , for LCT

d af fd

c c c

1 2 0

2 tan( / 2) 2 csc, , for FRFTfd d

{a, b, c, d} = {cosα, sinα, -sinα, cosα}

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Implementation FRFT/LCT by Optical Systems (cont.)

• If we want to have shorter length in the optical implementation of LCT, then case1 is preferred.

• If we want to retrench the number of lenses we use, or avoid placing the lenses contacting to the input and output, then case2 is preferred.

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Conclusion and future works

• FRFT/LCT are more general and flexible than the FT. Then, We hope to find other applications of FRFT/LCT.

• Can we find more general functions? Then, we let the functions use in other

applications.

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References

[1] V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Maths. Applics., vol. 25, pp. 241-265, 1980.

[2] A. C. McBride and F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math., vol. 39, pp. 159-175, 1987.

[3] S. C. Pei, C. C. Tseng, and M. H. Yeh, “Discrete fractional Hartley and Fourier transform,” IEEE Trans. Circuits Syst., II: Analog and Digital Signal Processing, vol. 45, no. 6, pp. 665-675, June 1998.

[4] B. Santhanam and J. H. McClellan, “The DRFT—A rotation in time frequency space,” in Proc. ICASSP, May 1995, pp. 921–924.

[5] 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.

[6] B. W. Dickinson and K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-30, pp. 25–31, Feb. 1982.

[7] H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their rotation to chirp and wavelet transform,” J. Opt. Soc. Am. A, vol. 11, no. 2, pp. 547-559, Feb. 1994.

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References (cont.)

[8] Zayed, “A convolution and product theorem for the fractional Fourier transform,” IEEE Signal Processing Letters, vol. 5, no. 4, pp. 101-103, Apr. 1998.

[9] A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Amer. A, vol. 10, no. 10, pp.2181–2186, Oct. 1993.

[10]D. A. Mustard, “The fractional Fourier transform and the Wigner distribution,”J. Australia Math. Soc. B, vol. 38, pt. 2, pp. 209–219, Oct.1996.

[11]S. C. Pei and J. J. Ding, “Relations between the fractional operations and the Wigner distribution, ambiguity function,” IEEE Trans. Signal Process., vol. 49, no. 8, pp. 1638–1655, Aug. 2001.

[12]H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their rotation to chirp and wavelet transform,” J. Opt. Soc. Amer. A, vol. 11, no. 2, pp. 547–559, Feb. 1994

[13]A. Sahin, M. A. Kutay, and H. M. Ozaktas, ‘Nonseparable two-dimensional fractional Fourier transform`, Appl. Opt., vol. 37, no. 23, p 5444-5453, Aug 1998.

[14]S. C. Pei, and J. J. Ding “Two-Dimensional affine generalized fractional Fourier transform,” IEEE Trans. Signal Processing, Vol.49, No. 4, April 2001.

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