The Fractional Fourier Transform and Its Applications

The Fractional Fourier Transform and Its Applications

Presenter:

Pao-Yen LinResearch Advisor:

Jian-Jiun Ding , Ph. D.Assistant professor

Digital Image and Signal Processing LabGraduate Institute of Communication Engineering

National Taiwan University

2

Outlines • Introduction • Fractional Fourier Transform (FrFT)• Linear Canonical Transform (LCT)• Relations to other Transformations• Applications

04/22/2023Digital Image and Signal Processing Lab Graduate Institute of Communication

Engineering National Taiwan University

3

Introduction• Generalization of the Fourier

Transform• Categories of Fourier Transform

a) Continuous-time aperiodic signal b) Continuous-time periodic signal (FS)c) Discrete-time aperiodic signal (DTFT)d) Discrete-time periodic signal (DFT)



4

Fractional Fourier Transform (FrFT)• Notation

• is a transform of

• is a transform of



T f x F u

F T f T f x F u

F T f

5

Fractional Fourier Transform (FrFT) (cont.)• Constraints of FrFT① Boundary condition

② Additive property



0T f x f u

1T f x F u

T T f x T f x

6

Definition of FrFT• Eigenvalues and Eigenfunctions of

FT

• Hermite-Gauss Function



exp 2f x F v f x i vx dx

2 24 2 1 / 2 0f x n x f x

1 4

22 2 exp2 !

n nnx H x x

n

7

Definition of FrFT (cont.)• Eigenvalues and Eigenfunctions of

FT



,n n nx x 2in

n e

0

,n nn

f x A x

n nA x f x dx

2

0

e inn n

n

f x A x

8

Definition of FrFT (cont.)• Eigenvalues and Eigenfunctions of

FrFTUse the same eigenfunction but α order

eigenvalues



2i nn nx e x

2

0

i nn n

n

f x x A e x

9

Definition of FrFT (cont.)• Kernel of FrFT



,f x x B x x f x dx

0

1 2 2 2

2

0

,

2 exp

2 22 !

n n nn

i n

n nnn

B x x x x

x x

e H x H xn

10

Definition of FrFT (cont.)



2 2

cot cot csc2 21 cot

if is not a multiple of 2

if is a multiple of 2

if + is a multiple of 2

,

u tj j jutje x t e dt

t

t

X u x t K t u dt

11

Properties of FrFT• Linear.• The first-order transform

corresponds to the conventional Fourier transform and the zeroth-order transform means doing no transform.

• Additive.



12

Linear Canonical Transform (LCT)• Definition

where



2 2

, , ,, , ,

2 21 for 02

a b c dF a b c d

j d j j au ut tb b b

O g t G u

e e e g t dt bj b

2

, , , 2 for 0cdj ua b c d

FO g t de g du b

1ad bc

13

Linear Canonical Transform (LCT) (cont.)• Properties of LCT1. When , the LCT

becomes

FrFT.2. Additive property

where



cos sinsin cos

a bc d

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

2 2 1 1

2 2 1 1

a b a be fc d c dg h

14

Relation to other Transformations

• Wigner Distribution• Chirp Transform• Gabor Transform• Gabor-Wigner Transform• Wavelet Transform• Random Process



15

Relation to Wigner Distribution• Definition

• Property



,

2 2 exp 2

W f x W x v

f x x f x x j vx dx

2,f x W x v dv

2,F v W x v dx

Total energy ,f x W x v dxdv

16

Relation to Wigner Distribution



x

x

2

f u

17

Relation to Wigner Distribution (cont.)• WD V.S. FrFT

• Rotated with angle



, cos sin , sin cosf fW x v W x v x v

18

Relation to Wigner Distribution (cont.)• Examples



exp 2 ,c cf x j v x W x v v v

,f x x c W x v x c

22 1 0

2 1

exp 2 2

,

f x j b x b x b

W x v b x b v

slope= 2b

v

x

1b

19

Relation to Chirp Transform• for

Note thatis the same as rotated by



0 0 0cf x x x

1 2

2 20 0

ˆexp 4 2

sin

exp cot 2 csc cotc c

jf x

j x x x x

0 1 0 1 0, cos sin cW x x x x x

0 0cx x

20

Relation to Chirp Transform (cont.)

• Generally,



0 0 0 0f x f x f x x x dx 0 0f x f x x x dx

f x f x x x dx

,f x f x B x x dx

21

Relation to Gabor Transform (GT)

• Special case of the Short-Time Fourier Transform (STFT)

• Definition



2( ) ( )

2 2, 1/2t tj

fG t e e f d

22

Relation to Gabor Transform (GT) (cont.)• GT V.S. FrFT

• Rotated with angle



, cos sin , sin cosF fG u v G u v u v

23

Relation to Gabor Transform (GT) (cont.)• Examples

(a)GT of (b)GT of (c)GT of (d)WD of



-10 0 10-10

0

10

-10 0 10-10

0

10

-10 0 10-10

0

10

-10 0 10-10

0

10

s t r t f t

2

22

exp 10 3 for 9 1, 0 otherwise,

exp 2 6 exp 4 10

s t jt j t t s t

r t jt j t t

f t s t r t

f t

24

GT V.S. WD• GT has no cross term problem

• GT has less complexity

• WD has better resolution• Solution: Gabor-Wigner Transform



2exp 2 0.0001 when 4.2919x x

25

Relation to Gabor-Wigner Transform (GWT)• Combine GT and WD with arbitrary

function



,p x y

, ( , ), ( , )f f fC t p G t W t

26

Relation to Gabor-Wigner Transform (GWT) (cont.)• Examples1. In (a) 2. In (b)



, ( , ) ( , )f f fC t G t W t

2, min ( , ) , ( , )f f fC t G t W t

-10 0 10-10

-5

0

5

10

-10 0 10-10

-5

0

5

10(a) (b)

27

Relation to Gabor-Wigner Transform (GWT) (cont.)• Examples3. In (c) 4. In (d)



, ( , ) ( , ) 0.25f f fC t W t G t

2.6 0.6( , ) ( , ) ( , )f f fC t G t W t

-10 0 10-10

-5

0

5

10

-10 0 10-10

-5

0

5

10(c) (d)

28

Relation to Wavelet Transform• The kernels of Fractional Fourier

Transform corresponding to different values of can be regarded as a wavelet family.



2 2

2

1 2

exp sinsec

exptan

yg y f C j y

y xj f x dx

secy x

29

Relation to Random Process• Classification 1. Non-Stationary Random Process2. Stationary Random Process Autocorrelation function, PSD are

invariant with time t



30

Relation to Random Process (cont.)• Auto-correlation function

• Power Spectral Density (PSD)



, 2 2GR u E G u G u

, 2 , , jg g gS t FT R t R t e d

31

Relation to Random Process (cont.)• FrFT V.S. Stationary random process

• Nearly stationary



cot

sin cos, ,cos cos

uju j

G geR u e R

, sec secG gR u R

arg , tanGR u u

cos 0

32

Relation to Random Process (cont.)• FrFT V.S. Stationary random process for



cos 0

, , when 2 1 2G gR u S u H

, , when 2 3 2G gR u S u H

33

Relation to Random Process (cont.)• FrFT V.S. Stationary random process PSD:



, sin cos , when cos 0G gS u v S u v

, , when cos 0G gS u v S u

34

Relation to Random Process (cont.)• FrFT V.S. Non-stationary random

process Auto-correlation function

PSD

rotated with angle



2 2 33

cot csc cotcsc 2 22 3 2 3, 2,

4 sin

jju j t t tjutG g

eR u e e e R t t dt dt

, cos sin , sin cosG gS u v S u v u v

35

Relation to Random Process (cont.)• Fractional Stationary Random

Process If is a non-stationary random process

but is stationary and the

autocorrelation function of is independent of , then we call the -order fractional stationary random process.



g t

G u

G u u g t

36

Relation to Random Process (cont.)• Properties of fractional stationary

random process1. After performing the fractional filter, a white

noise becomes a fractional stationary random process.

2. Any non-stationary random process can be expressed as a summation of several fractional stationary random process.



37

Applications of FrFT• Filter design• Optical systems• Convolution • Multiplexing • Generalization of sampling theorem



38

Filter design using FrFT• Filtering a known noise

• Filtering in fractional domain



signal

noise

noise

u

x

u

39

Filter design using FrFT (cont.)• Random noise removal If is a white noise whose

autocorrelation function and PSD are:

After doing FrFT

Remain unchanged after doing FrFT!



g t

,gR gS

,GR gS

40

Filter design using FrFT (cont.)• Random noise removal

• Area of WD ≡ Total energy



signal

x

u

signal

x

u

41

Optical systems• Using FrFT/LCT to Represent Optical

Components• Using FrFT/LCT to Represent the

Optical Systems• Implementing FrFT/LCT by Optical

Systems



42

Using FrFT/LCT to Represent Optical Components1. Propagation through the cylinder

lens with focus length

2. Propagation through the free space (Fresnel Transform) with length



f

z

1 02 1

a bc d f

1 20 1

a b zc d

43

Using FrFT/LCT to Represent the Optical Systems



input output

2f1f

0d

0

1 2

0 2 0

00 1

1 2 1 2

1 0 1 01 22 1 2 10 1

1 2 2 1 1 1

a b df fc d

d f d

d d ff f f f

44

Implementing FrFT/LCT by Optical Systems• All the Linear Canonical Transform can be

decomposed as the combination of the chirp multiplication and chirp convolution and we can decompose the parameter matrix into the following form



1 0 1 01

if 01 1 1 10 1

a b bb

d b a bc d

1 01 1 1 1 if 0

10 1 0 1a b a c d c

cc d c

45

Implementing FrFT/LCT by Optical Systems (cont.)



input output

1f

0d 1d

The implementation of LCT with 1 cylinder lens and 2 free spaces

input output

2f1f

0d

The implementation of LCT with 2 cylinder lenses and 1 free space

46

Convolution • Convolution in domain

• Multiplication in domain



g x g f h f x h x

g f h

g x g f h f x h x

g f h

47

Convolution (cont.)



+1+1 +1 +1 +1

-1-1 -1 -1 -1

+1+1 +1 +1 +1

-1-1 -1 -1 -1

f h f h f h

f h f h f h

f h f h f h

f h f h f h

1 1

1 1

f h f h f h

f h f h f h

48

Multiplexing using FrFT



x

u

TDM FDM

x

u

49

Multiplexing using FrFT



x

u

Inefficient multiplexing Efficient multiplexing

x

u

50

Generalization of sampling theorem• If is band-limited in some

transformed domain of LCT, i.e.,

then we can sample by the interval as



f t

, , , 0, for and for some value of , , ,a b c dF u u a b c d

f t

b

51

Generalization of sampling theorem (cont.)



u

x

u

52

Conclusion and future works • Other relations with other

transformations• Other applications



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References [1] Haldun M. Ozaktas and M. Alper Kutay, “Introduction to the Fractional Fourier

Transform and Its Applications,” Advances in Imaging and Electron Physics, vol. 106, pp. 239~286.

[2] V. Namias, “The Fractional Order Fourier Transform and Its Application to Quantum Mechanics,” J. Inst. Math. Appl., vol. 25, pp. 241-265, 1980.

[3] Luis B. Almeida, “The Fractional Fourier Transform and Time-Frequency Representations,” IEEE Trans. on Signal Processing, vol. 42, no. 11, November, 1994.

[4] H. M. Ozaktas and D. Mendlovic, “Fourier Transforms of Fractional Order and Their Optical Implementation,” J. Opt. Soc. Am. A 10, pp. 1875-1881, 1993.

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

[6] A. W. Lohmann, “Image Rotation, Wigner Rotation, and the Fractional Fourier Transform,” J. Opt. Soc. Am. 10,pp. 2181-2186, 1993.

[7] S. C. Pei and J. J. Ding, “Relations between Gabor Transform and Fractional Fourier Transforms and Their Applications for Signal Processing,” IEEE Trans. on Signal Processing, vol. 55, no. 10, pp. 4839-4850, Oct. 2007.



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References [8] Haldun M. Ozaktas, Zeev Zalevsky, M. Alper Kutay, The fractional Fourier transform

with applications in optics and signal processing, John Wiley & Sons, 2001.[9] M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onural, “Optimal Filter in Fractional

Fourier Domains,” IEEE Trans. Signal Processing, vol. 45, no. 5, pp. 1129-1143, May 1997.

[10] J. J. Ding, “Research of Fractional Fourier Transform and Linear Canonical Transform,” Doctoral Dissertation, National Taiwan University, 2001.

[11] C. J. Lien, “Fractional Fourier transform and its applications,” National Taiwan University, June, 1999.

[12] Alan V. Oppenheim, Ronald W. Schafer and John R. Buck, Discrete-Time Signal Processing, 2nd Edition, Prentice Hall, 1999.

[13] R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed., Boston, McGraw Hill, 2000.



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Chenquieh!



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The Fractional Fourier Transform and Its Applications

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