The Fractional Fourier Transform and Its Applications Presenter: Pao-Yen Lin Research Advisor:...

The Fractional Fourier Transform and Its

ApplicationsPresenter:

Pao-Yen LinResearch Advisor:

Jian-Jiun Ding , Ph. D.Assistant professor

Digital Image and Signal Processing LabGraduate Institute of Communication Engineering

National Taiwan University

Outlines

• Introduction

• Fractional Fourier Transform (FrFT)

• Linear Canonical Transform (LCT)

• Relations to other Transformations

• Applications

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Digital Image and Signal Processing Lab Graduate Institute of Communication

Engineering National Taiwan University

Introduction

• Generalization of the Fourier

Transform

• Categories of Fourier Transforma) Continuous-time aperiodic signal b) Continuous-time periodic signal (FS)c) Discrete-time aperiodic signal (DTFT)d) Discrete-time periodic signal (DFT)

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Fractional Fourier Transform (FrFT)• Notation

• is a transform of

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T f x F u

Fractional Fourier Transform (FrFT) (cont.)• Constraints of FrFT

①Boundary condition

②Additive property

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0T f x f u

1T f x F u

T T f x T f x

Definition of FrFT

• Eigenvalues and Eigenfunctions of

• Hermite-Gauss Function

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exp 2f x F v f x i vx dx

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

222 exp

2 !n nnx H x x

Definition of FrFT (cont.)

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,n n nx x 2in

f x A x

n nA x f x dx

e inn n

f x A x

Use the same eigenfunction but α order

eigenvalues

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2i nn nx e x

i nn n

f x x A e x

• Kernel of FrFT

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,f x x B x x f x dx

1 2 2 2

2 22 !

n n nn

B x x x x

eH x H x

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cot cot csc2 2

1 cotif is not a multiple of

if is a multiple of 2

if + is a multiple of 2

u tj j jutje x t e dt

X u x t K t u dt

Properties of FrFT

• Linear.

• The first-order transform

corresponds to the conventional

Fourier transform and the zeroth-

order transform means doing no

transform.

• Additive.

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Linear Canonical Transform (LCT)• Definition

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2 21 for 0

a b c dF a b c d

j d j j au ut t

O g t G u

e e e g t dt bj b

, , , 2 for 0cdj ua b c d

FO g t de g du b

1ad bc

Linear Canonical Transform (LCT) (cont.)• Properties of LCT

1.When , the LCT

becomes

2.Additive property

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cos sin

sin cos

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

a b a be f

c d c dg h

Relation to other Transformations• Wigner Distribution

• Chirp Transform

• Gabor Transform

• Gabor-Wigner Transform

• Wavelet Transform

• Random Process

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Relation to Wigner Distribution• Definition

• Property

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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

Relation to Wigner Distribution

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

• Rotated with angle

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

Relation to Wigner Distribution (cont.)• Examples

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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

exp 2 2

f x j b x b x b

W x v b x b v

slope= 2b

Relation to Chirp Transform

• for

Note that

is the same as rotated by

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0 0 0cf x x x

2 20 0

ˆexp 4 2

exp cot 2 csc cotc c

j x x x x

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

0 0cx x

Relation to Chirp Transform (cont.)

• Generally,

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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

Relation to Gabor Transform (GT)• Special case of the Short-Time

Fourier Transform (STFT)

• Definition

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( )2 2, 1/2t t

fG t e e f d

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

• Rotated with angle

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

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

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

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-10 0 10-10

s t r t f t

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

GT V.S. WD

• GT has no cross term problem

• GT has less complexity

• WD has better resolution

• Solution: Gabor-Wigner Transform

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2exp 2 0.0001 when 4.2919x x

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

function

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,p x y

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

Relation to Gabor-Wigner Transform (GWT) (cont.)• Examples

1.In (a)

2.In (b)

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

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

-10 0 10-10

10(a) (b)

Relation to Gabor-Wigner Transform (GWT) (cont.)• Examples

3.In (c)

4.In (d)

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, ( , ) ( , ) 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

10(c) (d)

Relation to Wavelet Transform• The kernels of Fractional Fourier

Transform corresponding to different

values of can be regarded as a

wavelet family.

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exp sinsec

exptan

yg y f C j y

y xj f x dx

secy x

Relation to Random Process

• Classification

1.Non-Stationary Random Process

2.Stationary Random Process

Autocorrelation function, PSD are

invariant with time t

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

• Power Spectral Density (PSD)

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, 2 2GR u E G u G u

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

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

• Nearly stationary

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sin cos, ,cos cos

eR u e R

, sec secG gR u R

arg , tanGR u u

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

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

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

, , when cos 0G gS u v S u

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

process

Auto-correlation function

rotated with angle

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2 2 33

cot csc cotcsc 2 22 3 2 3, 2,

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

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

If is a non-stationary random process

is stationary and the autocorrelation

function of is independent of , then

we call the -order fractional stationary

random process.

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G u u g t

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

random process

1.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.

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Applications of FrFT

• Filter design

• Optical systems

• Convolution

• Multiplexing

• Generalization of sampling theorem

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Filter design using FrFT

• Filtering a known noise

• Filtering in fractional domain

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signal

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!

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,gR gS

,GR gS

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

• Area of WD ≡ Total energy

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signal

Optical systems

• Using FrFT/LCT to Represent Optical

Components

• Using FrFT/LCT to Represent the

Optical Systems

• Implementing FrFT/LCT by Optical

Systems

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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

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Using FrFT/LCT to Represent the Optical Systems

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input output

1 2 1 2

1 0 1 01 2

2 1 2 10 1

2 1 11

f fc d

f f f f

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

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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 1

a b a c d cc

Implementing FrFT/LCT by Optical Systems (cont.)

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input output

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

input output

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

Convolution

• Convolution in domain

• Multiplication in domain

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g x g f h f x h x

Convolution (cont.)

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+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

Multiplexing using FrFT

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TDM FDM

Multiplexing using FrFT

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Inefficient multiplexing Efficient multiplexing

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

transformed domain of LCT, i.e.,

then we can sample by the

interval as

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

Generalization of sampling theorem (cont.)

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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,

[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

[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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