Presenter:

Hong Wen-Chih

112/04/19 1

OutlineIntroduction Definition of fractional fourier transformLinear canonical transformImplementation of FRFT/LCT

The Direct ComputationDFT-like MethodChirp Convolution Method

Discrete fractional fourier transformConclusion and future work

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IntroductionDefinition of fourier transform:

Definition of inverse fourier transform:

1

2j wtF e f t dt

1

2j wtf t e F dt

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Introduction In time-frequency representation

Fourier transform: rotation π/2+2k πInverse fourier transform: rotation -π/2+2k πParity operator: rotation –π+2k πIdentity operator: rotation 2k π

And what if angle is not multiple of π/2 ?

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Introduction

t

v

u

( , )u v( , )t w

Time-frequency plane and a set of coordinates

rotated by angle α relative to the original coordinates

.

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Fractional Fourier Transform Generalization of FTuse to represent FRFTThe properties of FRFT:

Zero rotation:Consistency with Fourier transform:Additivity of rotations: 2π rotation:

Note: do four times FT will equal to do nothing

0F I/2F F

F F F 2F I

F

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Fractional Fourier Transform Definition:

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

( ) ( ) ( , )F u x t K t u dt

2 2

cot cot csc2 21 cot( )

2

u tj j jutje x t e e dt

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Linear Canonical Transform Generalization of FRFTDefinition:

when b≠0

when b=0a constraint: must be satisfied.

2 2

2 2( , , , )

1( ) ( )

2

jd j jau ut tb b b

a b c dF u e e e f t dtj b

2

2( ,0, , ) ( ) ( )

jcd u

a c dF u d e f du

1ad bc

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Linear Canonical Transform Additivity property:

where

Reversibility property:

where

tfOtfOO hgfeF

dcbaF

dcbaF

),,,(),,,(),,,( 11112222

2 2 1 1

2 2 1 1

a b a be f

c d c dg h

tftfOO dcbaF

acbdF ),,,(),,,(

d b a bI

c a c d

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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.

)()()0,1,1,0( tfFTjtfOF

)()()0,1,1,0( FIFTjFOF

tfOetfO Fj

F 2/1)cos,sin,sin,(cos

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Implementation of FRFT/LCTConventional Fourier transform

Clear physical meaningfast algorithm (FFT)Complexity : (N/2)log2N

LCT and FRFTThe Direct ComputationDFT-like MethodChirp Convolution Method

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Implementation of FRFT/LCTThe Direct Computationdirectly sample input and output

2 2

2 2( , , , )

12

u d ut t aj j jb b b

a b c dY u e e e x t dtj b

2 2 2 2

2

1

2 2( , , , )

12

u u t tm d mn n anj j jb b b

a b c d u t tn n

Y m e e e x nj b

t u

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Implementation of FRFT/LCTThe Direct Computation

Easy to designNo constraint expect forDrawbacks

lose many of the important propertiesnot be unitary no additivity Not be reversible lack of closed form properties

applications are very limited

1ad bc

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Implementation of FRFT/LCTChirp Convolution Method

Sample input and output as and

tp uq2 2

2 2( , , , )

1( ) ( )

2

jd j jau ut tb b b

a b c dF u e e e f t dtj b

2 2 2 2

2 2( , , , )

1( ) ( )

2

u u t tj d j j aMq p q pb b b

a b c d u tp M

F q e e e f pj b

M

Mpt

pb

ajpq

b

jq

b

dj

udcba pfbj

qFttuu eee22222 1

2

1

2,,, 2

1

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Implementation of FRFT/LCTChirp Convolution Methodimplement 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

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Implementation of FRFT/LCTDFT-like Method

constraint on the product of t and u

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

Put /2

1/

01

0

0/1

01

10

1/

01

bab

b

bddc

ba

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Implementation of FRFT/LCTDFT-like Method Chirp multiplication:

Scaling:

Fourier transform:

Chirp multiplication:

tfbjattf 2/exp 21

2

22 1

abj t

f t f bb teb t bf

dttfeuF tuj

j

23 2

1

uFbjduuF 32

4 2/exp

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Implementation of FRFT/LCTDFT-like Method

For 3rd step

Sample the input t and output u as pt and qu

Put /2

dttfeuF tuj

j

23 2

1

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Implementation of FRFT/LCTDFT-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/LCTCompare

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 Put /2

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Discrete fractional fourier transformDirect form of DFRFTImproved sampling type DFRFTLinear combination type DFRFTEigenvectors decomposition type DFRFTGroup theory type DFRFTImpulse train type DFRFTClosed form DFRFT

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Discrete fractional fourier transform

Direct form of DFRFTsimplest way sampling the continuous FRFT and computing it

directly

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Discrete fractional fourier transformImproved sampling type DFRFTBy Ozaktas, ArikanSample the continuous FRFT properlySimilar to the continuous caseFast algorithm

Kernel will not be orthogonal and additiveMany constraints

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Discrete fractional fourier transformLinear combination type DFRFTBy Santhanam, McClellan

Four bases: DFT IDFT Identity Time reverse

nFAnfAnFAnfAnF 3210

4

1

2

4

1

k

kqj

q eA

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Discrete fractional fourier transformLinear combination type DFRFT

transform matrix is orthogonaladditivity propertyreversibility property

very similar to the conventional DFT or the identity operation

lose the important characteristic of ‘fractionalization’

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Discrete fractional fourier transformLinear combination type DFRFTDFRFT 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.

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(a) = 0.01 (b) = 0.05 (c) = 0.2 (d) = 0.4 (e) = π/4 (f) = π/2

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Discrete fractional fourier transformEigenvectors decomposition type DFRFT

DFT : F=Fr – j Fi

Search eigenvectors set for N-points DFT

t tr iF U U U U

( ) tr iF U U

( ) tr iF U U

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Discrete fractional fourier transformEigenvectors decomposition type DFRFT

Good in removing chirp noiseBy Pei, Tseng, Yeh, Shyucf. : DRHT can be H Fr Fi

T1N

T1

Τ0

1N10

d

d

d

dddF

)1(00

0

0

001

Nj

j

e

e

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Discrete fractional fourier transformEigenvectors decomposition type DFRFTDFRFT 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

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Discrete fractional fourier transformGroup theory type DFRFTBy 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

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Discrete fractional fourier transformImpulse train type DFRFT By Arikan, Kutay, Ozaktas, Akdemir

special case of the continuous FRFTf(t) is a periodic, equal spaced impulse trainN = 2 , tanα = L/Mmany properties of the FRFT exists

many constraints not be defined for all values of

0 5 10 15 20 25-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

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Discrete fractional fourier transformClosed form DFRFTBy Pei, Ding

further improvement of the sampling type of DFRFTTwo types

digital implementing of the continuous FRFT practical applications about digital signal

processing

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Discrete fractional fourier transformType I Closed form DFRFT

Sample input f(t) and output Fa(u)

Then

Matrix form:

tnfny Δ umFmY Δαα

N

Nn

tnj

tumnjum

j

nyeetj

mY e2222 Δαcot

2ΔΔαcscΔαcot

2α Δ

π2

αcot1

N

Nn

nynmFmY ,αα

M

Mm

N

Nk

kykmFnmFny ,, αα

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Discrete fractional fourier transformType I Closed form DFRFT

Constraint:

M

Mm

N

Nk

tuknmjtnk

j

kyt ee ΔΔαcsc

Δαcot2

2 222

αsinπ2

Δ

M

Mm

N

Nk

kykmFnmFny ,, αα

12/αsinπ2ΔΔ MStu

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Discrete fractional fourier transformType I Closed form DFRFT

and

choose S = sgn(sin) = 1

2222 Δαcot

212

π2Δαcot

2α Δ

π2

αcot1,

tnj

M

mnSjum

j

eeetjnmF

M

Mm

N

Nk

kykmFnmF ,, αα nytM

2Δ

αsin)αsgn(sinπ2

12

2222 Δαcot

212

π2)αsin(sgnΔαcot

2α 12

αcos)αsin(sgnαsin,

tnj

M

mnjum

j

eeeM

jnmF

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Discrete fractional fourier transformType I Closed form DFRFT

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

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

2 2 2 22π

cot α Δ cot α Δ2 2 1 2

α

sin α cosα

2 1

j n m jNm u j n tM

n N

jF m y n

Me e e

2 2 2 22π

cot α Δ cot α Δ2 2 1 2

α

sin α cos α

2 1

j n m jNm u j n tM

n N

jF m y n

Me e e

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Discrete fractional fourier transformType 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

mnFnmF uttu ,, Δ,Δ,αΔ,Δ,α

α α πF m F m 2F m F m

F m F m

t u

22 ( / 2) logP P P

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Discrete fractional fourier transformType 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

22

212

)sgn(2

2),( 12

1,

nqj

M

mnbjmp

j

qp eeeM

nmF

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Discrete fractional fourier transformType 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)

adMqp 2)12/(2

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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))

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Discrete fractional fourier transform Type II Closed form DFRFT

2P+(P/2)log2P

No additive propertyConvertible

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Discrete fractional fourier transformThe relations between the DLCT of type 2 and its

special cases

DFRFT of type 2 p = q, s = 1

DFRFT of type 1 p = cotu2, q = cott2, s = sgn(sin)

DLCT of type 1 p = d/bu2, q = a/bt2, s = sgn(b)

DFT, IDFT p = q = 0, s = 1 for DFT, s = 1 for DFT

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Discrete fractional fourier transformComparison of Closed Form DFRFT and DLCT

with Other Types of DFRFTDirectly Improved Linear Eigenfxs. Group Impulse Proposed

Reversible *

Closed form

Similarity

Complexity P2 Plog2P+

2P

P2/2 Plog2P+

2P

Plog2P+

2P +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

PP

2log2

PP

2log2

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Conclusions and future work

Generalization of the Fourier transformApplications of the conventional FT can also be the

applications of FRFT and LCTMore flexibleUseful tools for signal processing

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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.

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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. Audio Electroacoust., vol. AU-20, pp. 66–74, Mar. 1972.

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