a new two-dimensional fractional fourier...

A New Two-Dimensional Fractional Fourier Transform

A New Two-Dimensional Fractional FourierTransform

Ahmed I. Zayed,Department of Mathematical Sciences,DePaul University, Chicago, IL 60614

Aspects of Time-Frequency Analysis, Turin, Italy,June 5-7, 2017

Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform


Outline

Outline

1 Introduction

2 The Mittag-Leffler Transform

3 A Practical Fractional Fourier Transform

4 Motivations and Applications

5 Wigner Distribution

6 N-Dimensional FrFT

7 Four-Dimensional Rotations

8 The Main Theorem



Introduction

The Fourier transform is a very valuable tool in many branchesof applied science such as physics, electrical engineering, andoptics. If we denote the Fourier transformation of a function f by

F [f ] (ω) = f̂ (ω) =1√2π

∫R

f (t) eiωtdt , (1)

then for an appropriate function f the inversion formula takes onthe form

F−1[f̂]

(t) = f (t) =1√2π

∫R

f̂ (ω) e−iωtdω, (2)



Introduction

There are many generalizations of the exponentialfunctions but some of them depend on a parameter α or qbetween zero and one and reduce to the exponentialfunction when the parameter is equal to one. That is whythe integral transforms whose kernels are those functionsare called Fraction Fourier Transforms.

The question is: What other properties do these functionsshare with the exponential functions? What properties dothese fractional Fourier transforms share with the classicalFourier transform?In this talk I will discuss some of these fractional Fouriertransforms and point out some of the similarities anddifferences between them.



Introduction

There are many generalizations of the exponentialfunctions but some of them depend on a parameter α or qbetween zero and one and reduce to the exponentialfunction when the parameter is equal to one. That is whythe integral transforms whose kernels are those functionsare called Fraction Fourier Transforms.The question is: What other properties do these functionsshare with the exponential functions?

What properties dothese fractional Fourier transforms share with the classicalFourier transform?In this talk I will discuss some of these fractional Fouriertransforms and point out some of the similarities anddifferences between them.



Introduction

There are many generalizations of the exponentialfunctions but some of them depend on a parameter α or qbetween zero and one and reduce to the exponentialfunction when the parameter is equal to one. That is whythe integral transforms whose kernels are those functionsare called Fraction Fourier Transforms.The question is: What other properties do these functionsshare with the exponential functions? What properties dothese fractional Fourier transforms share with the classicalFourier transform?

In this talk I will discuss some of these fractional Fouriertransforms and point out some of the similarities anddifferences between them.



Introduction

There are many generalizations of the exponentialfunctions but some of them depend on a parameter α or qbetween zero and one and reduce to the exponentialfunction when the parameter is equal to one. That is whythe integral transforms whose kernels are those functionsare called Fraction Fourier Transforms.The question is: What other properties do these functionsshare with the exponential functions? What properties dothese fractional Fourier transforms share with the classicalFourier transform?In this talk I will discuss some of these fractional Fouriertransforms and point out some of the similarities anddifferences between them.



The Mittag-Leffler Transform

The Mittag-Leffler Function

The Mittag-Leffler function is defined

Eα,A(x) =∞∑

k=0

(Ax)k

Γ(1 + αk).

It is an entire function of order 1/α and type A. In particular, forα = 1, we have

E1,A(x) = eAx .

We write Eα,1 = Eα.





The Mittag-Leffler Transform of f (x) is defined as

Fα,A(t) =

∫R

f (x)Eα,A(tx)dx ,

whenever the integral exists.

For α = 1,A = 1, andf (x) = 0, x < 0, the Mittag-Leffler Transform reduces to theLaplace transform.





The Mittag-Leffler Transform of f (x) is defined as

Fα,A(t) =

∫R

f (x)Eα,A(tx)dx ,

whenever the integral exists. For α = 1,A = 1, andf (x) = 0, x < 0, the Mittag-Leffler Transform reduces to theLaplace transform.




G. Jumarie in a series of papers (2008-2012) combined thegeneralized Taylor expansion of fractional order and theMittag-Leffler function, to get

f (x + h) = Eα (hαDαx ) f (x) =

[ ∞∑k=0

(hαDαx )k

Γ(1 + αk)

]f (x),

where 0 < α ≤ 1 and Dα is a fractional derivative of order α.From the relation

Dαxγ = Γ(γ + 1)Γ−1(γ + 1− α)xγ−α,

we can show that

DαEα(λxα) = λEα(λxα).




More generally, the solution of the fractional order differentialequation

Dαx f (x) = λf (x), f (0) = A

isf (x) = AEα(λxα).

This is a generalization of

dfdx

= λf (x), f (0) = A,→ f (x) = Aeλx .




Jumarie showed that the Mittag-Leffler function Eα(λxα) hasthe following property

Eα(λxα)Eα(λyα) = Eα(λ(x + y)α), λ ∈ C.

Let Mα be the solution of the equation

Eα(i(Mα)α) = 1.

Then it follows that

Eα(ixα)Eα(i(Mα)α) = Eα(ixα) = Eα(λ(x + Mα)α), λ ∈ C,

that isEα(ixα)

is periodic with period Mα.




Now we define

Eα(ixα) = cosα xα + i sinα xα,

where

cosα xα =∞∑

k=1

(−1)k x2kα

(2kα)!),

and

sinα xα =∞∑

k=1

(−1)k x (2k+1)α

(2kα + α)!).

These functions are periodic with period Mα and satisfy therelations

Dα cosα xα = − sinα xα and Dα sinα xα = cosα xα.



A Practical Fractional Fourier Transform

The next FRFT was introduced in 1980 by V. Namias in‘The fractional order Fourier transform and its applicationto quantum mechanics,’ J. Inst. Math. Appl., (1980).

His results were later refined by A. McBride and F. Kerr"On Namias’s fractional Fourier Transforms,“ IMA J. Appl.Math., (1987), who, among other things, also developed anoperational calculus for the FRFT.




The next FRFT was introduced in 1980 by V. Namias in‘The fractional order Fourier transform and its applicationto quantum mechanics,’ J. Inst. Math. Appl., (1980).His results were later refined by A. McBride and F. Kerr"On Namias’s fractional Fourier Transforms,“ IMA J. Appl.Math., (1987), who, among other things, also developed anoperational calculus for the FRFT.




The fractional Fourier transform gained very much popularity inthe early 1990s because of its numerous applications in signalanalysis and optics.

Some of the early pioneers in the field are L. Almeida, M. Kutay,A. Lohmann, D. Mendlovic, D. Mustard, H. Ozaktas, and Z.Zalevsky. Journals of IEEE , and Opt. Soc. Amer., andAustralian Math. Soc.The Fractional Fourier Transform with Applications in Opticsand Signal Processing, H. Ozaktas, Z. Zalevsky, and M. Kutay,Wiley (2001)




The fractional Fourier transform gained very much popularity inthe early 1990s because of its numerous applications in signalanalysis and optics.Some of the early pioneers in the field are L. Almeida, M. Kutay,A. Lohmann, D. Mendlovic, D. Mustard, H. Ozaktas, and Z.Zalevsky. Journals of IEEE , and Opt. Soc. Amer., andAustralian Math. Soc.The Fractional Fourier Transform with Applications in Opticsand Signal Processing, H. Ozaktas, Z. Zalevsky, and M. Kutay,Wiley (2001)




The Fractional Fourier transform may also be viewed as a(family of bounded operators) Fα, with 0 ≤ α ≤ 1, such that

F0(f ) = f , F1 = f̂ .

In practice, it is indexed by an angle 0 ≤ θ ≤ 2π so that

F0(f ) = f , Fπ/2 = f̂ , Fπ (f (x)) = f (−x), F2π = f .




Fθ [x ] (ω) = x̂θ(ω) =

∫ ∞−∞

x(t)Kθ (t , ω) dt (3)

where

Kθ(t , ω) =

c(θ) · eia(θ)(t2+ω2)−ib(θ)ωt , θ 6= pπδ(t − ω), θ = 2pπδ(t + ω), θ = (2p − 1)π

(4)




is the transformation kernel with c(θ) =√

1−i cot θ2π ,

a(θ) = cot θ/2, and b(θ) = csc θ. The kernel Kθ(t , ω) isparameterized by an angle θ ∈ R and p is some integer.

Notice that when

θ = π/2, a(π/2) = 0, b(π/2) = 1, c(π/2) =1√2π,

and the kernel is reduced to Kθ(t , ω) = 1√2π

e−itω.




is the transformation kernel with c(θ) =√

1−i cot θ2π ,

a(θ) = cot θ/2, and b(θ) = csc θ. The kernel Kθ(t , ω) isparameterized by an angle θ ∈ R and p is some integer.Notice that when

θ = π/2, a(π/2) = 0, b(π/2) = 1, c(π/2) =1√2π,

and the kernel is reduced to Kθ(t , ω) = 1√2π

e−itω.




1 Linearity: The Fractional Fourier Transform is linear, i.e.,

Fθ [αf + βg] = αFθ[f ] + βFθ[g]

where α and β are constants.2 Additivity: FθFφ = Fθ+φ,3 Commutativity: FθFφ = FφFθ,4 Associativity: Fθ1

(Fθ2Fθ3

)= (Fθ1Fθ2)Fθ3 .

5 Inverse: (Fθ)−1 = F−θ.



Motivations and Applications

In an optical system with several lenses and using a pointsource for illumination, one observes the Fourier transform(the absolute value) of the object at the image of the pointsource. In the simplest case, the Fourier transform isobserved at the focal plane.

Whatever is being observedhalfway between the lens and the focal plane may becalled the ( one half Fourier transform) !For light propagation in quadratic graded-index media(fiber optics), it is known that the Fourier transform isproduced at a certain distance d0 that depends on themedium. Thus, it is reasonable to call the light distributionat distance ad0,0 < a ≤ 1, the fractional Fourier transformof order a.




In an optical system with several lenses and using a pointsource for illumination, one observes the Fourier transform(the absolute value) of the object at the image of the pointsource. In the simplest case, the Fourier transform isobserved at the focal plane. Whatever is being observedhalfway between the lens and the focal plane may becalled the ( one half Fourier transform) !

For light propagation in quadratic graded-index media(fiber optics), it is known that the Fourier transform isproduced at a certain distance d0 that depends on themedium. Thus, it is reasonable to call the light distributionat distance ad0,0 < a ≤ 1, the fractional Fourier transformof order a.




In an optical system with several lenses and using a pointsource for illumination, one observes the Fourier transform(the absolute value) of the object at the image of the pointsource. In the simplest case, the Fourier transform isobserved at the focal plane. Whatever is being observedhalfway between the lens and the focal plane may becalled the ( one half Fourier transform) !For light propagation in quadratic graded-index media(fiber optics), it is known that the Fourier transform isproduced at a certain distance d0 that depends on themedium. Thus, it is reasonable to call the light distributionat distance ad0,0 < a ≤ 1, the fractional Fourier transformof order a.




Let us look at the FrFR of f (x) = χ[−1,1](x), the characteristicfunction of [−1,1].

Since the FrFT is complex-valued, we will plot the modulus ofFθ for different θ.




Let us look at the FrFR of f (x) = χ[−1,1](x), the characteristicfunction of [−1,1].Since the FrFT is complex-valued, we will plot the modulus ofFθ for different θ.




-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1.0

Figure: Zero FrFT




-6 -4 -2 0 2 4 6

0.5

1.0

1.5

2.0

Figure: One Quarter FT, θ = π/8




-6 -4 -2 0 2 4 6

0.5

1.0

1.5

2.0

Figure: One half FT θ = π/4




-6 -4 -2 0 2 4 6

0.5

1.0

1.5

2.0

Figure: One FT θ = π/2




DefinitionThe cross–ambiguity function Af ,g(u, v) of two functions f andg is defined by

Af ,g(u, v) =

∫ ∞−∞

f(

t +u2

)g(

t − u2

)e−ivtdt ,




The Wigner distribution of a signal f is defined as

Wf (u, v) =

∫R

f (u + x/2)f ∗(u − x/2)e−2πivxdx .

It is related to the Radar ambiguity function by

Wf ,g(u, v) = 2Af ,h(2u,2v),

where h(z) = g(−z).




The Wigner distribution of Wf̂ (u, v) is obtained from Wf (u, v) bya rotation of π/2.

Wf̂ (u, v) = Wf (−v ,u).

What does correspond to a rotation by an angle π/4? Whateverit is, we call it the one half Fourier transform.





Wf̂ (u, v) = Wf (−v ,u).

What does correspond to a rotation by an angle π/4?

Whateverit is, we call it the one half Fourier transform.





Wf̂ (u, v) = Wf (−v ,u).

What does correspond to a rotation by an angle π/4? Whateverit is, we call it the one half Fourier transform.




More generally, what does correspond to a rotation by an angleθ? ,i.e., Find g such that

Wg(u, v) = Wf (u cos θ − v sin θ,u sin θ + v cos θ).

g is the fractional Fourier transform with angle θ.



N-Dimensional FrFT

Namias’s original idea is the observation that the Hermitefunctions hn(x) = e−x2/2Hn(x) are the eigenfunctions of theFourier transform with eigenvalues einπ/2, that is

Fπ/2[hn(x)] = F [hn(x)] (ω) = einπ/2hn(ω), (5)

where Hn(x) is the Hermite polynomial of degree n.

Namais looked for a family of integral transforms {Fθ} indexedby a parameter θ such that

Fθ [hn(x)] (ω) = einθhn(ω). (6)

When θ = π/2, Eq.(6) reduces to Eq. (5).



N-Dimensional FrFT

Namias’s original idea is the observation that the Hermitefunctions hn(x) = e−x2/2Hn(x) are the eigenfunctions of theFourier transform with eigenvalues einπ/2, that is

Fπ/2[hn(x)] = F [hn(x)] (ω) = einπ/2hn(ω), (5)

where Hn(x) is the Hermite polynomial of degree n.Namais looked for a family of integral transforms {Fθ} indexedby a parameter θ such that

Fθ [hn(x)] (ω) = einθhn(ω). (6)

When θ = π/2, Eq.(6) reduces to Eq. (5).



N-Dimensional FrFT

The n-Dimensional Fractional Fourier Transform:

The fractional Fourier transform in n-variables is defined bytaking the tensor product of n copies of the one dimensionalfractional Fourier transforms. That is

Fθ1,··· ,θn (ω1, · · · , ωn) =∫R· · ·∫R

Kθ1 (t1, ω1) · · ·Kθn (tn, ωn) f (t1, · · · , tn)dt1 · · · dtn,

where Kθi (ti , ωi) , i = 1,2, · · · ,n, is the kernel of theone-dimensional fractional Fourier transform



N-Dimensional FrFT

In particular, in two dimensions we have

Fθ1,θ2(ω1, ω2) = ∫R

∫R

Kθ1 (t1, ω1) Kθ2 (t2, ω2) f (t1, t2)dt1dt2,

The eigenvalue equation takes the form

Fθ1,θ2

[hk1,k2(t1, t2)

]=[ei(θ1k1)hk1(t1)

] [ei(θ2k2)hk2(t2)

]. (7)



N-Dimensional FrFT

It leads to the 4-dimensional rotation of the Wigner distributionattained by the matrix

cos θ1 0 − sin θ1 00 cos θ2 0 − sin θ2sin θ1 0 cos θ1 00 sin θ2 0 cos θ2

. (8)



N-Dimensional FrFT

The complex Hermite polynomials Hm,n(z, z) introduced by M.Ismail in Trans. Amer. Math. Soc. in 2016

Hm,n(z1, z2) =m∧n∑k=0

(−1)kk !

(mk

)(nk

)zm−k

1 zn−k2 . (9)

Their generating function is given by

∞∑m,n=0

Hm,n(z1, z2)tm

m!

sn

n!= etz1+sz2−ts,



N-Dimensional FrFT

ei(p+q)π/2hq,p(z1, z1) =1

2π

∫R2

hq,p(z2, z2)ei(ux+vy)dxdy , (10)

wherehq,p(z, z) = Hq,p(z, z)e−|z|

2/2

is the Hermite function of two variables. Eq. (10 ) shows thatthe Hermite functions are the eigenfunctions of the twodimensional Fourier transform with eigenvalues(i)p+q = ei(p+q)π/2;



N-Dimensional FrFT

k (z1, z1, z2, z2; s, t)

= d(γ) exp{−a(γ)

(x2 + y2 + u2 + v2

)(11)

+ b(γ, δ) (ux + vy) + c(γ, δ) (vx − uy)} , (12)

where s = eiα, t = eiβ, γ = (α + β)/2, δ = (α− β)/2,

a(γ) = icot γ

2, b(γ, δ) =

i cos δsin γ

(13)

c(γ, δ) =i sin δsin γ

, d(γ) =ie−iγ

2π sin γ. (14)



N-Dimensional FrFT

Definition

For a function f ∈ L2 (R2) we define the two-dimensionalfractional Fourier transform with angles α and β as

Fα,β (z1, z1) =

∫R2

k (z1, z1, z2, z2;α, β) f (z2, z2)dz2. (15)

orFα,β (u, v) =

∫R2

k (x , y ,u, v ;α, β) f (x , y)dxdy , (16)

where z1 = u + iv , z2 = x + iy .



N-Dimensional FrFT

The auto-Wigner distribution function of f (or the Wignerdistribution function of f for short) is defined as

Wf (u1,u2; v1, v2) =

∫R2

f (u1+x2,u2+

y2

)f (u1−x2,u2−

y2

)ei(xv1+yv2)dxdy



Four-Dimensional Rotations

This new two-dimensional FrFT has almost all the properties ofthe standard FrFT, including its effect on the Wigner distribution.

It is known that each 4-dimensional rotation can bedecomposed into two matrices representing left andright-multiplication by a unit quaternion. In fact, let

ML =

a −b −c −db a −d cc d a −bd −c b a

, MR =

p −q −r −sq p s −rr −s p qs r −q p

.

(17)where

a2 + b2 + c2 + d2 = 1, p2 + q2 + r2 + s2 = 1,

then a general 4D rotation is attained by

A = MLMR.



Four-Dimensional Rotations

This new two-dimensional FrFT has almost all the properties ofthe standard FrFT, including its effect on the Wigner distribution.It is known that each 4-dimensional rotation can bedecomposed into two matrices representing left andright-multiplication by a unit quaternion. In fact, let

ML =

a −b −c −db a −d cc d a −bd −c b a

, MR =

p −q −r −sq p s −rr −s p qs r −q p

.

(17)where

a2 + b2 + c2 + d2 = 1, p2 + q2 + r2 + s2 = 1,

then a general 4D rotation is attained by

A = MLMR.



The Main Theorem

TheoremThe Wigner distribution WFα,β of the two-dimensional fractionalFourier transform Fα,β of a function f (x , y) is obtained from theWigenr distribution Wf of f by a 4-dimensional rotation throughthe matrix

A =

cos γ cos δ cos γ sin δ − sin γ cos δ − sin γ sin δ− cos γ sin δ cos γ cos δ sin γ sin δ − sin γ cos δsin γ cos δ sin γ sin δ cos γ cos δ cos γ sin δ− sin γ sin δ sin γ cos δ − cos γ sin δ cos γ cos δ

.

The matrix A is a genuine 4-dimensional rotation matrix.



The Main Theorem

Which is much more general than the 4-dimensional rotationwe saw before

cos θ1 0 − sin θ1 00 cos θ2 0 − sin θ2sin θ1 0 cos θ1 00 sin θ2 0 cos θ2

. (18)

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