A New Two-Dimensional Fractional Fourier...
Transcript of 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
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
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
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
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
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
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
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
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
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
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
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
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
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
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
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
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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)
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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)
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
The Mittag-Leffler Transform
The Mittag-Leffler 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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
The Mittag-Leffler Transform
The Mittag-Leffler 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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
The Mittag-Leffler 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).
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
The Mittag-Leffler Transform
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 .
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
The Mittag-Leffler Transform
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 .
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
The Mittag-Leffler Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
The Mittag-Leffler Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
A Practical Fractional Fourier Transform
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)
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
A Practical Fractional Fourier Transform
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)
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
A Practical Fractional Fourier Transform
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 .
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
A Practical Fractional Fourier Transform
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 .
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
A Practical Fractional Fourier Transform
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)
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
A Practical Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
A Practical Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
A Practical Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1.0
Figure: Zero FrFT
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
-6 -4 -2 0 2 4 6
0.5
1.0
1.5
2.0
Figure: One Quarter FT, θ = π/8
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
-6 -4 -2 0 2 4 6
0.5
1.0
1.5
2.0
Figure: One half FT θ = π/4
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
-6 -4 -2 0 2 4 6
0.5
1.0
1.5
2.0
Figure: One FT θ = π/2
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
-6 -4 -2 0 2 4 6
0.5
1.0
1.5
2.0
Figure: One FT θ = π/2
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
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 ,
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
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).
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
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).
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
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).
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
Motivations and Applications
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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).
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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).
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-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)
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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)
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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,
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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;
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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)
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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 .
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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.
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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)
Thanks for listening
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform
A New Two-Dimensional Fractional Fourier Transform
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)
Thanks for listening
Ahmed I. Zayed A New Two-Dimensional Fractional Fourier Transform