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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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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
• is a transform of
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T f x F u
F T f
T f x F u
F T f
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
FT
• 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
1 4
222 exp
2 !n nnx H x x
n
Definition of FrFT (cont.)
• Eigenvalues and Eigenfunctions of
FT
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,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
Definition of FrFT (cont.)
• Eigenvalues and Eigenfunctions of
FrFT
Use the same eigenfunction but α order
eigenvalues
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2i nn nx e x
2
0
i nn n
n
f x x A e x
Definition of FrFT (cont.)
• Kernel of FrFT
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,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
eH x H x
n
Definition of FrFT (cont.)
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2 2
cot cot csc2 2
1 cotif 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
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
where
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2 2
, , ,
, , ,
2 21 for 0
2
a b c dF a b c d
j d j j au ut t
b 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
Linear Canonical Transform (LCT) (cont.)• Properties of LCT
1.When , the LCT
becomes
FrFT.
2.Additive property
where
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cos sin
sin cos
a b
c 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 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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x
x
2
f u
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
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
Relation to Chirp Transform
• for
Note that
is the same as rotated by
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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
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 2, 1/2t t
j
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
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
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
-5
0
5
10
-10 0 10-10
-5
0
5
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
-5
0
5
10
-10 0 10-10
-5
0
5
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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2 2
2
1 2
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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cot
sin cos, ,cos cos
uju j
G g
eR u e R
, sec secG gR u R
arg , tanGR u u
cos 0
Relation to Random Process (cont.)• FrFT V.S. Stationary random process
for
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cos 0
, , when 2 1 2G gR u S u H
, , when 2 3 2G gR u S u H
Relation to Random Process (cont.)• FrFT V.S. Stationary random process
PSD:
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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
PSD
rotated with angle
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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
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.
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g t
G u
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
noise
noise
u
x
u
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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g t
,gR gS
,GR gS
Filter design using FrFT (cont.)• Random noise removal
• Area of WD ≡ Total energy
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signal
x
u
signal
x
u
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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f
z
1 0
2 1
a b
c d f
1 2
0 1
a b z
c d
Using FrFT/LCT to Represent the Optical Systems
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input output
2f1f
0d
0
1 2
0 2 0
00 1
1 2 1 2
1 0 1 01 2
2 1 2 10 1
1 2
2 1 11
a b d
f fc d
d f d
dd f
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
c d c
Implementing FrFT/LCT by Optical Systems (cont.)
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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
Convolution
• Convolution in domain
• Multiplication in domain
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g x g f h f x h x
g f h
g x g f h f x h x
g f h
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
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
Multiplexing using FrFT
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x
u
TDM FDM
x
u
Multiplexing using FrFT
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x
u
Inefficient multiplexing Efficient multiplexing
x
u
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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f t
, , , 0, for and for some value of , , ,a b c dF u u a b c d
f t
b
Generalization of sampling theorem (cont.)
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u
x
u
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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