Transcript The Fractional Fourier Transform and Its Applications
The Fractional Fourier Transform and Its Applications
Presenter:
Pao-Yen Lin
Research Advisor:
Jian-Jiun Ding , Ph. D.
Assistant professor
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University
Digital Image and Signal Processing Lab Graduate 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 Transform a) 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 •
F
•
F
T
T
f
f
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Fractional Fourier Transform (FrFT) (cont.)
• Constraints of FrFT Boundary condition
T
0
T
1 Additive property
T
T
T
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Definition of FrFT
• Eigenvalues and Eigenfunctions of FT • exp Hermite-Gauss Function
i
2
f
4 2 2
n
x
2 0
n
2 1 4 2
n n
!
H n
2
x
exp
x
2
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Definition of FrFT (cont.)
• Eigenvalues and Eigenfunctions of FT
n
n n
,
n
e
in
2
n
0
A n
n
,
A n
n
n
0
A n
e
in
2
n
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Definition of FrFT (cont.)
• Eigenvalues and Eigenfunctions of FrFT Use the same eigenfunction but α order eigenvalues
n
e
n
0
A e n
2
n
2
n
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Definition of FrFT (cont.)
• Kernel of FrFT
B
n
0
n
n
1 2 2 exp
n
0
e
2
n n
!
2
B
x
2
H n
x
2 2 2
x
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Definition of FrFT (cont.)
X
1 2
j
cot
e j u
2 2 cot
j t
2 2 cot
jut
csc
dt
if is a multiple of 2
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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
O F
G
1
j
2
b e
2
j d b u
2
e
b j ut e
2
j a t
2
b O
F
de j cd
2
u
2 for
b
0 for
b
0 where
ad
bc
1
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Linear Canonical Transform (LCT) (cont.)
• 1.
Properties of LCT When , the LCT becomes
c d
cos sin cos
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2.
FrFT.
Additive property
O F
, 2 , 2 ,
d
O
F
1 where
e g f h
1
a
2
c
2
b
2
d
2
a
1
c
1
O
F b
1
d
1
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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
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• • Definition
x
2 Property Total energy 2 2
x
x
j
2
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Relation to Wigner Distribution
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f
x
2
x
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Relation to Wigner Distribution (cont.)
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• WD V.S. FrFT
W f
W f
x
cos
v
• Rotated with angle
v
cos
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Relation to Wigner Distribution (cont.)
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• Examples exp
j
x
2
v x c c
x
v
c
v c v
slope=
b
2 exp
j
2
b x
2 2
b x
2 2
b x
1 1
v
b
0
b
1
x
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Relation to Chirp Transform
• for
f
0
x
0
x
0
c
exp
j
ˆ 4 2 sin 1 2
j
x
2 cot 2
x x
0
c
csc
x
0 2
c
cot Note that 0 , 1
x
0 cos
x
1 sin
x
0
c
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Relation to Chirp Transform (cont.)
f
f
f
0 0
f
0
f
x
f
0
x
x
x
0
x
0
x
dx
• Generally,
f
f
x
B
dx
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Relation to Gabor Transform (GT)
• Special case of the Short-Time Fourier Transform (STFT) • Definition
G f
1/2
e
(
t
) 2 2
e
j
t
2 )
f
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Relation to Gabor Transform (GT) (cont.)
• GT V.S. FrFT •
G F
G f
u
cos
v
Rotated with angle
v
cos
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Relation to Gabor Transform (GT) (cont.)
• Examples exp
jt
2 10 exp
jt
2 2
t
1, 0 otherwise, 4 2 10
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10 0 10 0 10 0 10 0 -10 -10 0 10 -10 -10 0 -10 10 -10 0 -10 10 -10 0 10 (a)GT of (b)GT of (c)GT of (d)WD of
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GT V.S. WD
• GT has no cross term problem • GT has less complexity exp
x
2 2 0.0001 when
x
4.2919
• WD has better resolution • Solution: Gabor-Wigner Transform
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Relation to Gabor-Wigner Transform (GWT)
• Combine GT and WD with arbitrary function
C f
f
f
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Relation to Gabor-Wigner Transform (GWT) (cont.)
• Examples 1.
In (a) 2.
In (b)
C f C f
f
min
f
G t f
2
f
(a) 10 5 0 -5 -10 -10 10 (b) 10 5 0 -5 -10 -10 0 0 10
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Relation to Gabor-Wigner Transform (GWT) (cont.)
• Examples 3.
In (c) 4.
In (d)
C f f
f
G f
2.6
f
W f
0.6
0.25
(c) 10 5 0 -5 -10 -10 10 (d) 10 5 0 -5 -10 -10 0 0 10
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Relation to Wavelet Transform
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• The kernels of Fractional Fourier Transform corresponding to different wavelet family.
f
y
sec
C
exp
j
y
x
tan 1 2 exp 2
y
x
sec 2 sin 2
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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
R G
2 2 • Power Spectral Density (PSD)
S g
2
FT g
g
e
j
d
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Relation to Random Process (cont.)
• FrFT V.S. Stationary random process
R G
ju
cot
e
cos
e
j
sin
u
cos
R g
cos , cos 0
R G
, arg
R G
• Nearly stationary sec
R g
sec
u
tan
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Relation to Random Process (cont.)
• FrFT V.S. Stationary random process for cos 0
R G
R G
g u
, when 2
H
1 2
g
u
, when 2
H
3 2
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Relation to Random Process (cont.)
• FrFT V.S. Stationary random process PSD:
S G
S g
u
sin
v
cos , when cos
S G
S g
, when cos 0 0
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Relation to Random Process (cont.)
• FrFT V.S. Non-stationary random process
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Auto-correlation function
R G
,
e ju
cot
e
jut
3 csc
e
j t
2 2 csc
e
2
j
PSD
S G
S g
u
cos
v
cot 2 2, 2 3
v
cos rotated with angle
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Relation to Random Process (cont.)
• Fractional Stationary Random Process
G
is stationary and the autocorrelation
u
random process.
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Relation to Random Process (cont.)
• Properties of fractional stationary random process 1.
2.
After performing the fractional filter, a white noise becomes a fractional stationary random process. 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
u u
noise
x
noise signal • Filtering in fractional domain
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Filter design using FrFT (cont.)
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• Random noise removal function and PSD are:
R g
,
S g
After doing FrFT
R G
,
S g
Remain unchanged after doing FrFT!
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Filter design using FrFT (cont.)
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• Random noise removal
u u x
signal • Area of WD
≡
Total energy signal
x
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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 Components
1.
2.
Propagation through the cylinder lens with focus length
f
a c b d
1 2
f
0 1 Propagation through the free space (Fresnel Transform) with length
z
a c b d
1 0
z
1 2
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Using FrFT/LCT to Represent the Optical Systems
a c b d
1 2
f
1 1
d
0 2
d
0
f f
1 2 0 1 1 0 1
f
2
f
1 1
f
2
d
0 1 2 1 2 1
d
0
d
0 2
f
1
f
2
f
1
f
2 0 1 input
d
0 output
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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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a c
a c b d
d
1 1
b b d
1 0 0 1 1 0
a
1 1
c
1
c b
1
a
1 1
b
0 1 1 0 0 1 if
b
0
d
1 1
c
if
c
0
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Implementing FrFT/LCT by Optical Systems (cont.)
f
1
f
2 input
d
0 output The implementation of LCT with 2 cylinder lenses and 1 free space
f
1 input
d
0
d
1 output
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The implementation of LCT with 1 cylinder lens and 2 free spaces
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Convolution
g
g
f
g
g
f
h
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Convolution (cont.)
+1
f
h
+1 -1
f
+1
f
-1
f
h
-1
h
+1
h
-1 +1 -1 +1 -1 1
f
+1
f
-1
f
+1
h
-1
h
f
+1
f
-1 1
h f
+1
h
-1
h
f
1
h f
1
h
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Multiplexing using FrFT
u u x
TDM FDM
x
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Multiplexing using FrFT
u u x
Inefficient multiplexing Efficient multiplexing
x
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Generalization of sampling theorem
• If is band-limited in some transformed
F
domain of LCT, i.e., 0, for
u
and for some value of then we can sample by the interval as
b
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Generalization of sampling theorem (cont.)
u u
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x
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Conclusion and future works
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• 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, 2 nd Edition, Prentice Hall, 1999.
[13] R. N. Bracewell, The Fourier Transform and Its Applications, 3 rd ed., Boston, McGraw Hill, 2000.
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Chenquieh!
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