Transcript Introduction of Fractional Fourier Transform (FRFT)

Introduction of
Fractional Fourier Transform (FRFT)
Speaker: Chia-Hao Tsai
Research Advisor: Jian-Jiun Ding
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering
National Taiwan University
10/23/2009
1
Outlines
• Introduction of Fractional Fourier
Transform (FRFT)
• Introduction of Linear Canonical
Transform (LCT)
• Introduction of Two-Dimensional
Affine Generalized Fractional Fourier
Transform (2-D AGFFT)
• Relations between Wigner
Distribution Function (WDF), Gabor
Transform (GT), and FRFT
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2
Outlines (cont.)
• Implementation Algorithm of FRFT
/LCT
• Closed Form Discrete FRFT/LCT
• Advantages of FRFT/LCT contrast
with FT
• Optics Analysis and Optical
Implementation of the FRFT/LCT
• Conclusion and future works
• References
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3
Introduction of Fractional Fourier
Transform (FRFT)
• The FRFT: a rotation in time-frequency plane.

R
•
: a operator
• Properties of R 
- R 0 =I: zero rotation
 /2
R
: FT operator
- R : time-reverse operator
- R3 /2 : inverse FT operator
2
R
- =I: 2π rotation
-R R  R  : additivity
(I: identity operator)
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4
Introduction of FRFT (cont.)
• Definition of FRFT:


R x  X  (u)  O F ( x(t ))

 
x(t ) K  (t, u)dt
2
 1  j cot  j u 2 cot  
j t cot   jut csc 
dt , if  isn't a multiple of 
e 2
e
 x(t )e 2

2



x(t ), if  is multiple of 2

x(t ), if    is multiple of 2


( K  (t , u): Kernel of FRFT)
10/23/2009
5
Introduction of FRFT (cont.)
• When   0 : identity
• When   0.5 : FT
• When  isn’t equal to a multiple of   0.5 , the
FRFT is equivalent to doing  / (0.5 ) times of the FT.
-when   0.2  doing the FT 0.4 times.
-when   0.25  doing the FT 0.5 times.
-when    / 3  doing the FT 2/3 times.
10/23/2009
6
Introduction of FRFT (cont.)
• An Example for the FRFT of a rectangle
(Blue line: real part, green line: imaginary part)
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7
Introduction of Linear Canonical
Transform (LCT)
• The LCT is more general than the FRFT. The FRFT has
one free parameter (), but the LCT has four
parameters (a, b, c, d) to adjust the signal.
• The LCT can use some specific value to change into the
FRFT.
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8
Introduction of LCT (cont.)
• Definition of LCT:

F a ,b,c,d (u )  OF
a ,b , c , d 
x  t 
jd 2 
1
ja 2

1

j
ut
e 2 b u  e b e 2 b u x(t )dt , b  0

  j 2 b
j

cd u 2
d e2
x(du ), b  0

with the constraint ad – bc = 1
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9
Introduction of LCT (cont.)
• Additivity property:
 a2 ,b2 ,c2 ,d2 
OF
O
 a1 ,b1 ,c1 ,d1 
F
f   a2


h   c2
e
where  g

• Reversibility property:
 d ,  b ,  c ,a 
OF
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O
 a ,b,c,d 
F

 x t   OF
 e , f , g ,h 
x t 
b2   a1 b1 
d 2   c1 d1 

 x t   x(t )
10
Introduction of LCT (cont.)
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11
Fractional / Canonical Convolution
and Correlation
• FT for convolution:
y (t )  IFT  FT  x(t )  FT  h(t )  
• FT for correlation :

y(t )  IFT FT  x(t )  FT  h(t ) 
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
12
Fractional / Canonical Convolution
and Correlation (cont.)
• Fractional convolution (FRCV):
y (t )  O F  OF  x(t )   OF  h(t )  
 x(t )h(t )
 transformed domain
Y  (u)  X  (u)  H  (u)
• Fractional correlation (FRCR) :
P1, P2.P3( x(t ), h(t ))  P3
type1: Ocorr
O F O FP1( x(t ))  O FP2(h(t ))


P1, P 2.P3( x(t ), h(t ))  P3
type2: Ocorr
O F  O PF1( x(t ))  O FP 2(h(t )) 
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Fractional / Canonical Convolution
and Correlation (cont.)
• Canonical convolution (CCV):

y(t )  O (Fd ,b,c,a ) O (Fa,b,c,d )  x(t )   O (Fa,b,c,d )  h(t ) 

 x(t )(a,b,c,d )h(t )
 transformed domain
Y (a,b,c,d )(u)  X (a,b,c,d )(u)  H (a,b,c,d )(u)
• Canonical correlation (CCR) :
type1:
( a,b,c,d ),( e, f , g ,h ),( m,n, s ,v )
Ocorr
type2:

m, n , s , v )
( a,b,c,d )
( e, f , g , h )
( x(t ), h(t ))  O(corr
(h(t ))
Ocorr ( x(t ))  Ocorr


a ,b,c ,d ),( e, f , g ,h ),( m,n, s ,v )
m, n , s , v )
a ,b , c , d )
e, f , g , h )
( x(t ), h(t ))  O(corr
( x(t ))  O(corr
(h(t ))
O (corr
O(corr
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
Introduction of Two-Dimensional Affine
Generalized Fractional Fourier Transform (2-D
AGFFT)
• The 2-D AGFFT that it can be regarded as
generalization of 2-D FT, 2-D FRFT, and 2-D LCT.
• Definition of 2-D AGFFT:
where
,
,
,
, and
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Introduction of 2-D AGFFT (cont.)
where
,
• Reversibility property:
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16
Introduction of 2-D AGFFT (cont.)
• When
cos   cos  2
A
 cos   sin  2
 sin   cos  1
 cos   
 cos   sin  1
1  2
,

B

cos   cos  1 
 sin   sin  1

 cos  1   2 
 sin   sin  2 
cos  1   2  
,
sin   cos  2 

cos  1   2  
 cos   cos  1
 cos   
1 2
D
 cos   sin  1

 cos  1 2 
 cos   sin  1 
cos  1 2  
,
cos   cos  1 

cos  1 2  
 sin   cos  2 sin   sin  1 
C
,

  sin   sin  2 sin   cos  1
the 2-D AGFFT can become the 2-D unseparable FRFT
which was introduced by Sahin et al.
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Introduction of 2-D AGFFT (cont.)
2-D AGFFT
a12=a21=b12=b21=c12
=c21=d12=d21=0
2-D separable
canonical
a11=a22=d11=d22=1
c11=c22=0
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B=C=0
Sahin’s 2-D
unseparable
FRFT
Geometric
twisting
(a11,b11,c11,d11)=(cosα,sinα,-sinα,cosα)
(a22,b22,c22,d22)=(cosβ,sinβ,-sinβ,cosβ)
2-D separable
Fresnel
2-D separable
FRFT
α = β = π/2
α = β = -π/2
2-D forward
Fourier
2-D inverse
Fourier
α=β=0
Identity
operation
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2-D Affine Generalized Fractional
Convolution/ Correlation
• 2-D Affine Generalized Fractional Convolution (2-D
AGFCV):
DT,-BT,-CT, AT 

z  x, y   O F
O(FA,B,C,D)  f  x, y   O(FA,B,C,D)  g  x, y 
-applications of 2-D AGFCV:
filter design, generalized Hilbert transform, and mask
• 2-D Affine Generalized Fractional Correlation (2-D
AGFCR):


*
z  x, y   O F
f
x
,
y
g


 x, y 


-applications of 2-D AGFCR: 2-D pattern recognition
 E,F,G,H 
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( A,B,C,D)
OF
( A' ,B' ,C' ,D' )
 OF

19
Relation Between FRFT and Wigner
Distribution Function (WDF)
• Definition of WDF:
1 
W f (t , w) 

2 
 t 
f
 2
 *  t     jw
 f 
 e d

 2 
• The property of the WDF:
-high clarity
-with cross-term problem
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Relation Between FRFT and WDF
(cont.)
• Why does the WDF have a cross-term problem?
*
f
((
t


)
/
2)
((t  ) / 2)
f
• Ans: autocorrelation-term
of the WDF is existed
• If f (t )  s(t )  r (t ) , its WDF will become
W f (t, w)  W s(t, w)  W r (t, w).
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Relation Between FRFT and WDF
(cont.)
• Clockwise-Rotation Relation:
W F (u, v)  W f (u cos   v sin  , u sin   v cos  )
• The FRFT with parameter  is equivalent to the
clockwise-rotation operation with angle  for WDF.
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Relation Between FRFT and Gabor
Transforms (GT)
• Definition of GT:
1   t 2/2  jw (t /2) 
f ( )d
G f (t , w) 
e
 e
2
• The property of the GT:
-with clarity problem
-avoid cross-term problem
-cost less computation time
2



x
exp 
  0.0001, when x  4.2919
 2 


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23
Relation Between FRFT and GT (cont.)
• Why can the GT avoid the cross-term problem?
• Ans: the GT no have the autocorrelation-term
f ((t   ) / 2) f *((t  ) / 2)
• If f (t )  s(t )  r (t ) , its GT will become
G f (t, w)  G s(t, w)  Gr (t, w).
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Relation Between FRFT and GT (cont.)
• Clockwise-Rotation Relation:
G F (u, v)  G f (u cos   v sin  , u sin   v cos  )
• The FRFT with parameter  is equivalent to the
clockwise-rotation operation with angle  for GT.
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Relations Between FRFT, WDF, and GT
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26
Relations Between FRFT, WDF, and GT
(cont.)
• Definition of GWT:
C f (t, w)  p(G f (t, w),W f (t, w))
• Clockwise-Rotation Relation:
C F (u, v)  C f (u cos   v sin  , u sin   v cos  )
• Ex1: if p( x, y)  xy, then C f (t, w)  G f (t, w)W f (t, w).
• Ex2: if p( x, y)  x  y, then C f (t, w)  G f (t, w)  W f (t, w).
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Implementation Algorithm of FRFT/LCT
• Two methods to implement FRFT/LCT:
- Chirp Convolution Method
- DFT-Like Method
• To implement the FT, we need to use  N 2  log2N
multiplications.
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Implementation Algorithm of FRFT/LCT
(cont.)
Chirp Convolution Method:
• For LCT, we sample t-axis and u-axis as pt and q u ,
then the continuous LCT becomes
F ( a,b,c,d )(qu) 

jd 2 2 M
j
ja 2 2
1
 pqu t
q u
 e b
e2 b
e 2 b p t f ( pt )
j 2 b
p  M
j d 1 2 2 M
j
2 j a 1 2 2
1
p u q t 
q u

 e 2b
e2 b
e 2 b p t f ( pt ).
j 2 b
p  M
chirp multiplication
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chirp convolution
29
Implementation Algorithm of FRFT/LCT
(cont.)
• To implement the LCT, we need to use 2 chirp
multiplications and 1 chirp convolution.
• To implement 1 chirp convolution, we need to use
requires 2 DFTs.
• Complexity:
2P (2 chirp multiplications) + P  log 2P (2 DFTs)
 P  log 2P
(P = 2M+1 = the number of sampling points)
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Implementation Algorithm of FRFT/LCT
(cont.)
• This is 2 times of complexity of FT.
2
P
• To implement the LCT directly, we need to use
multiplications.
• So, we use Chirp Convolution Method to implement
the LCT that it can improve the efficiency of the LCT.
• For FRFT, its complexity is the same as LCT.
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Implementation Algorithm of FRFT/LCT
(cont.)
DFT-Like Method:
a b   1 0  0 1 1 b 0  1 0
 c d   d b 1 1 0  0 b a b 1 , b  0

 




(chirp multi.) (FT) (scaling) (chirp multi.)
Step.1: chirp multi.
Step.2: scaling
Step.3: FT
Step.4: chirp multi.
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f 1(t )  exp( jat 2 2b) f (t )
f 2(t )  b f 1(bt )  b e
j
ab 2
t
2 f
(bt )
1
  jut
f 2(t )dt
 e
j 2
2
F 4(u)  exp( jd u 2b) F 3(u)
F 3(u) 
32
Implementation Algorithm of FRFT/LCT
(cont.)
• We can implement the LCT:
F ( a,b,c,d )(s u )  t
jd 2 2 M
2 pq  j ab p 2 t 2

b
j
f ( pb t ) 
q u
e 2

 e
e2 b
P 

j 2 b
p  M
where t u  2 P
• To implement the LCT, we need to use 2 M-points
multiplication operations and 1 DFT.
(P = 2M+1 = the number of sampling points)
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Implementation Algorithm of FRFT/LCT
(cont.)
• Complexity:
2P (2 multiplication operations) + ( P 2)  log2P (1 DFT)
 ( P 2)  log 2P
• This is only half of the complexity of Chirp
Convolution Method.
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Implementation Algorithm of FRFT/LCT
(cont.)
• When using Chirp Convolution Method, the sampling
interval is free to choose, but it needs to use 2 DFTs.
• When using the DFT-like Method, although it has
some constraint for the sampling intervals, but we
only need 1 DFT to implement the LCT.
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Closed Form Discrete FRFT/LCT
• DFRFT/DLCT of type 1:
applied to digital implementing of the continuous
FRFT
• DFRFT/DLCT of type 2:
applied to the practical applications about digital
signal processing
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Closed Form Discrete FRFT/LCT (cont.)
• Definition of DFRFT of type 1:
Y  ( m) 
2 nm j
cot  m2 u 2 N
sin   j cos 
j
cot  n 2 t 2
 e 2 M 1 e 2
2
 y ( n)
e
2 M 1
n  N
j
when   2D+(0, ), D is integer (sin > 0)
Y  ( m) 
2 nm j
cot  m2 u 2 N
 sin   j cos 
j
cot  n 2 t 2

2

 y ( n)
2 M 1 e 2
e
e
2 M 1
n  N
j
when   2D+(, 0), D is integer (sin < 0)
with the constraint M  N and u t  2  sin  / (2M 1)
(2N+1, 2M+1 are the number of points in the time,
frequency domain)
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Closed Form Discrete FRFT/LCT (cont.)
• when   D   and D are integer, because we can’t find
proper choice for u and t that can’t use as above.
Y  (m)  y(m), when  = 2D
Y  (m)  y(m), when  = (2D+1)
• The DFRFT of type 1 is efficient to calculate and
implement.
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Closed Form Discrete FRFT/LCT (cont.)
• Complexity:
2P (2 chirp multiplications)+ ( P 2)  log2P (1 FFT)
 ( P 2)  log 2P
• But it doesn’t match the continuous FRFT, and lacks
many of the characteristics of the continuous FRFT.
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Closed Form Discrete FRFT/LCT (cont.)
• Definition of DLCT of type 1:
N
d
2 nm
a
1
j m2 u 2
j
j n 2 t 2
 e 2b
 y(n)
 e 2 M 1  e 2b
Y ( a,b,c,d )(m) 
2M  1
n N
N
d
2 nm
a
1
j m2 u 2
j
j n 2 t 2
 e 2b
 y(n)
 e 2 M 1  e 2b
Y ( a,b,c,d )(m) 
2M  1
n N
(b>0)
(b<0)
with the constraint u t  2  b / (2M 1)
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Closed Form Discrete FRFT/LCT (cont.)
• When b = 0, we can’t also find proper choice for u
and t that can’t use as above.
j
cd 
Y ( a,0,c,d )(m)  d  e 2 m u y(d  m) (b=0, d is integer)
2
2
1 j c m2 u 2 N M j 2 sgn( a )k m  j 2 nk
 e 2a
 e 2 N 1  y(n)
  e
Y ( a,0,c,d )(m) 
2 M 1
R
n  N k  M
(b=0, d isn’t integer)
where R= (2M+1)(2N+1) and
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u  (2N 1)  a t / (2M 1)
41
Closed Form Discrete FRFT/LCT (cont.)
• Definition of DLCT of type 2:
From DLCT of type 1, we let p = (d/b)u2, q =
(a/b)t2
N
j
2 snm
j 2
1
 pm2
j
e2
 e 2 M 1  e 2 qn  y(n)
Y ( p,q,s )(m) 
2M  1
n  N
where M  N
(2N+1, 2M+1 are the number of points in the time,
frequency domain), s is prime to M
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42
Closed Form Discrete FRFT/LCT (cont.)
• By setting p = q and s = 1, we can define the DFRFT
from the DLCT and get the formula of DFRFT of type
2.
• Definition of DFRFT of type 2:
N
j 2
2 nm
j 2
1
m  p
j
e2
 e 2 M 1  e 2 n  p  y(n)
Y ( p )(m) 
2M  1
n  N
where M  N
• Complexity of DFRFT/DLCT of type 2 is the same as
complexity of DFRFT/DLCT of type 1.
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Advantages of FRFT/LCT contrast with
FT
• The FRFT/LCT are more general and flexible than the
FT.
• The FRFT/LCT can be applied to partial differential
equations (order n > 2). If we choice appropriate
parameter , then the equation can be reduced
order to n-1.
• The FT only deal with the stationary signals, we can
use the FRFT/LCT to deal with time-varying signals.
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44
Advantages of FRFT/LCT contrast with
FT (cont.)
• Using the FRFT/LCT to design the filters, it can reduce
the NMSE. Besides, using the FRFT/LCT, many noises
can be filtered out that the FT can’t remove in optical
system, microwave system, radar system, and
acoustics.
• In encryption, because the FRFT/LCT have more
parameter than the FT, it’s safer in using the
FRFT/LCT than in using the FT.
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45
Advantages of FRFT/LCT contrast with
FT (cont.)
• In signal synthesis, using the transformed domain of
the FRFT/LCT to analyze some signal is easier than
using the time domain or frequency domain to
analyze signals.
• In multiplexing, we can use multiplexing in fractional
domain for super-resolution and encryption.
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46
Applications of FRFT/LCT
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47
Use FRFT/LCT to Represent Optical
Components
• Propagation through the cylinder lens with focus
length f:
1
a b  
 c d     2  f

 
0
1
• Propagation through the free space (Fresnel
Transform) with length z:
 a b  1  z 2 
 c d   0

1

 

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48
Implementation FRFT/LCT by Optical
Systems
• Case1:
1
0  1 b  
1
0
a b  
 c d   (d  1) / b 1 0 1 (a 1) / b 1 , b  0

 



Free space
Cylinder lens
Cylinder lens
Input
f1
f2
Output
d0
The
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implementing of LCT (b≠0) with 2 cylinder lenses and 1 free space.
49
Implementation FRFT/LCT by Optical
Systems (cont.)
2 b
2 b
2 b
, d0 
, f2
, for LCT
f1
 (1  a)

 (1  d )
{a, b, c, d} = {cosα, sinα, -sinα, cosα}
f1 f 2
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2 cot( / 2)

, d0 
2 sin 

, for FRFT
50
Implementation FRFT/LCT by Optical
Systems (cont.)
• Case2:
a b  1 (a  1) / c  1 0 1 (d 1) / c 
, c0
 c d   0





1
1

 
 c 1  0

Free space Cylinder lens Free space
Input
f0
d1
The implementing
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Output
d2
of LCT (c ≠ 0) with 1 cylinder lenses and 2 free space.
51
Implementation FRFT/LCT by Optical
Systems (cont.)
2 (d  1)
2
2 ( a  1)
, f0
, f2
, for LCT
d1 
c
c
c
{a, b, c, d} = {cosα, sinα, -sinα, cosα}
d1  d 2 
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2 tan( / 2)

, f0
2 csc 

, for FRFT
52
Implementation FRFT/LCT by Optical
Systems (cont.)
• If we want to have shorter length in the optical
implementation of LCT, then case1 is preferred.
• If we want to retrench the number of lenses we use,
or avoid placing the lenses contacting to the input
and output, then case2 is preferred.
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53
Conclusion and future works
• FRFT/LCT are more general and flexible than
the FT. Then, We hope to find other
applications of FRFT/LCT.
• Can we find more general functions?
Then, we let the functions use in other
applications.
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54
References
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1998.
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55
References (cont.)
[8] Zayed, “A convolution and product theorem for the fractional Fourier transform,” IEEE Signal
Processing Letters, vol. 5, no. 4, pp. 101-103, Apr. 1998.
[9] A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J.
Opt. Soc. Amer. A, vol. 10, no. 10, pp.2181–2186, Oct. 1993.
[10]D. A. Mustard, “The fractional Fourier transform and the Wigner distribution,”J. Australia
Math. Soc. B, vol. 38, pt. 2, pp. 209–219, Oct.1996.
[11]S. C. Pei and J. J. Ding, “Relations between the fractional operations and the Wigner
distribution, ambiguity function,” IEEE Trans. Signal Process., vol. 49, no. 8, pp. 1638–1655,
Aug. 2001.
[12]H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, “Convolution, filtering, and
multiplexing in fractional Fourier domains and their rotation to chirp and wavelet transform,”
J. Opt. Soc. Amer. A, vol. 11, no. 2, pp. 547–559, Feb. 1994
[13]A. Sahin, M. A. Kutay, and H. M. Ozaktas, ‘Nonseparable two-dimensional fractional Fourier
transform`, Appl. Opt., vol. 37, no. 23, p 5444-5453, Aug 1998.
[14]S. C. Pei, and J. J. Ding “Two-Dimensional affine generalized fractional Fourier transform,”
IEEE Trans. Signal Processing, Vol.49, No. 4, April 2001.
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