Transcript Fractional fourier transform

Presenter:
Hong Wen-Chih
2015/7/21
1
Outline




Introduction
Definition of fractional fourier transform
Linear canonical transform
Implementation of FRFT/LCT
 The Direct Computation
 DFT-like Method
 Chirp Convolution Method
 Discrete fractional fourier transform
 Conclusion and future work
2015/7/21
2
Introduction
 Definition of fourier transform:
F   
1

2


e
 j w t
f  t   dt
 Definition of inverse fourier transform:
f t  
2015/7/21
1
2



e
j w t
F     dt
3
Introduction
In time-frequency representation
 Fourier transform: rotation π/2+2k π
 Inverse fourier transform: rotation -π/2+2k π
 Parity operator: rotation –π+2k π
 Identity operator: rotation 2k π
And what if angle is not multiple of π/2 ?
2015/7/21
4
.
Introduction

v

u

t
Time-frequency plane and a set of coordinates ( u , v )
rotated by angle α relative to the original coordinates ( t , w )
2015/7/21
5
Fractional Fourier Transform
 Generalization of FT
 use F  to represent FRFT
 The properties of FRFT:
 Zero rotation: F 0  I
 Consistency with Fourier transform:
 Additivity of rotations: F  F   F   
 2π rotation: F 2   I
F
 /2
 F
Note: do four times FT will equal to do nothing
2015/7/21
6
Fractional Fourier Transform
 Definition:
F ( u ) 




x ( t ) K  ( t , u ) dt
1  j cot 
2
j
e
u
2
2
cot 



j
x (t ) e
t
2
2
cot 
e
jut csc 
dt
 Note: when α is multiple of π, FRFTs degenerate
into parity and identity operator
2015/7/21
7
Linear Canonical Transform
 Generalization of FRFT
 Definition:
F( a , b , c , d ) ( u ) 
1
j 2 b
jd
 e 2b
j
F( a ,0 , c , d ) ( u )  d  e 2
cd u
u
2

j


e
b
ut
ja
t
2
e 2 b f ( t )  dt
when b≠0
2
f ( du )
when b=0
 a constraint: ad  bc  1 must be satisfied.
2015/7/21
8
Linear Canonical Transform
 Additivity property:
( a 2 ,b2 , c 2 , d 2 )
OF
O
( a1 , b1 , c1 , d 1 )
F
where
 f t   O F( e , f , g , h )  f t 
e

g
f   a2

h   c2
b 2   a1

d 2   c1
b1 

d1 
 Reversibility property:
( d , b , c ,a )
OF
O
( a ,b , c , d )
F
where
2015/7/21
 f t     f t 
 d
I 
 c
b   a
 
a  c
b

d
9
Linear Canonical Transform
 Special cases of LCT:
 {a, b, c, d} = {0, 1, 1, 0}:
 {a, b, c, d} = {0, 1, 1, 0}:
 f (t )  
( 0 ,  1 ,1 , 0 )
 F ( )  
OF
( 0 ,1,  1, 0 )
OF
 j  FT  f ( t ) 
j  IFT  F ( ) 
 {a, b, c, d} = {cos, sin, sin, cos}:
(cos  , sin  ,  sin  , cos  )
OF
 f t   
e
 j

1/ 2

OF
 f t  
 {a, b, c, d} = {1, z/2, 0, 1}: LCT becomes the 1-D Fresnel
transform
 {a, b, c, d} = {1, 0, , 1} : LCT becomes the chirp multiplication
operation
 {a, b, c, d} = {, 0, 0, 1}: LCT becomes the scaling operation.
2015/7/21
10
Implementation of FRFT/LCT
 Conventional Fourier transform
 Clear physical meaning
 fast algorithm (FFT)
 Complexity : (N/2)log2N
 LCT and FRFT
 The Direct Computation
 DFT-like Method
 Chirp Convolution Method
2015/7/21
11
Implementation of FRFT/LCT
 The Direct Computation
 directly sample input and output
2
Y( a , b , c , d )  u  
1 e
j 2 b
j
u d
2b

t
2015/7/21
1 e
j 2 b
e
2
ut
b
j
e
t a
2b
x  t  dt
u
2
Y( a , b , c , d )  m  u  
 j

j
2
m u d
2b
n2
e
 j
2
mn u  t
b
j
e
2
n t a
2b
x  n t   t
n  n1
12
Implementation of FRFT/LCT
 The Direct Computation
 Easy to design
 No constraint expect for ad  bc  1
 Drawbacks
 lose many of the important properties
 not be unitary
 no additivity
 Not be reversible
 lack of closed form properties
 applications are very limited
2015/7/21
13
Implementation of FRFT/LCT
 Chirp Convolution Method
p t
 Sample input and output as
F( a , b , c , d ) ( u ) 
F( a , b , c , d ) ( q  u ) 
F  a ,b ,c , d   q  u  
2015/7/21
jd
1
j 2 b
 e 2b
j 2 b
j 2 b
2

jd
1
1
u
e
e2b
 j


2
e
2
q u
ut
b
ja
t
2
e 2 b f ( t )  dt
M

q u
and

e
j
b
p  q  u  t
ja
e2b
2
2
p t
f ( p t )
p M
j d 1 2 2
q u
2 b
M
j
 eb
pM
 q  u  p  t 2
e
j a 1 2 2
p t
2 b
f  p t 
14
Implementation of FRFT/LCT
 Chirp Convolution Method
 implement by


2 chirp multiplications
1 chirp convolution
 complexity

2P (required for 2 chirp multiplications) + Plog2P (required
for 2 DFTs)
 Plog2P (P = 2M+1 = the number of sampling points)
 Note: 1 chirp convolution needs to 2DFTs
2015/7/21
15
Implementation of FRFT/LCT
 DFT-like Method
 constraint on the product of t and u

a

c
b  1
  
d  d / b
0  0

1  1
1  1 / b

0  0
0  1

b  a / b
 t  u  2 / P
0

1
(chirp multi.) (FT) (scaling) (chirp multi.)
2015/7/21
16
Implementation of FRFT/LCT
 DFT-like Method
 Chirp multiplication:

f 1 t   exp jat
2

/ 2 b  f t 
 Scaling:
j
ab
t
2
f2 t  
b  f1  b  t  
b e
F3 u  
1
 f 2 t   dt
2
 f  bt 
 Fourier transform:
 Chirp multiplication:



j 2

F 4 u   exp jdu
2015/7/21
e
2
 j u t

/ 2 b  F 3 u 
17
Implementation of FRFT/LCT
 DFT-like Method
 For 3rd step
F3 u  
1
j 2



e
 j u t
 f 2 t   dt
 Sample the input t and output u as pt and qu
  t  u  2 / P
2015/7/21
18
Implementation of FRFT/LCT
 DFT-like Method
 Complexity



2015/7/21
2 M-points multiplication operations
1 DFT
2P (two multiplication operations) + (P/2)log2P (one DFT) 
(P/2)log2P
19
Implementation of FRFT/LCT
 Compare
 Complexity



Chirp convolution method: Plog2P (2-DFT)
DFT-like Method:
(P/2)log2P (1-DFT)
DFT:
(P/2)log2P (1-DFT)
 trade-off:

chirp. Method: sampling interval is FREE to choice

DFT-like method: some constraint for the sampling
intervals
 t  u  2 / P
2015/7/21
20
Discrete fractional fourier transform
 Direct form of DFRFT
 Improved sampling type DFRFT
 Linear combination type DFRFT
 Eigenvectors decomposition type DFRFT
 Group theory type DFRFT
 Impulse train type DFRFT
 Closed form DFRFT
2015/7/21
21
Discrete fractional fourier transform
 Direct form of DFRFT
 simplest way
 sampling the continuous FRFT and computing it
directly
2015/7/21
22
Discrete fractional fourier transform
 Improved sampling type DFRFT
 By Ozaktas, Arikan
 Sample the continuous FRFT properly
 Similar to the continuous case
 Fast algorithm
 Kernel will not be orthogonal and additive
 Many constraints
2015/7/21
23
Discrete fractional fourier transform
 Linear combination type DFRFT
 By Santhanam, McClellan
 Four bases:




DFT
IDFT
Identity
Time reverse
 F  n   A0    f  n   A1    F  n   A2    f   n   A3    F   n 
 A q   
2015/7/21
1
4
4
e
 

j    q k
2 

k 1
24
Discrete fractional fourier transform
 Linear combination type DFRFT
 transform matrix is orthogonal
 additivity property
 reversibility property
 very similar to the conventional DFT or the identity
operation
 lose the important characteristic of ‘fractionalization’
2015/7/21
25
Discrete fractional fourier transform
 Linear combination type DFRFT
 DFRFT of the rectangle window function for various angles :
 (top left) α= 0:01,
 (top right) α = 0:05,
 (middle left) α = 0:2,
 (middle right) α = 0:4,
 (bottom left) α =π/4,
 (bottom right) α =π/2.
2015/7/21
26
 (a) = 0.01
 (b) = 0.05
 (c) = 0.2
 (d) = 0.4
 (e) = π/4
 (f) = π/2
2015/7/21
27
Discrete fractional fourier transform
 Eigenvectors decomposition type DFRFT
 DFT : F=Fr – j Fi
 Search eigenvectors set for N-points DFT
 F  U  r U t  U  iU t
 F  U (  r   i )U t

2015/7/21
F


 U ( r   i ) U
t
28
Discrete fractional fourier transform
 Eigenvectors decomposition type DFRFT
 Good in removing chirp noise
 By Pei, Tseng, Yeh, Shyu



 cf. : DRHT can be H  F r  F i
F
2015/7/21

 d 0
d1

1

0

d N 1    


 0
0
e
 j 










0

Τ
  d0 

  d T 
 1 

   
0

T
 
d
 j  ( N 1 )
  N  1 
e
0
29
Discrete fractional fourier transform
 Eigenvectors decomposition type DFRFT
 DFRFT of the rectangle window function for various angles :
 (top left) α= 0:01,
 (top right) α = 0:05,
 (middle left) α = 0:2,
 (middle right) α = 0:4,
 (bottom left) α =π/4,
 (bottom right) α =π/2
2015/7/21
30
Discrete fractional fourier transform
 Group theory type DFRFT
 By Richman, Parks
 Multiplication of DFT and the periodic chirps
 Rotation property on the Wigner distribution
 Additivity and reversible property
 Some specified angles
 Number of points N is prime
2015/7/21
31
Discrete fractional fourier transform
 Impulse train type DFRFT
 By Arikan, Kutay, Ozaktas, Akdemir
 special case of the continuous FRFT
 f(t) is a periodic, equal spaced impulse train
 N = 2 , tanα = L/M
 many properties of the FRFT exists
1
0.8
0.6
0.4
0.2
0
-0.2
 many constraints
 not be defined for all values of
-0.4

-0.6
-0.8
-1
2015/7/21
0
5
10
15
20
25
32
Discrete fractional fourier transform
 Closed form DFRFT
 By Pei, Ding
 further improvement of the sampling type of DFRFT
 Two types


2015/7/21
digital implementing of the continuous FRFT
practical applications about digital signal processing
33
Discrete fractional fourier transform
 Type I Closed form DFRFT
 Sample input f(t) and output Fa(u)
y n   f n  Δ t 
 Then
Yα  m  
 Matrix form:
1  j  cot α
2π
Yα  m   Fα  m  Δ u 
j
Δ t  e
2
2
N
e
j
 j csc α  n  m Δ u Δ t
e
2
 cot α  n Δ t
2
2
 y n 
n N
Yα  m  
N
 Fα  m , n   y  n 
n N
y n  
M

m M
2015/7/21
2
 cot α  m Δ u
N

 Fα  m , n   Fα  m , k   y  k 
k  N
34
Discrete fractional fourier transform
 Type I Closed form DFRFT

y n  
M

m M

Δt
N

 Fα  m , n   Fα  m , k   y  k 
k  N
2
2π sin α
M

m M
N
j
e2

2
 cot α  k  n
2
Δ t 2
e
j  csc α  m   n  k Δ u Δ t
 y k 
k  N
 Constraint:
Δ u  Δ t  S  2π  sin α /  2 M  1 
2015/7/21
35
Discrete fractional fourier transform
 Type I Closed form DFRFT
Fα  m , n  
 and
M

1  j  cot α
2π
j
Δ t  e 2
2
 cot α  m Δ u
2
m M k  N
S  2π  n  m
j
e2
2 M 1
2
 cot α  n Δ t
2M  1
N

 Fα  m , n   Fα  m , k   y  k  
e
 j
2
 Δ t  y n 
2
2π sgn(sin α )  sin α
 choose S = sgn(sin) = 1
Fα  m , n  
2015/7/21
sin α  j sgn ( sin α ) cos α
2M  1
j
e2
2
 cot α  m Δ u
2
e
 j
sgn ( sin α )  2π  n  m
2 M 1
j
e2
2
 cot α  n Δ t
36
2
Discrete fractional fourier transform
 Type I Closed form DFRFT

Fα  m  
sin α  j cos α
2M  1
j
e
2
 cot α  m Δ u
2
2
N
e
 j
2 π n m
2 M 1
j
e
2
 cot α  n Δ t
2
2
 y n
n N
when   2D+(0, ), D is integer (i.e., sin > 0)

Fα  m  
 sin α  j cos α
2M  1
j
e
2
2
 cot α  m Δ u
2
N
e
j
2 π n m
2 M 1
j
e
2
2
 cot α  n Δ t
2
 y n
n N
when   2D+(, 0), D is integer (i.e., sin < 0)
2015/7/21
37
Discrete fractional fourier transform
 Type I Closed form DFRFT
 Some properties






2015/7/21
1
2
3
4
5
6
F α ,Δ u ,Δ t  m , n   Fα ,Δ t ,Δ u  n , m 

and F  m   F  m 
Conjugation property: F  m   F   m  if y(n) is real
No additivity property
When  is small,  t and  u also become very small
Complexity 2 P  ( P / 2)  log 2 P
Fα   m   Fα  π  m 

  2


38
Discrete fractional fourier transform
 Type II Closed form DFRFT
 Derive from transform matrix of the DLCT of type 1
 Type I has too many parameters
 Simplify the type I
 Set p = (d/b)u2, q = (a/b)t2

2015/7/21
F( p ,q ) m , n  
1
2M  1
j
e
2
 p m
2
e
j
2  sgn( b )  n  m
2 M 1
j
e
q n
2
2
39
Discrete fractional fourier transform
 Type II Closed form DFRFT
 from tu = 2|b|/(2M+1), we find
p  q   2 /( 2 M  1)   ad
2
 a, d : any real value
 No constraint for p, q, and p, q can be any real value.
 3 parameters p, q, b without any constraint,
 Free dimension of 3 (in fact near to 2)
2015/7/21
40
Discrete fractional fourier transform
 Type II Closed form DFRFT
 p=0: DLCT becomes a CHIRP multiplication
operation followed by a DFT
 q=0: DLCT becomes a DFT followed by a chirp
multiplication
 p=q: F(p,p,s)(m,n) will be a symmetry matrix (i.e.,
F(p,p,s)(m,n) = F(p,p,s)(n,m))
2015/7/21
41
Discrete fractional fourier transform
 Type II Closed form DFRFT
 2P+(P/2)log2P
 No additive property
 Convertible
2015/7/21
42
Discrete fractional fourier transform
 The relations between the DLCT of type 2 and its
special cases
2015/7/21
DFRFT of type 2
p = q, s = 1
DFRFT of type 1
p = cotu2, q = cott2, s = sgn(sin)
DLCT of type 1
p = d/bu2, q = a/bt2, s = sgn(b)
DFT, IDFT
p = q = 0, s = 1 for DFT, s = 1 for DFT
43
Discrete fractional fourier transform
 Comparison of Closed Form DFRFT and DLCT
with Other Types of DFRFT
Directly
Improved
Linear
Eigenfxs.
Group
Impulse
Proposed
Reversible

*





Closed form







Similarity






Complexity
2015/7/21
P2
Plog2P+
P
2P
2
P2/2
log
2
P
Plog2P+
Plog2P+
2P
2P

P
2
log
2
P
+2P
FFT

2 FFT
1 FFT

2 FFT
2 FFT
1 FFT
Constraints
Less
Middle
Unable
Less
Much
Much
Less
All orders







Properties
Less
Middle
Middle
Less
Many
Many
Many
Adv./Cvt.
No
Convt.
Additive
Additive
Additive
Additive
Convt.
DSP







44
Conclusions and future work
 Generalization of the Fourier transform
 Applications of the conventional FT can also be the
applications of FRFT and LCT
 More flexible
 Useful tools for signal processing
2015/7/21
45
References
[1] V. Namias , ‘The fractional order Fourier transform and its
application to quantum mechanics’, J. Inst. Maths Applies. vol. 25,
p. 241-265, 1980.
[2] L. B. Almeida, ‘The fractional Fourier transform and time-frequency
representations’. IEEE Trans. Signal Processing, vol. 42, no. 11, p.
3084-3091, Nov. 1994.
[3] J. J. Ding, Research of Fractional Fourier Transform and Linear
Canonical Transform, Ph. D thesis, National Taiwan Univ., Taipei,
Taiwan, R.O.C, 1997
[4] H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier
Transform with Applications in Optics and Signal Processing, 1st Ed.,
John Wiley & Sons, New York, 2000.
2015/7/21
46
References
[5] S. C. Pei, C. C. Tseng, M. H. Yeh, and J. J. Shyu,’ Discrete fractional
Hartley and Fourier transform’, IEEE Trans Circ Syst II, vol. 45, no.
6, p. 665–675, Jun. 1998.
[6] H. M. Ozaktas, O. Arikan, ‘Digital computation of the fractional
Fourier transform’, IEEE Trans. On Signal Proc., vol. 44, no. 9,
p.2141-2150, Sep. 1996.
[7] B. Santhanam and J. H. McClellan, “The DRFT—A rotation in time
frequency space,” in Proc. ICASSP, May 1995, pp. 921–924.
[8] J. H. McClellan and T. W. Parks, “Eigenvalue and eigenvector
decomposition of the discrete Fourier transform,” IEEE Trans. Audio
Electroacoust., vol. AU-20, pp. 66–74, Mar. 1972.
2015/7/21
47