Fractional fourier transform
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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
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Introduction
Definition of fourier transform:
F
1
2
e
j w t
f t dt
Definition of inverse fourier transform:
f t
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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 ?
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.
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 )
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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
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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
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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.
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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
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f t f t
d
I
c
b a
a c
b
d
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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.
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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
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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
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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
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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
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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
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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
pM
q u p t 2
e
j a 1 2 2
p t
2 b
f p t
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Implementation of FRFT/LCT
Chirp Convolution Method
implement by
2 chirp multiplications
1 chirp convolution
complexity
2P (required for 2 chirp multiplications) + Plog2P (required
for 2 DFTs)
Plog2P (P = 2M+1 = the number of sampling points)
Note: 1 chirp convolution needs to 2DFTs
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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.)
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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
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e
2
j u t
/ 2 b F 3 u
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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 pt and qu
t u 2 / P
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Implementation of FRFT/LCT
DFT-like Method
Complexity
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2 M-points multiplication operations
1 DFT
2P (two multiplication operations) + (P/2)log2P (one DFT)
(P/2)log2P
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Implementation of FRFT/LCT
Compare
Complexity
Chirp convolution method: Plog2P (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
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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
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Discrete fractional fourier transform
Direct form of DFRFT
simplest way
sampling the continuous FRFT and computing it
directly
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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
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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
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1
4
4
e
j q k
2
k 1
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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’
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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.
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(a) = 0.01
(b) = 0.05
(c) = 0.2
(d) = 0.4
(e) = π/4
(f) = π/2
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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
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F
U ( r i ) U
t
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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
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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
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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
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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
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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
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0
5
10
15
20
25
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Discrete fractional fourier transform
Closed form DFRFT
By Pei, Ding
further improvement of the sampling type of DFRFT
Two types
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digital implementing of the continuous FRFT
practical applications about digital signal processing
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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
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2
cot α m Δ u
N
Fα m , n Fα m , k y k
k N
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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
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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
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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
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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)
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Discrete fractional fourier transform
Type I Closed form DFRFT
Some properties
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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
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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
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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 tu = 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)
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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))
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Discrete fractional fourier transform
Type II Closed form DFRFT
2P+(P/2)log2P
No additive property
Convertible
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Discrete fractional fourier transform
The relations between the DLCT of type 2 and its
special cases
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DFRFT of type 2
p = q, s = 1
DFRFT of type 1
p = cotu2, q = cott2, s = sgn(sin)
DLCT of type 1
p = d/bu2, q = a/bt2, 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
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P2
Plog2P+
P
2P
2
P2/2
log
2
P
Plog2P+
Plog2P+
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
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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.
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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.
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