Abstract

A joint fractional domain signal representation is proposed based on an intuitive understanding from a time-frequency distribution of signals that designates the joint time and frequency energy content. The joint fractional signal representation (JFSR) of a signal is so designed that its projections onto the defining joint fractional Fourier domains give the modulus square of the fractional Fourier transform of the signal at the corresponding orders. We derive properties of the JFSR, including its relations to quadratic time-frequency representations and fractional Fourier transformations, which include the oblique projections of the JFSR. We present a fast algorithm to compute radial slices of the JFSR and the results are shown for various signals at different fractionally ordered domains.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21-67 (1992).
    [CrossRef]
  2. L. Cohen, Time-Frequency Analysis (Prentice Hall, 1995).
  3. S. Mann and S. Haykin, “The chirplet transform: physicalconsiderations,” IEEE Trans. Signal Process. 43, 2745-2761 (1995).
    [CrossRef]
  4. R. G. Baraniuk and D. L. Jones, “Matrix formulation of the chirplet transform,” IEEE Trans. Signal Process. 44, 3129-3135 (1996).
    [CrossRef]
  5. R. G. Baraniuk, “Beyond time-frequency analysis: energy density in one and many dimensions,” IEEE Trans. Signal Process. 46, 2305-2314 (1998).
    [CrossRef]
  6. O. Akay and G. F. Boudreaux-Bartels, “Joint fractional signal representations,” J. Franklin Inst. 337, 365-378 (2000).
    [CrossRef]
  7. L. B. Almedia, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084-3091 (1994).
    [CrossRef]
  8. H. Ozaktas and O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119-124 (1995).
    [CrossRef]
  9. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).
  10. H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547-559 (1994).
    [CrossRef]
  11. B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32-36 (1997).
    [CrossRef]
  12. M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129-1143 (1997).
    [CrossRef]
  13. M. A. Kutay and H. M. Ozaktas, “Optimal image restoration with the fractional Fourier transform,” J. Opt. Soc. Am. A 15, 825-833 (1998).
    [CrossRef]
  14. M. F. Erden and H. M. Ozaktas, “Synthesis of general linear systems with repeated filtering in consecutive fractional Fourier domains,” J. Opt. Soc. Am. A 15, 1647-1657 (1998).
    [CrossRef]
  15. A. K. Özdemir and O. Arikan, “Efficient computation of the ambiguity function and the Wigner distribution on arbitrary line segments,” IEEE Trans. Signal Process. 49, 381-393 (2001).
    [CrossRef]
  16. L. Durak and O. Arikan, “Short-time Fourier transform: two fundamental properties and an optimal implementation,” IEEE Trans. Signal Process. 51, 1231-1242 (2003).
    [CrossRef]
  17. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
    [CrossRef]
  18. L. M. Bernardo and O. D. D. Soares, “Fractional Fourier transform and optical systems,” Opt. Commun. 110, 517-522 (1994).
    [CrossRef]
  19. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743-751 (1995).
    [CrossRef]
  20. O. Aytür and H. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transform,” Opt. Commun. 120, 166-170 (1995).
    [CrossRef]
  21. A. W. Lohmann and B. H. Soffer, “Relationships between the Radon-Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798-1801 (1994).
    [CrossRef]
  22. S. Shinde and V. M. Gadre, “An uncertainty principle for real signals in the fractional Fourier transform domain,” IEEE Trans. Signal Process. 49, 2545-2548 (2001).
    [CrossRef]
  23. H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).
    [CrossRef]
  24. L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its applications,” Bell Syst. Tech. J. 48, 1249-1292 (1969).
  25. A. K. Ozdemir, S. Karakas, E. D. Cakmak, D. I. Tufekci, and O. Arikan, “Time-frequency component analyzer and its application to analysis of brains oscillatory activity,” J. Neurosci. Methods 145, 107-125 (2005).
    [CrossRef] [PubMed]
  26. http://www.dsp.rice.edu/software/TFA/RGK/BAT/batsig.bin.Z.

2005 (1)

A. K. Ozdemir, S. Karakas, E. D. Cakmak, D. I. Tufekci, and O. Arikan, “Time-frequency component analyzer and its application to analysis of brains oscillatory activity,” J. Neurosci. Methods 145, 107-125 (2005).
[CrossRef] [PubMed]

2003 (1)

L. Durak and O. Arikan, “Short-time Fourier transform: two fundamental properties and an optimal implementation,” IEEE Trans. Signal Process. 51, 1231-1242 (2003).
[CrossRef]

2001 (2)

A. K. Özdemir and O. Arikan, “Efficient computation of the ambiguity function and the Wigner distribution on arbitrary line segments,” IEEE Trans. Signal Process. 49, 381-393 (2001).
[CrossRef]

S. Shinde and V. M. Gadre, “An uncertainty principle for real signals in the fractional Fourier transform domain,” IEEE Trans. Signal Process. 49, 2545-2548 (2001).
[CrossRef]

2000 (1)

O. Akay and G. F. Boudreaux-Bartels, “Joint fractional signal representations,” J. Franklin Inst. 337, 365-378 (2000).
[CrossRef]

1998 (3)

1997 (2)

B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32-36 (1997).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129-1143 (1997).
[CrossRef]

1996 (2)

R. G. Baraniuk and D. L. Jones, “Matrix formulation of the chirplet transform,” IEEE Trans. Signal Process. 44, 3129-3135 (1996).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).
[CrossRef]

1995 (4)

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743-751 (1995).
[CrossRef]

O. Aytür and H. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transform,” Opt. Commun. 120, 166-170 (1995).
[CrossRef]

S. Mann and S. Haykin, “The chirplet transform: physicalconsiderations,” IEEE Trans. Signal Process. 43, 2745-2761 (1995).
[CrossRef]

H. Ozaktas and O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119-124 (1995).
[CrossRef]

1994 (4)

1992 (1)

F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21-67 (1992).
[CrossRef]

1980 (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

1969 (1)

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its applications,” Bell Syst. Tech. J. 48, 1249-1292 (1969).

Bell Syst. Tech. J. (1)

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its applications,” Bell Syst. Tech. J. 48, 1249-1292 (1969).

IEEE Signal Process. Mag. (1)

F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21-67 (1992).
[CrossRef]

IEEE Trans. Signal Process. (9)

L. B. Almedia, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084-3091 (1994).
[CrossRef]

S. Mann and S. Haykin, “The chirplet transform: physicalconsiderations,” IEEE Trans. Signal Process. 43, 2745-2761 (1995).
[CrossRef]

R. G. Baraniuk and D. L. Jones, “Matrix formulation of the chirplet transform,” IEEE Trans. Signal Process. 44, 3129-3135 (1996).
[CrossRef]

R. G. Baraniuk, “Beyond time-frequency analysis: energy density in one and many dimensions,” IEEE Trans. Signal Process. 46, 2305-2314 (1998).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129-1143 (1997).
[CrossRef]

A. K. Özdemir and O. Arikan, “Efficient computation of the ambiguity function and the Wigner distribution on arbitrary line segments,” IEEE Trans. Signal Process. 49, 381-393 (2001).
[CrossRef]

L. Durak and O. Arikan, “Short-time Fourier transform: two fundamental properties and an optimal implementation,” IEEE Trans. Signal Process. 51, 1231-1242 (2003).
[CrossRef]

S. Shinde and V. M. Gadre, “An uncertainty principle for real signals in the fractional Fourier transform domain,” IEEE Trans. Signal Process. 49, 2545-2548 (2001).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).
[CrossRef]

J. Franklin Inst. (1)

O. Akay and G. F. Boudreaux-Bartels, “Joint fractional signal representations,” J. Franklin Inst. 337, 365-378 (2000).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

J. Neurosci. Methods (1)

A. K. Ozdemir, S. Karakas, E. D. Cakmak, D. I. Tufekci, and O. Arikan, “Time-frequency component analyzer and its application to analysis of brains oscillatory activity,” J. Neurosci. Methods 145, 107-125 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32-36 (1997).
[CrossRef]

L. M. Bernardo and O. D. D. Soares, “Fractional Fourier transform and optical systems,” Opt. Commun. 110, 517-522 (1994).
[CrossRef]

O. Aytür and H. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transform,” Opt. Commun. 120, 166-170 (1995).
[CrossRef]

Signal Process. (1)

H. Ozaktas and O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119-124 (1995).
[CrossRef]

Other (3)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

L. Cohen, Time-Frequency Analysis (Prentice Hall, 1995).

http://www.dsp.rice.edu/software/TFA/RGK/BAT/batsig.bin.Z.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

a 0 th-order fractional Fourier domain makes an angle of ϕ 0 = a 0 ( π 2 ) on the time-frequency plane.

Fig. 2
Fig. 2

Value of the time-frequency distribution at a point P contributes to the energy densities of the fractional Fourier domains u and v at points u = u 0 and v = v 0 , respectively.

Fig. 3
Fig. 3

JFSRs of x ( t ) = rect ( t 10 ) e j 2 π ( 0.5 t 2 + t ) at joint fractional Fourier domains with orders (a) ( a 1 , a 2 ) = ( 0 , 0.25 ) , (b) ( a 1 , a 2 ) = ( 0 , 0.5 ) , (c) ( a 1 , a 2 ) = ( 0 , 0.75 ) , (d) ( a 1 , a 2 ) = ( 0 , 1 ) .

Fig. 4
Fig. 4

JFSRs of x ( t ) = e π ( ( t 3 ) 2 + 0.3 j t 3 ) at joint fractional Fourier domains with orders (a) ( a 1 , a 2 ) = ( 0 , 0.25 ) , (b) ( a 1 , a 2 ) = ( 0 , 0.5 ) , (c) ( a 1 , a 2 ) = ( 0 , 0.75 ) , (d) ( a 1 , a 2 ) = ( 0 , 1 ) .

Fig. 5
Fig. 5

(a) 2.5 ms echolocation pulse emitted by a large brown bat, eptesicus fuscus. The JFSRs at joint fractional Fourier domains with orders (b) ( a 1 , a 2 ) = ( 0 , 1 ) , (c) ( a 1 , a 2 ) = ( 0 , 1.2 ) , (d) ( a 1 , a 2 ) = ( 0 , 1.3 ) .

Fig. 6
Fig. 6

(a) One of the components extracted from the bat signal. (b) The WD of the component and (c) its JFSR corresponding to the fractional Fourier order pairs of ( 0 , 1.2 ) . (d) The computed instantaneous frequency from the JFSR shown in (c).

Equations (44)

Equations on this page are rendered with MathJax. Learn more.

x a ( t ) { F a x } ( t ) = B a ( t , t ) x ( t ) d t , 2 < a < 2 ,
B a ( t , t ) = e j ( π sgn ( a ) 4 + ϕ 2 ) sin ϕ 1 2 e j π ( t 2 cot ϕ 2 t t csc ϕ + t 2 cot ϕ )
D ( t , f ) d f = x ( t ) 2 ,
D ( t , f ) d t = X ( f ) 2 ,
D ( t , f ) d t d f = x 2 ,
W x ( t , f ) = x ( t + τ 2 ) x * ( t τ 2 ) e j 2 π τ f d τ .
E x a ( u , v ) d v = x a 1 ( u ) 2 ,
E x a ( u , v ) d u = x a 2 ( v ) 2 ,
E x a ( u , v ) d u d v = x 2 .
W x ( u cos ϕ v sin ϕ , u sin ϕ + v cos ϕ ) d v = x a ( u ) 2 .
E x a ( u , v ) = C W x ( P ( t ( u , v ) , f ( u , v ) ) ) ,
[ cos ϕ 1 sin ϕ 1 cos ϕ 2 sin ϕ 2 ] [ t f ] = [ u v ] ,
C = csc ( ϕ 12 ) ,
csc ( ϕ 12 ) x ( u sin ϕ 2 v sin ϕ 1 sin ϕ 12 + τ 2 ) x * ( u sin ϕ 2 v sin ϕ 1 sin ϕ 12 τ 2 ) × exp ( j 2 π τ u cos ϕ 2 + v cos ϕ 1 sin ϕ 12 ) d τ .
W x a 1 ( u , v ) = W x ( u cos ϕ 12 v sin ϕ 12 , u sin ϕ 12 + v cos ϕ 12 ) ,
E x a ( u , v ) = x a 1 ( u + τ sin ϕ 12 2 ) x a 1 * ( u τ sin ϕ 12 2 ) × e j 2 π ( v u cos ϕ 12 ) τ d τ ,
E x y a ( u , v ) = csc ( ϕ 12 ) W x y ( P ( t ( u , v ) , f ( u , v ) ) ) = x a 1 ( u + τ sin ϕ 12 2 ) y a 1 * ( u τ sin ϕ 12 2 ) × e j 2 π ( v u cos ϕ 12 ) τ d τ .
E x a ( u , v ) = ( E x a ) * ( u , v ) .
E x a ( u , v ) d v = x a 1 ( u ) 2 ,
E x a ( u , v ) d u = x a 2 ( v ) 2 .
E x a ( u , v ) d u d v = x ( t ) 2 d t ,
E x a ( u , v ) = csc ϕ 12 W x a 1 ( u , v u cos ϕ 12 sin ϕ 12 ) = csc ϕ 12 W x ( u sin ϕ 2 v sin ϕ 1 sin ϕ 12 , u cos ϕ 2 + v cos ϕ 1 sin ϕ 12 ) .
E x a a ( u , v ) = E x a + a ( u , v ) ,
E x a a ( u , v ) = csc ϕ 12 W x a ( u sin ϕ 2 v sin ϕ 1 sin ϕ 12 , u cos ϕ 2 + v cos ϕ 1 sin ϕ 12 ) .
E x a a ( u , v ) = csc ϕ 12 W x ( u , v ) ,
u = u sin ( ϕ + ϕ 2 ) v sin ( ϕ + ϕ 1 ) sin ϕ 12 ,
v = u cos ( ϕ + ϕ 2 ) + v cos ( ϕ + ϕ 1 ) sin ϕ 12 ,
P ϕ [ E x a ] ( r ) = x ( a ) ( r M ) 2 , a = 2 ϕ π ,
ϕ = arctan 2 ( cos ϕ 1 + cos ϕ + cos ϕ 2 sin ϕ , sin ϕ 1 cos ϕ + sin ϕ 2 sin ϕ ) ,
M = ( 1 + sin 2 ϕ cos ϕ 12 ) 1 2 .
P ϕ [ E x a ] ( r ) = F x ( ζ cos ϕ , ζ sin ϕ ) e j 2 π ζ r d ζ ,
F x a ( ζ , η ) = E x a ( u , v ) e j 2 π ( ζ u + η v ) d u d v ,
F x a ( ζ , η ) = A x ( ζ cos ϕ 1 + η cos ϕ 2 , ζ sin ϕ 1 + ζ sin ϕ 2 ) .
F x a ( ζ cos ϕ , ζ sin ϕ ) = A x ( ζ M cos ϕ , ζ M sin ϕ ) ,
A x ( ζ cos ϕ , ζ sin ϕ ) = x ( a ) ( r ) 2 e j 2 π ζ r d r .
σ a 1 σ a 2 sin [ π ( a 1 a 2 ) 2 ] 4 π ,
σ a 1 , 2 = [ ( u a 1 , 2 η a 1 , 2 ) 2 x a 1 , 2 ( u a 1 , 2 ) 2 d u a 1 , 2 ] 1 2 x ,
η a 1 , 2 = [ u a 1 , 2 x a 1 , 2 ( u a 1 , 2 ) 2 d u a 1 , 2 ] x 2 ,
E x a a ( r cos ϕ , r sin ϕ ) = csc ϕ 12 W x ( r cos ϕ , r sin ϕ ) ,
ϕ = arctan ( cos ϕ cos ϕ 2 + sin ϕ cos ϕ 1 , cos ϕ sin ϕ 2 sin ϕ cos ϕ 1 ) ,
W x ( r cos ϕ , r sin ϕ ) = x ( a 1 ) ( λ 2 ) x ( a 1 ) * ( λ 2 ) e j 2 π r λ d λ .
E x a a ( r cos ϕ , r sin ϕ ) = csc ϕ 12 x ( a 1 ) ( λ 2 ) x ( a 1 ) * ( λ 2 ) × e j 2 π r λ d λ .
E x a a ( r Δ x cos ϕ , r Δ x sin ϕ ) = csc ϕ 12 Δ x k = N N 1 q [ k ] e ( j 2 π r k Δ x ) ,
f a 1 , a 2 ( u ) = v W a 1 , a 2 ( u , v ) d v W a 1 , a 2 ( u , v ) d v ,

Metrics