Abstract

Through the optimization of the main interval of the fractional order, an improved fast algorithm for numerical calculation of the fractional Fourier transforms is proposed. With this improved algorithm, the fractional Fourier transforms of a rectangular function and a Gaussian function are calculated. Its calculation errors are compared with those calculated with the previously published algorithm, and the results show that the calculation accuracy of the improved algorithm is much higher.

© 2004 Optical Society of America

PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  2. B. Yurke, W. Schleich, D. F. Walls, “Quantum superposition generated by quantum nondemolition measurements,” Phys. Rev. A 42, 1703–1711 (1990).
    [CrossRef] [PubMed]
  3. W. Vogel, W. Schleich, “Phase distribution of a quantum state without using phase states,” Phys. Rev. A 44, 7642–7646 (1991).
    [CrossRef] [PubMed]
  4. M. Beck, M. G. Raymer, I. A. Walmsley, V. Kong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).
    [CrossRef] [PubMed]
  5. M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  6. 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 11, 547–559 (1994).
    [CrossRef]
  7. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  8. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  9. H. M. Ozaktas, D. Mendlovic, “The fractional Fourier transform as a tool for analyzing beam propagation andspherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
    [CrossRef] [PubMed]
  10. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
  11. L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transform and Fourier optical systems,” Opt. Commun. 110, 517–522 (1994).
    [CrossRef]
  12. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: part I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  13. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: part II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  14. H. M. Ozaktas, O. Arikan, M. A. Kutay, “Digital computation of the fractional Fourier transform,” Signal Process. 44, 2141–2150 (1996).
  15. X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).
    [CrossRef]
  16. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
    [CrossRef]
  17. X. Deng, B. Bihari, J. Gan, F. Zhao, R. T. Chen, “Fast algorithm for chirp transforms with zooming-in ability and its applications,” J. Opt. Soc. Am. A 17, 762–771 (2000).
    [CrossRef]
  18. A. Bultheel, H. E. Martinez Sulbaran, “Computation of the fractional Fourier transform” (2003); http://www.cs.kuleuven.ac.be/cwis/research/nalag/papers/ade/frftcomp/ccomp.pdf .
  19. S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
    [CrossRef]
  20. F. J. Marinho, L. M. Bernardo, “Numerical calculation of fractional Fourier transforms with a single fast-Fourier-transform algorithm,” J. Opt. Soc. Am. A 15, 2111–2116 (1998).
    [CrossRef]
  21. J. Garcia, D. Mas, R. G. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,” Appl. Opt. 35, 7013–7018 (1996).
    [CrossRef] [PubMed]
  22. H. M. Ozaktas, Z. Zalevski, M. A. Kutay, The Fractional Fourier Transform (Wiley, New York, 2001), Chap. 6, pp. 218–220.
  23. J. O’Neill, “Discrete TFDs (fracft.m)”: a collection of Matlab files for time-frequency analysis (1999); ftp.mathworks.com/pub/contrib/v5/signal/DiscreteTFDs/.
  24. M. A. Kutay, “FracF: Fast computation of the fractional Fourier transform (1996)”; http://www.ee.bilkent.edu.tr/∼haldun/fracF.m .
  25. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]

2000 (1)

1999 (2)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

1998 (1)

1997 (1)

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).
[CrossRef]

1996 (2)

H. M. Ozaktas, O. Arikan, M. A. Kutay, “Digital computation of the fractional Fourier transform,” Signal Process. 44, 2141–2150 (1996).

J. Garcia, D. Mas, R. G. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,” Appl. Opt. 35, 7013–7018 (1996).
[CrossRef] [PubMed]

1994 (6)

1993 (4)

1991 (1)

W. Vogel, W. Schleich, “Phase distribution of a quantum state without using phase states,” Phys. Rev. A 44, 7642–7646 (1991).
[CrossRef] [PubMed]

1990 (1)

B. Yurke, W. Schleich, D. F. Walls, “Quantum superposition generated by quantum nondemolition measurements,” Phys. Rev. A 42, 1703–1711 (1990).
[CrossRef] [PubMed]

1980 (2)

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

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

Arikan, O.

H. M. Ozaktas, O. Arikan, M. A. Kutay, “Digital computation of the fractional Fourier transform,” Signal Process. 44, 2141–2150 (1996).

Barshan, B.

Beck, M.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, V. Kong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).
[CrossRef] [PubMed]

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

F. J. Marinho, L. M. Bernardo, “Numerical calculation of fractional Fourier transforms with a single fast-Fourier-transform algorithm,” J. Opt. Soc. Am. A 15, 2111–2116 (1998).
[CrossRef]

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

Bihari, B.

Chen, R. T.

Deng, X.

Dorsch, R. G.

Fan, D.

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).
[CrossRef]

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Gan, J.

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

J. Garcia, D. Mas, R. G. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,” Appl. Opt. 35, 7013–7018 (1996).
[CrossRef] [PubMed]

Kong, V.

Kutay, M. A.

H. M. Ozaktas, O. Arikan, M. A. Kutay, “Digital computation of the fractional Fourier transform,” Signal Process. 44, 2141–2150 (1996).

H. M. Ozaktas, Z. Zalevski, M. A. Kutay, The Fractional Fourier Transform (Wiley, New York, 2001), Chap. 6, pp. 218–220.

Li, Y.

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).
[CrossRef]

Lohmann, A. W.

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Marinho, F. J.

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

J. Garcia, D. Mas, R. G. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,” Appl. Opt. 35, 7013–7018 (1996).
[CrossRef] [PubMed]

McAlister, D. F.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Mendlovic, D.

Namias, V.

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

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

Onural, L.

Ozaktas, H. M.

Pei, S.-C.

S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

Pellat-Finet, P.

Qiu, Y.

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).
[CrossRef]

Raymer, M. G.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, V. Kong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).
[CrossRef] [PubMed]

Schleich, W.

W. Vogel, W. Schleich, “Phase distribution of a quantum state without using phase states,” Phys. Rev. A 44, 7642–7646 (1991).
[CrossRef] [PubMed]

B. Yurke, W. Schleich, D. F. Walls, “Quantum superposition generated by quantum nondemolition measurements,” Phys. Rev. A 42, 1703–1711 (1990).
[CrossRef] [PubMed]

Soares, O. D. D.

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

Tseng, C.-C.

S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

Vogel, W.

W. Vogel, W. Schleich, “Phase distribution of a quantum state without using phase states,” Phys. Rev. A 44, 7642–7646 (1991).
[CrossRef] [PubMed]

Walls, D. F.

B. Yurke, W. Schleich, D. F. Walls, “Quantum superposition generated by quantum nondemolition measurements,” Phys. Rev. A 42, 1703–1711 (1990).
[CrossRef] [PubMed]

Walmsley, I. A.

Yeh, M.-H.

S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

Yurke, B.

B. Yurke, W. Schleich, D. F. Walls, “Quantum superposition generated by quantum nondemolition measurements,” Phys. Rev. A 42, 1703–1711 (1990).
[CrossRef] [PubMed]

Zalevski, Z.

H. M. Ozaktas, Z. Zalevski, M. A. Kutay, The Fractional Fourier Transform (Wiley, New York, 2001), Chap. 6, pp. 218–220.

Zhao, F.

Appl. Opt. (2)

IEEE Trans. Signal Process. (1)

S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

J. Inst. Math. Appl. (2)

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

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

J. Opt. Soc. Am. A (6)

Opt. Commun. (3)

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

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (2)

B. Yurke, W. Schleich, D. F. Walls, “Quantum superposition generated by quantum nondemolition measurements,” Phys. Rev. A 42, 1703–1711 (1990).
[CrossRef] [PubMed]

W. Vogel, W. Schleich, “Phase distribution of a quantum state without using phase states,” Phys. Rev. A 44, 7642–7646 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Signal Process. (1)

H. M. Ozaktas, O. Arikan, M. A. Kutay, “Digital computation of the fractional Fourier transform,” Signal Process. 44, 2141–2150 (1996).

Other (4)

A. Bultheel, H. E. Martinez Sulbaran, “Computation of the fractional Fourier transform” (2003); http://www.cs.kuleuven.ac.be/cwis/research/nalag/papers/ade/frftcomp/ccomp.pdf .

H. M. Ozaktas, Z. Zalevski, M. A. Kutay, The Fractional Fourier Transform (Wiley, New York, 2001), Chap. 6, pp. 218–220.

J. O’Neill, “Discrete TFDs (fracft.m)”: a collection of Matlab files for time-frequency analysis (1999); ftp.mathworks.com/pub/contrib/v5/signal/DiscreteTFDs/.

M. A. Kutay, “FracF: Fast computation of the fractional Fourier transform (1996)”; http://www.ee.bilkent.edu.tr/∼haldun/fracF.m .

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.


Metrics