Abstract

The optical and digital implementations of general linear systems are costly. Through several examples we show that either exact realizations or useful approximations of these systems may be implemented in the form of repeated-filtering operations in consecutive fractional Fourier domains. These implementations are much cheaper than direct implementations of general linear systems. Thus we may significantly decrease the implementation costs of general linear systems with little or no decrease in performance by synthesizing them with the proposed repeated-filtering method.

© 1998 Optical Society of America

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    [CrossRef]
  2. M. A. Kutay, H. M. Ozaktas, L. Onural, O. Arikan, “Optimal filtering in fractional Fourier domains,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE, Piscataway, N.J., 1995), pp. 937–940.
  3. M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
    [CrossRef]
  4. M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
    [CrossRef]
  5. H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  28. D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
    [CrossRef]
  29. J. Garcia, D. Mendlovic, Z. Zalevsky, A. Lohmann, “Space-variant simultaneous detection of several objects by the use of multiple anamorphic fractional-Fourier-transform filters,” Appl. Opt. 35, 3945–3952 (1996).
    [CrossRef] [PubMed]
  30. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
    [CrossRef] [PubMed]
  31. A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
    [CrossRef]
  32. J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon–Wigner transform,” IEEE Trans. Signal Process. 42, 2105–2111 (1994).
    [CrossRef]
  33. Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  35. B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
    [CrossRef]
  36. A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1997).
    [CrossRef]
  37. A. Sahin, “Two-dimensional fractional Fourier transform and its optical implementation,” M.S. thesis (Bilkent University, Department of Electrical and Electronics Engineering, Ankara, Turkey, 1996).
  38. H. M. Ozaktas, “Repeated fractional Fourier domain filtering is equivalent to repeated time and frequency domain filtering,” Signal Process. 54, 81–84 (1996).
    [CrossRef]
  39. H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional Fourier domain filtering,” Appl. Opt. 35, 3167–3170 (1996).
    [CrossRef] [PubMed]
  40. A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
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    [CrossRef] [PubMed]

1997 (5)

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

H. M. Ozaktas, M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

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

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1997).
[CrossRef]

1996 (8)

H. M. Ozaktas, “Repeated fractional Fourier domain filtering is equivalent to repeated time and frequency domain filtering,” Signal Process. 54, 81–84 (1996).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional Fourier domain filtering,” Appl. Opt. 35, 3167–3170 (1996).
[CrossRef] [PubMed]

Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
[CrossRef] [PubMed]

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

J. Garcia, D. Mendlovic, Z. Zalevsky, A. Lohmann, “Space-variant simultaneous detection of several objects by the use of multiple anamorphic fractional-Fourier-transform filters,” Appl. Opt. 35, 3945–3952 (1996).
[CrossRef] [PubMed]

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

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

1995 (4)

1994 (7)

S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
[CrossRef]

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almedia, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon–Wigner transform,” IEEE Trans. Signal Process. 42, 2105–2111 (1994).
[CrossRef]

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

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]

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

1993 (6)

1987 (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1982 (1)

1981 (1)

1979 (1)

1978 (1)

Abe, S.

Agullo-Lopez, F.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almedia, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Alieva, T.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almedia, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Almedia, L. B.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almedia, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Almeida, L. B.

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

Arikan, O.

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

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

M. A. Kutay, H. M. Ozaktas, L. Onural, O. Arikan, “Optimal filtering in fractional Fourier domains,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE, Piscataway, N.J., 1995), pp. 937–940.

Barry, D. T.

J. C. Wood, D. T. Barry, “Linear signal synthesis using the Radon–Wigner transform,” IEEE Trans. Signal Process. 42, 2105–2111 (1994).
[CrossRef]

Barshan, B.

Bastiaans, M. J.

Beck, M.

Bernardo, L. M.

Bonnet, G.

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Bozdagi, G.

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

Case, S. K.

Clarke, L.

Dias, A. R.

Dorsch, R. G.

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

Erden, M. F.

H. M. Ozaktas, M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

M. F. Erden, M. A. Kutay, H. M. Ozaktas, “Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration,” IEEE Trans. Signal Process. (to be published).

Garcia, J.

Goodman, J. W.

Haugen, P. R.

Horner, J. L.

J. L. Horner, Optical Signal Processing (Academic, London, 1987).

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Kutay, M. A.

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

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

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

M. A. Kutay, H. M. Ozaktas, L. Onural, O. Arikan, “Optimal filtering in fractional Fourier domains,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE, Piscataway, N.J., 1995), pp. 937–940.

M. F. Erden, M. A. Kutay, H. M. Ozaktas, “Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration,” IEEE Trans. Signal Process. (to be published).

Løberge, O. J.

Lohmann, A.

Lohmann, A. W.

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
[CrossRef] [PubMed]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional order Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

Lopez, V.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almedia, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Mayer, A.

McAlister, D. F.

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mendlovic, D.

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

J. Garcia, D. Mendlovic, Z. Zalevsky, A. Lohmann, “Space-variant simultaneous detection of several objects by the use of multiple anamorphic fractional-Fourier-transform filters,” Appl. Opt. 35, 3945–3952 (1996).
[CrossRef] [PubMed]

Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
[CrossRef] [PubMed]

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional Fourier domain filtering,” Appl. Opt. 35, 3167–3170 (1996).
[CrossRef] [PubMed]

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

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

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
[CrossRef] [PubMed]

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]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Multistage optical interconnection architectures with the least possible growth of system size,” Opt. Lett. 18, 296–298 (1993).
[CrossRef] [PubMed]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Optical-coordinate transformation methods and optical-interconnection architectures,” Appl. Opt. 32, 5119–5124 (1993).
[CrossRef] [PubMed]

Nazarathy, M.

Onural, L.

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

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]

M. A. Kutay, H. M. Ozaktas, L. Onural, O. Arikan, “Optimal filtering in fractional Fourier domains,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE, Piscataway, N.J., 1995), pp. 937–940.

Ozaktas, H. M.

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

H. M. Ozaktas, M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1997).
[CrossRef]

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

H. M. Ozaktas, “Repeated fractional Fourier domain filtering is equivalent to repeated time and frequency domain filtering,” Signal Process. 54, 81–84 (1996).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional Fourier domain filtering,” Appl. Opt. 35, 3167–3170 (1996).
[CrossRef] [PubMed]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

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

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

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
[CrossRef] [PubMed]

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]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Multistage optical interconnection architectures with the least possible growth of system size,” Opt. Lett. 18, 296–298 (1993).
[CrossRef] [PubMed]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Optical-coordinate transformation methods and optical-interconnection architectures,” Appl. Opt. 32, 5119–5124 (1993).
[CrossRef] [PubMed]

M. F. Erden, M. A. Kutay, H. M. Ozaktas, “Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration,” IEEE Trans. Signal Process. (to be published).

M. A. Kutay, H. M. Ozaktas, L. Onural, O. Arikan, “Optimal filtering in fractional Fourier domains,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE, Piscataway, N.J., 1995), pp. 937–940.

Pellat-Finet, P.

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Raymer, M. G.

Sahin, A.

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

A. Sahin, “Two-dimensional fractional Fourier transform and its optical implementation,” M.S. thesis (Bilkent University, Department of Electrical and Electronics Engineering, Ankara, Turkey, 1996).

Saleh, B. E. A.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Appl. Opt. (7)

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
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[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Configuration for repeated filtering in consecutive fractional Fourier domains.

Fig. 2
Fig. 2

Repeated filtering in consecutive (a) one-dimensional (1D) quadratic-phase systems, (b) two-dimensional (2D) quadratic-phase systems.

Fig. 3
Fig. 3

Canonical forms for (a) the 1D repeated-filtering configuration in Fig. 2(a), (b) the 2D repeated-filtering configuration in Fig. 2(b).

Fig. 4
Fig. 4

Normalized error en versus number of filters in the repeated-filtering configuration for the signal restoration example.

Fig. 5
Fig. 5

Same as Fig. 4, but for the moment generation example.

Fig. 6
Fig. 6

(a) Sinusoidal function, (b) moment space representation of the sine function in (a), (c) approximation of the moment space representation in (b) obtained with one filter (M=1), (d) approximation of the moment space representation in (b) obtained with five filters (M=5).

Fig. 7
Fig. 7

Reverse perfect shuffle interconnection architecture.

Fig. 8
Fig. 8

64 points are mapped to 64 points. The points lying in the first quadrant are mapped to the points in the third quadrant, which are symmetric to them with respect to the origin. The points lying in the fourth quadrant are similarly mapped to points in the second quadrant. However, the points lying in the second quadrant are mapped to points in the first quadrant, which are symmetric to them with respect to the y axis. The points in the third quadrant are likewise mapped to the fourth quadrant.

Equations (48)

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pa(u)=(1-j cot ϕ)1/2-exp[jπ(u2 cot ϕ-2uu csc ϕ+u2 cot ϕ)]p(u)du,
h(x, x)=K exp(jπx2/λR)expjπs2x2M2cot ϕ-2 xxMcsc ϕ+x2 cot ϕ,
pout(x)=-Q(x, x)pin(x)dx,
Q(x, x)=Kq expjπs2(αx2-2βxx+γx2),
fout(u)=Td(u, u)fin(u)du,
fout(u, v)=Td(u, v, u, v)fin(u, v)dudv,
T(u, u)=du1duM-1 h1(u)×h2(u1)hM(uM-1)hM+1(u)×QM(u, uM-1)QM-1(uM-1, uM-2)Q1(u1, u).
e=|Td(u, u)-T(u, u)|2 dudu.
Qk(x, y, x, y)=Kk expjπs2(akx2+bky2+ckx2+dky2+2ekxy+2 fkxx)×expjπs2(2gkxy+2hkyx+2mkyy+2nkxy),
T(u, v, uv)=du1dv1duM-1dvM-1×h1(u, v)h2(u1, v1)hM+1(u, v)×QM(u, v, uM-1,  vM-1)Q1(u1, v1, u, v).
e=|Td(u, v, u, v)-T(u, v, u, v)|2 dudvdudv.
f¯out=Tˆf¯in,
Tˆ=ΛˆM+1FˆΛˆMFˆFˆΛˆ2FˆΛˆ1,
e=k=1Nl=1N|(Tˆd)kl-Tˆkl|2.
fˆout=T˜fˆin,
T˜=Λ˜M+1F˜2DΛ˜MF˜2DF˜2DΛ˜2F˜2DΛ˜1,
(Λ˜k)lmjk=(hˆk)lmifl=jandm=k0otherwise.
(K˜fˆ )uv=lmK˜uvlmfˆlm
(K˜L˜)uvjk=lmK˜uvlmL˜lmjk.
e=u=1Nv=1Nj=1Nk=1N|(T˜d)uvjk-T˜uvjk|2.
Aˆ=ΛˆM+1FˆFˆΛˆk+1Fˆ,
Bˆ=FˆΛˆk-1FˆFˆΛˆ1,
Tˆ=AˆΛˆkBˆ.
hkm=hkmr+jhkmi.
δeδhkmr=0,δeδhkmi=0,m=1, 2,, N.
Dˆh¯k=c¯.
Dˆ=(AˆHAˆ)(BˆBˆH)T,
c¯l=(AˆHTˆdBˆH)ll.
A˜=Λ˜M+1F˜2DF˜2DΛ˜k+1F˜2D,
B˜=F˜2DΛ˜k-1F˜2DF˜2DΛ˜1
T˜=A˜Λ˜kB˜.
D˜hˆk=cˆ.
D˜=(A˜HA˜)(B˜B˜H)T,
cˆlm=(A˜HT˜dB˜H)lmlm.
(h¯k)l=(AˆHTˆdBˆH)ll.
(h¯k)l|(h¯k)l|(h¯k)l,l, m=1, 2,, N.
(hˆk)lm=(A˜HT˜dB˜H)lmlm
(hˆk)lm|(hˆk)lm|(hˆk)lm,l, m=1, 2,, N.
y¯=Hˆx¯+n¯.
σ2=1NE[(x¯-x¯e)H(x¯-x¯e)].
x¯e=[ΛˆM+1FˆΛˆMFˆFˆΛˆ2FˆΛˆ1]y¯
x¯e=Gˆopty¯.
h(x; x)=exp[-πα2(x)(x-x)2],
en=e/Tˆd2,
Mi=-11fin(x)xi dx.
M¯=Tˆdf¯in,
Mi,l=-11-11fin(x, y)xiyl dxdy.
Mˆ=T˜dfˆin,

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