Abstract

The fractional Fourier transform is a useful mathematical operation that generalizes the well-known continuous Fourier transform. Several discrete fractional Fourier transforms (DFRFT’s) have been developed, but their results do not match those of the continuous case. We propose a new DFRFT. This improved DFRFT provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.

© 1997 Optical Society of America

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References

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  1. A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
    [Crossref]
  2. R. G. Dorsch, A. W. Lohmann, Y. Bitran, and D. Mendlovic, Appl. Opt. 33, 7599 (1994).
    [Crossref] [PubMed]
  3. A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
    [Crossref]
  4. D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, Appl. Opt. 34, 303 (1995).
    [Crossref] [PubMed]
  5. A. W. Lohmann and B. H. Soffer, J. Opt. Soc. Am. A 11, 1798 (1994).
    [Crossref]
  6. D. Mendlovic, Z. Zalevsky, A. W. Lohmann, and R. G. Dorsch, Opt. Commun. 126, 14 (1996).
    [Crossref]
  7. G. Sansone, Orthogonal Function, Vol. 9 of Pure and Applied Mathematics (Interscience, New York, 1959).
  8. B. W. Dickinson and K. Steiglitz, IEEE Trans. Acoust. Speech Signal Process ASSP-30, 25 (1982).
    [Crossref]
  9. C. C. Shih, Opt. Commun. 118, 495 (1995).
    [Crossref]
  10. B. Santhanam and J. H. McClellan, IEEE Trans. Signal Process. 42, 994 (1996).
    [Crossref]
  11. H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, IEEE Trans. Signal Process. 44, 2141 (1996).
    [Crossref]
  12. Z. T. Deng, H. J. Caulfield, and M. Schamschula, Opt. Lett. 21, 1430 (1996).
    [Crossref] [PubMed]
  13. H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, J. Opt. Soc. Am. A 11, 547 (1994).
    [Crossref]
  14. J. H. McClellan and T. W. Parks, IEEE Trans. Audio Electroacoust. AU-20, 66 (1972).
    [Crossref]
  15. P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1416.

1996 (4)

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, and R. G. Dorsch, Opt. Commun. 126, 14 (1996).
[Crossref]

B. Santhanam and J. H. McClellan, IEEE Trans. Signal Process. 42, 994 (1996).
[Crossref]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, IEEE Trans. Signal Process. 44, 2141 (1996).
[Crossref]

Z. T. Deng, H. J. Caulfield, and M. Schamschula, Opt. Lett. 21, 1430 (1996).
[Crossref] [PubMed]

1995 (2)

1994 (3)

1993 (1)

1987 (1)

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[Crossref]

1982 (1)

B. W. Dickinson and K. Steiglitz, IEEE Trans. Acoust. Speech Signal Process ASSP-30, 25 (1982).
[Crossref]

1972 (1)

J. H. McClellan and T. W. Parks, IEEE Trans. Audio Electroacoust. AU-20, 66 (1972).
[Crossref]

Arikan, O.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, IEEE Trans. Signal Process. 44, 2141 (1996).
[Crossref]

Barshan, B.

Bitran, Y.

Bozdagi, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, IEEE Trans. Signal Process. 44, 2141 (1996).
[Crossref]

Caulfield, H. J.

Deng, Z. T.

Dickinson, B. W.

B. W. Dickinson and K. Steiglitz, IEEE Trans. Acoust. Speech Signal Process ASSP-30, 25 (1982).
[Crossref]

Dorsch, R. G.

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, and R. G. Dorsch, Opt. Commun. 126, 14 (1996).
[Crossref]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, and D. Mendlovic, Appl. Opt. 33, 7599 (1994).
[Crossref] [PubMed]

Feschbach, H.

P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1416.

Kerr, F. H.

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[Crossref]

Kutay, M. A.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, IEEE Trans. Signal Process. 44, 2141 (1996).
[Crossref]

Lohmann, A. W.

McBride, A. C.

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[Crossref]

McClellan, J. H.

B. Santhanam and J. H. McClellan, IEEE Trans. Signal Process. 42, 994 (1996).
[Crossref]

J. H. McClellan and T. W. Parks, IEEE Trans. Audio Electroacoust. AU-20, 66 (1972).
[Crossref]

Mendlovic, D.

Morse, P. M.

P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1416.

Onural, L.

Ozaktas, H. M.

Parks, T. W.

J. H. McClellan and T. W. Parks, IEEE Trans. Audio Electroacoust. AU-20, 66 (1972).
[Crossref]

Sansone, G.

G. Sansone, Orthogonal Function, Vol. 9 of Pure and Applied Mathematics (Interscience, New York, 1959).

Santhanam, B.

B. Santhanam and J. H. McClellan, IEEE Trans. Signal Process. 42, 994 (1996).
[Crossref]

Schamschula, M.

Shih, C. C.

C. C. Shih, Opt. Commun. 118, 495 (1995).
[Crossref]

Soffer, B. H.

Steiglitz, K.

B. W. Dickinson and K. Steiglitz, IEEE Trans. Acoust. Speech Signal Process ASSP-30, 25 (1982).
[Crossref]

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, and R. G. Dorsch, Opt. Commun. 126, 14 (1996).
[Crossref]

Appl. Opt. (2)

IEEE Trans. Acoust. Speech Signal Process (1)

B. W. Dickinson and K. Steiglitz, IEEE Trans. Acoust. Speech Signal Process ASSP-30, 25 (1982).
[Crossref]

IEEE Trans. Audio Electroacoust. (1)

J. H. McClellan and T. W. Parks, IEEE Trans. Audio Electroacoust. AU-20, 66 (1972).
[Crossref]

IEEE Trans. Signal Process. (2)

B. Santhanam and J. H. McClellan, IEEE Trans. Signal Process. 42, 994 (1996).
[Crossref]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, IEEE Trans. Signal Process. 44, 2141 (1996).
[Crossref]

IMA J. Appl. Math. (1)

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, and R. G. Dorsch, Opt. Commun. 126, 14 (1996).
[Crossref]

C. C. Shih, Opt. Commun. 118, 495 (1995).
[Crossref]

Opt. Lett. (1)

Other (2)

P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1416.

G. Sansone, Orthogonal Function, Vol. 9 of Pure and Applied Mathematics (Interscience, New York, 1959).

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Figures (3)

Fig. 1
Fig. 1

FRFT of the rectangular window.

Fig. 2
Fig. 2

Current DFRFT of the rectangular window.

Fig. 3
Fig. 3

Improved DFRFT of the rectangular window.

Tables (2)

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Table 1 Multiplicities of DFT Eigenvalues

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Table 2 Eigenvalue-Assignment Rule of DFRFT Kernel Matrix

Equations (11)

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Kαv, u=1-j cotα2π1/2×expjv2+u22 cotα-juv cscα
=n=0exp-jnαhnvhnu,
χαu=-xtKαt, udt.
Fnk=1Nexp-j2πknN 0n, kN-1.
F2α/π=a0αF0+a1αF1+a2αF2+a3αF3,
aiα=14k=14expjα-iπ2k.
F2α/πxn=a0αxn+a2αx-n+a1αXn+a3αX-n,
S=[21000112 cos ω1000012 cos 2ω100100012 cosN-1ω],
F=kE1vkvk*+kE2-jvkvk*+kE3-1vkvk*+kE4jvkvk*,
F2α/π=VD2α/πVT=k=0N-1exp-jkαvkvkT N oddk=0N-2exp-jkαvkvkT+exp-jNαvN-1vN-1T N even,
Xαn=F2α/πxn.

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