The discrete fractional fourier transform

IEEE Transactions on Signal Processing

Volumn 48, Issue 5, 2000, Pages 1329-1337

  Candan, Çagatay ^a   Alper Kutay, M ^b   Ozaktas, Haldun M ^c  

^a BILKENT UNIVERSITY   (Turkey)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^c Drexel University   (United States)

Author keywords

Chirplets; Discrete wigner distributions; Hermite gaussian functions; Time frequency analysis

Indexed keywords

EID: 0041072734     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/78.839980     Document Type: Article

References (51)

1. S. C. Pel and M. H. Yeh, Improved discrete fractional fourier transform, Opt. Lett., vol. 22, pp. 1047-1049, 1997.
2. S. C. Pei, C. C. Tseng, M. H. Yeh, and J. J. Shyu, Discrete fractional Hartley and Fourier transforms, IEEE Trans. Circuits Syst. II. vol. 45, pp. 665-675, 1998.
3. V. Namias, The fractional order Fourier transform and its application to quantum mechanics, J. Inst. Math. Appl., vol. 25, pp. 241-265, 1980.
4. A. C. McBride and F. H. Kerr, On Namias's fractional Fourier transform, IMA J. Appl. Math., vol. 39, pp. 159-175, 1987.
5. H. M. Ozaktas, M. A. Kutay, and D. Mendlovic, Introduction to the fractional Fourier transform and its applications, in Advances in Imaging Electronics and Physics. New York: Academic, 1999, ch. 4.
6. D. Mendlovic and H. M. Ozaktas, Fractional Fourier transforms and their optical implementation 1, J. Opt. Soc. Amer. A., vol. 10, pp. 1875-1881, 1993.
7. H. M. Ozaktas and D. Mendlovic, Fractional Fourier transforms and their optical implementation 2, J. Opt. Soc. Amer. A., vol. 10, pp. 2522-2531, 1994.
8. 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. Amer. A., vol. 11, pp. 547-559, 1994.
9. A. W. Lohmann, Image rotation, Wigner rotation, and the fractional Fourier transform, / Opt. Soc.Amer.A., vol. 10, pp. 2181-2186, 1993.
10. A. W. Lohmann and B. H. Soffer, Relationships between the RadonWigner and fractional Fourier transforms, J. Opt. Soc. Amer. A., vol. 11, pp. 1798-1811, 1994.
11. D. Mustard, The fractional Fourier transform and the Wigner distribution, J. Austral. Math. Soc. B-Appl. Math., vol. 38, pp. 209-219, 1996.
12. L. B. Almeida, The fractional Fourier transform and time-frequency representations, IEEE Trans. Signal Processing, vol. 42, pp. 3084-3091, Nov. 1994.
13. M. A. Kutay, H. M. Ozaktas, L. Onural, and O. Ankan, Optimal filtering in fractional Fourier domains, in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 2, 1995, pp. 937-941.
14. M. A. Kutay, H. M. Ozaktas, O. Ankan, and L. Onural, Optimal filtering in fractional Fourier domains, IEEE Trans. Signal Processing, vol. 45, pp. 1129-1143, July 1997.
15. _, Optimal image restoration with the fractional Fourier transform, J. Opt. Soc. Amer. A., vol. 15, pp. 825-833, 1998.
16. Z. Zalevsky and D. Mendlovic, Fractional Wiener filter, Appl. Opt., vol. 35, pp. 3930-3936, 1996.
17. M. F. Erden and H. M. Ozaktas, Synthesis of general linear systems with repeated filtering in consecutive fractional Fourier domains, J. Opt. Soc. Amer. A., vol. 15, pp. 1647-1657, 1998.
18. M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Ankan, O. Güleryüz, and Ç. Candan, Space-bandwidth efficient realizations of linear systems, Opt. Lett., vol. 23, pp. 1069-1071, 1998.
19. _, Cost-efficient approximation of linear systems with repeated and multi-channel filtering configurations, in Proc. IEEE Int. Conf. Acoust. Speech Signal Process., vol. 6, 1998, pp. 3433-3437.
20. M. F. Erden, M. A. Kutay, and H. M. Ozaktas, Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration, IEEE Trans. Signal Processing, vol. 47, pp. 1468-1462, May 1999.
21. M. A. Kutay, H. Ozaktas, M. F. Erden, H. M. Ozaktas, and O. Ankan, Solution and cost analysis of general multi-channel and multi-stage filtering circuits, in Proc. IEEE SP Symp. Time-Freq. Anal, vol. 1, 1998, pp. 481-484.
22. H. M. Ozaktas, N. Erkaya, and M. A. Kutay, Effect of fractional Fourier transformation on time-frequency distributions belonging to the cohen class, IEEE Signal Processing Lett., vol. 40, pp. 40-41, Feb. 1996.
23. O. Akay and G. F. Boudreaux-Bartels, Joint fractional representations, in Proc. IEEE SP Symp. Time-Freq. Anal., vol. 1, 1998, pp. 417-420.
24. H. M. Ozaktas and D. Mendlovic, Fractional Fourier optics, J. Opt. Soc. Amer. A., vol. 12, pp. 743-751, 1995.
25. H. M. Ozaktas and M. F. Erden, Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems, Opt. Commun., vol. 143, pp. 75-86, 1997.
26. D. Mendlovic, Z. Zalevsky, and H. M. Ozaktas, The applications of the fractional Fourier transform to optical pattern recognition, in Optical Pattern Recognition. New York: Academic, 1998, ch. 3.
27. A. I. Zayed, On the relationship between the Fourier and fractional Fourier transforms, IEEE Signal Processing Lett., vol. 3, pp. 310-311, Nov. 1996.
28. X. Xia, On bandlimited signals with fractional Fourier transform, IEEE Signal Processing Lett., vol. 3, pp. 72-74, Mar. 1996.
29. L. B. Almeida, Product and convolution theorems for the fractional Fourier transform, IEEE Signal Processing Lett., vol. 4, pp. 15-17, Jan. 1997.
30. A. I. Zayed, Convolution and product theorem for the fractional Fourier transform, IEEE Signal Processing Lett., vol. 5, pp. 101-103, Apr. 1998.
31. _, Hubert transform associated with the fractional Fourier transform, IEEE Signal Processing Lett., vol. 5, pp. 206-208, July 1998.
32. M. J. Bastiaans and A. J. van Leest, From the rectangular to the quincunx gabor lattice via fractional Fourier transformation, IEEE Signal Processing Lett., vol. 5, pp. 203-205, July 1998.
33. A. Sahin, M. A. Kutay, and H. M. Ozaktas, Non-separable two-dimensional fractional Fourier transform, Appl. Opt., vol. 37, pp. 5444-5453, 1998.
34. H. M. Ozaktas, O. Ankan, M. A. Kutay, and G. Bozdagi, Digital computation of the fractional Fourier transform, IEEE Trans. Signal Processing, vol. 44, pp. 2141-2150, Sept. 1996.
35. O. Ankan, M. A. Kutay, H. M. Ozaktas, and 0. K. Akdemir, Discrete fractional Fourier transformation, Proc. IEEE SP Symp. TimeFreq. Anal, vol. 4, pp. 205-207, 1996.
36. N. M. Atakishiyev and K. B. Wolf, Fractional Fourier-Kravchuk transform, J. Opt. Soc. Amer. A., vol. 14, pp. 1467-1477, 1997.
37. M. S. Richman, T. W. Parks, and R. G. Shenoy, Understanding discrete rotations, in Proc. IEEE Int. Conf. Acoust. Speech Signal Process., 1997.
38. B. W. Dickinson and K. Steiglitz, Eigenvectors and functions of the discrete Fourier transform, IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-30, pp. 25-31, Jan. 1982.
39. B. Santhanam and J. H. McClellan, The discrete rotational Fourier transform, IEEE Trans. Signal Processing, vol. 44, pp. 994-998, Apr. 1996.
40. G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, Multiplicity of fractional Fourier transforms and their relations, IEEE Trans. Signal Processing, vol. 48, pp. 227-241, Jan. 2000.
41. 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, 1972.
42. P. Beckmann, Orthogonal Polynomials for Engineers and Physicists. New York: Golem, 1973.
43. G. Birkhoff and G. C. Rota, Ordinary Differential Equations. New York: Wiley, 1989.
44. K. B. Wolf, Construction and properties of canonical transforms, in Integral Transforms in Science and Engineering. New York: Plenum, 1979, ch. 9.
45. J. H. Wilkinson, The Algebraic Eigenvalue Problem. Oxford, U.K.: Oxford, 1988.
46. F. Hildebrand, Finite-Difference Equations and Simulations. Englewood Cliffs, NJ: Prentice-Hall, 1968.
47. S. N. Elaydi, An Introduction to Difference Equations. New York: Springer, 1996.
48. Ç. Candan, The Discrete Fractional Fourier Transform, M.S. thesis, Bilkent Univ., Ankara, Turkey, 1998.
49. L. J. Barker, Ç. Candan, T. Hakioglu, M. A. Kutay, and H. M. Ozaktas, The harmonic oscillator, Harper's equation and the discrete fractional Fourier transform, J. Phys. A.: Math., vol. 33, pp. 2209-2222, Mar. 2000.
50. L. J. Barker, The discrete fractional Fourier transform and Harper's equation, preprint.
51. N. M. Atakishiyev and S. K. Suslov, Difference analogs of the harmonic oscillator, Theor. Math. Phys., vol. 85, pp. 1055-1062, 1991.