Closed-form discrete fractional and affine fourier transforms

IEEE Transactions on Signal Processing

Volumn 48, Issue 5, 2000, Pages 1338-1353

  Pel, Soo Chang ^a   Ding, Jian Jiun ^a  

^a NATIONAL TAIWAN UNIVERSITY   (Taiwan)

Author keywords

Affine fourier transform; Discrete affine fourier transform; Discrete fourier transform; Discrete fractional fourier transform; Fourier transform

Indexed keywords

EID: 33747799699     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/78.839981     Document Type: Article

References (24)

1. V. Namias, The fractional order Fourier transform and its application to quantum mechanics, J. Inst. Math. Appl., vol. 25, pp. 241-265, 1980.
2. L. B. Almeida, The fractional Fourier transform and time-frequency representations, IEEE Trans. Signal Processing, vol. 42, pp. 3084-3091, Nov. 1994.
3. S. Abe and J. T. Sheridan, Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation, Opt. Lett., vol. 19, no. 22, pp. 1801-1803, 1994.
4. M. Moshinsky and C. Quesne, Linear canonical transformations and their unitary representations, J. Math. Phys., vol. 12, no. 8, pp. 1772-1783, Aug. 1971.
5. H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, Digital computation of the fractional Fourier transform, IEEE Trans. Signal Processing, vol. 44, pp. 2141-2150, Sept. 1996.
6. 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.
7. B. Santhanam and J. H. McClellan, The DRFT-A rotation in time-frequency space, in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., 1995, pp. 921-924.
8. _, The discrete rotational Fourier transform, IEEE Trans. Signal Processing, vol. 42, pp. 994-998, Apr. 1996.
9. S. C. Pel, C. C. Tseng, and M. H. Yeh, Discrete fractional Hartley and Fourier transform, IEEE Trans. Circuits Syst. II, vol. 45, pp. 665-675, June 1998.
10. S. C. Pei and M. H. Yeh, Improved discrete fractional Fourier transform, Opt. Lett., pp. 1047-1049, July 1997.
11. S.C. Pei, M. H. Yeh, and C. C. Tseng, Discrete fractional Fourier transform based on orthogonal projection, IEEE Trans. Signal Processing. vol. 47, pp. 1335-1348, May 1999.
12. B. Z. Dong, Y. Zhang, B. Y Gu, and G. Z. Yang, Numerical investigation of phase retrieval in a fractional Fourier transform, J. Opt. Soc. Amer. A. vol. 14, no. 10, pp. 2709-2713, 1997.
13. O. Ankan, M. A. Kutay, H. M. Ozaktas, and O. K. Akdemir, The discrete fractional Fourier transformation, in Proc. IEEE Int. Symp. TimeFrequency Time-Scale Anal., 1996, pp. 205-207.
14. M. S. Richman and T. W. Parks, Understanding discrete rotations, in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 3, 1997, pp. 2057-2060.
15. M. L. Curtis, Matrix Groups, 2nd ed. New York: Springer-Verlag, 1979.
16. S. C. Pei and C. C. Tseng, New discrete fractional Fourier transform based on constrained eigendecomposition of DFT matrix by Lagrange multiplier method, in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 5, 1997, pp. 3965-3968.
17. J. J. Ding, Derivation and properties of orthogonal transform, M.S. thesis, Nat. Taiwan Univ., Taipei, Taiwan, R.O.C., 1997.
18. D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, Fractional correlation, Appl. Opt., vol. 34, pp. 303-309, 1994.
19. A. W. Lohmann, Z. Zalevsky, and D. Mendlovic, Synthesis of pattern recognition filters for fractional Fourier processing, Opt. Commun., vol. 128, pp. 199-204, July 1996.
20. N. Ahmed, T. Natarajan, and K. R. Rao, Discrete cosine transform, IEEE Trans. Comput., vol. C-23, pp. 90-93, Jan. 1974.
21. Z. Wang, Fast algorithm for the discrete W transform and for the discrete Fourier transform, IEEE. Trans. Acoust. Speech Signal Processing, vol. ASSP-32, pp. 803-816, Aug. 1984.
22. W. X. Cong, N. X. Chen, and B. Y Gu, Recursive algorithm for phase retrieval in the fractional Fourier transform domain, Appl. Opt., vol. 37, no. 29, pp. 6906-6910, Oct. 1998.
23. A. W. Lohmann, Image rotation, Wigner rotation, and the fractional Fourier transform,J. Opt. Soc. Amer. A, vol. 10,no. 10, pp. 2181-2186, Oct. 1993.
24. G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, A unified framework for the fractional Fourier transform, IEEE Trans. Signal Processing, vol. 46, pp. 3206-3219, Dec. 1998.