OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 21, Iss. 7 — Jul. 1, 2004
  • pp: 1179–1185

Fractional finite Fourier transform

Kedar Khare and Nicholas George  »View Author Affiliations

JOSA A, Vol. 21, Issue 7, pp. 1179-1185 (2004)

View Full Text Article

Enhanced HTML    Acrobat PDF (176 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We show that a fractional version of the finite Fourier transform may be defined by using prolate spheroidal wave functions of order zero. The transform is linear and additive in its index and asymptotically goes over to Namias’s definition of the fractional Fourier transform. As a special case of this definition, it is shown that the finite Fourier transform may be inverted by using information over a finite range of frequencies in Fourier space, the inversion being sensitive to noise. Numerical illustrations for both forward (fractional) and inverse finite transforms are provided.

© 2004 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(070.2590) Fourier optics and signal processing : ABCD transforms
(200.3050) Optics in computing : Information processing

Original Manuscript: November 17, 2003
Revised Manuscript: February 19, 2004
Manuscript Accepted: February 19, 2004
Published: July 1, 2004

Kedar Khare and Nicholas George, "Fractional finite Fourier transform," J. Opt. Soc. Am. A 21, 1179-1185 (2004)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. V. Namias, “The fractional order Fourier transform and its applications to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980). [CrossRef]
  2. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993). [CrossRef]
  3. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993). [CrossRef]
  4. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993). [CrossRef]
  5. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993). [CrossRef]
  6. P. Pellat-Finet, “Fresnel diffraction and the fractional order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994). [CrossRef] [PubMed]
  7. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995). [CrossRef]
  8. 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]
  9. C. Candan, M. A. Kutay, H. M. Ozaktas, “The discretefractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000). [CrossRef]
  10. S. Pei, J. Ding, “Simplified fractional Fourier transforms,” J. Opt. Soc. Am. A 17, 2355–2367 (2000). [CrossRef]
  11. H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chichester, UK, 2001; this book contains a comprehensive list of publications on this subject.)
  12. K. Khare, N. George, “Sampling theory approach to prolate spheroidal wave functions,” J. Phys. A 36, 10011–10021 (2003). [CrossRef]
  13. C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949). [CrossRef]
  14. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).
  15. J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (Wiley, New York, 1956).
  16. D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961). [CrossRef]
  17. B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on based on the use of prolate functions,” in Progress in Optics, Vol. IX, E. Wolf ed. (Elsevier, New York, 1971), pp. 311–407.
  18. M. Bertero, C. De Mol, “Super-resolution by data inversion,” in Progress in Optics, Vol. XXXVIE. Wolf, ed. (Elsevier, New York, 1996), pp. 129–178.
  19. D. Slepian, “Some asymptotic expansions for prolate spheroidal wave functions,” J. Math. Phys. 44, 99–140 (1965).
  20. P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, London, 1953), p. 781.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited