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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 19, Iss. 9 — Sep. 1, 2002
  • pp: 1763–1773

Wigner distribution moments in fractional Fourier transform systems

Martin J. Bastiaans and Tatiana Alieva  »View Author Affiliations

JOSA A, Vol. 19, Issue 9, pp. 1763-1773 (2002)

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It is shown how all global Wigner distribution moments of arbitrary order in the output plane of a (generally anamorphic) two-dimensional fractional Fourier transform system can be expressed in terms of the moments in the input plane. Since Wigner distribution moments are identical to derivatives of the ambiguity function at the origin, a similar relation holds for these derivatives. The general input–output relationship is then broken down into a number of rotation-type input–output relationships between certain combinations of moments. It is shown how the Wigner distribution moments (or ambiguity function derivatives) can be measured as intensity moments in the output planes of a set of appropriate fractional Fourier transform systems and thus be derived from the corresponding fractional power spectra. The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived. As an important by-product we get a number of moment combinations that are invariant under (anamorphic) fractional Fourier transformation.

© 2002 Optical Society of America

OCIS Codes
(030.5630) Coherence and statistical optics : Radiometry
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(110.6980) Imaging systems : Transforms
(120.4800) Instrumentation, measurement, and metrology : Optical standards and testing

Original Manuscript: February 20, 2002
Revised Manuscript: April 12, 2002
Manuscript Accepted: April 24, 2002
Published: September 1, 2002

Martin J. Bastiaans and Tatiana Alieva, "Wigner distribution moments in fractional Fourier transform systems," J. Opt. Soc. Am. A 19, 1763-1773 (2002)

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  1. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932). [CrossRef]
  2. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978). [CrossRef]
  3. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979). [CrossRef]
  4. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986). [CrossRef]
  5. W. Mecklenbräuker, F. Hlawatsch, eds., The Wigner Distribution—Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997).
  6. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light beams,” in Optics and Optoelectronics, Theory, Devices and Applications, Proceedings of ICOL’98, the International Conference on Optics and Optoelectronics, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narosa, New Delhi, 1998), pp. 101–115.
  7. International Organization for Standardization, Technical Committee/Subcommittee 172/SC9, “Lasers and laser-related equipment—test methods for laser beam parameters—beam widths, divergence angle and beam propagation factor,” (International Organization for Standardization, Geneva, 1999).
  8. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).
  9. G. Nemes, A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11, 2257–2264 (1994). [CrossRef]
  10. B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998). [CrossRef]
  11. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988). [CrossRef]
  12. C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997). [CrossRef]
  13. J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991). [CrossRef]
  14. J. Serna, F. Encinas-Sanz, G. Nemes, “Complete spatial characterization of a pulsed doughnut-type beam by use of spherical optics and a cylindrical lens,” J. Opt. Soc. Am. A 18, 1726–1733 (2001). [CrossRef]
  15. P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, London, 1953), Chap. 7.
  16. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974). [CrossRef]
  17. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), pp. 284–295.
  18. L. Cohen, Time-Frequency Signal Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1995).
  19. F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag., April1992, pp. 21–67.
  20. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993). [CrossRef]
  21. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994). [CrossRef]
  22. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Vol. 38, pp. 263–342.
  23. H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, pp. 239–291.
  24. H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform—with Applications in Optics and Signal Processing (Wiley, Chichester, UK, 2001).
  25. M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory IT-8, 179–187 (1962).
  26. T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 320–323 (2000). [CrossRef]
  27. N. G. de Bruijn, “Uncertainty principles in Fourier analysis,” in Inequalities, O. Shisha, ed. (Academic, New York, 1967), pp. 57–71.

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