Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Relationship between the ambiguity function coordinate transformations and the fractional Fourier transform

Relation entre la transformation de coordonnÉes de la fonction d’ambiguÏtÉ et la transformation de fourier fractionnaire

  • 148 Accesses


It has been shown that the fractional Fourier transform, recently very intensively investigated in mathematics, quantum mechanics, optics and signal processing, can be obtained as a special case of the earlier introduced linear coordinate transformations of the ambiguity function or Wigner distribution. Some applications of the generalized fractional transform on the time-frequency analysis are presented.


Il a été montré que la transformation de Fourier fractionnaire, qui a fait l’objet d’études récentes intensives en mathématique, en mécanique quantique, en optique et en traitement du signal, peut-être obtenue comme cas spécial des transformations de coordonnées linéaires de la fonction d’ambiguïté ou de la distribution de Wigner. Quelques applications de la transformation de Fourier fractionnaire à l’analyse temps-fréquence sont présentées.

This is a preview of subscription content, log in to check access.


  1. [1]

    Namias (V.): The fractional order Fourier transform and its application to quantum mechanics,J. Inst. Math. Appl. 25, pp. 241–265, (1980).

  2. [2]

    Lohmann (A. W.): Image rotation, Wigner rotation and fractional Fourier transform,J. Opt. Soc. Amer A. 10, pp. 2181–2186, (1993).

  3. [3]

    Lohmann (A. W.), Soffer (B. H.): Relation between the Radon-Wigner and fractional Fourier transform,J. Opt. Soc. Amer. A. 11, pp. 1798–1801, (1994).

  4. [4]

    Almeida (L. B.): The fractional Fourier transform and time-frequency representations,IEEE Trans. Signal Processing. 42, pp. 3084–3091,(1994).

  5. [5]

    Zayed (A. I.): On the relationship between the Fourier transform and fractional Fourier transform,IEEE Signal Processing Letters, 3, no.12, pp. 310–211, (1996).

  6. [6]

    Santhanam (B.), McClellan (J. H.): The discrete rational Fourier transform,IEEE Trans, on Signal Processing, 44, no. 4, (April 1996), pp. 994–997.

  7. [7]

    Mcbride (A. C), Keer (F. H.): The discrete rational Fourier transform,IMA J. Appl. Math.,39, pp. 159–175, (1987).

  8. [8]

    Almeida (L.B.),: Product and convolution theorems for the fractional Fourier transform,IEEE Signal Processing Letters, 4, no.1, (Jan. 1997), pp. 15–17.

  9. [9]

    Weyl (N.): Quantenmechanik und groupentheorie,Ztsch. f. Physik,46, pp. 1–47,(1927).

  10. [10]

    Wiener (N.): Hermintian polynomials and Fourier analysis,J. Math. Phys. MIT,8, pp. 70–73, (1929).

  11. [11]

    Harms (B.): Computing time-frequency distributions,IEEE Trans, on Signal Processing,39, no. 3, pp. 727–729, (March 1991).

  12. [12]

    RiSTic (B.), Boashash (B.): Kernel design for time-frequency signal analysis using the Radon transform,IEEE Trans, on Signal Processing,41, no. 5, pp. 1996–2008, (May 1995).

  13. [13]

    Wood (J. C), Barry (D. T.): Linear signal synthesis using the Radon-Wigner distribution,IEEE Trans, on Signal Processing,42, no. 8, pp. 2105–2111, (Aug. 1994).

  14. [14]

    Wood (J. C), Barry (D. T.): Radon transform of time-frequency distributions for analysis of multicomponent signals,IEEE Trans, on Signal Processing,42, no. 11, pp. 3166–3177, (Nov 1994)

  15. [15]

    Boashash (B.): Estimating and interpreting the instantaneous frequency of a signal-Part I Fundamentals,IEEE Proc. 80, no. 4, pp. 519–538, (April 1992).

  16. [16]

    Papoulis (A.): Ambiguity function in Fourier optics,J. opt. Soc. Amer.,64, no. 6, pp. 779–788, (June 1974).

  17. [17]

    Papoulis (A.): Signal analysis,McGraw Hill Book Company, New York, pp. 287–289, (1977).

  18. [18]

    Sadowsky (J.) : An application of the Weyl theory to signal processing, inProc. 23rd Asilomar Conf., Circuits, Syst., Compy, pp. 628–632, (1989).

  19. [19]

    Stankovic’ (S.): Wigner distribution in signal analysis Master thesis, University of Zagred. Croatia, pp. 20–25, (1992).

  20. [20]

    Stankovic’ (L. J.): An analysis of instantaneous frequency presentation using time-frequency distributions -Generalized Wigner distribution,IEEE Trans, on Signal Processing,43, no. 2, pp. 549–552, (Feb. 1995).

Download references

Author information

Correspondence to Igor DjuroviĆ or LJubiša StankoviĆ.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

DjuroviĆ, I., StankoviĆ, L. Relationship between the ambiguity function coordinate transformations and the fractional Fourier transform. Ann. Télécommun. 53, 316–319 (1998).

Download citation

Key words

  • Fourier transform
  • Fractional number
  • Mathematical transformation
  • Ambiguity function
  • Frequency time representation
  • Signal analysis

Mots clés

  • Transformation Fourier
  • Nombre fractionnaire
  • Transformation mathématique
  • Fonction ambiguïté
  • Représentation temps fréquence
  • Analyse signal