Signal, Image and Video Processing

, Volume 7, Issue 5, pp 911–923 | Cite as

Analysis of Dirichlet, Generalized Hamming and Triangular window functions in the linear canonical transform domain

  • Navdeep Goel
  • Kulbir Singh
Original Paper


Linear canonical transform is a four-parameter class of integral transform that plays an important role in many fields of optics and signal processing. Well-known transforms such as the Fourier transform, the fractional Fourier transform, and the Fresnel transform can be seen as the special cases of the linear canonical transform. This paper presents a new mathematical model for obtaining the linear canonical transforms of Dirichlet, Generalized “Hamming”, and Triangular window functions. The different window function parameters are also obtained from the simulations. By changing the value of four parameters and then changing the adjustable parameter, the main-lobe width, −3 dB bandwidth, −6 dB bandwidth and correspondingly, the minimum stop-band attenuation of the resulting window functions can be controlled. It has been shown that by using linear canonical transform, we are able to obtain all window parameters successfully as compared to fractional Fourier transform.


Linear canonical transform Window function Fresnel transform 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alieva T., Bastiaans M.J.: Powers of transfer matrices determined by means of eigenfunctions. J. Opt. Soc. Am. A 16(10), 2413–2418 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Moshinsky M., Quesne C.: Linear canonical transformations and their unitary representations. J. Math. Phys. 12(8), 1772–1783 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Nazarathy M., Shamir J.: First-order optics—a canonical operator representation: lossless systems. J. Opt. Soc. Am. 72(3), 356–364 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Pei S.C., Ding J.J.: Relations between fractional operations and time-frequency distributions, and their applications. IEEE Trans. Signal Process. 49(8), 1638–1655 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Pei S.C., Ding J.J.: Eigenfunctions of linear canonical transform. IEEE Trans. Acoust. Speech. Signal Process. 50(1), 11–26 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hennelly B.M., Sheridan J.T.: Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms. J. Opt. Soc. Am. A 22(5), 917–927 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Stern A.: Sampling of linear canonical transformed signals. Signal Process. 86(7), 1421–1425 (2006)zbMATHCrossRefGoogle Scholar
  8. 8.
    Tao R., Qi L., Wang Y.: Theory and Applications of the Fractional Fourier Transform. Tsinghua University Press, Beijing (2004)Google Scholar
  9. 9.
    Ozaktas H.M., Kutay M.A., Zalevsky Z.: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York (2000)Google Scholar
  10. 10.
    Almeida L.B.: The fractional Fourier transform and time-frequency representations. IEEE Trans. Signal Process. 42(11), 3084–3091 (1994)CrossRefGoogle Scholar
  11. 11.
    Barshan B., Ozaktas H.M., Kutay M.A.: Optimal filters with linear canonical transformations. Opt. Commun. 135(1–3), 32–36 (1997)CrossRefGoogle Scholar
  12. 12.
    Sharma K.K., Joshi S.D.: Signal separation using linear canonical and fractional Fourier transforms. Opt. Commun. 265(2), 454–460 (2006)CrossRefGoogle Scholar
  13. 13.
    Deng B., Tao R., Wang Y.: Convolution theorem for the linear canonical transform and their applications. Sci. China F 49(5), 592–603 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Li B.Z., Tao R., Wang Y.: New sampling formulae related to linear canonical transform. Signal Process. 87(5), 983–990 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Koc A., Ozaktas H.M., Candan C., Kutay M.A.: Digital computation of linear canonical transforms. IEEE Trans. Signal Process. 56(6), 2383–2394 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Healy J.J., Sheridan J.T.: Cases where the linear canonical transform of a signal has compact support or is band-limited. Opt. Lett. 33(3), 228–230 (2008)CrossRefGoogle Scholar
  17. 17.
    Tao R., Li B.Z., Wang Y., Aggrey G.K.: On sampling of band-limited signals associated with the linear canonical transform. IEEE Trans. Signal Process. 56(11), 5454–5464 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pei S.C., Ding J.J.: Closed-form discrete fractional and affine Fourier transforms. IEEE Trans. Signal Process. 48(5), 1338–1353 (2002)MathSciNetGoogle Scholar
  19. 19.
    Hennelly B.M., Sheridan J.T.: Fast numerical algorithm for the linear canonical transform. J. Opt. Soc. Am. A 22(5), 928–937 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Healy J.J., Sheridan J.T.: Sampling and discretization of the linear canonical transform. Signal Process. 89(4), 641–648 (2009)zbMATHCrossRefGoogle Scholar
  21. 21.
    Oktem F.S., Ozaktas H.M.: Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space-bandwidth product. J. Opt. Soc. Am. A 27(8), 1885–1895 (2010)CrossRefGoogle Scholar
  22. 22.
    Weisstein E.W.: CRC Concise Encyclopedia of Mathematics. CRC Press, Boca Raton (2003)Google Scholar
  23. 23.
    Harris, F.J.: On the use of windows for harmonic analysis with discrete Fourier transform. In: Proceedings of the IEEE, vol. 66(1), pp. 51–83 (1978)Google Scholar
  24. 24.
    Liu Y., Kou K., Ho I.: New sampling formulae for non-band limited signals associated with linear canonical transform and non linear Fourier atoms. Signal Process. 90, 933–945 (2010)zbMATHCrossRefGoogle Scholar
  25. 25.
    Alieva T., Bastiaans M.J.: Properties of the linear canonical integral transformation. J. Opt. Soc. Am. A. 24(11), 3658– 3665 (2007)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kumar S., Singh K., Saxena R.: Analysis of Dirichlet and generalized “Hamming” window functions in the fractional Fourier transform domains. Signal Process. 91, 600–606 (2010)CrossRefGoogle Scholar
  27. 27.
    James D.F.V., Agarwal G.S.: The generalized Fresnel transform and its applications to optics. Opt. Commun. 126(4–6), 207–212 (1996)CrossRefGoogle Scholar
  28. 28.
    Bernardo L.M.: ABCD matrix formalism of fractional Fourier optics. Opt. Eng. 35(3), 732–740 (1996)CrossRefGoogle Scholar
  29. 29.
    Abe S., Sheridan J.T.: Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation. Opt. Lett. 19(22), 1801–1803 (1994)CrossRefGoogle Scholar
  30. 30.
    Zhao J., Tao R., Wang Y.: Sampling rate conversion for linear canonical transform. Signal Process. 88, 2825–2832 (2008)zbMATHCrossRefGoogle Scholar
  31. 31.
    Dainty, J.C.: Current Trends in Optics. Academic Press, New York, Ch. 10, pp. 139–148 (1994)Google Scholar
  32. 32.
    Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, vol. 55 of National Bureau of Standards Applied Mathematics Series. US Government Printing Office, Washington, DC (1964)Google Scholar
  33. 33.
    Saxena R., Singh K.: Fractional Fourier transform—a review. IETE J. Educ. 48, 13–30 (2007)Google Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Electronics and Communication Engineering Section, YCOEPunjabi University Guru Kashi CampusTalwandi SaboIndia
  2. 2.Department of Electronics and Communication EngineeringThapar UniversityPatialaIndia

Personalised recommendations