Skip to main content

Advertisement

Springer Nature Link
Log in

Modified correlation theorem for the linear canonical transform with representation transformation in quantum mechanics

  • Original Paper
  • Published:
Signal, Image and Video Processing Aims and scope Submit manuscript

Abstract

As generalization of the fractional Fourier transform, the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties of the LCT are currently known including correlation theorem, but however, these do not generalize very nicely to the classical result for the FT. This paper presents a modified correlation theorem in the LCT domain with representation transformation in quantum mechanics. The derivation is direct and concise since the authors make full use of the Dirac’s representation theory. The proposed theorem is compared with the existing correlation theorem in the literature for the LCT and found to be better and befitting proposition. As an application, the proposed correlation theorem is used to determine the power spectral density of frequency-modulated (FM) signal, which shows that the same FM signal can be transmitted with less bandwidth requirement.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  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)

    Article  MathSciNet  Google Scholar 

  2. Moshinsky, M., Quesne, C.: Linear canonical transformations and their unitary representations. J. Math. Phys. 12(8), 1772–1783 (1771)

    Google Scholar 

  3. Nazarathy, M., Shamir, J.: First-order optics—a canonical operator representation: lossless systems. J. Opt. Soc. Am. 72(3), 356–364 (1982)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  5. Pei, S.C., Ding, J.J.: Eigenfunctions of linear canonical transform. IEEE Trans. Acoust. Speech Signal Process. 50(1), 11–26 (2002)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  7. Stern, A.: Sampling of linear canonical transformed signals. Signal Process. 86(7), 1421–1425 (2006)

    Article  MATH  Google Scholar 

  8. Tao, R., Qi, L., Wang, Y.: Theory and Applications of the Fractional Fourier Transform. Tsinghua University Press, Beijing (2004)

    Google Scholar 

  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. Almeida, L.B.: The fractional Fourier transform and time-frequency representations. IEEE Trans. Signal Process. 42(11), 3084–3091 (1994)

    Article  Google Scholar 

  11. Barshan, B., Ozaktas, H.M., Kutay, M.A.: Optimal filtering with linear canonical transformations. Opt. Commun. 135(1–3), 32–36 (1997)

    Article  Google Scholar 

  12. Sharma, K.K., Joshi, S.D.: Signal separation using linear canonical and fractional Fourier transforms. Opt. Commun. 265(2), 454–460 (2006)

    Article  Google Scholar 

  13. Goel, N., Singh, K.: Analysis of Dirichlet, generalized Hamming and Triangular window functions in the linear canonical transform domain. Signal Image Video Process. 7(5), 911–923 (2013)

    Google Scholar 

  14. Goel, N., Singh, K.: Modified convolution and product theorem for the linear canonical transform derived by representation transformation in quantum mechanics. Appl. Math. Comput. Sci. 23(3), 685–695 (2013)

    Google Scholar 

  15. Wolf, K.B.: Integral Transforms in Science and Engineering. Plenum Press, New York (1979)

    Book  MATH  Google Scholar 

  16. Deng, B., Tao, R., Wang, Y.: Convolution theorem for the linear canonical transform and their applications. Sci. China F 49(5), 592–603 (2006)

    Article  MathSciNet  Google Scholar 

  17. Li, B.Z., Tao, R., Wang, Y.: New sampling formulae related to linear canonical transform. Signal Process. 87(5), 983–990 (2007)

    Google Scholar 

  18. 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)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  Google Scholar 

  20. Tao, R., Li, B.Z., Wang, Y.: On sampling of band-limited signals associated with the linear canonical transform. IEEE Trans. Signal Process. 56(11), 5454–5464 (2008)

    Article  MathSciNet  Google Scholar 

  21. Pei, S.C., Ding, J.J.: Closed-form discrete fractional and affine Fourier transforms. IEEE Trans. Signal Process. 48(5), 1338–1353 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hennelly, B.M., Sheridan, J.T.: Fast numerical algorithm for the linear canonical transform. J. Opt. Soc. Am. A 22(5), 928–937 (2005)

    Article  MathSciNet  Google Scholar 

  23. Healy, J.J., Sheridan, J.T.: Sampling and discretization of the linear canonical transform. Signal Process. 89(4), 641–648 (2009)

    Article  MATH  Google Scholar 

  24. 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)

    Article  Google Scholar 

  25. Ozaktas, H.M., Barshan, B., Mendlovic, D.: Convolution, filtering, and multiplexing in fractional Fourier domains and their relationship to chirp and wavelet transforms”. J. Opt. Soc. Am. A 11(2), 547–559 (1994)

    Article  MathSciNet  Google Scholar 

  26. Almeida, L.B.: Product and convolution theorems for the fractional Fourier transform. IEEE Trans. Signal Process. Lett 4(1), 15–17 (1997)

    Article  MathSciNet  Google Scholar 

  27. Zayed, A.I.: A product and convolution theorems for the fractional Fourier transform. IEEE Trans. Signal Process. 5(4), 101–103 (1998)

    Article  Google Scholar 

  28. Sharma, K.K., Joshi, S.D.: Papoulis-like generalized sampling expansions in fractional Fourier domains and their application to super resolution. Opt. Commun. 278(1), 52–59 (2007)

    Article  MathSciNet  Google Scholar 

  29. Wei, D., Ran, Q., Li, Y.: A convolution and correlation theorem for the linear canonical transform and its application. Circuits Syst. Signal Process. 31(1), 301–312 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  30. Wei, D., Ran, Q., Li, Y., Ma, J., Tan, L.: A convolution and product theorem for the linear canonical transform. IEEE Signal Process. Lett. 16(10), 853–856 (2009)

    Article  Google Scholar 

  31. Singh, A.K., Saxena, R.: On convolution and product theorems for FRFT. Wirel. Pers. Commun. 65(1), 189–201 (2012)

    Article  MathSciNet  Google Scholar 

  32. Akay, O., Boudreaux-Bartels, G.F.: Fractional convolution and correlation via operator methods and application to detection of linear FM signals. IEEE Trans. Signal Process. 49(5), 979–993 (2001)

    Article  Google Scholar 

  33. Mendlovic, D., Bitran, Y., Dorsch, R.G., Lohmann, A.W.: Optical fractional correlation: experimental results. J. Opt. Soc. 12(8), 1665–1670 (1995)

    Article  MathSciNet  Google Scholar 

  34. Mendlovic, D., Ozaktas, H.M., Lohmann, A.W.: Fractional correlation. Appl. Opt. 34(2), 303–309 (1995)

    Article  Google Scholar 

  35. Torres, R., Pellat-Finet, P., Torres, Y.: Fractional convolution, fractional correlation and their translation invariance properties. Signal Process. 90(6), 1976–1984 (2010)

    Article  MATH  Google Scholar 

  36. James, D.F.V., Agarwal, G.S.: The generalized Fresnel transform and its applications to optics. Opt. Commun. 126(4–6), 207–212 (1996)

    Google Scholar 

  37. Palma, C., Bagini, V.: Extension of the Fresnel transform to ABCD systems. J. Opt. Soc. Am. 14(8), 1774–1779 (1997)

    Google Scholar 

  38. Bernardo, L.M.: ABCD matrix formalism of fractional Fourier optics. Opt. Eng. 35(3), 732–740 (1996)

    Article  Google Scholar 

  39. Collins, S.A.: Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am. 60(9), 1168–1174 (1970)

    Article  Google Scholar 

  40. Bastiaans, M.J.: Wigner distribution function and its application to first-order optics. J. Opt. Soc. Am. 69(12), 1710–1716 (1979)

    Article  Google Scholar 

  41. Hua, J., Liu, L., Li, G.: Extended fractional Fourier transforms. J. Opt. Soc. Am. 14(12), 3316–3322 (1997)

    Article  MathSciNet  Google Scholar 

  42. 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)

    Article  Google Scholar 

  43. Abe, S., Sheridan, J.T.: Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach. J. Phys. A 27, 4179–4187 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  44. Fan, H.Y., Yue, F.: New eigenmodes of propagation in quadratic graded index media and complex fractional Fourier transform. Commun. Theor. Phys. 39(1), 97–100 (2003)

    MathSciNet  Google Scholar 

  45. Fan, H.Y., Ren, H., Lu, H.L.: Convolution theorem of fractional Fourier transformation derived by representation transformation in quantum mechanics. Commun. Theor. Phys. 50(3), 611–614 (2008)

    Article  MathSciNet  Google Scholar 

  46. Kiusalaas, J.: Numerical Methods in Engineering with Python, Ch. 6. Cambridge University Press, New York (2010)

    Google Scholar 

  47. Borsellino, A., Poggio, T.: Convolution and correlation algebras. Kybernetik. 13(2), 113–122 (1973)

    Google Scholar 

Download references

Acknowledgments

The authors would like to thank the editor-in-chief and the anonymous referees for their valuable comments and suggestions that improved the clarity and quality of this manuscript.

Author information

Authors and Affiliations

  1. Electronics and Communication Engineering Section, Yadavindra College of Engineering, Punjabi University Guru Kashi Campus, Talwandi Sabo, 151302, Punjab, India

    Navdeep Goel

  2. Department of Electronics and Communication Engineering, Thapar University, Patiala, 147001, Punjab, India

    Kulbir Singh

Authors
  1. Navdeep Goel
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Kulbir Singh
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Navdeep Goel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goel, N., Singh, K. Modified correlation theorem for the linear canonical transform with representation transformation in quantum mechanics. SIViP 8, 595–601 (2014). https://doi.org/10.1007/s11760-013-0564-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11760-013-0564-9

Keywords