Skip to main content
SpringerLink
Log in

Sampling expansion in function spaces associated with the linear canonical transform

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

Abstract

In this paper, we investigate sampling expansion for the linear canonical transform (LCT) in function spaces. First, some properties of the function spaces related to the LCT are obtained. Then, a sampling theorem for the LCT in function spaces with a single-frame generator is derived by using the Zak Transform and its generalization to the LCT domain. Some examples are also presented.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

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

    Book  MATH  Google Scholar 

  2. Healy, J.J., Sheridan, J.T.: Fast linear canonical transforms. J. Opt. Soc. Am. A 27, 21–30 (2010)

    Article  MathSciNet  Google Scholar 

  3. Ozaktas, H.M., Zalevsky, Z., Kutay, M.A.: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York (2000)

    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, 1638–1655 (2001)

    Article  MathSciNet  Google Scholar 

  5. Healy, J.J., Sheridan, J.T.: The space-bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms. J. Opt. Soc. Am. A 28, 786–790 (2011)

    Article  Google Scholar 

  6. Shi, J., Sha, X., Zhang, Q., Zhang, N.: Extrapolation of bandlimited signals in linear canonical transform domain. IEEE Trans. Signal Process. 60, 1502–1508 (2012)

    Article  MathSciNet  Google Scholar 

  7. Shi, J., Liu, X., Zhang, N.: Generalized convolution and product theorems associated whit linear canonical transform. Signal Image Video Process. (2012) doi:10.1007/s11760-012-0348-7

  8. Shi, J., Liu, X., Zhang, N.: On uncertainty principles for linear canonical transform of complex signals via operator methods. Signal Image Video Process. (2013) doi:10.1007/s11760-013-0466-x

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Stern, A.: Sampling of compact signals in offset linear canonical transform domains. Signal Image Video Process. 1, 359–367 (2007)

    Article  MATH  Google Scholar 

  12. Tao, R., Li, B., Wang, Y., Aggrey, G.K.: On sampling of band-Limited signals associated with the linear canonical transform. IEEE Trans. Signal Process. 56, 5454–5464 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Sharma, K.K.: Vector sampling expansions and linear canonical transform. IEEE Signal Process. Lett. 18, 583–586 (2011)

    Article  Google Scholar 

  15. Wei, D., Ran, Q., Li, Y.: Multichannel sampling and reconstruction of bandlimited signals in the linear canonical transform domain. IET Signal Process. 5, 717–727 (2011)

    Article  MathSciNet  Google Scholar 

  16. Liu, Y.-L., Kou, K.-I., Ho, I.-T.: New sampling formulae for non-bandlimited signals associated with linear canonical transform and nonlinear Fourier atoms. Signal Process. 90, 933–945 (2010)

    Article  MATH  Google Scholar 

  17. Shi, J., Liu, X., Sha, X., Zhang, N.: Sampling and reconstruction of signals in function spaces associated with the linear canonical transform. IEEE Trans. Signal Process. 60, 6041–6047 (2012)

    Article  MathSciNet  Google Scholar 

  18. Christensen, Q.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)

    Book  MATH  Google Scholar 

  19. Janssen, A.J.E.M.: The Zak-transform and sampling theorem for wavelet subspaces. IEEE Trans. Signal Process. 41, 3360–3364 (1993)

    Article  MATH  Google Scholar 

  20. Unser, M., Aldroubi, A., Eden, M.: B-spline signal processing-part I: theory. IEEE Trans. Signal Process. 41, 821–833 (1993)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported in part by the National Basic Research Program of China (Grant No. 2013CB329003) and the National Natural Science Foundation of China (Grant No. 61171110).

Author information

Authors and Affiliations

  1. Communication Research Center, Harbin Institute of Technology, Harbin, 150001, China

    Xiaoping Liu, Jun Shi, Xuejun Sha & Naitong Zhang

  2. Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, 518055, China

    Naitong Zhang

Authors
  1. Xiaoping Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xuejun Sha
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Naitong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoping Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, X., Shi, J., Sha, X. et al. Sampling expansion in function spaces associated with the linear canonical transform. SIViP 8, 143–148 (2014). https://doi.org/10.1007/s11760-013-0507-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11760-013-0507-5

Keywords

Search

Navigation