DOI QR코드

DOI QR Code

ON UNIFORM SAMPLING IN SHIFT-INVARIANT SPACES ASSOCIATED WITH THE FRACTIONAL FOURIER TRANSFORM DOMAIN

  • Kang, Sinuk (Division of Mathemetics and Informational Statistics, Wonkwang University)
  • Received : 2016.07.11
  • Accepted : 2016.08.13
  • Published : 2016.09.25

Abstract

As a generalization of the Fourier transform, the fractional Fourier transform plays an important role both in theory and in applications of signal processing. We present a new approach to reach a uniform sampling theorem in the shift-invariant spaces associated with the fractional Fourier transform domain.

Keywords

Acknowledgement

Supported by : Wonkwang University

References

  1. A. Bhandari and A. I. Zayed, Shift-invariant and sampling spaces associated with the fractional Fourier transform domain, IEEE T. Signal Process. 60(4) (2012), 1627-1637. https://doi.org/10.1109/TSP.2011.2177260
  2. C. Candan and H. M. Ozaktas, Sampling and series expansion theorems for fractional Fourier and other transforms, Signal Process. 83(11) (2003), 2455-2457. https://doi.org/10.1016/S0165-1684(03)00196-8
  3. O. Christensen, Frames and bases, an introductory course, Birkhauser, Boston, 2008.
  4. A. G. Garcia, G. Perez-Villalon, and A. Portal, Riesz bases in $L^2$(0; 1) related to sampling in shift-invariant spaces, J. Math. Anal. Appl. 308(2) (2005), 703-713. https://doi.org/10.1016/j.jmaa.2004.11.058
  5. J. R. Higgins, Sampling theory in Fourier and signal analysis, Volume 1: Foundations, Oxford University Press, USA, 1996.
  6. A. J. Jerri, Shannon Sampling Theorem - Its Various Extensions and Applications - Tutorial Review, Proceedings of the IEEE 65(11) (1977), 1565-1596. https://doi.org/10.1109/PROC.1977.10771
  7. J. M. Kim and K. H. Kwon, Sampling expansion in shift invariant spaces, Int. J. Wavelets Multiresolut. Inf. Process. 6(2) (2008), 223-248. https://doi.org/10.1142/S021969130800232X
  8. V. A. Kotelnikov, On the transmission capacity of the 'ether' and of cables in electrical communications, Izd Red Upr Svyazzi RKKA Translated from the Russian by V. E. Katsnelson, Appl. Numer. Harmon. Anal., Modern sampling theory, Birkhauser Boston, Boston, MA, 2001. (1933), 27-45.
  9. A. C. McBride and F. H. Kerr, On Namias's Fractional Fourier Transforms, IMA J. Appl. Math. 39(2) (1987), 159-175. https://doi.org/10.1093/imamat/39.2.159
  10. V. Namias, The fractional order Fourier transform and its application to quantum mechanics, IMA J. Appl. Math. 25(3) (1980), 241-265. https://doi.org/10.1093/imamat/25.3.241
  11. E. Sejdic, I. Djurovic and L. Stankovic, Fractional Fourier transform as a signal processing tool An overview of recent developments, Signal Process. 91(6) (2011), 1351-1369. https://doi.org/10.1016/j.sigpro.2010.10.008
  12. C. E. Shannon, Communication in the presence of noise, Proc. IRE 37 (1949), 10-21.
  13. J. Shi, W. Xiang, X. Liu, and N. Zhang, A sampling theorem for the fractional Fourier transform without band-limiting constraints, Signal Process. 98 (2014), 158-165. https://doi.org/10.1016/j.sigpro.2013.11.026
  14. R. Tao, B. Deng, W. Q. Zhang and Y. Wang, Sampling and Sampling Rate Conversion of Band Limited Signals in the Fractional Fourier Transform Domain, IEEE T. Signal Process. 56(1) (2008), 158-171. https://doi.org/10.1109/TSP.2007.901666
  15. R. Torres, P. Pellat-Finet and Y. Torres, Sampling Theorem for Fractional Bandlimited Signals: A Self-Contained Proof. Application to Digital Holography, IEEE Signal Proc. Let. 13(11) (2006), 676-679. https://doi.org/10.1109/LSP.2006.879470
  16. M. Unser, Sampling - 50 years after Shannon, Proceedings of the IEEE 88(4) (2000), 569-587. https://doi.org/10.1109/5.843002
  17. E. T. Whittaker, On the functions which are represented by the expansions of the interpolation theory, Proc. Royal. Soc. Edinburgh 35 (1915), 181-194. https://doi.org/10.1017/S0370164600017806
  18. A. I. Zayed, On the relationship between the Fourier and fractional Fourier transforms, IEEE Signal Proc. Let. 3(12) (1996), 310-311.
  19. A. I. Zayed, A convolution and product theorem for the fractional Fourier transform, IEEE Signal Proc. Let. 5(4) (1998), 101-103.