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Signal, Image and Video Processing

, Volume 9, Issue 1, pp 211–220 | Cite as

Multiresolution analysis and orthogonal wavelets associated with fractional wavelet transform

  • Jun Shi
  • Xiaoping Liu
  • Naitong Zhang
Original Paper

Abstract

The fractional wavelet transform (FRWT), which generalizes the classical wavelet transform, has been shown to be potentially useful for signal processing. Many fundamental results of this transform are already known, but the theory of multiresolution analysis and orthogonal wavelets is still missing. In this paper, we first develop multiresolution analysis associated with the FRWT and then derive a construction of orthogonal wavelets for the FRWT. Several fractional wavelets are also presented. Moreover, some applications of the derived results are discussed.

Keywords

Time-frequency analysis Multiresolution analysis Orthogonal wavelets Fractional wavelet transform 

Notes

Acknowledgments

This work was completed in parts while Shi J. was visiting the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716 USA. The 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).

References

  1. 1.
    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
  2. 2.
    Almeida, L.B.: The fractional Fourier transform and time-frequency representations. IEEE Trans. Signal Process. 42, 3084–3091 (1994)CrossRefGoogle Scholar
  3. 3.
    Xia, X.-G., Owechko, Y., Soffer, B.H., Matic, R.M.: On generalized-marginal time-frequency distributions. IEEE Trans. Signal Process. 44, 2882–2886 (1996)CrossRefGoogle 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, 1638–1655 (2001)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Subramaniam, S.R., Ling, B.W.-K., Georgakis, A.: Filtering in rotated time-frequency domains with unknown noise statistics. IEEE Trans. Signal Process. 60, 489–493 (2012)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Xia, X.-G.: On bandlimited signals with fractional Fourier transform. IEEE Signal Process. Lett. 3, 72–74 (1996)CrossRefGoogle Scholar
  7. 7.
    Martone, M.: A multicarrier system based on the fractional Fourier transform for time-frequency-selective channels. IEEE Trans. Commun. 46, 1011–1020 (2001)CrossRefGoogle Scholar
  8. 8.
    Shi, J., Chi, Y., Zhang, N.: Multichannel sampling and reconstruction of bandlimited signals in fractional Fourier domain. IEEE Signal Process. Lett. 17, 909–912 (2010) Google Scholar
  9. 9.
    Shi, J., Sha, X., Song, X., Zhang, N.: Generalized convolution theorem associated with fractional Fourier transform. Wirel. Commun. Mob. Comput. (2012). doi: 10.1002/wcm.2254
  10. 10.
    Bhandari, A., Marziliano, P.: Sampling and reconstruction of sparse signals in fractional Fourier domain. IEEE Signal Process. Lett. 17, 221–224 (2010)CrossRefGoogle Scholar
  11. 11.
    Bhandari, A., Zayed, A.I.: Shift-invariant and sampling spaces associated with the fractional Fourier transform domain. IEEE Trans. Signal Process. 60, 1627–1637 (2012)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Sejdić, E., Djurović, I., Stanković, L.: Fractional Fourier transform as a signal processing tool: an overview of recent developments. Signal Process. 91, 1351–1369 (2011)CrossRefzbMATHGoogle Scholar
  13. 13.
    Stanković, L., Alieva, T., Bastiaans, M.J.: Time-frequency signal analysis based on the windowed fractional Fourier transform. Signal Process. 83, 2459–2468 (2003)CrossRefzbMATHGoogle Scholar
  14. 14.
    Tao, R., Lei, Y., Wang, Y.: Short-time fractional Fourier transform and its applications. IEEE Trans. Signal Process. 58, 2568–2580 (2010)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Shinde, S., Gadre, V.M.: An uncertainty principle for real signals in the fractional Fourier transform domain. IEEE Trans. Signal Process. 49, 2545–2548 (2001)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Shi, J., Liu, X., Zhang, N.: On uncertainty principle for signal concentrations with fractional Fourier transform. Signal Process. 92, 2830–2836 (2012)CrossRefGoogle Scholar
  17. 17.
    Mendlovic, D., Zalevsky, Z., Mas, D., García, J., Ferreira, C.: Fractional wavelet transform. Appl. Opt. 36, 4801–4806 (1997)CrossRefGoogle Scholar
  18. 18.
    Shi, J., Zhang, N., Liu, X.: A novel fractional wavelet transform and its applications. Sci. China Inf. Sci. 55, 1270–1279 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Prasad, A., Mahato, A.: The fractional wavelet transform on spaces of type S. Integral Transform. Spec. Funct. 23, 237–249 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Chen, L., Zhao, D.: Optical image encryption based on fractional wavelet transform. Opt. Commun. 254, 361–367 (2005)CrossRefGoogle Scholar
  21. 21.
    Bhatnagar, G., Raman, B.: Encryption based robust watermarking in fractional wavelet domain. Rec. Adv. Mult. Sig. Process. and Commun. 231, 375–416 (2009)CrossRefGoogle Scholar
  22. 22.
    Taneja, N., Raman, B., Gupta, I.: Selective image encryption in fractional wavelet domain. Int. J. Electron. Commun. 65, 338–344 (2011)CrossRefGoogle Scholar
  23. 23.
    Daubechies, I.: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia (1992)CrossRefzbMATHGoogle Scholar
  24. 24.
    Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)CrossRefzbMATHGoogle Scholar
  25. 25.
    Zayed, A.I.: On the relationship between the Fourier and the fractional Fourier transforms. IEEE Signal Process. Lett. 3, 310–311 (1996)CrossRefGoogle Scholar
  26. 26.
    Erseghe, T., Kraniauskas, P., Cariolaro, G.: Unified fractional Fourier transform and sampling theorem. IEEE Trans. Signal Process. 47, 3419–3423 (1999)CrossRefzbMATHGoogle Scholar
  27. 27.
    Flandrin, P.: Time-frequency and chirps. Proc. SPIE 4391, 161–175 (2001)CrossRefGoogle Scholar
  28. 28.
    Sharma, K.K., Joshi, S.D.: Time delay estimation using fractional Fourier transform. Signal Process. 87, 853–865 (2007)CrossRefzbMATHGoogle Scholar
  29. 29.
    Tao, R., Li, X.-M., Li, Y.-L., Wang, Y.: Time delay estimation of chirp signals in the fractional Fourier transform. IEEE Trans. Signal Process. 57, 2852–2855 (2009)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Cowell, D.M.J., Freear, S.: Separation of overlapping linear frequency modulated (LFM) signals using the fractional Fourier transform. IEEE Trans. Ultrason. Ferr. 57, 2324–2333 (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  1. 1.Communication Research CenterHarbin Institute of TechnologyHarbinChina
  2. 2.Shenzhen Graduate SchoolHarbin Institute of TechnologyShenzhenChina

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