Science China Information Sciences

, Volume 55, Issue 6, pp 1270–1279

A novel fractional wavelet transform and its applications

Research Paper

Abstract

The wavelet transform (WT) and the fractional Fourier transform (FRFT) are powerful tools for many applications in the field of signal processing. However, the signal analysis capability of the former is limited in the time-frequency plane. Although the latter has overcome such limitation and can provide signal representations in the fractional domain, it fails in obtaining local structures of the signal. In this paper, a novel fractional wavelet transform (FRWT) is proposed in order to rectify the limitations of the WT and the FRFT. The proposed transform not only inherits the advantages of multiresolution analysis of the WT, but also has the capability of signal representations in the fractional domain which is similar to the FRFT. Compared with the existing FRWT, the novel FRWT can offer signal representations in the time-fractional-frequency plane. Besides, it has explicit physical interpretation, low computational complexity and usefulness for practical applications. The validity of the theoretical derivations is demonstrated via simulations.

Keywords

time-frequency analysis wavelet transform multiresolution analysis fractional Fourier transform time-fractional-frquency analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Gargour C, Gabrea M, Ramachandran V, et al. A short introduction to wavelets and their applications. IEEE Circ Syst Mag, 2009, 2: 57–68CrossRefGoogle Scholar
  2. 2.
    Flandrin P. Time-frequency and chirps. P SPIE, 2001, 4391: 161–175CrossRefGoogle Scholar
  3. 3.
    Xia X G. On bandlimited signals with fractional Fourier transform. IEEE Signal Proc Let, 1996, 3: 72–74CrossRefGoogle Scholar
  4. 4.
    Mendlovic D, Zalevsky Z, Mas D, et al. Fractional wavelet transform. Appl Optics, 1997, 36: 4801–4806CrossRefGoogle Scholar
  5. 5.
    Bhatnagar G, Raman B. Encryption based robust watermarking in fractional wavelet domain. Rec Adv Mult Sig Proc Commun, 2009, 231: 375–416CrossRefGoogle Scholar
  6. 6.
    Chen L, Zhao D. Optical image encryption based on fractional wavelet transform. Opt Commun, 2005, 254: 361–367CrossRefGoogle Scholar
  7. 7.
    Huang Y, Suter B. The fractional wave packet transform. Multidim Syst Signal Process, 1998, 9: 399–402MATHCrossRefGoogle Scholar
  8. 8.
    Tao R, Deng B, Wang Y. Research progress of the fractional Fourier transform in signal processing. Sci China Ser F-Inf Sci, 2006, 49: 1–25MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Tao R, Li Y L, Wang Y. Short-time fractional Fourier transform and its applications. IEEE Trans Signal Proces, 2010, 58: 2568–2580MathSciNetCrossRefGoogle Scholar
  10. 10.
    Wood J C, Barry D T. Linear signal synthesis using the Radon-Wigner transform. IEEE Trans Signal Proces, 1994, 42: 2105–2111CrossRefGoogle Scholar
  11. 11.
    Dinç E, Baleanu D. New approaches for simultaneous spectral analysis of a complex mixture using the fractional wavelet transform. Commun Nonlinear Sci Numer Simul, 2010, 15: 812–818CrossRefGoogle Scholar
  12. 12.
    Shi J, Chi Y G, Zhang N T. Multichannel sampling and reconstruction of bandlimited signals in fractional domain. IEEE Signal Proc Let, 2010, 17: 909–912CrossRefGoogle Scholar
  13. 13.
    Candan C, Kutay M A, Ozaktas H M. The discrete fractional Fourier transform. IEEE Trans Signal Proces, 2000, 48: 1329–1337MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Tao R, Li X, Li Y, et al. Time-delay estimation of chirp signals in the fractional Fourier transform. IEEE Trans Signal Proces, 2009, 57: 2852–2855MathSciNetCrossRefGoogle Scholar
  15. 15.
    Akay O, Erözden E. Employing fractional autocorrelation for fast detection and sweep rate estimation of pulse compression radar waveforms. Signal Process, 2009, 89: 2479–2489MATHCrossRefGoogle Scholar
  16. 16.
    Cowell D M J, Freear S. Separation of overlapping linear frequency modulated (LFM) signals using the fractional Fourier transform. IEEE Trans Ultrason Ferr, 2010, 57: 2324–2333CrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Communication Research CenterHarbin Institute of TechnologyHarbinChina

Personalised recommendations