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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ozaktas, H.M., Zalevsky, Z., Kutay, M.A.: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York (2000)
Almeida, L.B.: The fractional Fourier transform and time-frequency representations. IEEE Trans. Signal Process. 42, 3084–3091 (1994)
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)
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)
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)
Xia, X.-G.: On bandlimited signals with fractional Fourier transform. IEEE Signal Process. Lett. 3, 72–74 (1996)
Martone, M.: A multicarrier system based on the fractional Fourier transform for time-frequency-selective channels. IEEE Trans. Commun. 46, 1011–1020 (2001)
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)
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
Bhandari, A., Marziliano, P.: Sampling and reconstruction of sparse signals in fractional Fourier domain. IEEE Signal Process. Lett. 17, 221–224 (2010)
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)
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)
Stanković, L., Alieva, T., Bastiaans, M.J.: Time-frequency signal analysis based on the windowed fractional Fourier transform. Signal Process. 83, 2459–2468 (2003)
Tao, R., Lei, Y., Wang, Y.: Short-time fractional Fourier transform and its applications. IEEE Trans. Signal Process. 58, 2568–2580 (2010)
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)
Shi, J., Liu, X., Zhang, N.: On uncertainty principle for signal concentrations with fractional Fourier transform. Signal Process. 92, 2830–2836 (2012)
Mendlovic, D., Zalevsky, Z., Mas, D., García, J., Ferreira, C.: Fractional wavelet transform. Appl. Opt. 36, 4801–4806 (1997)
Shi, J., Zhang, N., Liu, X.: A novel fractional wavelet transform and its applications. Sci. China Inf. Sci. 55, 1270–1279 (2012)
Prasad, A., Mahato, A.: The fractional wavelet transform on spaces of type S. Integral Transform. Spec. Funct. 23, 237–249 (2012)
Chen, L., Zhao, D.: Optical image encryption based on fractional wavelet transform. Opt. Commun. 254, 361–367 (2005)
Bhatnagar, G., Raman, B.: Encryption based robust watermarking in fractional wavelet domain. Rec. Adv. Mult. Sig. Process. and Commun. 231, 375–416 (2009)
Taneja, N., Raman, B., Gupta, I.: Selective image encryption in fractional wavelet domain. Int. J. Electron. Commun. 65, 338–344 (2011)
Daubechies, I.: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia (1992)
Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)
Zayed, A.I.: On the relationship between the Fourier and the fractional Fourier transforms. IEEE Signal Process. Lett. 3, 310–311 (1996)
Erseghe, T., Kraniauskas, P., Cariolaro, G.: Unified fractional Fourier transform and sampling theorem. IEEE Trans. Signal Process. 47, 3419–3423 (1999)
Flandrin, P.: Time-frequency and chirps. Proc. SPIE 4391, 161–175 (2001)
Sharma, K.K., Joshi, S.D.: Time delay estimation using fractional Fourier transform. Signal Process. 87, 853–865 (2007)
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)
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)
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shi, J., Liu, X. & Zhang, N. Multiresolution analysis and orthogonal wavelets associated with fractional wavelet transform. SIViP 9, 211–220 (2015). https://doi.org/10.1007/s11760-013-0498-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11760-013-0498-2