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On Inversion of Continuous Wavelet Transform
State Key Laboratory of Geodesy and Earth’s Dynamics, Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan, China
Shandong University of Technology, Zibo, China
State Key Laboratory of Geodesy and Earth’s Dynamics, Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan, China
- 1 State Key Laboratory of Geodesy and Earth’s Dynamics, Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan, China
- 2 Shandong University of Technology, Zibo, China
- 3 State Key Laboratory of Geodesy and Earth’s Dynamics, Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan, China
Open Journal of Statistics·Volume 05 (2015)·Pages 714–720·Published 11 December 2015·DOI10.4236/ojs.2015.57071
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Abstract
This study deduces a general inversion of continuous wavelet transform (CWT) with timescale being real rather than positive. In conventional CWT inversion, wavelet’s dual is assumed to be a reconstruction wavelet or a localized function. This study finds that wavelet’s dual can be a harmonic which is not local. This finding leads to new CWT inversion formulas. It also justifies the concept of normal wavelet transform which is useful in time-frequency analysis and time-frequency filtering. This study also proves a law for CWT inversion: either wavelet or its dual must integrate to zero.
KeywordsContinuous Wavelet TransformWavelet’s DualInversionNormal Wavelet TransformTime-Frequency Filtering
- http://scienceworld.wolfram.com/biography/Zweig.html
- Goupillaud, P., Grossman, A. and Morlet, J. (1984) Cycle-Octave and Related Transforms in Seismic Signal Analysis. Geoexploration, 23, 85-102. http://dx.doi.org/10.1016/0016-7142(84)90025-5
- Daubechies, I. (1992) Ten Lectures on Wavelets. SIAM, Philadelphia. http://dx.doi.org/10.1137/1.9781611970104
- Chui, C.K. (1992) An Introduction to Wavelets (Wavelet Analysis & Its Applications). Academic Press, Waltham.
- Mallat, S. (1998) A Wavelet Tour of Signal Processing. The Sparse Way.
- Holschneider, M. (1995) Wavelet: An Analysis Tool. Clarendon Press, Oxford.
- Gross, R.S. (1992) Correspondence between Theory and Observations of Polar Motion. Geophysical Journal International, 109, 162-170. http://dx.doi.org/10.1111/j.1365-246X.1992.tb00086.x
- Su, X.Q., Liu, L.T., Hsu, H. and, Wang, G.C. (2014) Long-Term Polar Motion Prediction Using Normal Time-Frequency Transform. Journal of Geodesy, 88, 145-155. http://dx.doi.org/10.1007/s00190-013-0675-7
- Liu, L.T. and, Hsu, H. (2012) Inversion and Normalization of Time-Frequency Transform. Applied Mathematics & Information Sciences, 6, 67-74.
- Stockwell, R.G., Mansinha, L. and Lowe, R.P. (1996) Localization of the Complex Spectrum: The S-Transform. IEEE Transactions on Signal Processing, 44, 998-1001. http://dx.doi.org/10.1109/78.492555
- Pinnegar, C. and Mansinha, L. (2003) The S-Transform with Windows of Arbitrary and Varying Shape. Geophysics, 68, 381-385. http://dx.doi.org/10.1190/1.1543223