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Introduction to Wavelet Transform and Time-Frequency Analysis

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Wavelet Applications in Chemical Engineering

Abstract

The wavelet transform has been developed in recent years and has attracted growing attention from mathematicians as well as engineers. In this introductory chapter, we would like to review briefly some basic concepts and methods of this new approach under the more general framework of time-frequency analysis methods. We will discuss the Fourier transform, the short-time Fourier transform and time-frequency distributions, followed by a discussion of wavelet theory and its variations.

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References

  • Alpert, B. K. “Wavelets and other bases for fast numerical linear algebra.” in Wavelets: A Tutorial in Theory and Applications Chui, C. K. ed., Academic Press, (1992): 181–216.

    Google Scholar 

  • Bacry, E., J. F. Muzy and A. Arneodo “ Singularity spectrum of fractal signals from wavelet analysis: exact results” Journal of Statistical Physics, 70, 3–4 (1993): 635–674.

    Article  Google Scholar 

  • Bakshi, B. R. and G. Stephanopoulos “ Wave-Net: a multiresolution, hierarchical neural network with localized learning.” AIChE J., 39, 1 (1993): 57–81.

    Article  CAS  Google Scholar 

  • Battle, G. “A block spin construction of ondelettes. Part I: Lemarie functions.” Comm. Math. Phys.110, (1987): 601–615.

    Article  Google Scholar 

  • Benzi, R. and M. Vergassola “Optimal wavelet analysis and its application to two-dimensional turbulence.” Fluid Dynamics Research 8, 1–4 (1991): 117–126.

    Article  Google Scholar 

  • Beylkin, G., R. Coifman and V. Rokhlin “Fast wavelet transforms and Numerical Algorithms.” Comm. Pure Appl. Math., 44, (1991): 141–183.

    Article  Google Scholar 

  • Boashash, B. and P. O’Shea “A methodology for detection and classification of some underwater acoustic signals using time-frequency analysis techniques.” IEEE Trans. Acoust., Speech, Signal Processing, 38, 11 (1990): 1829–1841.

    Google Scholar 

  • Boashash, B. “Time-frequency signal analysis.” Advances in Spectrum Analysis and Array Processing Volume I, Haykin, Simon. Ed. Prentice Hall, Englewood Cliffs, NJ., (1991): 418–517.

    Google Scholar 

  • Choi, H. I. and W. J. Williams “Improved time-frequency representation of multicomponent signals using exponential kernels.” IEEE Trans. Acoust., Speech, Signal Processing, 37, 6 (1989): 862–871.

    Google Scholar 

  • Choi, H. I., W. J. Williams and H. Zaveri “Analysis of event related potentials: time frequency energy distribution.” Biomedical Sciences Instrumentation, 23, (1987): 251–258.

    CAS  Google Scholar 

  • Chui, C. K. An Introduction to Wavelets. Academic Press, Boston, 1992a.

    Google Scholar 

  • Chui, C. K. ed. Wavelets: A tutorial in theory and applications. Academic Press, Boston, 1992b.

    Google Scholar 

  • Cohen, A. “Biothogonal wavelets.” Wavelets: A tutorial in theory and applications. Chui, C. K. ed. Academic Press, Boston, (1992): 123–152.

    Google Scholar 

  • Cohen, L. “Time-frequency distributions -- A review.” Proceedings of the IEEE, 77, 7 (1989): 941–981.

    Article  Google Scholar 

  • Cohen, L. “Generalized phase-space distribution functions.” J. Math. Phys., 7 (1966): 781–786.

    Article  Google Scholar 

  • Coifman, R. R., Y. Meyer, S. Quake and M. V. Wickerhauser “Signal processing and compression with wavelet packets.” Progress in Wavelet Analysis and Applications, Meyer and Roques eds., Editions Frontieres, Toulouse, France, (1992a): 77–93.

    Google Scholar 

  • Coifman, R. R. and M. V. Wickerhauser “Entropy-based algorithms for best basis selection.” IEEE Trans. Information Theory, 38, 2 (1992b): 713–718.

    Article  Google Scholar 

  • Combes, J. M. et al. eds. Wavelets,Time-Frequency Methods and Phase Space Springer-Verlag, Berlin, 1989.

    Google Scholar 

  • Crowe, J. A. et al. “Wavelet transform as a potential tool for ECG analysis and compression.” Journal of Biomedical Engineering, 14, 3 (1992): 268–272.

    Article  CAS  Google Scholar 

  • Daubechies, I. “Orthonormal basis of compactly supported wavelets.” Comm. Pure & Applied Math., 41 (1988): 909–996.

    Article  Google Scholar 

  • Daubechies, I. “The wavelet transform, time-frequency localization and signal analysis.” IEEE Trans. Information Theory, 36, 5 (1990): 961–1005.

    Article  Google Scholar 

  • Daubechies, I. “The wavelet transform: a method for time-frequency localization.” Advances in Spectrum Analysis and Array Processing Volume I, Haykin, Simon Ed. Prentice Hall, Englewood Cliffs, NJ., (1991): 366–417.

    Google Scholar 

  • Daubechies, I. Ten Lectures in Wavelets Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1992.

    Book  Google Scholar 

  • Daubechies, I. Wavelets Making Waves in Mathematics and Engineering. Videotape, American Mathematical Society, 1993.

    Google Scholar 

  • Daubechies, I., A. Grossmann and Y. Meyer “Painless non-orthogonal expansions.” J. Math. Phys., 27 (1986): 293–309.

    Article  Google Scholar 

  • Daubechies, I., S. Mallat and A. S. Willsky eds. Special Issue on Wavelet Transforms and Multiresolution Signal Analysis,IEEE Trans. Information Theory, 38, 2 (1992).

    Google Scholar 

  • Desai, M. and D. J. Shazeer “Acoustic transient analysis using wavelet decomposition.” IEEE Conference on Neural Networks for Ocean Engineering, (1991): 29–40.

    Google Scholar 

  • Donoho, David L. and Iain M. Johnstone “Ideal Spatial Adaptation by Wavelet Shrinkage.” Preprint, Department of Statistics, Stanford University, (1992a).

    Google Scholar 

  • Donoho, David L. “De-Noising by Soft-Thresholding.” Preprint, Department of Statistics, Stanford University, (1992b).

    Google Scholar 

  • Elias-Juarez, A. and J. C. Kantor “On the application of wavelets to model predictive control.” Proceedings American Control Conference, Chicago, (1992): 1582–1586.

    Google Scholar 

  • Farge, M. et al. “Improved predictability of two-dimensional turbulent flows using wavelet packet compression.” Fluid Dynamics Research, 10, 4–6 (1992): 229–250.

    Article  Google Scholar 

  • Flandrin, P. “Wavelets and related time-scale transforms.” SPIE Advanced Signal-Processing Algorithms, Architectures, and Implementation, 1348, (1990): 2–13.

    Google Scholar 

  • Forrester, B. D. “Use of the Wigner-Ville distribution in helicopter fault detection.” Proc. ASSPA 89, (1989): 78–82.

    Google Scholar 

  • Gabor, D. “Theory of communication.” J. Inst. Elec. Eng., 93, (1946): 429–441.

    Google Scholar 

  • Ghosh, J., L. Deuser and S. D. Beck “A neural network based hybrid system for detection, characterization, and classification of short-duration oceanic signals.” IEEE Journal of Oceanic Engineering, 17, 4 (1992): 351–363.

    Article  Google Scholar 

  • Goupil, M. J., M. Auvergne and A. Baglin “Wavelet analysis of pulsating white dwarfs” Astronomy and Astrophysics, 250, 1 (1991): 89–98.

    Google Scholar 

  • Grossmann, A. and J. Morlet “Decomposition of hardy functions into square integrable wavelets of constant shape.” SIAM J. Math. Anal., 5 (1984): 723–736.

    Article  Google Scholar 

  • Hjorth, P. G. et al. “Wavelet analysis of ‘doublet quasar’ flux data” Astronomy and Astrophysics, 255, 1–2 (1992): 20–23.

    Google Scholar 

  • Hlawatsch, F. and G. F. Boudreaux-Bartels “Linear and quadratic time-frequency signal representations.” IEEE Signal Processing Magazine, (Apr. 1992): 21–67.

    Google Scholar 

  • Hopper, T. and F. Preston “Compression of grey-scale fingerprint images.” DCC ‘82. Data Compression Conference, Storer, J. A. and Cohn, M. eds., (1992): 309–318.

    Google Scholar 

  • Huang, W. Y. and M. R. Solorzano, “Wavelet preprocessing of acoustic signals.” Conference Record of the Twenty-Fifth Asilomar Conference on Signals, Systems and Computers, 2, (1991): 1114–1118.

    Google Scholar 

  • Jawerth, B. and Wim Sweldens “An overview of wavelet based multiresolution analysis.” Research report 1993:1, Department of Mathematics, University of South Carolina (1993).

    Google Scholar 

  • Jones, D. L. and T. W. Parks “A high resolution data-adaptive time-frequency representation.” IEEE Trans. Acoust.,Speech, Signal Processing, 38, 12 (1990): 2127–2135.

    Google Scholar 

  • Khadra, L., M. Matalgah, B. El-Asir and S. Mawagdeh “The wavelet transform and its applications to photocardiogram signal analysis.” Med. Inform., 16, 3 (1991): 271–277.

    Article  CAS  Google Scholar 

  • Kikuchi, T. and S. Sato “Experimental studies on ultrasonic measurements of scattering media by using wavelet transform.” Japanese Journal of Applied Physics,Supplement, 31, 1 (1992): 115–117.

    Google Scholar 

  • Kikuchi, H. et al. “Fast wavelet transform and its application to detecting detonation.” IEICE Transactions on Fundamentals of Electronics,Communications and Computer Sciences, E75-A, 8 (1992): 980–987.

    Google Scholar 

  • Lee, J. H., M. S. Gelormino, and M. Morari “Model predictive control of Multi-rate sampled-data systems: A state-space approach.” International J. of Control, 55, 1 (1992): 153–191.

    Article  Google Scholar 

  • Mallat, S. and S. Zhong “Signal characterization from multiscale edges.” NYU, Comput. Sci. Tech. Rep., (Nov. 1991): 891–896.

    Google Scholar 

  • Mallat, S. “Multiresolution approach to wavelets in computer vision.” Wavelets,Time-Frequency Methods and Phase Space Combes, J. M. et al. eds. Springer-Verlag, (1989): 313–327.

    Google Scholar 

  • Mallat, S. “A theory for multiresolution signal decomposition: the wavelet representation.” IEEE Trans. Pattern Analysis and machine intelligence, 11, (1989): 674–693.

    Article  Google Scholar 

  • Mallat, S. and W. L. Hwang “Singularity detection and processing with wavelets.” IEEE Trans. Information Theory, 38, 2 (1992): 617–643.

    Article  Google Scholar 

  • Manjunath, B. S. and R. Chellappa “A unified approach to boundary perception: edges, textures, and illusory contours” IEEE Transactions on Neural Networks, 4, 1 (1993): 96–108.

    Article  CAS  Google Scholar 

  • McAulay, A. D. and Jian Li “Wavelet data compression for neural network preprocessing.” Proceedings of the SPIE - The International Society for Optical Engineering, 1699, (1992): 356–365.

    Article  Google Scholar 

  • Meneveau, C. “Analysis of turbulence in the orthonormal wavelet representation.” Journal of Fluid Mechanics, 232, (1991): 469–520.

    Article  Google Scholar 

  • Morlet, D. et al., “Time-scale analysis of high-resolution signal-averaged surface ECG using wavelet transformation.” Proceedings. Computers in Cardiology, (1991): 393–396.

    Google Scholar 

  • Morlet, J., G. Arens, I. Fourgeau, and D. Giard “Wave propagation and sampling theory.” Geophysics, 47, (1982): 203–236.

    Article  Google Scholar 

  • Page, C. H. “ Instantaneous power spectra.” J. Appl. Phys., 23, (1952): 103–106.

    Article  Google Scholar 

  • Pati, Y. C. and P. S. Krishnaprasad “Frames generated by subspace addition.” Technical report, University of Maryland, Systems Research Center, (1992).

    Google Scholar 

  • Rioul, O. and M. Vetterli “Wavelets and signal processing.” IEEE Signal Processing Magazine, (Oct. 1991): 14–38.

    Google Scholar 

  • Ruskai, M. B. et al. eds. Wavelets and their applications. Jones and Barlett, Boston, 1992.

    Google Scholar 

  • Slezak, E., A. Bijaoui and G. Mars “Identification of structures from galaxy counts: use of the wavelet transform.” Astronomy and Astrophysics, 227, (1990): 301–316.

    Google Scholar 

  • Strang, G. “Wavelet transforms versus Fourier transforms.” Bulletin of the American mathematical society, 28, 2 (1993): 288–305.

    Article  Google Scholar 

  • Stromberg, J. O. “A modified franklin system and higher order spline systems on R” as unconditional bases for hardy spaces.“ Conf. in Honor of A. Zygmund, Beckner, Wet al. eds. II (1981): 475–493.

    Google Scholar 

  • Sweldens, Wim et al. eds. Wavelet Digest Department of mathematics, University of South Carolina, wavelet@math.scarolina.edu. 1992.

    Google Scholar 

  • Titeur, F. B. “Wavelet transformations in signal detection.” Wavelets, Time-Frequency Methods and Phase Space Combes, J. M, et al. eds. Springer-Verlag, (1989): 133–139.

    Google Scholar 

  • Wickerhauser, M. V. Lectures on Wavelet Packet Algorithms Department of Mathematics, Washington University in St. Louis, (1991).

    Google Scholar 

  • Wickerhauser, M. V. “Acoustic signal compression with wavelet packets.” Wavelets -- A Tutorial in Theory and Applications Chui, C. K. ed. Academic Press, Boston, (1992): 679–700.

    Google Scholar 

  • Wickerhauser, M. V. “Best-adapted wavelet packet bases.” Different Perspectives on Wavelets Daubechies, I. ed. American Mathematical Society, San Antonio, Texas, (1993): 155–171.

    Google Scholar 

  • Wigner, E. P. “On the quantum correction for thermodynamic equilibrium.” Phys. Rev., 40, (1932): 749–759.

    Article  CAS  Google Scholar 

  • Wornell, G. W. and A. V. Oppenheim “Estimation of fractal signal from noisy measurements using wavelets.” IEEE Trans. Signal Processing, 40, 3 (1992): 611–623.

    Article  Google Scholar 

  • Yamada, M. and K. Ohkitani “Orthonormal wavelet analysis of turbulence.” Fluid Dynamics Research, 8, 1–4 (1991): 101–115.

    Article  Google Scholar 

  • Yang, X., K. Wang and S. A. Shamma “Auditory representations of acoustic signals” IEEE Transactions on Information Theory, 38, 2, pt.2 (1992): 824–839.

    Article  Google Scholar 

  • Young, Randy K. Wavelet theory and its application. Kluwer Academic Publishers, MA., 1993.

    Book  Google Scholar 

  • Zhang, Q. and A. Benveniste “Wavelet networks.” IEEE Trans. Neural Networks, 3, 6 (1992): 889–898.

    Article  CAS  Google Scholar 

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Dai, Xd., Joseph, B., Motard, R.L. (1994). Introduction to Wavelet Transform and Time-Frequency Analysis. In: Motard, R.L., Joseph, B. (eds) Wavelet Applications in Chemical Engineering. The Kluwer International Series in Engineering and Computer Science, vol 272. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2708-4_1

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  • DOI: https://doi.org/10.1007/978-1-4615-2708-4_1

  • Publisher Name: Springer, Boston, MA

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