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Composite Coherent States Approximation for One-Dimensional Multi-Phased Wave Functions

Published online by Cambridge University Press:  20 August 2015

Dongsheng Yin  and
Chunxiong Zheng
Dongsheng Yin*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P.R. China
Chunxiong Zheng*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P.R. China
*
Email:[email protected]
Corresponding author.Email:[email protected]
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Abstract

The coherent states approximation for one-dimensional multi-phased wave functions is considered in this paper. The wave functions are assumed to oscillate on a characteristic wave length 0(ε) with ε ≪ 1. A parameter recovery algorithm is first developed for single-phased wave function based on a moment asymptotic analysis. This algorithm is then extended to multi-phased wave functions. If cross points or caustics exist, the coherent states approximation algorithm based on the parameter recovery will fail in some local regions. In this case, we resort to the windowed Fourier transform technique, and propose a composite coherent states approximation method. Numerical experiments show that the number of coherent states derived by the proposed method is much less than that by the direct windowed Fourier transform technique.

Keywords

33F0568W25Coherent statehigh frequencywindowed Fourier transform
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 3 , March 2012 , pp. 951 - 984
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Almeida, L.B., The fractional Fourier transform and time-frequency representations, IEEE Trans. Signal Processing 42 (11) (1994), 3084–3091.CrossRefGoogle Scholar
[2]Arama, A., Boag, A. and Heyman, E., Matching pursuit algorithm for Gaussian beam decomposition, Antennas and Propagation Society International Symposium (2005), 272–275.Google Scholar
[3]Ariel, G., Engquist, B., Tanushev, N.M., Tsai, R., Gaussian beam decomposition of high frequency wave fields using expectation-maximization, 2010, preprint.Google Scholar
[4]Cerveny, V., Popov, M. and Psencik, I., Computation of wave fields in inhomogeneous media – Gaussian beam approach, Geophys. J.R. Astr. Soc. 70 (1982), 109–128.Google Scholar
[5]Gray, S.H. and Bleinstein, N., True-amplitude Gaussian-beam migration, Geophysics 74 (2) (2009), S11–S23.Google Scholar
[6]Heller, E.J., Time-dependent approach to semiclassical dynamics, J. Chem. Phys., 62 (1975), 1544–1555.Google Scholar
[7]Heller, E.J., Frozen Gaussians: a very simple semiclassical approximation, J. Chem. Phys. 75 (1981), 2923–2931.Google Scholar
[8]Herman, M.F. and Kluk, E., A semiclassical justification for the use of non-spreading wavepackets in dynamics calculations, Chem. Phys. 91 (1984), 27–34.CrossRefGoogle Scholar
[9]Hill, N.R., Prestack Gaussian-beam depth migration, Geophysics 66 (4) (2001), 1240–1250.CrossRefGoogle Scholar
[10]Kay, K., The Herman-Kluk approximation: derivation and semiclassical corrections, Chem. Phys. 322 (2006), 3–12.Google Scholar
[11]Mallat, S.G. and Zhang, Z., Matching pursuits with time-frequency dictionaries, IEEE Trans. Signal Processing 41 (1993), 3397–3415.Google Scholar
[12]Motamed, M. and Runborg, O., Taylor expansion and discretization errorsin Gaussian beam superposition, Wave Motion 47 (2010), 421–439.Google Scholar
[13]Ozaktas, H.M. and Erden, M.F., Relationships among ray optical, Gaussian beams, and fractional Fourier transform descriptions of first-order optical systems, Optics Communications 143 (1997), 75–86.CrossRefGoogle Scholar
[14]Porter, M.B. and Bucker, H.P., Gaussian beam tracing for computing ocean acoustic fields, J. Acoust. Soc. Am. 82 (4) (1987), 1349–1359.CrossRefGoogle Scholar
[15]Qian, J. and Ying, L., Fast Gaussian wavepacket transforms and Gaussian beams for the Schrödinger equation, J. Comput. Phys. (2010), doi: 10.1016/j.jcp.2010.06.043.CrossRefGoogle Scholar
[16]Shlivinski, A., Heyman, E., Boag, A. and Letrou, C., A phase-space beam summation formulation for ultrawide-band radiation, IEEE Trans. Antennas Propagation 52(8) (2004), 2042–2056.Google Scholar
[17]Tanushev, N.M., Superpositions and higher order Gaussian beams, Commun. Math. Sci. 6 (2008), 449–475.Google Scholar
[18]Tanushev, N.M., Engquist, B. and Tsai, R., Gaussian beam decomposition of high frequency wave fields, J. Comput. Phys. 228 (2009), 8856–8871.Google Scholar