Evolutionary Spectra Estimation Using Wavelets
Publication: Journal of Engineering Mechanics
Volume 130, Issue 8
Abstract
A wavelets-based method is developed to estimate the evolutionary power spectral density (EPSD) of nonstationary stochastic processes. The method relies on the property that the continuous wavelet transform of a nonstationary process can be treated as a stochastic process with EPSD given in terms of the EPSD of the process in a closed form. This yields an equation in the frequency domain relating the instantaneous mean-square value of the wavelet transform to the EPSD of the process. A number of these equations are considered, each related to a certain scale of the wavelet transform, in conjunction with representing the target EPSD as a sum of time-independent shape functions modulated by time-dependent coefficients; the squared moduli of the Fourier transforms of the wavelets associated with the selected scales are taken as shape functions. This leads to a linear system of equations which is solved to determine the unknown time-dependent coefficients; the same system matrix applies for all time instances. Numerical examples demonstrate the accuracy and computational efficiency of the proposed method.
Get full access to this article
View all available purchase options and get full access to this article.
References
Agrawal, O. D.(1998). “Application of wavelets in modeling stochastic dynamic systems.” J. Vibr. Acoust., 120, 763–769.
Alkemade, J. A. H. (1993). “The finite wavelet transform with an application to seismic processing.” Wavelets: An elementary treatment of theory and applications, Koornwinder, eds., World Scientific, N.J., 183–208.
Basu, B., and Gupta, V. K.(1997). “Nonstationary seismic response of MDOF systems by wavelet transform.” Earthquake Eng. Struct. Dyn., 26, 1243–1258.
Basu, B., and Gupta, V. K.(1998). “Seismic response of SDOF systems by wavelet modeling of nonstationary processes.” J. Eng. Mech., 124(10), 1142–1150.
Basu, B., and Gupta, V. K.(2000). “Stochastic seismic response of single-degree-of-freedom systems through wavelets.” Eng. Struct., 22, 1714–1722.
Carmona, R., Hwang, W. J., and Torresani, B. (1998). Practical time–frequency analysis, Academic, New York.
Cohen, L. (1995). Time-frequency analysis, Prentice–Hall, Englewood Cliffs, N.J.
Conte, J. P., and Peng, B. F.(1997). “Fully nonstationary analytical earthquake ground-motion model.” J. Eng. Mech., 123(1), 15–24.
Daubechies, I. (1992). Ten lectures on wavelets, Society for Industrial and Applied Mathematics, Philadelphia.
Deodatis, G., and Shinozuka, M.(1988). “Autoregressive model for nonstationary stochastic processes.” J. Eng. Mech., 114(11), 1995–2012.
Gaupillaud, P., Grossmann, A., and Morlet, J.(1984). “Cycle-octave andrelated transforms in seismic signal analysis.” Geoexploration, 23, 85–102.
Ghanem, R., and Romeo, F.(1999). “A wavelet based approach for the identification of linear time varying dynamical systems.” J. Sound Vib., 234(4), 555–576.
Grossmann, A., and Morlet, J.(1984). “Decomposition of Hardy function into square integrable wavelets of constant shape.” SIAM J. Math. Anal., 15, 723–736.
Iyama, J., and Kuwamura, H.(1999). “Applications of wavelets to analysis and simulation of earthquake records.” Earthquake Eng. Struct. Dyn., 28, 255–272.
Mallat, S.(1989). “Multiresolution approximation and wavelets.” Trans. Am. Math. Soc., 315, 69–88.
Newland, D. E. (1993). Random vibrations, spectral, and wavelet analysis, 3rd Ed., Longman, Harlow, and Wiley, New York.
Newland, D. E.(1994a). “Wavelet analysis of vibration. I: Theory.” J. Vibr. Acoust., 116, 409–416.
Newland, D. E.(1994b). “Wavelet analysis of vibration. II: Wavelet maps.” J. Vibr. Acoust., 116, 417–425.
Papadimitriou, C. (1990). “Stochastic characterization of strong ground motion and applications to structural response.” Rep. No. EERL 90-03, California Institute of Technology, Pasadena, Calif.
Priestley, M. B. (1981). Spectral analysis and time series, Academic, New York.
Saragoni, G. R., and Hart, G. C. (1972). “Nonstationary analysis and simulation of earthquake ground motions.” Rep. No. UCLA-ENG-7238, Earthquake Engineering and Structural Laboratory, Univ. of California at Los Angeles, Los Angeles.
Shinozuka, M., and Deodatis, G.(1988). “Stochastic process models for earthquake ground motion.” Probab. Eng. Mech., 3(3), 114–123.
Spanos, P. D., and Zeldin, B. A.(1997). “Wavelet concepts on computational stochastic mechanics.” IASSAR Rep., J. Probab. Eng. Mech., 12(4), 244–255.
Spanos, P. D., and Zeldin, B. A.(1998). “Monte Carlo treatment of random fields: A broad perspective.” Appl. Mech. Rev., 51(3), 219–237.
Spanos, P. D., Tein, W. Y., and Ghanem, R. G.(1996). “Heuristic spectral estimation of bivariate nonstationary processes.” Meccanica, 31, 207–218.
Staszewski, W. J.(1997). “Identification of damping in MDOF systems using time-scale decomposition.” J. Sound Vib., 203(3), 283–305.
Staszewski, W. J.(1998). “Identification of nonlinear systems using multistage ridges and skeletons of the wavelet transform.” J. Sound Vib., 211(5), 735–760.
Trifunac, M. D.(1971). “Response envelope spectrum and interpretation of strong earthquake ground motion.” Bull. Seismol. Soc. Am., 61, 343–356.
Ville, J.(1948). “Theorie et applications de la notion de signal analytique.” Cables Transm., 2, 61–74.
Wigner, E. P.(1932). “On the quantum correction for thermodynamic equilibrium.” Phys. Rev., 40, 749–759.
Yeh, C. H., and Wen, Y. K.(1990). “Modeling of nonstationary ground motion and analysis of inelastic structural response.” Struct. Safety, 8(1–4), 281–298.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 130 • Issue 8 • August 2004
Pages: 952 - 960
Copyright
Copyright © 2004 American Society of Civil Engineers.
History
Received: May 9, 2003
Accepted: Nov 26, 2003
Published online: Jul 15, 2004
Published in print: Aug 2004
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.