Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-15T11:16:51.922Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparison of Spectral and Wavelet Estimators of Transfer Function for Linear Systems

Published online by Cambridge University Press:  28 May 2015

M. A. A. Bakar ,
D. A. Green  and
A. V. Metcalfe
M. A. A. Bakar*
Affiliation:
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, UKM Bangi, Selangor, Malaysia School of Mathematical Sciences, University of Adelaide, South Australia, Australia
D. A. Green
Affiliation:
School of Mathematical Sciences, University of Adelaide, South Australia, Australia
A. V. Metcalfe
Affiliation:
School of Mathematical Sciences, University of Adelaide, South Australia, Australia
*
Corresponding author. Email: [email protected]
Get access

Abstract

We compare spectral and wavelet estimators of the response amplitude operator (RAO) of a linear system, with various input signals and added noise scenarios. The comparison is based on a model of a heaving buoy wave energy device (HBWED), which oscillates vertically as a single mode of vibration linear system. HBWEDs and other single degree of freedom wave energy devices such as oscillating wave surge convertors (OWSC) are currently deployed in the ocean, making such devices important systems to both model and analyse in some detail. The results of the comparison relate to any linear system. It was found that the wavelet estimator of the RAO offers no advantage over the spectral estimators if both input and response time series data are noise free and long time series are available. If there is noise on only the response time series, only the wavelet estimator or the spectral estimator that uses the cross-spectrum of the input and response signals in the numerator should be used. For the case of noise on only the input time series, only the spectral estimator that uses the cross-spectrum in the denominator gives a sensible estimate of the RAO. If both the input and response signals are corrupted with noise, a modification to both the input and response spectrum estimates can provide a good estimator of the RAO. A combination of wavelet and spectral methods is introduced as an alternative RAO estimator. The conclusions apply for autoregressive emulators of sea surface elevation, impulse, and pseudorandom binary sequences (PRBS) inputs. However, a wavelet estimator is needed in the special case of a chirp input where the signal has a continuously varying frequency.

Keywords

65M1078A48Heaving buoyspectral analysisdiscrete wavelet transformimpulse responseresponse amplitude operator
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 2 , Issue 3 , August 2012 , pp. 214 - 237
Copyright
Copyright © Global-Science Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Akaike, H.. A new look at the statistical model identification. Automatic Control, IEEE Transactions on, 19(6):716–723, 1974.Google Scholar
[2]Bendat, J. S. and Piersol, A. G.. Random Data: Analysis and Measurement Procedures, volume 729. Wiley, 2011.Google Scholar
[3]Box, G. E. P., Jenkins, G. M., and Reinsel, G. C.. Time Series Analysis. Holden-day San Francisco, 1976.Google Scholar
[4]Bracewell, R. N.. The Fourier Transform & Its Applications 3rd Ed. McGraw-Hill, 2000.Google Scholar
[5]Chen, S. L., Liu, J. J., and Lai, H. C.. Wavelet analysis for identification of damping ratios and natural frequencies. Journal of Sound and Vibration, 323(1-2):130–147, 2009.Google Scholar
[6]Cowpertwait, P. S. P., Metcalfe, A., and Metcalfe, A. V.. Introductory Time Series with R. Springer Verlag, 2009.Google Scholar
[7]Daubechies, I.. Ten Lectures on Wavelets, volume 61. Society for Industrial Mathematics, 1992.Google Scholar
[8]Donoho, D. L. and Johnstone, I. M.. Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(3):425–455, 1994.CrossRefGoogle Scholar
[9]Gouttebroze, S. and Lardies, J.. On using the wavelet transform in modal analysis. Mechanics Research Communications, 28(5):561–569, 2001.CrossRefGoogle Scholar
[10]Haar, A.. Zur theorie der orthogonalen funktionensysteme. Mathematische Annalen, 69(3): 331–371, 1910.CrossRefGoogle Scholar
[11]Hardle, W., Kerkyacharian, G., Picard, D., and Tsybakov, A.. Wavelets, Approximation and Statistical Applications. Springer-Verlag, New York, 1998.Google Scholar
[12]Hearn, G. E. and Metcalfe, A.. Spectral Analysis in Engineering: Concepts and Cases. Arnold, London, 1995.Google Scholar
[13]Hewlett, W. R.. Inventions ofOpportunity: Matching Technology With Market Needs. Hewlett Packard Co, 1983.Google Scholar
[14]Jenkins, G.M. and Watts, D.G.. Spectral Analysis and Its Applications. Holden-Day, 1968.Google Scholar
[15]Kim, Y. Y., Hong, J. C., and Lee, N. Y.. Frequency response function estimation via a robust wavelet de-noising method. Journal of Sound and Vibration, 244(4):635–649, 2001.Google Scholar
[16]Kitada, Y.. Identification of nonlinear structural dynamic systems using wavelets. Journal of Engineering Mechanics, 124(10):1059, 1998.CrossRefGoogle Scholar
[17]Lee, D. T. L. and Yamamoto, A.. Wavelet analysis: Theory and applications. Hewlett-Packard Journal, 45:44–54, 1994.Google Scholar
[18]Mallat, S.. A Wavelet Tour of Signal Processing. Academic Press, 1999.Google Scholar
[19]Masubuchi, M. and Kawatani, R.. Frequency response analysis of an ocean wave energy converter. Journal ofDynamic Systems, Measurement, and Control, 105(1):30–38, 1983.Google Scholar
[20]Mathews, J. H. and Fink, K. D.. Numerical Methods Using MATLAB, volume 31. Prentice Hall Upper Saddle River, 1999.Google Scholar
[21]McCusker, J. R., Danai, K., and Kazmer, D. O.. Validation of dynamic models in the time-scale domain. Journal of Dynamic Systems, Measurement, and Control, 132(6):061402, 2010.Google Scholar
[22]Metcalfe, A., Maurits, L., Svenson, T., Thach, R., and Hearn, G. E.. Modal analysis of a small ship sea keeping trial. Australian & New Zealand Industrial and Applied Mathematics Journal, 47:915–933, July 2007.Google Scholar
[23]Nason, G. P.. Wavelet Methods in Statistics with R. Springer-Verlag, New York, 2008.CrossRefGoogle Scholar
[24]Newland, D. E.. An Introduction to Random Vibrations, Spectral and Wavelet Analysis. Longman Scientific & Technical, 1993.Google Scholar
[25]Ocean Power Technologies, Inc. Making Waves in Power. http://www.oceanpowertechnolo-gies.com.Google Scholar
[26]Pernot, S. and Lamarque, C. H.. A wavelet-galerkin procedure to investigate time-periodic systems: Transient vibration and stability analysis. Journal ofSound and Vibration, 245(5):845–875, 2001.Google Scholar
[27]R Development Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2008.Google Scholar
[28]Ringwood, J.. The dynamics of wave energy. In Proceeding ofIrish Signal and Systems Conference, June 2006.Google Scholar
[29]Robertson, A. N., Park, K. C., and Alvin, K. F.. Extraction of impulse response data via wavelet transform for structural system identification. Journal of Vibration and Acoustics, 120(1):252–260, 1998.CrossRefGoogle Scholar
[30]Robertson, A. N., Park, K. C., and Alvin, K. F.. Identification of structural dynamics models using wavelet-generated impulse response data. Journal ofVibration and Acoustics, 120(1):261–266, 1998.CrossRefGoogle Scholar
[31]Staszewski, W. J.. Analysis of non-linear systems using wavelets. Proceedings of the Institution of Mechanical Engineers – Part C, Journal of Mechanical Engineering Science, 214(11):1339–1353, 2000.Google Scholar
[32]Whittaker, T. and Folley, M.. Nearshore oscillating wave surge converters and the development of oyster. Philosophical Transactions of The Royal Society A, 370:345–364, 2012.Google Scholar
[33]Yu, Y., Shenoi, R. Ajit, Zhu, H., and Xia, L.. Using wavelet transforms to analyze nonlinear ship rolling and heave-roll coupling. Ocean Engineering, 33(7):912–926, 2006.CrossRefGoogle Scholar