周期广义谐和小波变换及重构
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摘要
谐和小波和广义谐和小波皆为在频域上紧支且时域为无穷的正交小波,其频域分辨率很好但时域分辨率较差。虽然谐和小波在无穷时域上具有正交性,但其正交性在有限时域上却无法体现。针对这个缺点,在广义谐和小波的基础上,将广义谐和小波周期化后,进而提出了一种周期广义谐和小波(Periodic Generalized Harmonic Wavelet,PGHW)。PGHW的母小波在时域中可以表达为经平移后的若干谐和项之和,在频域中表现若干δ函数之和,为一种以待分析信号持时为基本周期且在其上正交的离散广义谐和小波。基于PGHW在频域内的简单性,利用快速Fourier变换(FFT)技术实现了PGHW的快速小波变换及逆变换。最后的算例给出了某人工合成地震波的周期广义谐和小波变换及其重构,说明了所提算法的高效性与PGHW的完全重构性。
The periodic generalized harmonic wavelet(PGHW) and the algorithms for its fast wavelet transformation(FWT) and inverse fast wavelet transformation(iFWT) are presented here.The PGHW can be represented as a sum of several translated harmonic terms in time domain,and a sum of several δ functions in frequency domain.Compared to the generalized harmonic wavelet(GHW) and the harmonic wavelet(HW),the PGHW is orthogonal and periodic,while the former lose the orthogonality in a finite time interval.Considering the simplicity of the PGHW in frequency domain,its FWT and iFWT are developed via the fast Fourier transformation(FFT) technique.Numerical examples demonstrated the computational efficiency of the algorithms and the perfect reconstruction of the PGHW.
The periodic generalized harmonic wavelet(PGHW) and the algorithms for its fast wavelet transformation(FWT) and inverse fast wavelet transformation(iFWT) are presented here.The PGHW can be represented as a sum of several translated harmonic terms in time domain,and a sum of several δ functions in frequency domain.Compared to the generalized harmonic wavelet(GHW) and the harmonic wavelet(HW),the PGHW is orthogonal and periodic,while the former lose the orthogonality in a finite time interval.Considering the simplicity of the PGHW in frequency domain,its FWT and iFWT are developed via the fast Fourier transformation(FFT) technique.Numerical examples demonstrated the computational efficiency of the algorithms and the perfect reconstruction of the PGHW.
引文
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[24]Tratskas P,Spanos P D.Linear multi-degree-of-freedomsystem stochastic response by using the harmonicwavelet transform[J].Journal of Applied Mechanics,2003,70(5):724-731.
[25]Liang J W,Chaudhuri S R,Shinozuka M.Simulation ofnonostationary stochastic process by spectral representation[J].Journal of Engineering Mechanics,2007,133(6):616-627.
[26]Shinozuka M.Simulation of stochastic processes by spectralrepresentation[J].Applied Mechanics Reviews,1991,44(4):191-204.
[2]Meyer Y.Principe d’incertitude,bases hilbertiennes etalgebres d’opérateurs[J].Seminaire Bourbaki,1985,622:1985-1986.
[3]Meyer Y.Ondelettes et operateurs,I.Ondelettes[M].Paris:Hermann,1990.
[4]Mallat S.Multiresolution approximations and waveletorthonormal bases of L2(R)[J].Transactions of theAmerican Mathematical Society,1989,315(1):69-87.
[5]Daubechies I.Ten lectures on wavelets[M].Philadelphia:Society for Industrial and Applied Mathematics,1992.
[6]Coifman R R.Wavelet analysis and signal processing,signalprocessing part I[M].Springer-Verlag,1990:59-68.
[7]Chui C K.An introduction to wavelets[M].San Diego:Academic Press Professional,Inc,1992.
[8]Newland D E.An introduction to random vibrations,spectraland wavelet analysis[M].New York:Longman Scientific&Technical,1993.
[9]Newland D E.Harmonic and musical wavelets[J].Proceedings of the Royal Society of London.Series A:Mathematical and Physical Sciences,1994,444(1922):605-620.
[10]Newland D E.Harmonic wavelet analysis[J].Proceedingsof the Royal Society of London.Series A:Mathematical andPhysical Sciences,1993,443(1917):203-225.
[11]Newland D E.Ridge and phase identification in the frequencyanalysis of transient signals by harmonic wavelets[J].Journal of Vibration and Acoustics,1999,121(2):149-155.
[12]Newland D E.Practical signal analysis:do wavelets makeany difference?[A].Proceedings of the 16th ASMEBiennial Conference on Vibration and Noise.Sacramento,1997.
[13]Newland D E.Wavelet analysis of vibrations,part II:waveletmaps[J].Journal of Vibration and Acoustics,1994,116(4):417-425.
[14]Newland D E.Wavelet analysis of vibration,part I:theory[J].Journal of Vibration and Acoustics,1994,116(4):409-416.
[15]Muniandy S V,Moroz I M.Galerkin modelling of the burgersequation using harmonic wavelets[J].Physics Letters A,1997,235(4):352-356.
[16]Cattani C.The wavelet-based technique in dispersive wavepropagation[J].International Applied Mechanics,2003,39(4):493-501.
[17]Cattani C.Harmonic wavelets towards the solution ofnonlinear PDE[J].Computers and Mathematics withApplications,2005,50(8-9):1191-1210.
[18]Cattani C,Kudreyko A.Harmonic wavelet method towardssolution of the fredholm type integral equations of the secondkind[J].Applied Mathematics and Computation,2010,215(12):4164-4171.
[19]Cattani C,Kudreyko A.Application of periodized harmonicwavelets towards solution of eigenvalue problems for integralequations[J].Mathematical Problems in Engineering,2010,2010:570136-570143.
[20]Spanos P,Giaralis A,Politis N.Time-frequency representationof earthquake accelerograms and inelastic structural responserecords using the adaptive chirplet decomposition andempirical mode decomposition[J].Soil Dynamics andEarthquake Engineering,2007,27(7):675-689.
[21]Spanos P D,Tezcan J,Tratskas P.Stochastic processesevolutionary spectrum estimation via harmonic wavelets[J].Computer Methods in Applied Mechanics and Engineering,2005,194(12-16):1367-1383.
[22]Tratskas P.Wavelet-based excitation Representation andresponse determination of linear and nonlinear systems[D].Houston:Rice University,2001.
[23]Spanos P D,Kougioumtzoglou I A.Harmonic wavelets basedstatistical linearization for response evolutionary powerspectrum determination[J].Probabilistic EngineeringMechanics,2012,27(1):57-68.
[24]Tratskas P,Spanos P D.Linear multi-degree-of-freedomsystem stochastic response by using the harmonicwavelet transform[J].Journal of Applied Mechanics,2003,70(5):724-731.
[25]Liang J W,Chaudhuri S R,Shinozuka M.Simulation ofnonostationary stochastic process by spectral representation[J].Journal of Engineering Mechanics,2007,133(6):616-627.
[26]Shinozuka M.Simulation of stochastic processes by spectralrepresentation[J].Applied Mechanics Reviews,1991,44(4):191-204.
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