多分辨希尔伯特—黄（Hilbert-Huang）变换方法的研究
|
摘要
Hilbert－Huang变换是一种新的分析非线性非平稳信号的时频分析方法。
这种方法的关键部分是经验模态分解方法，任何复杂信号都可以分解为有限数
目并且具有一定物理定义的固有模态函数，利用Hilbert变换，求解每一阶固
有模态函数的瞬时频率，从而得到信号的时频表示。
本文讨论了瞬时频率的定义，详细论述了连续和离散瞬时频率的计算，
同时比较了瞬时频率Fourier频率的异同点。由于瞬时频率是时间的函数，因
此可以获得信号任意时刻的频率分布，利用瞬时频率的局部化特征可以有效地
揭示信号的内在结构。瞬时频率只能对单分量估量才有意义，本文通过局部类
比三类正弦信号获得了具有单分量信号特征的固有模态函数。瞬时频率在统计
上与Fourier频率是相容的。
通过经验模态分解方法可以获得一系列固有模态函数。这种分解方法直
接从信号本身获取基函数的概念，因此具有自适应性和高效性，同时，也存在
计算量大和模态混叠的缺点。本文在经验模态分解方法的基础上引入了多分辨
分析技术，提出了分段固有模态函数，建立了多分辨经验模态分解方法，通过
可调的时间矩形窗，对信号进行筛分，实现了信号的多尺度分解，并且显著地
减小了计算量，增加了信号处理的实时性，有效地消除了固有模态函数中模态
混叠现象。由于多分辨经验模态分解方法是基于信号的局部时间尺度特征的，
因此该方法特别适合于分析非线性非平稳信号。利用Hilbert变换对分段固有
模态函数求解瞬时频率，可以获得以分段固有模态函数为基函数的信号表示形
式，它是一般化的Fourier级数形式。进一步可以得到信号的能量时频分布一
11 if bert谱。结合多分辨经验模态分解方法和Hilbert谱分析方法，从而建立
了多分辨Hilbert－Huang变换。由于引入了多分辨分析技术，使得Hilbert－
IJuang变换既保留了小波变换中时频局部化的优点，同时又因为不需要基函
数，克服了小坡变换中选择小波基的的困难。多分辨Hilbert－llunag变换对信
号具有良好的局部化、自适应和分析的结果的直观性。
本文结合Fourier变换和小波变换，从非线性系统和非平稳信号两方面
入手，列举了多分辨Hilbert－Huang变换在多个领域的应用，大量的实例说明
该方法的有效性。
Hilbert-Huang transform (HHT) is a new two-step time-frequency analytic method to analyze the nonlinear and the non-stationary signal. The key step of this method is the empirical mode decomposition (EMD) method with which any complicated data set can be decomposed into a finite and often small number of intrinsic mode functions (IMF). Using Hilbert transform to those IMF components can yield instantaneous frequency, the final presentation of this results is an energy-frequency-time distribution, designated as the Hilbert spectrum.
This paper discussed the definition of instantaneous frequency and it's calculation in continuous and discrete time. Instantaneous frequency is function of time that gives sharp identification of imbedded structures in signal. Obtaining meaningful instantaneous frequency hove to be restricted to the mono-component signal, for multi-component signal, it is necessary to decompose this signal into the combination of some mono-component signals. In this paper, based on the local properties of signal, define a class of function designated as intrinsic mode function i in which the instantaneous frequency can be defined everywhere.
The empirical mode decomposition Method is adaptive and highly efficient, meanwhile, this method is time consuming and may run into difficulties when the data contains intermittence which will cause mode mixing. This paper invokes the multi-resolution analytic technology into the EMD, develops the sectional intrinsic mode function and constructs multi-resolution EMD method (MEMD). Using a series of adjustable time windows to control the shifting process can yield a multi-scale decomposition of signal. The MEMD method can notable decrease the time consuming and efficiently overcome the mode mixing. Since this decomposition is based on the local characteristic time scale of the signal, it is applicable to nonlinear
r
and non-stationary process. Using Hilbert transform to those sectional IMF components can obtain the presentation of signal that represents a generalized Fourier expansion. Furthermore yield the Hilbert spectrum.
Combination of the MEMD method and the Hilbert spectrum method sets up
v
the multi-resolution Hilbert-Huang transform (MHHT). The MHHT not only provides a more precise definition of particular events in time-frequent space than wavelet analysis, but also provides a more physically meaningful interpretation of the underlying dynamic processes and overcomes the difficult to choose the wavelet in wavelet transform.
In this paper, examples from the numerical results of the classical nonlinear equation system and data representing natural phenomena are given to demonstrate the power of this new meshed .those results can clarity the advance and efficient of this method.
这种方法的关键部分是经验模态分解方法，任何复杂信号都可以分解为有限数
目并且具有一定物理定义的固有模态函数，利用Hilbert变换，求解每一阶固
有模态函数的瞬时频率，从而得到信号的时频表示。
本文讨论了瞬时频率的定义，详细论述了连续和离散瞬时频率的计算，
同时比较了瞬时频率Fourier频率的异同点。由于瞬时频率是时间的函数，因
此可以获得信号任意时刻的频率分布，利用瞬时频率的局部化特征可以有效地
揭示信号的内在结构。瞬时频率只能对单分量估量才有意义，本文通过局部类
比三类正弦信号获得了具有单分量信号特征的固有模态函数。瞬时频率在统计
上与Fourier频率是相容的。
通过经验模态分解方法可以获得一系列固有模态函数。这种分解方法直
接从信号本身获取基函数的概念，因此具有自适应性和高效性，同时，也存在
计算量大和模态混叠的缺点。本文在经验模态分解方法的基础上引入了多分辨
分析技术，提出了分段固有模态函数，建立了多分辨经验模态分解方法，通过
可调的时间矩形窗，对信号进行筛分，实现了信号的多尺度分解，并且显著地
减小了计算量，增加了信号处理的实时性，有效地消除了固有模态函数中模态
混叠现象。由于多分辨经验模态分解方法是基于信号的局部时间尺度特征的，
因此该方法特别适合于分析非线性非平稳信号。利用Hilbert变换对分段固有
模态函数求解瞬时频率，可以获得以分段固有模态函数为基函数的信号表示形
式，它是一般化的Fourier级数形式。进一步可以得到信号的能量时频分布一
11 if bert谱。结合多分辨经验模态分解方法和Hilbert谱分析方法，从而建立
了多分辨Hilbert－Huang变换。由于引入了多分辨分析技术，使得Hilbert－
IJuang变换既保留了小波变换中时频局部化的优点，同时又因为不需要基函
数，克服了小坡变换中选择小波基的的困难。多分辨Hilbert－llunag变换对信
号具有良好的局部化、自适应和分析的结果的直观性。
本文结合Fourier变换和小波变换，从非线性系统和非平稳信号两方面
入手，列举了多分辨Hilbert－Huang变换在多个领域的应用，大量的实例说明
该方法的有效性。
Hilbert-Huang transform (HHT) is a new two-step time-frequency analytic method to analyze the nonlinear and the non-stationary signal. The key step of this method is the empirical mode decomposition (EMD) method with which any complicated data set can be decomposed into a finite and often small number of intrinsic mode functions (IMF). Using Hilbert transform to those IMF components can yield instantaneous frequency, the final presentation of this results is an energy-frequency-time distribution, designated as the Hilbert spectrum.
This paper discussed the definition of instantaneous frequency and it's calculation in continuous and discrete time. Instantaneous frequency is function of time that gives sharp identification of imbedded structures in signal. Obtaining meaningful instantaneous frequency hove to be restricted to the mono-component signal, for multi-component signal, it is necessary to decompose this signal into the combination of some mono-component signals. In this paper, based on the local properties of signal, define a class of function designated as intrinsic mode function i in which the instantaneous frequency can be defined everywhere.
The empirical mode decomposition Method is adaptive and highly efficient, meanwhile, this method is time consuming and may run into difficulties when the data contains intermittence which will cause mode mixing. This paper invokes the multi-resolution analytic technology into the EMD, develops the sectional intrinsic mode function and constructs multi-resolution EMD method (MEMD). Using a series of adjustable time windows to control the shifting process can yield a multi-scale decomposition of signal. The MEMD method can notable decrease the time consuming and efficiently overcome the mode mixing. Since this decomposition is based on the local characteristic time scale of the signal, it is applicable to nonlinear
r
and non-stationary process. Using Hilbert transform to those sectional IMF components can obtain the presentation of signal that represents a generalized Fourier expansion. Furthermore yield the Hilbert spectrum.
Combination of the MEMD method and the Hilbert spectrum method sets up
v
the multi-resolution Hilbert-Huang transform (MHHT). The MHHT not only provides a more precise definition of particular events in time-frequent space than wavelet analysis, but also provides a more physically meaningful interpretation of the underlying dynamic processes and overcomes the difficult to choose the wavelet in wavelet transform.
In this paper, examples from the numerical results of the classical nonlinear equation system and data representing natural phenomena are given to demonstrate the power of this new meshed .those results can clarity the advance and efficient of this method.
引文
[1] Chui, C. K., Approximation theory and functional analysis, Boston , Academic press , 1991
[2] Champency , D.C., A handbook of Fourier theorems, Cambridge university press, 1987
[3] 涂汉生，何平，赵联文，应用统计，西南交通大学出版社，1994
[4] L.科恩，白居宪译，时频分析：理论与应用，西安交通大学出版社，PRENTLCEHALL，1998
[5] [美].崔锦泰著。小波分析导论，西安交通大学出版社，1995
[6] J. Bertrand and P. Bertrand, Affine time-frequency distributions, Time-frequency signal analysis-----methods and application, Ed B. Boashash, Longman-chesire, Melbourne Australia, 1991
[7] N. E. Huang, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. Lond. A(1998)454, 903-995
[8] 易光初，程俊，信号的广义时频表示，电子学报，Oct 1993, 21(10): 92-101
[9] L. Colen, Time-frequency distributions: a review, IEEE. Proc. July 1989,77(7):941-981
[10] O. Rioul and M. Vetterli, Wavelet and signal processing, IEEE. SP Magazine, Oct 1991 39(10): 14-88
[11] F. Hlawatsch and G. F. Boudreaux-Bartels, Linear and quadratic time-frequency signal representations, IEEE. SP Magazine, April 1992 40(4): 21-67
[12] S.Kadambe, G. F. Boudreaux-Bartels, A comparsion of the exstence of "cross term" in Wigne distribution and the squared magnitude of the wavelet transform and the short time Fourier transform, IEEE. Trans signal processing, Oct 1992,40(10): 2498-2515
[13] J. Jeong and J. W. Williams, Kernal design for reduced interference distribution, IEEE. Trans SP, Feb 1992, 40(2):402-411
[14] 刘令普，周洪玉等，谈傅里叶分析与小波分析，哈尔滨理工大学学报，Vol3.No.6, Dec, 1998: 79-83.
[15] Youn, O. H., Short time Fourier transform using a bank of low-pass filter, IEEE. Trans Acoust, Speech, Sig Proc., Feb 1985, 33(2): 182-185.
[16] Auslander, L. and Buffalano, C., A comparison of the Gabor and short-time Fourier transform for signal detection and feature extraction in noisyenvironment, Proc. SPIE. Conf., Nov 1990,1348:230-247,
[17] Battle, G., Heisenberg proof of the Ballian-low theorem, Lett. Math. Phys., July 1988, 15(7):175-177
[18] Ferdlander, B. and Porat, B., Detection of transient signals by Gabor representation, IEEE. Trans. Acoust Speech Sig Proc, Feb 1989, 37(2):169-180
[19] Ales, R. A., Wide-band proportional bandwidth Wigner-Ville analysis, IEEE Trans. Acoust. Speech Sig proc, June 1990, 38(6):1005-1012
[20] 吴品三，近世代数,高等教育出版社，1985年3月第一版
[21] 郑维行，王声望，实变函数与泛函分析概要（第1,2册）高等教育出版社，1980年1月
[22] 刘贵忠，邸双亮，小波分析及其应用，西安电子科大出版社，1992
[23] 王中德，快速小波变换的历史与现状，电子学报，1989,17（15）：105-111
[24] 秦树人，徐铭陶，陈志奎等，L2（R）的多尺度逼近与小波正交分解，重庆大学学报，1996，19（2）：1-9
[25] 李建平，小波分析与信号处理一理论应用及软件实现。重庆出版社，1997
[26] I. Daubechies, The wavelet transform, Time-frequency localization and signal analysis, IEEE Trans. Informatiom theory, 1990, 36(5):961-1005.
[27] F. Hlawatsch and G. F. Boudreaux-Bartels, Linear and quadratic Time-frequency signal representation, Proc. IEEE, April 1992, 42(1):21-56.
[28] 胡广书，数字信号处理一理论、算法及实现，清华大学出版社，1997
[29] Portnoff, M. R., Short-time Fourier analysis of sampled speech, IEEE Trans. Acoust. Speech. Sig. Proc., June 1981, 29(2):364-373.
[30] 郑君里，杨为理，应启珩，信号与系统，上下册，高等教育出版社，1997年12月，第1版
[31] P. Loughlin, J. Pittor and L. E. Atlas, Bilinear time-frequency representation: new insights and proerties, IEEE Trans. Sig. Proc., 1993 41(4):750-767.
[32] M. G. Amin, Introducing the spectral diversity, IEEE Trans. Sig. Proc., 1993, 41(1):185-193.
[33] M. G. Amin, Spectral decomposition of the time-frequency distribution kernels, IEEE Trans. Sig. Proc., 1994, 42(6):1156-1994.
[34] Feldman, M., Seibold, S., Damage diagnosis of rotors: application of hilbert transform and multi-hypothesis testing, J. Vibration and control, 1995, 210(5):421-445.
[35] Feldman, M., Non-linear free vibration identification via the Hilbert transform, J. Sound and Vibration, Dec 1997, 208(3): 475-489.
[36]H. Helson, Harmonic analysis, The Wadworth & Brooks, Addison-Wesley, 1991.
[37] Nurnbergger, G., Approximation by spline function, New York, Springer-Verlag, 1989.
[38] -李岳生，黄友谦，数值逼近，人民教育出版社，1979
[39] -李岳生，齐车旭，样条函数方法，科学出版社，1981
[40] -汤宝平，秦树人，基于快速小波变换的信号分析系统，重庆大学学报，2001年1月23（1）：
[41] B. Boashash, Estimating and interpreting the instantaneous frequency of a signal, Part I: Fundamentals, Proc. IEEE, April 1992, 80(3)520-538.
[42] B. Boashash, Estimating and interpreting the instantaneous frequency of a signal, Part II: Algorithms and applications Proc. IEEE, April 1992, 80(3): 539-567.
[43] L. Cohen, Time-frequency distributions-A review, Proc. IEEE, 1989, 77(5): 941-981.
[44] L. Cohen, C. Lee, Instantaneous frequency, its standard deviation and multicomponent signals, Proc. SPIE, 1988, 975(1)186-208.
[45] L. Mandel, Interpretation of instantaneous frequency, Amer. J. Phys., 1974, 42(4): 840-846.
[46] S. O. Rice, Envelope of narrow-band signal, Proc. IEEE, July 1982, 70(4): 692-699.
[47] R. F. Barrett and R. Streit, Frequency tracking using hidden Markov models, IEEE Trans. Acoust. Speech. Sig. Proc., 1990, 38(3): 586-598.
[48] L. B. White and B. Boashash, Estimating the instantaneous frequency of a Gassian random process, IEEE Trans. Acoust. Speech. Sig. Proc., Sept 1988, 36(8): 1518-1521.
[49] B. Boashash, P. Oshea and M. J. Arnold, algorithms for instantaneous frequency estimation: a comparative study, Proc. SPIE, Int Soc. Opt. Eng., 1990, 1348.
[50] R. G. Niley, W. R. Carmichal, A practical procedure for estimation of instantaneous frequency, IEEE Trans. Instrum. Meas., 1981, 30(1): 76-79.
[51] M. Sun, C. C. Li, Efficient computation of discrete Pseudo Wigner distribution, IEEE Trans. Acoust. Speech. Sig. Proc., 1989, 37(11): 1135-11142.
[52] H. Al-Nash, Phase unwrapping of digital signals, IEEE Trans. Acoust. Speech. Sig. Proc., 1989, 37(11): 1693-1702.
[53] D. G. Long, Exact computation of the wrapped phase of a finite-length timeseries, IEEE Trans. Acoust. Speech. Sig. Proc., 1988,36(11): 1787-1790.
[54] B. Boashash, B. Lovell and P. Kootsookos, Time-frequency signal analysis and instantaneous frequency estimation methodology relationship and implementations, IEEE Proc., 1989, 77(11): 1237-1240.
[55] M. Sun, Discrete-time instantaneous frequency and its computation, IEEE Trans. Sig. Proc., May 1993, 41(5): 1867-1879.
[56] B. Bhanu, J. H. Mcclellan, On the computation of the complex cepstrum, IEEE Trans. Sig. Proc., 1980,28(2): 583-585.
[57] R. Mcgowan, R. Kuc, A direct relation between a signal time series and its unwrapped phase, IEEE Trans. Acoust. Speech. Sig. Proc., 1982, 30(2): 719-726.
[58] P. J. Kootsookos, A unified approach to the STFT, TED and instantaneous frequency, IEEE Trans Sig. Proc., Aug 1992, 40(8):1971-1982.
[59] B. C. Lovell, R.C. Williamson, B. Boashash, The relationship between instantaneous frquency and time-frequency representation, IEEE Trans Sig. Proc., 1993, 41(1): 230-254.
[60] L. B. White, B. Boashash, Cross-spectum analysis of non-stationary random processes, IEEE Trans. Information theory, July 1990, 36(7): 830-835.
[61] Y. Zhao, L. E. Atlas, and R. J. Marks, The use of cone-shaped kernels for generalization time-frequency representation of non-stationary signal, IEEE Trans. Acoust. Speech. Sig. Proc., July 1990, 38(7): 1084-1091.
[62] L. E. Atlas, W. Kooiman, New non-stationary techniques for the analysis and display of speech transients, Proc. IEEE, 1990, ICASSP-90:380-385.
[63] L. Cohen, Instantaneous frequency anything, Proc. IEEE ICASSP-93, 1993, 5(1): 105-108.
[64] L. Cohen, The scale represntation, IEEE Trans. Sig. Proc. 1993, 5(8): 3275-3293.
[65] O. D. Grace, Instantaneous power spectra, J. Aoust. Soc. AM, 1981, 69(1): 91-198.
[66] R. Hippenstiel and D. Oliveira, Time-varying spectral estimation using instantaneous power spectrum(IPS), IEEE Trans Sig. Proc., 1990, 38(5):1752-1759.
[67] G. Jones and B. Boashash, Instantaneous frequency, time delay, instantaneous bandwidth and the analysis of the multicomponent signals, Proc.IEEE, 1990, ICASSP-90: 2467-2470.
[68]R. Kumaresan, A. G. Sadasiv, Instantaneous non-linear operators for tracking multicomponent signal parameters, Proc. IEEE-SP, 1992, Int. Sympos. Time-frequency analysis and time-scale analysis: 404-407.
[69] R. Kumaresan, C. S. Ramalirgam, On separating voiced-speech into its components, Proc. of Twenty-seventh annual Asilomar conf. On signal, system and computer, 1993, 2:1041-1046.
[70] B. Lovell and R. Williamson, The statistical performance of some instantaneous frequency estimators, IEEE Trans. Sig. Proc., 1992, 40(6): 1708-1723.
[71] B. Lovell and R. Williamson and B. Boashash, The relationship between instantaneous frequency and time -frequency representation, IEEE Trans. Sig. Proc., 1993, 41(6): 1458-1469.
[72] T. W. Park and R. O. Shenoy, Time-frequency concentrated basis functions, Proc. IEEE, 1990, icassp-90: 2459-2462.
[73] M. Poletti, Instantaneous frequency and conditional moments in time-frequency plane, IEEE Trans. Sig. Proc., 1991, 39(3): 755-759.
[74] M. Poletti, The development of instantaneous banwidth via local signal expansion, Signal processing, 1993, 31(2): 273-281.
[75] L. Stankovic, A method for time-frequency analysis, IEEE Trans. Sig. Proc., 1994, 42(1): 225-229.
[76] N. Yen, Wave packet decomposition, J. Acoust. Soc. Am., 1994, 95(4): 889-896.
[77] O. Rioul and P. Flandrin, Time-scale energy distributions: a general class extending wavelet transform, IEEE Trans.Sig. Proc. 1992, 40(7): 1746-1757.
[78] N. E. Huang, H. H. Shin, S. R. Long, The ages of large amplitude coastal seiches on the caribbean coast of Puerto Rice, J. Phy. Oceanography, Aug 2000, 30(8): 2001-2012.
[79] N. E. Huang, S. R. Long and Z. Shen, The mechanism for frequency downshift in nonlinear wave evolution, Advance in applied mechanics, Academic press, 59-118, 1996.
[80] N. E. Huang, S. R. Long and C. C. Tung, The local properties of transient stochastic mechanics, 1993, 1(1): 253-279.
[81] S. A. Mallat, A theory for multi-resolution signal decomposition: the wavelet representation, IEEE Trans. PAMI, 1989, 11(7): 674-693.
[82] S. A. Mallet, Multi-frequency channel decomposition of the images and wavelet models, IEEE Trans. Acoust. Speech. Sig. Proc., 1989, 37(12):2091-2110.
[83] Pilippe, D., Benoit, M., Dirk, T. M., Signal-adapted multi-resolution transform for image coding, IEEE Trans. Information theory, 1992, 38(2): 897-904.
[84] Devore, R. A., Jawerth, B. and Lucier, P. J., Image compression through wavelet transform coding, IEEE Trans. Information theory, 1992,38(2): 719-746.
[85] 李建平，陈延槐，张万华等，Mallat算法的数学原理，后勤工程学院学报，1996年，12（2）：35-39
[86] Mallat, S. A., Multi-resolution approximation and wavelet orthonormal bases of L2(R), Trans on the American mathematical society, 1989, 315(1):210-221.
[87] M. J. Smith and T. P. Barnwell, Exact reconstruction techniques for tree-structured subband coders, IEEE Trans. ASSP., 1986, 34(2): 434-441.
[88] P. P. Vaidyanathan, Quadrate mirror filter banks, m-band extensions and perfect-reconstruction techniques, IEEE SP. Magazine, 1987, : 4-20.
[89] M. Vellerli and C. Herley, Wavelets and filter bank: theory and design, IEEE Trans. ASSP., 1992, 40(9): 2207-2232.
[90] A. Grossmann and J. Morlet, Decomposition of Hardy function into square integrable wavelet of constant shape, SIAM. J. Math., 1984, 15(3):723-736.
[91] Mallat, S. A., A compactly multi-resolution representation: the wavelet mode, Proc. IEEE Workshop compute vision, Dec 1987, 33(12):1789-1820.
[92] D. Marr and T. Poggio, A theory of human stereo vision, Proc. Royal Soc. London. B, 1979, 204: 301-328.
[93] J. W Woods and S. D. O'Neil, Subband coding of images, IEEE Trans. Acoust. Speech. Sig. Proc., Oct 1986, 34(10): 2101-2120.
[94] Olivier Rioul, A discrete-time multi-resolution theory, IEEE Trans. Sig. Proc., Aug 1993, 41(8): 2591-2605.
[95] O.Rioul and P. Duhanel, Fast algorithms for discrete and continuous wavelet transforms, IEEE Trans. on Inform. theory, May 1992,38(2): 569-586.
[96] M. J. Shensa, The discrete wavelet transform, wedding the atrous and Mallat algorithms, IEEE Trans. Signal processing, Oct,1992,40(10):2464-2482.
[97] M, vetterli, Multi-dimensional subhand coding: some theory and algorithms, Signal processing, Feb 1984, 6(2):pp97-112.
[98] 刘贵忠，冯轶岭，宗涛，样条多分辨分析，电子学报, July 1996, 24(7): 72-78.
[99] Chui, C. K, and J. Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math Soc., 1991,.113(3): 785-793
[100] Chui. C .K and J. K. Wang, On compactly supported spline wavelet and aduality principle, Trans. Amer. Math. Soc., 1991, 113(4):981-1005,
[101] Daubechies, I.; The wavelet transform, time frequency Localization and signal analysis, IEEE Trans Information Theory, 1990, 30(3): 961-1005.
[102] Chui, C. K, and J. Z. Wang, Fast B-spline transforms for continuous image represent and Interpolation, IEEE Trans on pattern analysis and machine Intelligence , Mach, 1991, 13(3):
769-774.
[103] P. V. Sankar and L. A. Ferroui, Simple algorithms and architectures for B-spline interpolation, IEEE Trans on pattern analysis and Machine intell., 1998, 10(1): 271-176.
[104] A. Carcaterra and A. Sestieri, Complex envelope displacement analysis: A quasi-static approach to vibrations, Journal of sound and vibration,1997,201(2): 205-233.
[105] B. Friedlander and B. Porat, Detection of transient signals by the Gabor representation, IEEE Trans. on ASSP., 1989, 37(2):278-289.
[106] Olivier Rioul, A discrete-Time Multi-resolution theory, IEEE Trans. on signal processing, August, 1993, 41(8): 1792-1820.
[107] P. J. Burt and E. H. Adelson, The Laplacian pyramid as a compact image code, IEEE Trans. Comun., April, 1983, 31(4): 789-804.
[108] I. Daubechies and J. C. Lagarias, Two scale difference equations existence and global regularity of solutions, SIMJ. Math. Anal, Sept, 1991, 22(5): 1338-1410.
[109] A. H. Tewfik, D. Sinha and P. Jorgensen, On the optimal choice of a wavelet for signal representation, IEEE Trans. on Information theory, 1992, 38(2): 747-765.
[110] Hehong Zopu, A. H..Tewik,. Parametrization of compactly supported orthonormal wavelets, IEEE Trans. on Signal processing, 1993, 41(3): 1428-1431
[111] J. D. Villasenor, B. Belzer and J. Liao Wavelet filter evaluation for image compression, IEEE Trans. on Image processing, 1995, 4(8): 1053-1060
[112] 沈建军，党亮，沈振康，一种最优正交小波基选择的遗传算法方法，信号处理，1999,12, 15(4): 335-340.
[113] Philippe Steeghs, Fast computation of the sliding window Radon transform applied to 3-D Seismic attribute extraction, Rice university, Houston, Tx, 1999
[114] 严中洪，小波滤波器的统一构造方法与软件，重庆大学学报，Jul 2000,23(1): 9-14.
[115]彭玮仁，陈明基，彭康，一种具有消失矩的小波基的构造，应用数学与力学，1999, 20(3): 235-240.
[116] 郑建明，李言，伸缩窗口短时Fourier分析，振动、测试与诊断，Dec 2000, 20(4): 254-268.
[117] Haldun, M. O., Orhan, A.,. Digital computation of the fractional Fourier transform, IEEE Trans. signal processing, 1996, 44(9): 2141-2150.
[118] N. E..Huang,. S. .R. .Long and Z. Shen, A new view of nonlinear water waves: The Hilbert spectrum, Annu.. Rev.. Fluid. Mech, 1999, 31: 417-457.
[119] N. E. .Huang, S..R..Long, Wave fusion as a mechanism for nonlinear evolution of water waves, J. .Fluid. Mech., 1995, 27: 254-280.
[120] N..E..Huang, S..R..Long. The local properties of ocean wave, by the phase-time method, Geophys. Res. Lett., 1992, 19: 685-688.
[121] Huang, N. E.,.Tung, C..C. and .S. R. Long, Wave spectra, The sea, 1990, 9: 197-237.
[122] Spedding, G..R.,.Bronand, F.K. Huang, N. E. and Long, S. R., A 2D complex wavelet analysis of an unsteady wind-generated surface wave field, Dyn. Atmos. Oceans. 1993, 20: 55-77.
[123] Yen, N. C., Wave packet decomposition, J. Acoust. Soc. Am., 1994, 95: 889-896.
[124] O. Rioul and P. Flandtin, Time-scale energy distributions: a general class extending wavelet transforms, IEEE Trans. Sig. Proc., 1992, 40: 1746-1757.
[125] T. E. Posch,, Wavelet transform and time-frequency distributions, Proc. SPIC, 1998, 1152: 472-482.
[126] W. J. Pielemeier and G. H. Wakefield, Time-frequency and time-scale analysis for musical transcription, Proc. IEEE-SP, Int. Sympos., Time frequency and time-scale analying, 1992, 421-424.
[127] Farge, M., Wavelet transforms and their application to turbulence, Annu. Rev. Fluid Mech., 1992, 24: 395-457
[128] Feldman, M., Method for determination of vibratory system modal parameters using Hilbert transform, Application for patent no: 698985, Israel, 28, July 1991.
[129] Feldman, M., Nonlinear system vibration analysis using Hilbert transform free vibration analysis method, FREEVIB. Mech. Syst. signal processing, 1994a, 8(2): 119-127.
[130] Feldman, M., Nonlinear system vibration analysis using Hilbert transform: II Forced vibration analysis method, FORCEVIB, Mech. Syst. signal processing,1994b, 8(2): 309-318.
[131] Feldman, M., Nonlinear free vibration identification via the Hillert transform, J. Sound vib., 1997, 208(3): 475-495.
[132] Feldaman, M., Braun, S., Identification of nonlinear system parameters via the instantaneous frequency: application of the Hilbert Transform and Wigner-Ville techniques, Proc, Int Modal analysis Conf., 1993, 13th: 637-642.
[133] Hara. T., Mei, C. C., Frequency downshift in narrowband surface waves under the influence of wind, J. Fluid. Mech., 1995, 2301: 429-477
[134] Long, S. R., Huang, N. E., Tung, C. C., Wu, M.L.,. The Hilbert techniques an alternate approach for non-steady time series analysis, IEEE Geosci,.Remote sensing Soc,. Lett, 1993, 3: 6-11.
[135] Nelkin, M., Universality and scaling in fully developed turbulence, Adu. Phys., 1994, 43: 143-181.
[136] Uchiyama, Y., Kawahara, T., A possible mechanism for frequency down-shift in non-linear wave modulation, Wave Motion, 1994, 20: 99-110.
[137] W. Hang, Z. Shen, N. E. Huang and Y. C. Fung, Engineering analysis of biological variable: An example of blood pressure over 1 day, Proc. Natl. Acad. Sci. USA., April, 1998, 95:4816-4821.
[138] Zhu, X., .Shen, Z.,. Echermam, S. D., Gravity wave characteristics in the middle atmosphere derived from the empirical mode decomposition meted, J. Geophysical Research, 1997, 102(14): 545-561.
[139] Liu, P. L., Is the wind wave frequency spectrum outdated, Ocean engineering, 2000, 27: 557-588.
[140] Balpck, T. E., Swan, C., Extreme waves in shallow and intermediate water depths, Coastal engineering, 1996, 27: 21-46.
[141] Kang, J. H., L. Lon, Y. S., Chan, E. S., Laboratory study of deep water breaking waves, Ocean engineering, 1988.25(8): 657-676.
[142] T. Schlurmann, S. Schimmels, T. Dose, Spectral analysis of Freak waves using wavelet spectra (Morlet) and Hilbert spectra (EMD), Proc. of 2nd German-Chinese Joint seminar on Recent development in coastal engineering, Sept. 1999: 65-80.
[143] Qin. Suren, et al, The research of wavelet transform analysis system for engineering signals, Chinese Journal of Mechanical Engineering, 2000, 13(2): 491-423.
[144] Qin Shuren, et al, Sampling principle and technology in wavelet analysis for signal, Chinese Journal of Mechanical engineering, 1998, 11(4): 259-263.
[145]秦树人，汤宝平，陈志奎等。工程信号小波变换分析仪系统的研究，中国机械工程， 1999，10（4）：443-446
[146] 沈远彤，叶碧泉，旭明。非线性摄动方程的小波算法，贵州工业大学学报，1997，10，26（增刊）：89-90
[147] 柯熙政，吴振森。一类高阶非齐次发展方程的小波基数值分析，数学研究与评论， 1999，8，19（3）：587-592。
[148] 向裕民。Duffing方程的振动解，四川轻化工学院学报，1996，9，（3）：42-44
[149] 方见树。参数微扰Duffing 振动子的求解与分析，湖南教育学院学报，2000，4，18（2）：36-40
[150] Kapitania, K.T., Analytical method of controlling chaos in Dufting's oscillator, J. Sound vib., 1993, 163:182-187
[151] Yang. J., Qu. Z. Hu, Dufting equation with two periodic forcings: The phase eftect, Phys. Rev. E., 1996, 53: 4402-4413.
[152] 曹庆杰，张天德，李久平。Dufting方程的静态与动态分岔特征研究，应用数学与力学，1999, 20（12）：1309-1315
[153] 裴韬，鲍征宇。傅立叶变换在地球化学数据处理中的应用，长春科技大学学报，1999，1，29（1）：88-92
[154] 马晶，谭立英，由启文，光学小波滤波理论初探，中国激光，1999，4，26（4）：343-346
[155] 刘曙光，殷明，何铖。子波消噪技术及其应用。西北纺织工学院学报，1999,3, 13（1）:16-20
[156] Mallat, S., Zero-crossing of a wavelet transform, IEEE Trans. on IT, 1991, 37(4): 1019-1033.
[157] Mallat, S., Huang, W. L., Singularity detection and processing with wavelets, IEEE Trans. Information theory, 1992, 38(2): 617-643.
[158] Mallat, S., Zhong, S., Characterization of signals from multi-scale edge, IEEE Trans. Pattern Anal. Mech. Intel, 1992, 14(7): 710-723.
[159] Cetin, A.E., Ansari, R., Signal recovery from wavelet transform maxima, IEEE Trans. signal processing, 1994, 42(7): 1845-1848
[160] 何正友，钱清泉。基于小波变换的信号奇异性指数计算方法及其应用，电力自动化设备，2000.6, 20(3): 12-15
[161] 刘希强，用蕙兰等。基于小波包变换的滤波方法，西北地震学报，1999.9,21(3): 248-253
[162] 韩其玉。地震信号时频表示，石油大学学报，1994，18(3): 26-29
[163] 范中，田立生。利用子波变换检测瞬时信号，电子学报，1996,24(1):78-82
[164]沈亚强，金洪震。一种基于子波变换的语言增强方法，科学通报，2000.5, 16(3): 206-211
[165] Donoho, D., Denoising by soft-thresholding, IEEE Trans. on IT, 1995,4(3): 613-627.
[166] 潘旭峰，谢波，李晓雷。小波变换理论及其在机械故障诊断中的应用，振动与冲击，1998，17（1）：14-19
[167] 姚成，吴小培。小波变换与生物医学信号处理，生物学杂志，2000,2,17(1): 25-26
[168] 杨福生，高上凯。生物医学信号处理，高等教育出版社，1989,5
[169] 李光林，昌维雪。基于小波变换的心电信号的分析与处理，浙江大学学报，Jan,1998,32(1): 83-86
[2] Champency , D.C., A handbook of Fourier theorems, Cambridge university press, 1987
[3] 涂汉生，何平，赵联文，应用统计，西南交通大学出版社，1994
[4] L.科恩，白居宪译，时频分析：理论与应用，西安交通大学出版社，PRENTLCEHALL，1998
[5] [美].崔锦泰著。小波分析导论，西安交通大学出版社，1995
[6] J. Bertrand and P. Bertrand, Affine time-frequency distributions, Time-frequency signal analysis-----methods and application, Ed B. Boashash, Longman-chesire, Melbourne Australia, 1991
[7] N. E. Huang, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. Lond. A(1998)454, 903-995
[8] 易光初，程俊，信号的广义时频表示，电子学报，Oct 1993, 21(10): 92-101
[9] L. Colen, Time-frequency distributions: a review, IEEE. Proc. July 1989,77(7):941-981
[10] O. Rioul and M. Vetterli, Wavelet and signal processing, IEEE. SP Magazine, Oct 1991 39(10): 14-88
[11] F. Hlawatsch and G. F. Boudreaux-Bartels, Linear and quadratic time-frequency signal representations, IEEE. SP Magazine, April 1992 40(4): 21-67
[12] S.Kadambe, G. F. Boudreaux-Bartels, A comparsion of the exstence of "cross term" in Wigne distribution and the squared magnitude of the wavelet transform and the short time Fourier transform, IEEE. Trans signal processing, Oct 1992,40(10): 2498-2515
[13] J. Jeong and J. W. Williams, Kernal design for reduced interference distribution, IEEE. Trans SP, Feb 1992, 40(2):402-411
[14] 刘令普，周洪玉等，谈傅里叶分析与小波分析，哈尔滨理工大学学报，Vol3.No.6, Dec, 1998: 79-83.
[15] Youn, O. H., Short time Fourier transform using a bank of low-pass filter, IEEE. Trans Acoust, Speech, Sig Proc., Feb 1985, 33(2): 182-185.
[16] Auslander, L. and Buffalano, C., A comparison of the Gabor and short-time Fourier transform for signal detection and feature extraction in noisy
[17] Battle, G., Heisenberg proof of the Ballian-low theorem, Lett. Math. Phys., July 1988, 15(7):175-177
[18] Ferdlander, B. and Porat, B., Detection of transient signals by Gabor representation, IEEE. Trans. Acoust Speech Sig Proc, Feb 1989, 37(2):169-180
[19] Ales, R. A., Wide-band proportional bandwidth Wigner-Ville analysis, IEEE Trans. Acoust. Speech Sig proc, June 1990, 38(6):1005-1012
[20] 吴品三，近世代数,高等教育出版社，1985年3月第一版
[21] 郑维行，王声望，实变函数与泛函分析概要（第1,2册）高等教育出版社，1980年1月
[22] 刘贵忠，邸双亮，小波分析及其应用，西安电子科大出版社，1992
[23] 王中德，快速小波变换的历史与现状，电子学报，1989,17（15）：105-111
[24] 秦树人，徐铭陶，陈志奎等，L2（R）的多尺度逼近与小波正交分解，重庆大学学报，1996，19（2）：1-9
[25] 李建平，小波分析与信号处理一理论应用及软件实现。重庆出版社，1997
[26] I. Daubechies, The wavelet transform, Time-frequency localization and signal analysis, IEEE Trans. Informatiom theory, 1990, 36(5):961-1005.
[27] F. Hlawatsch and G. F. Boudreaux-Bartels, Linear and quadratic Time-frequency signal representation, Proc. IEEE, April 1992, 42(1):21-56.
[28] 胡广书，数字信号处理一理论、算法及实现，清华大学出版社，1997
[29] Portnoff, M. R., Short-time Fourier analysis of sampled speech, IEEE Trans. Acoust. Speech. Sig. Proc., June 1981, 29(2):364-373.
[30] 郑君里，杨为理，应启珩，信号与系统，上下册，高等教育出版社，1997年12月，第1版
[31] P. Loughlin, J. Pittor and L. E. Atlas, Bilinear time-frequency representation: new insights and proerties, IEEE Trans. Sig. Proc., 1993 41(4):750-767.
[32] M. G. Amin, Introducing the spectral diversity, IEEE Trans. Sig. Proc., 1993, 41(1):185-193.
[33] M. G. Amin, Spectral decomposition of the time-frequency distribution kernels, IEEE Trans. Sig. Proc., 1994, 42(6):1156-1994.
[34] Feldman, M., Seibold, S., Damage diagnosis of rotors: application of hilbert transform and multi-hypothesis testing, J. Vibration and control, 1995, 210(5):421-445.
[35] Feldman, M., Non-linear free vibration identification via the Hilbert transform, J. Sound and Vibration, Dec 1997, 208(3): 475-489.
[36]
[37] Nurnbergger, G., Approximation by spline function, New York, Springer-Verlag, 1989.
[38] -李岳生，黄友谦，数值逼近，人民教育出版社，1979
[39] -李岳生，齐车旭，样条函数方法，科学出版社，1981
[40] -汤宝平，秦树人，基于快速小波变换的信号分析系统，重庆大学学报，2001年1月23（1）：
[41] B. Boashash, Estimating and interpreting the instantaneous frequency of a signal, Part I: Fundamentals, Proc. IEEE, April 1992, 80(3)520-538.
[42] B. Boashash, Estimating and interpreting the instantaneous frequency of a signal, Part II: Algorithms and applications Proc. IEEE, April 1992, 80(3): 539-567.
[43] L. Cohen, Time-frequency distributions-A review, Proc. IEEE, 1989, 77(5): 941-981.
[44] L. Cohen, C. Lee, Instantaneous frequency, its standard deviation and multicomponent signals, Proc. SPIE, 1988, 975(1)186-208.
[45] L. Mandel, Interpretation of instantaneous frequency, Amer. J. Phys., 1974, 42(4): 840-846.
[46] S. O. Rice, Envelope of narrow-band signal, Proc. IEEE, July 1982, 70(4): 692-699.
[47] R. F. Barrett and R. Streit, Frequency tracking using hidden Markov models, IEEE Trans. Acoust. Speech. Sig. Proc., 1990, 38(3): 586-598.
[48] L. B. White and B. Boashash, Estimating the instantaneous frequency of a Gassian random process, IEEE Trans. Acoust. Speech. Sig. Proc., Sept 1988, 36(8): 1518-1521.
[49] B. Boashash, P. Oshea and M. J. Arnold, algorithms for instantaneous frequency estimation: a comparative study, Proc. SPIE, Int Soc. Opt. Eng., 1990, 1348.
[50] R. G. Niley, W. R. Carmichal, A practical procedure for estimation of instantaneous frequency, IEEE Trans. Instrum. Meas., 1981, 30(1): 76-79.
[51] M. Sun, C. C. Li, Efficient computation of discrete Pseudo Wigner distribution, IEEE Trans. Acoust. Speech. Sig. Proc., 1989, 37(11): 1135-11142.
[52] H. Al-Nash, Phase unwrapping of digital signals, IEEE Trans. Acoust. Speech. Sig. Proc., 1989, 37(11): 1693-1702.
[53] D. G. Long, Exact computation of the wrapped phase of a finite-length time
[54] B. Boashash, B. Lovell and P. Kootsookos, Time-frequency signal analysis and instantaneous frequency estimation methodology relationship and implementations, IEEE Proc., 1989, 77(11): 1237-1240.
[55] M. Sun, Discrete-time instantaneous frequency and its computation, IEEE Trans. Sig. Proc., May 1993, 41(5): 1867-1879.
[56] B. Bhanu, J. H. Mcclellan, On the computation of the complex cepstrum, IEEE Trans. Sig. Proc., 1980,28(2): 583-585.
[57] R. Mcgowan, R. Kuc, A direct relation between a signal time series and its unwrapped phase, IEEE Trans. Acoust. Speech. Sig. Proc., 1982, 30(2): 719-726.
[58] P. J. Kootsookos, A unified approach to the STFT, TED and instantaneous frequency, IEEE Trans Sig. Proc., Aug 1992, 40(8):1971-1982.
[59] B. C. Lovell, R.C. Williamson, B. Boashash, The relationship between instantaneous frquency and time-frequency representation, IEEE Trans Sig. Proc., 1993, 41(1): 230-254.
[60] L. B. White, B. Boashash, Cross-spectum analysis of non-stationary random processes, IEEE Trans. Information theory, July 1990, 36(7): 830-835.
[61] Y. Zhao, L. E. Atlas, and R. J. Marks, The use of cone-shaped kernels for generalization time-frequency representation of non-stationary signal, IEEE Trans. Acoust. Speech. Sig. Proc., July 1990, 38(7): 1084-1091.
[62] L. E. Atlas, W. Kooiman, New non-stationary techniques for the analysis and display of speech transients, Proc. IEEE, 1990, ICASSP-90:380-385.
[63] L. Cohen, Instantaneous frequency anything, Proc. IEEE ICASSP-93, 1993, 5(1): 105-108.
[64] L. Cohen, The scale represntation, IEEE Trans. Sig. Proc. 1993, 5(8): 3275-3293.
[65] O. D. Grace, Instantaneous power spectra, J. Aoust. Soc. AM, 1981, 69(1): 91-198.
[66] R. Hippenstiel and D. Oliveira, Time-varying spectral estimation using instantaneous power spectrum(IPS), IEEE Trans Sig. Proc., 1990, 38(5):1752-1759.
[67] G. Jones and B. Boashash, Instantaneous frequency, time delay, instantaneous bandwidth and the analysis of the multicomponent signals, Proc.IEEE, 1990, ICASSP-90: 2467-2470.
[68]
[69] R. Kumaresan, C. S. Ramalirgam, On separating voiced-speech into its components, Proc. of Twenty-seventh annual Asilomar conf. On signal, system and computer, 1993, 2:1041-1046.
[70] B. Lovell and R. Williamson, The statistical performance of some instantaneous frequency estimators, IEEE Trans. Sig. Proc., 1992, 40(6): 1708-1723.
[71] B. Lovell and R. Williamson and B. Boashash, The relationship between instantaneous frequency and time -frequency representation, IEEE Trans. Sig. Proc., 1993, 41(6): 1458-1469.
[72] T. W. Park and R. O. Shenoy, Time-frequency concentrated basis functions, Proc. IEEE, 1990, icassp-90: 2459-2462.
[73] M. Poletti, Instantaneous frequency and conditional moments in time-frequency plane, IEEE Trans. Sig. Proc., 1991, 39(3): 755-759.
[74] M. Poletti, The development of instantaneous banwidth via local signal expansion, Signal processing, 1993, 31(2): 273-281.
[75] L. Stankovic, A method for time-frequency analysis, IEEE Trans. Sig. Proc., 1994, 42(1): 225-229.
[76] N. Yen, Wave packet decomposition, J. Acoust. Soc. Am., 1994, 95(4): 889-896.
[77] O. Rioul and P. Flandrin, Time-scale energy distributions: a general class extending wavelet transform, IEEE Trans.Sig. Proc. 1992, 40(7): 1746-1757.
[78] N. E. Huang, H. H. Shin, S. R. Long, The ages of large amplitude coastal seiches on the caribbean coast of Puerto Rice, J. Phy. Oceanography, Aug 2000, 30(8): 2001-2012.
[79] N. E. Huang, S. R. Long and Z. Shen, The mechanism for frequency downshift in nonlinear wave evolution, Advance in applied mechanics, Academic press, 59-118, 1996.
[80] N. E. Huang, S. R. Long and C. C. Tung, The local properties of transient stochastic mechanics, 1993, 1(1): 253-279.
[81] S. A. Mallat, A theory for multi-resolution signal decomposition: the wavelet representation, IEEE Trans. PAMI, 1989, 11(7): 674-693.
[82] S. A. Mallet, Multi-frequency channel decomposition of the images and wavelet models, IEEE Trans. Acoust. Speech. Sig. Proc., 1989, 37(12):
[83] Pilippe, D., Benoit, M., Dirk, T. M., Signal-adapted multi-resolution transform for image coding, IEEE Trans. Information theory, 1992, 38(2): 897-904.
[84] Devore, R. A., Jawerth, B. and Lucier, P. J., Image compression through wavelet transform coding, IEEE Trans. Information theory, 1992,38(2): 719-746.
[85] 李建平，陈延槐，张万华等，Mallat算法的数学原理，后勤工程学院学报，1996年，12（2）：35-39
[86] Mallat, S. A., Multi-resolution approximation and wavelet orthonormal bases of L2(R), Trans on the American mathematical society, 1989, 315(1):210-221.
[87] M. J. Smith and T. P. Barnwell, Exact reconstruction techniques for tree-structured subband coders, IEEE Trans. ASSP., 1986, 34(2): 434-441.
[88] P. P. Vaidyanathan, Quadrate mirror filter banks, m-band extensions and perfect-reconstruction techniques, IEEE SP. Magazine, 1987, : 4-20.
[89] M. Vellerli and C. Herley, Wavelets and filter bank: theory and design, IEEE Trans. ASSP., 1992, 40(9): 2207-2232.
[90] A. Grossmann and J. Morlet, Decomposition of Hardy function into square integrable wavelet of constant shape, SIAM. J. Math., 1984, 15(3):723-736.
[91] Mallat, S. A., A compactly multi-resolution representation: the wavelet mode, Proc. IEEE Workshop compute vision, Dec 1987, 33(12):1789-1820.
[92] D. Marr and T. Poggio, A theory of human stereo vision, Proc. Royal Soc. London. B, 1979, 204: 301-328.
[93] J. W Woods and S. D. O'Neil, Subband coding of images, IEEE Trans. Acoust. Speech. Sig. Proc., Oct 1986, 34(10): 2101-2120.
[94] Olivier Rioul, A discrete-time multi-resolution theory, IEEE Trans. Sig. Proc., Aug 1993, 41(8): 2591-2605.
[95] O.Rioul and P. Duhanel, Fast algorithms for discrete and continuous wavelet transforms, IEEE Trans. on Inform. theory, May 1992,38(2): 569-586.
[96] M. J. Shensa, The discrete wavelet transform, wedding the atrous and Mallat algorithms, IEEE Trans. Signal processing, Oct,1992,40(10):2464-2482.
[97] M, vetterli, Multi-dimensional subhand coding: some theory and algorithms, Signal processing, Feb 1984, 6(2):pp97-112.
[98] 刘贵忠，冯轶岭，宗涛，样条多分辨分析，电子学报, July 1996, 24(7): 72-78.
[99] Chui, C. K, and J. Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math Soc., 1991,.113(3): 785-793
[100] Chui. C .K and J. K. Wang, On compactly supported spline wavelet and a
[101] Daubechies, I.; The wavelet transform, time frequency Localization and signal analysis, IEEE Trans Information Theory, 1990, 30(3): 961-1005.
[102] Chui, C. K, and J. Z. Wang, Fast B-spline transforms for continuous image represent and Interpolation, IEEE Trans on pattern analysis and machine Intelligence , Mach, 1991, 13(3):
769-774.
[103] P. V. Sankar and L. A. Ferroui, Simple algorithms and architectures for B-spline interpolation, IEEE Trans on pattern analysis and Machine intell., 1998, 10(1): 271-176.
[104] A. Carcaterra and A. Sestieri, Complex envelope displacement analysis: A quasi-static approach to vibrations, Journal of sound and vibration,1997,201(2): 205-233.
[105] B. Friedlander and B. Porat, Detection of transient signals by the Gabor representation, IEEE Trans. on ASSP., 1989, 37(2):278-289.
[106] Olivier Rioul, A discrete-Time Multi-resolution theory, IEEE Trans. on signal processing, August, 1993, 41(8): 1792-1820.
[107] P. J. Burt and E. H. Adelson, The Laplacian pyramid as a compact image code, IEEE Trans. Comun., April, 1983, 31(4): 789-804.
[108] I. Daubechies and J. C. Lagarias, Two scale difference equations existence and global regularity of solutions, SIMJ. Math. Anal, Sept, 1991, 22(5): 1338-1410.
[109] A. H. Tewfik, D. Sinha and P. Jorgensen, On the optimal choice of a wavelet for signal representation, IEEE Trans. on Information theory, 1992, 38(2): 747-765.
[110] Hehong Zopu, A. H..Tewik,. Parametrization of compactly supported orthonormal wavelets, IEEE Trans. on Signal processing, 1993, 41(3): 1428-1431
[111] J. D. Villasenor, B. Belzer and J. Liao Wavelet filter evaluation for image compression, IEEE Trans. on Image processing, 1995, 4(8): 1053-1060
[112] 沈建军，党亮，沈振康，一种最优正交小波基选择的遗传算法方法，信号处理，1999,12, 15(4): 335-340.
[113] Philippe Steeghs, Fast computation of the sliding window Radon transform applied to 3-D Seismic attribute extraction, Rice university, Houston, Tx, 1999
[114] 严中洪，小波滤波器的统一构造方法与软件，重庆大学学报，Jul 2000,23(1): 9-14.
[115]
[116] 郑建明，李言，伸缩窗口短时Fourier分析，振动、测试与诊断，Dec 2000, 20(4): 254-268.
[117] Haldun, M. O., Orhan, A.,. Digital computation of the fractional Fourier transform, IEEE Trans. signal processing, 1996, 44(9): 2141-2150.
[118] N. E..Huang,. S. .R. .Long and Z. Shen, A new view of nonlinear water waves: The Hilbert spectrum, Annu.. Rev.. Fluid. Mech, 1999, 31: 417-457.
[119] N. E. .Huang, S..R..Long, Wave fusion as a mechanism for nonlinear evolution of water waves, J. .Fluid. Mech., 1995, 27: 254-280.
[120] N..E..Huang, S..R..Long. The local properties of ocean wave, by the phase-time method, Geophys. Res. Lett., 1992, 19: 685-688.
[121] Huang, N. E.,.Tung, C..C. and .S. R. Long, Wave spectra, The sea, 1990, 9: 197-237.
[122] Spedding, G..R.,.Bronand, F.K. Huang, N. E. and Long, S. R., A 2D complex wavelet analysis of an unsteady wind-generated surface wave field, Dyn. Atmos. Oceans. 1993, 20: 55-77.
[123] Yen, N. C., Wave packet decomposition, J. Acoust. Soc. Am., 1994, 95: 889-896.
[124] O. Rioul and P. Flandtin, Time-scale energy distributions: a general class extending wavelet transforms, IEEE Trans. Sig. Proc., 1992, 40: 1746-1757.
[125] T. E. Posch,, Wavelet transform and time-frequency distributions, Proc. SPIC, 1998, 1152: 472-482.
[126] W. J. Pielemeier and G. H. Wakefield, Time-frequency and time-scale analysis for musical transcription, Proc. IEEE-SP, Int. Sympos., Time frequency and time-scale analying, 1992, 421-424.
[127] Farge, M., Wavelet transforms and their application to turbulence, Annu. Rev. Fluid Mech., 1992, 24: 395-457
[128] Feldman, M., Method for determination of vibratory system modal parameters using Hilbert transform, Application for patent no: 698985, Israel, 28, July 1991.
[129] Feldman, M., Nonlinear system vibration analysis using Hilbert transform free vibration analysis method, FREEVIB. Mech. Syst. signal processing, 1994a, 8(2): 119-127.
[130] Feldman, M., Nonlinear system vibration analysis using Hilbert transform: II Forced vibration analysis method, FORCEVIB, Mech. Syst. signal processing,
[131] Feldman, M., Nonlinear free vibration identification via the Hillert transform, J. Sound vib., 1997, 208(3): 475-495.
[132] Feldaman, M., Braun, S., Identification of nonlinear system parameters via the instantaneous frequency: application of the Hilbert Transform and Wigner-Ville techniques, Proc, Int Modal analysis Conf., 1993, 13th: 637-642.
[133] Hara. T., Mei, C. C., Frequency downshift in narrowband surface waves under the influence of wind, J. Fluid. Mech., 1995, 2301: 429-477
[134] Long, S. R., Huang, N. E., Tung, C. C., Wu, M.L.,. The Hilbert techniques an alternate approach for non-steady time series analysis, IEEE Geosci,.Remote sensing Soc,. Lett, 1993, 3: 6-11.
[135] Nelkin, M., Universality and scaling in fully developed turbulence, Adu. Phys., 1994, 43: 143-181.
[136] Uchiyama, Y., Kawahara, T., A possible mechanism for frequency down-shift in non-linear wave modulation, Wave Motion, 1994, 20: 99-110.
[137] W. Hang, Z. Shen, N. E. Huang and Y. C. Fung, Engineering analysis of biological variable: An example of blood pressure over 1 day, Proc. Natl. Acad. Sci. USA., April, 1998, 95:4816-4821.
[138] Zhu, X., .Shen, Z.,. Echermam, S. D., Gravity wave characteristics in the middle atmosphere derived from the empirical mode decomposition meted, J. Geophysical Research, 1997, 102(14): 545-561.
[139] Liu, P. L., Is the wind wave frequency spectrum outdated, Ocean engineering, 2000, 27: 557-588.
[140] Balpck, T. E., Swan, C., Extreme waves in shallow and intermediate water depths, Coastal engineering, 1996, 27: 21-46.
[141] Kang, J. H., L. Lon, Y. S., Chan, E. S., Laboratory study of deep water breaking waves, Ocean engineering, 1988.25(8): 657-676.
[142] T. Schlurmann, S. Schimmels, T. Dose, Spectral analysis of Freak waves using wavelet spectra (Morlet) and Hilbert spectra (EMD), Proc. of 2nd German-Chinese Joint seminar on Recent development in coastal engineering, Sept. 1999: 65-80.
[143] Qin. Suren, et al, The research of wavelet transform analysis system for engineering signals, Chinese Journal of Mechanical Engineering, 2000, 13(2): 491-423.
[144] Qin Shuren, et al, Sampling principle and technology in wavelet analysis for signal, Chinese Journal of Mechanical engineering, 1998, 11(4): 259-263.
[145]
[146] 沈远彤，叶碧泉，旭明。非线性摄动方程的小波算法，贵州工业大学学报，1997，10，26（增刊）：89-90
[147] 柯熙政，吴振森。一类高阶非齐次发展方程的小波基数值分析，数学研究与评论， 1999，8，19（3）：587-592。
[148] 向裕民。Duffing方程的振动解，四川轻化工学院学报，1996，9，（3）：42-44
[149] 方见树。参数微扰Duffing 振动子的求解与分析，湖南教育学院学报，2000，4，18（2）：36-40
[150] Kapitania, K.T., Analytical method of controlling chaos in Dufting's oscillator, J. Sound vib., 1993, 163:182-187
[151] Yang. J., Qu. Z. Hu, Dufting equation with two periodic forcings: The phase eftect, Phys. Rev. E., 1996, 53: 4402-4413.
[152] 曹庆杰，张天德，李久平。Dufting方程的静态与动态分岔特征研究，应用数学与力学，1999, 20（12）：1309-1315
[153] 裴韬，鲍征宇。傅立叶变换在地球化学数据处理中的应用，长春科技大学学报，1999，1，29（1）：88-92
[154] 马晶，谭立英，由启文，光学小波滤波理论初探，中国激光，1999，4，26（4）：343-346
[155] 刘曙光，殷明，何铖。子波消噪技术及其应用。西北纺织工学院学报，1999,3, 13（1）:16-20
[156] Mallat, S., Zero-crossing of a wavelet transform, IEEE Trans. on IT, 1991, 37(4): 1019-1033.
[157] Mallat, S., Huang, W. L., Singularity detection and processing with wavelets, IEEE Trans. Information theory, 1992, 38(2): 617-643.
[158] Mallat, S., Zhong, S., Characterization of signals from multi-scale edge, IEEE Trans. Pattern Anal. Mech. Intel, 1992, 14(7): 710-723.
[159] Cetin, A.E., Ansari, R., Signal recovery from wavelet transform maxima, IEEE Trans. signal processing, 1994, 42(7): 1845-1848
[160] 何正友，钱清泉。基于小波变换的信号奇异性指数计算方法及其应用，电力自动化设备，2000.6, 20(3): 12-15
[161] 刘希强，用蕙兰等。基于小波包变换的滤波方法，西北地震学报，1999.9,21(3): 248-253
[162] 韩其玉。地震信号时频表示，石油大学学报，1994，18(3): 26-29
[163] 范中，田立生。利用子波变换检测瞬时信号，电子学报，1996,24(1):78-82
[164]
[165] Donoho, D., Denoising by soft-thresholding, IEEE Trans. on IT, 1995,4(3): 613-627.
[166] 潘旭峰，谢波，李晓雷。小波变换理论及其在机械故障诊断中的应用，振动与冲击，1998，17（1）：14-19
[167] 姚成，吴小培。小波变换与生物医学信号处理，生物学杂志，2000,2,17(1): 25-26
[168] 杨福生，高上凯。生物医学信号处理，高等教育出版社，1989,5
[169] 李光林，昌维雪。基于小波变换的心电信号的分析与处理，浙江大学学报，Jan,1998,32(1): 83-86