局域波时频分析方法的理论研究与应用
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摘要
本文对时频分析技术领域的发展和现状进行了综述。在认真研究与
总结前人成果的基础之上，提出了局域波时频分析方法的一些新理论，
从理论分析、数值计算、应用研究等多方面加以论述，并对设备诊断中
遇到的非平稳非线性数据现象进行了研究。
用局域波分解方法和Hilbert时频谱分析非平稳非线性数据是本文
研究的核心问题。该方法是基于数据时域局部特征的，可把复杂的数据
分解成有限的，通常是少量的几个内蕴模式函数分量。由于分解是基于
信号时域局部特征的，因此它特别适合用来分析非平稳非线性过程。用
Hilbert变换对每个内蕴模式函数变换，可得到瞬时频率和瞬时频率密
度，构成能量一时间一频率三维分布的Hilbert时频分布谱图。它能清
晰地分辨出交叠复杂数据的内蕴模式。该分解方法是自适应的和正交的，
因此非常有效。
在非平稳非线性信号分析的技术和理论方法方面，本文着重从时频
域分析的角度进行了广泛深入的研究，在理论与应用上都作出了几点创
新：提出了广义线性局域波时频表示的新概念和它的数学分解表达式；
提出了符合局域波时频分析理论的极值域均值模式分解算法（EMMD）；
用波形匹配法解决了数据分解过程中的边界效应问题；提出了局域波时
频分析的频率多分辨分析的概念；提出了适合非平稳非线性信号的自适
应滤波器组，从而解释了局域波法的物理意义；分析了信号分解的自适
应广义基的意义，对EMMD分解的完备性与正交性做出了有效解释。以
上创新发展了局域波时频分析方法的理论，为分析非平稳非线性信号提
供了新的工具。
局域波时频分析方法的主要意义在于引入了基于信号局部特征的内
蕴模式函数，这在分析非平稳非线性信号时可得到有意义的瞬时频率，
克服了传统方法用无意义的谐波分量来表示非平稳非线性信号的缺点；
并可得到极高的时频分辨率。通过典型信号的仿真分析和实际应用结果
表明局域波时频分析方法非常有效。
实际应用表明利用而不是忽略信号中的非线性与非平稳特征，我们
可以得出更有物理意义的分析结果。
人连理工大学博上学位论文
An overview of Time-Frequency analysis theory in the pass and now is given in this paper. Based on the study of former achievements and by using the newest Time-Frequency analysis method, I have developed a lot of new theory on Local Wave Time-Frequency analysis method by means of theory analysis, numerical computation and experiment.
A core problem discussed in this paper is using Local Wave decomposition method and Hilbert spectrum to analyse nonlinear and non-stationary data. The key part of the method is the 'empirical mode decomposition ' method with which any complicated data set can be decomposed into a finite and often small number of 'intrinsic mode functions' that admit well-behaved Hilbert transforms. This decomposition method is adaptive, and, therefore, highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it is applicable to nonlinear and non-stationary processes. With the Hilbert transform, the 'intrinsic mode functions' yield instantaneous frequencies as functions of time that give sharp identifications of imbedded structures. The final presentation of the results is an energy-frequency-time distribution, designated as the Hilbert spectrum.
In the theoretical aspect, this paper pays special attention to Time-Frequency domain analysis method, and achieved some improvements both in theory and application. I first point out that the Local Wave Time-Frequency analysis method is a broad sense linear expression and deduce a mathematical decomposition formula of the Local Wave Method; Then I present an Extremum field Mean Mode Decomposition (EMMD) method based on the mean value theorem in a local ; And then I develop wave matching method to solve the end effect of the decomposition. In this paper, I present some new theory. First, I put forward a new concept of multi-frequency distinguish analysis of Local Wave Method, this method highly improves the resolution and precision of decomposition and is especially useful for the extraction of fault characteristics from nonlinear or non-stationary vibration signal. Next I presnt a adaptive filter group concept for filtering nonlinear and non-stationary data, thus explain the physical significance of the Local Wave Method ; Finally , I analyse the sense of adaptive broad sense radix, and explain the completeness and orthogonality
of the Local Wave Method.
The main conceptual innovations are the introduction of intrinsic mode functions'based on local properties of the signal, which makes the instantaneous frequency meaningful; and the introduction of the instantaneous frequencies for complicated data sets, which eliminate the need for squrious harmonics to represent nonlinear and non-stationary signals.
Practical application imply that the utilization of nonlinear and non-stationary character of signals is better then the ignorance of it, we can extract more fault characteristic of vibration signals by that, and obtain more physical explanation.
总结前人成果的基础之上，提出了局域波时频分析方法的一些新理论，
从理论分析、数值计算、应用研究等多方面加以论述，并对设备诊断中
遇到的非平稳非线性数据现象进行了研究。
用局域波分解方法和Hilbert时频谱分析非平稳非线性数据是本文
研究的核心问题。该方法是基于数据时域局部特征的，可把复杂的数据
分解成有限的，通常是少量的几个内蕴模式函数分量。由于分解是基于
信号时域局部特征的，因此它特别适合用来分析非平稳非线性过程。用
Hilbert变换对每个内蕴模式函数变换，可得到瞬时频率和瞬时频率密
度，构成能量一时间一频率三维分布的Hilbert时频分布谱图。它能清
晰地分辨出交叠复杂数据的内蕴模式。该分解方法是自适应的和正交的，
因此非常有效。
在非平稳非线性信号分析的技术和理论方法方面，本文着重从时频
域分析的角度进行了广泛深入的研究，在理论与应用上都作出了几点创
新：提出了广义线性局域波时频表示的新概念和它的数学分解表达式；
提出了符合局域波时频分析理论的极值域均值模式分解算法（EMMD）；
用波形匹配法解决了数据分解过程中的边界效应问题；提出了局域波时
频分析的频率多分辨分析的概念；提出了适合非平稳非线性信号的自适
应滤波器组，从而解释了局域波法的物理意义；分析了信号分解的自适
应广义基的意义，对EMMD分解的完备性与正交性做出了有效解释。以
上创新发展了局域波时频分析方法的理论，为分析非平稳非线性信号提
供了新的工具。
局域波时频分析方法的主要意义在于引入了基于信号局部特征的内
蕴模式函数，这在分析非平稳非线性信号时可得到有意义的瞬时频率，
克服了传统方法用无意义的谐波分量来表示非平稳非线性信号的缺点；
并可得到极高的时频分辨率。通过典型信号的仿真分析和实际应用结果
表明局域波时频分析方法非常有效。
实际应用表明利用而不是忽略信号中的非线性与非平稳特征，我们
可以得出更有物理意义的分析结果。
人连理工大学博上学位论文
An overview of Time-Frequency analysis theory in the pass and now is given in this paper. Based on the study of former achievements and by using the newest Time-Frequency analysis method, I have developed a lot of new theory on Local Wave Time-Frequency analysis method by means of theory analysis, numerical computation and experiment.
A core problem discussed in this paper is using Local Wave decomposition method and Hilbert spectrum to analyse nonlinear and non-stationary data. The key part of the method is the 'empirical mode decomposition ' method with which any complicated data set can be decomposed into a finite and often small number of 'intrinsic mode functions' that admit well-behaved Hilbert transforms. This decomposition method is adaptive, and, therefore, highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it is applicable to nonlinear and non-stationary processes. With the Hilbert transform, the 'intrinsic mode functions' yield instantaneous frequencies as functions of time that give sharp identifications of imbedded structures. The final presentation of the results is an energy-frequency-time distribution, designated as the Hilbert spectrum.
In the theoretical aspect, this paper pays special attention to Time-Frequency domain analysis method, and achieved some improvements both in theory and application. I first point out that the Local Wave Time-Frequency analysis method is a broad sense linear expression and deduce a mathematical decomposition formula of the Local Wave Method; Then I present an Extremum field Mean Mode Decomposition (EMMD) method based on the mean value theorem in a local ; And then I develop wave matching method to solve the end effect of the decomposition. In this paper, I present some new theory. First, I put forward a new concept of multi-frequency distinguish analysis of Local Wave Method, this method highly improves the resolution and precision of decomposition and is especially useful for the extraction of fault characteristics from nonlinear or non-stationary vibration signal. Next I presnt a adaptive filter group concept for filtering nonlinear and non-stationary data, thus explain the physical significance of the Local Wave Method ; Finally , I analyse the sense of adaptive broad sense radix, and explain the completeness and orthogonality
of the Local Wave Method.
The main conceptual innovations are the introduction of intrinsic mode functions'based on local properties of the signal, which makes the instantaneous frequency meaningful; and the introduction of the instantaneous frequencies for complicated data sets, which eliminate the need for squrious harmonics to represent nonlinear and non-stationary signals.
Practical application imply that the utilization of nonlinear and non-stationary character of signals is better then the ignorance of it, we can extract more fault characteristic of vibration signals by that, and obtain more physical explanation.
引文
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[2] R. K. Potter, et al. Visible. Van Nostrand, New York, NY, 1947
[3] J. Ville. Theorie, et Applications de la notion de signal analvtigue. Cables et Transmissions, 1948,2A:61-74
[4] E. P. Wigner. On the quantum correction for thermodynamic equilibrium. Physical Review, 1932,40:749-759
[5] Koenig R, Dram H K, Lacy L Y. The sound spectrograh. J. Acoust. Soc. Amer., 1946,18:19-49
[6] Potter R K, Kopp G,Green H C.Visible speech. New York: Van Nostrand, 1947
[7] J. Morlet, et al. Wave propogation and samplin theory-partz: sampling theory and complex waves. Ceophysics, 1982,47(2) : 222-236
[8] Mann S,Haykin S. Chirplets and wavelets. Novel tme-frequency methods.Electronic Letters, 1992,28: 114-116
[9] C. H. Page. Instantaneous power spectra. J. Apple. Phys., 1952,23: 103-106
[10] L. Cohen. Time-Frequency Analysis: Theory and Applications. Englewood Cliffs, Nj: Prentice-Hall, 1995
[11] C. H. M. Tumer. On the concept of an instantaneous spectrum and its relationship to the autocorrelation function, Jour. Appl. Phys., vol. 1954, 25: 1347-1351
[12] H. Margenau and R. N. Hill. Correlation between measurements in quantum theory. Prog. Theoret. Phys., vol. 26, pp. 722-738, 1961
[13] M. J. Levin. Instantaneos Spectra and Ambiguity Functions. IEEE Trans. Inform. Theory, vol.10, pp. 95-97,1964
[14] A. Rihaczek. IEEE. Trans. Inform. Theory, 1968,14: 369-374
[15] M. H. Ackroyd. Radio Eng. 1970,239: 145-152
[16] L. Cohen. Generalized phase-space distribution functions. Jour. Math. Phys. vol. 7,pp.781-786,1966
[17] S. Qian, D. Chen. Joint Time-Frequency Analysis: Methods and Applications Englewood Cliffs, NJ: Prentice-Hall, 1996
[18] O. Rioul and P. Flandrin. Time-scale energy distributions: a general class extending wavelet transforms. IEEE Trans on Signal Processing, vol.40, pp. 1746-1757, 1992
[19] P. Flandrin, et al. Appl. Comp. Harm. Anal, 1996,3(1) : 10-39
[20] K. Kodera, C.de Villedary and R.gendria A new method for the numerical analysis of nonstationary signals. Physics of the Earth and Planetary Interiors, vol. 12, pp. 142-150,1976
[21] Kodera K, Gendrin R, De Villedary C. Analysis of time-varying signals with small BT values. IEEE Trans. Acoust., Speech, Signal Processing, 1986, 34: 64-76
[22] Auger F, Flandrin P. Improving the readability of time-frequency and time-scale representations by the reassignment method. IEEE Trans. Signal Processing,
1995, 43, 1068-1089
[23] R.G. Baraniuk and D. L. Jones. Signal-dependent time-frequency analysis using a radially Gaus-sian kernel. Signal Processing, vol.32, pp. 263-284, 1993
[24] S. Qian and D.Chen. Decomposition of the Wigner-Ville Distribution. IEEE Trans. on Signal Processing, 1994
[25] S.Mallat, Z.Zhang. IEEE Trans. SP, 1993, 41(12): 3397-3415
[26] 邹红星，周小波，李衍达．Dopplerlet变换在不同特性噪声背景下的信号恢复中的应用．2000，28(9)：1-4
[27] S. Mann, S. Haykin. Vistion Interface'91, 1991, 6:3-7
[28] Mann S, Haykin S. Chirplets and wavelets: Novel time-frequency methods. Electronic Letters, 1992, 28:114-116
[29] Mihovic D, Bracewell R N. Adaptive chirplet representation of signals on time-frequency plane. Electronics Letters, 1991, 27:1159-1161
[30] Mihovilovic D, Bracewell R N. Whistler analysis in the time-frequency plane using chirplets. J. Geophysical Research, 1992, 97: 17, 199-17, 204
[31] Mann S, Haykin S. The chirplet transform: physical considerations. IEEE Trans. Signal Processing, 1995, 43: 2745-2761
[32] Baraniuk R G, Jones D L. Wigner-based formulation of the cherplet transform. IEEE Trans. Signal Processing, 44:3129-3135
[33] 殷勤业，等．电子学报，1997，25(4)：52-58
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