Hilbert-Huang变换及其在信号处理中的应用
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摘要
传统分析和处理非平稳信号的方法主要有：短时Fourier变换、Wigner-Ville分布和小波变换等等。但是这三种方法本质上都是以Fourier变换为基础的，因此具有Fourier分析的缺点，不能从根本上摆脱Fourier分析的局限性。
Hilbert-Huang Transform是一种新的非平稳信号处理技术，它是由N.E.Huang于1998年提出的。该方法由经验模式分解与Hilbert谱分析两部分组成。任意的非平稳信号首先经过EMD方法处理后被分解为一系列具有不同特征尺度的数据序列，每一个序列称为一个内在模式函数，然后对每个IMF分量进行Hilbert谱分析得到相应分量的Hilbert谱，汇总所有Hilbert谱就得到了原信号的谱图。该方法从本质上讲是对非平稳信号进行平稳化处理，将信号中真实存在的不同尺度波动或趋势逐级分解出来，最终用瞬时频率和能量来表征原信号的频率含量，而不是Fourier谱分析中的全局频率和能量，避免了Fourier变换中需要使用许多虚假谐波表达非线性、非平稳信号的不足。
本文在以下几个方面展开研究：
第一，本文深入研究了Hilbert-Huang变换的基本实现原理，并通过仿真验证了此方法的有效性和正确性。HHT能把信号分解成具有一定物理意义的一系列IMF分量，进而通过Hilbert变换求得各IMF分量的瞬时频率和瞬时幅度，得到信号的Hilbert谱。
第二，本文结合幅度归一化方法，提出了基于HHT方法的信号调制方式自动识别方法。在对AM、FM、2ASK、2FSK和2PSK等5种信号进行调制方式识别时，本文首先利用特征参数γ进行幅度调制和非幅度调制的分类，并且对AM和2ASK进行了很好的识别；然后利用HHT变换方法得到特征参数β和α，把FM、2FSK和2PSK从中识别出来，在信噪比10dB以上，都能取得很好的识别效果。
第三，本文在趋势项的去除和提取中，分别运用ARIMA模型、最小二乘法和EMD方法进行仿真分析比较。结果显示EMD方法能自适应地提取出趋势项，与其它两种的方法相比，算法简单，容易实现。本文并把EMD方法提取得到的趋势项应用在股票分析上，可以对股票的走势做出很好的判断。
第四，本文分别应用EMD尺度滤波去噪、EMD阈值去噪和EMD与小波结合去噪等三种方法，对仿真信号进行去噪处理，并且进行了比较分析。然后把这三种去噪方法应用于EP信号中进行去噪处理，效果都比较好，其中EMD阈值去噪效果最好，并且方法简单。
In modern signal processing, non-linear, non-Gaussian and non-stable signals are usually the analysed and processed objects, especially non-stable signals. The conventional ways to analyze and process non-stable signals are: short time Fourier Tansform, Wigner-Ville distribution, Wavelet Transform and so on. But the above three algorithms are all based on Fourier Transform, so they all have the shortcomings of Fourier Analysis and can not get rid of the localization of it.
Hilbert-Huang Transform is a new non-stable signal processing technology, proposed by N.E.Huang in 1998. It is composed of Empirical Mode Decomposition (referred to as EMD) and Hilbert Spectral Analysis (referred to as HSA). After EMD processing, any non-stable signal will be decomposed to a series of data sequences with different eigenscales. Each sequence is called an Intrinsic Mode Function (referred to as IMF). And then the energy distribution plot of the original non-stable signal can be found by summing all the Hilbert spectrums of each IMF. In essence, this algorithm makes the non-stable signals become stable and decomposes the fluctuations and tendencies of different scales by degrees and at last describes the frequency components with instantaneous frequency and energy instead of the total frequency and energy in Fourier Spectral Analysis. In this case, the shortcoming of using many fake harmonic waves to describe non-linear and non-stable signals in Fourier Transform can be avoided.
This paper researches in the following parts:
First, this paper deeply researches on the basic realization principles of Hilbert-Huang transform and confirms its validity by simulations. HHT can decompose the signal to a series of useful, physical meaningful IMF components, and calculate the instantaneous frequencies and amplitudes of IMF components by Hilbert transform to get Hilbert spectrum of the signal.
Second, combing with amplitude normalization method, this paper proposes the automatic recognition algorithm for modulations signals based on HHT algorithm. When recognizing AM、FM、2ASK、2FSK and 2PSK five kinds of signals, this paper first makes use of character parameterγto classify amplitude modulation and non-amplitude modulations, and it can recognize AM and 2ASK well. Then, this paper takes advantage of HHT to get character parametersβandα, so that FM、2FSK and 2PSK can be recognized. When signal-noise-ratio (referred to as SNR) is beyond 10dB, the results are well.
Third, this paper separately uses ARIMA model, LS and EMD to the elimination and extraction of tendency parts and makes comparisons. The results show that compared with the other two algorithms, EMD algorithm can adaptively extract tendency parts, and is easy to realize. This paper applies the tendency parts extracted by EMD algorithm to stock analysis and can judge the tendency of stock well.
Fourth, this paper separately uses EMD scale filter algorithm, EMD threshold algorithm and EMD with wavelet algorithm to remove the noise in the simulation signals and makes comparisons. And then these three algorithms ate applied to remove the noise in EP signals. The results are good and EMD is the simplest and best one.
Hilbert-Huang Transform是一种新的非平稳信号处理技术，它是由N.E.Huang于1998年提出的。该方法由经验模式分解与Hilbert谱分析两部分组成。任意的非平稳信号首先经过EMD方法处理后被分解为一系列具有不同特征尺度的数据序列，每一个序列称为一个内在模式函数，然后对每个IMF分量进行Hilbert谱分析得到相应分量的Hilbert谱，汇总所有Hilbert谱就得到了原信号的谱图。该方法从本质上讲是对非平稳信号进行平稳化处理，将信号中真实存在的不同尺度波动或趋势逐级分解出来，最终用瞬时频率和能量来表征原信号的频率含量，而不是Fourier谱分析中的全局频率和能量，避免了Fourier变换中需要使用许多虚假谐波表达非线性、非平稳信号的不足。
本文在以下几个方面展开研究：
第一，本文深入研究了Hilbert-Huang变换的基本实现原理，并通过仿真验证了此方法的有效性和正确性。HHT能把信号分解成具有一定物理意义的一系列IMF分量，进而通过Hilbert变换求得各IMF分量的瞬时频率和瞬时幅度，得到信号的Hilbert谱。
第二，本文结合幅度归一化方法，提出了基于HHT方法的信号调制方式自动识别方法。在对AM、FM、2ASK、2FSK和2PSK等5种信号进行调制方式识别时，本文首先利用特征参数γ进行幅度调制和非幅度调制的分类，并且对AM和2ASK进行了很好的识别；然后利用HHT变换方法得到特征参数β和α，把FM、2FSK和2PSK从中识别出来，在信噪比10dB以上，都能取得很好的识别效果。
第三，本文在趋势项的去除和提取中，分别运用ARIMA模型、最小二乘法和EMD方法进行仿真分析比较。结果显示EMD方法能自适应地提取出趋势项，与其它两种的方法相比，算法简单，容易实现。本文并把EMD方法提取得到的趋势项应用在股票分析上，可以对股票的走势做出很好的判断。
第四，本文分别应用EMD尺度滤波去噪、EMD阈值去噪和EMD与小波结合去噪等三种方法，对仿真信号进行去噪处理，并且进行了比较分析。然后把这三种去噪方法应用于EP信号中进行去噪处理，效果都比较好，其中EMD阈值去噪效果最好，并且方法简单。
In modern signal processing, non-linear, non-Gaussian and non-stable signals are usually the analysed and processed objects, especially non-stable signals. The conventional ways to analyze and process non-stable signals are: short time Fourier Tansform, Wigner-Ville distribution, Wavelet Transform and so on. But the above three algorithms are all based on Fourier Transform, so they all have the shortcomings of Fourier Analysis and can not get rid of the localization of it.
Hilbert-Huang Transform is a new non-stable signal processing technology, proposed by N.E.Huang in 1998. It is composed of Empirical Mode Decomposition (referred to as EMD) and Hilbert Spectral Analysis (referred to as HSA). After EMD processing, any non-stable signal will be decomposed to a series of data sequences with different eigenscales. Each sequence is called an Intrinsic Mode Function (referred to as IMF). And then the energy distribution plot of the original non-stable signal can be found by summing all the Hilbert spectrums of each IMF. In essence, this algorithm makes the non-stable signals become stable and decomposes the fluctuations and tendencies of different scales by degrees and at last describes the frequency components with instantaneous frequency and energy instead of the total frequency and energy in Fourier Spectral Analysis. In this case, the shortcoming of using many fake harmonic waves to describe non-linear and non-stable signals in Fourier Transform can be avoided.
This paper researches in the following parts:
First, this paper deeply researches on the basic realization principles of Hilbert-Huang transform and confirms its validity by simulations. HHT can decompose the signal to a series of useful, physical meaningful IMF components, and calculate the instantaneous frequencies and amplitudes of IMF components by Hilbert transform to get Hilbert spectrum of the signal.
Second, combing with amplitude normalization method, this paper proposes the automatic recognition algorithm for modulations signals based on HHT algorithm. When recognizing AM、FM、2ASK、2FSK and 2PSK five kinds of signals, this paper first makes use of character parameterγto classify amplitude modulation and non-amplitude modulations, and it can recognize AM and 2ASK well. Then, this paper takes advantage of HHT to get character parametersβandα, so that FM、2FSK and 2PSK can be recognized. When signal-noise-ratio (referred to as SNR) is beyond 10dB, the results are well.
Third, this paper separately uses ARIMA model, LS and EMD to the elimination and extraction of tendency parts and makes comparisons. The results show that compared with the other two algorithms, EMD algorithm can adaptively extract tendency parts, and is easy to realize. This paper applies the tendency parts extracted by EMD algorithm to stock analysis and can judge the tendency of stock well.
Fourth, this paper separately uses EMD scale filter algorithm, EMD threshold algorithm and EMD with wavelet algorithm to remove the noise in the simulation signals and makes comparisons. And then these three algorithms ate applied to remove the noise in EP signals. The results are good and EMD is the simplest and best one.
引文
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[2] L．科恩．时-频分析：理论与应用．西安：西安交通大学出版社，1998．
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[9] Antoni Homs-Corbera, Jose Antonio, et al. Time-frequency detection andanalysis of wheezes during forced exhalation. IEEE Transactions on Biomedical Engineering. 2004, 51(1): 182-186.
[10] Cedric Fevotte, Christian Doncarli. Two contributions to blind sourceseparation using time-frequency distributions. IEEE Signal ProcessingLetters. 2004, 11(3):386-389.
[11] B. Barkat, B. Boashash. Reply to "Comments on 'A High-Resolution Quadratic Time-Frequency Distribution for Multicomponent Signals Analysis'". IEEE Transactions on Signal Processing. 2004, 52(1):290-291.
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[26] P. Gloerson, N.E. Huang. Comparison of Interannual Intrinsic Mode in Hemispheric Sea Ice Covers and Other Geophysical Parameters. IEEE Trans. on Geosci. and Remote Sens. 2003, 41 (5): 1062-1074.
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[29] E. C. Tichmarsch. Introduction to the theory of Fourier Integrals. Oxford: Oxford University Press, 1948.
[30] D. Gabor. Theory of communication. Proc. IEEE, 1946, 93:429-457.
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[2] L．科恩．时-频分析：理论与应用．西安：西安交通大学出版社，1998．
[3] S. Qian, D. Chen. Joint Time-Frequency Analysis:Methods and Applications. Englewood Cliffs, NJ:Prentice Hall, 1996.
[4] S.Mallat. A theory for multi-resolution signal decomposition the wavelet representation. IEEE Trans.On Pattern Analysis and Machine Intell.1989, 11(7):674-693.
[5] 王宏禹．非平稳随机信号分析与处理．北京：国防工业出版社，1999．
[6] J. Patrick, Lough Lin, Leon Cohen. The uncertainty principle: Global, Local, orBoth. IEEE Transactions Signal Processing. 2004, 52(5):1218-1227.
[7] J. Ville. Theorie et application de la notion de signal analytique. Cables et Transmission, Pairs. 1948,21(1):61-74.
[8] Masoud Karimi-Ghartemani, Alirera K Ziarani A nonlinear time-frequency analysis method. IEEE Transactions on Signal Processing. 2004, 52(6):1585-1595.
[9] Antoni Homs-Corbera, Jose Antonio, et al. Time-frequency detection andanalysis of wheezes during forced exhalation. IEEE Transactions on Biomedical Engineering. 2004, 51(1): 182-186.
[10] Cedric Fevotte, Christian Doncarli. Two contributions to blind sourceseparation using time-frequency distributions. IEEE Signal ProcessingLetters. 2004, 11(3):386-389.
[11] B. Barkat, B. Boashash. Reply to "Comments on 'A High-Resolution Quadratic Time-Frequency Distribution for Multicomponent Signals Analysis'". IEEE Transactions on Signal Processing. 2004, 52(1):290-291.
[12] J. Morlet, G. Arens, E. Fourgeau et al. Wave Propagation and Sampling Theory-part Ⅱ: Sampling Theory and Complex Waves. Geophysics, 1982, 47(2):222-236.
[13] 秦前清，杨宗凯．实用小波分析．西安：西安电子科技大学出版社，1994．
[14] N.E. Huang, Z. Shen, S.R. Long et al. The empirical mode decompositionand the Hilbert spectrum for nonlinear and non-stationary time seriesanalysis. Proc Roy Soc, London A. 1998, 454:903-995.
[15] N. E. Huang, Z. Shen, S. R. Long. A New View of Nonlinear Water Waves: The Hilbert Spectrum. Ann Rev Fluid Mech. 1999, 31:417-457.
[16] N. E. Huang. Computer Implicated Empirical Mode Decomposition Method, Apparatus, and Article of Manufacture. USA, Provisional Application. 1999, 5:983-988.
[17] W. Huang, Z. Shen, N.E. Huang et al. Engineering Analysis of Biological Variables: An Example of Blood Pressure over one Day. Proceeding of the National Academy of Science, USA. 1998, 95:4816-4821.
[18] H. Liang, Z. Lin, R.W. McCallum. Artifact Reduction in Electrogastrogram Based on the Empirical Mode Decomposition Method. Med. Biol. Eng. Comput. 2000, 38(1):35-41.
[19] H. Liang, Z. Lin. Multiresolution signal decomposition and its application to electrogastric signals. Recent research developments in biomedical engineering, Kerala, India. 2002,21: 15-31.
[20] R. Balocchi, D. Menicucci, M. Varanini. Empirical mode decomposition to approach the problem of detecting sources from a reduced number of mixtures. Proceedings of the 25 Annual International Conference of the IEEE EMBS, Mexico. 2003, 12: 17-21.
[21] H. L. Liang, S. L. Bressler, R. Desimone et al. Empirical Mode Decomposition: A Method for Analyzing Neural Data. Neurocomputing. 2005, 65-66: 801-807.
[22] H.L. Liang, Q.H. Lin, J.D.Z. Chen. Application of the Empirical Mode Decomposition to the Analysis of Esophageal Manometric Data in Gastroesophageal Reflux Disease. IEEE Transactions on Biomedical Engineering. 2005, 52(10):1692-1701.
[23] H. L. Liang, S.L. Bressler, E.A. Buffalo et al. Empirical mode decomposition of local field potentials from macaque V4 in visual spatial attention. Biological Cybernetics. 2005, 92 (6): 380-392.
[24] N. E. Huang, Z. Shen, S.R. Long et al. The empirical mode decomposition and the nilbert spectrum for nonlinear and non-stationary time series analysis Proc. of the Royal Society, London, 1998, 454(A):903-995.
[25] M.L. Wu, S. Schubert, N.E. Huang. The Development of the South Asian Summer Monsoon and the Intraseasonal Oscillatio. Journal of Climate. 1999, 12:2054-2075.
[26] P. Gloerson, N.E. Huang. Comparison of Interannual Intrinsic Mode in Hemispheric Sea Ice Covers and Other Geophysical Parameters. IEEE Trans. on Geosci. and Remote Sens. 2003, 41 (5): 1062-1074.
[27] G. Leisk, N. Hsu, N.E. Huang. Application of the Hilbert-Huang Transform to machine tool condition/health monitoring. AIP conference proceeding, Brunswick. 2002, 615(1):1711-1718.
[28] B. Boashash. Estimating and interpreting the instantaneous frequency of a signal, Part Ⅰ Fundamentals. Proc. IEEE, 1992, 80(4):520-538.
[29] E. C. Tichmarsch. Introduction to the theory of Fourier Integrals. Oxford: Oxford University Press, 1948.
[30] D. Gabor. Theory of communication. Proc. IEEE, 1946, 93:429-457.
[31] E.T. Copson. Asymptotic expansions. Cambridge:Cambridge University Press, 1967.
[32] 何鸣，刘光斌等．一种基于模态分解的时频分析方法．现代电子技术．2003，18：15-17．
[33] 常鸣，袁慎芳．一种新型数字信号处理技术的研究．计算机测量与控制．2005，13(4)：356-362．
[34] K.T. Coughlin, K.K. Tung. 11-Year solar cycle in the stratosphere extracted by the empirical mode decomposition method. Advances in Space Research. 2004, 34:323-329.
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