基于固有模态分解的时频分析技术研究与应用
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摘要
非平稳信号是日常生活和科学研究中经常观察到的现象,这些信号往往持续时间有限,存在产生和消亡。传统的信号频域分析法一傅立叶分析只适用于分析信号组成分量的频率不随时间变化的平稳信号,因此人们针对非平稳信号的分析引入了在时频二维平面上表征信号的新方法一时频分析方法。随后,各种时频分析技术,例如STFT、小波分析等,也都得到了快速地发展。其中,上世纪末,Huang等人首次提出的一种新的信号分析理论—希尔伯特-黄变换(HilbertHuang Transform)被公认为是对时频分析技术的一个突破。分析和改进HHT变换是本论文研究的核心问题。
本文在深入分析了HHT变换的基础上,指出了HHT目前存在的一些缺点:在EMD分解的过程中,细小特征尺度可能会丢失;对与存在跳跃性变化的信号的分解存在模态混叠现象;以及HHT变换不能实现信号的即时分析,即不具备边提供数据边进行时频分析的能力。针对以上缺点,本文提出了如下改进1)提出了Cur-EMD算法,在EMD分解中采用曲率极值点代替幅度极值点作为信号的特征时间尺度,克服了原始EMD中细小特征尺度分解不完全的缺点,提高了EMD分解和HHT分析的精度;2)对EMD分解中筛选过程进行了改进,提出了信号的特征时间尺度平稳度的概念和检测信号突变点的方法,当信号的时间尺度存在着跳跃性变化时,较好的解决了原方法中严重的模态混叠现象;3)传统上,因为HHT变换涉及到整个信号的反复样条插值,因此HHT分析必须针对一个完整的信号。本文采用阿克玛插值法替代三次样条插值,并提出了一种“分而治之”的递归模态分解法,探讨了将HHT应用于即时信号分析的可能性,当信号在连续输入的时,经过一个有限的延迟,该算法就可以给出当前的时频分析结果。本文通过编程(Matlab和Java)实现以上各算法,并完成了一系列数值试验,直观的给出了不同时频分析方法得到的结果。数值试验的结果证明改进后的算法具有分析精度更高,有效消除模态混叠等优点,证实了改进的有效性。最后,本文介绍时频分析技术在作者参与的一个实际项目中的应用。在这个项目中,时频分析技术主要被用于信号去噪以及图像纹理分析两个方面。
Non-stationary signal can be often observed in our daily life and scientific research.For those signals,they are generated suddenly and wither away in a limited time.However,traditional frequency-analysis like Fourier transform is applicable only when the components of the signal last forever and have frequencies that are constant.So,the time-frequency analysis has been introduced as a new method to analysis the non-stationary signal.The new method represents signals in a two-dimensional plane of time and frequency.Different kind of time-frequency technologies including Sort Time Fourier Transform and Wavelet Analysis has been well developed since then.Among them,the new proposed Hilbert-Huang Transform (HHT) is commonly considered as a breakthrough to time-frequency analysis.This thesis will focus on discussions and improvements base on it.
Base on the summary and a relatively thorough analysis of HHT,The thesis point out its deficiencies:There is possibility of losing tiny components in the process of Empirical Mode Decomposition;Aliasing phenomenon arises if the characteristic scales of signal suddenly changed;And,HHT cannot analyze the real-time signal. That means it is unable to analyze the signal the same time it being provided. Against those deficiencies,several improvements have been proposed here:1) the thesis proposes the Cur-EMD algorithm which adopts curvature extremes instead of amplitude extremes as the characteristic scales in the decomposition.Comparing to the traditional EMD algorithm,Cur-EMD boasts a higher precision.2) The thesis defines the conception of Mutation Point and the way to detect it.Thanks to this conception,an improved decomposition method of HHT has been proposed in this thesis.The new method can effectively avoid the aliasing phenomenon caused by the sudden change of the characteristic scales.3) Traditionally,because of HHT involves the repeated spline interpolate of an entire signal,the signal that HHT analyzes must be integrated.This thesis gives a discussion of a new improved decomposition base on the notion of "divide and rule".In additional,the new method adopts Akima interpolate instead of spline interpolate.All those together make it possible for using HHT to analyze real-time signal.That is the algorithm can give the current time-frequency analysis result of a consecutive input signal after a limited delay.In this thesis we implement the algorithms mentioned above by programming(Matlab and Java),and make a series of numerical experimentation.Based on the clearly presented results comes after different Time-Frequency analysis methods,this thesis gives an analysis and proves the efficiency of the improved algorithm.Finally,this thesis introduces an application of the proposed Time-Frequency analysis technology methods in a project that the author participated in.In this project,the Time-Frequency analysis technology was mainly used in aspects of signal de-noising and texture analysis for images.
本文在深入分析了HHT变换的基础上,指出了HHT目前存在的一些缺点:在EMD分解的过程中,细小特征尺度可能会丢失;对与存在跳跃性变化的信号的分解存在模态混叠现象;以及HHT变换不能实现信号的即时分析,即不具备边提供数据边进行时频分析的能力。针对以上缺点,本文提出了如下改进1)提出了Cur-EMD算法,在EMD分解中采用曲率极值点代替幅度极值点作为信号的特征时间尺度,克服了原始EMD中细小特征尺度分解不完全的缺点,提高了EMD分解和HHT分析的精度;2)对EMD分解中筛选过程进行了改进,提出了信号的特征时间尺度平稳度的概念和检测信号突变点的方法,当信号的时间尺度存在着跳跃性变化时,较好的解决了原方法中严重的模态混叠现象;3)传统上,因为HHT变换涉及到整个信号的反复样条插值,因此HHT分析必须针对一个完整的信号。本文采用阿克玛插值法替代三次样条插值,并提出了一种“分而治之”的递归模态分解法,探讨了将HHT应用于即时信号分析的可能性,当信号在连续输入的时,经过一个有限的延迟,该算法就可以给出当前的时频分析结果。本文通过编程(Matlab和Java)实现以上各算法,并完成了一系列数值试验,直观的给出了不同时频分析方法得到的结果。数值试验的结果证明改进后的算法具有分析精度更高,有效消除模态混叠等优点,证实了改进的有效性。最后,本文介绍时频分析技术在作者参与的一个实际项目中的应用。在这个项目中,时频分析技术主要被用于信号去噪以及图像纹理分析两个方面。
Non-stationary signal can be often observed in our daily life and scientific research.For those signals,they are generated suddenly and wither away in a limited time.However,traditional frequency-analysis like Fourier transform is applicable only when the components of the signal last forever and have frequencies that are constant.So,the time-frequency analysis has been introduced as a new method to analysis the non-stationary signal.The new method represents signals in a two-dimensional plane of time and frequency.Different kind of time-frequency technologies including Sort Time Fourier Transform and Wavelet Analysis has been well developed since then.Among them,the new proposed Hilbert-Huang Transform (HHT) is commonly considered as a breakthrough to time-frequency analysis.This thesis will focus on discussions and improvements base on it.
Base on the summary and a relatively thorough analysis of HHT,The thesis point out its deficiencies:There is possibility of losing tiny components in the process of Empirical Mode Decomposition;Aliasing phenomenon arises if the characteristic scales of signal suddenly changed;And,HHT cannot analyze the real-time signal. That means it is unable to analyze the signal the same time it being provided. Against those deficiencies,several improvements have been proposed here:1) the thesis proposes the Cur-EMD algorithm which adopts curvature extremes instead of amplitude extremes as the characteristic scales in the decomposition.Comparing to the traditional EMD algorithm,Cur-EMD boasts a higher precision.2) The thesis defines the conception of Mutation Point and the way to detect it.Thanks to this conception,an improved decomposition method of HHT has been proposed in this thesis.The new method can effectively avoid the aliasing phenomenon caused by the sudden change of the characteristic scales.3) Traditionally,because of HHT involves the repeated spline interpolate of an entire signal,the signal that HHT analyzes must be integrated.This thesis gives a discussion of a new improved decomposition base on the notion of "divide and rule".In additional,the new method adopts Akima interpolate instead of spline interpolate.All those together make it possible for using HHT to analyze real-time signal.That is the algorithm can give the current time-frequency analysis result of a consecutive input signal after a limited delay.In this thesis we implement the algorithms mentioned above by programming(Matlab and Java),and make a series of numerical experimentation.Based on the clearly presented results comes after different Time-Frequency analysis methods,this thesis gives an analysis and proves the efficiency of the improved algorithm.Finally,this thesis introduces an application of the proposed Time-Frequency analysis technology methods in a project that the author participated in.In this project,the Time-Frequency analysis technology was mainly used in aspects of signal de-noising and texture analysis for images.
引文
[1]张贤达.非平稳信号分析与处理.北京:国防:工业出版社,1998.
[2]邹红星,周小波,李衍达.时频分析:回溯与前瞻.电子学报,2000,28(9):78 84.
[3]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
[4]C.D.Champency,A Handbook of Fourier Theorems,Cambridge university press,1987
[5]Shie Qian and Dapang Chen,Joint Time-Frequency Analysis,IEEE signal Processing Magazine,1999,16(2):52-67
[6]邹红星.参数化时频信号表示方法研究[博士学位论文].北京:清华大学,2002
[7]张海勇.一种新的非平稳信号分析方法--局域波分析[J].电子与信息学报,2003,25(10):1327-1332.
[8]刘庆云,李志舜,刘朝晖.时频分析技术及研究现状,计算机工程,2004,30(1):171-173
[9]R.K.Potter,et al.Visible.Van Nostrand,New York,NY,1947
[10]Mallat S,A theory for multi-resolution signal decomposition the wavelet representation.IEEE Trans on Pattern Analysis and Machine inell,Vol.11,No.7,July1989
[11]Danbechies L,Orthonormal Bases of Compactly Supported Wavelets,Comm.In Pure and Applied Math,VOl.41,No.7,Sept.1990
[12]Coifman R.R and Wickerhanser M.V.,Entroy-based Algorithms for Best Basis Selection IEEE Tran.On Information Theory,Vol.38.NO.2,Mar,1992
[13]崔锦泰著,程正兴译,小波分析导论,西安:西安电子科技大学出版社,1994
[14]Chui C.K.,Wang J.Z.,A Cardinal spline Approach to Wavelet,proc.Amer.Math.Soc.,1991(113):785-793.
[15]Special issue on adaptive wavelet transforms,Opt.Eng.,1994,33(7).
[16]Youming Zhong,Shuren Qin and Baoping Tang,A Virtual Wavelet Analyzer Based on the Direct Algorithm,The 5~(th) ISMTII(international conference),Kailuo,Sept.2001:325-340
[17]钟佑明,汤宝平,秦树人.基于直接算法的虚拟式小波变换信号分析仪重庆大学学报(自然科学版)2002 25(3)10-14
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[21]Boashash B.Estimating and Interpreting the Instantaneous Frequency of a Signal--Part 1:Fundamentals[J].Proc.IEEE,1992,80(4):520-538.
[22]L.Mandel,Interpretation of instantaneous frequency,Amer.J.Phys.,1974,42(4):840-846.
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[25]Cauchy-Riemann equations.Wikipedia.http://en.wikipedia.org/wiki/Cauchy-Riemann_equations.27 April 2009
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[27]陈忠,郑时雄.EMD信号分析方法边缘效应的分析[J].数据采集与处理,2003,18(1):114-118.
[28]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
[29]R.S.Long,et al,The Hilbert Techniques:an Alternate Approach for Non-steady Time Series Analysis,IEEE Geoscience Remote Sensing Soc.Lett.,1995 3:6-11
[30]胡聿贤,张郁山,梁建文.应用HHT方法从竖向地震动记录识别场地液化的物理过程[J].地震工程与工程振动,2004,24(3):1-11.
[31]公茂盛,谢礼立.HHT方法在地震工程中的应用之初步探讨[J].世界地震工程,2003,19(3):39-43.
[32]Li,Cheng-Ping.A seismic source inversion study and non-stationary nonlinear seismic signal processing[D].UNIVERSITY OF SOUTHERN CALIFORNIA.1999.
[33]Huang,Norden E.,Chern,Ching C.,Huang,Kang,et al.A New Spectral Representation of Earthquake Data:Hilbert Spectral Analysis of Station[J].Bulletin of the Seismological Society of America,2001,91(5):1310-1338.
[34]Zhang Ray Ruichong,Ma,Shuo,Safak Erdal,et al.Hilbert-Huang Transform Analysis of Dynamic and Earthquake Motion Recordings[J].Journal of engineering mechanics,2003,129(8):861-876.
[35]Duffy,Dean G.The Application of Hilbert-Huang Transforms to Meteorological Dataset[J].Journal of Atmospheric and Oceanic Technology,2004,21(4):599-611
[36]Echeverria J C,Crowe J A,Woolfson M S.Application of empirical mode decomposition to heart rate variability analysis[J].Medical & biological engineering & computing,2001,39(4):471-480.
[37]全海燕,王威廉.Hilbert-Huang变换及其在心音信号分析中的应用[A].第十一届全国信号处理学术年会论文集.北京:中国电子学会,2003.8.
[38]赵治栋.基于舒张期心音信号分析与特征提取的冠心病无损诊断研究[D].杭州:浙江大学,2004.
[39]杨志华,齐东旭,江力,等.一种基于EMD分解的睡眠脑电图梭形波自动识别方法[A].第一届中国情感计算及智能交互学术会议论文集.北京:中国白动化学会,2003.12.
[40]DraZin,P.G.,1992.Nonlinear Systems,Cambridge University Press,Cambridge 12
[41]N.E.huang,Z.Shen,S.R.Long,A new View of Nonlinear Water Waves:the Hilbert Spectrum,J.Annu.Rev.Fluid Mech.,1999,31:417-457
[42]S.Q.Rice,Mathematical Analysis of random Noise.Ⅲ.Statistical Properties of Random Noise Currents,J.Bell Sys.Tech.,1945,24:46-108.
[43]S.Q.Rice,Mathematical Analysis of Random Noise IV.Noise through Nonlinear Devices,J.Bell Sys.Tech.,1945,24:109-156.
[44]B.C.Lovell,R.C.Williamson,B.Boashash,The relationship between instantaneous frequency and time-frequency representation,IEEE Trans Sig.Proc.,1993,41(1):230-254.
[45]N.E.Huang,S.R.Long.The local properties of ocean wave,by the phase-time method,Geophys.Res.Lett.,1992,19:685-688.
[46]Farge,M.,Wavelet transforms and their application to turbulence,Annu.Rev.Fluid Mech.,1992,24:395-457
[47]谭善文,多分辨希尔伯特黄(Hilbert Huang)变换方法的研究,重庆大学博士论文,2005[48]Mathias Johansson,The Hilbert transform,Master Thesis,V(A|¨)axj(A|¨)o University,2003
[49]Hahn Stefan L.,Hilbert transforms in signal processing,Artech House,Inc.,Boston,1996.
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[2]邹红星,周小波,李衍达.时频分析:回溯与前瞻.电子学报,2000,28(9):78 84.
[3]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
[4]C.D.Champency,A Handbook of Fourier Theorems,Cambridge university press,1987
[5]Shie Qian and Dapang Chen,Joint Time-Frequency Analysis,IEEE signal Processing Magazine,1999,16(2):52-67
[6]邹红星.参数化时频信号表示方法研究[博士学位论文].北京:清华大学,2002
[7]张海勇.一种新的非平稳信号分析方法--局域波分析[J].电子与信息学报,2003,25(10):1327-1332.
[8]刘庆云,李志舜,刘朝晖.时频分析技术及研究现状,计算机工程,2004,30(1):171-173
[9]R.K.Potter,et al.Visible.Van Nostrand,New York,NY,1947
[10]Mallat S,A theory for multi-resolution signal decomposition the wavelet representation.IEEE Trans on Pattern Analysis and Machine inell,Vol.11,No.7,July1989
[11]Danbechies L,Orthonormal Bases of Compactly Supported Wavelets,Comm.In Pure and Applied Math,VOl.41,No.7,Sept.1990
[12]Coifman R.R and Wickerhanser M.V.,Entroy-based Algorithms for Best Basis Selection IEEE Tran.On Information Theory,Vol.38.NO.2,Mar,1992
[13]崔锦泰著,程正兴译,小波分析导论,西安:西安电子科技大学出版社,1994
[14]Chui C.K.,Wang J.Z.,A Cardinal spline Approach to Wavelet,proc.Amer.Math.Soc.,1991(113):785-793.
[15]Special issue on adaptive wavelet transforms,Opt.Eng.,1994,33(7).
[16]Youming Zhong,Shuren Qin and Baoping Tang,A Virtual Wavelet Analyzer Based on the Direct Algorithm,The 5~(th) ISMTII(international conference),Kailuo,Sept.2001:325-340
[17]钟佑明,汤宝平,秦树人.基于直接算法的虚拟式小波变换信号分析仪重庆大学学报(自然科学版)2002 25(3)10-14
[18]Morlet wavelet.Wikipedia.http://en.wikipedia.org/wiki/Morlet_wavelet.2 April 2009
[19]成礼智,王红霞,等.小波理论与应用[M].北京:科学出版社,2004.
[20]Special issue on Wavelet transforms and multi-resolution analysis,IEEE Trans.Inform.Theory,1992:38(2).
[21]Boashash B.Estimating and Interpreting the Instantaneous Frequency of a Signal--Part 1:Fundamentals[J].Proc.IEEE,1992,80(4):520-538.
[22]L.Mandel,Interpretation of instantaneous frequency,Amer.J.Phys.,1974,42(4):840-846.
[23]S.O.Rice,Envelope of narrow-band signal,Proc.IEEE,July 1982,70(4):692-699.
[24]L.Cohen,C.Lee,Instantaneous frequency,its standard deviation and multi-component signals,Proc.SPIE,1988,975(1)186-208.
[25]Cauchy-Riemann equations.Wikipedia.http://en.wikipedia.org/wiki/Cauchy-Riemann_equations.27 April 2009
[26]罗奇峰,石春香.Hilbert-Huang变换理论及其计算中的问题[J].同济大学学报(自然科学版),2003,31(6):637-640.
[27]陈忠,郑时雄.EMD信号分析方法边缘效应的分析[J].数据采集与处理,2003,18(1):114-118.
[28]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
[29]R.S.Long,et al,The Hilbert Techniques:an Alternate Approach for Non-steady Time Series Analysis,IEEE Geoscience Remote Sensing Soc.Lett.,1995 3:6-11
[30]胡聿贤,张郁山,梁建文.应用HHT方法从竖向地震动记录识别场地液化的物理过程[J].地震工程与工程振动,2004,24(3):1-11.
[31]公茂盛,谢礼立.HHT方法在地震工程中的应用之初步探讨[J].世界地震工程,2003,19(3):39-43.
[32]Li,Cheng-Ping.A seismic source inversion study and non-stationary nonlinear seismic signal processing[D].UNIVERSITY OF SOUTHERN CALIFORNIA.1999.
[33]Huang,Norden E.,Chern,Ching C.,Huang,Kang,et al.A New Spectral Representation of Earthquake Data:Hilbert Spectral Analysis of Station[J].Bulletin of the Seismological Society of America,2001,91(5):1310-1338.
[34]Zhang Ray Ruichong,Ma,Shuo,Safak Erdal,et al.Hilbert-Huang Transform Analysis of Dynamic and Earthquake Motion Recordings[J].Journal of engineering mechanics,2003,129(8):861-876.
[35]Duffy,Dean G.The Application of Hilbert-Huang Transforms to Meteorological Dataset[J].Journal of Atmospheric and Oceanic Technology,2004,21(4):599-611
[36]Echeverria J C,Crowe J A,Woolfson M S.Application of empirical mode decomposition to heart rate variability analysis[J].Medical & biological engineering & computing,2001,39(4):471-480.
[37]全海燕,王威廉.Hilbert-Huang变换及其在心音信号分析中的应用[A].第十一届全国信号处理学术年会论文集.北京:中国电子学会,2003.8.
[38]赵治栋.基于舒张期心音信号分析与特征提取的冠心病无损诊断研究[D].杭州:浙江大学,2004.
[39]杨志华,齐东旭,江力,等.一种基于EMD分解的睡眠脑电图梭形波自动识别方法[A].第一届中国情感计算及智能交互学术会议论文集.北京:中国白动化学会,2003.12.
[40]DraZin,P.G.,1992.Nonlinear Systems,Cambridge University Press,Cambridge 12
[41]N.E.huang,Z.Shen,S.R.Long,A new View of Nonlinear Water Waves:the Hilbert Spectrum,J.Annu.Rev.Fluid Mech.,1999,31:417-457
[42]S.Q.Rice,Mathematical Analysis of random Noise.Ⅲ.Statistical Properties of Random Noise Currents,J.Bell Sys.Tech.,1945,24:46-108.
[43]S.Q.Rice,Mathematical Analysis of Random Noise IV.Noise through Nonlinear Devices,J.Bell Sys.Tech.,1945,24:109-156.
[44]B.C.Lovell,R.C.Williamson,B.Boashash,The relationship between instantaneous frequency and time-frequency representation,IEEE Trans Sig.Proc.,1993,41(1):230-254.
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