希尔伯特—黄变换局瞬信号分析理论的研究
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摘要
希尔伯特-黄变换（HHT）是上世纪末Huang等人首次提出的一种新的信号分析理论。它的主要创新是固有模态（IMF）概念的提出和经验筛法（EMD）的引入。通过EMD，将信号分解成IMF（一般为有限数目）的和，对每个IMF进行Hilbert变换就可以获得有意义的瞬时频率，从而给出频率变化的精确表达。信号最终可以被表示为时频平面上的能量分布，称为Hilbert谱。进而还可以得到信号的边际谱。HHT是基于信号局部特征的和自适应的，因而是高效的。它特别适用于分析大量频率随时间变化的非线性、非平稳信号。变化的频率是现实生活中人们经常直观感觉到的现象，HHT的根本目的就是描述和揭示这种时变频率现象及其规律，即求信号的瞬时频率。但为了获得信号某一时刻的瞬时频率值，HHT自适应地利用了信号在该时刻的局部信息。本文将这种表现为瞬时值，实际上自适应地隐含了信号局部信息的量称为局瞬量，将这种获得局瞬量的信号分析理论称为局瞬信号分析。为了体现这一特点，本文将HHT进一步称为希尔波特-黄变换局瞬信号分析（HHT-LISA）。
HHT-LISA具有重要的理论价值和广阔的应用前景，已在一些实际工程领域中获得了有效的应用。但HHT的第一篇公开文献直到1998年才发表，因而这一理论出现的时间还很短暂，其完善和发展还有诸多工作要做。本文在Huang等人的前期研究工作基础上，采用物理意义、理论推导和编程验证相结合的研究方法，对HHT-LISA展开了较为深入和全面地研究，取得了一定的研究成果，将HHT-LISA理论向前推进了一步。
HHT-LISA是现阶段一个全新的研究课题。对变化频率的研究虽然很早就已经开始，但后来的工作大都转向通过对信号的时频联合分析间接揭示这一现象，且都采用积分方法，它们的最终理论依据都基于傅里叶分析。傅里叶分析是发展最早和最成熟的信号分析理论，也是首先采用频谱分析信号的方法。但傅里叶分析中的频率是用全局的正弦波定义的，与时间无关。用傅里叶变换分析时变频率的信号会出现虚假信号和假频等缺陷。用基于傅里叶分析理论的时频联合分析也必然遭受同样的局限。并且，由于受Heisenberg不确定原理的限制，时频分析不能达到精确描述频率随时间变化的目的。HHT-LISA直接研究瞬时频率和对其规律进行精确地描述，且采用微分方法，因而其应用价值大为提高。当然研究难度也大为增加。为了对这一全新的课题进行彻底而全面的研究，在对HHT-LISA进行直接
介绍和研究之前，本文首先对信号分析、傅里叶分析和时频分析进行了回顾。深入分析了傅里叶分析和时频分析与人们直观感觉上的差异及其原因。这些工作表明了人们对瞬时频率分析理论的渴望和研究的困难。
之后，本文对HHT提出的历史背景进行了较详细地回顾。全面地介绍了HHT的主要依据、基本概念、基本方法、分析步骤、和分解的理想模型等；分析了HHT的明显创新性。在这些回顾的基础上，提出了“局瞬”这一新概念，从而深刻和准确地揭示了瞬时频率概念的本质属性，暗示了瞬时频率分析的方法和途径，阐释了长期以来人们感到瞬时频率概念难以把握的根源，也体现了HHT的根本特点。针对IMF的描述性定义，提出了IMF应满足的本征条件，从而初步用数学关系式给出了IMF的数学模型，基于该模型较为成功地论证了IMF局部对称性的必要性和用极值点拟合IMF包络线的合理性。针对经验筛法的理论还不够完善问题，提出和论证了Hilbert变换的局部乘积定理，该定理进一步给出了经验筛法的理论依据。提出了“自适应多分辨”和“恒定多分辨”概念，这两个概念概括了经验筛法和小波变换的根本区别。另外还总结了HHT的十四个特点；给出了Hilbert谱的进一步表示；改善了经验筛法的终止条件；对瞬时频率的算法作了比较，提出了选择恰当方法的意见等。
包络线和均值曲线的拟合是HHT-LISA的关键问题，它在很大程度上将影响到新理论和新方法的成熟和推广应用，因而对其研究具有重要意义。本文首先分析出了HHT-LISA中曲线拟合所要求的性质；总结了文献中采用的三次样条插值法的不足，因而引入了一种光滑性虽然更差，但自适应更好的插值法——阿克玛法。然后基于抛物线参数样条拟合法的原理，提出了一种新的插值法——分段幂函数法，用三种插值法的仿真试验和比较说明了新插值法的优点。但以上这些曲线拟合法都只具有低阶光滑性，不能提高光滑性，或者，为了提高光滑性，就会造成更为严重的过冲和欠冲问题；这些方法也不是从频率要求方面出发提出的。为了更深入地研究和解决HHT-LISA中的曲线拟合问题，本文对非均匀采样信号的重构算法进行了研究，在分析了文献中的一些非均匀采样信号重构算法的不足之后，基于内插函数，提出了一定带宽条件下非均匀采样信号的一种重构算法——解方程组法,这一方法简单有效。仿真试验验证了这一重构算法的正确性。
筛法是HHT-LISA的核心问题，它需要得到信号的均值曲线。Huang等人在提出HHT时首先采用的方法是用三次样条曲线分别拟合信号的极大值和极小值点作为上、下包络线，对上、下包络线求平均得到均值曲线。这种通过拟合包络线得到均值曲线的方法每次筛选都需要经过两次曲线拟合，因而很费机时，也容易造
成过
The Hilbert-Huang transform (HHT) is a new theory, which is first developed by Huang et al at the end of last century, for the signal analysis. The main innovations embodied in this method are the present of the concept of the intrinsic mode (IMF) and introduction of the method of empirical sifting method (EMD). A signal is first decomposed into often finite IMFs by the EMD, and then, with the Hilbert transform, the meaning instantaneous frequencies of every IMF is obtained which give precise description of the varying frequency. The final presentation of the result is an energy-frequency-time distribution, designated as the Hilbert spectrum. We can also further get marginal spectrum of the signal. The HHT is highly efficient since it is based on the local characteristic of a signal and adaptive. It is very applicable to analyze nonlinear and non-stationary signal which frequency is variable with time. The varying frequency is a natural phenomenon in the actual life that is often felt by people. The essential aim of the HHT is to describe this phenomenon and uncover its law, that is, to look for the instantaneous frequency of a signal. However, the HHT adaptively takes advantage of the local information of a signal at a certain time to obtain the instantaneous frequency of the signal at that time. The amount, which appears an instantaneous value but in fact has adaptive local information of a signal, is called Local-instantaneous amount in this paper, and the signal analysis theory by which some Local-instantaneous amounts is obtained is called Local-instantaneous signal analysis. For showing this characteristic, this paper further calls the HHT as the Hilbert-Huang transform Local-instantaneous signal analysis (HHT-LISA).
The HHT-LISA has important theoretical value and widely applying perspective. It has effectively been applied in some actual engineering problems. However, this theory appears very late since its first public literature wasn't published until 1998. Many works need to be done to perfect and improve it. Based on the former works done by Huang et al, using the study method of jointing physical meaning, theoretical induction and program validation, this paper relatively deeply and thorough studied the HHT-LISA. Some achievements are achieved and the HHT-LISA was further perfected.
The HHT-LISA is a new problem in all at present moment. Although the studies on the varying frequency were begun very early, most of these subsequent works are translated to indirectly uncover this phenomenon by time-frequency joint analysis and
adopt integral method to analyze it. These works are all based on the Fourier analysis theory. The Fourier theory is earliest and most perfect one for signal analysis, and it also first adopted frequency spectrum analysis method. However, the frequency of the Fourier analysis is independent of time since it is defined by a whole sine signal. Fourier analysis will suffer the shortcomings such as dummy signal and dummy frequency. Certainly, the time-frequency joint analysis of a signal based on the Fourier analysis theory will also suffer the same shortcomings. They can't also arrive at the aim to precisely describe the law of varying frequency with time because of the Heisenberg indefinite principle. The HHT-LISA directly studies instantaneous frequency and precisely describes its law. Furthermore, it adopts differential method to analyze it, so its applied value is largely increased. Of course the difficult degree is also largely increased. In order to deeply and detailedly study this all-new problem, this paper first gives a introduction to the historical perspective of signal analysis, Fourier analysis and time-frequency analysis before direct introduction and study on the HHT-LISA. The difference between people's straight feel and the result of Fourier analysis and time-frequency analysis and its reason are deeply analyzed in this paper. These works shows people's thirst for the instantaneous frequency analysis theory and the difficulty to obtain it.
Subsequent
HHT-LISA具有重要的理论价值和广阔的应用前景，已在一些实际工程领域中获得了有效的应用。但HHT的第一篇公开文献直到1998年才发表，因而这一理论出现的时间还很短暂，其完善和发展还有诸多工作要做。本文在Huang等人的前期研究工作基础上，采用物理意义、理论推导和编程验证相结合的研究方法，对HHT-LISA展开了较为深入和全面地研究，取得了一定的研究成果，将HHT-LISA理论向前推进了一步。
HHT-LISA是现阶段一个全新的研究课题。对变化频率的研究虽然很早就已经开始，但后来的工作大都转向通过对信号的时频联合分析间接揭示这一现象，且都采用积分方法，它们的最终理论依据都基于傅里叶分析。傅里叶分析是发展最早和最成熟的信号分析理论，也是首先采用频谱分析信号的方法。但傅里叶分析中的频率是用全局的正弦波定义的，与时间无关。用傅里叶变换分析时变频率的信号会出现虚假信号和假频等缺陷。用基于傅里叶分析理论的时频联合分析也必然遭受同样的局限。并且，由于受Heisenberg不确定原理的限制，时频分析不能达到精确描述频率随时间变化的目的。HHT-LISA直接研究瞬时频率和对其规律进行精确地描述，且采用微分方法，因而其应用价值大为提高。当然研究难度也大为增加。为了对这一全新的课题进行彻底而全面的研究，在对HHT-LISA进行直接
介绍和研究之前，本文首先对信号分析、傅里叶分析和时频分析进行了回顾。深入分析了傅里叶分析和时频分析与人们直观感觉上的差异及其原因。这些工作表明了人们对瞬时频率分析理论的渴望和研究的困难。
之后，本文对HHT提出的历史背景进行了较详细地回顾。全面地介绍了HHT的主要依据、基本概念、基本方法、分析步骤、和分解的理想模型等；分析了HHT的明显创新性。在这些回顾的基础上，提出了“局瞬”这一新概念，从而深刻和准确地揭示了瞬时频率概念的本质属性，暗示了瞬时频率分析的方法和途径，阐释了长期以来人们感到瞬时频率概念难以把握的根源，也体现了HHT的根本特点。针对IMF的描述性定义，提出了IMF应满足的本征条件，从而初步用数学关系式给出了IMF的数学模型，基于该模型较为成功地论证了IMF局部对称性的必要性和用极值点拟合IMF包络线的合理性。针对经验筛法的理论还不够完善问题，提出和论证了Hilbert变换的局部乘积定理，该定理进一步给出了经验筛法的理论依据。提出了“自适应多分辨”和“恒定多分辨”概念，这两个概念概括了经验筛法和小波变换的根本区别。另外还总结了HHT的十四个特点；给出了Hilbert谱的进一步表示；改善了经验筛法的终止条件；对瞬时频率的算法作了比较，提出了选择恰当方法的意见等。
包络线和均值曲线的拟合是HHT-LISA的关键问题，它在很大程度上将影响到新理论和新方法的成熟和推广应用，因而对其研究具有重要意义。本文首先分析出了HHT-LISA中曲线拟合所要求的性质；总结了文献中采用的三次样条插值法的不足，因而引入了一种光滑性虽然更差，但自适应更好的插值法——阿克玛法。然后基于抛物线参数样条拟合法的原理，提出了一种新的插值法——分段幂函数法，用三种插值法的仿真试验和比较说明了新插值法的优点。但以上这些曲线拟合法都只具有低阶光滑性，不能提高光滑性，或者，为了提高光滑性，就会造成更为严重的过冲和欠冲问题；这些方法也不是从频率要求方面出发提出的。为了更深入地研究和解决HHT-LISA中的曲线拟合问题，本文对非均匀采样信号的重构算法进行了研究，在分析了文献中的一些非均匀采样信号重构算法的不足之后，基于内插函数，提出了一定带宽条件下非均匀采样信号的一种重构算法——解方程组法,这一方法简单有效。仿真试验验证了这一重构算法的正确性。
筛法是HHT-LISA的核心问题，它需要得到信号的均值曲线。Huang等人在提出HHT时首先采用的方法是用三次样条曲线分别拟合信号的极大值和极小值点作为上、下包络线，对上、下包络线求平均得到均值曲线。这种通过拟合包络线得到均值曲线的方法每次筛选都需要经过两次曲线拟合，因而很费机时，也容易造
成过
The Hilbert-Huang transform (HHT) is a new theory, which is first developed by Huang et al at the end of last century, for the signal analysis. The main innovations embodied in this method are the present of the concept of the intrinsic mode (IMF) and introduction of the method of empirical sifting method (EMD). A signal is first decomposed into often finite IMFs by the EMD, and then, with the Hilbert transform, the meaning instantaneous frequencies of every IMF is obtained which give precise description of the varying frequency. The final presentation of the result is an energy-frequency-time distribution, designated as the Hilbert spectrum. We can also further get marginal spectrum of the signal. The HHT is highly efficient since it is based on the local characteristic of a signal and adaptive. It is very applicable to analyze nonlinear and non-stationary signal which frequency is variable with time. The varying frequency is a natural phenomenon in the actual life that is often felt by people. The essential aim of the HHT is to describe this phenomenon and uncover its law, that is, to look for the instantaneous frequency of a signal. However, the HHT adaptively takes advantage of the local information of a signal at a certain time to obtain the instantaneous frequency of the signal at that time. The amount, which appears an instantaneous value but in fact has adaptive local information of a signal, is called Local-instantaneous amount in this paper, and the signal analysis theory by which some Local-instantaneous amounts is obtained is called Local-instantaneous signal analysis. For showing this characteristic, this paper further calls the HHT as the Hilbert-Huang transform Local-instantaneous signal analysis (HHT-LISA).
The HHT-LISA has important theoretical value and widely applying perspective. It has effectively been applied in some actual engineering problems. However, this theory appears very late since its first public literature wasn't published until 1998. Many works need to be done to perfect and improve it. Based on the former works done by Huang et al, using the study method of jointing physical meaning, theoretical induction and program validation, this paper relatively deeply and thorough studied the HHT-LISA. Some achievements are achieved and the HHT-LISA was further perfected.
The HHT-LISA is a new problem in all at present moment. Although the studies on the varying frequency were begun very early, most of these subsequent works are translated to indirectly uncover this phenomenon by time-frequency joint analysis and
adopt integral method to analyze it. These works are all based on the Fourier analysis theory. The Fourier theory is earliest and most perfect one for signal analysis, and it also first adopted frequency spectrum analysis method. However, the frequency of the Fourier analysis is independent of time since it is defined by a whole sine signal. Fourier analysis will suffer the shortcomings such as dummy signal and dummy frequency. Certainly, the time-frequency joint analysis of a signal based on the Fourier analysis theory will also suffer the same shortcomings. They can't also arrive at the aim to precisely describe the law of varying frequency with time because of the Heisenberg indefinite principle. The HHT-LISA directly studies instantaneous frequency and precisely describes its law. Furthermore, it adopts differential method to analyze it, so its applied value is largely increased. Of course the difficult degree is also largely increased. In order to deeply and detailedly study this all-new problem, this paper first gives a introduction to the historical perspective of signal analysis, Fourier analysis and time-frequency analysis before direct introduction and study on the HHT-LISA. The difference between people's straight feel and the result of Fourier analysis and time-frequency analysis and its reason are deeply analyzed in this paper. These works shows people's thirst for the instantaneous frequency analysis theory and the difficulty to obtain it.
Subsequent
引文
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