经验模式分解的数学理论研究
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摘要
信号分析是对信号基本性质的研究和表征。从传统观点来看,信号的不同表示通常是将信号在函数空间的完备正交基集上展开。但是,由于基函数的形式固定,往往难以适应现实信号的复杂性,因此需要一种更加灵活的信号分解方法。于是,经验模式分解(EMD)方法应运而生,它不是采用某种正交基对信号进行分解,而是通过特殊的迭代规则将信号分解为一系列本征模态函数(IMF)之和,这种方法摆脱了传统信号分解方法中基函数的束缚,从而能够更加灵活的发掘信号特征。但是,这种灵活性是以迭代规则的复杂性为代价的,因此经验模式分解数学理论研究的重点在于对迭代规则性质的分析。
经验模式分解中的迭代过程称为筛过程,其基本步骤如下:首先找到信号的极值点,然后构造信号的上包络和下包络,最后从原有信号中减去上下包络的均值。重复这个迭代过程,直到信号的上下包络均值为零,就得到了信号中包含的本征模态函数。从前面的步骤可以看出,筛过程包括两个关键步骤:(1)信号极值点的定位;(2)上下包络的构造。本文正是从这两个关键点出发,提出了一种精确的信号极值点定位方法和一种解析性质良好的包络构造方法。
信号极值点定位算法主要建立在插值理论的基础上,分别提出了基于代数插值和三角插值的极值点定位算法。其中,代数插值采用了三次样条插值和分段Hermite插值,分别介绍了它们的插值公式的构造方法和插值误差估计的表达式。对于三角插值,不仅给出了插值公式,并首次给出了三角插值误差估计公式及其上界的推导方法。
包络构造算法首先定义了一组信号包络必须满足的规则,然后采用二次多项式和三次多项式来构造满足规则的包络,最后将这种方法推广到n次多项式。同时,证明了这种包络的一个重要性质:采用这种包络进行迭代时,信号包络的波动会逐渐变小,最后收敛到恒包络信号。因此,采用这种包络的经验模式分解算法,是一种恒包络信号分解方法。在分析包络性质的过程中,发现这种包络通过若干关键节点,从而提出了一种基于分段Hermite插值的等效包络构造方法。
本文针对目前经验模式分解中存在的理论空白开展了研究,解决了低采样率情况下信号中极值点定位的问题,同时设计了一种产生恒包络本征模态函数的包络构造方法。仿真结果表明,在对于恒包络组成的信号进行分解时,该方法优于现有的经验模式分解算法。因此,本文的研究工作具有较高的理论价值和一定的实际意义。
Signal analysis is used to research and represent the basic property of the signal. In traditional view, the different representations of the signal are made by the signal's extention onto different orthogonal basises. However, the basis function is fixed in an application so that it can not accommodate the special need of the practical signal. Empirical Mode Decomposition (EMD) is just to solve the problem. The signal decomposites into Intrinsic Mode Functions (IMFs) not by some orthogonal basis but by some iterative rule. The new method throws off the chains of the basis and finds the feature of the signal adaptively. However, the flexibility is at the cost of the complexity of the iterative rule. Therefore, the mathematical theory research of EMD focuses on the analysis to the property of iterative rule.
The iterative process in EMD is called the sifting process, which has the following steps: finding the extrema of the signal, constructing the upper and lower envelopes of the signal by the extrema and subtracting the mean of the two envelopes from original signal. The steps are repeated until the mean envelope is close to zero and the residue signal is called IMF. From the steps, the two key parts are the location of the extream and the construction of the envelope. This dissertation focuses on these two points and proposes a method to locate the extrema and a theory to design the envelopes.
The location algorithm is based on the interpolation theory including algebraic interpolation and trigonometric interpolation. Algebraic interpolation mainly composes of cubic interpolation and piecewise Hermite interpolation, and their interpolation formula and errors are introduced. For the trigonometric interpolation, the interpolation error is discussed and its the upper bound is also given.
The envelope design defines the property of the undetermined envelope, and then constructs the envelope by quadratic polynomial, cubic polynomial and n-order polynomial. The important property of these envelopes is that the fluctuation of the envelope will be decreased in the iterative process and the iterative process will converge to a constant-envelope signal. Therefore, the EMD using the new envelope is a method that decomposites the signal into the constant-envelope signals. In the analysis of the envelopes, we find the envelopes pass through some special points and propose an equivalent construction method by piece-wise Hermite interpolation.
The dissertation intends tofill the gaps in the theoretical research of Empirical Mode Decomposition. We solve the extrema location problem under low sampling rate and propose an envelope construction method that leads to a constant-envelope signal decomposition. Simulations show that the new algorithm is better than the traditional algorithm in the constant-envelope signal decomposition. Therefore, the works of this dissertation have strong theoretical and practical significance.
经验模式分解中的迭代过程称为筛过程,其基本步骤如下:首先找到信号的极值点,然后构造信号的上包络和下包络,最后从原有信号中减去上下包络的均值。重复这个迭代过程,直到信号的上下包络均值为零,就得到了信号中包含的本征模态函数。从前面的步骤可以看出,筛过程包括两个关键步骤:(1)信号极值点的定位;(2)上下包络的构造。本文正是从这两个关键点出发,提出了一种精确的信号极值点定位方法和一种解析性质良好的包络构造方法。
信号极值点定位算法主要建立在插值理论的基础上,分别提出了基于代数插值和三角插值的极值点定位算法。其中,代数插值采用了三次样条插值和分段Hermite插值,分别介绍了它们的插值公式的构造方法和插值误差估计的表达式。对于三角插值,不仅给出了插值公式,并首次给出了三角插值误差估计公式及其上界的推导方法。
包络构造算法首先定义了一组信号包络必须满足的规则,然后采用二次多项式和三次多项式来构造满足规则的包络,最后将这种方法推广到n次多项式。同时,证明了这种包络的一个重要性质:采用这种包络进行迭代时,信号包络的波动会逐渐变小,最后收敛到恒包络信号。因此,采用这种包络的经验模式分解算法,是一种恒包络信号分解方法。在分析包络性质的过程中,发现这种包络通过若干关键节点,从而提出了一种基于分段Hermite插值的等效包络构造方法。
本文针对目前经验模式分解中存在的理论空白开展了研究,解决了低采样率情况下信号中极值点定位的问题,同时设计了一种产生恒包络本征模态函数的包络构造方法。仿真结果表明,在对于恒包络组成的信号进行分解时,该方法优于现有的经验模式分解算法。因此,本文的研究工作具有较高的理论价值和一定的实际意义。
Signal analysis is used to research and represent the basic property of the signal. In traditional view, the different representations of the signal are made by the signal's extention onto different orthogonal basises. However, the basis function is fixed in an application so that it can not accommodate the special need of the practical signal. Empirical Mode Decomposition (EMD) is just to solve the problem. The signal decomposites into Intrinsic Mode Functions (IMFs) not by some orthogonal basis but by some iterative rule. The new method throws off the chains of the basis and finds the feature of the signal adaptively. However, the flexibility is at the cost of the complexity of the iterative rule. Therefore, the mathematical theory research of EMD focuses on the analysis to the property of iterative rule.
The iterative process in EMD is called the sifting process, which has the following steps: finding the extrema of the signal, constructing the upper and lower envelopes of the signal by the extrema and subtracting the mean of the two envelopes from original signal. The steps are repeated until the mean envelope is close to zero and the residue signal is called IMF. From the steps, the two key parts are the location of the extream and the construction of the envelope. This dissertation focuses on these two points and proposes a method to locate the extrema and a theory to design the envelopes.
The location algorithm is based on the interpolation theory including algebraic interpolation and trigonometric interpolation. Algebraic interpolation mainly composes of cubic interpolation and piecewise Hermite interpolation, and their interpolation formula and errors are introduced. For the trigonometric interpolation, the interpolation error is discussed and its the upper bound is also given.
The envelope design defines the property of the undetermined envelope, and then constructs the envelope by quadratic polynomial, cubic polynomial and n-order polynomial. The important property of these envelopes is that the fluctuation of the envelope will be decreased in the iterative process and the iterative process will converge to a constant-envelope signal. Therefore, the EMD using the new envelope is a method that decomposites the signal into the constant-envelope signals. In the analysis of the envelopes, we find the envelopes pass through some special points and propose an equivalent construction method by piece-wise Hermite interpolation.
The dissertation intends tofill the gaps in the theoretical research of Empirical Mode Decomposition. We solve the extrema location problem under low sampling rate and propose an envelope construction method that leads to a constant-envelope signal decomposition. Simulations show that the new algorithm is better than the traditional algorithm in the constant-envelope signal decomposition. Therefore, the works of this dissertation have strong theoretical and practical significance.
引文
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