【论文翻译】（摘要及引言）The Fourier decomposition method for nonlinear and non-stationary time series analysis

【Abstract】For many decades, there has been a general perception in the literature that Fourier methods are not suitable for the analysis of nonlinear and non-stationary data. In this paper, we propose a novel and adaptive Fourier decomposition method (FDM), based on the Fourier theory , and demonstrate its efficacy for the analysis of nonlinear and non-stationary time series. The proposed FDM decomposes any data into a small number of ‘Fourier intrinsic band functions’ (FIBFs). The FDM presents a generalized Fourier expansion with variable amplitudes and variable frequencies of a time series by the Fourier method itself. We propose an idea of zero-phase filter bank-based multivariate FDM (MFDM), for the analysis of multivariate nonlinear and non-stationary time series, using the FDM. We also present an algorithm to obtain cut-off frequencies for MFDM.The proposed MFDM generates a finite number of band-limited multivariate FIBFs (MFIBFs). The MFDM preserves some intrinsic physical properties of the multivariate data, such as scale alignment, trend and instantaneous frequency. The proposed methods provide a time–frequency–energy (TFE) distribution that reveals the intrinsic structure of a data. Numerical computations and simulations have been carried out and comparison is made with the empirical mode decomposition algorithms.

【摘要】几十年来，文献中普遍认为傅立叶方法不适用于非线性和非平稳数据的分析。本文基于傅里叶理论，提出了一种新颖的自适应傅里叶分解方法(FDM)，并证明了其对非线性和非平稳时间序列分析的有效性。所提出的FDM将任何数据分解成少量的“傅里叶本征带函数”(FIBFS)。FDM通过傅里叶方法本身给出了时间序列的具有可变振幅和可变频率的广义傅里叶展开。我们提出了基于零相位滤波器组的多元FDM (MFDM)的思想，用于分析多元非线性和非平稳时间序列。本文还提出了一种获得MFDM截止频率的算法，该算法产生有限个带限的多元FIBFs（MFIBFs）。MFDM保留了多变量数据的一些固有物理特性，如尺度对齐、趋势和瞬时频率。所提出的方法提供了揭示数据内在结构的时间-频率-能量分布。进行了数值计算和仿真，并与经验模态分解算法进行了比较。

Keywords: Fourier decomposition method, Fourier intrinsic band functions, analytic Fourier intrinsic band functions, zero-phase filter bank-based multivariate Fourier decomposition method, empirical mode decomposition

关键词:傅里叶分解法，傅里叶本征带函数，解析傅里叶本征带函数，基于零相位滤波器组的多元傅里叶分解法，经验模态分解

Introduction
The time–frequency representation (TFR) of a signal is a well-established powerful tool for the analysis of time series signals. It maps a one-dimensional signal of time into a two-dimensional signal of time and frequency , generally , by using time-varying frequency representations. There exist three types of TFRs [1]: linear (e.g. the short-time Fourier transform (FT) and wavelet transform), quadratic (like the Wigner distribution and the ambiguity function), nonlinear and non-quadratic (e.g. Cohen’s non-negative distribution) TFRs. TFR is obtained by a formulation often referred to as time–frequency distribution (TFD) and provides insight into the complex structure of a signal consisting of several components. There are various methods for non-stationary data processing and analysis such as the spectrogram, wavelet analysis, the Wigner-Ville distribution, evolutionary spectrum [2] and the empirical orthogonal function expansion (EOF) (or principal component analysis or singular value decomposition). Although these approaches have many useful applications, however, the analysis of non-stationary signals are not well presented by these methods.

信号的时频表示（TFR）是分析时间序列信号的有力工具。它通常使用时变频率表示法将一维时间信号映射为二维时间和频率信号。TFRs有三种类型[1]：线性（如短时傅立叶变换（STFT）和小波变换）、二次型（如Wigner分布和模糊函数）、非线性和非二次型（如Cohen非负分布）。TFR是通过一种通常被称为时频分布（TFD）的公式获得的，它提供了对由多个分量组成的信号的复杂结构的洞察。对于非平稳数据的处理和分析有多种方法，如谱图、小波分析、Wigner-Ville分布、演化谱[2]和经验正交函数展开（EOF）（或主成分分析或奇异值分解）等。虽然这些方法有许多有用的应用，但是这些方法并不能很好地用于非平稳信号的分析。

The recently proposed empirical mode decomposition (EMD) [3] provides a general method for examining the TFD. The EMD is an adaptive signal decomposition algorithm for the analysis of non-stationary and nonlinear signals (i.e. signals generated from nonlinear systems). The EMD has become an established method for signal and other data analysis in various applications such as medical studies [4–7], meteorology [3], geophysical studies [8] and image analysis [9]. The EMD decomposes any given data into a set of finite number of narrow band intrinsic mode functions (IMFs) which are derived directly from the data, whereas other signal decomposition techniques such as Fourier and wavelet transforms incorporate predefined fixed basis for signal modelling and analysis. The ensemble EMD (EEMD) is a noise-assisted data analysis method developed in [10] to overcome the time-scale separation problem of EMD. The MEMD developed in [11] i s a generalization of the EMD for multichannel data analysis. The compact EMD (CEMD) algorithm is proposed in [12] to reduce mode mixing, end effect and detrend uncertainty present in the EMD. The IMFs generated by the EMD algorithm are dependent on distribution of local extrema of signal and the type of spline used for upper and lower envelope interpolation. The traditional EMD uses the cubic spline for upper and lower envelopes interpolation. The EMD algorithm, proposed in [13] to reduce mode mixing and detrend uncertainty , uses non-polynomial cubic spline interpolation to obtain upper and lower envelopes which improves orthogonality among IMFs [14].

最近提出的经验模式分解(EMD) [3]提供了一种检查TFD的通用方法。EMD是一种自适应信号分解算法，用于分析非平稳和非线性信号（即非线性系统产生的信号）。EMD已成为信号和其他数据分析的一种既定方法，在各种应用中，如医学研究[4–7]，气象学[3]，地球物理研究[8]和图像分析[9]。EMD将任何给定的数据分解为一组有限数量的窄带本征模态函数(IMFs)，这些本征模态函数直接从数据中导出，而其他信号分解技术，如傅里叶和小波变换，结合了信号建模和分析的预定义固定基。集成EMD（EEMD）是文献[10]中提出的一种噪声辅助数据分析方法，用来克服EMD的时间尺度分离问题。[11]中提出的MEMD是EMD在多通道数据分析中的推广。在[12]中提出了紧凑的经验模态分解(CEMD)算法，以减少EMD中存在的模态混叠、端点效应和趋势不确定性。由EMD算法生成的IMFs依赖于信号的局部极值分布和上、下包络插值样条的类型。传统的EMD采用三次样条函数进行上下包络的插值。文献[13]中提出的EMD算法用于减少模式混合和趋势不确定性，使用非多项式三次样条插值来获得上下包络，这改善了IMFs之间的正交性[14]。

The property of energy preservation is important for any kind of transformation, and it is obtained by the orthogonal decomposition of a signal in various transforms such as the Fourier, wavelet and Fourier–Bessel representation. The energy preserving property is especially important for the accurate and faithful analysis of three-dimensional TFE distribution of a signal. In order to preserve energy in the signal decomposition, energy preserving EMD (EPEMD) algorithms are proposed in [15] which ensure orthogonality among IMFs or generate a set of IMFs which are linearly independent, non-orthogonal yet energy preserving (LINOEP) vectors.

能量保持特性对于任何类型的变换都很重要，它是通过在各种变换(如傅立叶变换、小波变换和傅立叶-贝塞尔表示)中对信号进行正交分解而获得的。能量保持特性对于准确可靠地分析信号的三维TFE分布尤为重要。为了在信号分解中保持能量，文献[15]提出了能量保持EMD（EPEMD）算法，该算法保证IMFs之间的正交性，或者生成一组线性无关的、非正交但能量保持（LINOEP）的向量。

Despite considerable success, all of the EMD algorithms are based on empirical, heuristic and ad hoc procedures that make them hard to analyse mathematically , and EMD may suffer from mode mixing, detrend uncertainty , aliasing and end effect artefacts [16]. There is also a lack of mathematical understanding of the EMD algorithms, e.g. dependence of IMFs on the number of sifting, and the stopping criteria, convergence property and stability to noise perturbation. Despite all these limitations, the EMD is a widely used non-stationary data analysis method. Therefore, in this paper, EMD is used as a reference to establish the validity , reliability and calibration of the proposed methods.

尽管取得了相当大的成功，但所有EMD算法都是基于经验、启发式和特殊程序的，这使得它们很难进行数学分析，并且EMD可能会受到模态混叠、趋势不确定性、混叠和端点效应伪影的影响[16]。也缺乏对EMD算法的数学理解，如IMFs对筛选次数的依赖性、停止准则、收敛性和对噪声扰动的稳定性等。尽管有这些局限性，EMD是一种广泛使用的非平稳数据分析方法。因此，本文以EMD为参考，验证了所提方法的有效性、可靠性和可校正性。

There has been a general understanding in the literature (e.g. [3,16,17]) for many decades that Fourier methods are not suitable for the analysis of nonlinear and non-stationary data, and various reasons (e.g. linearity , periodicity or stationarity) are provided to support it. The FT is valid under very general Dirichlet conditions (i.e. the signal is absolutely integrable with finite number of maxima and minima, and finite number of finite discontinuities in any finite interval) and thus includes analysis of nonlinear and non-stationary signals as well. Therefore, in this study , we explore and provide algorithms to analyse nonlinear and non-stationary data by the Fourier method termed the Fourier decomposition method (FDM), which generates a set of a small number of band limited Fourier intrinsic band functions (FIBFs). It is already well established that Fourier theory-based methods are the best tool for spectrum analysis, and this study demonstrates that the Fourier methods are also the best tool for time–frequency analysis and processing of any signal. The power of the FT can also be realized from the fact that the analytic representation and, hence, the Hilbert transform (HT) of a signal are inherently present in the complex Fourier representation.

几十年来，文献（如[3,16,17]）普遍认为傅立叶方法不适合分析非线性和非平稳数据，并给出了各种原因（如线性、周期性或平稳性）来支持这一观点。FT在非常一般的Dirichlet条件下是有效的（即信号是绝对可积的，具有有限个极大值和极小值，以及有限个有限区间内的有限个有限间断），因此也包括非线性和非平稳信号的分析。因此，在本研究中，我们探索并提供了通过称为傅里叶分解法（FDM）的傅里叶方法分析非线性和非平稳数据的算法，该方法生成一组少量的带限傅里叶本征带函数（FIBFs）。众所周知，基于Fourier理论的方法是频谱分析的最佳工具，本研究表明Fourier方法也是对任何信号进行时频分析和处理的最佳工具。傅立叶变换的能力也可以从信号的解析表示以及希尔伯特变换（HT）固有地存在于复傅立叶表示中这一事实来实现。

We also propose a method which captures the essential features of the MEMD, using a zero-phase filter bank (ZPFB) approach to construct MFIBFs and the residue components. This multivariate FDM (MFDM) generates frequency band-matched MFIBFs and residue through zero-phase filtering (ZPF). Thus, we obtain an adaptive, data-driven, ZPFB-based time–frequency analysis method.

本文还提出了一种捕捉MEMD基本特征的方法，使用零相位滤波器组(ZPFB)的方法来构建多元傅里叶本征带函数（MFIBFs）和残差分量。这种多变量FDM (MFDM)通过零相位滤波(ZPF)产生频带匹配的多元傅里叶本征带函数（MFIBFs）和残差分量。因此，我们获得了一种自适应的、数据驱动的、基于ZPFB的时频分析方法。

For the adaptive data analysis approach, the most difficult challenge has been to establish a general adaptive decomposition method without a priori bases. In this study , we propose the FDM and MFDM general adaptive data analysis methods that are inspired by the EMD algorithms and their filter bank properties [18,19]. The main contributions of this study are as follows:

对于自适应数据分析方法，最困难的挑战是建立一种无先验基的通用自适应分解方法。本文提出了FDM和MFDM通用自适应数据分析方法，该方法受EMD算法及其滤波器组性质的启发[18，19]。本研究的主要贡献如下：

(i) Introduction of the FIBFs which are complete, adaptive, local, orthogonal and uncorrelated by the virtue of construction.
(ii) Introduction of a novel FDM, completely based on the Fourier theory , to decompose given data into a set of analytic FIBFs.
(iii) Introduction of a novel MFDM, which is based on the zero-phase filter-bank approach that can be realized by the Fourier as well as filter theory, for multichannel data analysis.
(iv) An algorithm is also presented to obtain cut-off frequencies required for the decomposition of data into a set of FIBFs via MFDM.

（i）介绍了完整的、自适应的、局部的、正交的、由于构造的优点而不相关的FIBFs。
（ii）介绍了一种完全基于傅里叶理论的新的FDM，将给定数据分解为一组解析FIBFs。
（iii）为多通道数据分析引入了一种新的MFDM方法，该方法基于零相位滤波器组方法，可通过傅里叶和滤波器理论实现。
（iv）还提出了一种算法，通过MFDM将数据分解成一组FIBFs所需的截止频率。

Thus, in this study a generalized Fourier expansion of a signal is obtained by the Fourier method itself. The representation of a signal by a generalized Fourier expansion is also the main objective of all the EMD algorithms and other data analysis methods such as synchrosqueezed wavelet transforms (SSWTs) [20], variational mode decomposition (VMD) [21], eigenvalue decomposition (EVD) [22], sparse TFR [23], time-varying vibration decomposition [24] and resonance-based signal decomposition approach [25].

因此，在本研究中，信号的广义傅里叶展开式由傅里叶方法本身获得。用广义傅里叶展开表示信号也是所有EMD算法和其他数据分析方法的主要目标，如同步压缩小波变换(SSWTs)[20]、变分模式分解(VMD)[21]、特征值分解(EVD)[22]、稀疏TFR[23]、时变振动分解[24]和基于共振的信号分解方法[25]。

The rest of this paper is organized as follows: In §2, the analytic signal and EMD algorithm are briefly presented. The proposed FDM is explained in §3. We propose the ZPFB-based MFDM algorithm in §4. Simulation results are presented in §5. Finally, conclusions are presented in §6.

本文的其余部分组织如下：§2简要介绍了解析信号和EMD算法。提出的FDM在§3中进行了解释。§4提出了基于ZPFB的MFDM算法。§5展示了仿真结果。最后，在§6中给出了结论。

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