【论文翻译】（第二部分）The Fourier decomposition method for nonlinear and non-stationary time series analysis
- Brief overview of the analytic signal and empirical mode decomposition algorithm
2.简要概述了解析信号和经验模态分解算法
The EMD is a well-established adaptive signal decomposition algorithm that decomposes a nonlinear and non-stationary data into a set of finite band-limited IMFs and residue through a sifting process. The decomposition of a signal x(t) is expressed as the sum of l IMFs and residue
x(t)=∑i=1lyi(t)+rl(t)=∑i=1l+1yi(t)（2.1）x(t)=\displaystyle\sum_{i=1}^{l} y_i(t)+r_l(t)=\displaystyle\sum_{i=1}^{l+1} y_i(t) （2.1） x(t)=i=1∑lyi(t)+rl(t)=i=1∑l+1yi(t)（2.1）
where yi(t)y_i(t)yi(t) is the ith IMF and rl(t)=yl+1(t)r_l(t)=y_{l+1}(t)rl(t)=yl+1(t)is final residue.
EMD是一种成熟的自适应信号分解算法，通过筛选过程将非线性非平稳数据分解为一组有限带限IMFs和残差。x(t) 信号的分解可以表示为l组IMFs和残差的和
x(t)=∑i=1lyi(t)+rl(t)=∑i=1l+1yi(t)（2.1）x(t)=\displaystyle\sum_{i=1}^{l} y_i(t)+r_l(t)=\displaystyle\sum_{i=1}^{l+1} y_i(t) （2.1） x(t)=i=1∑lyi(t)+rl(t)=i=1∑l+1yi(t)（2.1）
其中yi(t)y_i(t)yi(t)是第i个IMF，rl(t)=yl+1(t)r_l(t)=y_{l+1}(t)rl(t)=yl+1(t)是最后的残差。
All the IMFs, {yi(t)}i=1l\{ y_i(t) \} _{i=1}^l{yi(t)}i=1l, must satisfy two basic conditions [3]: (i) in the complete duration of time series, the number of extrema (i.e. maxima and minima) and the number of zero crossings are equal or differ at most by one. (ii) At any point of time in the complete duration of time series, the average of the upper and lower envelopes, obtained by the interpolation of local maxima and the local minima, is zero. The first condition ensures that IMFs are narrow band signals and the second condition is necessary to ensure that the IF does not fluctuate excessively because of asymmetry of waveforms [3]. Thus, the IMFs admit amplitude–frequency modulated (AM–FM) representation [20] (i.e. yi(t)≈ai(t)cos(ϕi(t))y_i(t)≈a_i(t)cos(\phi_i(t))yi(t)≈ai(t)cos(ϕi(t)), with ai(t),dϕi(t)dt=ϕi′(t)>0,∀ta_i(t), \frac{d\phi_i(t)}{dt}=\phi_i^{'}(t)>0 ,\forall{t}ai(t),dtdϕi(t)=ϕi′(t)>0,∀t) and well-behaved HT [3]. For any IMF yi(t), its HT y^i(t) is defined as convolution of yi(t) and 1/πt, i.e. , y^i(t)=(1/π)p.v.∫−∞∞yi(τ)t−τdτ\hat{y}_i(t)=(1/π)p.v .\int_{-\infty}^\infty\frac{y_i(\tau)}{t − \tau} d\tauy^i(t)=(1/π)p.v.∫−∞∞t−τyi(τ)dτ,where p.v. stands for the Cauchy principal value of the integral. It is to be noted that the HT is a non-causal and unstable (i.e. not absolutely integrable) linear time-invariant (LTI) system model. The HT is an ideal operator that in practice cannot be physically realizable. Practically , the quadrature method [26], which under certain conditions is equivalent to the HT, using the FT obtain analytic signal. Although the HT is global, it emphasizes the properties of the function at the local time t. Thus, the HT is used to examine and reveal the local properties of the function yi(t) and hence signal x(t) in (2.1). An analytic signal zi(t) can be represented by zi(t)=yi(t)+jy^i(t)=ai(t)ejϕi(t)z_i(t)=y_i(t)+j\hat{y}_i(t)=a_i(t)e^{j\phi_i(t)}zi(t)=yi(t)+jy^i(t)=ai(t)ejϕi(t), where ai(t)=[yi2(t)+y^i2(t)]12a_i(t)=[y_i^2(t)+\hat{y}_i^2(t)]^{\frac12}ai(t)=[yi2(t)+y^i2(t)]21 and ϕi(t)=tan−1[y^i(t)yi(t)]\phi_i(t)=tan^{−1}[ \frac{\hat{y}_i(t)}{y_i(t)}]ϕi(t)=tan−1[yi(t)y^i(t)] are instantaneous amplitude (IA) and instantaneous phase (IP) of yi(t), respectively . The instantaneous frequency (IF) of yi(t) is defined as the time derivative of IP, i.e. ωi(t)=ϕi′(t)=(y^i′(t)yi(t)−y^i(t)yi′(t))(y^i2(t)+yi2(t))\omega_i(t)=\phi_i^{'}(t)=\frac{(\hat{y}_i^{'}(t) y_i(t)-\hat{y}_i(t)y_i^{'}(t) )}{(\hat{y}_i^2(t) + y_ i^2(t))}ωi(t)=ϕi′(t)=(y^i2(t)+yi2(t))(y^i′(t)yi(t)−y^i(t)yi′(t)). The physical meaning of IF ωi(t) constrains that φi(t) must be a mono-component function (i.e. an increasing or a non-decreasing function) of time. The Bedrosian and Nuttall theorems [27,28] further impose non-overlapping spectra constraints on the pair [ai(t), cos(φi(t))] of a signalyi(t)=ai(t)cos(ϕi(t))y_i(t)=a_i(t)cos(\phi_i(t))yi(t)=ai(t)cos(ϕi(t)).
所有的本征模态函数IMFs,{yi(t)}i=1l\{ y_i(t) \} _{i=1}^l{yi(t)}i=1l，必须满足两个基本条件[3]：
（i）在时间序列的完整持续时间内，极值的个数（即极大值和极小值）和过零的个数相等或最多相差一个。(ii）在时间序列的完全持续时间内的任意时刻，由局部极大值和局部极小值插值得到的上下包络的平均值为零。
第一个条件确保IMFs是窄带信号，第二个条件确保IF不会因为波形不对称而过度波动[3]。因此，IMFs允许振幅-频率调制（AM–FM）表示[20]（即
yi(t)≈ai(t)cos(ϕi(t))y_i(t)≈a_i(t)cos(\phi_i(t))yi(t)≈ai(t)cos(ϕi(t))
其中
ai(t),dϕi(t)dt=ϕi′(t)>0,∀ta_i(t), \frac{d\phi_i(t)}{dt}=\phi_i^{'}(t)>0 ,\forall{t}ai(t),dtdϕi(t)=ϕi′(t)>0,∀t
和表现良好的HT[3]。对于任何IMFs yi(t)y_i(t)yi(t)，其希尔伯特变换HT y^i(t)\hat{y}_i(t)y^i(t) 定义为 yi(t)y_i(t)yi(t) 和1/πt的卷积， 即
y^i(t)=(1/π)p.v.∫−∞∞yi(τ)t−τdτ\hat{y}_i(t)=(1/π)p.v .\int_{-\infty}^\infty\frac{y_i(\tau)}{t − \tau} d\tauy^i(t)=(1/π)p.v.∫−∞∞t−τyi(τ)dτ
其中p.v.表示积分的柯西主值。需要注意的是，HT是一个非因果和不稳定（即不完全可积）线性时不变（LTI）系统模型。HT是一个理想的算子，在实际中是物理不可实现的。实际上，正交方法[26]在一定条件下等价于利用FT得到解析信号的HT。虽然HT是全局的，但它强调函数在局部时间t的性质。因此，HT被用来检查和揭示函数yi(t)y_i(t)yi(t)的局部性质，从而揭示（2.1）中的信号 x(t)x(t)x(t) 。解析信号zi(t)z_i(t)zi(t)可以由zi(t)=yi(t)+jy^i(t)=ai(t)ejϕi(t)z_i(t)=y_i(t)+j\hat{y}_i(t)=a_i(t)e^{j\phi_i(t)}zi(t)=yi(t)+jy^i(t)=ai(t)ejϕi(t)
表示，其中
ai(t)=[yi2(t)+y^i2(t)]12a_i(t)=[y_i^2(t)+\hat{y}_i^2(t)]^{\frac12}ai(t)=[yi2(t)+y^i2(t)]21
ϕi(t)=tan−1[y^i(t)yi(t)]\phi_i(t)=tan^{−1}[ \frac{\hat{y}_i(t)}{y_i(t)}]ϕi(t)=tan−1[yi(t)y^i(t)]
分别是yi(t)y_i(t)yi(t)的瞬时振幅（IA）和瞬时相位（IP）。将 yi(t)y_i(t)yi(t) 的瞬时频率（IF）定义为IP的时间导数，即ωi(t)=ϕi′(t)=(y^i′(t)yi(t)−y^i(t)yi′(t))(y^i2(t)+yi2(t))\omega_i(t)=\phi_i^{'}(t)=\frac{(\hat{y}_i^{'}(t) y_i(t)-\hat{y}_i(t)y_i^{'}(t) )}{(\hat{y}_i^2(t) + y_ i^2(t))}ωi(t)=ϕi′(t)=(y^i2(t)+yi2(t))(y^i′(t)yi(t)−y^i(t)yi′(t))。IF ωi(t)\omega_i(t)ωi(t)的物理意义在于ϕi(t)\phi_i(t)ϕi(t) 必须是时间的单分量函数（即递增函数或非递减函数）。Bedrosian定理和Nuttall定理[27,28]进一步对信号yi(t)=ai(t)cos(ϕi(t))y_i(t)=a_i(t)cos(\phi_i(t))yi(t)=ai(t)cos(ϕi(t)) 的谱[ai(t), cos(φi(t))] 施加了非重叠谱约束。
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