Empirical mode decomposition (EMD), introduced by N.E. Huang et al. in 1998, is perhaps the most popular data-driven computational scheme for the decomposition of a non-stationary signal or time series $f(t)$, with time-domain $\mathbb{R}:=(-\infty ,\infty )$, into finitely many oscillatory components $\{f_1(t),\cdots ,f_K(t)\}$, called intrinsic mode functions (IMFs), and some “almost monotone” remainder $r(t)$, called the trend of $f(t)$. The core of EMD is the iterative “sifting process” applied to each function $m_{k-1}(t)$ to compute $f_k(t)$, for $k=1,\cdots ,K$, where $m_0(t):=f(t)$ and $m_k(t):=m_{k-1}(t)-f_k(t)$, with trend $r(t):=m_K(t)$. For the computation of each IMF, the sifting process depends on cubic spline interpolation of the local maxima and local minima for computing the upper and lower envelopes, respectively, and on subtracting the mean of the two envelopes from the result of the previous iterative step. Since it is not feasible to search for all local extrema in the entire time-domain $(-\infty ,\infty )$, implementation of the sifting process is commonly performed on some desired truncated bounded interval $[a,b]$. The main objective of this paper is to introduce and develop four “cubic spline manipulation engines”, called “quasi-interpolation (QI)”, “enhanced quasi-interpolation (EQI)”, “local interpolation (LI)”, and “improved global interpolation (IGI)” cubic spline manipulation engines, in order to significantly improve the performance of EMD on the truncated time-domains with minimal boundary artifacts, computational efficiency, accuracy, and consistency. Introduction and construction of the “fundamental quasi-interpolation” (FQI) splines as basis functions of the QI manipulation engine eliminates the need of matrix inversion for computing (global) cubic spline interpolation, since the local maximum values and local minimum values are used as coefficients of their FQI spline series representations, respectively. For the EQI spline manipulation engine, the FQI functions are formulated in terms of the same cubic B-spline basis for both the upper and lower envelopes; and for the LI spline manipulation engine, the “cubic spline blending” operation is applied to further modify the FQI splines to enable true cubic spline interpolation by “correcting the approximate interpolation error” of the EQI engine. As a consequence, the EQI and LI manipulation engines have the common property that in computing the means of the upper and lower envelopes, the only computation is averaging the B-spline coefficients, instead of computing the upper and lower envelopes separately. Furthermore, fast cubic spline pre-processing of the given $f(t)$ is also introduced to assure numerical stability in the computation of the Hilbert transform of the first IMF $f_1(t)$ on the truncated time-domain. The theory, along with methods and explicit formulas, developed in this paper are intended for other applications beyond EMD.

经验模态分解 (EMD)，由 NE Huang 等人提出。1998 年，可能是用于分解非平稳信号或时间序列的最流行的数据驱动计算方案$F（t）$，时域$\mathbb{右}:=（-无穷大,无穷大）$，分解为有限多个振荡分量$\{F_1（t）,\cdots ,F_K（t）\}$，称为本征模态函数(IMF)，以及一些“几乎单调”的余数$r（t）$，称为趋势$F（t）$。EMD 的核心是应用于每个函数的迭代“筛选过程”${米}_{k-1}（t）$计算$F_k（t）$， 为了$k=1,\cdots ,K$， 在哪里${米}_0（t）:=F（t）$和${米}_k（t）:={米}_{k-1}（t）-F_k（t）$, 有趋势$r（t）:={米}_K（t）$。对于每个 IMF 的计算，筛选过程取决于局部最大值和局部最小值的三次样条插值，分别计算上包络和下包络，并从上一迭代步骤的结果中减去两个包络的平均值。由于在整个时域内搜索所有局部极值是不可行的$（-无穷大,无穷大）$，筛选过程的实现通常在某些所需的截断有界区间上执行$[A,乙]$。本文的主要目的是介绍和开发四种“三次样条操纵引擎”，分别称为“准插值（QI）”、“增强准插值（EQI）”、“局部插值（LI）”和“改进的准插值（LI）”。全局插值（IGI）”三次样条操纵引擎，以显着提高截断时域上的 EMD 性能，同时最大限度地减少边界伪影、计算效率、准确性和一致性。引入和构建“基本准插值”（FQI）样条作为 QI 操纵引擎的基函数，消除了计算（全局）三次样条插值时矩阵求逆的需要，因为局部最大值和局部最小值被用作分别是其 FQI 样条级数表示的系数。对于 EQI 样条操纵引擎，FQI 函数是根据上包络线和下包络线的相同三次B样条基础来制定的；对于LI样条操纵引擎，应用“三次样条混合”操作来进一步修改FQI样条，以通过“校正EQI引擎的近似插值误差”来实现真正的三次样条插值。因此，EQI 和 LI 操纵引擎具有共同的属性，即在计算上下包络的均值时，唯一的计算是对 B样条系数进行平均，而不是分别计算上下包络。此外，给定的快速三次样条预处理$F（t）$还引入了确保第一 IMF 希尔伯特变换计算中的数值稳定性$F_1（t）$在截断的时域上。本文开发的理论以及方法和显式公式旨在用于 EMD 之外的其他应用。