The empirical mode decomposition (EMD) is a method that is commonly applied to extract the intrinsic mode functions (IMFs) of a signal by a sifting process, which requires imposing the extended extrema at both ends of the signal (i.e., the end condition). The imposition of extended extrema can cause an error, which is often presented by the changing shapes of original envelopes and distort extracted IMFs, which is described as the end effect. An important issue during the application of the EMD is restricting the end effect. This paper reveals the decisive factors that can restrict the end effect by determining the uniqueness of the envelope, and provides an interpretation of the end effect in terms of the differences between the original envelope and the extended envelope based on the cubic spline theory. Two principles that are important to the design of an end condition method are provided. The first principle is that the domain of the extended envelope needs to cover the original signal; the second is that the ordinate value of the extended local maxima is greater than or equal to that of the extended local minima. Following these two principles, a new end condition method, the cubic spline based method (CSBM), is proposed in this study. The novelty of the CSBM is that the extended envelope maintains the shape of the original envelope in their intersection domain, and the end effect can be restricted in a limited domain during the sifting process of EMD for each different input signal. Six signals are used to demonstrate the performance of the CSBM by comparing them with two other end condition methods, the extreme method and the improved slope based method (ISBM). The six signals include: a damped sinusoid signal, four monovariate signals with various amplitude modulation-frequency modulation (AM-FM) behaviors, and a one-channel functional near-infrared spectroscopy (fNIRS) signal. Results show that the CSBM in general performs better than the other two methods.

经验模式分解（EMD）是一种通常用于通过筛选过程提取信号的固有模式函数（IMF）的方法，该过程需要在信号的两端（即结束条件）施加扩展的极值。 。强加极端可能会导致错误，该错误通常由原始信封的形状变化和提取的IMF变形引起，这被描述为最终效果。EMD的应用过程中的一个重要问题是限制最终效果。本文揭示了可以通过确定包络的唯一性来限制最终效果的决定性因素，并基于三次样条理论对原始包络和扩展包络之间的差异进行了解释。提供了对结束条件方法的设计很重要的两个原理。第一个原则是扩展包络的域需要覆盖原始信号。第二个是扩展局部最大值的坐标值大于或等于扩展局部最小值的坐标值。遵循这两个原理，本研究提出了一种新的最终条件方法，即基于三次样条的方法（CSBM）。CSBM的新颖之处在于，扩展包络在其交集域中保持原始包络的形状，并且在针对每个不同输入信号的EMD筛选过程中，可以将最终效果限制在有限的域中。通过将六个信号与其他两种最终条件方法进行比较来证明CSBM的性能，极端方法和改进的基于坡度的方法（ISBM）。这六个信号包括：阻尼正弦信号，具有各种幅度调制-频率调制（AM-FM）行为的四个单变量信号，以及一个单通道功能近红外光谱（fNIRS）信号。结果表明，CSBM总体上优于其他两种方法。