Eliminating The End Effect Of Empirical Mode Decomposition Using A Cubic Spline Based Method

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 (AMFM) 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. (C) 2020 Elsevier Inc. All rights reserved.