| dc.description.abstract | Empirical mode decomposition (EMD) and disturbance-assisted EMD (D-A EMD) algorithms are adaptive and nonlinear methods that decompose a nonstationary signal into several intrinsic mode functions (IMFs) through the sifting process. From an algorithmic point of view, they are adaptive and nonlinear methods. Compared with short-time Fourier transform (STFT) and wavelet, the important physical meaning of the wave: phase and amplitude can be completely preserved, especially
the latter can be well extracted. Therefore, in many fields, EMD and D-A EMD algorithms have been applied, they have been proved to be superior to Fourier and wavelet in processing non-stationary signals, and the IMFs have strong physical or physiological significance.
However, EMD sometimes suffers mode-mixing effect. Therefore, D-A EMD algorithms can be applied to solve the problem. But there is no method that can solve all the problems encountered by EMD. Because EMD is a method that its theory is far behind its application. This study will explore three important applications of EMD. (a) When the D-A EMD algorithms are applied to decompose a signal polluted with noise in some regions, the noise will spread into the clean region and introduce errors in the IMFs. (b) During real-time computation of a D-A EMD algorithm, the signal is partitioned into a series of overlapping time windows. Points outside the window are discarded, they act as an error source and the error will spread into the current window. We will prove that the above two problems are mathematically identical. (c) The cubic spline interpolation which is an important step of EMD’s sifting procedure, it requires two boundary conditions at the two ends of the domain. The natural BC and not-a-knot BC are often applied. These two BCs are combined with mirror BC, linear extrapolation, modified linear extrapolations to further reduce errors. But the jury is still out on witch method is best for EMD.
Due to the complex nature of the sifting iteration, a mathematical theory for the error analysis of the boundary effect is still lacking. Previous studies mainly rely on simulations, and the results show that the error seems to be confined locally near the boundary. Beginning with one sifting iteration, which is the kernel of an EMD, we will theoretically analyze the boundary effects completely based on the sifting process without any approximations, prove that the error will propagate into the interior domain and may be amplified and thus obscure the meaning of an IMF. Then, prove a sufficient condition for the exponential decay of the error. When the error is amplified, we propose a method to resolve this problem. Finally, numerical experiments are conducted to analyze multiple siftings and the result of setting different boundary conditions. | en_US |