| dc.description.abstract | The empirical mode decomposition (EMD) method is an adaptive decomposition method for non-stationary signals. It can decompose a signal into several sub-signals. Due to the recursively sifting process it can decompose a time signal into several well separated Intrinsic Mode Functions (IMF) on the time-frequency plane. From a mathematical point of view, EMD is a non-linear and adaptive method. Finally, these two properties are different from Fourier transform and wavelet methods. Compared with short-time Fourier transform (STFT) and wavelet, EMD can usually completely retain the important physical parameters of wave: phase and wave height, so IMF has strong physical or physiological significance. In many applications, it has been proved that EMD is superior to Fourier transform and wavelet in processing non-stationary signals, so EMD has been widely used in many fields such as mechanical engineering and so on. But EMD still has many problems, so many improved versions have been developed one after another. But currently there is no single algorithm that can solve all problems. The root cause is that EMD is a method whose theoretical development lags far behind its application.
Early study has proved the Impulse Response Theorem of Sifting Operator: the sifting operator (iteration) is equal to the linear combination of the nonlinear impulse response corresponding to each (extreme) point, and the extreme value is exactly its coefficient. Therefore, the properties of the sifting operator can be obtained by the properties of impulse response, and then obtain the properties of the IMF. Give an impulse signal with an amplitude of one at a certain point in time of the signal, and obtain the impulse response curve through the cubic spline. The time domain can be divided into far-field and near-field according to the distance from the impulse signal. The near-field refers to the interval adjacent to the node where the impulse signal is located, and the rest of intervals are the far-field. Locality of the far field means that the amplitude of the impulse response decreases as the distance from the impulse signal increases. Otherwise, if the amplitude becomes larger as the distance increase called non-local. The properties of far-field have been obtained by early study. The distance between two adjacent maximum (minimum) values is defined as the scale, and the ratio of the two adjacent scales is called the scale ratio, its value is always greater than or equal to one. Early study has proved that local and non-local sufficient condition for the far-field are as follows. When the scale ratio is less than 2.4, the far-field impulse response is local; The scale ratio is greater than 3.1, the far-field impulse response is nonlocal.
The mathematical properties of Near-field are more complicated rather than far-field, so there is no relevant theory of the near-field properties at present, make local properties are not yet complete. The purpose of this paper is to analyze the properties of the near-field. In signal processing, ideally amplitude of the impulse response should be smaller than the amplitude of the impulse signal. Later chapters will prove that is not true in EMD. Therefore, we must redefine the local conditions in the vicinity of near-field. When the amplitude of IR is less than certain threshold called local, and vice versa, it called nonlocal. The research and analysis of this study found that the local sufficient conditions of the near-field are the same as those of the far-field.
Finally, I will explore how locality of the signal affects the decomposition performance to examine the applicability of EMD. It is a very important question: give a signal to know whether it is suitable for EMD analysis.
When the variety in amplitude modulation (AM) is small and local, the decomposition efficiency of EMD is better; Conversely, when the change in AM is large or nonlocal, the decomposition efficiency of EMD is poor. At last, I will give a few examples to illustrate this phenomenon. | en_US |