Theoretical Analysis of Empirical Mode Decomposition

Abstract

This work suggests a theoretical principle about the oscillation signal decomposition, which is based on the requirement of a pure oscillation component, in which the mean zero is extracted from the signal. Using this principle, the validity and robustness of the empirical mode decomposition (EMD) method are first proved mathematically. This work also presents a modified version of EMD by the interpolation solution, which is able to improve the frequency decomposition of the signal. The result shows that it can provide a primary theoretical basis for the development of EMD. The simulation signal verifies the effectiveness of the EMD algorithm. At the same time, compared with the existing denoising algorithm, it has achieved good results in the denoising of rolling bearing fault signals. It contributes to the development and improvement of adaptive signal processing theory in the field of fault diagnosis. It provides practical value research results for the rapid development of adaptive technology in the field of fault diagnosis. then theoretically explained briefly how EMD operates on harmonic functions and why it selects the highest frequency oscillation, leaving the lower frequency oscillation in the signal, and also derived the obtainable frequency resolutions of the method and the existence of a critical frequency limit that allows separation of the closest harmonics. In order to improve the frequency resolution and the mode mixing effect [1,18], many efforts have been made for these years. Meanwhile, it has been found that methods associated with the local mean decomposition are more benefit to this purpose [19][20][21].Unfortunately, due to the lack of a complete and generally accepted theoretical framework, the EMD method still gives rise to puzzles, because the local mean of the signal depends on its characteristic local time-scales. To this end, this work suggested a theoretical principle involving the oscillation signal decomposition, which is based on the requirement of the pure oscillation component with mean zero extracted from the signal in it. The principle not only firstly demonstrates the validity and robustness of EMD mathematically, but also provides a theoretical framework for the analysis of EMD. Theoretical Principle of Oscillation Signal Decomposition Theoretical PrincipleConsider an oscillation signal x(t) varying with time t and use this signal as a signal for analysis, as shown in Figure 1. Assuming that this signal is composed of a pure oscillation component c(t) of proper rotation, with mean zero and a residual term (the trend or the baseline signal) r(t) from which the oscillation component is removed.Since c(t) is an oscillatory function with mean zero, its integral in the interval of two local maxima (or minima) points t k and t k + 2 should be equal to zero. Therefore, from Equation (1), one has

Optical Fiber FP Sensor for Simultaneous Measurement of Refractive Index and Temperature Based on the Empirical Mode Decomposition Algorithm

“…In our case, we will use the Empirical Mode Decomposition (EMD) which is a widely used technique for signal analysis processing since it is a powerful tool to decompose an original signal into a set of AM/FM zero-mean components called Intrinsic Mode Functions (IMF). Additionally, a remainder function with non-zero mean will be obtained [23,24]. The EMD algorithm is explained with detail in elsewhere [23][24][25], but for the sake of the discussion of our application, we will just to list the most important steps of the EMD procedure: 1.-A counter is initialized as j=1 and the overall R spectrum is defined as the input signal S j =R; 2.-The peaks and valleys of S j are localized; 3.-One vector with the peaks and another one with the valleys points are formed.…”

“…Additionally, a remainder function with non-zero mean will be obtained [23,24]. The EMD algorithm is explained with detail in elsewhere [23][24][25], but for the sake of the discussion of our application, we will just to list the most important steps of the EMD procedure: 1.-A counter is initialized as j=1 and the overall R spectrum is defined as the input signal S j =R; 2.-The peaks and valleys of S j are localized; 3.-One vector with the peaks and another one with the valleys points are formed. 4.-A cubic spline interpolation of the peaks points (Pk) of each one of the vectors is performed in order to obtain the superior envelope (S SEj ) and the lower envelope (S LEj ) (Figure 6a); 5.-The mean of the envelopes (Res j ) is calculated (Figure 6a); 7.-The function F j =S j −Res j is calculated and if F j is a zero-mean function, the step 8 is executed, otherwise it jumps to step 9; 8.-The function F j is accepted as an IMF, therefore set IMFj=F j (Figure 6b), j=j+1, as the new input signal set S j =Res j−1 and repeat the procedure from step 2 until Res j get as a function which is either constant, monotonic or it has only one maximum and a minimum, from which no an IMF can be extracted (Figure 6b); 9.-Set S j =F j and repeat the procedure from step 2.…”

“…Finally, it is important to mention that one advantage of the EMD algorithm is its capability to process noisy signals. In that case, the noise is separated from the spectrum and returned as an additional high frequency IMF [23][24][25].…”

A Hybrid Energy Feature Extraction Approach for Ship-Radiated Noise Based on CEEMDAN Combined with Energy Difference and Energy Entropy

“…After Empirical mode decomposition (EMD) was first proposed as a classical mode decomposition approach, it has become widely used [4,5]. The research history and the current status of EMD mainly include two parts.…”

Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems

“…What can we say about the asymptotic behavior (as t → ∞) of solutions of perturbed system (1)? This question represents one of the fundamental problems in the area of robust stability and robustness of the systems in general and so the effect of (known or unknown) perturbations on the solutions of nominal system as a potential source of instability attracts the attention and interest of scientific community for a long time in the various contexts, recently for example [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. A comprehensive overview of the most significant results on robust control theory as a stand-alone subfield of control theory and its history is presented in [17,18].…”