Research On The Theory And Approach Of Key Issues For Empirical Mode Decomposition

With the development of signal detection and processing technology, non-stationary and non-linear signals are widely used in the field of fault diagnosis, system identification and biomedicine. The ability of extracting non-stationary characteristic usually affects the performance of the entire system. Time-frequency distribution is a non-stationary and nonlinear feature that have gained more and more attention. Hilbert-huang transform (HHT) provides an adaptive and effective means to extract the time-frequency feature of the input signal. The core of the HHT method is the empirical mode decomposition (EMD) algorithm. Since the lack of mathematical framework, EMD has some problems affecting the quality of decomposition. This dissertation focus on improving the theoretical framework of EMD, aims to solve the frequency resolution, modal aliasing and low sampling rate problem. The main contributions of our work are as follows:1. A local mean calculation algorithm is proposed using the zero crossings of higher order derivative. Instead of generating the upper and lower envelopes, the local mean signal is obtained by interpolating the feature points. Two theoretical propositions are discussed. First, the zero crossings of even order derivative are related to the ideal local mean signal. Second, increasing the derivative order can improve the frequency separation of EMD. Theoretical analysis shows that the zero crossings of higher order derivative reflects the local oscillation. Experimental results show that the proposed algorithm is able to effectively improve the separation performance of the EMD method for linear signals. The numerical results are consistent with the theoretical expectation.2. A nonlinear filter based on B-spline fitting of non-equally spaced knot is designed. Then an adaptive sifting algorithm based on this nonlinear filter is proposed. The sifting algorithm is based on the following propositions. First, for a signal of a symmetric or approximately symmetry envelopes, the local time scale of the ideal local mean signal can be obtained from the time scale of its envelope. Second, for a signal of a asymmetric envelopes, the inflections of the input signal is related to the time scale of the ideal mean signal. Third, the least square fitting (B-spline fitting) of equal spaced knot behaves similar to a low pass filter, and the cutoff frequency of the filter is determined by the distance between the knots. Fourth, non-equally spaced knot B-spline fitting behaves similar to a time-varying low-pass filter, and the local cutoff frequency is determined by the local knot spacing. Based on the above proposition, an adaptive algorithm for estimating the local mean is proposed. Experimental results show that the algorithm has a high separation perfomance on the non-stationary signal.3. A preprocessing algorithm is proposed to solve the mode mixing problem. This algorithm adaptively calculate the local mean according to the extrema distribution. The proposed algorithm is compared with the ensemble empirical mode decomposition (EEMD). Although EEMD is able to guarantee the resulting integrity of time scale in a certain degree, they lost the advantages of locality. Moreover, they are even time consuming. Comparison results show that the proposed algorithm is able to effectively eliminate the interference of the noise and preserve the locality of EMD. Inspired by the properties of the B-spline filter, An algorithm based on global time scale is proposed. The proposed method adopts a three-trial-timescales framework that calculates the local mean. Theoretical analysis shows that the proposed method is able to achieve a good convergence. The experimental results show that this method has higher frequency resolution compared with EEMD. Besides it demonstrate a better performance of noise resistance.4. A local mean estimation method based on extrema re-sampling is proposed. In compare with the EMD method, the proposed method does not rely on an exact location and value of extrema. As a result, our method is less susceptible to low sampling rates. Experiments show that proposed method is able to achieve a high performance under sampling rates close to the Nyquist rate. It also demonstrates that the proposed method is superior to existing interpolation methods in separation performance. An additional definition to intrinsic mode function (IMF) is also developed under low sampling rates. One of the necessary conditions for defining the intrinsic mode function (IMF) is that the envelope of the signal must be symmetric about the time axis. In this work, we found that this necessary conditions are not always valid under low sampling rates. Then we propose a definition of the IMF based on instantaneous bandwidth, so that IMF can work under low sample rates. Experimental results confirm the the definition.