Time-Frequency Representation Based on Robust Local Mean Decomposition

Fourier transform based frequency representation makes an underlying assumption of stationarity and linearity for the target signal whose spectrum is to be computed, and thus it is unable to track time varying characteristics of non-stationary signals that also widely exist in the physical world. Time-frequency representation (TFR) is a technique to reveal useful information included in the signals, and thus the TFR methods are very attractive to the scientific and engineering world. Local mean decomposition (LMD) is a TFR technique used in many fields, e.g. machinery fault diagnosis. Similar to Hilbert-Huang transform, it is an alternative approach to demodulate amplitude-modulation (AM) and frequency-modulation (FM) signals into a set of components, each of which is the product of an instantaneous envelope signal and a pure FM signal. TFR can then be derived by the instantaneous envelope signal and the pure FM signal. However, LMD based TFR technique still has two limitations, i.e. the end effect and the mode mixing problems. Solutions for the two limitations greatly depend on three critical parameters of LMD that are boundary condition, envelope estimation, and sifting stopping criterion. Most reported studies aiming to improve performance of LMD have focused on only one parameter a time, and thus they ignore the fact that the three parameters are not independent to each other, and all of them are needed to address the end effect and the mode mixing problems in LMD. In this paper, a robust optimization approach is proposed to improve performance of LMD through an integrated framework of parameter selection in terms of boundary condition, envelope estimation, and sifting stopping criterion. The proposed optimization approach includes three components. First, the mirror extending method is employed to deal with the boundary condition problem. Second, moving average is used as the smooth algorithm for envelope estimation of local mean and local magnitude in LMD. The fixed subset size is the only parameter that usually needs to be predefined with a prior knowledge. In this step, a self-adaptive method based on the statistics theory is proposed to automatically determine a fixed subset size of moving average for accurate envelope estimation. Third, based on the first and the second steps, a soft sifting stopping criterion is proposed to enable LMD to achieve a self-adaptive stop for each sifting process. In this last step, we define an objective function that considers both global and local characteristics of a target signal. Based on the objective function, a heuristic mechanism is proposed to automatically determine the optimal number of sifting iterations in the sifting process. Finally, numerical simulation results show the effectiveness of the robust LMD in terms of mining time-frequency representation information.