Abstract
Recently proposed Hilbert–Huang transform (HHT), consisting of empirical mode decomposition (EMD) and Hilbert spectrum analysis (HSA), has been proved to be an effective approach in both scientific researches and engineering applications. However, this method is still empirical because of lacking rigorously mathematical foundation. This paper primarily focuses on providing a mathematical contribution on its sifting characteristics and instantaneous features. Firstly, the theory of the original methods as well as their advantages and restrictions are briefly reviewed. Secondly, we conduct in-depth investigations of the instantaneous parameters (IPs) and sifting ability. Thirdly, we proposed a new EMD stopping criterion, determined an optimal number of sifting iterations, employed a new masking signal to fix the mode mixing problem and investigated into the sifting property according to the extremum distribution. Finally, comparative studies, simulations and real data analyses depending on the proposed method are presented to demonstrate the validity of the novel research. The simulations illustrate that the typically defined intrinsic mode functions (IMFs) are not perfectly symmetric with zero-mean, there is still no rigorous mathematical standard to determine the “watershed” between mono and multi-component IMFs. The comparative researches indicate that unlike the prism property of the Fourier transform (FT) and the mathematical microscope property of the wavelet transform (WT), the ultimate goal of the HHT is to work as raindrops.
Highlights
The extraction of hidden physical meanings is critical for determining the underlying mechanisms involved in any given signal
Frequency analysis based on the Fourier transform (FT), time–frequency analysis such as the STFT and Cohenclass quadratic distributions, and time-scale analysis based on the wavelet transform (WT) are often applied to investigate the hidden properties of real signals
The FT can represent the meaningful spectrum property of only linear and stationary processes; the STFT and WT require a predefined basis; these integral transform techniques suffer from the Heisenberg uncertainty principal, which limits their ability to accurately measure time and frequency properties simultaneously
Summary
The extraction of hidden physical meanings is critical for determining the underlying mechanisms involved in any given signal. The key part is EMD, by which any multi-component signal can be decomposed into a set of intrinsic mode functions (IMFs) that admit a well-defined Hilbert transform (HT). Unlike the conventional TF methods, EMD is intuitive, direct and adaptive, with an entirely data-driven property [Huang and Shen (2005)] It is suitable for analyzing signals from nonstationary and nonlinear processes. It is limited to mono-component signals [Huang et al (1998)], which is the main reason why Huang et al proposed the EMD to decompose data into IMFs. For multi-component signals with more than one frequency component at a given time, directly estimating the IF will lead to a meaningless result. Generalization of the Fourier spectrum [Wu and Huang (2010)], due to the additional dimension t
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