This paper develops new theory and algorithms for 1D general mode decompositions. First, we introduce the 1D synchrosqueezed wave packet transform and prove that it is able to estimate the instantaneous information of well-separated modes from their superposition accurately. The synchrosqueezed wave packet transform has a better resolution than the synchrosqueezed wavelet transform in the time-frequency domain for separating high frequency modes. Second, we present a new approach based on diffeomorphisms for the spectral analysis of general shape functions. These two methods lead to a framework for general mode decompositions under a weak well-separation condition and a well different condition. Numerical examples of synthetic and real data are provided to demonstrate the fruitful applications of these methods.
There are three other research lines to address the mode decomposition problems of the form . The first one is the empirical mode decomposition (EMD) method initialized by in @cite_20 and refined in @cite_17 . To improve the noise resistance of the EMD methods, some variants have been proposed in @cite_10 @cite_8 . It has been shown that the EMD methods can decompose some signals into more general components of the form instead of the form in some cases (see Figure left) in @cite_19 . In this sense, the EMD methods are able to reflect the nonlinear evolution of the physically meaningful oscillations using general shape functions. However, this advantage is not stable and consistent as illustrated in Figure right. It is worth more effort to understand the EMD methods on general mode decompositions.
@cite_8 @cite_19 @cite_10 @cite_20 @cite_17
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A new Ensemble Empirical Mode Decomposition (EEMD) is presented. This new approach consists of sifting an ensemble of white noise-added signal (data) and treats the mean as the final true result. Finite, not infinitesimal, amplitude white noise is necessary to force the ensemble to exhaust all possible solutions in the sifting process, thus making the different scale signals to collate in the proper intrinsic mode functions (IMF) dictated by the dyadic filter banks. As EEMD is a time–space analysis method, the added white noise is averaged out with sufficient number of trials; the only persistent part that survives the averaging process is the component of the signal (original data), which is then treated as the true and more physical meaningful answer. The effect of the added white noise is to provide a uniform reference frame in the time–frequency space; therefore, the added noise collates the portion of the signal of comparable scale in one IMF. With this ensemble mean, one can separate scales naturall... In this paper, we present some general considerations about data analysis from the perspective of a physical scientist and advocate the physical, instead of mathematical, analysis of data. These considerations have been accompanying our development of novel adaptive, local analysis methods, especially the empirical mode decomposition and its major variation, the ensemble empirical mode decomposition, and its preliminary mathematical explanations. A particular emphasis will be on the advantages and disadvantages of mathematical and physical constraints associated with various analysis methods. We argue that, using data analysis in a given temporal domain of observation as an example, the mathematical constraints imposed on data may lead to difficulties in understanding the physics behind the data. With such difficulties in mind, we promote adaptive, local analysis method, which satisfies fundamental physical principle of consequent evolution of a system being not able to change the past evolution of the system. We also argue, using the ensemble empirical mode decomposition as an example, that noise can be helpful to extract physically meaningful signals hidden in noisy data. In this paper, we propose a variant of the Empirical Mode Decomposition method to decompose multiscale data into their intrinsic mode functions. Under the assumption that the multiscale data satisfy certain scale separation property, we show that the proposed method can extract the intrinsic mode functions accurately and uniquely. A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the ‘empirical mode decomposition’ method with which any complicated data set can be decomposed into a finite and often small number of ‘intrinsic mode functions’ that admit well-behaved Hilbert transforms. This decomposition method is adaptive, and, therefore, highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it is applicable to nonlinear and non-stationary processes. With the Hilbert transform, the ‘instrinic mode functions’ yield instantaneous frequencies as functions of time that give sharp identifications of imbedded structures. The final presentation of the results is an energy-frequency-time distribution, designated as the Hilbert spectrum. In this method, the main conceptual innovations are the introduction of ‘intrinsic mode functions’ based on local properties of the signal, which make the instantaneous frequency meaningful; and the introduction of the instantaneous frequencies for complicated data sets, which eliminate the need for spurious harmonics to represent nonlinear and non-stationary signals. Examples from the numerical results of the classical nonlinear equation systems and data representing natural phenomena are given to demonstrate the power of this new method. Classical nonlinear system data are especially interesting, for they serve to illustrate the roles played by the nonlinear and non-stationary effects in the energy-frequency-time distribution. Instantaneous frequency (IF) is necessary for understanding the detailed mechanisms for nonlinear and nonstationary processes. Historically, IF was computed from analytic signal (AS) through the Hilbert transform. This paper offers an overview of the difficulties involved in using AS, and two new methods to overcome the difficulties for computing IF. The first approach is to compute the quadrature (defined here as a simple 90° shift of phase angle) directly. The second approach is designated as the normalized Hilbert transform (NHT), which consists of applying the Hilbert transform to the empirically determined FM signals. Additionally, we have also introduced alternative methods to compute local frequency, the generalized zero-crossing (GZC), and the teager energy operator (TEO) methods. Through careful comparisons, we found that the NHT and direct quadrature gave the best overall performance. While the TEO method is the most localized, it is limited to data from linear processes, the GZC method is the m...