Data-driven Time-frequency Analysis Research Articles

Multicomponent signal analysis: Interwoven Fourier decomposition method

Fourier decomposition method (FDM) is an adaptive data-driven time-frequency analysis tool developed recently for nonlinear and non-stationary time series. This method intrinsically decomposes any signal into a small number of band-limited zero-mean orthogonal functions called as Fourier intrinsic band functions (FIBFs) via zero-phase ideal band-pass filters, and can represent the signal by these functions completely. As a result of using ideal band-pass filters, however, the FDM is not appropriate for analyzing multicomponent signals with intersecting instantaneous frequencies. It results in dispersed time-frequency representation (TFR) of such signals. To offer a solution to this inadequacy of the FDM, a generalized version of the FDM, termed interwoven FDM (IWFDM), is proposed in this paper. Unlike FDM, the IWFDM decomposes signal into analytical FIBFs through the zero-phase non-rectangular band-pass filters. The characteristics of these filters are determined such that the sum of all FIBFs guarantees the full reconstruction of signal. Similar to FDM, the IWFDM is an adaptive data-driven method. Through the IWFDM, clear TFRs of the multicomponent signals are provided. Comparisons made with the FDM and the empirical mode decomposition method on several multicomponent signals confirm the success of the IWFDM.

Data-driven time-frequency analysis method based on variational mode decomposition and its application to gear fault diagnosis in variable working conditions

The data-driven time-frequency analysis (DDTFA)-method-based initial phase selection directly influences the convergence and calculation results of the algorithm. The central frequency accuracy of each component by means of variational mode decomposition (VMD) is high, the calculation speed of VMD is fast, and the method shows satisfactory noise resistance, but the decomposed components are sensitive to noise. A VMD method that is complementary to the DDTFA method is introduced in this paper to estimate the initial phase of DDTFA, and the VMD-DDTFA method is proposed for time-varying non-stationary signals. This method first analyzes the time-varying non-stationary signal through VMD and estimates the initial phase of each signal component, then uses DDTFA to sparsely decompose the signal after phase smoothing. To examine the analytical ability of VMD-DDTFA on time-varying non-stationary signals, the five aspects of the VMD-DDTFA method, accuracy, noise resistance, efficiency, applicability and anti-mode-mixing ability, are analyzed. The VMD-DDTFA method compares with the current commonly used signal analysis methods of VMD and ensemble empirical mode decomposition (EEMD) and the comparison results confirm that VMD-DDTFA has a superior decomposition accuracy, satisfactory noise resistance and efficiency. In addition, VMD-DDTFA features strong anti-mode mixing and applicability even under strong noise. The VMD-DDTFA method is applied to the fault diagnosis of measured gear crack and broken tooth in variable working conditions. In addition, the results of the VMD-DDTFA method are compared with those obtained by VMD and EEMD; the results verify the effectiveness and superiority of the method.

Sinusoidal Signal Assisted Multivariate Empirical Mode Decomposition for Brain-Computer Interfaces.

A brain-computer interface (BCI) is a communication approach that permits cerebral activity to control computers or external devices. Brain electrical activity recorded with electroencephalography (EEG) is most commonly used for BCI. Noise-assisted multivariate empirical mode decomposition (NA-MEMD) is a data-driven time-frequency analysis method that can be applied to nonlinear and nonstationary EEG signals for BCI data processing. However, because white Gaussian noise occupies a broad range of frequencies, some redundant components are introduced. To solve this leakage problem, in this study, we propose using a sinusoidal assisted signal that occupies the same frequency ranges as the original signals to improve MEMD performance. To verify the effectiveness of the proposed sinusoidal signal assisted MEMD (SA-MEMD) method, we compared the decomposition performances of MEMD, NA-MEMD, and the proposed SA-MEMD using synthetic signals and a real-world BCI dataset. The spectral decomposition results indicate that the proposed SA-MEMD can avoid the generation of redundant components and over decomposition, thus, substantially reduce the mode mixing and misalignment that occurs in MEMD and NA-MEMD. Moreover, using SA-MEMD as a signal preprocessing method instead of MEMD or NA-MEMD can significantly improve BCI classification accuracy and reduce calculation time, which indicates that SA-MEMD is a powerful spectral decomposition method for BCI.

Sparse Time-Frequency Decomposition for Multiple Signals with Same Frequencies

In this paper, we consider multiple signals sharing the same instantaneous frequencies. This kind of data is very common in scientific and engineering problems. To take advantage of this special structure, we modify our data-driven time-frequency analysis by updating the instantaneous frequencies simultaneously. Moreover, based on the simultaneous sparsity approximation and the Fast Fourier Transform, we develop several efficient algorithms to solve this problem. Since the information of multiple signals is used, this method is very robust to the perturbation of noise and it is applicable to the general nonperiodic signals even with missing samples or outliers. Several synthetic and real signals are used to demonstrate the robustness of this method. The performances of this method seems quite promising.

A two-level method for sparse time-frequency representation of multiscale data

Based on the recently developed data-driven time-frequency analysis (Hou and Shi, 2013), we propose a two-level method to look for the sparse time-frequency decomposition of multiscale data. In the two-level method, we first run a local algorithm to get a good approximation of the instantaneous frequency. We then pass this instantaneous frequency to the global algorithm to get an accurate global intrinsic mode function (IMF) and instantaneous frequency. The two-level method alleviates the difficulty of the mode mixing to some extent. We also present a method to reduce the end effects.

Extracting a shape function for a signal with intra-wave frequency modulation.

In this paper, we develop an effective and robust adaptive time-frequency analysis method for signals with intra-wave frequency modulation. To handle this kind of signals effectively, we generalize our data-driven time-frequency analysis by using a shape function to describe the intra-wave frequency modulation. The idea of using a shape function in time-frequency analysis was first proposed by Wu (Wu 2013 Appl. Comput. Harmon. Anal. 35, 181-199. (doi:10.1016/j.acha.2012.08.008)). A shape function could be any smooth 2π-periodic function. Based on this model, we propose to solve an optimization problem to extract the shape function. By exploring the fact that the shape function is a periodic function with respect to its phase function, we can identify certain low-rank structure of the signal. This low-rank structure enables us to extract the shape function from the signal. Once the shape function is obtained, the instantaneous frequency with intra-wave modulation can be recovered from the shape function. We demonstrate the robustness and efficiency of our method by applying it to several synthetic and real signals. One important observation is that this approach is very stable to noise perturbation. By using the shape function approach, we can capture the intra-wave frequency modulation very well even for noise-polluted signals. In comparison, existing methods such as empirical mode decomposition/ensemble empirical mode decomposition seem to have difficulty in capturing the intra-wave modulation when the signal is polluted by noise.

Time-Frequency Analysis of Nonstationary Process Based on Multivariate Empirical Mode Decomposition

Currently, empirical mode decomposition (EMD) has become a popular data-driven time-frequency analysis method for nonstationary and nonlinear data. However, it is still limited to univariate data due to the number and/or scale misalignment for multivariate data. A newly developed multivariate EMD (MEMD) scheme decomposes multivariate data simultaneously and thus leads to mode alignment and minimizes mode mixing. In addition, an improved amplitude and frequency modulation (AM-FM) decomposition algorithm presented here provides an estimation of a more meaningful instantaneous amplitude and frequency than the widely used Hilbert transform (HT). Both of these facilitate development of a time-frequency analysis framework for multivariate nonstationary and nonlinear data analysis. In this paper, MEMD-based scalogram and coscalogram, and instantaneous frequency spectra and cospectra are proposed to characterize a multivariate nonstationary process. The scalogram and instantaneous frequency spectra capture spectral evolution of each component while the coscalogram and instantaneous frequency cospectra reveal embedded intermittent correlation between two components. Compared with scale-based scalogram and coscalogram, frequency-based instantaneous frequency spectra and cospectra provide a more detailed portrait of multivariate data. The effectiveness of the proposed MEMD-based time-frequency analysis framework is demonstrated by numerical examples of a thunderstorm downburst and an earthquake ground motion. Also, the results from the MEMD-based approach are compared with those based on a continuous wavelet transform, which further reinforces the effectiveness of the proposed framework.

Sparse time-frequency representation of nonlinear and nonstationary data

Adaptive data analysis provides an important tool in extracting hidden physical information from multiscale data that arise from various applications. In this paper, we review two data-driven time-frequency analysis methods that we introduced recently to study trend and instantaneous frequency of nonlinear and nonstationary data. These methods are inspired by the empirical mode decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {ɑ(t) cos(θ(t))}, where a is assumed to be less oscillatory than cos(θ(t)) and θ′ ⩾ 0. This problem can be formulated as a nonlinear l^0 optimization problem. We have proposed two methods to solve this nonlinear optimization problem. The first one is based on nonlinear basis pursuit and the second one is based on nonlinear matching pursuit. Convergence analysis has been carried out for the nonlinear matching pursuit method. Some numerical experiments are given to demonstrate the effectiveness of the proposed methods

Data-driven time–frequency analysis

In this paper, we introduce a new adaptive data analysis method to study trend and instantaneous frequency of nonlinear and nonstationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t)cos(θ(t))}, where a∈V(θ), V(θ) consists of the functions smoother than cos(θ(t)) and θ′⩾0. This problem can be formulated as a nonlinear l0 optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit for the l0 optimization problem. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide an error analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the state-of-the-art methods. We also apply our method to study data without scale separation, and data with incomplete or under-sampled data.

Sparse time frequency representations and dynamical systems

In this paper, we establish a connection between the recently developed data-driven time-frequency analysis \cite{HS11,HS13-1} and the classical second order differential equations. The main idea of the data-driven time-frequency analysis is to decompose a multiscale signal into a sparsest collection of Intrinsic Mode Functions (IMFs) over the largest possible dictionary via nonlinear optimization. These IMFs are of the form $a(t) \cos(\theta(t))$ where the amplitude $a(t)$ is positive and slowly varying. The non-decreasing phase function $\theta(t)$ is determined by the data and in general depends on the signal in a nonlinear fashion. One of the main results of this paper is that we show that each IMF can be associated with a solution of a second order ordinary differential equation of the form $\ddot{x}+p(x,t)\dot{x}+q(x,t)=0$. Further, we propose a localized variational formulation for this problem and develop an effective $l^1$-based optimization method to recover $p(x,t)$ and $q(x,t)$ by looking for a sparse representation of $p$ and $q$ in terms of the polynomial basis. Depending on the form of nonlinearity in $p(x,t)$ and $q(x,t)$, we can define the degree of nonlinearity for the associated IMF. %and the corresponding coefficients for the associated highest order nonlinear terms. This generalizes a concept recently introduced by Prof. N. E. Huang et al. \cite{Huang11}. Numerical examples will be provided to illustrate the robustness and stability of the proposed method for data with or without noise. This manuscript should be considered as a proof of concept.