Original Empirical Mode Decomposition Research Articles

Precise analysis of gravitational waves from binary neutron star coalescence using Hilbert–Huang transform based on Akima spline interpolation

Abstract The equation of state (EOS) information of neutron stars (NSs) can be obtained by analyzing the post-merger phases of gravitational waves resulting from the coalescence of NS binaries. In a previous study, we proposed a method to discriminate the EOS using the Hilbert–Huang transform (HHT). The HHT comprises empirical mode decomposition (EMD) and Hilbert spectrum analysis. An essential aspect of the EMD involves the generation of envelopes through interpolating extrema values. The original EMD (used in the previous study) utilizes cubic spline (CS) interpolation. However, the CS occasionally produces pseudo oscillations and overshoots that may decrease the performance of the EMD. In this study, we propose an extended version of the HHT by substituting Akima spline interpolation for the CS. We compared the ability of the original HHT (based on the CS) and the proposed HHT (based on the Akima spline) to discriminate the EOS. The results reveal that the proposed HHT yields a more precise analysis than the original HHT. With the proposed HHT, the number of events for discriminating the EOS is enhanced by a factor of 11.4.

A Fast and Adaptive Empirical Mode Decomposition Method and Its Application in Rolling Bearing Fault Diagnosis

Although the ensemble empirical mode decomposition (EEMD) method and the complementary EEMD (CEEMD) method can greatly improve the mode mixing of the original empirical mode decomposition (EMD) method, the difference in white noise amplitude and the number of ensemble trials have a great influence on the decomposition results, and the low computational efficiency cannot meet the requirements of the online monitoring system. To solve these problems, a fast and adaptive EMD (FAEMD) method was proposed in this article, which combines the advantages of the order statistics filter (OSF) with the original EMD. First, the upper envelope and lower envelope of the original signal were drawn by using the OSF. Then, the original signal was decomposed into a series of intrinsic mode functions (IMFs) by the mean envelopes. Finally, the fault feature information was obtained by using the envelope spectrum analysis. The simulation signal and two groups of test fault signals were taken as examples to verify the effectiveness of the proposed method. Compared with EMD, CEEMD, and CEEMDAN methods, FAEMD can effectively extract the key feature information of fault signals and has strong practicability because of the low calculation cost.

Non-Contact Detection of Vital Signs Based on Improved Adaptive EEMD Algorithm (July 2022).

Non-contact vital sign detection technology has brought a more comfortable experience to the detection process of human respiratory and heartbeat signals. Ensemble empirical mode decomposition (EEMD) is a noise-assisted adaptive data analysis method which can be used to decompose the echo data of frequency modulated continuous wave (FMCW) radar and extract the heartbeat and respiratory signals. The key of EEMD is to add Gaussian white noise into the signal to overcome the mode aliasing problem caused by original empirical mode decomposition (EMD). Based on the characteristics of clutter and noise distribution in public places, this paper proposed a static clutter filtering method for eliminating ambient clutter and an improved EEMD method based on stable alpha noise distribution. The symmetrical alpha stable distribution is used to replace Gaussian distribution, and the improved EEMD is used for the separation of respiratory and heartbeat signals. The experimental results show that the static clutter filtering technology can effectively filter the surrounding static clutter and highlight the periodic moving targets. Within the detection range of 0.5 m~2.5 m, the improved EEMD method can better distinguish the heartbeat, respiration, and their harmonics, and accurately estimate the heart rate.

Empirical Mode Decomposition: Theory and Applications in Underwater Acoustics

Empirical mode decomposition (EMD) is a signal processing method that produces a data-driven time-frequency representation suited to characterize time-varying and nonlinear phenomena. In EMD, intrinsic mode functions (IMF) are sequentially estimated from the signal of interest to represent different intrinsic oscilation modes and produce an orthogonal representation of the original information. Different algorithms have been proposed for EMD estimation to deal with limitations such as mode-mixing and noise sensitivity. To obtain a frequency-domain representation, EMD is usually associated with the Hilbert transform, in this case, the method is referred to as the Hilbert-Huang transform (HHT). This paper presents a theoretical review of the fundamental aspects of both EMD and HHT, such as IMF estimation procedure and IMF orthogonality. Variations of the original EMD algorithm are also presented. Both simulated and experimental underwater acoustic signals are used to illustrate the efficiency of EMD/HHT in revealing relevant characteristics from time-varying and nonlinear information.

Micro-Motion Parameter Extraction for Ballistic Missile with Wideband Radar Using Improved Ensemble EMD Method

Micro-motion parameters extraction is crucial in recognizing ballistic missiles with a wideband radar. It is known that the phase-derived range (PDR) method can provide a sub-wavelength level accuracy. However, it is sensitive and unstable when the signal-to-noise ratio (SNR) is low. In this paper, an improved PDR method is proposed to reduce the impacts of low SNRs. First, the high range resolution profile (HRRP) is divided into a series of segments so that each segment contains a single scattering point. Then, the peak values of each segment are viewed as non-stationary signals, which are further decomposed into a series of intrinsic mode functions (IMFs) with different energy, using the ensemble empirical mode decomposition with the complementary adaptive noise (EEMDCAN) method. In the EEMDCAN decomposition, positive and negative adaptive noise pairs are added to each IMF layer to effectively eliminate the mode-mixing phenomenon that exists in the original empirical mode decomposition (EMD) method. An energy threshold is designed to select proper IMFs to reconstruct the envelop for high estimation accuracy and low noise effects. Finally, the least-square algorithm is used to do the ambiguous phases unwrapping to obtain the micro-curve, which can be further used to estimate the micro-motion parameters of the warhead. Simulation results show that the proposed method performs well with SNR at −5 dB with an accuracy level of sub-wavelength.

Real-time chatter detection via iterative Vold-Kalman filter and energy entropy

Real-time chatter detection is important in improving the surface quality of workpieces in milling. Since the process from stable cutting to chatter is characterized by the progressive variation of the vibration energy distribution, entropy has been utilized to capture the decreasing randomness of vibration signals when chatter occurs. To make such an index more sensitive to transitions of the cutting state, the entropy can be computed based on signal components obtained through signal decomposition techniques. However, the classic empirical mode decomposition (EMD) is difficult to put into practice due to its weak robustness to noises. The up-to-date variational mode decomposition (VMD) has strict requirements on a priori information about the signal and thus is not applicable either. In this paper, a novel method named the iterative Vold-Kalman filter (I-VKF) is proposed under the framework of the greedy algorithm, where the Vold-Kalman filter (VKF), a classic order tracker for rotating machinery, is improved to recursively extract each signal component. In the meantime, a spectrum concentration index–based technique is developed for the estimation of the instantaneous chatter frequency to adaptively determine the filter parameter. Numerical examples demonstrate the superiority of the I-VKF over the original VKF, EMD, and VMD, especially in the presence of strong noises. Combined with the energy entropy of extracted components and an automatically calculated threshold, the proposed strategy greatly helps in timely chatter detection, which has been verified by dynamic simulation and experiments.

Data-driven and coarse-to-fine baseline correction for signals of analytical instruments

Baseline correction is an indispensable step in the signal processing of chemical analysis instruments. With the increasing demand for on-site applications, a variety of analytical instruments require a more friendly, rapid and adaptive baseline correction method. In this paper, a data-driven and coarse-to-fine (DD-CF) baseline correction scheme mainly based on the empirical mode decomposition (EMD) algorithm is proposed. For eliminating the mode-mixing effect of the original EMD, the proposed method firstly obtains a coarse baseline estimation using automatic peak detection, elimination and interpolation; and the EMD is applied on the coarse baseline to get a fine baseline finally. We have compared this method with the adaptive iteratively reweighted Penalized Least Squares algorithm (airPLS) and the sparse representation baseline correction methods using simulated signals and experimental signals from different analytical instruments. Results indicate that the proposed DD-CF scheme can effectively estimate the baseline more accurate than the comparing methods for varies of analytical signals such as mass spectrometer, ion mobility spectrometer, gas chromatograph, etc. Furthermore, with signals of different length, different peak distributions and even from totally different instruments, the proposed method requires minimal user intervention, in which the parameters of the comparing methods should be adjusted for a wide range.

Reducing the residual error of multibeam sounding data through empirical mode decomposition based on cubic Bessel interpolation

The systematic residual errors present in multibeam echo-sounding data cause the areas of overlap between adjacent swaths to become distorted. A method is proposed in this paper to reduce the residual error of multibeam sounding data through empirical mode decomposition (EMD) based on cubic Bessel interpolation. Numerical experiments confirm that the discrepancy in the overlap between two swaths is significantly reduced after applying EMD improved by cubic Bessel interpolation compared with both the original water depth data and with data processed using conventional EMD based on cubic spline interpolation. The mean square error of the improved method is decreased by 67% and 29% compared with that of the original and conventional EMD cases, respectively. Therefore, EMD with cubic Bessel interpolation can efficiently reduce the residual error of multibeam echo-sounding data.

Analysis of Functional Brain Network in MDD Based on Improved Empirical Mode Decomposition With Resting State EEG Data.

At present, most brain functional studies are based on traditional frequency bands to explore the abnormal functional connections and topological organization of patients with depression. However, they ignore the characteristic relationship of electroencephalogram (EEG) signals in the time domain. Therefore, this paper proposes a network decomposition model based on Improved Empirical Mode Decomposition (EMD), it is suitable for time-frequency analysis of brain functional network. On the one hand, it solves the problem of mode mixing on original EMD method, especially on high-density EEG data. On the other hand, by building brain function networks on different intrinsic mode function (IMF), we can perform time-frequency analysis of brain function connections. It provides a new insight for brain function connectivity analysis of major depressive disorder (MDD). Experimental results found that the IMFs waveform decomposed by Improved EMD was more stable and the difference between IMFs was obvious, it indicated that the mode mixing can be effectively solved. Besides, the analysis of the brain network, we found that the changes in MDD functional connectivity on different IMFs, it may be related to the pathological changes for MDD. More statistical results on three network metrics proved that there were significant differences between MDD and normal controls (NC) group. In addition, the aberrant brain network structure of MDDs was also confirmed in the hubs characteristic. These findings may provide potential biomarkers for the clinical diagnosis of MDD patients.

Mean-optimized mode decomposition: An improved EMD approach for non-stationary signal processing

As an effective signal separation method of non-stationary signal, empirical mode decomposition (EMD) has been widely used in the data or time series analysis of many engineering fields. However, the decomposing result of EMD often is affected by the fitting in mean curve construction and the sifting process. In this paper, the mean-optimized mode decomposition (MOMD) procedure is proposed to enhance the performance of the original EMD in mean curve construction. Also, the proposed MOMD algorithm is compared with original EMD through analyzing two artificial signals and the analysis results demonstrate that MOMD has much more significantly improvement in decomposition performance and precision than the original EMD. Last, MOMD is introduced to the signal processing stemming from the faulty rolling bearing and the rotor system with failure. Also, the comparison of the proposed MOMD method with EMD was made and the analysis results show that MOMD obtains much more accurate IMFs and fault diagnostic effect than the original EMD method.

Empirical mode decomposition with shape-preserving spline interpolation

Empirical mode decomposition (EMD) is a popular, novel, user-friendly algorithm to decompose a given signal into its constituting components, utilizing spline interpolation. In this paper, we equip EMD with a shape-preserving interpolation scheme based on quadratic B-splines. Using numerical experiments, we show that our scheme, which we coin Geometric EMD, or GEMD, outperforms the original EMD, both qualitatively and quantitatively.

Financial time series forecasting model based on CEEMDAN and LSTM

In order to improve the accuracy of the stock market prices forecasting, two hybrid forecasting models are proposed in this paper which combine the two kinds of empirical mode decomposition (EMD) with the long short-term memory (LSTM). The financial time series is a kind of non-linear and non-stationary random signal, which can be decomposed into several intrinsic mode functions of different time scales by the original EMD and the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN). To ensure the effect of historical data onto the prediction result, the LSTM prediction models are established for all each characteristic series from EMD and CEEMDAN deposition. The final prediction results are obtained by reconstructing each prediction series. The forecasting performance of the proposed models is verified by linear regression analysis of the major global stock market indices. Compared with single LSTM model, support vector machine (SVM), multi-layer perceptron (MLP) and other hybrid models, the experimental results show that the proposed models display a better performance in one-step-ahead forecasting of financial time series.

Using filter bank property to simplify the calculations of Empirical Mode Decomposition

The Empirical Mode Decomposition (EMD) is a method to analyze non-linear non-stationary signals. EMD was proved to behave like a dyadic filter bank when operating on a broadband signal. This means that the frequency content of the Intrinsic Mode Functions (IMFs) calculated decreases from one IMF to the following one. As the frequency contents of the signal decreases, the number of points needed to represent them and the calculations they require should decrease similar to what happens in the Fast Wavelet Transform. The sifting operation of the original EMD algorithm does not benefit from this property. It evaluates the upper and lower envelopes for all data points regardless of the decrease in frequency content that occurs from one IMF to the other. In this work, we present the Computationally Adaptive Empirical Mode Decomposition (CAEMD). This method approximates data using a piecewise-cubic polynomial (PCP) having a minimal number of knots. The sifting and all the steps of the EMD are modified to be applicable to a PCP. After the calculation of each IMF, the PCP is reapproximated to one with fewer knots. Hence, this method benefits from the frequency decrease between IMFs and simplifies the required calculations and reduces the memory needed to store the IMFs. The proposed method was implemented and has demonstrated an improvement in execution time and required storage when compared to EMD. This method, although it involves approximation, can be set to produce a negligible reconstruction error comparable to the rounding error obtained from the EMD method.

ECG Signal De-noising and Baseline Wander Correction Based on CEEMDAN and Wavelet Threshold.

A novel electrocardiogram (ECG) signal de-noising and baseline wander correction method based on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and wavelet threshold is proposed. Although CEEMDAN is based on empirical mode decomposition (EMD), it represents a significant improvement of the original EMD by overcoming the mode-mixing problem. However, there has been no previous study on using CEEMDAN to de-noise ECG signals, to the authors’ best knowledge. In the proposed method, the original noisy ECG signal is decomposed into a series of intrinsic mode functions (IMFs) sorted from high to low frequency by CEEMDAN. Each IMF is then analyzed by the autocorrelation method to find out the first few high frequency IMFs containing random noise, and these IMFs should be de-noised by the wavelet threshold. The zero-crossing rate (ZCR) of all IMFs, including final residue, are computed, and the IMFs with ZCR less than a certain value are removed. Finally, the remaining IMFs are reconstructed to obtain the clean ECG signal. The proposed algorithm is validated through experiments using the MIT–BIH ECG databases, and the results show that the random noise in the ECG signal can be effectively suppressed, and at the same time the baseline wander can be corrected efficiently.

An improved EMD method based on the multi-objective optimization and its application to fault feature extraction of rolling bearing

An improved Empirical Mode Decomposition (EMD) method based on the multi-objective optimization is proposed in this paper and is applied to extract the fault feature of rolling bearing with inner and outer race fault. Firstly, a new rational spline interpolation which has a shape controlling parameter is introduced based on cubic spline interpolation. Secondly, the selection criterion concluding time domain objective and frequency objective are considered simultaneously rather than separately. Then, the Particle Swarm Optimization (PSO) is used to find out the optimal IMF and determine the optimal shape controlling parameter. The procedure of the improved EMD based on multi-objective optimization (MO-EMD) is described in this paper. The effectiveness of MO-EMD is validated by the simulation signal and the robustness of MO-EMD to noise is also investigated. At last, the MO-EMD is employed to process the vibration signals of rolling bearing with fault. By comparison with original EMD, EEMD and improved CEEMDAN, it can be found that the MO-EMD has better decomposition capability and can restrain the mode mixing apparently. Combined with envelope spectrum analysis, more fault feature information can be extracted by MO-EMD.

Hierarchical decomposition based on a variation of empirical mode decomposition

Adaptive methods of signal analysis have proved a very useful tool for analysis of non-stationary signals. This is due to the ability of these methods to adapt to the local structures of the signals being analysed, as these methods are not constrained by a fixed basis. Empirical mode decomposition (EMD) is among the more recent data-adaptive signal decomposition methods, which decomposes a given signal into modes which are hierarchically arranged based on their frequency content. In this paper, we will present a novel adaptive hierarchical decomposition scheme based on a novel modification of EMD, namely empirical mode decomposition-modified peak selection (EMD-MPS). EMD-MPS allows a time-scale-based signal decomposition, thereby allowing control over the decomposition process, not possible in the original EMD algorithm. Using time-scale-based decomposition and the properties of EMD-MPS, a given signal can be decomposed into octave frequency bands, with the centre frequency of the separated modes given by the frequency separation criterion of EMD-MPS. The spectral limits of the separated bands are established, and their relation with the centre frequency derived empirically. The method is validated by its application to simulated and real signals.

Fault Diagnosis of a Hydraulic Pump Based on the CEEMD-STFT Time-Frequency Entropy Method and Multiclass SVM Classifier

The fault diagnosis of hydraulic pumps is currently important and significant to ensure the normal operation of the entire hydraulic system. Considering the nonlinear characteristics of hydraulic-pump vibration signals and the mode mixing problem of the original Empirical Mode Decomposition (EMD) method, first, we use the Complete Ensemble EMD (CEEMD) method to decompose the signals. Second, the time-frequency analysis methods, which include the Short-Time Fourier Transform (STFT) and time-frequency entropy calculation, are applied to realize the robust feature extraction. Third, the multiclass Support Vector Machine (SVM) classifier is introduced to automatically classify the fault mode in this paper. An actual hydraulic-pump experiment demonstrates the procedure with a complete feature extraction and accurate mode classification.

An adaptively fast ensemble empirical mode decomposition method and its applications to rolling element bearing fault diagnosis

Ensemble empirical mode decomposition (EEMD) represents a significant improvement over the original empirical mode decomposition (EMD) method for eliminating the mode mixing problem. However, the added white noises generate some tough problems including the high computational cost, the determination of the two critical parameters (the amplitude of the added white noise and the number of ensemble trials), and the contamination of the residue noise in the signal reconstruction. To solve these problems, an adaptively fast EEMD (AFEEMD) method combined with complementary EEMD (CEEMD) is proposed in this paper. In the proposed method, the two critical parameters are respectively fixed as 0.01 times standard deviation of the original signal and two ensemble trials. Instead, the upper frequency limit of the added white noise is the key parameter which needs to be prescribed beforehand. Unlike the original EEMD method, only two high-frequency white noises are added to the signal to be investigated with anti-phase in AFEEMD. Furthermore, an index termed relative root-mean-square error is employed for the adaptive selection of the proper upper frequency limit of the added white noises. Simulation test and vibration signals based fault diagnosis of rolling element bearing under different fault types are utilized to demonstrate the feasibility and effectiveness of the proposed method. The analysis results indicate that the AFEEMD method represents a sound improvement over the original EEMD method, and has strong practicability.

An unconstrained optimization approach to empirical mode decomposition

Empirical mode decomposition (EMD) is an adaptive (data-driven) method to decompose non-linear and non-stationary signals into AM-FM components. Despite its well-known usefulness, one of the major EMD drawbacks is its lack of mathematical foundation, being defined as an algorithm output. In this paper we present an alternative formulation for the EMD method, based on unconstrained optimization. Unlike previous optimization-based efforts, our approach is simple, with an analytic solution, and its algorithm can be easily implemented. By making no explicit use of envelopes to find the local mean, possible inherent problems of the original EMD formulation (such as the under- and overshoot) are avoided. Classical EMD experiments with artificial signals overlapped in both time and frequency are revisited, and comparisons with other optimization-based approaches to EMD are made, showing advantages for our proposal both in recovering known components and computational times. A voice signal is decomposed by our method evidencing some advantages in comparison with traditional EMD and noise-assisted versions. The new method here introduced catches most flavors of the original EMD but with a more solid mathematical framework, which could lead to explore analytical properties of this technique.

An improved ensemble empirical mode decomposition method and its application to pressure pulsation analysis of hydroelectric generator unit

The noise-assisted method of ensemble empirical mode decomposition represents a significant improvement over the original empirical mode decomposition for eliminating the mode mixing problem. However, the ensemble empirical mode decomposition method will generate some additional problems, including the contamination of the residue noise in the signal reconstruction and the high computational cost. In this work, an improved ensemble empirical mode decomposition method, combining the complementary ensemble empirical mode decomposition and a time-saving ensemble empirical mode decomposition method by over-sampling the investigated signal, is proposed to solve these problems. By using the proposed method, the residue of the added white noise in the signal reconstruction can be eliminated completely by adding white noises in pairs with positive and negative signs to the targeted signal, and the computational cost can be saved drastically by processing the original signal with the cubic spline interpolation technique. Two simulation signals have been used to demonstrate the effectiveness of the proposed method in this article. The analysis results indicate that this method has good performance in eliminating the residue noise and reducing the costing time, which also provides more accurate decomposition results than the original ensemble empirical mode decomposition. Finally, the application to the feature extraction of pressure pulsation signal of hydroelectric generator unit shows that the proposed method has strong practicability.