Variational Nonlinear Chirp Mode Decomposition Research Articles

Time-varying characteristics analysis of bridge under moving vehicle using a modified time-frequency method with limited sensors

Investigating the bridge dynamic characteristics under operational traffic load is critical for bridge health monitoring. As vehicles traverse the bridge, the bridge frequencies present time-varying characteristics owing to the effect of vehicle-bridge dynamic interaction. The recently introduced variational nonlinear chirp mode decomposition (VNCMD) demonstrates significant benefits in identifying structural instantaneous frequencies (IFs) from the measured responses. However, its practical application is limited by the requirement for artificially setting priori parameters (i.e., upper bound of noise level and number of modal components). In this study, a modified VNCMD (MVNCMD) is proposed to overcome this limitation and is employed to identify the IFs of the bridge under moving vehicle using single sensor data. First, by modifying the optimization function, the proposed MVNCMD adopts a novel algorithm framework that eliminates the necessity for presetting the upper bound of noise level. An autoregressive spectrum-based method is then introduced to preset the number of modal components. The proposed MVNCMD is evaluated using a synthetic non-stationary signal, demonstrating its effectiveness in identifying IFs and overcoming the limitations of the original VNCMD. Furthermore, the numerical simulation involving different parameter analysis and laboratory experiments of the coupled vehicle-bridge system are conducted to validate the effectiveness of the proposed method. It is to be shown that the proposed MVNCMD method is capable of identifying the IFs of the bridge under moving vehicle using single sensor data and outperforms other time-frequency methods in terms of identification accuracy, which provides a robust tool for analyzing the time-varying characteristics of the vehicle-bridge interaction system.

Study on the frequency modulation phenomenon in the rotor system with blade-casing rub-impact fault

In an aero-engine rotor system, a blade-casing rub-impact fault can lead to a frequency modulation phenomenon of the vibration signal. This paper discusses the extraction of the modulation information in the fault signal using the variational nonlinear chirp mode decomposition (VNCMD) method to detect early rub-impact faults. The regularities of frequency modulation are revealed in both the simulated and experimental blade-casing rubbing signals. According to the frequency modulation law, an empirical formula is summarized to judge the rub-impact fault type. Then, we propose an evaluation index to judge the fault severity based on the energy theory by using the advantage of the VNCMD method that can reconstruct the decomposed fault signal. Finally, the fault severity in the simulated and experimental signals with different parameters is compared. The results show the correctness and reliability of this evaluation index. The research in this paper provides a theoretical basis for the detection and judgment of the rubbing severity of blade-casing rub-impact faults.

Wind turbine gearbox fault diagnosis via adaptive IMFogram

Wind turbines usually operate in mountainous, offshore and other field environments. As the wind speed is constantly changing, the gearboxes of wind turbines operate under variable operating conditions, resulting in a non-stationery and time-varying signal characteristic. For existing time–frequency analysis (TFA) methods, they undergo low time–frequency energy concentration as well as noise interference during the fault feature extraction. IMFogram, a recently proposed time–frequency representation (TFR) method, calculates local frequency and amplitude information of signals based on fast iterative filtering (FIF). Moreover, it is able to remove uncertainty on the time–frequency plane, allowing a long enough time window to reduce the local boundary effects associated with the signal decomposition methods. Therefore, it has potential advantages for TFA. However, the actual performance of IMFogram calculation largely depends on the decomposition effect provided by FIF and the selection of model parameters. With the purpose of obtaining a more distinct and focused TFR of the wind turbine vibration signal, a novel TFA method, namely Adaptive IMFogram (AIMFogram), is proposed in this paper. Firstly, a range of intrinsic mode functions (IMFs) are gained after the decomposition of vibration signals using an improved variational nonlinear chirp mode decomposition, which is treated as the input to AIMFogram to solve the unsatisfied result by traditional FIF under strong noise. Secondly, Renyi entropy-based particle swarm optimization is conducted to perform adaptive parameter optimization of the AIMFogram. Later, using the optimal parameter, the proposed AIMFogram achieves a high-resolution TFR of complex multi-component vibration signals. To verify whether the proposed AIMFogram is effective or not, the method was successfully employed to the fault diagnosis of the rolling bearing in an experimental bench and the gearbox in a 850-kW wind turbine.

Automated Variational Nonlinear Chirp Mode Decomposition for Bearing Fault Diagnosis

The variational non-linear chirp mode decomposition (VNCMD) requires initialization of number of modes (NMs) and instantaneous frequency (IF). This paper proposes an automated method for NM selection and IF initialization which works on the scale-space representation based automated boundary detection in magnitude spectrum (MS). The proposed automated VNCMD (AVNCMD) method is applied for bearing fault detection in which the kurtosis based dominant mode selection method is recommended. The instantaneous amplitude (IA) and IF with spectral entropy are computed from the dominant mode. Features are given to feed forward neural network classifier. Methodology is investigated on two datasets for inner race, outer race, and ball race faults detection. The proposed method classifies inner race, outer race, and ball race bearing faults with 97.52% accuracy and classifies inner race and outer race bearing fault with 100% accuracy. Efficacy of the proposed method is compared with the existing methods to justify the superiority.

An adaptive VNCMD and its application for fault diagnosis of industrial sewing machines

Due to the non-linearity and non-stationarity of sound signals from complex transmission structure and frequent start-stop of industrial equipment, such as sewing machine, its incipient faults cannot be accurately identified with classical fault diagnosis methods. Therefore, a new fault diagnosis method with an adaptive variational nonlinear chirp mode decomposition (AVNCMD) is proposed. Firstly, a pre-processing step of convex optimization with generalized minimax-concave penalty function and spline-kernelled chirplet transform is applied to obtain the time–frequency distribution (TFD) with energy concentration, and a peak frequency extension method is proposed to adaptively search initial instantaneous frequencies of AVNCMD on the TFD; then the Pearson correlation coefficient is introduced as the termination condition of the adaptive signal decomposition. Secondly, multi-domain features are extracted from the decomposed nonlinear chirp modes, and an optimal features set is constructed by the feature preference method of an improved ReliefF. Finally, the fault classification is realized by the extreme learning machine, whose input weights and biases are optimized via the salp swarm algorithm. The AVNCMD is testified to be accurate in simulation with nonlinear amplitude modulation–frequency modulation signals and be effective in actual sound signals of industrial sewing machines on four different states. Through the detailed analysis of the measured sound signals in factory, the proposed fault diagnosis method can identify four states of the industrial sewing machines with an overall identification accuracy of 95.0%.

A novel variational nonlinear chirp mode decomposition-based critical brain-region investigation for automatic emotion recognition

The field of affective computing that deals with emotion recognition from physiological information, particularly electroencephalography (EEG), is becoming more and more important. Researchers have employed these signals to identify emotions, but the majority of them only distinguish a limited number of emotional states (e.g. happiness, fear, sadness, calmness, anger, etc.). This study outlines the proposed model for classifying nine emotional states from the DREAMER database into arousal, valence, and dominance dimensions using a discrete scale. To create it, we studied variational non-linear chirp mode decomposition (VNCMD) to decompose the EEG signals into five distinct modes. 11 entropy features were extracted from each mode. Then a comparative analysis of six popular machine learning-based classifiers was performed. The VNCMD's third mode with a rotational random forest classifier gave the best performance parameters for this application. This systematic analysis revealed that the best brain region, i.e., the anterior frontal area using only 2 channels, provided the highest accuracy and F1-score for arousal: 96.07% and 97.30%, valence: 94.49% and 95.58%, and dominance: 95.83% and 97.19% categories. Results, using the DREAMER dataset, show that our model can have the highest predictive ability with AUCs of 0.98, 0.98, and 0.97 for the arousal, valence, and dominance categories, respectively. The findings of this study demonstrate that the features, brain regions, and machine learning models commonly employed in emotion classification tasks may be used in more difficult tasks, like the prediction of precise values for arousal, valence, and dominance.

Machine-learning-enabled adaptive signal decomposition for a brain-computer interface using EEG

Background and objectiveThe use of adaptive signal decomposition methods and machine learning (ML) algorithms have gained interest in biomedical applications. Brain-computer interfaces (BCIs) are one such application that uses electroencephalogram (EEG) signals to assist people with severe brain disabilities to communicate with their external environment. This paper uses recent adaptive signal decomposition methods such as empirical mode decomposition, empirical wavelet transform, variational mode decomposition, and variational non-linear chirp mode decomposition to decompose EEG signals into various modes. Three publicly available EEG datasets are employed. EEG datasets are decomposed into modes using the aforementioned methods. Linear and non-linear time-domain features are extracted from the modes. A feature set is obtained by selecting highly discriminant features using the Kruskal-Wallis test. Classification is performed using four recent ML-based algorithms. ResultsThe maximum classification accuracy achieved is 89.6 ± 4.6% and 61.1 ± 5.1% in binary and multiclass (seven classes) signals, respectively, for dataset 1. The effectiveness of the proposed model is measured by evaluating the performance parameters such as parameters recall (REC), specificity (SPEC), precision (PREC), F1-score, and area under the curve (AUC). For binary classification, REC, SPEC, PREC, F1-score, and AUC are 95%, 94%, 38%, 55%, and 89% (95% CI: [0.871–0.937]), respectively. Also, for multiclass classification, REC, SPEC, PREC, F1-score, and AUC are achieved as 60%, 93%, 60%, 60%, and 89% (95% CI: [0.873–0.971]), respectively. ConclusionsThe results show that the employed algorithms achieve excellent accuracy for the available datasets. The results also strengthen the possibility of using EEG signals for silent communication.

Variational nonlinear component decomposition for fault diagnosis of planetary gearboxes under variable speed conditions

Planetary gearboxes are extensively used in various heavy equipment and their failures may cause equipment breakdown. Since planetary gearboxes under variable speed conditions will induce time-varying complicated characteristics, which are often contaminated by strong noise and other irrelevant vibration components, fault diagnosis of varying-speed planetary gearboxes is still a challenging task. Although variational nonlinear chirp mode decomposition (VNCMD) can deal with nonstationary signals, it requires some prior knowledge, which limits its practical applications. In this paper, a novel scheme called variational nonlinear component decomposition (VNCD) is proposed as an improved version of the VNCMD to overcome the limitations of the VNCMD. Firstly, the VNCD adopts a new algorithmic framework by modifying the optimization function of the VNCMD to eliminate an upper bound determined by noise, which makes the VNCD more adaptive in practical applications. Secondly, in order to automatically determine initial frequencies and the number of signal components, we propose a novel initial frequencies estimation method based on optimizing a spectrum concentration index and a resampling technique. The initial frequencies estimation method is potential and effective under a strong noise environment, which is suitable to deal with very close time-frequency components. In addition, a high-resolution time-frequency distribution is generated to precisely exhibit time-varying fault features of planetary gearboxes and it improves the signal-to-noise ratio of raw vibration signals. Both our simulated and experimental studies have demonstrated the effectiveness of the proposed VNCD method in fault diagnosis and weak fault feature detection of planetary gearboxes under variable speed conditions.

Nonlinear Chirp Component Decomposition: A Method Based on Elastic Network Regression

Variational nonlinear chirp mode decomposition (VNCMD) is a recently proposed time-frequency analysis method that combines variational mode decomposition with the sparse representation framework. It can effectively decompose crossed and adjacent nonlinear chirp components. However, due to its optimization target's ridge regression characteristic, VNCMD cannot accurately reconstruct amplitude mutation components in fast-time-varying signals. Based on the framework of VNCMD and the sparsity and smoothness of Elastic Network Regression, we propose the nonlinear chirp component decomposition (NCCD) algorithm to decompose nonlinear chirps with strong piecewise linear characteristics accurately. We establish our new joint optimization model by adding weighted l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm regulation term to the optimized objective function of VNCMD and solves the model by alternating direction method of multipliers. Meanwhile, we use the dynamic path optimization (DPO) algorithm to extract the time-frequency distribution ridge and initialize the instantaneous frequency. Simulation signals suggest that our method has higher decomposition accuracy for fast-time-varying nonstationary signals, smaller end effect, better convergence and noise robustness, and lower computational complexity. Furthermore, experimental signals of rotor rubbing suggest that our method can be more effectively applied to process strong modulation vibration signals than other methods such as synchrosqueezing transform (SST), synchro-extracting transform (SET), time reassigned-SST (TSST), and VNCMD.

Detecting and Analyzing Nonlinearity-Caused Oscillations in Process Control Systems Using an Improved VNCMD

Nonlinearity-caused oscillations are a frequent issue in process control systems. Its incidence degrades the product quality, stability and safety of the plant. Therefore, it is important to detect and analyze the nonlinearity-caused oscillations to maintain the control performance. In this study, we propose a novel oscillation detection and analyze method based on an improved variational nonlinear chirp mode decomposition (VNCMD) algorithm. Specifically, the original VNCMD needs to manually set the mode number in advance, which is a challenging task in practice. To tackle this problem, an improved VNCMD (IVNCMD) is proposed by utilizing the approximate entropy of instantaneous frequency. Then a novel IVNCMD-based detector is developed to detect and analyze the nonlinearity-induced oscillations by revealing the harmonic content of process variable. Besides detecting the nonlinearity problem, the IVNCMD-based method can contribute in locating the root cause for nonlinearity-caused unit-wide oscillations. The proposed method is model-free and data-driven thus requiring no prior knowledge about the process dynamics. Compared with the latest related methods, the proposed method is able to process nonstationary oscillations and providing corresponding time-frequency information. The effectiveness and advantages are demonstrated through simulations as well as industrial applications.

Iterative nonlinear chirp mode decomposition: A Hilbert-Huang transform-like method in capturing intra-wave modulations of nonlinear responses

Intra-wave modulations are a class of inherent nonlinear characteristics, exhibiting a fast oscillating instantaneous frequency and/or amplitude of responses. With the aid of the well-known Hilbert-Huang transform (HHT), such phenomena have been observed and utilized in many practical nonlinear systems including mechanical, power, ocean and even human biological systems. However, the empirical nature of the HHT makes the results physically uninterpretable and sensitive to perturbations of noise. Variational nonlinear chirp mode decomposition (VNCMD) is a recently proposed tool for analyzing wideband multicomponent signals, including intra-wave modulated responses. On the other hand, the VNCMD has strict requirements on the priori information of the signal. In this paper, we combine the framework of the VNCMD with that of the HHT, by replacing the joint-optimization scheme of the VNCMD with a recursive procedure adopted in the HHT. In this way, the new method becomes more adaptive without losing the rigorous mathematical foundation. This construction leads to a descendant of VNCMD, named the iterative nonlinear chirp mode decomposition (INCMD). Through dynamic simulations and applications to real data, it is demonstrated that the INCMD considerably outperforms state-of-the-art techniques of the same class. Using the INCMD, intra-wave modulations can be captured with high accuracy and strong noise-robustness. Extracted modulation features by the INCMD greatly help to detect and identify nonlinear systems.

MMT-HEAR: Multiple Moving Targets Heartbeats Estimation and Recovery Using IR-UWB Radars.

Populations around the world are rapidly ageing. Age-friendly environments address the significance of continuous inhome vital sign monitoring. Impulse Radio Ultra-WideBand (IR-UWB) radar serves as a household healthcare assistance, providing non-contact vital sign monitoring without privacy issues and illumination limitation. However, the body movements bring difficulty in extracting heartbeat from radar signals, let alone obtaining complete information with body occlusions among multiple targets. This paper proposes a Multiple Moving Targets Heartbeat Estimation And Recovery (MMT-HEAR) approach to extract vital signs using IR-UWB radars. CLEAN and Joint Probability Data Association (JPDA) algorithms are firstly performed on each radar to estimate target-to-antenna distances of multiple targets. Considering signal obstruction and attenuation for targets occluded by others, the location-based distance optimization is proposed to refine these distances by combining information from all radars. Then the mapping from signal amplitudes to refined distances is introduced and combined with the Variational Nonlinear Chirp Mode Decomposition (VNCMD) to extract vital signs with body movements. To the best of our knowledge, this is the first attempt to monitor vital signs of multiple moving targets with radars. The averaging accuracy for two moving targets heartbeat monitoring during a 20-minutes observation is 85.93% with MMT-HEAR. Compared to two other conventional methods, the MMT-HEAR approach yields improvements of 16.11% and 10.16%, revealing the efficiency and robustness of this proposed approach.

Rotating Angle Estimation for Hybrid Stepper Motors With Application to Bearing Fault Diagnosis

Hybrid stepper motors, which convert the input pulse train into rotating angle increments by using an open-loop control strategy, are widely used in industrial applications. The bearing is a key and vulnerable component in stepper motors, and the bearing faults affect the positioning accuracy and output torque of motors. Order analysis is an effective method in bearing fault diagnosis under varying speed conditions. The rotating angle curve is necessary for order analysis, but angle or speed sensors are not installed on hybrid stepper motors. In this article, a method of estimating the rotating angle from motor current is proposed and applied to bearing fault diagnosis. First, a variational nonlinear chirp mode decomposition method is introduced to decompose the current signal and extract a dominant mode that has a strong correlation with motor rotation. Second, the rotating angle is calculated, and the synchronous vibration signal is resampled using the rotating angle curve. Finally, the bearing fault type is identified from the envelope order spectrum of the resampled signal. The proposed method can accurately diagnose bearing faults and has potential applications in condition monitoring of stepper motors.

An Optimization Tendency Guiding Mode Decomposition Method for Bearing Fault Detection Under Varying Speed Conditions

Tacholess order tracking techniques based on time-frequency (TF) ridge detection have been extensively used in bearing fault diagnosis under varying speed conditions for decades. However, the signal components of a fault bearing related to shaft rotational frequency (SRF) is difficult to be accurately extracted by these methods because of TF resolution limitation and strong noise interference. A new TF decomposing algorithm, that is, variational nonlinear chirp mode decomposition (VNCMD) is effective to extract the time-varying feature under limited TF resolution. However, its performance is influenced by prior knowledge of initial parameters. Besides, ridge information hidden in noise is difficult to be mined effectively, which increases the difficulty of ridge extraction. In this study, a feature isolation technology is proposed to enhance fault-related features and reduce the interference of noise and irrelevant components. Then inspired by the decomposing properties research on the convergence characteristics of VNCMD, an optimization tendency guiding mode decomposition (OTGMD) method is proposed to track the instantaneous frequency (IF) of fault-related mode, which can alleviate the personnel experience requirement and is not affected by the set of TF resolution. The proposed method mainly consists of three steps. First, SRF-related information is highlighted through low-pass filtering, and the dominant IF is achieved through ridge detection method. Subsequently, for the convenience of mode extraction, the fault characteristic is augmented through iterative envelope analysis. Then, the OTGMD optimization strategy is developed to gradually decompose the target mode on the basis of the above process. Finally, a stopping criterion based on characteristic frequency ratios (CFRs) is constructed to adaptively terminate the iteration process. Simulation and experiments demonstrate that the proposed method is effective and suitable for bearing fault diagnosis under varying speed conditions.

Detection and identification of cutting chatter based on improved variational nonlinear chirp mode decomposition

During the turning process of the lathe, the cutting chatter may be generated with the increase of cutting vibration amplitude, which is harmful to machining precision of the turning process. In order to effectively avoid the influence of cutting chatter during the turning process, the improved variational nonlinear chirp mode decomposition (VNCMD) algorithm is presented for the detection and identification of cutting chatter. The wideband and weak characteristic of the vibration signal are considered by the improved algorithm, which cannot only overcome the mode mixing and pseudo-component problems of the empirical mode decomposition (EMD) algorithm but also overcome the drawbacks of the wideband signal extraction. To eliminate the influence of uncertainty of the number of signal components on the decomposition of the VNCMD algorithm, the cross-correlation coefficient method is used to determine the optimal number of signal intrinsic mode functions. And the improved algorithm is further applied to the detection of the cutting chatter, where the fourth-order cumulant, permutation entropy, and instantaneous frequency of the signal intrinsic mode functions are extracted as the multi-feature vector for the cutting chatter. As the results show, it can effectively identify the existence and extent of cutting chatter.

A Coarse TF Ridge-Guided Multi-Band Feature Extraction Method for Bearing Fault Diagnosis Under Varying Speed Conditions

Currently, most of the vibration signal analysis methods for bearing fault diagnosis under varying speed conditions are based on the resampling technology with shaft rotational frequency (SRF). However, the SRF obtained by a fixed tachometer or time-frequency (TF) ridge detection reduces measuring flexibility or introduces errors inevitably. In this paper, a multi-band feature extraction method using coarse TF ridge-guided variational nonlinear chirp mode decomposition (VNCMD) is proposed for bearing fault diagnosis under varying speed conditions. Specifically, the proposed method is conducted as follows. First, the low-frequency component (LFC) and resonance component are extracted by the low-pass filtering and the fast kurtogram method, respectively, to alleviate the noise interference. Second, the coarse TF ridges are identified by a tractable ridge estimation method that is based on the TF representation for preliminary selection of the initial instantaneous frequency. Third, the coarse TF ridge-guided VNCMD is constructed to track the SRF and instantaneous fault characteristic frequency (IFCF) from the envelope signals of the LFC and the resonance component, respectively. Finally, the characteristic frequency ratio is computed on the basis of the values of SRF and IFCF to determine the fault type of ball bearing without resampling. The simulation studies and experimental verifications confirm that the proposed method can accurately locate bearing defect types and outperforms some existing methods.

Detection of rub-impact fault for rotor-stator systems: A novel method based on adaptive chirp mode decomposition

For rotating machineries, rub-impact between the surfaces of the rotor and the stator during the rotation is a common and severe fault. The rub-impact fault will lead to a frequency-modulated (FM) vibration signal with a fast fluctuating instantaneous frequency (IF). It is difficult to extract the fast fluctuating IF feature due to the limited resolution of the current time-frequency (TF) methods. The recently proposed variational nonlinear chirp mode decomposition (VNCMD) method shows promising advantages in analyzing strongly FM signals. However, the VNCMD solves a joint-estimation problem which requires the prior information of the number of the signal components and may cause instability issues. In this paper, a tractable version of the VNCMD, called adaptive chirp mode decomposition (ACMD), is introduced to extract the fast fluctuating IF of the vibration signal from the rub-impact rotor. The ACMD employs a greedy algorithm to catch each signal component individually. Moreover, using the estimated instantaneous amplitude and the IF by ACMD, we obtain a high-resolution adaptive TF spectrum which can clearly represent the rub-impact feature of the vibration signal. The effectiveness of the fault detection method based on ACMD is demonstrated by dynamic simulations at first. Then, the method is applied to vibration signals of a heavy oil catalytic cracking machine set indicating its usefulness in early fault detection and multi-feature extraction.

HEAR: Approach for Heartbeat Monitoring with Body Movement Compensation by IR-UWB Radar.

Further applications of impulse radio ultra-wideband radar in mobile health are hindered by the difficulty in extracting such vital signals as heartbeats from moving targets. Although the empirical mode decomposition based method is applied in recovering waveforms of heartbeats and estimating heart rates, the instantaneous heart rate is not achievable. This paper proposes a Heartbeat Estimation And Recovery (HEAR) approach to expand the application to mobile scenarios and extract instantaneous heartbeats. Firstly, the HEAR approach acquires vital signals by mapping maximum echo amplitudes to the fast time delay and compensating large body movements. Secondly, HEAR adopts the variational nonlinear chirp mode decomposition in extracting instantaneous frequencies of heartbeats. Thirdly, HEAR extends the clutter removal method based on the wavelet decomposition with a two-parameter exponential threshold. Compared to heart rates simultaneously collected by electrocardiograms (ECG), HEAR achieves a minimum error rate 4.6% in moving state and 2.25% in resting state. The Bland–Altman analysis verifies the consistency of beat-to-beat intervals in ECG and extracted heartbeat signals with the mean deviation smaller than 0.1 s. It indicates that HEAR is practical in offering clinical diagnoses such as the heart rate variability analysis in mobile monitoring.

Adaptive chirp mode pursuit: Algorithm and applications

Abstract Signal decomposition has drawn growing interest in various applications these days. Some recent decomposition methods, like the variational mode decomposition (VMD) and the variational nonlinear chirp mode decomposition (VNCMD), employ a joint-optimization scheme to accurately estimate all the signal modes underlying a signal. Some existing issues for these methods are: requiring prior knowledge of the number of the signal modes, empirically setting the bandwidth parameter to a fixed value, and lacking of an effective initialization scheme for the optimization algorithm. To address these issues, this paper presents a new decomposition approach called adaptive chirp mode pursuit (ACMP). Similar to the matching pursuit method, the ACMP captures signal modes one by one in a recursive framework. In addition, an adaptive bandwidth parameter updating rule and an instantaneous frequency initialization method based on Hilbert transform are incorporated into the ACMP. Several examples including simulated signals as well as real-life applications are provided to show the effectiveness and advantages of the ACMP.

Nonlinear Chirp Mode Decomposition: A Variational Method

Variational mode decomposition (VMD), a recently introduced method for adaptive data analysis, has aroused much attention in various fields. However, the VMD is formulated based on the assumption of narrow-band property of the signal model. To analyze wide-band nonlinear chirp signals (NCSs), we present an alternative method called variational nonlinear chirp mode decomposition (VNCMD). The VNCMD is developed from the fact that a wideband NCS can be transformed to a narrow-band signal by using demodulation techniques. Our decomposition problem is, thus, formulated as an optimal demodulation problem, which is efficiently solved by the alternating direction method of multipliers. Our method can be viewed as a time–frequency filter bank, which concurrently extracts all the signal modes. Some simulated and real data examples are provided showing the effectiveness of the VNCMD in analyzing NCSs containing close or even crossed modes.