Quadratic Phase Coupling Research Articles

Nonlinear vibro-acoustic modulation for microcrack detection of steel strands based on S-transform bispectrum

Fatigue crack detection is an important issue to ensure the safety of steel strands. The key to solve this problem is to extract the nonlinear response generated by fatigue cracks. In this paper, the nonlinear VAM method is used to detect the structural cracks. The structure with fatigue cracks is excited using low-frequency pumping and high-frequency probing, and the power spectrum analysis of the modulated signal is carried out. In view of the shortcomings of spectral analysis, a new method combining S-transform and bispectrum is proposed, which is called S-transform bispectrum. S transform contains the phase factor, which can retain the absolute phase characteristics of each frequency, and has good time–frequency multi-scale focusing performance. The bispectrum can suppress Gaussian noise, retain phase information, and quantitatively describe the quadratic phase coupling in the signal. Then the simulation and experiment of damaged straight rod, helical rod, and steel strands are carried out. The results show that the proposed method can effectively detect the nonlinear features by using the sideband peaks in the S-transform bispectrum three-dimensional plot, and the nonlinear features are important to identify the structure with damage. At the same time, in order to prove the ability of S-transform bispectrum, an S-transform bispectrum detector is used to verify it, which is superior to the spectrum in terms of its ability to localize modulation sidebands. The proposed S-transform bispectrum has a good application prospect and provides a new tool for structural damage detection.

Motor Current Time-Varying Quadratic Phase Coupling Analysis and Its Application in Traction Motor Fault Detection Under Varying-Speed Condition

Motor Current Time-Varying Quadratic Phase Coupling Analysis and Its Application in Traction Motor Fault Detection Under Varying-Speed Condition

Detection of quadratic phase coupling by cross-bicoherence and spectral Granger causality in bifrequencies interactions

Quadratic Phase Coupling (QPC) serves as an essential statistical instrument for evaluating nonlinear synchronization within multivariate time series data, especially in signal processing and neuroscience fields. This study explores the precision of QPC detection using numerical estimates derived from cross-bicoherence and bivariate Granger causality within a straightforward, yet noisy, instantaneous multiplier model. It further assesses the impact of accidental statistically significant bifrequency interactions, introducing new metrics such as the ratio of bispectral quadratic phase coupling and the ratio of bivariate Granger causality quadratic phase coupling. Ratios nearing 1 signify a high degree of accuracy in detecting QPC. The coupling strength between interacting channels is identified as a key element that introduces nonlinearities, influencing the signal-to-noise ratio in the output channel. The model is tested across 59 experimental conditions of simulated recordings, with each condition evaluated against six coupling strength values, covering a wide range of carrier frequencies to examine a broad spectrum of scenarios. The findings demonstrate that the bispectral method outperforms bivariate Granger causality, particularly in identifying specific QPC under conditions of very weak couplings and in the presence of noise. The detection of specific QPC is crucial for neuroscience applications aimed at better understanding the temporal and spatial coordination between different brain regions.

Parkinsonian Tremor Detection with Compact Convolutional Transformer from Bispectrum Representation of tri-Axial Accelerometer Signals.

After the breakthroughs of Transformer networks in Natural Language Processing (NLP) tasks, they have led to exciting progress in visual tasks as well. Nonetheless, there has been a parallel growth in the number of parameters and the amount of training data, which led to the conclusion that Transformers are not suited for small datasets. This paper is the first to convey the feasibility of Compact Convolutional Transformers (CCT) for the prediction of Parkinsonian postural tremor based on the Bispectrum (BS) representation of IMU accelerometer time series. The dataset includes tri-axial accelerometer signals collected unobtrusively in-the-wild while subjects are on a phone call, and labelled by neurologists and signal processing experts. The BS is a noise-immune, higher-order representation that reflects a signal's deviation from Gaussianity and measures quadratic phase coupling. We performed comparative classification experiments using the CCT, pre-trained CNNs such as VGG-16 and ResNet-50, and the conventional Vision Transformer (ViT). Our model achieves competitive prediction accuracy and F1 score of 96% with only 1.016 M trainable parameters, compared to the ViT with 21.659 M trainable parameters, in a five-fold cross-validation scheme. Our model also outperforms pre-trained CNNs such as VGG-16 and ResNet-50. Furthermore, we show that the performance gains are maintained when training on a larger dataset of BS images. Our effort here is motivated by the hypothesis that data-efficient transformers outperform transfer learning using pre-trained CNNs, paving the way for promising deep learning architecture for small-scale, novel and noisy medical imaging datasets.Clinical relevance- Novel deep learning model for unobtrusive prediction of Parkinsonian Postural Tremor from Bispectrum image representation of tri-axial accelerometer signals collected in-the-wild.

Dynamic response of floating offshore wind turbines under freak waves with large crest and deep trough

Floating offshore wind turbines (FOWTs) are at high risk being attacked by freak waves with profiles of large crest or deep trough, which could damage the structures or weaken the efficiency of power generation. This study developed a numerical model to address the dynamic response of an in-place FOWT under the action of freak waves with large crest and deep trough. Based on the modulation method of freak wave profiles, various random wave trains embedded freak waves were firstly generated in the numerical wave tank. Noted that these waves generated match Gaussian seas. Then, the effect of freak waves was investigated extensively through the FOWT motions, axial acceleration on nacelle, tether forces as well as wave loads and wave fields in time domain and time-frequency domain. The results show that the occurrence of freak waves significantly amplified the dynamic response of FOWT and the impact lasted for approximately 20 spectral peak periods. Under freak waves, the effect of quadratic phase coupling on nacelle acceleration was first observed. The large crest led to a larger horizontal excursion. While the deep trough caused a larger pitch inducing a larger axial acceleration on nacelle. In addition, the nonlinear interaction (between FOWT and freak waves) caused by the large crest was stronger than that caused by deep trough.

Evolution of turbulent kinetic energy during the entire sandstorm process

Abstract. An adaptive segmented stationary method for non-stationary signal is proposed to reveal the turbulent kinetic energy evolution during the entire sandstorm process observed at the Qingtu Lake Observation Array. Sandstorms, which are a common natural disaster, are mechanically characterized by a particle-laden two-phase flow experiencing wall turbulence, with an extremely high Reynolds number and significant turbulent kinetic energy. Turbulence energy transfer is important to the understanding of sandstorm dynamics. This study indicates that large-scale and very large-scale coherent structures originally exist in the rising stage of sandstorms with a streamwise kinetic energy of 75 % (at z/δ=0.2) rather than gradually forming. In addition to carrying a substantial portion of energy, the very large-scale motions are active structures with strong nonlinear energy transfer. These structures gain energy from strong nonlinear interaction. As a sandstorm evolves, these large structures are gradually broken by quadratic phase coupling, with the energy fraction reducing to 40 % (at z/δ=0.2) in the declining stage. The nonlinear process in the steady and declining stages weakens and maintains a balanced budget of energy. The systematic bispectrum results provide a new perspective for further insight into sandstorms.

Evolution of turbulent kinetic energy during the entire sandstorm process

Abstract. An adaptive segmented stationary method for non-stationary signal is proposed to reveal the turbulent kinetic energy evolution during the entire sandstorm process observed at the Qingtu Lake Observation Array. Sandstorm which is a common natural disaster is mechanically characterized by a particle-laden two-phase flow experiencing wall turbulence, with an extremely high Reynolds number and significant turbulent kinetic energy. Turbulence energy transfer is important to the understanding of sandstorm dynamics. This study indicates that large-/very-large-scale coherent structures originally exist in the rising stage of sandstorms with a streamwise kinetic energy of 75 % rather than gradually forming. In addition to carrying a substantial portion of energy, the very-large-scale-motions are active structures with strong nonlinear energy transfer. These structures gain energy from strong nonlinear interaction. As sandstorm evolves, these large structures are gradually broken by quadratic phase coupling, with the energy fraction reducing to 40 % in the declining stage. The nonlinear process in the steady and declining stages weakens and maintains a balanced budget of energy. The systematic bispectrum results provide a new perspective for further insight of sandstorms.

Locomotive Gear Fault Diagnosis Based on Wavelet Bispectrum of Motor Current

The motor current signature analysis (MCSA) provides a nondestructive method for gear fault detection. The motor current in the faulty gear system not only involves the frequency information related to the fault but also the electric supply frequency and gear meshing-related frequency, which not only contaminates the fault characteristics but also increases the difficulty of fault extraction. To extract the fault characteristic frequency effectively, an innovative method based on the wavelet bispectrum (WB) is proposed. Bispectrum is an effective tool for identifying the fault-related quadratic phase coupling (QPC). However, it requires a large amount of data averaging, which is not suitable for short data analysis. In this paper, the wavelet bispectrum is introduced to motor current analysis and the problem of QPC extraction under variable speed conditions is preliminarily solved. Furthermore, a fault diagnostic approach for locomotive gears using the wavelet bispectrum and wavelet bispectral entropy is suggested. The presented method was effectively applied to the locomotive online running operations, and faults of the drive gear were successfully diagnosed.

Age-dependent cross frequency coupling features from children to adults during general anesthesia

BackgroundThe frequency coupling characteristics in electroencephalogram (EEG) induced by anesthetics have been well studied in adults, but the investigation of the age-dependent cross frequency coupling features from children to adults is still lacking. MethodsWe analyzed EEG signals recorded from pediatric to adult patients (n = 131), separated into six age groups: <1 year (n = 15), 1–3 years (n = 23), 3–6 years (n = 19), 6–12 years (n = 18), 12–18 years (n = 16), and 18–45 years (n = 40). Age related EEG power and cross frequency coupling analysis (phase amplitude coupling (PAC) and quadratic phase coupling) of data from maintenance of a surgical state of anesthesia (MOSSA) was conducted. Also, for patients of ages less than 6 years, we analyzed the performance of cross frequency coupling derived indices in distinguishing the states of wakefulness, MOSSA, and recovery of consciousness (ROC). Results(1) During MOSSA, EEG power substantially increased with age from infancy to 3–6 years then decreased with age in the theta-gamma frequency bands. The infant group (<1 year) had the highest slow oscillation (SO) power among all age groups. (2) The distinct PAC pattern is absent in patients less than 1 year of age both in SO-alpha and delta-alpha frequency band coupling during propofol induced unconsciousness. The modulation index between delta and alpha oscillations in MOSSA increased with age. (3) Wavelet bicoherence derived indices reach their peaks in the 3–6 years group and then decrease with age growth. (4) The Diag_En index (normalized entropy of the diagonal bicoherence entries of the bicoherence matrix) performed the best at distinguishing different states for ages less than 6 years (p<0.05). ConclusionsThe combination of propofol induction and sevoflurane maintenance exhibited age-dependent EEG power spectra, PAC, and bicoherence, likely related to brain development. These observations suggest new rules for infant and child brain state monitoring during general anesthesia are needed.

Bispectral mode decomposition of nonlinear flows

Triadic interactions are the fundamental mechanism of energy transfer in fluid flows. This work introduces bispectral mode decomposition as a direct means of educing flow structures that are associated with triadic interactions from experimental or numerical data. Triadic interactions are characterized by quadratic phase coupling which can be detected by the bispectrum. The proposed method maximizes an integral measure of this third-order statistic to compute modes associated with frequency triads, as well as a mode bispectrum that identifies resonant three-wave interactions. Unlike the classical bispectrum, the decomposition establishes a causal relationship between the three frequency components of a triad. This permits the distinction of sum- and difference-interactions, and the computation of interaction maps that indicate regions of nonlinear coupling. Three examples highlight different aspects of the method. Cascading triads and their regions of interaction are educed from direct numerical simulation data of laminar cylinder flow. It is further demonstrated that linear instability mechanisms that attain an appreciable amplitude are revealed indirectly by their difference-self-interactions. Applicability to turbulent flows and noise-rejection is demonstrated on particle image velocimetry data of a massively separated wake. The generation of sub- and ultra-harmonics in large eddy simulation data of a transitional jet is explained by extending the method to cross-bispectral information.

Tonal-timbral based bicoherence analysis of a viola da gamba suite for wolf tone characterization

Wolf tonal and timbral modulation of excited strings on viola da gambas are an annoying occurrence having a serious impact on the overall delivery of a musical composition. Fourier spectral analysis of this phenomenon obfuscates deep understanding through its exclusive focus on Gaussianity and disregard of frequency and quadratic phase coupling. With the desire to exhume nonlinear acoustical structure, an experimental study was conducted to investigate the efficacy of bicoherence analysis in understanding wolf tonal and timbral structure existing in two viola da gambas of similar dimension but different construction tuned to A 415 Hz. Bicoherence analysis of digitally recorded notes played on the C string and sampled at 44.1 kHz revealed significant but medium bicoherence spectral levels in variable patterns surpassing 50%. These levels appeared on unique notes for each instrument, D# and F, and were independent of amplitude providing evidence of non-Gaussian wolf tone warbling which was not always clearly audible. This was possibly due to phase coupling at very low and high frequencies lying on the perimeter of the audible music range. Results have implications for improved understanding of the relationship between viola da gamba construction and acoustical characteristics.

Correlation analysis between forced oscillation modes caused by wind power

With the increase of the grid-connected capacity of wind power, the influence of power system forced oscillation caused by wind power is becoming bigger and bigger, so it is necessary to understand its production mechanism and general characteristics. In this paper, the nonlinear and non-stationary signal generated by the unstable wind speed is analyzed to speculate the characteristics of wind-induced oscillation. Secondly, the bispectrum analysis and the bicoherence analysis are introduced. The obtained bispectrum and bicoherence coefficients are used to analyze the oscillation caused by wind power and the quadratic phase coupling relationship between oscillation modes. Finally, the simulation is operated considering small wind farm connected to a 4-generator-2-area (4G2A) system. The simulation results verify the correctness of the proposed speculation and the effectiveness of the proposed method.

Anaesthesia-dependent oscillatory EEG features in the super-elderly

ObjectiveAlthough the characteristics of electroencephalograms (EEGs) have been reported to change with age, anaesthesia-dependent oscillatory features and reactivity of the super-elderly EEG to anaesthesia have not been examined in detail. MethodsParticipants comprised 20 super-elderly patients (age; mean ± standard deviation, 87.1 ± 3.8 years) and 20 young adult patients (35.5 ± 8.5 years). At three levels of sevoflurane anaesthesia (minimum alveolar concentration [MAC] of 0.3, 0.7, and 1.4), oscillatory features of the frontal EEG were examined by analysing quadratic phase coupling (bicoherence) and power spectrum in α and δ-θ areas and compared in an anaesthesia-dependent manner, using the Friedman test. ResultsAmong super-elderly individuals, bicoherences in the δ-θ area showed anaesthesia-dependent increases (median [interquartile range], 12.9% [5.2%], 19.2% [9.1%], 23.3% [8.7%]; 0.3, 0.7, 1.4 MAC sevoflurane, p = 0.000), whereas bicoherence in the α area did not change at these different anaesthesia levels (11.2% [3.9%], 12.5% [4.4%], 14.1% [5.7%], respectively; p = 0.142), counter to the results found in young adult patients, where both δ-θ and α bicoherences changed with anaesthesia. ConclusionsIn the super-elderly, δ-θ bicoherence of EEG shows anaesthesia- dependent changes, whereas α activity remains small irrespective of anaesthesia level. SignificanceQuantification of δ-θ bicoherence is a candidate for anaesthesia monitoring in the super-elderly.

Characterizing the orientation selectivity in V1 and V4 of macaques by quadratic phase coupling

Objective. Orientation selectivity is one of the significant characteristics of neurons in the primary visual cortex (V1). Some neurons in extrastriate visual cortical areas also exhibit certain orientation selectivity. But it is still not well understood that how the orientation selectivity generates. Most previous studies about the orientation selectivity are based on the spike firing rate. However, the spikes are prone to be biased by the detection and sorting algorithms. Then, in this paper, the local field potential (LFP) is adopted to investigate the mechanism of orientation selectivity. Approach. We used the quadratic phase coupling (QPC), which was calculated by wavelet bicoherence, to describe the characteristics of orientation selectivity available in V1 and V4. The raw wideband neural signals were recorded by two chronically implanted multi-electrode arrays, which were placed in V1 and V4 respectively in two macaques performing a selective visual attention task. Main results. There is a strong correlation between the total bicoherence (TotalBic), which is a quantization for the overall QPC of frequency pairs in gamma band, and the grating orientation. Furthermore, the QPC distribution at the non-preferred orientation is mainly concentrated in the low frequencies (30–40 Hz) of gamma; while the QPC distribution at the preferred orientation concentrates in both the low frequencies and high frequencies (60–80 Hz) of gamma. In addition, the TotalBic of the gamma-band LFP between V1 and V4 varies with the grating orientations, indicating that the QPC is available in the feedforward link and the gamma-band LFP in V1 modulates the QPC in V4. Significance. The QPC reflects the orientations of the sinusoidal grating and describes the interaction of gamma-band LFP between different brain regions. Our results suggest that the QPC is an alternative avenue to explore the mechanism for generating orientation selectivity of visual neurons effectively.

Experimental study on the wave run-up and air-gap response of a three-column semi-submersible platform

The hydrodynamic responses, characterized by 6 degrees of freedom (DOF) motions, wave run-up and air-gap of a three-column semi-submersible platform in random sea are experimentally investigated. Irregular waves representative of a 100-yr typhoon were used as the incident wave field and the actions of wind and current are included. It was found that extremely large slow-drift surge motions occur under irregular waves, while the low-frequency pitch motions are relatively small which were believed to be remarkable for the deep-draft semi-submersible. A wavelet cross bicoherence analysis was applied to investigate the extreme wave run-up responses. The high-frequency components in the extreme wave run-ups were found to be from the quadratic phase coupling between the dominant response frequencies of the incident waves, heave motion, and pitch motion.

Bicoherence Interpretation in EEG Requires Signal to Noise Ratio Quantification: An Application to Sensorimotor Rhythms

In the electroencephalogram (EEG) the quadratic phase coupling (QPC) phenomenon indicates the presence of non-linear interactions among brain rhythms that could affect the interpretation of their physiological meaning. We propose the use of the bicoherence as a QPC quantification method to understand the nature of brain rhythm interplay. We firstly provide a simulation study to show under which condition of signal to noise ratio (SNR) the bicoherence is a reliable QPC quantifier and how to interpret the results. Secondly, in the light of the simulation results, we applied the bicoherence analysis to real EEG data acquired on thirteen volunteers during a cue-paced reaching motor task to quantify coupling and decoupling between mu and beta rhythms. An inter-trial averaging procedure was adopted in order to allow the correct calculation of the bicoherence during a motor task. Simulations demonstrated that SNR has a strong impact on the correct quantification of bicoherence and that a reliable detection of QPC is possible when the SNR is favorable (>-5dB). Results from EEG data demonstrated a QPC between mu and beta rhythms during the resting state and its fading during movement planning and execution, providing valuable information for the interpretation of their dynamics. The bicoherence was proven to be an effective tool for the investigation of coupling between the sensorimotor rhythms during all the phases of a motor task. This was assessed in relation to the physiological changing of the SNR characterizing the frequency components of interest.

Complementary Techniques for the Accelerometric Environment Characterization of Thermodiffusion Experiments on the ISS

The accelerometric environment of IVIDIL and DCMIX experiments were successively monitored not only to identify the main disturbances that could affect the experiments but as well to ensure the correct interpretation of the experimental results. To do so, the conventional techniques used by NASA have been complemented by new tools developed and adapted to help the surveillance of the runs. A summary of these main new techniques is presented further. To show the potentiality of all these techniques, moderate and strong disturbance episodes such as berthings, dockings and reboostings were analyzed by using acceleration signals that came from three different sensors located in the Destiny, Columbus and JEM/Kibo modules, respectively. The first technique proposed is based on the Shannon entropy concept in both time (TEN) and frequency (SEN) domains. It has been found, that SEN technique is a fast and easy tool to detect the different disturbances registered throughout the experiments. The second technique suggested by the authors is based on the one-third octave frequency band RMS values and is called RMS warning map. It is a visual tool which was demonstrated to be very efficient in detecting the range of the frequencies that surpasses the ISS limits requirements, especially when a sudden disturbance occurs. Finally, in order to identify nonlinearities in the frequency domain within a signal, bispectrum and trispectrum functions have been applied. Quadratic and cubic phase couplings have been detected with these techniques only between high frequencies and especially for the signals coming from JEM/Kibo module.

Higher order spectral analysis of scalp EEG activity reveals non-linear behavior during rhythmic visual stimulation

Objective. Flickering visual stimulation is known to evoke rhythmic oscillations in the electroencephalographic (EEG) activity, called steady-state visually evoked potentials (SSVEP). The presence of harmonic components in the EEG signals during SSVEP suggests the non-linearity of the visual-system response to rhythmic stimulation, but the nature of this behavior has not been deeply understood. The aim of this study is the quantitative evaluation and characterization of this non-linear phenomenon and its interference with the physiological alpha rhythm by means of spectral and higher order spectral analysis. Approach. EEG signals were acquired in a group of 12 healthy subjects during a pattern-reversal stimulation protocol at three different driving frequencies (7.5 Hz, 15 Hz and 24 Hz). Spectral power values were estimated, after Laplacian spatial filtering, to quantitatively evaluate the changes in the power of the individual alpha and stimulation frequencies related harmonic components. Bicoherence measure were employed to assess the presence of quadratic phase coupling (QPC) at each channel location. Main results. Our analysis confirmed a strong non-linear response to the rhythmic stimulus principally over the parieto-occipital channel locations and a simultaneous significant alpha power suppression during 7.5 Hz and 15 Hz stimulation. A prominent sub-harmonic component characterized the resonance behavior of the 24 Hz stimulation. Significance. The findings presented suggest that bicoherence is a useful tool for the identification of QPC interactions between stimulus-related frequency components within the same signal and the characterization of the non-linearity of SSVEP-induced harmonics generation. In addition, the applied methodology demonstrates the presence of coupled EEG rhythms (harmonics of the main oscillation) both in resting condition and during stimulation, with different characteristics in the distinct brain areas.

Study on Nonlinear Characteristics of Freak-Wave Forces with Different Wave Steepness

The nonlinear wave forces on vertical cylinders induced by freak wave trains were experimentally investigated. A series of freak wave trains with different wave steepness were modeled in a wave flume. The corresponding wave forces on vertical cylinders of different diameters were measured. The experimental wave forces were also compared with the predicted results based on Morison formula. Particular attentions were paid to the effects of wave steepness on the dimensionless peak forces, asymmetry characteristics of the impact forces and high-frequency force components. Wavelet-based analysis methods were employed in revealing the local energy structures and quadratic phase coupling in the freak wave forces.

Electroencephalography quadratic phase self-coupling correlates with consciousness states and restoration in patients with disorders of consciousness

ObjectiveThe objective of this study was to explore the role for quadratic phase coupling within electroencephalography (EEG) oscillations in the diagnosis of consciousness and consciousness restoration for disorders of consciousness (DOC). MethodsFifty-one DOC patients were enrolled in this study. For each patient, a Coma Recovery Scale-Revised (CRS-R) score and 20-min resting-state EEG were recorded. Consciousness recovery was assessed with a CRS-R score at a three-month follow-up. Twenty healthy subjects were included as controls. General harmonic wavelet transform-based bicoherence was used to quantify the quadratic phase coupling characteristics of the EEG oscillations. ResultsQuadratic phase self-coupling (QPSC) at the delta (QPSC_delta), theta (QPSC_theta) and alpha (QPSC_alpha) bands were closely correlated with patient CRS-R scores. Particularly, the QPSC_theta value could significantly differentiate between vegetative state (VS) patients, minimally conscious state (MCS) patients and healthy control subjects. As compared to VS patients, patients with MCS had a lower QPSC_theta value on the left as well as a higher QPSC_alpha value in right frontal regions. The frontal QPSC_theta value showed significant differences between recovered and unrecovered patients. ConclusionQPSC characteristics could differentiate between consciousness states and show a predictive ability for the recovery of consciousness in DOC patients. SignificanceChanges in QPSC accompany consciousness injury and restoration in DOC patients. A QPSC assessment is helpful in the diagnosis and prognosis of DOC patients.