Higher Order Spectra Research Articles

Decomposition of higher-order spectra for blind multiple-input deconvolution, pattern identification and separation

Like the ordinary power spectrum, higher-order spectra (HOS) describe signal properties that are invariant under translations in time. Unlike the power spectrum, HOS retain phase information from which details of the signal waveform can be recovered. Here we consider the problem of identifying multiple unknown transient waveforms which recur within an ensemble of records at mutually random delays. We develop a new technique for recovering filters from HOS whose performance in waveform detection approaches that of an optimal matched filter, requiring no prior information about the waveforms. Unlike previous techniques of signal identification through HOS, the method applies equally well to signals with deterministic and non-deterministic HOS. In the non-deterministic case, it yields an additive decomposition, introducing a new approach to the separation of component processes within non-Gaussian signals having non-deterministic higher moments. We show a close relationship to minimum-entropy blind deconvolution (MED), which the present technique improves upon by avoiding the need for numerical optimization, while requiring only numerically stable operations of time shift, element-wise multiplication and averaging, making it particularly suited for real-time applications. The application of HOS decomposition to real-world signals is demonstrated with blind denoising, detection and classification of normal and abnormal heartbeats in electrocardiograms.

Characterizing non-Gaussian vibration loading using the trispectrum

This paper addresses the use of higher-order spectra to study the non-Gaussian nature of random vibration loading. Since the power spectral density is only a full description for stationary Gaussian processes, specifying non-Gaussian random vibration loading requires a sophisticated statistical description. In recent research higher-order statistical moments such as skewness and kurtosis have been used to define non-Gaussian properties of vibration loading. However, useful information contained in the spectral representation of these moments is neglected. This paper introduces the trispectrum as a tool for analyzing vibration loading. It is the spectral representation of the fourth-order moment and thus extends the information content of the kurtosis. For demonstration several common methods for generating non-Gaussian loading are reviewed and used to derive loads that reproduce the power spectral density and kurtosis of a real in-service loading. These loads are analyzed using Fatigue Damage Spectra and trispectra to relate structural response behavior to their non-Gaussian nature. The results suggest that the trispectrum is a valuable tool for analyzing and classifying non-Gaussian random loading.

Multichannel Interference in Resonancelike Enhancement of High-Order Above-Threshold Ionization.

The intensity-dependent resonancelike enhancement phenomenon in the high-order above-threshold ionization spectrum is a typical quantum effect for atoms or molecules in an intense laser field, which has not been well understood. The calculations of the time-dependent Schrödinger equation (TDSE) are in remarkable agreement with the experimental data, but they cannot clarify the contributions of the bound states. The semiclassical approach of strong field approximation, in which no excited states are involved, can obtain a similar phenomenon, but the laser intensities of enhanced regions predicted by the strong field approximation are inconsistent with the results of the TDSE. In this Letter, a new fully quantum model is established from the TDSE without any excited states. Two types of enhanced structures, unimodal and multimodal structures, are found in the results of the TDSE and model. In addition, the calculations of the model reproduce the key features of the results of the TDSE. It shows that the excited states are not the key factor in the resonancelike enhancements in our calculated system, since there are no excited states in our model. Based on the calculations of our model, we show that such resonancelike enhancements are caused by the constructive interference of different momentum transfer channels. Last, the Fano-like line shapes are also discussed for the features of multichannel interference.

Characterization of fibromyalgia using sleep EEG signals with nonlinear dynamical features

Fibromyalgia is an intense musculoskeletal pain causing sleep, fatigue, and mood problems. Sleep studies have suggested that 70%–80% of fibromyalgia patients complain of non-restorative sleep. The abnormalities in sleep have been implicated as both a cause and effect of the disease. In this paper, the electroencephalogram (EEG) signals of sleep stages 2 and 3 are used to classify the normal and fibromyalgia classes automatically. We have used various nonlinear parameters, namely sample entropy (SampEn), fractal dimension (FD), higher order spectra (HOS), largest Lyapunov exponent (LLE), Kolmogorov complexity (KC), Hurst exponent (HE), energy, and power in various frequency bands from the EEG signals. Then these features are subjected to Student's t-test to select the clinically significant features, and are classified using the support vector machine (SVM) classifier. Our proposed method can classify normal and fibromyalgia subjects using the stage 2 sleep EEG signals with an accuracy of 96.15%, sensitivity and specificity of 96.88% and 95.65%, respectively. Performance of the developed system can be improved further by adding more subjects in each class, and can be employed for clinical use.

Effect Analysis of Antenna Vibration on GEO SAR Image

Existing effect analyses of antenna vibration (AV) on synthetic aperture radar (SAR) image, which are all aimed for low-earth-orbit SAR and based on the linear trajectory, second-order azimuth spectrum, and range time-invariant as well as azimuth time-variant antenna gain, are not feasible in geosynchronous SAR (GEO SAR), because GEO SAR has an obviously curved trajectory, high-order azimuth spectrum, and two-dimensional (2-D) time-variant antenna gain. To overcome these problems and analyze the effects of AV on GEO SAR image, a modified analysis is proposed in this paper. First, based on the curved trajectory, the high-order ideal echo signal model and the ideal point spread function (PSF) of GEO SAR are, respectively, established and deducted, and the AV that consists of antenna translational vibration (ATV) and antenna rotation motion (ARV) and the 2-D time-variant antenna gain in squint mode are also, respectively, modeled. Then, based on the series expansion, the paired echo theory, and the aforementioned models, the echo signal model affected by ATV, the corresponding PSF affected by ATV, the antenna gain model affected by ARV, and the corresponding PSF affected by ARV are obtained, and the effects of ATV and ARV on GEO SAR image are, respectively, analyzed in detail. Compared with existing effect analyses of AV on SAR image, the modified analysis considers the influences of curved trajectory, high-order azimuth spectrum, and 2-D time-variant antenna gain simultaneously, and it has much more accuracy and adaptability correspondingly. Finally, computer simulations validate the correctness of the proposed analysis.

Fatigue life due to non-Gaussian excitation – An analysis of the Fatigue Damage Spectrum using Higher Order Spectra

The fatigue damage spectrum (FDS) is an established tool for the assessment of the fatigue damage potential of any kind of vibration loading exerted on vibratory structures. It is especially useful for describing the fatigue damage potential of non-Gaussian random vibration signals, because for this type of signal no comprehensive, theoretically based statistical description is available. As the current definition of the FDS is based on the assumption of single degree of freedom vibration systems, the presented work analyses the resulting limitations for general vibration systems with multiple resonant frequencies. Improvements based on findings from higher-order spectra are proposed.

Research on real-time network\xa0data mining technology for big data

The data distribution in big data environment is very different, and it is difficult to mine the data because of the strong interference of redundant data and frequent items. The traditional data mining algorithm uses closed frequent item feature extraction algorithm. Due to the uneven distribution of web data in big data environment, the mining accuracy of closed frequent item feature extraction is not high. A real-time web data mining model is proposed based on high order spectral feature fuzzy neural network learning in big data environment. The transmission channel model and statistical time series model of web data under big data environment are constructed, and the redundant information flow is removed and reprocessed, and the web data after redundant filtering is analyzed by fusion clustering. The feature of high order spectrum is extracted, and the optimal mining of web data is realized by using fuzzy neural network learning classification method. The simulation results show that this web data mining method has good timeliness, high mining precision, and superior performance.

Parity-time-symmetry–enhanced sideband generation in an optomechanical system

Optical high-order sideband generation, creating new frequencies with equal spectral separation, has developed potential applications in optical communication and metrology. Here, we investigate the generation of optomechanically induced high-order optical sideband spectra in an optical parity-time ($\mathcal{PT}$) -symmetric system. The system consists of an active optical mode, a passive optical mode, and a mechanical mode, where the mechanical mode is optomechanically coupled with the passive optical mode. Based on numerical analysis, the $\mathcal{PT}$-symmetric system enhances sideband spectra in both amplitude and number of sidebands. With certain parameters, forty sidebands head up on both sides of the control field near the exceptional point. The capability to generate optical high-order sidebands and control light propagation in micro- and nanomechanical systems may open up a promising perspective for complex integrated $\mathcal{PT}$ devices.

Rotor Fault Detection in Induction Motors Based on Time-Frequency Analysis Using the Bispectrum and the Autocovariance of Stray Flux Signals

The aim of this work is to find out, through the analysis of the time and frequency domains, significant differences that lead us to obtain one or several variables that may result in an indicator that allows diagnosing the condition of the rotor in an induction motor from the processing of the stray flux signals. For this, the calculation of two indicators is proposed: the first is based on the frequency domain and it relies on the calculation of the sum of the mean value of the bispectrum of the flux signal. The use of high order spectral analysis is justified in that with the one-dimensional analysis resulting from the Fourier Transform, there may not always be solid differences at the spectral level that enable us to distinguish between healthy and faulty conditions. Also, based on the high-order spectral analysis, differences may arise that, with the classical analysis with the Fourier Transform, are not evident, since the high order spectra from the Bispectrum are immune to Gaussian noise, but not the results that can be obtained using the one-dimensional Fourier transform. On the other hand, a second indicator based on the temporal domain that is based on the calculation of the square value of the median of the autocovariance function of the signal is evaluated. The obtained results are satisfactory and let us conclude the affirmative hypothesis of using flux signals for determining the condition of the rotor of an induction motor.

Electromagnetic Signal Classification Based on Deep Sparse Capsule Networks

In complex electromagnetic environments, electromagnetic signal classification rates are low as long time have to be the cost to extract features. To cope with the issue, in this paper, an electromagnetic signal classification method is proposed based on deep sparse capsule networks. In the proposed method, received signals are frequency reduced and sampled processing first. Subsequently, a cross ambiguity function based on linear canonical transformation, a cross ambiguity function based on linear canonical domain, and higher-order spectrum are estimated, respectively. The maximum value of each section of the cross ambiguity function is combined with the maximum value of equally spaced cross sections of higher order amplitude spectrum to obtain the two-dimensional feature information. Finally, electromagnetic signals are classified by the deep sparse capsule networks. The simulation results show that the proposed method not only has good classification performance but also can automatically get a hierarchical feature representation by learning. Moreover, the corresponding time cost can be effectively reduced.

Feature Extraction and Similarity of Movement Detection during Sleep, Based on Higher Order Spectra and Entropy of the Actigraphy Signal: Results of the Hispanic Community Health Study/Study of Latinos.

The aim of this work was to develop a new unsupervised exploratory method of characterizing feature extraction and detecting similarity of movement during sleep through actigraphy signals. We here propose some algorithms, based on signal bispectrum and bispectral entropy, to determine the unique features of independent actigraphy signals. Experiments were carried out on 20 randomly chosen actigraphy samples of the Hispanic Community Health Study/Study of Latinos (HCHS/SOL) database, with no information other than their aperiodicity. The Pearson correlation coefficient matrix and the histogram correlation matrix were computed to study the similarity of movements during sleep. The results obtained allowed us to explore the connections between certain sleep actigraphy patterns and certain pathologies.

Higher-order moments, cumulants, and spectra of continuous quantum noise measurements

We present general quantum mechanical expressions for higher order moments, cumulants, and spectra of continuously measured quantum systems with applications in spin noise spectroscopy, quantum transport, and measurement theory in general. Starting from the so-called stochastic master equation of continuous measurement theory, we find that the leading orders of the fluctuating detector output $z(t)$ with respect to the measurement strength $\beta$ are a white shot noise background, a constant measurement offset, and the leading order quantum noise of the measured operator $A$. Starting from quantum expressions for the multi-time moments $\langle z(t_n)\cdots z(t_1) \rangle$ we derive three- and four-time cumulants that are valid in all orders of $\beta$ covering the full regime between the weak and strong measurement limit (Zeno-limit). Intriguingly, quantum expressions for the cumulants were found that exhibit the same simple structure as those for the moments after introduction of only a slightly modified system propagator. Very compact expressions for the cumulant-based third and fourth order spectra (bispectrum and trispectrum) follow naturally. We illustrate the usefulness of higher order spectra by treating a real world two-spin system with strong hyperfine interaction.Moreover, spin noise spectroscopy is shown to have the potential for investigating the transition from weak measurements to the famous quantum Zeno regime for realistic probe laser intensities.

Characterization of a 3DOF aeroelastic system with freeplay and aerodynamic nonlinearities – Part I: Higher-order spectra

Abstract The identification of nonlinear systems in aeroelasticity poses a significant challenge for practitioners, often hampered by the complex nature of aeroelastic response data which may contain multiple forms of nonlinearity. Characterizing and quantifying nonlinearities is further hampered when the response is obtained at a location which is away from the nonlinear source and/or the response is contaminated by noise. In the present paper, a three-degree-of-freedom airfoil with a freeplay nonlinearity located in the control surface and exposed to transonic flow is investigated. In this Part I paper the main form of analysis is via higher-order spectra techniques to unveil features of the nonlinear mechanism which result from i) structural nonlinearities (freeplay) in isolation and ii) freeplay with Euler derived nonlinear inviscid aerodynamic phenomena (transition between Tijdeman Type-A and Type-B shock motion). It is shown that the control surface structural freeplay nonlinearity is characterized by strong cubic phase-coupling between linear and nonlinear modes. On the other hand, nonlinear inviscid flow phenomena are shown to be characterized by quadratic phase-coupling between linear and nonlinear modular modes, the strength of which is related to the strength of the aerodynamic nonlinearity (amplitude of the shock motion). The nonlinear inviscid flow phenomena do not appear to affect the identification of the freeplay nonlinearity. Conjectures are made which address the transition between aperiodic, quasi-periodic and periodic behavior (pre-flutter), further physical support towards these conjectures is provided in Part II [1] . The limitations of the higher-order spectra approach are assessed, in particular, the analysis demonstrates the difficulty in extracting natural frequencies with this approach.

Study on a Novel Fault Damage Degree Identification Method Using High-Order Differential Mathematical Morphology Gradient Spectrum Entropy.

A damage degree identification method based on high-order difference mathematical morphology gradient spectrum entropy (HMGSEDI) is proposed in this paper to solve the problem that fault signal of rolling bearings are weak and difficult to be quantitatively measured. In the HMGSEDI method, on the basis of mathematical morphology gradient spectrum and spectrum entropy, the changing scale influence of structure elements to damage degree identification is thoroughly analyzed to determine its optimal scale range. The high-order difference mathematical morphology gradient spectrum entropy is then defined in order to quantitatively describe the fault damage degree of bearing. The discrimination concept of fault damage degree is defined to quantitatively describe the difference between the high-order differential mathematical entropy and the general mathematical morphology entropy in order to propose a fault damage degree identification method. The vibration signal of motors under no-load and load states are used to testify the effectiveness of the proposed HMGSEDI method. The experiment shows that high-order differential mathematical morphology entropy can more effectively identify the fault damage degree of bearings and the identification accuracy of fault damage degree can be greatly improved. Therefore, the HMGSEDI method is an effective quantitative fault damage degree identification method, and provides a new way to identify fault damage degree and fault prediction of rotating machinery.

Noisy texture analysis based on higher order spectra

Texture is described in several approaches by 1st and 2nd order statistics which cannot preserve phase information carried by the Fourier spectrum. Besides, these statistics are very sensitive to noise. In this paper, we study features derived from higher order spectra, especially the third order spectrum, namely the bispectrum, known to offer a high noise immunity and to recover Fourier phase information. In this paper, we exploit phase preservation property by using bispectrum phase. We propose wrapped Cauchy distribution to model phase. Wrapped Cauchy parameters are estimated by maximizing the log-likelihood function. Experiments show that the wrapped Cauchy distribution fits our phase information well. Hence, their parameters are used to feed our feature vector in order to classify textures corrupted by Gaussian noise. Classification results using the proposed approach show a good noise immunity compared to a statistical model based on Gabor phase.

Continuous wavelet transform and higher-order spectrum: combinatory potentialities in breath sound analysis and electroencephalogram-based pain characterization.

The combination of the continuous wavelet transform (CWT) with a higher-order spectrum (HOS) merges two worlds into one that conveys information regarding the non-stationarity, non-Gaussianity and nonlinearity of the systems and/or signals under examination. In the current work, the third-order spectrum (TOS), which is used to detect the nonlinearity and deviation from Gaussianity of two types of biomedical signals, that is, wheezes and electroencephalogram (EEG), is combined with the CWT to offer a time–scale representation of the examined signals. As a result, a CWT/TOS field is formed and a time axis is introduced, creating a time–bifrequency domain, which provides a new means for wheeze nonlinear analysis and dynamic EEG-based pain characterization. A detailed description and examples are provided and discussed to showcase the combinatory potential of CWT/TOS in the field of advanced signal processing.This article is part of the theme issue ‘Redundancy rules: the continuous wavelet transform comes of age’.

Asymptotic Theory for Estimators of High-Order Statistics of Stationary Processes

High-order cumulants and high-order spectra play an important role in the theory of stationary processes. For nonlinear processes, it is quite challenging to develop an asymptotic theory for their estimators. This paper presents a systematic asymptotic theory for estimators of high-order moments for a general class of stationary processes, using the framework of functional dependence measures. In particular, we prove the asymptotic normality of the estimators and establish a uniform convergence rate. We also provide a sufficient condition for the summability of high-order cumulants. Based on the latter, we prove consistency and asymptotic normality of the third-order spectra or bispectrum estimators under mild and easily verifiable conditions.

Experimental characterization of non-linear interfacial wave interaction in stratified gas-liquid pipe flow

We present a statistical and spectral study of interfacial waves in stratified turbulent gas-liquid flow in a horizontal pipe. The main objective of this paper is to report on experimental findings with respect to the influence of the gas flow rate on statistical and spectral properties of interfacial waves. Emphasis is put on the assessment of non-linear interfacial mode interactions by means of higher-order spectra. Interfacial elevation time series were acquired using conductance probes with sampling frequency Fs = 500 Hz. The probes were placed at the pipe centerline and 270D downstream of the pipe inlet. Local wave parameters such as wavelengths, heights, and phase speeds were extracted by means of a zero-crossing technique. The evolution of their statistical properties is assessed for a range of flow conditions in which the liquid flow rate was kept constant at Usl = 0.1 m/s, whilst the gas flow rate was gradually increased from 1.3 to 4 m/s. The mode wave speed is compared to linear and weakly non-linear theory. Interfacial mode interactions are investigated through an assessment of first and second-order spectra, i.e., power spectral density and bispectrum. The results indicate that weakly non-linear (triadic) interactions in the form of overtone and sub-harmonic interactions play an important role and may partially be responsible for the observed transition to the wave saturation regime reported in our previous studies. Finally, a flow map is presented in terms of a Ursell-vs-Froude number plot, in which observed wave regimes are classified according to underlying linear/non-linear mechanisms.

High-order harmonic generation of CO and N2 molecules under linearly- and bi circularly-polarized laser pulses by TD-DFT

We present a method for high-order harmonics generation of N2 and CO molecules under two-color circularly polarized counter-rotating laser pulses at frequencies of and 2. Pulse envelope in this investigation is sin-squared and the intensity of each laser beam is with ten-optical cycle (o.c.). We show that an isolated pulse with a pulse duration shorter than 20 attosecond from the superposition of several harmonics can be generated. Both two-color linearly- and bicircularly-polarized laser pulses are considered. Our results have also been compared with the outcomes of the previous theoretical works as well as experiment observations. It is found that for CO molecule, the bicircularly-polarized laser pulses are superior and more efficient, and it can generate narrower attosecond pulses than the linearly-polarized pulses. While for N2 molecule, the two-color linearly-polarized pulses are more efficient, and it can generate narrower attosecond pulses than the bicircularly-polarized pulses. Furthermore, in order to demonstrate the origin of red- and blue-shifts in high-harmonic spectra, the effect of pulse duration on the high-order harmonics spectra is investigated. In addition, to obtain imaging on the temporal dependence of the electron densities, the time dependent electron localization function is used. Moreover, in order to study of the quantum trajectory of electrons, time-frequency analysis is utilized.

Radar emitter identification with bispectrum and hierarchical extreme learning machine

Radar Emitter Identification (REI) has been broadly used in military and civil fields. In this paper, a novel method is proposed for radar emitter signal identification, where the bispectrum estimation of radar signal is extracted and the recent hierarchical extreme learning machine (BS + H-ELM) is adopted for further feature learning and recognition. Conventional REI methods generally rely on the time-difference-of-arrival, carrier frequency, pulse width, pulse amplitude, direction-of-arrival, etc., for signal representation and recognition. However, the increasingly violent electronic confrontation and the emergence of new types of radar signals generally degrade the recognition performance. With this objective, we explore radar emitter signal representation and classification method with the high order spectrum and deep network based H-ELM. After extracting the bispectrum of radar signals, the sparse autoencoder (AE) in H-ELM is employed for feature learning. Simulations on four representative radar signals, namely, the continuous wave (CW), linear frequency modulation wave(LFM), nonlinear frequency modulation wave(NLFM) and binary phase shift keying wave (BPSK), are conducted for performance validation. In comparison to the existing multilayer ELM algorithm and the popular histogram of gradient (HOG) based feature extraction method are proved that the proposal is feasible and potentially applicable in real applications.