Matrix-based higher order spectral analysis for three-wave couplingprocesses

Composite linear and quadratic systems produce three-wave coupling when stimulated by random phase input sinusoids. Due to the nonlinearity of the system, the output frequencies are arithmetically related to the input. Using third-order cumulant statistics and their associated bispectrum, techniques are devised based on phase-insensitive matrix structures for detection and frequency estimation of coupling frequencies. The separation of the third-order cumulant series into symmetric and skew-symmetric portions allows one to exploit their characteristic eigendecompositions for frequency estimation. After symmetrization, biphases can be easily extracted as coefficients of the cumulant sequence. Using a generalized eigenvector representation, one can relate symmetric and skew-symmetric bases by a subspace rotation algorithm. Biphases can be estimated directly from generalized eigenvalues of the matrix pencil formed by symmetric and skew-symmetric matrices. The dimensionality of the matrices can be reduced through the use of cumulant projections that yield a slice of the bispectrum. The Radon transform procedure is related to bispectral processing through an isotropic radial-slice Volterra filter. The compact third-order Kronecker product matrix formulation and algorithms for coupling frequency estimation can also be converted for use in biphase estimation. Simulations showing the performance of the above procedures are also presented for both synthetic and biomedical time series     

Blind reconstruction of non-minimum-phase systems from 1-D obliqueslices of bispectrum

The author considers the problem of identifying a non-minimum-phase signal from one-dimensional slices of its output bispectra. It is proved that any single slice of the bispectrum carries sufficient information to estimate the impulse response of a complex valued system within a time shift, as long as the chosen slice is not parallel to any one of the frequency axes or to the diagonal at 135 degrees. The author also derives identifiability criteria associated with complex valued signals that admit finite-dimensional ARMA representations. One-dimensional techniques are proposed for signal reconstruction from bispectrum slices and their performance is investigated through Monte-Carlo simulations. The slices required for the proposed method can be estimated on a polar raster directly from observations, avoiding the heavy computational burden associated with cumulant estimates. The freedom to choose arbitrarily oriented and shifted slice(s) allows bispectrum regions dominated by larger estimation variance and higher noise to be avoided     

Bispectrum estimation: A digital signal processing framework

It is the purpose of this tutorial paper to place bispectrum estimation in a digital signal processing framework in order to aid engineers in grasping the utility of the available bispectrum estimation techniques, to discuss application problems that can directly benefit from the use of the bispectrum, and to motivate research in this area. Three general reasons are behind the use of bispectrum in signal processing and are addressed in the paper: to extract information due to deviations from normality, to estimate the phase of parametric signals, and to detect and characterize the properties of nonlinear mechanisms that generate time series.     

Improved bispectrum based tests for Gaussianity and linearity

The classical bispectrum based tests for linearity of time series are based on Gaussian asymptotics and a suboptimal smoothing in the bispectral domain. We show that the resulting classical tests may lead to vastly incorrect significance levels for non-Gaussian time series. This implies that a non-Gaussian linear time series may incorrectly be classified as non-linear. The purpose of this paper is to propose simple yet accurate tests for Gaussianity and linearity. The improved tests are derived through: (1) an optimal hexagonal smoothing in the bispectral domain, (2) the construction of simple and intuitive bispectrum based test statistics, and (3) determination of correct significance levels through a new skewness preserving scheme for linear surrogate data. The superiority of the proposed tests is demonstrated through extensive Monte Carlo simulations using relevant synthetic data.     

Higher order spectral estimation for random fields

This paper is concerned with the nonparametric estimation of the higher order cumulant spectra of vector-valued stationary random fields onZ ^d by smoothing the periodograms, whereZ is the space of integers and the dimensiond1. We derive the asymptotic cumulant properties of the spectral estimates, and consider an application to multidimensional nonlinear systems identification. Numerical examples with simulated data are provided.     

A time-domain test for some types of nonlinearity

The bispectrum and third-order moment can be viewed as equivalent tools for testing for the presence of nonlinearity in stationary time series. This is because the bispectrum is the Fourier transform of the third-order moment. An advantage of the bispectrum is that its estimator comprises terms that are asymptotically independent at distinct bifrequencies under the hypothesis of linearity. An advantage of the third-order moment is that its values in any subset of joint lags can be used in the test, whereas when using the bispectrum the entire (or truncated) third-order moment is required to construct the Fourier transform. We propose a test for nonlinearity based upon the estimated third-order moment. We use the phase scrambling bootstrap method to give a nonparametric estimate of the variance of our test statistic under the hypothesis. Using a simulation study, we demonstrate that the test obtains its target significance level, with large power, when compared to an existing standard parametric test that uses the bispectrum. Further we show how the proposed test can be used to identify the source of nonlinearity due to interactions at specific frequencies. We also investigate implications for heuristic diagnosis of nonstationarity.     

高阶累积量在红外热像损伤检测信号分析中的应用

针对常规红外检测信号的脉冲相位法没有考虑测量噪音,简单的Fourier变换对数据信息的挖掘不足、利用率不高的问题,提出在处理中引入非高斯特性分析,应用三阶累积量和双谱估计对信号进行分析,提取发生损伤时信号的特性信息,得到分辨完好区域与损伤区域的特征判据。给出信号分析中三阶累积量和双谱的定义,分析其幅值和相位的对角切片谱在提取损伤特征方面的优势,最后对比分析两者在实际检测中的效果。结果表明:利用该方法具有良好的抗高斯和非高斯对称分布型噪音的优点,且运算量小、计算速度快。同时,弥补了传统脉冲相位法中幅值谱不宜使用的缺点。     

Development and application of a statistically based feature extraction algorithm for monitoring tool wear in Circuit Board Assembly

Vibroacoustic signals of rotating machinery are composed of sums of modulated periodicities, broadband random components, and occasionally a set of transient responses. These signals are not ergodic as the modulated periodicities are partially coherent. Progressive wear of the rotating machine causes the nonlinear structure of the received signal to intensify, and nonlinearity results in transfer of energy between harmonics of the signal's periodic components. Statistics developed from bispectrum and second-order cumulant spectrum estimates of the measured signal are combined with power spectrum amplitudes as feature inputs for standard multivariate classifiers. The higher-order statistics measure, respectively, the extent of nonlinearity and intermodulation of the received signal. Classification results of simulated and actual incipient wear data collected from a controlled experiment drilling circuit boards illustrate the potential of this novel statistical signal processing approach.     

The higher order theory of generalized almost-cyclostationary timeseries

In this paper, the class of generalized almost-cyclostationary (GACS) time series is introduced. Time series belonging to this class are characterized by multivariate statistical functions that are almost-periodic functions of time whose Fourier series expansions can exhibit coefficients and frequencies depending on the lag shifts of the time series. Moreover, the union over all the lag shifts of the lag-dependent frequency sets is not necessarily countable. Almost-cyclostationary (ACS) time series turn out to be the subclass of GACS time series for which the frequencies do not depend on the lag shifts and the union of the above-mentioned sets is countable. The higher order characterization of GACS time series in the strict and wide sense is provided. It is shown that the characterization in terms of cyclic moment and cumulant functions is inadequate for those GACS time series that are not ACS. Then, generalized cyclic moment and cumulant functions (in both the time and frequency domains) are introduced. Finally, the problem of estimating the introduced generalized cyclic statistics is addressed, and two examples of GACS time series are considered     

Time delay estimation using the cross bispectrum

The cross bispectrum phase can be effectively used to estimate the time required for a nonGaussian signal to propagate between a pair of spatially separated sensors in the presence of highly correlated Gaussian noise. The authors present a consistent estimator of the phase of the cross bispectrum, derive the exact distribution of the phase of a complex Gaussian sample bispectrum, and show that in most cases the exact distribution can be approximated by a Gaussian distribution. Using this Gaussian approximation, the authors derive the variance of the time delay estimate computed from the sample cross bispectrum of a signal in additive correlated noise. These results allow the performance of time delay estimators based on the cross bispectrum phase to be quantified as a function of the sample size, the skewness of the signal, the signal-to-noise ratio (SNR), and the noise correlation     

基于加权累积量的纳米颗粒反演算法

结合传统累积量法和Frisken累积量法,提出了一种加权累积量纳米颗粒反演算法,以减小反演过程中相关时间对测量的影响。采用动态光散射纳米颗粒测量实验平台对90 nm和190 nm的乳胶颗粒进行了测量,通过传统累积量法、Frisken累积量法和文中提出的加权累积量法对颗粒粒径进行反演。实验结果表明:传统累积量法在相关时间变长、相关函数到达零位附近后,颗粒粒径误差和标准方差变大趋势明显;Frisken累积量法在相关时间较小的情况下,其反演的颗粒粒径误差和标准方差较大;而加权累积量法继承了传统累积量运算量小的特点,同时也克服了以上2种算法的缺点,在全局均有较好的反演结果。     

Characterization of sleep spindles using higher order statistics and spectra

This work characterizes the dynamics of sleep spindles, observed in electroencephalogram (EEG) recorded from humans during sleep, using both time and frequency domain methods which depend on higher order statistics and spectra. The time domain method combines the use of second- and third-order correlations to reveal information on the stationarity of periodic spindle rhythms to detect transitions between multiple activities. The frequency domain method, based on normalized spectrum and bispectrum, describes frequency interactions associated with nonlinearities occurring in the observed EEG.     

System reconstruction from higher order spectra slices

We consider the problem of system reconstruction from higher order spectra (HOS) slices. We establish that the impulse response of a complex system can be reconstructed up to a scalar and a shift based on any pair of HOS slices, as long as the distance between the two slices satisfies a certain condition. One slice is sufficient for the reconstruction in the case of a real system. We propose a cepstrum-based method for system reconstruction. We also propose a new method for the reconstruction of the system Fourier phase based on the phase of any odd-indexed bispectrum slice. Being able to choose the slices to be used in the reconstruction allows us to avoid bispectrum regions dominated by noise     

Improved phase estimation based on complete bispectrum and modified group delay

In this paper, a new method for extracting the system phase from the bispectrum of the system output has been proposed. This is based on the complete bispectral data computed in the frequency domain and modified group delay. The frequency domain bispectrum computation improves the frequency resolution and the modified group delay reduces the variance preserving the frequency resolution. The use of full bispectral data also reduces the variance as it is used for averaging. For the proposed method at a signal to noise ratio of 5dB, the reduction in root mean square error is in the range of 1.5–7 times over the other methods considered.     

Performance Study of Adaptive Filtering in Bispectrum Signal Reconstruction

The problem of reconstructing a signal waveform when the observed realizations are corrupted by intensive noise and random shifts is considered in this paper. Several ways of performing bispectrum filtering are proposed and investigated. First, it is shown that the signal reconstruction is more efficient if one applies smoothing to the recovered real and imaginary parts of the Fourier spectrum separately instead of filtering the magnitude and phase spectra recovered from a bispectrum estimate. Second, several nonadaptive filters are studied, and it is demonstrated that the proper choice of the filter type and its parameters is critical. Some adaptive filtering techniques based on the Z-parameter and on local polynomial approximation (LPA)-intersection of confidence intervals (ICI) are discussed. The performances of nonadaptive and adaptive filtering techniques in the bispectrum-based signal reconstruction are studied using the mean-squared error as the criterion. It is shown that the use of LPA-ICI and other adaptive filters provides improvement of signal reconstruction in comparison to the conventional bispectrum method and the combined bispectrum filtering methods proposed earlier for nonadaptive filters. The benefits achieved are mainly observed for low (smaller than unity) signal-to-noise ratios.     

Sensitivity Study of the Cumulant Method for Evaluating Reliability Measures of Two Interconnected Systems

The rationale for using the cumulant method to take advantage of its computational efficiency is well known among power system planners. However, although an analysis of the sensitivity of the univariate Gram-Charlier series has been investigated, an equivalent analysis of the sensitivity of the bivariate Gram-Charlier series has not yet been reported in the literature. This paper investigates the sensitivity of the bivariate Gram-Charlier series in the evaluation of reliability for several types of interconnected systems. The impact of different number of terms in the series on the accuracy of the results as well as on the computational requirements is also investigated. Load correlation between the interconnected systems is considered. As anticipated, the cumulant method is much faster than the commonly used recursive method. However, the reliability indexes, obtained using this method for interconnected systems with low reserve margin and with units of low forced outage rates can not be trusted. The relative error in the calculation of the loss of load probabilities increases with the increase of tie line capacity. However, the error is greatly reduced if the systems have units of higher forced outage rate. The use of additional terms in the bivariate Gram-Charlier series increases somewhat the accuracy of the results but also increases the computational time.     

基于双谱理论的雷达辐射源指纹特征提取

针对雷达辐射源指纹特征提取困难的问题,提出了基于双谱切片法和双谱围线积分法,提取辐射源指纹特征的方法。介绍了二相编码信号的无意调制,仿真了理想情况和含有相位噪声的二相编码信号的时域波形。阐述了含相位噪声雷达信号的双谱估计,仿真了含相位噪声的二相编码信号的双谱图及等高图。最后利用双谱切片法和双谱围线积分法,提取了辐射源的指纹特征,通过计算机仿真实验验证了该方法的可实施性。     

Modified Wigner bispectrum and its generalizations

The Wigner bispectrum of multicomponent signals is studied, and its modified and reduced forms are introduced. A generalization of the presented forms to the Wigner higher-order spectra (WHOS), in the case of multicomponent signals, is provided. From our previous work it is known that cross terms removal (reduction) is possible for odd-order spectra with equal numbers of conjugated and nonconjugated terms. Here, we extend the analysis to even-order spectra. The theory is illustrated by examples.     

On a condition of higher order spectra factorization

This paper presents a necessary and sufficient condition that guarantees that a higher order cumulant spectrum is factorizable in an appropriate product form. This condition provides a way to check whether or not a given higher order spectrum is compatible with the stronger hypothesis of “white generation”, where the analyzed signal is obtained by exciting a linear time invariant system with a white stationary non-Gaussian noise. The condition is based on symmetries of higher order spectra and on a generalization of a formula proposed by Marron et al. (1990) for third-order spectra phase unwrapping. It is expressed, at any order, as an identity relating six higher order spectrum values. This test can be applied directly on higher order spectral data and does not require transformations of the data or application of elaborate algorithms     

AR modelling of skewed signals using third-order cumulants

Algorithms for selecting the order and estimating the parameters of an AR process, which is driven by noise having an underlying non-Gaussian distribution, from the observed noisy time series are presented. The order selection algorithm makes use of the growing memory covariance predictive least-squares (GMCPLS) criterion together with diagonal slices of the third-order cumulant plane. A triangular region of the third-order cumulant plane is used to estimate the model parameters. Extensive simulation results are presented and based on these trends, one of which has been verified using real data obtained from a rotating machine, recommendations are made on the efficacy of methods for AR order selection and parameter estimation problems