Spectral analysis of non-linear systems involving square-law operations

This paper contains a number of useful theoretical formulas to analyze the frequency domain properties of Gaussian input data passing through non-linear square-law systems. Special bispectral density functions are defined and applied that are functions of a single variable. From measurements of input data and output data only, results are obtained to identify the separate frequency response functions for two models of linear systems in parallel with non-linear square-law systems. Non-linear coherence functions are defined from these models which determine the proportion of the output spectrum due to the non-linear operations. Together with ordinary coherence functions, a measured output spectrum for these models can be decomposed into three components representing the linear operations, the non-linear operations, and the remaining uncorrelated noise effects. This material indicates also how to analyze other types of non-linear models by employing similar techniques.     

Frequency domain analysis and identification of block-oriented nonlinear systems

Block-oriented nonlinear models including Wiener models, Hammerstein models and Wiener-Hammerstein models, etc. have been extensively applied in practice for system identification, signal processing and control. In this study, analytical frequency response functions including generalized frequency response functions (GFRFs) and nonlinear output spectrum of block-oriented nonlinear systems are developed, which can demonstrate clearly the relationship between frequency response functions and model parameters, and also the dependence of frequency response functions on the linear part of the model. The nonlinear part of these models can be a more general multivariate polynomial function. These fundamental results provide a significant insight into the analysis and design of block-oriented nonlinear systems. Effective algorithms are therefore proposed for the estimation of nonlinear output spectrum and for parametric or nonparametric identification of nonlinear systems. Compared with some existing frequency domain identification methods, the new estimation algorithms do not necessarily require model structure information, not need the invertibility of the nonlinearity and not restrict to harmonic inputs. Simulation examples are given to illustrate these new results.     

The critical excitation and response of simple dynamic systems

A method for defining the critical excitations and responses of dynamic systems is examined. The critical excitations are those functions which maximize some response norm with respect to the constraints placed on the admissible excitations. A class of critical responses for linear, elastoplastic and hysteretic single degree of freedoms systems is studied, showing the frequency and amplitude relations for these solutions. For linear systems it is shown that the critical excitations producing either a maximum displacement response or maximum energy input are harmonic and derivable from the harmonically excited response functions for the same linear system. The critical excitations for elastoplastic systems, however, are not harmonic and at low frequencies the response is significantly larger than the harmonically excited response. The critical response solutions require higher multiple frequency components to exist. Both periodic and inelastic offset types of critical response are examined for a hysteretic, elastoplastic system and the response characteristics for these solutions are discussed.     

Use of the Hilbert transform in modal analysis of linear and non-linear structures

An initial study into the application of the Hilbert transform in modal analysis procedures is presented. It is shown that typical structural non-linearities such as non-linear damping and stiffness can be detected and identified directly without the need to generate explicit models. No assumptions regarding the degree of non-linearity are made, which is a restriction in the classical methods for dealing with non-linearities. The properties of the Hilbert transform are discussed with respect to linear and non-linear dynamical systems, and a discrete transform, developed from the continuous functions, is derived in the frequency domain and adapted to modal analysis data in the form of mobility transfer functions. Truncation effects arising from limited frequency ranges of the mobility transfer functions are accounted for by employing correction terms in the frequency domain. Several examples are studied of single and multi-mode systems with non-linearities such as friction, clearance and non-linear stiffness. These examples indicate that the Hilbert transform offers a new method for extending modal analysis to the domains of non-linear systems.     

一种基于最优频段的X射线脉冲星累积轮廓时延估计方法

为提高X射线脉冲星导航中累积脉冲轮廓的时间延迟估计精度, 分析了X射线脉冲星累积脉冲轮廓的频谱特性和现有Taylor快速傅立叶变换时延估计算法的缺陷, 提出了一种基于最优频段的累积轮廓时延估计算法, 并通过建立不同信噪比下时延估计误差与所采用频段之间的关系以确定最优频段. 数值及实测数据实验结果表明: 在短时观测或光子流量较小时, 该算法优于常用的近似最大似然 (FAML)、相关 (CC)、最小二乘 (NLS) 及加权最小二乘 (WNLS) 方法; 在观测时间较长或光子流量较大时, 该算法的估计精度与CC及NLS方法相当, 但其运算量低于NLS, FAML 及WNLS方法. 本文所提算法适用于短时观测脉冲轮廓或低流量脉冲星的高精度时延估计.     

A computational algorithm for computing nonlinear auditory frequency selectivity.

Computational algorithms that mimic the response of the basilar membrane must be capable of reproducing a range of complex features that are characteristic of the animal observations. These include complex input output functions that are nonlinear near the site's best frequency, but linear elsewhere. This nonlinearity is critical when using the output of the algorithm as the input to models of inner hair cell function and subsequent auditory-nerve models of low- and high-spontaneous rate fibers. We present an algorithm that uses two processing units operating in parallel: one linear and the other compressively nonlinear. The output from the algorithm is the sum of the outputs of the linear and nonlinear processing units. Input to the algorithm is stapes motion and output represents basilar membrane motion. The algorithm is evaluated against published chinchilla and guinea pig observations of basilar membrane and Reissner's membrane motion made using laser velocimetry. The algorithm simulates both quantitatively and qualitatively, differences in input/output functions among three different sites along the cochlear partition. It also simulates quantitatively and qualitatively a range of phenomena including isovelocity functions, phase response, two-tone suppression, impulse response, and distortion products. The algorithm is potentially suitable for development as a bank of filters, for use in more comprehensive models of the peripheral auditory system.     

具有非线性阻尼涨落的线性谐振子的随机共振

较之于线性噪声, 非线性噪声更广泛地存在于实际系统中, 但其研究远不能满足实际情况的需要. 针对作为非线性阻尼涨落噪声基本构成成分的二次阻尼涨落噪声, 本文考虑了周期信号与之共同作用下的线性谐振子, 关注这类具有基本意义的阻尼涨落噪声的非线性对系统共振行为的影响. 利用Shapiro-Loginov公式和Laplace变换推导了系统稳态响应振幅的解析表达式, 并分析了稳态响应振幅的共振行为, 且以数值仿真验证了理论分析的有效性. 研究发现: 系统稳态响应振幅关于非线性阻尼涨落噪声系数具有非单调依赖关系, 特别是非线性阻尼涨落噪声比线性阻尼涨落噪声更有助于增强系统对外部周期信号的响应程度; 而且, 非线性阻尼涨落噪声比线性阻尼涨落噪声使得稳态响应振幅关于噪声强度具有更为丰富的共振行为; 同时, 二次阻尼涨落噪声使得稳态响应振幅关于系统频率出现真正的共振现象; 而在这些现象和性质中, 非线性噪声项的非线性性质对共振行为起着关键的作用. 显然, 以二次阻尼涨落作为基本形式引入的非线性阻尼涨落噪声, 可以有助于提高微弱周期信号检测的灵敏度和实现对周期信号的频率估计.     

Statistical errors for non-linear system measurements involving square-law operations

This paper develops normalized random error formulas for special bispectra estimates and associated frequency response function estimates in finite memory square-law systems. Error formulas are also derived for output spectrum estimates from these non-linear systems and for associated non-linear coherence functions. These formulas are useful to evaluate such measured non-linear results as well as to design experimental programs.     

A dynamic Lagrangian frequency-time method for the vibration of dry-friction-damped systems

A new frequency-time domain procedure, the dynamic Lagrangian mixed frequency-time method (DLFT), is proposed to calculate the non-linear steady state response to periodic excitation of structural systems subject to dry friction damping. In this formulation, the dynamic Lagrangians are defined as the non-linear contact forces obtained from the equations of motion in the frequency domain, with the adjunction of a penalization on the difference between the interface displacements calculate by the non-linear solver in the frequency domain and those calculated in the time domain from the non-linear contact forces, thus accounting for Coulomb friction and non-penetration conditions. The dynamic Lagrangians allow one to solve for the non-linear forces between two points in contact without using artifacts such as springs. The new DLFT method is thus particularly well suited to handling finite element models of structures in frictional contact, as it does not require a special model for the contact interface. Dynamic Lagrangians are also better suited to frequency-domain friction problems than the traditional time-domain method of augmented Lagrangians. Furthermore, a reduction of the non-linear system to relative interface displacements is introduced to decrease the computation time. The DLFT method is validated for a beam in contact with a flexible dry friction element connected to ground, for frictional constraints that feature two-dimensional relative motion. Results are also obtained for a large-scale structural system with a large number of one-dimensional dry-friction dampers. The DLFT method is shown to be accurate and fast, and it does not suffer from convergence problems, at least in the examples studied.     

Flexible polyurethane foam modelling and identification of viscoelastic parameters for automotive seating applications

A hereditary model and a fractional derivative model for the dynamic properties of flexible polyurethane foams used in automotive seat cushions are presented. Non-linear elastic and linear viscoelastic properties are incorporated into these two models. A polynomial function of compression is used to represent the non-linear elastic behavior. The viscoelastic property is modelled by a hereditary integral with a relaxation kernel consisting of two exponential terms in the hereditary model and by a fractional derivative term in the fractional derivative model. The foam is used as the only viscoelastic component in a foam-mass system undergoing uniaxial compression. One-term harmonic balance solutions are developed to approximate the steady state response of the foam-mass system to the harmonic base excitation. System identification procedures based on the direct non-linear optimization and a sub-optimal method are formulated to estimate the material parameters. The effects of the choice of the cost function, frequency resolution of data and imperfections in experiments are discussed. The system identification procedures are also applied to experimental data from a foam-mass system. The performances of the two models for data at different compression and input excitation levels are compared, and modifications to the structure of the fractional derivative model are briefly explored. The role of the viscous damping term in both types of model is discussed.     

Linearization of non-linear stochastically excited dynamic systems

A “second order” linearization method for non-linear stochastically excited dynamic systems is described. Second order probabilistic functions are used to evaluate the linearized stiffness or damping coefficients of the non-linear elements. The values of these coefficients are frequency dependent, in contrast to the nowadays generally used “first order” linearization methods. Principles of both “first order” and “second order” methods are discussed. So-called “linearization functions” of typical non-linearities greatly simplifying practical computations are introduced. The actual linearization procedure is discussed. A simple approximate criterion for the validity of the linearization is established.     

Linear and non-linear energy barriers in systems of interacting single-domain ferromagnetic particles

A comparative analysis between linear and non-linear energy barriers used for modeling statistical thermally-excited ferromagnetic systems is presented. The linear energy barrier is obtained by new symmetry considerations about the anisotropy energy and the link with the non-linear energy barrier is also presented. For a relevant analysis we compare the effects of linear and non-linear energy barriers implemented in two different models: Preisach-Néel and Ising-Metropolis. The differences between energy barriers which are reflected in different coercive field dependence of the temperature are also presented.     

The use of moment equations for calculating the mean square response of a linear system to non-stationary random excitation

The moment equations approach is used to calculate the mean square response of a linear system to non-stationary random excitation which is expressed as a product of a deterministic envelope function and a Gaussian stationary non-white noise. The moment equations are derived by performing single integrations in the time domain and are solved numerically by digital computer. Numerical examples are given for the response of single and two degree-of-freedom systems which are excited by noise with an exponentially decaying harmonic correlation function. It is shown that an overshoot, in the sense that the transient response exceeds its stationary value, may occur even in the case of an exponential envelope function, but that the response does not exhibit overshoot when the natural frequency of the system is almost coincident with the dominant frequency of the input.     

System identification from multiple input/output data

For stationary random or transient data representing multicorrelated (multicoherent) input/output data occurring in physical systems, iterative computational algorithms are developed to identify the frequency response functions of optimum constant parameter linear systems connecting this data. Results are obtained from Fourier transform methods and optimum least-squares prediction techniques by changing original arbitrary input records into ordered sets of conditioned input records. These procedures provide the basis for efficient digital computer analysis of general multiple input/output problems. The Appendix contains useful error analysis results for the optimum frequency response estimates determined from measured data.     

Sparse Estimation Strategies in Linear Mixed Effect Models for High-Dimensional Data Application

In a host of business applications, biomedical and epidemiological studies, the problem of multicollinearity among predictor variables is a frequent issue in longitudinal data analysis for linear mixed models (LMM). We consider an efficient estimation strategy for high-dimensional data application, where the dimensions of the parameters are larger than the number of observations. In this paper, we are interested in estimating the fixed effects parameters of the LMM when it is assumed that some prior information is available in the form of linear restrictions on the parameters. We propose the pretest and shrinkage estimation strategies using the ridge full model as the base estimator. We establish the asymptotic distributional bias and risks of the suggested estimators and investigate their relative performance with respect to the ridge full model estimator. Furthermore, we compare the numerical performance of the LASSO-type estimators with the pretest and shrinkage ridge estimators. The methodology is investigated using simulation studies and then demonstrated on an application exploring how effective brain connectivity in the default mode network (DMN) may be related to genetics within the context of Alzheimer’s disease.     

傅里叶变换红外光谱（FTIR）非线性多点定标方法研究

给出了线性定标和非线性多点定标的具体算法,并且应用加拿大BOMEM公司的MR-154傅里叶变换红外光谱仪,液氮制冷型中红外MCT探测器测量了中国科学院上海技术物理研究所研制的HFY-300A型黑体在几种不同温度下的辐射谱,其中的一些辐射谱用于定标,用其中的一个特定温度下的辐射谱来检验定标结果。结果表明这种非线性多点定标方法非常可靠,理论辐射谱与定标后辐射谱相对偏差集中在0.2%附近,最大相对偏差仍小于0.4%。     

Extending the articulation index to account for non-linear distortions introduced by noise-suppression algorithms

The conventional articulation index (AI) measure cannot be applied in situations where non-linear operations are involved and additive noise is present. This is because the definitions of the target and masker signals become vague following non-linear processing, as both the target and masker signals are affected. The aim of the present work is to modify the basic form of the AI measure to account for non-linear processing. This was done using a new definition of the output or effective SNR obtained following non-linear processing. The proposed output SNR definition for a specific band was designed to handle cases where the non-linear processing affects predominantly the target signal rather than the masker signal. The proposed measure also takes into consideration the fact that the input SNR in a specific band cannot be improved following any form of non-linear processing. Overall, the proposed measure quantifies the proportion of input band SNR preserved or transmitted in each band after non-linear processing. High correlation (r?=?0.9) was obtained with the proposed measure when evaluated with intelligibility scores obtained by normal-hearing listeners in 72 noisy conditions involving noise-suppressed speech corrupted in four different real-world maskers.     

基于平移不变多小波方法的磁共振去噪研究

为了提高影像的质量,将一种新颖的基于平移不变原理的多小波方法引入到磁共振的复数信号去噪中.通过对多小波向量系数在一定范围内的循环平移求平均估计子,获得一个改进的阈值估计子,并在不同尺度的高频子带内取双元硬阈值以实现最小化期望风险.实验结果显示,该方法相对于目前几类高效算法和传统的平移不变单小波方法,在消除噪音和抑制伪影的同时能更好地保留边缘特征,并显著改善了信噪比和均方根误差.     

Ground induced non-stationary response of vehicles

A vehicle moving with variable velocity is subjected to non-stationary base excitation. The equations of motion governing the vibration of the vehicle are transformed to the space co-ordinates and the solution is expressed in terms of a Stieltjes integral containing a space dependent frequency response function and an orthogonal function related to the power spectral density of the ground unevenness. Response statistics are computed by using the evolutionary spectra approach. Results obtained by using the Monte Carlo method for the non-linear model, and the time domain and evolutionary spectra approaches for the equivalent linear models are compared.     

Two-Stream Free Electron Lasers: Physical and Project Analysis of the Multiharmonical Models

The multi-harmonic models of two-stream superheterodyne free electron lasers (TSFELs) are proposed and analyzed. It is shown that such systems can have a number of unique features. Including, they are the generation of signals with complex multi-harmonic spectrum, amplification of complex multi-harmonic signals with small distortions, etc. The nonlinear theory of the multi-harmonic TSFELs is constructed and analyzes of two possible projects for their experimental realization is accomplished.