Cross Power Spectral Density Matrix Research Articles

Eigenanalysis of vector autoregressive model for optimal fitting of a predefined cross power spectral density matrix: Application to numeric generation of stationary homogeneous isotropic/anisotropic turbulent wind fields

Turbulent wind fields representing realistic atmospheric conditions are required for wind turbine load calculations. For a statistically stationary situation, the wind loads on a wind turbine depend, among other wind characteristics, on the spectra and spatial coherences of the velocity field in the rotor plane (i.e. the cross-power spectral density matrix). In this study, a novel approach is proposed to optimally calibrate the parameters of a vector autoregressive model for reproducing a predefined target cross-power spectral density matrix to be used for the synthetic generation of turbulent wind fields. This approach is based on the eigenanalysis of the companion matrix of the vector autoregressive model. In this work, the cross-power spectral density matrix, for different turbulent wind conditions, is considered as targets. This approach is compared to the state-of-the-art approaches for calibration of parameters of vector autoregressive models. The discussion covers the implications derived from approaching the vector autoregressive model determination in the frequency domain or in the time domain.

A novel eigenvalue-based iterative simulation method for multi-dimensional homogeneous non-Gaussian stochastic vector fields

The generation of sample functions of stochastic fields or processes is a prerequisite to use the Monte Carlo method in stochastic mechanics and structural reliability analysis. When it comes to non-Gaussian characteristics such as the spatial variability of material or structural properties, the simulation of multi-dimensional non-Gaussian stochastic vector fields becomes necessary. However, the conventional Cholesky decomposition-based iterative simulation method requires three times of decomposition operations at each iteration and a complex procedure of random shuffle, which inevitably augment the model complexity. To this end, this study develops a novel eigenvalue-based iterative simulation method which only requires one decomposition operation in the whole simulation process. Central to this method is to represent the cross power spectral density matrix with its eigenvalue matrix during the iteration procedure, which avoids the cumbersome Cholesky decomposition and random shuffle procedure. Meanwhile, it can yield unique convergence result and present a powerful ability in addressing cases with the complex-valued spectral matrix. Numerical example demonstrates that the proposed method has a similar accuracy as that of the Cholesky decomposition-based iterative simulation method. Meanwhile, the estimated cumulative distribution function from one simulated sample and the estimated ensemble correlation function from 1000 simulated samples are all consistent with their respective targets. In addition, the sensitive analysis is performed to demonstrate that the proposed method has a relatively wide applicability. Therefore, this method may have a great potential to simulate multi-dimensional homogeneous non-Gaussian stochastic vector fields in practice.

Simulation of stationary non-Gaussian stochastic vector processes using an eigenvalue-based iterative translation approximation method

The simulation of stationary non-Gaussian stochastic vector processes is of great importance in the time-domain response analysis of structures under non-Gaussian excitations. When incompatibility between the target cross power spectral density matrix and the marginal probabilistic density function of the stochastic vector process occurs in the sense of translation processes, an iterative simulation scheme must be required. Currently, the Cholesky decomposition-based iterative translation approximation method is widely utilized. However, the use of this method still encounters some difficulties. First, it requires performing Cholesky decompositions three times at each iteration and is therefore cumbersome. Second, a complicated random shuffling procedure is necessary, which renders the final iteration result non-unique. Third, it is very difficult for this method to handle cases with complex-valued spectral matrices. Lastly, it can generally be used to simulate stochastic vector processes with very limited components. To overcome these difficulties, an eigenvalue-based iterative translation approximation method is proposed in this study. This method can not only avoid the cumbersome Cholesky decomposition and complex random shuffling, but can also handle cases with complex-valued and large spectral matrices more easily. Four numerical examples with no incompatibility, significant incompatibility, complex-valued coherence and a large number of component processes are respectively employed to demonstrate the effectiveness of the proposed method. The results show that the proposed method has similar accuracy to the Cholesky decomposition-based iterative method in the first two examples. For the last two examples, the simulation accuracy is also satisfactory. The above results verify that the proposed method is superior to the Cholesky decomposition-based iterative translation approximation method overall.

Joint cardiac tissue conductivity and activation time estimation using confirmatory factor analysis

Mathematical models of the electrophysiology of cardiac tissue play an important role when studying heart rhythm disorders like atrial fibrillation. Model parameters such as conductivity, activation time, and anisotropy ratio are useful parameters to determine the arrhythmogenic substrate that causes abnormalities in the atrial tissue. Existing methods often estimate the model parameters separately and assume some of the parameters to be known as a priori knowledge. In this work, we propose an efficient method to jointly estimate the parameters of interest from the cross power spectral density matrix (CPSDM) model of the electrograms. By applying confirmatory factor analysis (CFA) to the CPSDMs of multi-electrode electrograms, we can make use of the spatial information of the data and analyze the relationship between the desired resolution and the required amount of data. With the reasonable assumptions that the conductivity parameters and the anisotropy parameters are constant across different frequencies and heart beats, we estimate these parameters using multiple frequencies and multiple heart beats simultaneously to easier satisfy the identifiability conditions in the CFA problem. Results on the simulated data show that using multiple heart beats decreases the estimation errors of the conductivity and the estimated activation time parameters. The experimental results on clinical data show that using multiple heart beats for parameter estimation can reduce the reconstruction errors of the clinical electrograms, which further demonstrates the robustness of the proposed method.

Efficient simulation of fully non-stationary random wind field based on reduced 2D hermite interpolation

The spectral representation method (SRM) has been widely used to simulate stationary or non-stationary wind fields for engineering structures. Although several attempts have been made to realize the invoking of Fast Fourier Transform (FFT), the SRM is still very inefficient to simulate the fully non-stationary wind field with a time-varying coherence due to the extremely time-consuming Cholesky decompositions and large memory requirement. In this paper, a reduced 2D Hermite interpolation-enhanced approach is developed to further improve the efficiency of SRM in simulating fully non-stationary wind fields. Central to this approach is the interpolation procedure which requires Cholesky decompositions and storage of cross power spectral density matrix (CEPSD) elements only at interpolation knots. Thus the computational costs of Cholesky decompositions and memory requirement are dramatically decreased. The number of Cholesky decompositions is then fixed with no relation to the segments of frequency and duration of wind samples, which eliminates the Cholesky decomposition as a cause that affects the simulation efficiency. Meanwhile, each element in the decomposed CEPSD matrix is decoupled into products of time- and frequency-dependent functions by the reduced 2D Hermite interpolation, so the FFT can be used to expedite the summation of trigonometric terms. Apart from using FFT, another merit of the proposed approach is that an accelerated FFT algorithm can be incorporated to further improve the simulation efficiency based on the specific decoupled expression of frequency-dependent functions. The parametric analysis shows that the proposed approach is very efficient in comparison with the existing method using proper orthogonal decomposition (POD), and it provides a desired level of simulation accuracy when appropriate interpolation interval is selected. The case study in simulating the fully non-stationary wind field of a long-span cable-stayed bridge demonstrates the effectiveness of the proposed approach with verifications on both evolutionary power spectra and correlation functions.

An approach for predicting multi-support seismic underground motions in layered saturated soil under surface water

Simulation of multi-support (i.e. spatially variable) seismic underground motions in sea areas plays a significant role in the seismic analysis of cross-sea structures such as cross-sea bridges or subsea tunnels. However, existing approaches for predicting multi-support seismic motions mainly focus on the dry site soils without overlying surface water. This paper proposes an approach for predicting multi-support seismic underground motions in layered saturated half space under surface water, subjected to oblique incident P waves. The transfer function in saturated soil under surface water, as the theoretical basis of the subsequent numerical simulation, is first derived based on wave propagation theory and the calculated reflection coefficients of P wave–induced P1, P2, SV waves in saturated soils. The derived transfer function is further employed to deduce and obtain the underground (sub-seabed) power spectral density function and response spectrum function. The two obtained functions, combined with the additional cross-coherence function, are subsequently employed to construct the cross power spectral density matrix and thus to simulate multi-support seismic underground motions. The solutions are validated against the target power spectral density, target response spectrum and target cross-coherence functions. A parametric analysis is presented where the effects of the soil thickness, the incident angle and the overlying water depth are investigated. Results show that the soil thickness, incident angle and overlying water depth have significant influences on the amplitude of transfer functions, which further affect the ratios between seismic ground and underground motions.

A Low-Cost Robust Distributed Linearly Constrained Beamformer for Wireless Acoustic Sensor Networks With Arbitrary Topology

We propose a new robust distributed linearly constrained beamformer that utilizes a set of linear equality constraints to reduce the cross power spectral density matrix to a block-diagonal form. The proposed beamformer has a convenient objective function for use in arbitrary distributed network topologies while having identical performance to a centralized implementation. Moreover, the new optimization problem is robust to relative acoustic transfer function (RATF) estimation errors and to target activity detection (TAD) errors. Two variants of the proposed beamformer are presented and evaluated in the context of multimicrophone speech enhancement in a wireless acoustic sensor network, and are compared with other state-of-the-art distributed beamformers in terms of communication costs and robustness to RATF estimation errors and TAD errors.

Modal analysis of 2-D sedimentary basin from frequency domain decomposition of ambient vibration array recordings

Frequency domain decomposition (FDD) is a well-established spectral technique used in civil engineering to analyse and monitor the modal response of buildings and structures. The method is based on singular value decomposition of the cross-power spectral density matrix from simultaneous array recordings of ambient vibrations. This method is advantageous to retrieve not only the resonance frequencies of the investigated structure, but also the corresponding modal shapes without the need for an absolute reference. This is an important piece of information, which can be used to validate the consistency of numerical models and analytical solutions. We apply this approach using advanced signal processing to evaluate the resonance characteristics of 2-D Alpine sedimentary valleys. In this study, we present the results obtained at Martigny, in the Rhône valley (Switzerland). For the analysis, we use 2 hr of ambient vibration recordings from a linear seismic array deployed perpendicularly to the valley axis. Only the horizontal-axial direction (SH) of the ground motion is considered. Using the FDD method, six separate resonant frequencies are retrieved together with their corresponding modal shapes. We compare the mode shapes with results from classical standard spectral ratios and numerical simulations of ambient vibration recordings.

Assessing small failure probability by importance splitting method and its application to wind turbine extreme response prediction

This study presents an effective simulation framework with importance splitting (ISp) method for estimating small failure probabilities of dynamic structures with multi-correlated stochastic excitations, and addresses its application to predictions of large wind turbine extreme responses. The ISp method, also referred to as subset simulation with splitting, splits important sample paths into multiple branches at various stages in the simulation. It permits the estimation of a small failure probability of a rare event through estimations of conditional probabilities of intermediate subset events. The framework presented in this study combines the ISp method with multivariate autoregressive (MAR) modeling of stochastic excitations. The MAR model of excitations is established based on their cross power spectral density matrix, which transfers the stochastic excitations as the output of a loading system with a vector-valued uncorrelated white noise process as input. This scheme is very efficient in generating offsprings of loading and response time histories conditional on the intermediate events with very low rejection rate, which facilities the application of ISp method to different kinds of stochastic single and multiple excitations. The effectiveness and accuracy of the proposed new scheme are verified by a reliability problem of earthquake-excited 5-story building, and by the estimation of extreme responses of a 5MW onshore wind turbine with very small exceeding probabilities. Finally, this framework is applied to validate the extrapolation procedure of estimating wind turbine long-term extreme responses with various mean recurrence intervals from short-term simulations of turbine response histories, which is mandated by current wind turbine design standards.

Spline-Interpolation-Based FFT Approach to Fast Simulation of Multivariate Stochastic Processes

The spline-interpolation-based fast Fourier transform (FFT) algorithm, designated as the SFFT algorithm, is proposed in the present paper to further enhance the computational speed of simulating the multivariate stochastic processes. The proposed SFFT algorithm first introduces the spline interpolation technique to reduce the number of the Cholesky decomposition of a spectral density matrix and subsequently uses the FFT algorithm to further enhance the computational speed. In order to highlight the superiority of the SFFT algorithm, the simulations of the multivariate stationary longitudinal wind velocity fluctuations have been carried out, respectively, with resorting to the SFFT-based and FFT-based spectral representation SR methods, taking into consideration that the elements of cross-power spectral density matrix are the complex values. The numerical simulation results show that though introducing the spline interpolation approximation in decomposing the cross-power spectral density matrix, the SFFT algorithm can achieve the results without a loss of precision with reference to the FFT algorithm. In comparison with the FFT algorithm, the SFFT algorithm provides much higher computational efficiency. Likewise, the superiority of the SFFT algorithm is becoming more remarkable with the dividing number of frequency, the number of samples, and the time length of samples going up.

An efficient computational procedure for random vibro-acoustic simulations

We discuss a new method for handling random acoustic excitations in finite-element models. The idea is to approximate the cross power spectral density matrix of the response by a low rank matrix. We illustrate the low rank approximation can be computed efficiently by the implicitly restarted block Lanczos method. The novelty lies in applying the Lanczos method to the cross power spectral density matrix of the response rather than the excitation. We give a theoretical explanation for finite-element models with modal damping.

Reduced complexity modeling and reproduction of colored textures

An unsupervised color texture synthesis-by analysis method is described. The texture is reproduced to appear perceptually similar to a given prototype by copying its statistical properties up to the second order. The synthesized texture is obtained at the output of a single-input three-output nonlinear system driven by a realization of a white Gaussian random field. Significant complexity reduction is gained by exploiting the rank deficiency of the cross power spectral density matrix of the color texture samples.

Active Control of Transmission Tower under Stochastic Wind

An active mass driver has been designed for installation on the upper observation deck of the 310 m Nanjing TV transmission tower in China in order to reduce the acceleration response under strong wind gusts. In this paper, the linear quadratic Gaussian (LQG), H∞, and continuous sliding mode control (CSMC) strategies have been applied to the active mass driver system based on the acceleration feedbacks. Emphasis is placed on the practical applications of LQG, H∞, and CSMC controllers, including limitations on the actuator peak force and stroke, the limited number of sensors, and noise pollution. Further, a method for modulating the control effort for the CSMC controller is proposed in this paper to satisfy the practical constraints on the actuator. A state reduced-order system has been established to design the dynamic output feedback controllers. The wind loads acting on the tower are defined by the Davenport cross-power spectral density matrix. Simulation results demonstrate that, when designed appropriately, the LQG, H∞, and CSMC control strategies are suitable for full-scale implementations of the active mass driver, and their control performances are quite remarkable.

Reduced-order H∞ and LQR control for wind-excited tall buildings

Because of the large number of degrees-of-freedom involved in modeling wind-excited tall buildings, three reduced-order control methods are presented and their performances are investigated systematically. These include the state reduced-order control, critical-mode control and simple modal control. The important issues associated with reduced-order control, control spillover and observation spillover have been addressed. Both the robust H∞ and linear quadratic regulator ( LQR) control algorithms have been used. Applications of static output feedback controllers, that utilize only a limited number of sensors without an observer, are also investigated for practical implementations of control devices on wind-excited structures. The spatially correlated wind loads, as defined by a cross-power spectral density matrix, have been used to compute the stochastic response of the full-order tall building to demonstrate the performance of each control method. Numerical results indicate that as long as enough modes are considered in the reduced-order system, the performance for both state reduce-dorder control and critical-mode control are comparable to that of the full-order control. However, the performance of the simple modal control for reducing the acceleration response of wind-excited buildings is not satisfactory because of the spillover effects. Simulation results further indicate that static output controllers for reduced-order systems provide a possibility for reducing the required number of sensors. From the practical implementation standpoint, the state reduced-order control may be preferable to the critical-mode control, since it does not require an observer.

Stochastic analysis of mechanical vibrations in reactors

A FORTRAN code VIBRPI has been developed to compute the generalized cross power spectral density matrix for linear motions of a mechanical system with N connected elements. The code is applied to analysis of the parallel flow and cross flow induced vibrations of BWR instrument guide tubes. The guide tubes, and the neutron detectors contained in them, are regarded as probes to investigate the properties of the flow and the mechanical system. The computations are compared to noise measurements made on operating BWRs. Added mass and drag coefficients are inferred as are parameters of the cross flow induced effect.