Power Spectral Density Matrix Research Articles

Strain-based Multiaxial Fatigue Life Evaluation Using Spectral Method

To describe the impact of multiaxial service loads a power spectral density matrix of the strain tensor components has been taken into account. To determine the fatigue life some selected multiaxial fatigue failure criteria have been used. The main focus of this paper is set on the comparison of the cycle counting method and the spectral method. An algorithm used to calculate the fatigue life in the time domain as well as in the frequency domain has been introduced. The proposed algorithm has been verified on the basis of fatigue tests of cruciform specimens made out of the S355J2WP steel.

Multi-channel post-filtering based on spatial coherence measure

A multi-channel post-filtering algorithm using the proposed spatial coherence measure is derived. The spatial coherence measure evaluates the similarity between the measured signal fields using power spectral density matrices. In the proposed post-filter, the assumption of homogeneous sound fields is relaxed. Besides, multi-rank signal models can be easily adopted. Under this measure, the bias term due to the similarity of the desired signal field and the noise field is further investigated and a solution based on bias compensation is proposed. It can be shown that the compensated solution is equivalent to the optimal Wiener filter if the bias or the noise power spectral density matrix is perfectly measured. Simulations with incoherent, diffuse, and coherent noise fields and a local scattered desired source were conducted to evaluate the algorithms. The results demonstrate the superiority of the proposed bias compensated post-filter across different types of noise fields with a more accurate signal model.

Reformulation in the frequency domain of a critical plane-based multiaxial fatigue criterion

In the present paper, a new computationally-efficient frequency domain formulation of the critical plane-based Carpinteri–Spagnoli (C–S) criterion is proposed to evaluate the fatigue lives of smooth metallic structures subjected to multiaxial random loading. The critical plane orientation is here proposed to depend on the Power Spectral Density (PSD) matrix of the stress tensor. Then, the PSD function of an equivalent normal stress is defined by considering a linear combination of the PSD functions of the normal stress and the projected shear stress along the direction of maximum variance, with such stresses acting on the critical plane. Such an equivalent PSD function allows us to apply the Tovo–Benasciutti method to estimate the fatigue life of the structural components. The present frequency domain formulation of the C–S criterion is applied to some relevant fatigue tests related to smooth specimens under non-proportional bending and torsion random loading.

Transmit Beampattern Synthesis with Constant Beamwidth and Sidelobe Control for Wideband MIMO Radar

A beampattern synthesis approach is proposed to design the power spectral density matrix (PSDM), which is chosen to achieve a given transmit beampattern in wideband multiple-input multiple-output (MIMO) radar systems. The proposed approach focuses on transmit beampattern synthesis with constant beamwidth and sidelobe control. Moreover, the design problem is further converted to a convex optimization problem, which is solved efficiently via the modeling system CVX. In comparison to these recently developed wideband MIMO beampattern synthesis methods, the proposed approach maintains a constant beamwidth across the entire frequency band and provides a great improvement in sidelobe control. Numerical simulation results are obtained to validate the effectiveness of this approach.

ISF와 Floquet 벡터에 기초한 발진기 위상잡음 이론의 등가성에 대한 해석적 증명

This paper analytically proves the equivalence between two main oscillator phase noise theories, which are based on the ISF, and Floquet vector, respectively. For this purpose, this study obtains the power spectral density matrix from the ISF-based phase noise theory. As a result, one can prove that the power spectral density matrix obtained from the ISF-based phase noise theory is essentially equivalent to the power spectral density matrix presented by the Floquet vector-based phase noise theory, which manifests the equivalence of the two main theories. This study is intended to provide deeper insight into the relations between the two main theories.

Application of Power Spectral Density on Estimation of the Number of Source Signals

Blind source separation and estimation of the number of sources usually demand that the number of sensors should be greater than or equal to that of the sources, which, however, is very difficult to satisfy for the complex systems. A new estimating method based on power spectral density (PSD) is presented. When the relation between the number of sensors and that of sources is unknown, the PSD matrix is first obtained by the ratio of PSD of the observation signals, and then the number of source signals can be estimated by clustering the column vectors of PSD matrix. The effectiveness of the proposed method is verified by theoretical analysis and experiments.

New formulation of Cholesky decomposition and applications in stochastic simulation

Monte Carlo simulation plays a significant role in the mechanical and structural analysis due to its versatility and accuracy. Classical spectral representation method is based on the direct decomposition of the power spectral density (PSD) or evolutionary power spectral density (EPSD) matrix through Cholesky decomposition. This direct decomposition of complex matrix usually results in large computational time and storage memory.In this study, a new formulation of the Cholesky decomposition for the EPSD/PSD matrix and corresponding simulation scheme are presented. The key idea to this approach is to separate the phase from the complex EPSD/PSD matrix. The derived real modulus matrix evidently expedites decomposition compared to the direct Cholesky decomposition of the complex EPSD/PSD matrix. In the proposed simulation scheme, the separated phase can be easily assembled. The modulus of EPSD/PSD matrix could be further decomposed into the modulus of coherence matrix (or lagged coherence matrix), which describes the basic coherence structure of stochastic process. The lagged coherence matrix is independence of time and thus remarkably improves the Cholesky decomposition efficiency.The application of the proposed schemes to Gaussian stochastic simulations is presented. Firstly, the previous closed-form wind speed simulation algorithm for equally-spaced locations is extended to a more general situation. Secondly, the proposed approach facilitates the application of interpolation technique in stochastic simulation. The application of interpolation techniques in the wind field simulation is studied as an example.

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.

Conditional Simulation of Nonstationary Fluctuating Wind Speeds for Long-Span Bridges

This paper presents a computationally efficient method for the conditional simulation of nonstationary fluctuating wind speeds for the buffeting analysis of long-span bridges. The fluctuating wind speeds, with part of their time histories measured at some points, are regarded as a zero-mean nonstationary Gaussian vector process characterized by an evolutionary power spectral density (EPSD) matrix. The Kriging method is then applied to the random vector of the Fourier coefficients of the evolutionary vector process for conditional simulation. A mathematical proof of this conditional simulation is provided. A fast algorithm of the conditional simulation is further derived, by converting the Cholesky decomposition of a large size covariance matrix of the Fourier coefficients into the Cholesky decomposition of a series of small-size coherence matrices with much less computational time. The procedure for implementing the fast algorithm-based conditional simulation method is detailed. Finally, this method is applied for the conditional simulation of typhoon-induced nonstationary wind speed time histories for the buffeting analysis of Stonecutters Bridge in Hong Kong. The time domain buffeting responses of the bridge obtained using the conditionally simulated wind speed time histories indicate that the proposed simulation method is feasible and applicable.

Error Assessment for the Coherency Matrix-Based Spectral Representation Method in Multivariate Random Processes Simulation

Multivariate random processes are usually simulated by the spectral representation method (SRM). According to the matrix for decomposition, the SRM has two main types, that is, the SRM based on the decomposition of the power spectral density (PSD) matrix denoting the PSD matrix-based SRM, and the SRM based on the decomposition of the coherency matrix denoting the coherency matrix based-SRM. The stochastic errors of the PSD for the PSD matrix-based SRM have been given. This paper presents the stochastic errors of the PSD for the coherency matrix-based SRM, and makes a comparison of these errors for the PSD matrix-based SRM. For the random amplitudes formulas and random phase formula and Cholesky decomposition method, the stochastic errors of the PSDs for the PSD matrix-based SRM are the same as or the coherency matrix-based SRM, whereas for the random phases formula and eigendecomposition method and random phases formula and root decomposition method, they are different. However, the differences are slight when taking into account the sum of the PSD functions’ stochastic errors.

Stochastic optimal control analysis of a piezoelectric shell subjected to stochastic boundary perturbations

The stochastic optimal control for a piezoelectric spherically symmetric shell subjected to stochastic boundary perturbations is constructed, analyzed and evaluated. The stochastic optimal control problem on the boundary stress output reduction of the piezoelectric shell subjected to stochastic boundary displacement perturbations is presented. The electric potential integral as a function of displacement is obtained to convert the differential equations for the piezoelectric shell with electrical and mechanical coupling into the equation only for displacement. The displacement transformation is constructed to convert the stochastic boundary conditions into homogeneous ones, and the transformed displacement is expanded in space to convert further the partial differential equation for displacement into ordinary differential equations by using the Galerkin method. Then the stochastic optimal control problem of the piezoelectric shell in partial differential equations is transformed into that of the multi-degree-of-freedom system. The optimal control law for electric potential is determined according to the stochastic dynamical programming principle. The frequency-response function matrix, power spectral density matrix and correlation function matrix of the controlled system response are derived based on the theory of random vibration. The expressions of mean-square stress, displacement and electric potential of the controlled piezoelectric shell are finally obtained to evaluate the control effectiveness. Numerical results are given to illustrate the high relative reduction in the root-mean-square boundary stress of the piezoelectric shell subjected to stochastic boundary displacement perturbations by the optimal electric potential control.

Electric potential response analysis of a piezoelectric shell under random micro-vibrationexcitations

The response characteristics of a spherically symmetric piezoelectric shell under randomboundary micro-vibration excitations are analyzed and calculated. The equation for electricpotential is integrated radially to obtain the electric potential as a function ofdisplacement, so that the differential equations for the piezoelectric shell with electrical andmechanical coupling are converted into an equation only for the displacement. Thedisplacement transformation is constructed to convert the random boundary conditionsinto homogeneous ones, and the transformed displacement is expanded in space to furtherconvert the partial differential equation for the displacement into ordinary differentialequations using the Galerkin method. The equations represent a multi-degree-of-freedomdynamic system with an asymmetric stiffness matrix under random micro-vibrationexcitations. The frequency-response function matrix, power spectral density matrix andcorrelation function matrix of the system response are derived from these equations basedon the theory of random vibration. The expressions of mean-square displacement,stress and electric potential of the piezoelectric shell are finally obtained andillustrated by numerical results for random micro-vibration excitations. The randomelectrical and mechanical coupling properties, in particular the relations betweenboundary electric potential responses and micro-displacement excitations, areexplored.

The Two-Dimensional Power Spectral Density: A Connection Between 2-D Rational Functions and Linear Systems

A derivation of the power spectral density (PSD) matrix of a continuous-time, linear, constant-coefficient system with white Gaussian noise input is given. The assumption of wide-sense stationarity is not made. By transforming the two-dimensional (2-D) autocorrelation function (ACF), a 2-D spectral density is obtained that takes the system's transient behavior into account. The 2-D PSD is a non-separable rational function. The results are applicable to second-order Volterra series representations of nonlinear systems. Simulations are presented for a fourth order system that compare the 2-D PSD amplitude (both theoretical and empirical) with its 1-D counterpart on 50 realizations of a short data segment containing an initial transient.

A method for the random analysis of linear systems subjected to non-stationary multi-correlated loads

The paper shows a procedure for evaluating the correlation matrix and the evolutionary power spectral density matrix of the response of linear structural systems subjected to random non-stationary multi-correlated vector processes. The approach reduces this problem to the solution of some corresponding stationary problems. It is shown that the assumption of a modulating matrix function, whose elements are the sum of exponential functions, allows to transform the initial non-stationary problem into a stationary one. This stationary problem can be solved by well-known unconditionally stable step-by-step numerical procedures in the time domain, and, in closed form, in the frequency domain. The comparison of the results obtained using the proposed approach with those obtained by Monte Carlo Simulations (MCS) has revealed a very good level of accuracy.

Detection of Spatially Correlated Gaussian Time Series

This work addresses the problem of deciding whether a set of realizations of a vector-valued time series with unknown temporal correlation are spatially correlated or not. For wide sense stationary (WSS) Gaussian processes, this is a problem of deciding between two different power spectral density matrices, one of them diagonal. Specifically, we show that for arbitrary Gaussian processes (not necessarily WSS) the generalized likelihood ratio test (GLRT) is given by the quotient between the determinant of the sample space-time covariance matrix and the determinant of its block-diagonal version. Furthermore, for WSS processes, we present an asymptotic frequency-domain approximation of the GLRT which is given by a function of the Hadamard ratio (quotient between the determinant of a matrix and the product of the elements of the main diagonal) of the estimated power spectral density matrix. The Hadamard ratio is known to be the GLRT detector for vector-valued random variables and, therefore, what this paper shows is how frequency-dependent Hadamard ratios must be merged into a single test statistic when the vector-valued random variable is replaced by a vector-valued time series with temporal correlation. For bivariate time series, the derived frequency domain detector can be rewritten as a function of the well-known magnitude squared coherence (MSC) spectrum, which suggests a straightforward extension of the MSC spectrum to the general case of multivariate time series. Finally, the performance of the proposed method is illustrated by means of simulations.

Invariant features of deep ocean ambient noise.

Very low‐frequency, deep ocean ambient noise is generated by non‐linear interactions (collisions) between surface waves with nearly equal frequency, propagating in nearly opposite direction. It is shown that the usual “standing wave” approximation used to compute this noise predicts that the ratios of entries in the power spectral density matrix, obtained from auto and cross‐correlations of acoustic velocity and pressure fluctuations, are universal constants. This prediction is too strong and not observed. A careful consideration of bottom effects invalidates the standing wave approximation. Nevertheless a weak version of the standing wave approximation still holds and has interesting implications. While the ratios of entries in the power spectral density matrix due to surface wave generated noise are not universal constants, they are insensitive to the details of the surface wave spectrum. This theoretical prediction is borne out by data from the Hawaii‐2 Observatory. [Work supported by ONR.]

DAMAGE DETECTION OF ACTUAL BUILDING STRUCTURES THROUGH SINGULAR VALUE DECOMPOSITION OF POWER SPECTRAL DENSITY MATRICES OF MICROTREMOR RESPONSES

Damage detection through microtremor measurements was applied to two actual building structures in which walls and a beam were damaged artificially. The responses recorded in the time domain are transformed into the power spectral density matrix at each discrete frequency. The matrix is decomposed into a set of auto spectral density functions by the singular value decomposition. The changes of the resonant frequencies and the corresponding modal shapes are searched by comparing the state after damage with the state before damage. Structural damage locations are found through the largest change of the modal shape.

Three-dimensional reverberation modeling.

Previously, reverberation modeling at sonar and torpedo frequencies was based on spectral factorization, which does not allow correct broadband, non-zero Doppler simulation in real time for a multi-channel receiver. This paper presents a new approach which avoids these limitations. Instead of transmitting the pulse itself, the method transmits a superposition of many copies of the pulse, each with a random amplitude and phase. This “noiselet” mimics the point-scatterer method without requiring a prohibitively large number of scattering points. We refer to it as the early randomization method (ERM). The propagation model used, the GRAB ray model, contains the bathymetry information, enabling the ERM to produce a statistically realistic reverberation field with a minimal number of rays. The computational load increases linearly with the number of receiver elements instead of with its cube, since there is no power spectral density matrix to factor. Since the randomness is introduced directly into the noiselet, the ERM is not limited to a particular statistical model, allowing non-Gaussian cases to be treated. Previously, this approach has shown excellent results for the two-dimensional, range-dependent case. Here, the method is generalized to also allow cross-range dependence. Comparisons with the CASS/GRAB model are shown.

Information Theoretic Bounds for Compound MIMO Gaussian Channels

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, achievable rates for compound Gaussian multiple-input–multiple-output (MIMO) channels are derived. Two types of channels, modeled in the frequency domain, are considered when: 1) the channel frequency response matrix <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$H$</tex> </formula></emphasis> belongs to a subset of <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$H^{\infty}$</tex></formula></emphasis> normed linear space, and 2) the power spectral density (PSD) matrix of the Gaussian noise belongs to a subset of <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$L_1$</tex></formula></emphasis> space. The achievable rates of these two compound channels are related to the maximin of the mutual information rate. The minimum is with respect to the set of all possible <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$H$</tex> </formula></emphasis> matrices or all possible PSD matrices of the noise. The maximum is with respect to all possible PSD matrices of the transmitted signal with bounded power. For the compound channel modeled by the set of <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$H$</tex> </formula></emphasis> matrices, it is shown, under certain conditions, that the code for the worst case channel can be used for the whole class of channels. For the same model, the water-filling argument implies that the larger the set of matrices <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$H$</tex></formula></emphasis>, the smaller the bandwidth of the transmitted signal will be. For the second compound channel, the explicit relation between the maximizing PSD matrix of the transmitted signal and the minimizing PSD matrix of the noise is found. Two PSD matrices are related through a Riccati equation, which is always present in Kalman filtering and liner-quadratic Gaussian control problems. </para>

Response analysis of piezoelectric shells in plane strain under random excitations

The random response of a piezoelectric thick shell in plane strain state under boundary random excitations is studied and illustrated with a piezoelectric cylindrical shell. The differential equation for electric potential is integrated radially to obtain the electric potential as a function of displacement. The random stress boundary conditions are converted into homogeneous ones by transformation, which yields the electrical and mechanical coupling differential equation for displacement under random excitations. Then this partial differential equation is converted into ordinary differential equations using the Galerkin method and the Legendre polynomials, which represent a random multi-degree-of-freedom system with asymmetric stiffness matrix due to the electrical and mechanical coupling and the transformed boundary conditions. The frequency-response function matrix and response power spectral density matrix of the system are derived based on the theory of random vibration. The mean-square displacement and electric potential of the piezoelectric shell are finally obtained, and the frequency-response characteristics and the electrical and mechanical coupling properties are explored.