Spectral Density Function Research Articles

Topography stitching in the spatial frequency domain for the representation of mid-spatial frequency errors.

Sub-aperture fabrication techniques such as diamond turning, ion beam figuring, and bonnet polishing are indispensable tools in today's optical fabrication chain. Each of these tools addresses different figure and roughness imperfections corresponding to a broad spatial frequency range. Their individual effects, however, cannot be regarded as completely independent from each other due to the concurrent formation of form and finish errors, particularly in the mid-spatial frequency (MSF) region. Deterministic Zernike polynomials and statistical power spectral density (PSD) functions are often used to represent form and finish errors, respectively. Typically, both types of surface errors are treated separately when their impact on optical performance is considered: (i) wave aberrations caused by figure errors and (ii) stray light resulting from surface roughness. To fill the gap between deterministic and statistical descriptions, a generalized surface description is of great importance for bringing versatility to the entire optical fabrication chain by enabling easy and quick exchange of surface topography data between three disciplines: optical design, manufacturing, and characterization. In this work, we present a surface description by stitching the amplitude and unwrapped phase spectra of several surface topography measurements at different magnifications. An alternative representation of surface errors at different regimes is proposed, allowing us to bridge the gap between figure and finish as well as to describe the well-known MSF errors.

Response spectrum method for fatigue damage assessment of mechanical systems

In this paper, a new response spectrum method is proposed for high-cycle fatigue damage assessment of a linear multi-degree-of-freedom system subjected to random base acceleration. Mode damage response can be accessed by separating the modal participation factor and stress mode from the contribution of a mode to the total damage, assessed by the Single Moment method. The fatigue damage response spectrum is defined as the family of curves formed by the mode damage response with the variation of the frequency and modal damping ratio, which is irrelevant to the spatial characteristics of the load and structure. Three calculation formats of the response spectrum method are formed by considering different cross-correlation hypotheses of each mode (i.e. the Complete Quadratic Combination format, the Square Root of the Sum of Squares format, and the Modified Square Root of the Sum of Squares format). The proposed method has higher computational efficiency than other frequency-domain methods because of avoiding the calculation of power spectral density function and spectral moment of stress responses. The spatial and frequency information of the structure are decoupled from each other in the implementation, which can effectively reduce the computational cost of reanalysis. Compared to the result of the Dirlik method (self-compiled program and MSC.Patran/Nastran) and the Single Moment method, the correctness and efficiency of the proposed method are verified, and the influences of the material and load spectrum form on the response spectrum method are investigated.

Power spectral density-based fractal analyses of sputtered yttria-stabilized zirconia thin films

This study provides information about the surface morphology of sputtered yttria-stabilized zirconia (YSZ) thin films from the atomic force microscope (AFM) spectral analyses using the power spectral density (PSD) function at varying annealing temperatures. Applying fractal and k-correlation fitting models to the PSD data, fractal dimension, Hurst exponent, correlation length, and equivalent root mean square roughness are quantified. The PSDs of the films exhibit an inverse power-law variation at high spatial frequency, which points to the existence of the fractal components in the film’s surface. The annealing temperatures up to 900 ∘C decreased fractal dimension from 2.60 to 2. The surface roughness increased from 0.10 to 13.92 nm and from 0.04 to 3.95 nm, obtained from the statistical analyses of AFM images and the k-correlation model. The films annealed from 500 ∘C to 800 ∘C showed fine grain size morphology with Hurst exponent values from 0.40 to 0.53, indicating a homogeneous spatial roughness distribution. While the film annealed at 900 ∘C exhibited large aggregate grains morphology. The growth of a sample annealed at the temperature of 900 ∘C is more likely to be ruled by the step-edge barrier-induced mound growth and inhomogeneous spatial distribution of roughness. In contrast, normal self-affine behaviour is observed at lower annealing temperatures.

PERFORMANCE COMPARISON OF STRUCTURAL DAMAGE DETECTION METHODS BASED ON FREQUENCY RESPONSE FUNCTION AND POWER SPECTRAL DENSITY

Recent catastrophic events have aroused great interest in the scientific community regarding the evaluation and prediction of the structural response along the life cycle of infrastructures. Efforts are put into developing adequate health monitoring systems to help prevent future human life and economic losses. Here, two non-destructive damage detection methods are presented: the Frequency Response Function-based and the Spectral Density Function-based methods. The damage detection performance of both methods is compared through a particular case study, where different damage scenarios are analyzed in a 2D truss bridge. The reliability of each method is studied in terms of different prediction errors. Numerical results show that the PSD method for damage detection on a steel truss bridge structure provides more accurate and robust results when compared to that based on FRF. Keywords: Structural Health Monitoring, Power Spectral Density Function, Frequency Response Function, Construction, Structures, Damage detection, Non-destructive

Use of transform pairs to represent and simulate nonstationary non-Gaussian process with applications

We extend the use of the iterative power and amplitude correction (IPAC) algorithm to simulate nonstationary non-Gaussian processes by considering five transform pairs, including the redundant and non-redundant transform pairs that could provide time–frequency or time-scale decomposition. We compare the performance of the IPAC algorithm to that of the spectral representation method (SRM) in terms of matching the prescribed power spectral density function (PSDF) and the marginal probability distribution function. We also extend and examine the IPAC algorithm to simulate concatenated nonstationary non-Gaussian processes. It is emphasized that the IPAC algorithm can be used with a power or energy distribution defined in a transform domain corresponding to a selected transform pair and without directly relying on the definition of the evolutionary process. Unlike SRM, the use of the IPAC algorithm to simulate the nonstationary non-Gaussian process does not require evaluating the underlying Gaussian PSDF. This is especially advantageous for cases when one is interested in estimating the reliability or performance of a structural system during its design life that is subjected to a number of nonstationary non-Gaussian excitations, each defined by a different PSDF.

Linear Vortex Symmetrization: The Spectral Density Function

We investigate solutions of the 2d incompressible Euler equations, linearized around steady states which are radially decreasing vortices. Our main goal is to understand the smoothness of what we call the spectral density function associated with the linearized operator, which we hope will be a step towards proving full nonlinear asymptotic stability of radially decreasing vortices. The motivation for considering the spectral density function is that it is not possible to describe the vorticity or the stream function in terms of one modulated profile. There are in fact two profiles, both at the level of the physical vorticity and at the level of the stream function. The spectral density function allows us to identify these profiles, and its smoothness leads to pointwise decay of the stream function which is consistent with the decay estimates first proved in Bedrossian-Coti Zelati-Vicol in [5].

Model parameter estimations of the multi-channel turbulence response from flutter flight tests based on autoregressive spectra

Signal processing of flutter flight test data enables verification of aircraft flutter design, and the signal from a flutter flight test excited by atmospheric turbulence is a particularly important form of the flutter test. Owing to the randomness of atmospheric turbulence excitation, multi-channel analysis of turbulence responses at various positions in the same component can improve the analytical accuracy of flutter signal processing. The relationship between the maximum singular value of the multi-channel turbulence response power spectral density matrix and the system self-power spectral density function is elucidated herein using a frequency domain decomposition method. However, there is a contradiction in the power spectral density function between the spectral line density and the spectral line smoothing calculated based on the periodogram of the frequency domain decomposition. By applying an autoregressive spectral model, the power spectral density function of the turbulence response is calculated to achieve spectral line smoothing and sufficient spectral line density. Additionally, the power spectral density function is then used to construct the power spectral density function matrix of the multi-channel turbulence response, and the maximum singular value curve is calculated based on the singular value decomposition of each spectral pin. Finally, the modal parameters of the turbulence response signal are estimated via multi-modal frequency domain fitting. The developed approach is validated based on simulations and flutter flight test turbulence response signals.

Conditional simulation of non-stationary spatially variable ground motions for long-span bridges across non-uniform site conditions

Non-stationary spatially variable ground motions (SVGMs) are commonly modelled as multivariate oscillatory processes based on evolutionary power spectral density (EPSD) functions. The existing conditional simulation algorithms require the known EPSD functions. The EPSD functions are usually assumed to be identical for all locations, which is unreasonable for long-span bridges because variable soil conditions are practically observed at different bridge piers. This paper proposes a conditional simulation algorithm for non-stationary SVGMs in consideration of non-uniform site conditions. The spatial interpolation tool, termed inverse-distance-weighted (IDW) interpolation, is introduced to estimate the EPSD functions at sites without ground motion measurement. Subsequently, the covariance matrix of the random Fourier coefficients of the multivariate oscillatory processes can be calculated. The Kriging estimation is adopted to obtain the unknown random Fourier coefficients, from which the time histories of the non-stationary SVGMs can be conditionally simulated. The proposed conditional simulation algorithm is first validated through a numerical example, in which the EPSD functions of non-uniform sites are represented by a non-stationary Kanai-Tajimi spectrum with different soil parameters. Then, the algorithm is applied to the Jiuzhou Channel Bridge, a navigation channel bridge of the Hong Kong-Zhuhai-Macau Bridge (HZMB), with complex soil and water conditions. Based on the limited in-situ seismic measurement data, the site characteristics in the bridge area are analysed, and the ground motion time histories at all piers can be generated.

The probabilistic turbulence profiles of tropical cyclones in open and flat terrain

The fluctuating winds are usually characterized in the frequency domain by the power spectral density (PSD) function and the coherence function, in which the turbulence intensity and turbulence integral scale are key parameters to describe the gust characteristics of turbulent winds. These parameters estimated from tropical cyclone (TC) observations show significant scattering. Starting from a critical review of the state of the art, the Von Karman spectra are chosen for the TC winds. The turbulence intensity and the turbulence integral scale used in wind spectral functions are discussed in detail. By introducing randomness into the empirical models, the probabilistic longitudinal turbulence profiles of TCs are derived from a wide collection of field measurements with open and flat terrain in the literature. Moreover, the probabilistic models of the lateral and vertical turbulence components are obtained by assuming proportionality between these two components and the longitudinal one. The mean and standard deviation of the developed models for TCs are compared with those defined in the seven major wind loading codes worldwide as well as the model given by Solari to testify these models and illustrate their differences to synoptic winds.

A Computational Model to Simulate Thunderstorm Downbursts for Wind Turbine Loads Analysis

Abstract The generation of wind fields is of interest in the study of the structural performance of wind turbines in critical events, such as thunderstorm downbursts. Various methods ranging from the use of empirical data to employing computational simulations are typically adopted to study the response of wind turbines in downburst flow fields. While the former approach is limited in the ability to account for accurate and spatially resolved details of the flow field, the latter is expensive and, therefore, has limitations in its use. As an alternative, in this work, we propose a paused downburst model in which a snapshot of a time-dependent computational fluid dynamics (CFD) simulation is used to generate “mean” wind fields during thunderstorm downbursts. The developed model for the mean wind field is validated against recorded downburst data in the literature. The turbulent component of the wind field is generated using computationally inexpensive techniques based on Fourier-based power spectral density functions and coherence functions. In an illustrative example, the combined mean and turbulence wind fields are generated and applied on a utility-scale wind turbine to study structural load characteristics during a downburst event.

Bubble Behaviors in a Dense Gas‐Solids Fluidized Bed: Analysis of Pressure Fluctuation

AbstractThe pressure signals were analyzed by time‐domain and frequency‐domain methods, including standard deviation, autocorrelation function, and power spectral density function, to investigate the bubble behavior and determine the regime transition. Results indicated that the autocorrelation function of pressure signals derived from bubbles was periodic fluctuation. The periodic nature of the autocorrelation function disappeared and the dominant frequency increased when the transition from bubbling to the turbulent regime occurred. The decrease of static bed height facilitated the explosion of bubbles, which accelerated the transition from the bubbling to the turbulent regime. The fluidization index was used as an indicator for the bubble state in the dense fluidized bed.

Edgeworth expansion of the t-statistic of the whittle MLE for linear regression processes with long-memory disturbances

This paper establishes an Edgeworth expansion for the t-statistic of the Whittle Maximum Likelihood Estimator (WMLE) of a linear regression model whose residual component is stationary, Gaussian, and strongly dependent time series. Under the widely used set of assumptions and two more mild additional conditions on the spectral density function and the parametric values, an Edgeworh expansion of the t-statistic of arbitrarily large order of the process is proved to have an error of o ( n 1 − s / 2 ) where s is a positive integer.

An adaptive multi-taper spectral estimation for stationary processes

The multi-taper spectral estimation method is effective in reducing both bias and variance of the spectrum of stationary stochastic processes. The number of tapers (NoT) in the method controls the bias and variance trade-off of the spectrum. In general, the NoT is empirically determined and constant at all frequencies, limiting the adjustment of the local bias and variance. This paper develops an adaptive multi-taper approach with the varying NoT at each frequency point for estimating the power spectral density (PSD) and coherence function of multivariate stationary processes. An iterative procedure with a stopping criterion is proposed to optimize the NoT at each frequency without manual tunning. The sine taper with a simple analytical expression and satisfactory leakage protection is employed. The proposed adaptive multi-taper approach is applied to three numerical examples, the structural response of a building model, a wind speed process, and an autoregressive process, which exhibit different spectral characteristics. The results of all examples show that the proposed approach outperforms Welch’s and traditional multi-taper methods in estimating the PSD and coherence with a smaller bias and variance.

Quantum metrology with multimode Gaussian states of multiple point sources

A unified theoretical framework is developed for measuring multimode Gaussian states of multiple point sources. The spatial and temporal mode functions are found analytically by aid of the circular symmetry of the object plane and aperture, as well as by the rectangular spectral density function of the background. They relate the modal operators and statistical moments to the locations of the point sources through the prolate spheroidal functions. An experimentally straightforward implementation is proposed, which consists of a nonabsorptive polarization- and phase-modulating spatial light modulator (SLM), three lenses, and a polarization beam splitter. Interestingly, it is found that the spatial modal annihilation operators are related to the modal field operators in the image plane by a simple complex-valued scale factor. The quantum Fisher information matrix and quantum Cram\'er-Rao bound (QCRB) are also discussed. The numerical calculation of QCRB is completed for the case where the spatially multimode aperture fields are prepared by the spatial propagation from temporally single-mode thermal states, coherent states, or quadrature squeezed states of the point sources in the object plane. The results show that the Rayleigh limit can always be surpassed by means of increasing the mean photon rate or the squeeze parameter.

Superdense beaming of axion dark matter in the vicinity of the light cylinder of pulsars

In this article we treat the non-adiabatic photon-to-axion resonant conversion of curvature radiation, synchrotron emission and inverse Compton scattering dominating the spectral density function of pulsars. First, we introduce emission models and benchmark observational data. We adopt a state-of-the-art density profile that relieves tension with the quantum electrodynamics vacuum polarization effect in highly magnetic stars, leading to efficient mixing. Then, we estimate the dark matter flux induced by photon-axion oscillation across the light cylinder of the neutron star. We find that pulsars might produce axion overdensities many orders of magnitude over the occupation number of dark matter in the Galactic halo within a broad parameter space. We point out possible new methods for axion detection derived from these results and other future lines of work.

Downlink Precoding for DP-UPA FDD Massive MIMO via Multi-Dimensional Active Channel Sparsification

In this paper, we consider user selection and downlink precoding for an over-loaded single-cell massive multiple-input multiple-output (MIMO) system in frequency division duplexing (FDD) mode, where the base station is equipped with a dual-polarized uniform planar array (DP-UPA) and serves a large number of single-antenna users. Due to the absence of uplink-downlink channel reciprocity and the high-dimensionality of channel matrices, it is extremely challenging to design downlink precoders using closed-loop channel probing and feedback with limited spectrum resource. To address these issues, a novel methodology – active channel sparsification (ACS) – has been proposed recently in the literature for uniform linear array (ULA) to design sparsifying precoders, which substantially reduces channel feedback overhead. Pushing forward this line of research, we aim to facilitate the potential deployment of ACS in practical FDD massive MIMO systems, by extending it from ULA to DP-UPA with explicit user selection and making the current ACS implementation simplified. To this end, by leveraging Toeplitz matrix theory, we start with the spectral properties of channel covariance matrices from the lens of their matrix-valued spectral density function. Inspired by these properties, we extend the original ACS using scalar-weight bipartite graph representation to the matrix-weight counterpart. Building upon such matrix-weight bipartite graph representation, we propose a multi-dimensional ACS (MD-ACS) method, which is a generalization of original ACS formulation and is more suitable for DP-UPA antenna configurations. The nonlinear integer program formulation of MD-ACS can be classified as a generalized multi-assignment problem (GMAP), for which we propose a simple yet efficient greedy algorithm to solve it. Simulation results demonstrate the performance improvement of the proposed MD-ACS with greedy algorithm over the state-of-the-art methods based on the QuaDRiGa channel models.

Data-driven reliability assessment of dynamic structures based on power spectrum classification

The power spectral density function is a widely used tool to determine the frequency components and amplitudes of environmental processes, such as earthquakes or wind loads. It is an important technique especially in the engineering field of vibration analysis and in determining the response of structures. When using a large amount of data, a load model can be generated, which describes the characteristics of the underlying stochastic process. This load model enables artificially generated excitations to be created within the framework of Monte Carlo simulations. If multiple data records are utilised, a problem that can occur is that the individual records have a high variance in the frequency domain and are therefore too dissimilar from each other, even though they appear to be similar in the time domain. A load model derived from this data does not represent the entire data set, because not the whole spectral range is covered. Therefore, every attempt must be made to group the records according to their characteristics and thus combine similar data to derive two or more load models accordingly. In this work, an approach is proposed to classify real earthquake ground motion records using the k-means algorithm based on similarities within the data ensemble as determined by the Bhattacharyya distance. The silhouette method enables the identification of the optimal number of groups for the classification. The classified data thus form a subset of the entire data set from which load models can be generated and can be applied separately to the structure under investigation, leading to more accurate simulation results. The advantages of this classification approach are illustrated by means of an academic example and a seismic-isolated bridge pier model as a non-linear dynamic system.

Spin-wave theory in a randomly disordered lattice: A Heisenberg ferromagnet

Starting from the hamiltonian for the Heisenberg ferromagnet which comprise randomly distributed nonmagnetic ions as impurities in a Bravais lattice, we express the spin operators by means of the Dyson–Maleev transformation in terms of the Bose operators of the second quantization. Then by using methods of quantum statistical field theory, we derive the partition function and the free energy for the system. We adopt the Matsubara thermal perturbation method to a portion of the hamiltonian which describes the interaction between magnons and the stationary field of nonmagnetic ions. Upon averaging over all possible distributions of impurities, we express the free energy of the system as a function of the mean impurity concentration. Subsequently, we set up the double-time single particle Green function at temperature T in the momentum space in terms of magnon operators and derive the equation of motion for the Green function through the Heisenberg equation of motion and then solve the resulting equation. From this, we calculate the self-energy and then the spectral density function for the system. We apply the formalism to the case of the simple cubic lattice and compute the density of states, the spectral density function and the lifetime of the magnons as a function of energy for several values of the mean concentration of nonmagnetic ions in the ferromagnetic lattice. We calculate the magnon energy spectrum as a function of the average impurity concentration fraction c, which shows that for low lying states, the excitation energy increases continuously with c in the studied range 0.1≤c≤0.7. We also use the spectral density function to compute some thermodynamical quantities through the magnon occupation number. We have obtained closed form expressions for the configurationally averaged physical quantities of interest in a unified fashion as functions of the mean concentration of nonmagnetic impurities c to any order of c applicable below a critical percolation concentration cp. The quantities of interest comprise the thermodynamic potential (free energy), the spin-wave self-energy and the spectral density function from which other quantities can be derived.

Heart rate estimation and validation algorithm for fetal phonocardiography

Objective. Fetal heart rate (FHR) is an important parameter for assessing fetal well-being and is usually measured by doppler ultrasound. Fetal phonocardiography can provide non-invasive, easy-to-use and passive alternative for fetal monitoring method if reliable FHR measurements can be made even in noisy clinical environments. Therefore, this work presents an automatic algorithm to determine FHR from the fetal heart sound recordings in a noisy clinical environment. Approach. Using an electronic stethoscope fetal heart sounds were recorded from the expecting mother’s abdomen, during weeks 30–40 of their pregnancy. Of these, 60 recordings were analyzed manually by two observers to obtain reference heart rate measurement. An algorithm was developed to determine FHR using envelope detection and autocorrelation of the signals. Algorithm performance was improved by implementing peak validation algorithm utilizing knowledge of valid FHR from prior windows and power spectral density function. The improvements in accuracy and reliability of algorithm were measured by mean absolute error (MAE) and positive percent agreement. Main results. By including the validation step, the MAE reduced from 11.50 to 7.54 beats per minute and positive percent agreement improved from 81% to 87%. Significance. We classified the recordings into good, moderate and poor quality to understand how the algorithm works in each of the case. The proposed algorithms provide good accuracy overall but are sensitive to the noises in recording environment that influence the quality of the signals.

The Algorithm Accuracy of Mathematical Model to Improve the Transmission Speed of E-commerce Platform

Abstract After the input signal of e-commerce network information enters the network, it will inevitably be polluted by network delay. There is an error between the network output signal obtained by the receiver and the input signal of the transmitter. We combine control combination algorithms with mathematical models. And the power spectral density function is introduced into the signal transmission mathematical model of the e-commerce platform. Experiments prove that the new model is closer to the transmission efficiency change of the actual network communication signal than the traditional model.