Correlation Function Matrix Research Articles

Theoretical model for correlation analysis of wideband signal received by two vertical arrays and retrieval of modal dispersion curves in shallow water

Goal of this work is to carry out theoretical modeling extraction of characteristics of layered bottom (sound speeds c ;2 in layer, c in water, c ;b in half-space and layer’s thickness h using construction of correlation function (matrix) of wideband synthetic sound signal radiated by the moving source and imitating shipping noise. This work's motivation was experiment carried out by authors in shallow water. Combination of two vertical line arrays acts as a diffraction grating with 2N hydrophones that uses interference pattern in vertical and horizontal directions and gives more information than one-dimensional array or single receiver. Our waveguide model constitutes water layer above low-speed thin layer (c ;2 ≪ c, h is about a few meters) and half space (c ;b > c). In this waveguide there are two sorts of modes: in dependence on frequency and thickness of sediment’s layer– modes with vertical dependence distributed in water layer and modes trapped in the near bottom layer. It is shown that correlation function constructed in coordinates (phase speed, frequency) allows us to retrieve dispersion curves and to estimate critical frequencies—determining creation of trapped modes, which in turn give parameters of layered bottom. [Work was supported by ISF, Grant No. 946/20.]

Accuracy Analysis for Shack–Hartmann Wave Front Sensing with Extended Sources

Correlating the Shack–Hartmann wave front sensor (SH-WFS) with extended targets is widely used in solar adaptive optics systems. This paper aims to introduce a theoretical analysis that evaluates the accuracy of the SH-WFS on extended sources, with a specific focus on the implementation of the Normalized Cross-correlation (NCC) algorithm. To obtain an accurate error description, we utilized the calculation formula of the NCC algorithm to directly express the coordinates of the maximum value in the correlation function matrix. Furthermore, we determined the variance of the centroid position through the error transfer function, which quantifies the measurement error. In comparison with the previous findings of Michau et al., our result exhibits a coefficient disparity, specifically obtaining results 1.5 times higher than their work. The extensive solar granulation simulation and experimental results validate the theoretical error formulas. These error formulas can effectively estimate the accuracy of the SH-WFS, providing a theoretical foundation for the design of optical systems.

Analysis of the integration development of enterprise refinement management and financial integration in the context of informationization

Abstract In the background of economic informationization, the integration and development of enterprise refinement management and financial integration have become the mainstream management direction of modern enterprise management. In this paper, the correlation function and judgment matrix are firstly determined by the entropy-weighted elementary topologizable evaluation model, while the initial judgment matrix V is standardized, and the differences in the correlation degree of indicators of different natures should be noted. The entropy power method is applied to calculate the change of relevant evaluation indexes, and the value of information utility in the evaluation system is determined through the change of entropy value. Then the initial data are obtained by means of questionnaires, and data analysis is performed on the first-level indicators in the evaluation system of enterprise financial management capability. The results show that the three indicators of financial risk control capability, financial management informationization level and financial personnel capability, and total budget management level have more room for improvement, but the score of financial risk control capability is only 5.04 points different from the total score, indicating that the specific situation of the enterprise itself for specific analysis. This study improves the level of financial management of enterprises and promotes their development of enterprises.

Simulating Multivariate Multidimensional Homogenous Non-Gaussian Field Based on Unified Hermite Polynomial Model

Reasonable simulation of stochastic fields is a key prerequisite for addressing stochastic mechanics problems with Monte Carlo simulation solver. This paper presents a novel method to simulate m-variate and n-dimensional (mV-nD) homogenous non-Gaussian fields according to the prescribed power spectral density matrix (PSDM) and first four moments. In the proposed method, the unified Hermite polynomial model (UHPM) is first extended to mV-nD homogenous non-Gaussian fields. Then, based on the extended UHPM, a complete transformation model from the normalized non-Gaussian correlation function matrix (CFM) into the underlying Gaussian CFM with its applicable range is derived. In addition, two types of potential incompatibility of the transformation arising from the prescribed PSDM and first four moments are defined and neatly solved. Finally, a unified simulation framework for mV-nD homogenous non-Gaussian fields is presented, where the fast Fourier transform technique can be embedded in both spectral representation method and Wiener-Khintchine transformation to speed up the simulation progress. Two numerical examples including the simulations of non-Gaussian wind fields and non-Gaussian fields of material properties are presented to demonstrate the effectiveness of the proposed method.

Isospin 0 and 2 two-pion scattering at physical pion mass using all-to-all propagators with periodic boundary conditions in lattice QCD

A study of two-pion scattering for the isospin channels, $I=0$ and $I=2$, using lattice QCD is presented. M\"obius domain wall fermions on top of the Iwasaki-DSDR gauge action for gluons with periodic boundary conditions are used for the lattice computations which are carried out on two ensembles of gauge field configurations generated by the RBC and UKQCD collaborations with physical masses, inverse lattice spacings of 1.023 and 1.378 GeV, and spatial extents of $L=4.63$ and 4.58 fm, respectively. The all-to-all propagator method is employed to compute a matrix of correlation functions of two-pion operators. The generalized eigenvalue problem (GEVP) is solved for a matrix of correlation functions to extract phase shifts with multiple states, two pions with a non-zero relative momentum as well as two pions at rest. Our results for phase shifts for both $I=0$ and $I=2$ channels are consistent with and the Roy Equation and chiral perturbation theory, though at this preliminary stage our errors for $I=0$ are large. An important outcome of this work is that we are successful in extracting two-pion excited states, which are useful for studying $K\to\pi\pi$ decay, on physical-mass ensembles using GEVP.

A new OMA method to perform structural dynamic identification: numerical and experimental investigation

Operational modal analysis (OMA) methods are nowadays common in civil, mechanical and aerospace engineering to identify and monitor structural systems without any knowledge on the structural excitation provided that the latter is due to ambient vibrations. For this reason, OMA methods are embedded with stochastic concepts and then it is difficult for users that have no-knowledge in signal analysis and stochastic dynamics. In this paper an innovative method useful for structural health monitoring (SHM) is proposed. It is based on the signal filtering and on the Hilbert transform of the correlation function matrix. Specifically, the modal shapes are estimated from the correlation functions matrix of the filtered output process and then the frequencies and the damping ratios are estimated from the analytical signals of the mono-component correlation functions: a complex signals in which the real part represents the correlation function and the imaginary part is its Hilbert transform. This method is very simple to use since requires only few interactions with the users and thus it can be used also from users that are not experts in the aforementioned areas. In order to prove the reliability of the proposed method, numerical simulations and experimental tests are reported also considering comparisons with the most popular OMA methods.

Оптимізація поляризаційно-доплерівської селекції малорозмірних об’єктів на тлі підстильної поверхні

Using the method of maximum likelihood, the optimal algorithm of polarization-Doppler selection of objects on the background of the underlying surface, hydrometeors, urban buildings from aerospace carriers of radio electronic equipment has been synthesized. To solve the problem polarimetric properties of the scattered electromagnetic radiation of natural environments and anthropogenic objects were analyzed.The functional-deterministic mathematical model of the useful signalis determined. When solving the optimization problem, the method of maximum likelihood functionality and likelihood functionality for correlated processes were used, which contains the inverse matrix of inverse correlation functions of the observation equation. The obtained signal processing algorithm includes operations of spectral resection of passive interference. Polarization compensation of passive interference is performed by a combination of interchannel subtraction of the reflected signals of different polarizations. The quasi-optimal features of the synthesized algorithm, which has an obvious physical value and is optimal in the absence of internal noise, are considered. Based on the obtained results, a block diagram of the polarization-Doppler noise compensator and selection of useful signals against the background of noise reflections from the earth's surface has been developed. The study of the physical characteristics of the elements of the polarization covariance matrix of interferences at a high level of correlation of reflections of passive interferences of different polarizations is conducted.Based on the obtained results, it is proposed to introduce indicators of efficiency of passive interference compensation and selection of useful signals. The results of the analysis of the obtained indicators depending on the coefficient of the ratio of variances of passive interference indicate the need to use the polarization differences between signals and interference for selecting objects against the background of the underlying surface. The results obtained can be used to create ground-based and aerospace-based radar systems for detecting moving objects and objects with pronounced polarization features.

An Improved High-Dimensional Kriging Surrogate Modeling Method through Principal Component Dimension Reduction

The Kriging surrogate model in complex simulation problems uses as few expensive objectives as possible to establish a global or local approximate interpolation. However, due to the inversion of the covariance correlation matrix and the solving of Kriging-related parameters, the Kriging approximation process for high-dimensional problems is time consuming and even impossible to construct. For this reason, a high-dimensional Kriging modeling method through principal component dimension reduction (HDKM-PCDR) is proposed by considering the correlation parameters and the design variables of a Kriging model. It uses PCDR to transform a high-dimensional correlation parameter vector in Kriging into low-dimensional one, which is used to reconstruct a new correlation function. In this way, time consumption of correlation parameter optimization and correlation function matrix construction in the Kriging modeling process is greatly reduced. Compared with the original Kriging method and the high-dimensional Kriging modeling method based on partial least squares, the proposed method can achieve faster modeling efficiency under the premise of meeting certain accuracy requirements.

Best linear approximation of nonlinear and nonstationary systems using Operational Modal Analysis

Operational Modal Analysis (OMA) is a widely used class of tools to identify the modal properties of existing civil engineering structures and mechanical systems under ambient vibrations or normal operating conditions by only employing measured responses. The OMA techniques, however, are confined to estimate modal properties of linear and time-invariant systems, which do not always comply with real life applications. Additionally, even though structures are found to be nonlinear and nonstationary (i.e., time-varying), it is often engineering practice to apply linear models due to their straightforward insight into the dynamic behaviour of the considered system and mathematical properties when compared to more time consuming consideration of nonlinearities and nonstationarities. With that point of view, the concept of evaluating a best linear approximation of a nonlinear response is well established within Experimental Modal Analysis, i.e., deterministic modal analysis where both the system excitation and the corresponding response are measured. Along these lines, the paper takes the first step towards extending the concept of a best linear approximation to correlation-driven OMA in terms of estimated natural frequencies, damping ratios, and mode shapes. This paper proves that the correlation function matrix calculated from the measured response of a nonlinear and nonstationary system can be approximated by a set of free decays of the best linear approximation. In other words, the paper derives and shows through appropriately designed numerical simulations that the linear modal model, estimated using the output-only framework of conventional correlation-driven (i.e., covariance-driven) OMA, leads to suboptimal minimisation of the difference between the true response of a nonlinear and/or nonstationary system and the response of the linear modal model in a least squares sense. Moreover, the paper demonstrates, using an additional numerical simulation case study, that the output-only framework causes an underestimation of the error that is inevitably involved when estimating linear modal properties from a nonlinear and nonstationary response.

ОПТИМІЗАЦІЯ СТРУКТУРИ БАГАТОКАНАЛЬНОГО БОРТОВОГО РАДАРУ З СИНТЕЗУВАННЯМ АПЕРТУРИ АНТЕНИ ТА АЛГОРИТМОМ СЕЛЕКЦІЇ ЛІНІЙ ЕЛЕКТРОПЕРЕДАЧ НА ТЛІ ЗЕМНОЇ ПОВЕРХНІ

The task of synthesizing optimal signal processing in synthetic aperture radars equipped with a selection mode of power lines was solved by the maximum likelihood method. The distinguishing features of power lines from the earth's surface selected polarization and Doppler differences. The useful signal at the output of the coherent receiver specified in an analytical form with a complex envelope and high-frequency harmonic oscillations. The background radiation of the underlying surface is presented in a stochastic form and is completely determined by the matrix of correlation functions. The internal noise of a multichannel receiver is described by a delta-correlated random Gaussian process with the same power spectral density in different observation channels. Since the background radiation of the earth's surface is correlated in time and at different polarizations functional likelihood recorded with the inverse matrix inverse correlation functions. The obtained optimal algorithm consists of suppression of passive interference in decor-leveling filters, optimal coherent accumulation of trajectory signals for synthesizing the on-board antenna aperture, and their optimal filtering in filters matched with the reference signal. Simultaneous use of Doppler and polarization differences between wanted signals and passive interference allows you to effectively allocate power lines against the background of reflections from the ground in multichannel synthetic aperture radars. The proposed sequence of operations on the received oscillations can be a functional addition to the operating modes of aerospace-based cognitive radars, adapting by the covariance scattering matrix of the underlying surface known in advance or measured during remote sensing. The block diagram of a multichannel onboard polarization radar with a synthesized antenna aperture is synthesized based on the obtained results. The development of such a radar is relevant for a collision warning system for helicopter pilots, especially in bad weather conditions at any time of the year, day and night.

СИНТЕЗ ОПТИМАЛЬНОГО АЛГОРИТМУ ПОЛЯРИЗАЦІЙНОЇ СЕЛЕКЦІЇ КОРИСНИХ СИГНАЛІВ НА ФОНІ ПАСИВНИХ ЗАВАД В РАДАРІ З СИНТЕЗУВАННЯМ АПЕРТУРИ

Using the method of maximum likelihood, the optimal algorithm of polarization selection of objects on the background of the underlying surface, hydrometeors, urban buildings from aerospace carriers of radio electronic equipment has been synthesized. To solve the problem polarimetric properties of the scattered electromagnetic radiation of different natural environments and anthropogenic objects were analyzed. Has been determined the functional-determined mathematical model of the useful signal, the structure and correlation characteristics of scattered background radiation and white noise inside the multichannel receiver. It is assumed that the wanted signals are measured with phase accuracy in a coherent receiver. When solving the optimization problem, the method of the maximum likelihood functional and the likelihood functional for correlated processes were used, contains the inverse matrix of inverse correlation functions of the observation equation. The obtained signal processing algorithm, in addition to the classical operations of accumulating the trajectory signal with the preservation of the phase structure and its matched filtering, provides for polarization and frequency selection in the line of decorrelation filters. In the case when the internal noise is very small in comparison with external interference and the correlation between the channels of vertical and horizontal polarization approaches unity, almost complete compensation of passive interference occurs. Further spectral rejection of the reflected signals is practically unnecessary and it is possible to restrict ourselves only to matched filtering of the useful signal. In the absence of background radiation correlation, the channels for receiving oscillations of different polarizations become independent and the main operation for the selection of useful signals is spectral rejection of passive interference and matched filtering of the useful signal. The developed algorithm can be implement in cognitive radars due to its adaptability to the parameters of background radiation in accordance with the change in the scattering covariance matrix of the underlying surface. Based on the results obtained, a block diagram of a multichannel polarization radar with synthesized antenna aperture has been developed.

Simulating Stationary Non-Gaussian Processes Based on Unified Hermite Polynomial Model

In determining structural responses under random excitation using the Monte Carlo method, simulation of non-Gaussian processes is always a challenge. Various methods have been developed to simulate non-Gaussian processes but literature suggests that the complete expression of the Gaussian correlation function matrix (CFM) and the corresponding applicable range of the target non-Gaussian CFM have not been attempted in the transformation based on the Hermite polynomial model (HPM) to date. The intention of this paper is to derive a complete transformation model of CFM from non-Gaussian to Gaussian processes with the applicable range of the target non-Gaussian CFM based on the unified Hermite polynomial model (UHPM). An efficient procedure for simulating stationary non-Gaussian processes is also presented for easy application. It is found in the paper that the complete transformation model of Gaussian CFM is necessary because HPMs are not always monotonic and that the proposed method can simulate stationary non-Gaussian processes efficiently and accurately. The proposed method can equip researchers and engineers with a more efficient and accurate tool to handle non-Gaussian processes frequently encountered in engineering practices.

Анализ динамики нелинейных моделей конструкций ракетно-космической техники при нестационарных случайных нагрузках

In aerospace engineering, it is customary to employ stochastic analysis methods at the design stage to investigate how the mechanical system responds to random external forces. This is relevant due to high reliability requirements for spacecraft. We developed a method for probabilistic estimation of the dynamic properties of a structure subjected simultaneously to external (additive) and parametric (multiplicative) vibrations. An ordinary non-linear vector differential equation describes the vibrations in the elastic structure. Non-linear position and velocity properties of kinematic pairs may have cusps and discontinuities. We assume that the probabilistic dispersions of respective phase coordinates are close to the normal distribution of probability density. The initial non-linear vibration equations are statistically linearised. The system of differential equations is not rewritten in the canonical form, which means that it is possible to carry out the probabilistic analysis of the system for any external non-steady-state effect. The fundamental matrix of the linearised system is used to find the expected value vector and the correlation function matrix of the phase coordinate vector. The solution consists of a matrix integro-power series containing linear and quadratic terms. Using the method makes it possible to assess the contribution of each external force component to the total result. We consider an example of a non-linear system responding to a stepwise non-steady-state external influence.

The Bridge Between Chiral Lagrangians and QCD Sum-Rules

Properties of the scale-factor matrices forming a bridge between the mesonic fields of chiral Lagrangians and quark-level structures of QCD sum-rules are reviewed. The scale-factor matrices combined with mixing angles provide a physical projection of a QCD correlation function matrix that disentangles the mesonic states. This methodology is illustrated for the isotriplet a0(980)-a0(1450) system, and the scale factors are determined from the combined inputs of QCD sum-rules and chiral Lagrangians. The resulting scale factors are shown to be in good agreement with the vacuum expectation values in the chiral Lagrangian framework. The sensitivity of the scale factors on the gluon condensate QCD sum-rule parameter is explored.

The statistical errors in the estimated correlation function matrix for operational modal analysis

Given the random vibration of a linear and time-invariant system, the correlation function matrix is equivalent to free decays when the system is excited by Gaussian white noise. Correlation-driven Operational Modal Analysis utilises these properties to identify modal parameters from systems in operation based on the response only. Due to the finite length of the system response, the correlation function matrix must be estimated and this introduces statistical errors. This article focuses on the statistical errors due to this estimation process and the effect it has on the envelope and zero crossings of the estimated correlation function matrix. It is proven that the estimated correlation function matrix is a Gaussian stochastic process. Furthermore, it is proven that the envelope of the modal correlation function matrix is Rice distributed. This causes the tail region of the correlation function to become erroneous - called the noise tail. The zero crossings are unbiassed, but the random error related to the crossings increases fast in the noise tail. The theory is tested on a simulated case and there is a high agreement between theory and simulation. A new expression for the minimal time length is introduced based on the bias error on the envelope.

Automated reduction of statistical errors in the estimated correlation function matrix for operational modal analysis

In operational modal analysis, the correlation function matrix is treated as multiple free decays from which system parameters are extracted. The finite time length of the measured system response, however, introduces statistical errors into the estimated correlation function matrix. These errors cause both random and bias errors that transfer to an identification process of the modal parameters. The bias error is located on the envelope of the modal correlation functions, thus violating the assumption that the correlation function matrix contains multiple free decays. Therefore, the bias error transmits to the damping estimates in operational modal analysis. In this paper, we show an automated algorithm that reduces the bias error caused by the statistical errors. This algorithm identifies erratic behaviour in the tail region of the modal correlation function and reduces this noise tail. The algorithm is tested on a simulation case and experimental data of the Heritage Court Building, Canada. Based on these studies, the algorithm reduces bias error and uncertainty on the damping estimates and increases stability in the identification process.

A first model-independent radial BAO constraint from the final BOSS sample

ABSTRACT Using almost one million galaxies from the final Data Release 12 of the SDSS’s Baryon Oscillation Spectroscopic Survey (BOSS), we have obtained, albeit with low significance, a first model-independent determination of the radial baryon acoustic oscillation (BAO) peak with 9 per cent error: ΔzBAO(zeff = 0.51) = 0.0456 ± 0.0042. In order to obtain this measurement, the radial correlation function was computed in 7700 angular pixels, from which the mean correlation function and covariance matrix were obtained, making the analysis completely model-independent. This novel method of obtaining the covariance matrix was validated via comparison with 500 BOSS mock catalogues. This ΔzBAO determination can be used to constrain the background expansion of exotic models for which the assumptions adopted in the standard analysis cannot be satisfied. Future galaxy catalogues from J-PAS, DESI, and Euclid are expected to significantly increase the quality and significance of model-independent determinations of the BAO peak, possibly determined at various redshift and angular positions. We stress that it is imperative to test the standard paradigm in a model-independent way in order to test its foundations, maximize the extraction of information from the data, and look for clues regarding the poorly understood dark energy and dark matter.

Modal Participation Estimated from the Response Correlation Matrix

In this paper, we are considering the case of estimating the modal participation vectors from the operating response of a structure. Normally, this is done using a fitting technique either in the time domain using the correlation function matrix or in the frequency domain using the spectral density matrix. In this paper, a more simple approach is proposed based on estimating the modal participation from the correlation matrix of the operating responses. For the case of general damping, it is shown how the response correlation matrix is formed by the mode shape matrix and two transformation matrices T1 and T1 that contain information about the modal parameters, the generalized modal masses, and the input load spectral density matrix Gx. For the case of real mode shapes, it is shown how the response correlation matrix can be given a simple analytical form where the corresponding real modal participation vectors can be obtained in a simple way. Finally, it is shown how the real version of the modal participation vectors can be used to synthesize empirical spectral density functions.

Direct TDOA geolocation of multiple frequency‐hopping emitters in flat fading channels

The classic two-step approach for time difference of arrival (TDOA) geolocation is suboptimal since the TDOA measurements have not followed the constraint that all measurements should be consistent for a geolocation of a single emitter. In this study, the direct TDOA geolocation approach is proposed for frequency-hopping (FH) emitters. It makes use of the sparsity of the FH signals in frequency domain, and constructs a cross correlation function (CCF) matrix in frequency domain, then the location estimate is obtained by searching the maximum eigenvalue of the CCF matrix in a two dimensional grid. The Cramer–Rao lower bound has been derived. The resolution for a single FH signal geolocation is also analysed. Further, an extension of the new method for multiple FH emitters direct TDOA geolocation has been presented. The performance comparison between the direct approach and the conventional two-step method has been made by simulations. The results demonstrated that the proposed method outperforms the conventional two-step method. The simulations also demonstrated the effectiveness of the new method in locating multiple FH emitters.

On the application of correlation function matrices in OMA

In this paper the theoretical solution for the correlation function matrix of the random response of a structural system is re-visited. It is shown that using the classical definition of the correlation functions, the row space is defined by the mode shapes of the system, whereas the column space is defined by the modal participation vectors. This means that only the rows can be used for unbiased modal identification in operational modal analysis and if the columns are used for identification, then bias will be introduced on the mode shape estimates. It is pointed out that the mode shape bias is strongly dependent on the frequency distance between the modes, i.e. bias will significantly increase in case of closely spaced modes. The identification errors on the estimated biased and unbiased mode shapes are studied in a simulation example.