Set Of Orthonormal Basis Functions Research Articles

Hofstadter-like spectrum and magnetization of artificial graphene constructed with cylindrical and elliptical quantum dots

A comparative study of the electronic and magnetic properties of quasi-two-dimensional electrons in an artificial graphene-like superlattice composed of circular and elliptical quantum dots is presented. A complete orthonormal set of basis functions has been implemented for calculation of the energy dispersions, Hofstadter spectra, density of states and orbital magnetization of the considered systems. Our calculations indicate a topological change in the miniband structure due to the ellipticity of the quantum dots, and non-trivial modifications of the electron energy dispersion surfaces with the change of the magnetic flux per unite cell. The ellipticity of the quantum dots leads to an opening of a gap and considerable modifications of the Hofstadter spectrum. The orbital magnetization is shown to reveal significant oscillations with the change of the magnetic flux. The ellipticity of quantum dots has a qualitative impact on the dependencies of the magnetization on both the magnetic flux and the temperature.

Data-driven modal analysis of nonlinear quantities in turbulent plasmas using multi-field singular value decomposition

Nonlinear dynamics in the two-dimensional multi-component plasma turbulence described by the Hasegawa–Wakatani equation is investigated by using a data-driven modal analysis with the singular value decomposition (SVD). The conventional SVD is extended to ‘multi-field SVD’ which can decompose multiple turbulence fields simultaneously by a single set of orthonormal basis functions without imposing a priori scale separations. Then, in addition to the mode amplitude labeled by the singular value, the information on the phase relations in the nonlinear quantities such as a transport flux or a triad energy transfer is extracted in the mode space. Through applications to the two-dimensional plasma turbulence, it is revealed that the multi-field SVD can extract the dominant spatial structures for the turbulent transport and the nonlinear energy transfer, preserving the multi-scale nature of the original turbulent fields. It is also demonstrated that one can reduce the dimensionality or information using the multi-field SVD through comparisons with the conventional Fourier decomposition.

Optimal pupil basis set for telescope-coronagraph design and perturbation analysis based on the method of moments.

This paper concerns optimization and analysis of telescope-coronagraph systems for direct imaging of exoplanets. In this paper, the coronagraph system, with arbitrary pupil geometry, is theoretically considered, and the governing equation for the pupil design is derived. The method of moments is applied to solve the generalized energy-concentration eigenvalue problem to obtain the optimal pupil apodization and complete sets of orthonormal basis functions for arbitrary pupil geometries. The method yields eigenvalues indicating the fraction of starlight energy encircled in the area of the focal-plane mask (FPM), where starlight can be occulted and/or nulled. In other words, a higher eigenvalue implies less leakage/spillover of light outside of the FPM region and into the planet-discovery zone. Thus, a higher eigenvalue supports better starlight suppression for a given type of coronagraph. This methodology is useful for semi-quantitatively ranking different modes of perturbation with respect to energy spillage in the focal plane independent of coronagraph design details. A model-order-reduction-based sensitivity analysis is conducted to investigate the coupling between different pupil modes induced by aberrations. A pupil mode recovery scheme is presented to offer a theoretically rigorous and computationally efficient approach to reconstruct the optimal pupil mode under an arbitrary phase perturbation. The reconstruction coefficients and recovery-effectiveness factors are derived theoretically and demonstrated numerically. Several numerical examples, including the LUVOIR A and B pupils, are provided to validate and demonstrate the applicability of the proposed methods. The reported methodology enables model-order reduction based on degree of focal-plane energy concentration and reconstruction of optimal pupil apodization vis-á-vis phase aberrations using a precomputed basis set. These features should enhance computational efficiency for coronagraph design and sensitivity analysis.

Reduced order modelling for a rotor-stator cavity using proper orthogonal decomposition

The paper describes the Reduced Order Modelling (ROM) for the turbulent flow in a rotor-stator cavity, a configurations commonly encountered in secondary air system of aircraft engines. The proper orthogonal decomposition (POD) that uses data from Large Eddy Simulations (LES) is here considered in order to identify a set of orthonormal basis functions for the Galerkin projection of the Navier-Stokes equations. The POD-Galerkin procedure has been validated for the relative simple turbulent shear flow of a plane Couette flow. LES statistics and experimental measurements have been used as a benchmark and overall we may claim that the models studied reasonably well predict the turbulence phenomenon for the rotor-stator flow. Two novelty are introduced in this work: (i) the extension of the estimation of the turbulent eddy viscosity proposed by Aubry et al. [1] to problems with high Reynolds numbers and rotating flows; (ii) the derivation of a nonintrusive reduced order model for turbulent rotor-stator cavity flows.

Magnetic Anomaly Detection Using Full Magnetic Gradient Orthonormal Basis Function

Magnetic anomaly detection is an effective method for detecting ferromagnetic targets. The practical application of target detection by measuring the magnetic gradient has been confirmed in many studies, and its suppression of background noise has been widely recognized. However, the variation of the unidirectional magnetic field gradient is very small in a short distance, and the amplitude of the signal is also affected by the direction of the target’s magnetic moment and the shaking of the sensor platform itself. These make it difficult to extract the signal from complex background interference. In order to solve these problems, this paper is devoted to use the variation of three axial magnetic gradients in the geomagnetic coordinate system caused by the presence of magnetic target (we call it full magnetic gradient) to construct the magnetic anomaly characteristic of the target. A set of orthonormal basis functions is further derived to linearly describe this characteristic, which improves the signal-to-noise ratio (SNR) of the signal from the perspective of signal energy. The present work main contribution is to propose a full magnetic gradient orthonormal basis function (FMG-OBF) method to effectively characterize magnetic anomaly of magnetic targets. Compared with the traditional one-direction magnetic gradient orthonormal basis function method, the SNR of this method is greatly improved, and it is not affected by a change in the target magnetic moment’s direction nor by the shaking of the sensor carrier platform. Therefore, it has great value in the practical application of aeromagnetic detection.

Optimum Combiner for Spatially Correlated Nakagami-m Fading Channels

This paper proposes a novel optimum combiner that raises the time-averaged signal-to-interference-plus-noise ratio (SINR) for maximal-ratio combining (MRC) reception over spatially correlated Nakagami-m fading channels. By exploiting the principal component analysis (PCA) with eigenvalue decomposition (EVD) of the covariance matrix analytically evaluated offline, a complete set of orthonormal basis functions can be obtained. A decorrelation process analyzes and then projects the received signal into the space spanned by the basis functions. In accordance with the theorem of irrelevance, a principal component selection combining (PCSC) method is proposed to remove components in a few dimensions in which SINRs are considered low to raise the resulting time-averaged SINR on the subsequent MRC reception. The proposed technique also avoids noise enhancement occurring with MRC reception in the scenario where noises on different branches are correlated. The SINR distribution, level-crossing rate (LCR) and average fade duration (AFD) are derived. Based on a novel scattering model interpretation, a simulator consisting of spatially correlated Nakagami-m fading channels is developed according to the analytically evaluated covariance matrix. Computer simulations show that the proposed optimum combiner not only decreases interference and noise from the irrelevant subspace to achieve higher time-averaged SINR and lower AFD but also significantly reduces the complexity required for subsequent signal processing.

Limitations of the Recall Capabilities in Delay-Based Reservoir Computing Systems

We analyse the memory capacity of a delay-based reservoir computer with a Hopf normal form as nonlinearity and numerically compute the linear as well as the higher order recall capabilities. A possible physical realization could be a laser with external cavity, for which the information is fed via electrical injection. A task-independent quantification of the computational capability of the reservoir system is done via a complete orthonormal set of basis functions. Our results suggest that even for constant readout dimension the total memory capacity is dependent on the ratio between the information input period, also called the clock cycle, and the time delay in the system. Optimal performance is found for a time delay about 1.6 times the clock cycle.

Gradient signals analysis of scalar magnetic anomaly using orthonormal basis functions

Magnetic anomaly detection is a passive and effective approach for detecting magnetic targets. During detection using one scalar magnetometer, temporal magnetic variations will affect the detection performance. In order to reduce the influence of magnetic variations on detection, we focus on the analysis of magnetic gradient signals and obtain four orthonormal basis functions (OBFs) of gradient signal. Based on the set of OBFs, we propose an improved method using the magnetic gradient signal. Meanwhile, we can also obtain the bearing of the target relative to the survey line in the method. The proposed method is validated by simulation and real-world magnetic noise. The results show that the proposed method has a good detection performance with low signal to noise ratio. Therefore, the proposed method can improve the detection probability of weak magnetic anomalies and also determine the bearing of targets relative to the survey line.

Series Estimation of Stochastic Processes: Recent Developments and Econometric Applications

This paper overviews recent developments in series estimation of stochastic processes and some of their applications in econometrics. Underlying this approach is the idea that a stochastic process may under certain conditions be represented in terms of a set of orthonormal basis functions, giving a series representation that involves deterministic functions. Several applications of this series approximation method are discussed. The first shows how a continuous function can be approximated by a linear combination of Brownian motions (BMs), which is useful in the study of the spurious regressions. The second application utilizes the series representation of BM to investigate the effect of the presence of deterministic trends in a regression on traditional unit-root tests. The third uses basis functions in the series approximation as instrumental variables (IVs) to perform efficient estimation of the parameters in cointegrated systems. The fourth application proposes alternative estimators of long-run variances in some econometric models with dependent data, thereby providing autocorrelation robust inference methods in these models. We review some work related to these applications and some ongoing research involving series approximation methods.

Slope-Shift Keying LoRa-Based Modulation

LoRa, a low-power long-range physical-layer communication method, is one of the emerging technologies in low-power wide-area networks (LPWANs). In LoRa, a linearly varying-frequency up chirp and its cyclic shifts act as an orthonormal basis set for message representation. The number of orthonormal basis functions dictates the achievable bit rate of a LoRa system. This article aims to increase the achievable data rate of a conventional LoRa system by adding a set of new orthonormal basis functions to the existing set. Specifically, we use a linearly varying-frequency down chirp and its cyclic shifts to generate the second set of orthonormal basis functions in the proposed scheme, named slope-shift-keying LoRa (SSK-LoRa) modulation. We also develop both low-complexity optimum coherent and noncoherent detection algorithms for the proposed SSK-LoRa modulation. We then obtain closed-form analytical and tight approximations for the bit and symbol error probabilities of noncoherent detection in a Rayleigh fading environment. Our proposed SSK-LoRa scheme is shown to outperform the state-of-the-art LoRa-based modulation schemes.

Character motion in function space

We address the problem of animated character motion representation and approximation by introducing a novel form of motion expression in a function space. For a given set of motions, our method extracts a set of orthonormal basis (ONB) functions. Each motion is then expressed as a vector in the ONB space or approximated by a subset of the ONB functions. Inspired by the static PCA, our approach works with the time-varying functions. The set of ONB functions is extracted from the input motions by using functional principal component analysis and it has an optimal coverage of the input motions for the given input set. We show the applications of the novel compact representation by providing a motion distance metric, motion synthesis algorithm, and a motion level of detail. Not only we can represent a motion by using the ONB; a new motion can be synthesized by optimizing connectivity of reconstructed motion functions, or by interpolating motion vectors. The quality of the approximation of the reconstructed motion can be set by defining a number of ONB functions, and this property is also used to level of detail. Our representation provides compression of the motion. Although we need to store the generated ONB that are unique for each set of input motions, we show that the compression factor of our representation is higher than for commonly used analytic function methods. Moreover, our approach also provides lower distortion rate.

An algorithm of the boundary value problem based on multiscale orthogonal compact base

The basis scope of this paper is to introduce a set of orthonormal basis functions and to present a new algorithm for solving the boundary value problem by defining the ε-approximate solution, where the ε-approximate solution is defined in this article. First, the basis functions are introduced and proved to be orthonormal. Second, the ε-approximate solution is defined and proved to be the optimal solution. Third, the algorithm is proposed and proved to be stable and uniformly convergent. Finally, numerical examples are presented to confirm the theoretical results and show that the proposed method has wide application foreground.

The planar Multipole Resonance Probe: a functional analytic approach

Active Plasma Resonance Spectroscopy (APRS) is a well known diagnostic method, where a radio frequency probe is immersed into a plasma and excites plasma oscillations. The response of the plasma is recorded as frequency dependent spectrum, in which resonance peaks occur. By means of a mathematical model plasma parameters like the electron density or the electron temperature can be determined from the detected resonances.The majority of all APRS probes have in common, that they are immersed into the plasma and perturb the plasma due to the physical presence of the probe. Thus, they are invasive and can at least influence the homogeneity of the plasma. To overcome this problem, the planar Multipole Resonance Probe (pMRP) was invented, which can be integrated into the chamber wall of a plasma reactor.Within this paper, the first analytic model of the pMRP is presented, which is based on a cold plasma description of the electrons. The general admittance of the probe-plasma system is derived by means of functional analytic methods and a complete orthonormal set of basis functions. Explicit spectra for an approximated admittance including a convergence study are shown. The determined resonance frequencies are in good agreement with former simulation results.

A general radiation model for sound fields and nearfield acoustical holography in wedge propagation spaces.

In this work an expression for the solution of the Helmholtz equation for wedge spaces is derived. Such propagation spaces represent scenarios for many acoustical problems where a free field assumption is not eligible. The proposed sound field model is derived from the general solution of the wave equation in cylindrical coordinates, using sets of orthonormal basis functions. The latter are modified to satisfy several boundary conditions representing the reflective behaviour of wedge-shaped propagation spaces. This formulation is then used in the context of nearfield acoustical holography (NAH) and to obtain the expression of the Neumann Green function. The model and its suitability for NAH is demonstrated through both numerical simulations and measured data, where the latter was acquired for the specific case of a loudspeaker on a hemi-cylindrical rigid baffle.

Model Order Reduction of Electrical Machines With Multiple Inputs

In this paper, proper orthogonal decomposition (POD) method is employed to build a reduced-order model from a high-order nonlinear permanent magnet synchronous machine model with multiple inputs. Three parameters are selected as the multiple inputs of the machine. These parameters are terminal current, angle of the terminal current, and rotation angle. To produce the lower-rank system, snapshots or instantaneous system states are projected onto a set of orthonormal basis functions with small dimension. The reduced model is then validated by comparing the vector potential, flux density distribution, and torque results of the original model, which indicates the capability of using the POD method in the multivariable input problems. The developed methodology can be used for fast simulations of the machine.

Cross-contamination of the fitting parameters in multidimensional tunneling treatments

In this paper we examine the two-dimensional tunneling formalism used previously to fit the hydrogen-transfer and internal-rotation splittings in the microwave spectrum of 2-methylmalonaldehyde in an effort to determine the origin of various counterintuitive results concerning the isotopic dependence of the internal-rotation splittings in that molecule. We find that the cause of the problem lies in a “parameter contamination” phenomenon, where some of the numerical magnitude of splitting parameters from modes with large tunneling splittings “leaks into” the parameters of modes with smaller tunneling splittings. We show that such parameter contamination, which greatly complicates the determination of barrier heights from the least-squares-fitted splitting parameters, will be a general problem in spectral fits using the multi-dimensional tunneling formalism, since it arises from subtle mathematical features of the non-orthogonal framework functions used to set up the tunneling Hamiltonian. Transforming to a physically less intuitive orthonormal set of basis functions allows us to give an approximate numerical estimate of the contamination of tunneling parameters for 2-methylmalonaldehyde by combining a dominant tunneling path hypothesis with results recently given for the hydrogen-transfer-internal-rotation potential function for this molecule.

Steerable Principal Components for Space-Frequency Localized Images.

As modern scientific image datasets typically consist of a large number of images of high resolution, devising methods for their accurate and efficient processing is a central research task. In this paper, we consider the problem of obtaining the steerable principal components of a dataset, a procedure termed "steerable PCA" (steerable principal component analysis). The output of the procedure is the set of orthonormal basis functions which best approximate the images in the dataset and all of their planar rotations. To derive such basis functions, we first expand the images in an appropriate basis, for which the steerable PCA reduces to the eigen-decomposition of a block-diagonal matrix. If we assume that the images are well localized in space and frequency, then such an appropriate basis is the prolate spheroidal wave functions (PSWFs). We derive a fast method for computing the PSWFs expansion coefficients from the images' equally spaced samples, via a specialized quadrature integration scheme, and show that the number of required quadrature nodes is similar to the number of pixels in each image. We then establish that our PSWF-based steerable PCA is both faster and more accurate then existing methods, and more importantly, provides us with rigorous error bounds on the entire procedure.

On the use of shapelets in modelling resolved, gravitationally lensed images

Lens modeling of resolved image data has advanced rapidly over the past two decades. More recently pixel-based approaches, wherein the source is reconstructed on an irregular or adaptive grid, have become popular. Generally, the source reconstruction takes place in a Bayesian framework and is guided by a set of sensible priors. We discuss the integration of a shapelets-based method into a Bayesian framework and quantify the required regularization. In such approaches, the source is reconstructed analytically, using a subset of a complete and orthonormal set of basis functions, known as shapelets. To calculate the flux in an image plane pixel, the pixel is split into two or more triangles (depending on the local magnification), and each shapelet basis function is integrated over the source plane. Source regularization (enforcement of priors on the source) can also be performed analytically. This approach greatly reduces the number of source parameters from the thousands to hundreds and results in a posterior probability distribution that is much less noisy than pixel-based approaches.

On the detection of chipless RFID through signal space representation

paper presents a novel approach to model and represent chipless radio-frequency identification (RFID) frequency signatures. The approach involves the geometrical representation of chipless RFID frequency signatures in a signal space. A small set of orthonormal basis functions is derived using singular value decomposition in order to represent the 2b possible tag signatures of a b-bit chipless tag. Each tag signature is represented as a point in an L-dimensional signal space, and minimum distance detection is used to extract the information bit sequence of the tag. Detection error probability is also examined through analytical derivations and Monte Carlo simulation. A set of 3-bit tags were fabricated to validate the approach. Experimental results show that the new approach is capable of accurately detecting information contained in chipless RFID tags. This approach offers a solid mathematical framework for developing novel detection methods for chipless tags.

Mode analysis with a spatial light modulator as a correlation filter

A procedure for the real-time analysis of laser modes using a phase-only spatial light modulator is outlined. The procedure involves encoding into digital holograms by complex amplitude modulation a set of orthonormal basis functions into which the initial field is decomposed. This approach allows any function to be encoded and refreshed in real time (60 Hz). We implement a decomposition of guided modes propagating in optical fibers and show that we can successfully reconstruct the observed field with very high fidelity.