Fourier Space Approach Research Articles

Fourier space approach for convolutional neural network (CNN) electrocardiogram (ECG) classification: A proof-of-concept study.

There has been a proliferation of machine learning (ML) electrocardiogram (ECG) classification algorithms reaching>85% accuracy for various cardiac pathologies. Although the accuracy within institutions might be high, models trained at one institution might not be generalizable enough for accurate detection when deployed in other institutions due to differences in type of signal acquisition, sampling frequency, time of acquisition, device noise characteristics and number of leads. In this proof-of-concept study, we leverage the publicly available PTB-XL dataset to investigate the use of time-domain (TD) and frequency-domain (FD) convolutional neural networks (CNN) to detect myocardial infarction (MI), ST/T-wave changes (STTC), atrial fibrillation (AFIB) and sinus arrhythmia (SARRH). To simulate interinstitutional deployment, the TD and FD implementations were also compared on adapted test sets using different sampling frequencies 50Hz, 100Hz and 250Hz, and acquisition times of 5s and 10s at 100Hz sampling frequency from the training dataset. When tested on the original sampling frequency and duration, the FD approach showed comparable results to TD for MI (0.92 FD - 0.93 TD AUROC) and STTC (0.94 FD - 0.95 TD AUROC), and better performance for AFIB (0.99 FD - 0.86 TD AUROC) and SARRH (0.91 FD - 0.65 TD AUROC). Although both methods were robust to changes in sampling frequency, changes in acquisition time were detrimental to the TD MI and STTC AUROCs, at 0.72 and 0.58 respectively. Alternatively, the FD approach was able to maintain the same level of performance, and, therefore, showed better potential for interinstitutional deployment.

Analytical Force and Flux for a 1-D Electromagnetic Vibration Energy Harvester

A vibration energy harvester (VEH) converts the kinetic energy of a moving source into electrical energy. In this article, we consider a 1-D electromagnetic vibration energy harvester (1-D EMVEH) that consists of three coaxial cylindrical permanent magnets enclosed in a tube such that the middle magnet is levitating. The resulting movement of the middle magnet can then induce an electromotive force (EMF) in one or more surrounding coils. Using an analytical model, we derive expressions for the 1-D-EMVEHs characteristic frequency and output power by using the Fourier space approach. First, the magnetostatic energy of the system as a function of the position of the levitating magnet is calculated. Its spatial gradient gives the force acting on a magnet, which drives its dynamics. Next, more accurate magnetic flux and EMF expressions are obtained. The results are compared with experimental measurements, revealing an excellent agreement.

Microstructural and magnetic properties of thin obliquely deposited films: A simulation approach

The relation between microstructural and magnetic properties of thin obliquely deposited films has been studied by means of numerical techniques. Using our developed simulation code based on ballistic deposition model and Fourier space approach, we have investigated dependences of magnetometric tensor components and magnetic anisotropy parameters on the deposition angle of the films. A modified Netzelmann approach has been employed to study structural and magnetic parameters of an isolated column in the samples with tilted columnar microstructure. Reliability and validity of used numerical methods is confirmed by a good agreement of the calculation results with each other, as well as with our experimental data obtained by the ferromagnetic resonance measurements of obliquely deposited thin Ni80Fe20 films. The combination of these numerical methods can be used to design a magnetic film with a desirable value of uniaxial magnetic anisotropy and to extract the obliquely deposited film structure from only magnetic measurements.

Fourier space derivation of the demagnetization tensor for uniformly magnetized objects of cylindrical symmetry

We derive an analytical solution for the demagnetization tensor of a cylinder of finite size and uniform magnetization in both Cartesian and cylindrical coordinate systems, by employing a recently developed Fourier space approach. Simple arguments are given to show that a previously published solution using this approach is incomplete, and a detailed and comprehensive derivation containing the necessary corrections is given. The corrected result is shown to be in agreement with older analytical solutions based on an electrostatic potential approach. We subsequently expand our solution to objects of general cylindrical symmetry and derive an expression for the demagnetization tensor that involves a single remaining integral over a finite domain, enabling the use of conventional numerical tools to efficiently calculate the demagnetization tensor.

The effect of the demagnetizing field in cylindrical samples in high transverse field μ+SR experiments

We investigate the spatial variation of the demagnetizing field for uniformly magnetized, cylindrical samples using a recently developed Fourier space approach. We show that the demagnetizing response of the sample leads to a position dependence of the magnetic field, which varies most strongly near the radial boundary of the cylinder. Furthermore, we demonstrate that the demagnetizing field leads to a subsequent broadening of the field distribution, as experienced by the muons implanted in the sample, with a characteristic shape including a low field tail and a sharp high-field cutoff. We present a detailed study of this field broadening as a function of the aspect ratio of the cylinder and find that it is significant and largest for aspect ratios typical for cylindrical samples grown in mirror furnaces. We identify two strategies to minimize this broadening: adding a degrader so that muons implant closer to the surface of the sample and using a circular mask to stop muons from implanting near the radial edge. This could help identify whether an experimentally observed broadening is caused by the demagnetization response or the intrinsic properties of the sample.

Fourier space approach to the classical density functional theory for multi-Yukawa and square-well fluids

We present a Fourier space density functional approach for hard particles with attractive interactions, which is based on a previously developed two-dimensional approach [S. Hlushak, W. Rżysko, and S. Sokołowski, J. Chem. Phys. 131, 094904 (2009)] for hard-sphere chains. The interactions are incorporated by means of a three-dimensional Fourier image of the direct correlation function that is obtained from the first-order mean-spherical approximation. In order to improve the computational efficiency, we make extensive use of fast Fourier transforms for calculating density convolution integrals. A two-dimensional implementation of the new density functional approach, based on the expansion of the functional around the bulk fluid density, is used to study structure and adsorption of two model fluids in narrow cylindrical pores. We also investigate two methods that improve the accuracy of the theory as compared to the conventional DFT approach, which expands the free energy functional around the bulk fluid density: One a variant of the reference fluid density functional theory used by Gillespie et al. [Phys. Rev. E 68, 031503 (2003)], and the second a weighted density approach with energy route thermodynamics. Results from these two methods are compared to the conventional approach and also to the results of Monte Carlo simulations. We find that the method of Gillespie et al. and the weighted density approach with energy route thermodynamics yield significant improvement over the conventional approach.

An alternative energy bound derivation for a generalized Hasegawa–Mima equation

We present an alternative derivation of the H 1 -boundedness of solutions to a generalized Hasegawa–Mima equation, first investigated by Grauer (1998) [2]. We apply a Lyapunov function technique similar to the one used for constructing energy bounds for the Kuramoto–Sivashinsky equation. Different from Grauer (1998) [2], who uses this technique in a Fourier space approach, we employ the physical space construction of the Lyapunov function, as developed in Bronski and Gambill (2006) [11]. Our approach has the advantage that it is more transparent in what concerns the estimates and the dominant terms that are being retained. A key tool of the present work, which replaces the algebraic manipulations on the Fourier coefficients from the other approach, is a Hardy–Rellich type inequality.

Numerical orbital Fourier space approach to polymer band‐structure calculations

AbstractThe periodic symmetry of polymer chains can be applied via Bloch functions to polymer band structure calculations (Delhalle and Harris, Phys Rev B 1985, 31, 6755). The possibility of replacing Gaussian orbitals with numerically defined orbitals in this approach is discussed. Two modifications of the theory to overcome problems with the cancellation of divergent terms are described. Result for a LiH chain are compared with analytic Gaussian orbital results. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Discovering Ordered Phases of Block Copolymers: New Results from a Generic Fourier-Space Approach

A generic Fourier-space approach to solve the self-consistent field theory of block copolymers is developed. This approach is based on the fact that, for any computational box with periodic boundary conditions, all spatially varying functions are spanned by the Fourier series determined by the size and shape of the box. The method reproduces all known diblock copolymer phases. The application of this method to a model "frustrated" triblock copolymer leads to a phase diagram with a number of new phases. Furthermore, the capability of the method to reproduce experimentally observed structures is demonstrated using the knitting pattern of triblock copolymers.

CMB polarization systematics due to beam asymmetry: Impact on inflationary science

Cosmic microwave background (CMB) polarization provides a unique window into cosmological inflation; the amplitude of the B-mode polarization from last scattering is uniquely sensitive to the energetics of inflation. However, numerous systematic effects arising from optical imperfections can contaminate the observed B-mode power spectrum. In particular, systematic effects due to the coupling of the underlying temperature and polarization fields with elliptical or otherwise asymmetric beams yield spurious systematic signals. This paper presents a nonperturbative analytic calculation of some of these signals. We show that results previously derived in real space can be generalized, formally, by including infinitely many higher-order corrections to the leading order effects. These corrections can be summed and represented as analytic functions when a fully Fourier-space approach is adopted from the outset. The formalism and results presented in this paper were created to determine the susceptibility of CMB polarization probes of the primary gravitational wave signal but can be easily extended to the analysis of gravitational lensing of the CMB.

Spatial Resolution and Information Transfer in Scanning Transmission Electron Microscopy

The relation between image resolution and information transfer is explored. It is shown that the existence of higher frequency transfer in the image is just a necessary but not sufficient condition for the achievement of higher resolution. Adopting a two-point resolution criterion, we suggest that a 10% contrast level between two features in an image should be used as a practical definition of resolution. In the context of scanning transmission electron microscopy, it is shown that the channeling effect does not have a direct connection with image resolution because sharp channeling peaks do not move with the scanning probe. Through a quantitative comparison between experimental image and simulation, a Fourier-space approach is proposed to estimate defocus and sample thickness. The effective atom size in Z-contrast imaging depends on the annular detector's inner angle. Therefore, an optimum angle exists for the highest resolution as a trade-off between reduced atom size and reduced signal with limited information transfer due to noise.

Local screening of a core hole: A real-space approach applied to hafnium oxide

An approach to screening a core hole's potential in solids for purposes such as theoretical near-edge spectrum calculations is developed. In this approach, the core hole's unscreened potential is decomposed into a short-range part and a long-range part. The short-range part is screened using a real-space screening calculation, while the long-range part can be screened using a model dielectric function. Screening of the core hole is computed for 1s holes in LiF, MgO, Si, solid argon, diamond, and two structures of hafnium oxide. Most of the screening results obtained using the approach presented here are compared to similar results obtained using a traditional Fourier-space approach. Because the approach presented here is a real-space approach, it should be very practical for systems with large unit cells.

Demagnetization factors of the general ellipsoid: An alternative to the Maxwell approach

A transparent, exhaustive, and self-contained method for the calculation of the demagnetization tensor of the uniformly magnetized ellipsoid is presented. The method is an alternative to the established Maxwell derivation and is based on a Fourier-space approach to the micromagnetics of magnetized bodies. The key to the success of the procedure lies in the convenient treatment of shape effects through the Fourier representation. The scaled form of the demagnetization factors which depends on two dimensionless aspect ratios is argued to be their natural integral representation. Amongst other advantages, it allows for the immediate implementation of symmetry arguments such that only one of the principal factors needs to be computed. The oblate and prolate ellipsoids of revolution are examined from the same general point of view. The demagnetization factors for these distinct types of spheroid are seen to be related by analytic continuation of well-known Gaussian hypergeometric functions.

Demagnetization factors for elliptic cylinders

The magnetometric (volume averaged) demagnetization factors for cylinders with elliptical cross section are computed using a Fourier-space approach and compared with similar results obtained with a different treatment. The demagnetization factors are given as a series expansion in the eccentricity ϵ of the elliptical cross section, where the terms up to order ϵ10 are given explicitly as a function of the cylinder aspect ratio. Other simplified expressions, valid in restricted regimes, are also given. Two different series expansions, obtained previously and valid in particular combinations of shape parameters, are recalled and compared with the new results. After the computation of the magnetostatic and exchange-energy terms associated with a vortex closure-domain state in the elliptic cylinder, the single-domain limit, or the critical size below which the structure can support quasi-uniform magnetization, is derived and discussed.

Phase diagram for magnetic nano-rings

The minimum-energy single-domain magnetization state in a magnetized nano-ring is determined as a function of material and shape parameters. A phase diagram is derived within the framework of a Fourier-space approach for magnetic computations, showing the expected position of the ground state for any given set of external degrees of freedom. A series of micromagnetic simulations for suitably chosen parameters, show excellent agreement with the obtained theoretical results. An electron holography experiment has been carried out as a test on phase diagram reliability. The validity of the treatment, in particular the simplification employed in choosing ideal uniform rather than more physical quasi-uniform single-domain states, is thoroughly discussed in order to establish clear boundaries of applicability of the phase diagram.

A Fourier-Space Approach for the Computation of Magnetostatic Interactions Between Arbitrarily Shaped Particles

A new formalism has been developed to describe the magnetostatic energy associated with particles of arbitrary shape and magnetization state. The formalism relies on a Fourier space description of the particle shape, through the so-called shape amplitude, which can be used to obtain explicit expressions for the demagnetization tensor field, magnetic field, magnetic induction and magnetostatic energy of a particle for a given magnetization state. Moreover, the interaction energy between particles, located at arbitrary positions in space, which may have different shapes and magnetization states can also be computed. These results may contribute to a deeper understanding of magnetostatic coupling in nanostructures and of the role of shape anisotropy.

On the magnetostatic interactions between nanoparticles of arbitrary shape

The general expression for the magnetostatic energy of two magnetized nanoparticles with arbitrary shape and magnetization state is derived within the framework of a Fourier space approach. It is shown how the standard dipole–dipole interaction, valid for large interparticle distances, should be modified in order to take into account the shape anisotropy of each particle. Explicit computations are given for a simple system of two interacting cylinders. For magnetic nanowires, i.e., cylinders with a very large aspect ratio, a simple derivation shows that the interaction is of monopolar, rather than dipolar, nature.

On the computation of the demagnetization tensor for particles of arbitrary shape

A new method is presented to compute the demagnetization tensor of particles of arbitrary shape. By means of a Fourier space approach it is possible to compute analytically the Fourier representation of the demagnetization tensor for a broad class of magnetic nanoparticles. Then, specifying the direction of the uniform magnetization, the demagnetizing field and the magnetostatic energy associated with the particle can be evaluated.

A new efficient method for density functional theory calculations of inhomogeneous fluids

The accurate computation of the effects of solvation on chemical systems can be done using density functional theories (DFT) for inhomogeneous multicomponent fluids. The DFT models of interest are non-local theories which accurately treat hard-sphere fluid mixtures; attractive inter-particle potentials (Lennard–Jones) are added as perturbations. In this paper, we develop and demonstrate a new efficient method for an accurate non-local DFT. The method described here differs from previous work in the use of fast fourier transform (FFT) methods to carry out the convolutions. As with our previous real space work (J. Comput. Phys. 159(2) (2000) 407, 425), we demonstrate that the Fourier space approach can be solved with a Newton–GMRES approach; however, we now employ a very efficient matrix-free algorithm. A simple but effective preconditioner is presented. The method is demonstrated with calculations performed for one-, two-, and three-dimensional systems, including problems with single and multicomponent fluids. Timing comparisons with previous implementations are given.

On the computation of the demagnetization tensor field for an arbitrary particle shape using a Fourier space approach

A method is presented to compute the demagnetization tensor field for uniformly magnetized particles of arbitrary shape. By means of a Fourier space approach it is possible to compute analytically the Fourier representation of the demagnetization tensor field for a given shape. Then, specifying the direction of the uniform magnetization, the demagnetizing field and the magnetostatic energy associated with the particle can be evaluated. In some particular cases, the real space representation is computable analytically. In general, a numerical inverse fast Fourier transform is required to perform the inversion. As an example, the demagnetization tensor field for the tetrahedron will be given.