Fast Fourier Transform

Micromagnetics is based on the one hand on a continuum approximation of exchange interactions, including boundary conditions, on the other hand on Maxwell equations in the nonpropagative (static) limit for the evaluation of the demagnetizing field. The micromagnetic energy is most often restricted to the sum of the exchange, demagnetizing or self-magnetostatic, Zeeman, and anisotropy energies. When supplemented with a time evolution equation, including field induced magnetization precession, damping and possibly additional torque sources, micromagnetics allows for a precise description of magnetization distributions within finite bodies both in space and time. Analytical solutions are, however, rarely available. Numerical micromagnetics enables the exploration of complexity in small size magnetic bodies. Finite difference methods are here applied to numerical micromagnetics in two variants for the description of both exchange interactions/boundary conditions and demagnetizing field evaluation. Accuracy in the time domain is also discussed and a simple tool provided in order to monitor time integration accuracy. A specific example involving large angle precession, domain wall motion as well as vortex/antivortex creation and annihilation allows for a fine comparison between two discretization schemes with as a net result, the necessity for mesh sizes well below the exchange length in order to reach adequate convergence.


Keywords:

micromagnetics;
finite differences;
boundary conditions;
Landau—Lifshitz—Gilbert;
magnetization dynamics;
approximation order

/pdf/numerical-micromagnetics-finite-difference-methods-4nzdcec14y.pdf

Numerical Micromagnetics: Finite Difference Methods

Novel methodologies have been recently developed to characterize the microgeometry of neural tissues and porous structures via diffusion MRI data. In line with these previous works, this article provides a detailed mathematical description of q-space in spherical coordinates that helps to highlight the differences and similarities between various related q-space methodologies proposed to date such as q-ball imaging (QBI), diffusion spectrum imaging (DSI), and diffusion orientation transform imaging (DOT). This formulation provides a direct relationship between the orientation distribution function (ODF) and the diffusion data without using any approximation. Under this relationship, the exact ODF can be computed by means of the Radon transform of the radial projection (in q-space) of the diffusion MRI signal. This new methodology, termed exact q-ball imaging (EQBI), was put into practice using an analytical ODF estimation in terms of spherical harmonics that allows obtaining model-free and model-based reconstructions. This work provides a new framework for combining information coming from diffusion data recorded on multiple spherical shells in q-space (hybrid diffusion imaging encoding scheme), which is capable of mapping ODF to a high accuracy. This represents a step toward a more efficient development of diffusion MRI experiments for obtaining better ODF estimates.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/mrm.21917

Mathematical description of q-space in spherical coordinates: exact q-ball imaging.

Airflows and their impact on microclimatic parameters were
studied in the two entranced Cisařska Cave (Moravian Karst,
Czech Republic). Basing on the airflow direction with respect
to the upper entrance (UE) and lower entrance (LE), three
ventilation modes were distinguished: upward airflow one (UAF
mode), downward airflow one (DAF mode), and transitional mode.
A switching between the UAF and DAF happens in the transitional
mode, in the range of external temperatures from 9.5 to 11.1 C.
Basing on the extent of cave ventilation, a period of active
ventilation and period of stagnant ventilation were recognised.
The cave air is intensively exchanged with external airs during
the active ventilation period when the cave persists in either
the DAF or UAF mode. During the stagnant ventilation period,
the single airflow modes switch during diurnal cycle and only
cave entering passages are vented (impacted). The airflows were
found to depend (i) linearly on the differences in cave-outdoor
air densities, (ii) nonlinearly on the external temperature,
and (iii) both linearly (LE) and non-linearly (UE) on the
temperature gradient. Majority of the airflows showed high
frequency oscillations with the period from tens of seconds to
minutes and with the amplitude up to 0.1 m3/s. Different
airflow spectra were found at the individual entrances. The
highest spectral densities were registered at low driving
forces. The air circulation changed the pattern of temperature
distribution depending on the ventilation mode and ventilation
period. The cave relaxation length varied in the extreme range
from 1 to 69.6 m. The asymptotic temperature and asymptotic
specific humidity varied from 7.5 to 11.8 C, and from 5.47 to
7.15 kg/kg, respectively. The impact of airflows on the cave
CO2 levels is discussed based on a dynamic model.

Air circulation and its impact on microclimatic variables in the Císařská Cave (Moravian Karst, Czech Republic)

The conversion of time domain data via the fast Fourier (FFT) and Hilbert-Huang (HHT) transforms is compared. The FFT treats amplitude vs. time information globally as it transforms the data to an amplitude vs. frequency description. The HHT is not constrained by the assumptions of stationarity and linearity, required for the FFT, and generates both amplitude and frequency information as a function of time. The behavior and flexibility of these two transforms are examined for a number of different time domain signal types.

/pdf/the-fast-fourier-and-hilbert-huang-transforms-a-comparison-1ntka7q53c.pdf

The Fast Fourier and Hilbert-Huang Transforms: A Comparison

The discrete Fourier transform (DFT) provides a means for transforming data sampled in the time domain to an expression of this data in the frequency domain. The inverse transform reverses the process, converting frequency data into time-domain data. Such transformations can be applied in a wide variety of fields, from geophysics to astronomy, from the analysis of sound signals to CO/sub 2/ concentrations in the atmosphere. Over the course of three articles, our goal is to provide a convenient summary that the experimental practitioner will find useful. In the first two parts of this article, we'll discuss concepts associated with the fast Fourier transform (FFT), an implementation of the DFT. In the third part, we'll analyze two applications: a bat chirp and atmospheric sea-level pressure differences in the Pacific Ocean.

/pdf/the-fast-fourier-transform-for-experimentalists-part-i-2dl5qeqq40.pdf

D. Donnelle

Papers

The fast Fourier transform for experimentalists. Part I. Concepts