Application Of Fourier Analysis Research Articles

On the discrepancies between POD and Fourier modes on aperiodic domains

The application of Fourier analysis in combination with the Proper Orthogonal Decomposition (POD) is investigated. In this approach to turbulence decomposition, which has recently been termed Spectral POD (SPOD), Fourier modes are considered as solutions to the corresponding Fredholm integral equation of the second kind along homogeneous-periodic or homogeneous coordinates. In the present work, the notion that the POD modes formally converge to Fourier modes for increasing domain length is challenged. A family of analytical correlation functions parametrized by the Taylor macro/micro scale ratio (MMSR) is investigated numerically. The results of the analysis indicate that the discrepancy between POD and Fourier modes along locally translationally invariant coordinates is coupled to the MMSR of the flow. Increasing discrepancies are observed for smaller MMSRs, which are characteristic of low Reynolds number flows. The asymptotic convergence rate of the eigenspectrum matches the corresponding convergence rate of the exact analytical Fourier spectrum of the kernel in question, even for extremely small domains and small MMSRs where the corresponding DFT spectra suffer heavily from windowing effects. The Taylor micro scales are consistently underestimated when reconstructed using Fourier modes—failing to converge to the correct value even if all Fourier modes are used—while these are accurately reconstructed using POD modes.

Fourier transformation in spherical systems as a tool of structural biology

Applications of the most common adaptation of Fourier analysis in spherical coordinate systems used to solve a number of problems in structural biology, namely, flat wave decomposition (flat waves are represented as spherical functions decomposition), are herein considered. Arguments in favor of this decomposition are compared with other decompositions in superposition of special functions. A more general justification for the correctness of this decomposition is obtained than that existing today. A method for representing groups of atoms in the form of a Fourier object is proposed. It is also considered what opportunities give such a representation. The prospects for the application of Fourier analysis in structural biophysics are discussed.

The large time behavior of strong solutions to a P1-approximation arising in radiating hydrodynamics

This paper is concerned with the study of the Cauchy problem to a multi-dimensional P1-approximation model. Based on a known global well-posedness (Danchin and Ducomet in J Evol Equ 14:155–195, 2014), in $$L^{2}$$-critical regularity framework the time decay rates of the constructed global strong solutions are obtained if the low frequencies of the data under a suitable additional condition. The proof mainly relies on an application of Fourier analysis to a mixed parabolic-hyperbolic system, and on a refined time-weighted energy functional. As a by-product, those time-decay rates of $$L^{q}$$–$$L^{r}$$ type are also captured in the critical framework.

Testing $k$-Monotonicity: The Rise and Fall of Boolean Functions

$ \newcommand{\ensuremath}[1]{#1} \newcommand{\eps}{\ensuremath{\varepsilon}} $ A Boolean $k$-monotone function defined over a finite poset domain $\mathcal{D}$ alternates between the values $0$ and $1$ at most $k$ times on any ascending chain in $\mathcal{D}$. Therefore, $k$-monotone functions are natural generalizations of the classical monotone functions, which are the $1$-monotone functions. Motivated by the recent interest in $k$-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of $k$-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are $k$-monotone (or are close to being $k$-monotone) from functions that are far from being $k$-monotone. Our results include the following. We demonstrate a separation between testing $k$-monotonicity and testing monotonicity, on the hypercube domain $\{0,1\}^d$, for $k\geq 3$; We demonstrate a separation between testing and learning on $\{0,1\}^d$, for $k=\omega(\log d)$: testing $k$-monotonicity can be performed with $\exp(O(\sqrt d \cdot \log d\cdot \log(1/\eps)))$ queries, while learning $k$-monotone functions requires $\exp(\Omega(k\cdot \sqrt d\cdot{1/\eps}))$ queries (Blais et al. (RANDOM 2015)); We present a tolerant test for $k$-monotonicity of functions $f\colon[n]^d\to \{0,1\}$ with complexity independent of $n$. The test implies a tolerant test for monotonicity of functions $f\colon[n]^d\to [0,1]$ in $\ell_1$ distance, which makes progress on a problem left open by Berman et al. (STOC 2014). Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid $[n]^d$, and draw connections to distribution testing techniques. ---------- An extended abstract of this paper appeared in the Proceedings of the 8th Innovations in Theoretical Computer Science conference, 2017.

Toward the use of wavelet scalograms in the diagnostic analysis of rotating machinery transient data

Analysis of the vibration data has historically, and in practice today, been accomplished primarily using Fourier analysis which allows the diagnostician to review vibrational amplitudes and corresponding periodic frequencies generated by the machine. The most common application of Fourier analysis provides very limited information about the frequency content versus time, as these data are lost during the transform. This article presents a comparison analysis between Fourier transform and the wavelet transform, presented via the multilevel wavelet graphical presentation, or scalogram, on startup data from a large, vertical power plant pump driven by an electric motor. Wavelet scalograms also provide interesting and unexpected information such as frequency undulations during startup as well as intermittency of specific frequency components. As each method has its strengths and limitations, using the two methods together potentially provides a more complete picture of the vibration characteristics of the machine. Results of these analyses are presented with subsequent diagnostic analyses.

A new series representation derived for periodic functions that can be represented by the Fourier series

The Fourier series are, as we know, widely used in a number of mathematical problems arising in various topics of Communication systems, Signal Processing, Image processing and many more. The main contribution through these series is to analyze and build periodic functions in the form of infinite or finite sums of sines and cosines. In this paper, a new series representation like the Fourier series has been derived for every periodic function which can be represented by the Fourier series, thereby increasing the extent of applications of Fourier analysis in the related fields.

Applications of Fourier analysis in homogenization of Dirichlet problem I. Pointwise estimates

In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex and smooth domain, and smooth operator and boundary data, we prove pointwise convergence results, namely|uε(x)−u0(x)|⩽Cκε(d−1)/21d(x)κ,∀x∈D,∀κ>d−1, where uε and u0 are solutions of respectively oscillating and homogenized Dirichlet problems, and d(x) is the distance of x from the boundary of D. As a corollary for all 1⩽p<∞ we obtain Lp convergence rate as well.

Application of Fourier Analysis to the Outline of Object

Precise measurement of shape parameters of gear components is of ultimate importance for performance and service life of machinery. The conventional manual methods with aid of measurement tools rely largely on experiences of testing personnel. This study aims at developing an efficient and automatic methods for gear measurements, based on Fourier analysis with image processing software OPTIMAS. The shape parameters of gear are conveniently derived from the polar coordinates and the Fourier coefficients obtained from the image analysis software. The automatic methods are applied to two types of gear for illustration. The macro written for this purpose is presented and the measurement results are compared with conventional methods. It has been proven that Fourier analysis is a reliable and cost effective method for gear measurement. In addition, the results of Fourier analysis can be used to determine other parameters to characterize shape and size of the gear.

Application of Fourier analysis of scanning electron microscopy images for studying variation features of reservoir properties of carbonate rocks at stages of catagenetic transformations

The applicability of the fast Fourier transform method to studying carbonate rocks using a scanning electron microscope is experimentally studied. The method was used to find an integral estimate of sizes of carbonate matrix crystals formed during recrystallization at catagenesis stages.

Handbook of Fourier analysis &amp; its applications

Fourier analysis has many scientific applications - in physics, number theory, combinatorics, signal processing, probability theory, statistics, option pricing, cryptography, acoustics, oceanography, optics and diffraction, geometry, and other areas. In signal processing and related fields, Fourier analysis is typically thought of as decomposing a signal into its component frequencies and their amplitudes. This practical, applications-based professional handbook comprehensively covers the theory and applications of Fourier Analysis, spanning topics from engineering mathematics, signal processing and related multidimensional transform theory, and quantum physics to elementary deterministic finance and even the foundations of western music theory. As a definitive text on Fourier Analysis, Handbook of Fourier Analysis and Its Applications is meant to replace several less comprehensive volumes on the subject, such as Processing of Multifimensional Signals by Alexandre Smirnov, Modern Sampling Theory by John J. Benedetto and Paulo J.S.G. Ferreira, Vector Space Projections by Henry Stark and Yongyi Yang and Fourier Analysis and Imaging by Ronald N. Bracewell. In addition to being primarily used as a professional handbook, it includes sample problems and their solutions at the end of each section and thus serves as a textbook for advanced undergraduate students and beginning graduate students in courses such as: Multidimensional Signals and Systems, Signal Analysis, Introduction to Shannon Sampling and Interpolation Theory, Random Variables and Stochastic Processes, and Signals and Linear Systems.

Nonlinear effects in the propagation properties of numerical schemes employed in computational hydraulics and computational hydrology

Fourier analysis is a powerful tool for the determination of the propagation properties of numerical schemes employed in computational hydraulics and computational hydrology. Nevertheless, its use is restricted to cases that involve linear and constant coefficient equations, subject to periodic boundary conditions, or to linear and constant coefficient pure initial value problems occurring in infinite spatial domains. Most equations that describe hydraulic and hydrologic phenomena are nonlinear and, therefore, such limitation prevents the rigorous application of Fourier analysis and may even lead to erroneous conclusions when a so-called “frozen coefficients approach” is used. The purpose of this paper is to present a general methodology that removes the above described limitation. In order to accomplish this task, the differential or difference operator of the evolution equation or system of equations, are expanded in a Taylor–Fréchet series around a reference solution, which is assumed to be slowly varying. When only the zeroth and first order terms in the series are kept, the resulting equation is linear in a perturbation function, even though it depends nonlinearly on the reference solution, thereby capturing the dominant nonlinear behavior of the system under analysis. The ensuing step consists of performing a local analysis of the resulting equation or system of equations. Such local analysis is asymptotic in a conveniently defined small parameter. The zeroth order asymptotic approximation of the solution is governed by a linear, pure initial value problem with constant coefficients. Thus, Fourier analysis is applicable to such problem. Examples drawn from computational hydrology, such as the stability and iteration convergence of schemes employed for the solution of Richards equation, that describes the motion of water in partially saturated porous media; and from computational hydraulics, such as the stability of the generalized Preissman scheme, and the consistency of the generalized wave continuity equation formulation of the shallow water equations, are included, showing the advantages of the proposed methodology to explain observations that have defied understanding, when a frozen coefficient approach is employed.

Optimal decay rates to diffusion wave for nonlinear evolution equations with ellipticity

We derive the optimal convergence rates to diffusion wave for the Cauchy problem of a set of nonlinear evolution equations with ellipticity and dissipative effects { ψ t = − ( 1 − α ) ψ − θ x + α ψ x x , θ t = − ( 1 − α ) θ + ν ψ x + ( ψ θ ) x + α θ x x , subject to the initial data with end states ( ψ , θ ) ( x , 0 ) = ( ψ 0 ( x ) , θ 0 ( x ) ) → ( ψ ± , θ ± ) as x → ± ∞ , where α and ν are positive constants such that α < 1 , ν < 4 α ( 1 − α ) . Introducing the auxiliary function to avoid the difference of the end states, we show that the solutions to the reformulated problem decay as t → ∞ with the optimal decay order. The decay properties of the solution in the L 2 -sense, which are not optimal, were already established in paper [C.J. Zhu, Z.Y. Zhang, H. Yin, Convergence to diffusion waves for nonlinear evolution equations with ellipticity and damping, and with different end states, Acta Math. Sinica (English ed.), in press]. The main element of this paper is to obtain the optimal decay order in the sense of L p space for 1 ⩽ p ⩽ ∞ , which is based on the application of Fourier analysis and interpolation inequality under some suitable restrictions on coefficients α and ν. Moreover, we discuss the asymptotic behavior of the solution to general system (1.1) at the end. However, the optimal decay rates of the solution to general system (1.1) remains unknown.

Application of fourier analysis to the laser based interferometric strain/displacement gage

The laser based interferometric Strain/Displacement Gage (ISDG) measures the in-plane surface deformation between two small reflecting surface markers. When illuminated with a coherent beam of light, the reflected beams from the two markers form an interference pattern, and monitoring the shift of the fringe pattern allows strain in the gage section of a specimen to be directly measured. A minimum on the fringe pattern can be isolated and tracked as the test proceeds, but this technique utilizes only a small part of the optical signal and often requires a complex programming scheme. This paper presents the application of Fourier transform and phase shifting techniques to the use of the ISDG during microsample tensile testing. The Fourier transform samples the entire fringe pattern and greatly improves the optical signal to noise ratio, and the phase shifting fringe pattern analysis has proven to be more robust and less affected by speckle or optical noise than fringe pattern minimum tracking. This results in the ability to measure larger deformations with a system resolution of ∼5 microstrain and an uncertainty of ±15.5 microstrain. An example involving the microsample tensile testing of a MEMS related LIGA Ni specimen is included to demonstrate the utility of these new techniques.

Harmonic analysis and Boolean function complexity

Any attempt to find connections between mathematical properties of functions and their computational complexity has strong relevance to theory of computation. Indeed, there is the hope that developing new mathematical techniques could lead to discovering properties that might be responsible for lower bounds. The current situation is that none of the known techniques has yet led to lower bounds in general models of computation. The subject of this paper is related to the above general arguments. More precisely, we study the Fourier Transform of Boolean functions, and analyze the extent to which mathematical techniques from the area of abstract harmonic analysis can provide some insight in our current understanding of Boolean circuit complexity. In addition to presenting new applications of Fourier analysis to circuit complexity, we give the necessary background on abstract harmonic analysis and review some work on the subject.

Fourier transform resampling: theory and application

One of the most challenging problems in medical imaging is the development of reconstruction algorithms for nonstandard geometries. The standard geometry is the parallel ray geometry of the conventional Radon transform. This work focuses on the resampling of a nonstandard geometry to obtain a data set in standard geometry. The approach is guided by the application of Fourier analysis to resampling. Fourier Transform Resampling (FTRS) offers potential improvement because the Modulation Transfer Function (MTF) of the process behaves like an ideal low pass filter. Simulated MTFs were obtained by projecting point sources at different transverse positions in the flat fan beam detector geometry. These MTFs were compared to the closed form expression for FTRS. Excellent agreement was obtained for frequencies at or below the estimated cutoff frequency. The resulting FTRS algorithm is applied to simulations with a symmetric fan beam geometry, an elliptical orbit and uniform attenuation, with a normalized root mean square error (NRME) of 0.036. FTRS is also compared to sine interpolation, and it is shown empirically that the two methods are not equivalent. General expressions are obtained for the transfer function, the MTF, the frequency map, and the resampled autocovariance function. A closed form expression is found for the frequency map associated with the circular arc fan beam geometry.

Mhd Leaky Waves in a Layered Plasma: II Mechanisms of Leakage

The application of Fourier analysis to study leaky waves has the advantage of simplicity, but it is not clear why the complex roots of the dispersion relations represent leaky waves, nor how the leakage occurs. We investigate the different kinds of leakage that can occur in a three layer plasma, and to which Fourier pseudomodes they are associated. We find four basic mechanisms, called “surface mode ...”, “single interface ...”, “trapped wave ...” and “lost insulation leakage”. These mechanisms appear in pure form only near certain limiting cases, in which the parameters of the problem take some special values. As soon as the parameters depart significantly from the limiting values, the behavior of the leaky wave complicates and mixtures of the mechanisms occur, in varying amounts. In consequence different points of the same complex branch of the spectrum may correspond to different mechanisms. All the complex branches of the spectrum correspond to leaky waves, but in general it is not possible to classify them according to type of leakage, except close to a limiting case. Since in a three layer configuration there are many of these, the spectrum of leaky waves is very complicated.

Low-Frequency Micromotion of DEP-Levitated Bioparticles, II: Spectral and Stability Analysis

In Part I of this series of papers the experimental observations regarding the phenomenon of micromotion observed during the levitation of biological cells has been described along with a simple model. In this paper we examine the inadequacies of the simple model in reproducing the experiment in the frequency region below 10 Hz. In particular a loss in the gain signal in this region is not predicted by the model of Part I due to the omission of some rather crucial terms in the linearization process. The interpretation of this region requires an extended model which is discussed in this paper. Moreover, the model discussed in Part I is only applicable to a single-frequency levitator, while in this paper we include a dual-frequency levitator. A good approximation to the exact equation of motion is first of all selected by employing a Runge-Kutta fifth- and sixth-order algorithm; this approximation is then examined by the application of Fourier analysis. The exact conditions for stabilization of the periodic motion to occur depends upon the particle charge and are presented in a quantitative form.

The inheritance of patterns of metameric variation in the mouse (Mus musculus) vertebral column

In this paper, we examine the patterns of metameric variation shown by different dimensions of vertebrae from the mouse vertebral column. We examine differences in metameric variation between variables within an inbred strain and compare patterns of metameric variation between inbred strains and between inbred strains and their F1 S. We make both visual and numerical comparisons, the latter being based upon a novel application of Fourier analysis. Comparisons between dimensions within a strain reveal two distinct patterns of metameric variation within the column. The first, shown only by the antero‐posterior (dorso‐ventral) diameter of the neural canal, is relatively simple: dimensions become progressively smaller caudally. The second is more complex, with two peaks, and appears more closely related to somatic structures. The second pattern may be modified according to the temporal pattern of development of different vertebral elements. Comparisons between mouse strains suggest that those variables which complete growth early, and which preserve the basic patterns of metameric variation in a relatively unmodified form, are best suited to discriminating between inbred strains. The Fl S between inbred strains show patterns of metameric variation which in some cases are more like those of one parental strain and in others more like those of the other parental strain. We consider the significance of these findings in relation to further studies of mammalian vertebral column genetics, development and evolution.

ANALYSIS OF THE SPECTRAL REFLECTANCE OF SOILS, AGRICULTURAL CROPS, AND FLOODPLAIN HAYMEADOWS

The article begins with an overview of the current state of knowledge in the USSR on applications of crop, soil, and pasture grass reflectance values in agricultural studies. It then describes an experiment in which digital multispectral information (spectral brightness coefficients) is used to ascertain the condition, productivity, and efficiency of use of pasture land in Lithuania. Models testing the relationships between spectral brightness coefficients and these characteristics are elaborated and support software described. A subsequent section examines the applicability of Fourier analysis to the study of spectral brightness curves of various soils and agricultural crops. Translated by Edward Torrey, Alexandria, VA 22308 from: G. V. Dobrovol'skiy and V. L. Andronikov, eds., Aerokosmicheskiye metody v pochvovedenii i ikh ispol'zovaniye v sel'skom khozyaystve: sbornik nauchnykh trudov [Remote Sensing Methods in Soil Science and Their Use in Agriculture: A Collection of Scientific Works]. Moscow: Nauka,...

Application of Fourier analysis to phase-Doppler-signals generated by rough metal particles

With the phase-Doppler-difference method the diameter and the velocity of spherical particles can be determined by measuring the frequency and the phase difference of burst signals detected by two or more photodetectors. A Fourier analysis method is described that extracts frequency and phase information from the Doppler signals. The system is designed around a TMS320C25 signal-processor board fitted into an AT computer. The performance of the frequency and phase estimator is examined. The method is particularly useful with noisy burst signals, such as those that occur in investigating a metal spraying process. >