Fourier Series Coefficients Research Articles

Estimation du coefficient de réflexion en canal à houle

A complex function F( t) containing the water surface levels in a fixed point in the real part and a function obtained from the slope of this surface in the imaginary part, is used to study the reflection wave in a flume. This complex function analysis by the discrete Fourier transform allows to separate the incident and the reflected wave amplitudes, which are respectively given by the amplitudes of the Fourier-series coefficients of positive and negative frequencies. Errors linked to the evaluation of the reflection coefficient by the measurement of the water surface level in three nearby points are numerically studied.

On the coefficients of multiple Fourier series in -spaces

In this paper we use new function spaces and interpolation methods to study the dependence of the properties of summable multiple Fourier series on their coefficients. We obtain theorems for multiple orthogonal series that reinforce the Hardy-Littlewood theorem for trigonometric series. We prove inequalities of Hardy-Littlewood-Paley type for multiple orthogonal series that refine certain well-known inequalities of this kind.

Spectral characterization of jitter, shimmer, and additive noise in synthetically generated voice signals.

Alteration of the harmonic structure in voice source spectra, taken over at least two periods of the waveform, may occur due to the presence of fundamental frequency (f0) perturbation, amplitude perturbation, additive noise, or changes within the glottal source signal itself. In order to make accurate inferences regarding glottal-flow dynamics or perceptual evaluations based on spectral measurements taken from the acoustic speech waveform, investigation of the spectral features of each aperiodic component is required. Based on a heuristic development involving a consideration of the partial sum of the Fourier series taken for two periods of a jittered, shimmered, and (additive, random) noise-contaminated signal, the corresponding spectral characteristics are hypothesized. Subsequent to this, the Fourier series coefficients are calculated for the two periods in order to test the hypotheses. Definite spectral differences are found for each aperiodic component; based on these findings differential quantitative spectral measurements are suggested. Further supportive evidence is obtained through use of Fourier transform and periodogram-averaged calculations. The analysis is carried out on synthetically generated glottal-pulse waveforms and on radiated speech waveforms. A discussion of the results is given in terms of voice aperiodicity in general and in terms of their implication for future studies involving human voice signals.

Structural Refinement of X-Ray Diffraction Profile for Artificial Superlattices

A new structural refinement process for BaTiO3/SrTiO3 (BTO/STO) artificial superlattices was proposed, taking into account the effect of interdiffusion between BTO and STO. The samples were prepared using the molecular beam epitaxy (MBE) method on SrTiO3(001) substrates at 600°C. The step model, the phonon model and the supercell model were employed for simulation of the X-ray diffraction (XRD) profiles of the superlattices. The supercell model can deal with arbitrary variations of chemical composition and lattice spacing. A discrepancy in the intensity of satellite peaks was observed when the interdiffusion between BTO and STO was not incorporated in the simulation. In the phonon model, the concentration profile due to interdiffusion was first calculated using Fick's second law, and the coefficient of Fourier series describing the lattice distortion and modulation of the structure factor was determined. The XRD profiles thus simulated almost agreed with the observed ones, but the intensity of Laue peaks was not consistent with the observation. The degree of interdiffusion at each interface was then changed throughout the superstructures using the supercell model, and it was shown that the accuracy of simulation improved markedly.

Convergence of Series of Fourier Coefficients of p-Absolutely Continuous Functions

In the paper, we generalize the well-known criteria of Bernstein and Stechkin on the absolute convergence in terms of best approximations and moduli of smoothness of continuous functions. We give conditions for the convergence of the series of Fourier coefficients raised to the power β in terms of best approximations in the space of p-absolutely continuous functions and in terms of fractional moduli of continuity with respect to this space. We also prove the sharpness of our conditions for 0 < β ≤ 1 with no restriction and for 1 < β ≤ 2 under some restrictions.

Polarimetric microwave brightness signatures of ocean wind directions

The sensitivities of wind direction signals in passive microwave brightness temperatures of sea surfaces to wind speed, incidence angle, polarization, and frequency are presented in this paper. The experimental data were acquired from a series of aircraft flights from 1993 through 1996 by the Jet Propulsion Laboratory (JPL) using JPL 19 and 37 GHz polarimetric radiometers (WINDRAD). Fourier analysis of the data versus mind direction was carried out and the coefficients of Fourier series are illustrated against the wind speed at 45/spl deg/, 55/spl deg/, and 65/spl deg/ incidence angles. There is a good agreement between the JPL aircraft flight data and Wentz's Special Sensor Microwave/Imager (SSM/I) geophysical model function for the vertically polarized brightness temperatures, but Wentz's SSM/I wind direction model for horizontal polarization shows a significantly stronger upwind and downwind asymmetry than the aircraft flight data. Comparison of the dual-frequency WINDRAD data show's that the wind direction signals are similar at 19 and 37 GHz, although the 37 GHz data have slightly stronger signals than the 19 GHz data. In general, the azimuthal variations of brightness temperatures increase with increasing wind speed from low to moderate winds, then level off and decrease at high minds. The only exception is the U measurements at 65/spl deg/ incidence angle, which have a stronger than expected signal at low winds. An exponential function was proposed to model the sensitivities of wind direction signals to wind speeds. The coefficients of the empirical model are provided in this paper and are useful for the simulation of ocean brightness temperatures and for the development of geophysical retrieval algorithms.

The DFT: An Owner's Manual for the Discrete Fourier Transform

Preface 1. Introduction. A Bit of History An Application Problems 2. The Discrete Fourier Transform (DFT). Introduction DFT Approximation to the Fourier Transform The DFT-IDFT pair DFT Approximations to Fourier Series Coefficients The DFT from Trigonometric Approximation Transforming a Spike Train Limiting Forms of the DFT-IDFT Pair Problems 3. Properties of the DFT. Alternate Forms for the DFT Basic Properties of the DFT Other Properties of the DFT A Few Practical Considerations Analytical DFTs Problems 4. Symmetric DFTs. Introduction Real sequences and the Real DFT (RDFT) Even Sequences and the Discrete Cosine Transform (DST) Odd Sequences and the Discrete Sine Transform (DST) Computing Symmetric DFTs Notes Problems 5. Multi-dimensional DFTs. Introduction Two-dimensional DFTs Geometry of Two-Dimensional Modes Computing Multi-Dimensional DFTs Symmetric DFTs in Two Dimensions Problems 6. Errors in the DFT. Introduction Periodic, Band-limited Input Periodic, Non-band-limited Input Replication and the Poisson Summation Formula Input with Compact Support General Band-Limited Functions General Input Errors in the Inverse DFT DFT Interpolation - Mean Square Error Notes and References Problems 7. A Few Applications of the DFT. Difference Equations - Boundary Value Problems Digital Filtering of Signals FK Migration of Seismic Data Image Reconstruction from Projections Problems 8. Related Transforms. Introduction The Laplace Transform The z- Transform The Chebyshev Transform Orthogonal Polynomial Transforms The Discrete Hartley Transform (DHT) Problems 9. Quadrature and the DFT. Introduction The DFT and the Trapezoid Rule Higher Order Quadrature Rules Problems 10. The Fast Fourier Transform (FFT). Introduction Splitting Methods Index Expansions (One ---> Multi-dimensional) Matrix Factorizations Prime Factor and Convolution Methods FFT Performance Notes Problems Glossary of (Frequently and Consistently Used) Notations References.

Lifting-Line Theory for Arbitrarily Shaped Wings

This paper presents a new method for solving the classical lifting-line equation. The spanwise lift (or circulation) distribution is to be determined when the spanwise chord distribution and spanwise twist distribution are resolved separately and arbitrarily into their Fourier-series representations. An infinite set of equations for the resulting circulation Fourier-series coefficients is obtained in terms of the Fourier coefficients for the wing-chord shape and the twist distribution. When the equations are expressed in matrix form, the contributions of planform shape and twist are readily perceived. The set of equations can be truncated arbitrarily for the first N number of unknown coefficients, and easily solved by various software packages. Convergence properties can then be readily established by examining changes associated with changes in levels of truncation. A five-by-five (N = 5) truncation system is found to be sufficient for a wide variety of problems. Several example problems are considered, and the results are compared. The new method converges faster and is more accurate for the same level of truncation. Moreover, in comparison with other methods, it is comprehensive, rigorous, and lends itself to insight regarding the interactions of lift distribution, planform shape, and twist.

Electromagnetic Behavior of Bent Wire in a Shielded Enclosure

A unique Fourier series approach to the problem of computing currents excited on a conducting bent wire enclosed in a shielded enclosure is presented. The wire is folded at right angle and oriented in a plane parallel to one of the cavity walls. Electromagnetic sources exterior to the cavity excite one end of the wire, while the other end of the wire within the cavity may have any arbitrary load conditions. The coupling between the two portions of the bent wire, which are perpendicular to each other, is taken into consideration. The formulation of this problem makes use of the theory of Fourier series to approximate the waveform of unknown current excited on the wire. Of course, the infinite Fourier series is approximated by a truncated Fourier cosine series in order to obtain an analytically simple solution. The number of Fourier series coefficients and the cavity modes are the key issues to the formulation. Selected numerical results in the form of current distributions are presented to illustrate the formulation. This formulation can easily handle any arbitrary load condition.

Computing coefficients of fourier series

In this paper, the following questions for computing coefficients of Fourier series are discussed: n-order Filon quadrature formula and its partial cases, some features of applying the Filon method in computing coefficients when the adaptive integration strategy is employed, the program implementation of 3-order and 5-order Filon quadra­ ture formulas, using the adaptive integration strategy, and the experimental results of applying them in computing coefficients of Fourier series.

A Fourier descriptor model of hysteresis loops for sinusoidal and distorted waveforms

This work is concerned with the use of the Fourier descriptor technique for reproducing the hysteresis loop for both sinusoidal and distorted flux-density waveforms that produce minor hysteresis loops. The hysteresis curve is characterized only at discrete points. Approximating the curve by a polygon with a limited number of sides, the Fourier series coefficients of the normalized cumulative angular function are computed. The shape of the hysteresis loop for a pure sinusoidal waveform can be reconstructed easily when these coefficients are determined. The procedure can be extended for the distorted waveform by taking account of the dips in the exiting waveform that cause minor loops in the hysteresis curve. The deviation between the original curve and the generated one depends upon the number of Fourier coefficients. As the number of the Fourier coefficients used in the synthesis is increased, the generated curve becomes closer to the original one. However, the use of a large number of the Fourier coefficients requires a greater execution time.

Absolute convergence of multiple series of Fourier coefficients with respect to multiplicative systems

In this paper, we consider series of the Fourier coefficients of functions of several variables with respect to multiplicative systems generalizing the Walsh system. We present a sufficient condition for the convergence of these series in terms of the complete group modulus of continuity. We prove that this condition cannot be improved in the function classes H~ ~,'2 ..... ,~.. Let us present the relevant definitions and notation. Suppose that {p )}k~=t (J = 1, 2, ..., n) are

Why Isn't My Professor Using Graphics in the Freshman Programming Course?

AbstractA new approach to teaching computer programming to freshmen is discussed. The course is designed to heighten student interest through the use of computer graphics programming. The students are given a relatively small number of programming tools and must create increasingly elaborate images on the screen as their knowledge of basic computer programming develops and their own personal collection of library routines grows. The lab problems culminate in a demonstration of a periodic waveform constructed from its Fourier Series coefficients.

Regional models of potential evaporation and reference evapotranspiration for the northeast USA

Multivariate regression models of average monthly reference crop evapotranspiration, E t( τ), and potential evaporation, E p( τ), are developed for the northeast USA. Average monthly values of daily E t and E p are estimated from the Penman-Monteith equation using monthly climate data from 34 National Oceanic and Atmospheric Administration (NOAA) First Order weather observatories and specific reference vegetation characteristics. The periodic seasonal behavior of E t( τ) and E t( τ) and average monthly temperature are approximated by Fourier series functions. Regional regression relationships are then developed which relate the E t( τ) and E t( τ) Fourier series coefficients with the average monthly temperature Fourier coefficients, station longitude and station elevation. The regional E t( τ) model is shown to be an improvement over the Linacre method (Linacre, 1977, Agric. Meteorol., 18: 409–424) and the Hargreaves and Samani method (Hargreaves and Samani, 1985, Appl. Eng. Agric., 1(2): 96–99), and the regional E p( τ) model is shown to be an improvement over the Linacre method.

The coefficients of multiple Fourier series with respect to periodic multiplicative price systems

The coefficients of multiple Fourier series with respect to periodic multiplicative price systems

Series expansions for excited states of quantum lattice models

We show that by means of connected-graph expansions one can effectively generate exact high-order series expansions which are informative of low-lying excited states for quantum many-body systems defined on a lattice. In particular, the Fourier series coefficients of elementary excitation spectra are directly obtained. The numerical calculations involved are straightforward extensions of those which have already been used to calculate series expansions for ground-state correlations and T = 0 susceptibilities in a wide variety of models.

Statistical characteristics of moving acoustic sources in ocean waveguides

The statistical characteristics of the response of an array of hydrophones to a moving narrow-band acoustic source in an ocean waveguide are investigated. The discrete Fourier transform of a solution of the wave equation with a moving source is calculated, and the sequence of the amplitudes of the Fourier coefficients at particular frequencies are shown to contain oscillatory terms that depend upon the environment, the source frequency, position and velocity, and the resolution of the DFT. This analysis suggests that Gaussian mixture densities may apply to time series of Fourier coefficients of data received from a moving source. This is verified by analyzing data from a deep water Navy experiment. Simulations are then carried out which show that the fluctuations caused by source motion are sufficient to favor Gaussian mixture densities over Gaussian densities and that the applicability of the Gaussian model, but not the Gaussian mixture model, depends upon the velocity and position of the source.

Evolution of airplane gust loads design requirements

Nomenclature a = lift curve slope, per rad #„, bn — Fourier series coefficients for e.g. acceleration c = wing mean geometric chord, ft FK — flight profile alleviation factor FKm = flight profile factor due to airplane weight FKZ = flight profile factor due to altitude g = acceleration due to gravity, 32.2 ft/s g;j, hn = Fourier series coefficients for true gust velocity H = gust gradient distance, ft K, KK = gust load alleviation factors L = scale of turbulence, ft M = Mach number q = dynamic pressure, psf /?, = maximum landing weight/maximum takeoff weight R2 = maximum zero fuel weight/maximum takeoff weight S = wing area, ft s = distance along flight path, ft T = local atmospheric temperature, °R 7,,, /„ = real and imaginary parts of the e.g. acceleration transfer function T() = ambient atmospheric temperature, °R U = gust velocity, fps, true airspeed (7dc = derived equivalent gust velocity, fps, equivalent airspeed (7ds = design gust velocity, fps, equivalent airspeed £/dt = derived true gust velocity, fps, true airspeed £/rct = design reference gust velocity, fps, equivalent airspeed Utr = power spectral scale factor V = aircraft velocity, fps, true airspeed VB = rough air penetration speed, Kt, equivalent airspeed Vc = design cruise speed, Kt, equivalent airspeed VD = design dive speed, Kt, equivalent airspeed W = aircraft weight, Ib Zmo = maximum operating altitude, ft y = ratio of specific heats Aft = incremental load factor, g fjif, = airplane mass parameter p = air density, slugs/ft Introduction S UBSONIC aircraft respond to atmospheric turbulent air motions or eddies of, approximately, 30-2000 ft in extent. Smaller eddies will generally be averaged out over the surface of the aircraft, larger eddies typically will not cause sharp or excessive aircraft accelerations or structural loads on the airplane. Aircraft in supersonic flight respond to ever longer wavelengths as the flight speed is increased. From the aircraft design standpoint, atmospheric turbulence may be separated into two categories: 1) turbulence, which contributes to aircraft structural fatigue and passenger inconvenience and discomfort, is generally associated with the less intense, smaller scales (small eddy size or higher spatial frequencies as characterized by the turbulence power spectrum); and 2) turbulence that can cause aircraft upset, passenger injury, and possibly structural damage or failure is associated with the more intense larger scales of the turbulence spectrum. Whereas the first category may be in the inertial subrange of the turbulence power spectrum where local homogeneity and stationarity may be assumed, the second category is definitely associated with the larger energy-containing scales of turbulence that are definitely nonstationary and inhomogeneous. In fact, the second type of atmospheric turbulence may not be turbulence at all, but may be a part of, or derive directly from, the ordered convective or geostrophic motions of the atmosphere. Frictionally induced turbulence in the planetary boundary layer is dependent on wind speed near the ground and the surface roughness. Convective turbulence in the planetary boundary layer is dependent on the lapse rate (the rate of temperature change with altitude) and the depth of the mixing layer. Thus, the wind speed and surface roughness and the lapse rate and mixing layer thickness largely determine the intensity and scale of the boundary-layer turbulence. This low altitude boundary-layer turbulence will affect landing and takeoff operations for the larger commercial aircraft that normally cruise at high altitude, and it will be the primary cause of disturbance for small aircraft that operate at low altitudes. Usually, it can be characterized as random, locally stationary, and Gaussian, and the turbulence scale at the surface is of the order of 1000 ft and increases with altitude. It is only natural that pilots avoid flying into areas of obviously rough air such as thunderstorms or even cumulus clouds;

Truncation artifact reduction in magnetic resonance imaging by Markov random field methods

A new statistical method is proposed for reduction of truncation artifacts when reconstructing a function by a finite number of its Fourier series coefficients. Following the Bayesian approach, it is possible to take into account both the errors induced by the truncation of the Fourier series and some specific characteristics of the function. A suitable Markov random field is used for modeling these characteristics. Furthermore, in applications like Magnetic Resonance Imaging, where these coefficients are the measured data, the experimental random noise in the data can also be taken into account. Monte Carlo Markov chain methods are used to make statistical inference. Parameter selection in the Bayesian model is also addressed and a solution for selecting the parameters automatically is proposed. The method is applied successfully to both simulated and real magnetic resonance images.

Milling force convolution modeling for identification of cutter axis offset

This paper discusses the application of a convolution integral force model to the identification of the geometry of cutter axis offset in milling operations. This analysis builds upon the basis of linear decomposition of elemental local cutting forces into a nominal component and an offset-induced component. The convolution of each elemental local cutting force component with the chip width density in the context of cutter angular position provides an integral expression for the total cutting forces. By virtue of the convolution integration property, the total cutting forces in the frequency domain can be derived as closed-form functions of the cutting pressure constants, various cutting conditions, as well as the cutter offset geometry. Subsequently, the magnitude and phase angle of cutter axis offset are shown to be algebraic and explicit functions of the Fourier series coefficients of cutting forces at the spindle frequency. Following the theoretical analysis, experimental study is discussed to illustrate the implementation procedure for offset identification, and frequency domain data are presented to verify the analytical results. Potential industrial applications of this work include the real-time monitoring of dynamic cutter runout and the in-process compensation for the loss of tolerance or finish using automatic controls based on the feedback information of offset magnitude and phase angle.