Calculation Of Fourier Series Research Articles

Further Discussion on the Calculation of Fourier Series

Fourier series is an important mathematical concept. It is well known that we need too much computation to expand the function into Fourier series. The existing literature only pointed that its Fourier series is sine series when the function is an odd function and its Fourier series is cosine series when the function is an even function. And on this basis, in this paper, according to the function which satisfies different conditions, we give the different forms of Fourier series and the specific calculation formula of Fourier coefficients, so as to avoid unnecessary calculation. In addition, if a function is defined on [0,a], we can make it have some kind of nature by using the extension method as needed. So we can get the corresponding form of Fourier series.

Time-Domain Analysis of the Periodically Discontinuously Forced Fractional Oscillators

A new method for the solution of non-sinusoidal periodic states in linear fractionally damped oscillators is presented. The oscillator is forced by a periodic discontinuous waveform and a viscous element is taken into account. The presented method avoids completely the Fourier series calculations of the input and output oscillator waveforms. In the proposed method, the steady-state response of fractionally damped oscillator is formulated directly in the time domain as a superposition of the zero-input and forced responses for each continuous piecewise segments of the forcing waveform, separately. The whole periodic response is reached by taking into account the continuity and periodicity conditions at instants of discontinuities of the excitation and then using the concatenation procedure for all segments. The method can be applied efficiently to discontinuous and continuous non-harmonic excitations equally well. Solutions are exact and there is no need to apply any of the widely up-to-date used frequency approaches. The Fourier series is completely cut out of the oscillator analysis.

Dynamically adaptive spectral-element simulations of 2D incompressible Navier–Stokes vortex decays

Vortex dynamics often involves a wide range of strongly non-linearly interacting scales, and many situations arise in which relatively few aspects of the spatial configuration of the vortices seem to play a more significant dynamical role, and contribute more significantly to the energy spectrum, than does the rest of the flow. This presents a challenge to both simulation and analysis, as follows. A simulation method must be as highly accurate and computationally efficient as are methods that rely on simple spatial boundaries and geometry, but should also enable complex boundaries and geometry, as well as combined spatial and spectral analyses. That is, it should enable geometric analysis of vortex spatial aspects of any dimension – (approximate) points, curves, surfaces or volumes; and enable spatially local spectral analysis, to characterise potential spatially localised fractional-dimensional aspects such as spectral power laws. The spectral-element method (SEM) for numerically solving partial differential equations combines the spatial-configuration flexibility and computational efficiency of the finite-element method with the high accuracy (exponential convergence) of the pseudo-spectral method (PSM). Several SEM formulations until now have included dynamic adaptive refinement and coarsening (DARe), but always specialized to the 2D or 3D cases. To enable better simulation and analysis of vortex dynamics and similar complex flows, the present SEM-DARe formulation includes topological structure in order to achieve a completely general treatment of the d-dimensional case for arbitrary positive integer d. With a few exceptions, most SEM applications have been to compressible flow or to incompressible flow at low Reynolds number, Re. A new simulation code, the geophysical–astrophysical spectral-element adaptive refinement (GASpAR) code, recently introduced for two-dimensional linear and non-linear advection-diffusion simulations by Rosenberg et al. (Geophysical–astrophysical spectral-element adaptive refinement (GASpAR): Object-oriented h-adaptive fluid dynamics simulation. J. Comp. Phys., 2006, 215, 59–80, Available from http://dx.doi.org/10.1016/j.jcp.2005.10.031), has now been extended to decaying incompressible Navier–Stokes flows at high Re. GASpAR employs DARe of non-conforming type h (mesh size), and fixed number N of expansion polynomials per element per coordinate. GASpAR simulation of bi-periodic flow at Re = 2 × 104 initialised with 3 Gaussian vortices, closely reproduces that by PSM with the same number of computational degrees of freedom. The high N enables a recently introduced SEM-customised Fourier analysis (Fournier, A., Exact calculation of Fourier series in non-conforming spectral-element methods. J. Comp. Phys., 2006, 215, 1–5, Available from http://dx.doi.org/10.1016/j.jcp.2005.11.023.) to find evidence of power-law-scaling regimes E k ∼ | k |−β over about 1 wavenumber decade, with exponents 3 ≲ β ≲ 4. Thus, GASpAR not only has several new and potent capabilities, but is also shown to reproduce and potentially enhance traditional PSM simulation and analysis of vortex dynamics.

Efficient Approach to Determine the Steady-State Energy in Networks with Periodic Discontinuous Excitations

A new method for the solution of non-sinusoidal periodic states in linear networks is presented. It avoids the Fourier series calculations of the input and output network quantities. In the proposed method, the steady-state network response is formulated directly in the time domain as a superposition of the zero-input and forced responses for each continuous piecewise segments of the input quantity, separately. The whole periodic response is reached by taking into account the continuity and periodicity conditions at instants of discontinuities of the input quantity and then using the concatenation procedure for all segments. The method can be applied efficiently to continuous and discontinuous input quantities equally well. When we divide the network into two parts, i.e. a supplying two-terminal source and a two-terminal load, this method can be applied more efficiently to determine one-period energy for each of these particular subnetworks. Solutions are exact and there is no need to apply any of the widely up-to-date used frequency approaches. The Fourier series is completely cut out of the network analysis.

Exact calculation of Fourier series in nonconforming spectral-element methods

In this note is presented a method, given nodal values on multidimensional nonconforming spectral elements, for calculating global Fourier-series coefficients. This method is ``exact'' in that given the approximation inherent in the spectral-element method (SEM), no further approximation is introduced that exceeds computer round-off error. The method is very useful when the SEM has yielded an adaptive-mesh representation of a spatial function whose global Fourier spectrum must be examined, e.g., in dynamically adaptive fluid-dynamics simulations.

Asymptotic solutions for mode II dynamic rupture with nonlinear damping

In this study, the nonlinear Rayleigh damping (A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1981, pp. 147–150, 216–230) is adopted, and the governing equations are two coupled nonlinear PDEs, their unknown functions being the amplitude components in x and y directions, respectively. By assuming the unknown functions to be proper vibrational pattern functions which fulfill all the boundary conditions, transforming the fixed coordinates into the moving coordinates by Galileo transformation and applying the calculation of Fourier series, the two nonlinear PDEs are reduced to four nonlinear ODEs. Among them, a couple are of order 1, the rest are of order 2. In order to be able to solve them analytically, the nonlinear terms in one ODE of order 1 for the secondary amplitude v are neglected in the first iterative calculation and the rupture velocity is assumed to be a constant value, and the coupled term in one of the order two ODEs to be 0. Then the remaining order two ODEs is reduced to be a homogeneous van der Pol equation. Its solution is readily obtained. Substituting this known function into the other order two ODEs for the unknown function u, it is reduced to be a nonhomogeneous van der Pol equation. Its solution is also unknown.

Rubredoxin from Desulfovibrio gigas: A molecular model of the oxidized form at 1.4 Å resolution

The crystal structure of rubredoxin from the sulfate-reducing bacterium Desulfovibrio gigas has been determined at 1.4 Å resolution (1 Å = 0.1 nm) by X-ray diffraction methods; starting with a model of the isostructural rubredoxin from Desulfovibrio vulgaris. Refinement of the molecular model has been carried out by restrained least-squares techniques and Fourier series calculations. The present model includes a formyl at the N-terminal end and 121 possible sites for solvent molecules with full or partial occupancy, which corresponds to the modeling of nearly all the solvent medium. The crystallographic R factor against the data with 10 A ̊ > d > 1.4 A ̊ with F > 2 sig(F) , is 0.136; and R = 0.140 when all the data are considered. The estimated average root-mean-square (r.m.s.) error on the positional parameters is about 0.12 Å. The overall structural features of this molecule are close to those of the two highly refined rubredoxins from Clostridium pasteurianum and D. vulgaris. Superposition of these two molecules on the rubredoxin from D. gigas shows in both cases an overall r.m.s. deviation of 0.5 Å for the atoms in the main-chain and of 0.4 Å for the atoms in the side-chains that make up the hydrophobic core. The iron atom is co-ordinated to four cysteine sulfur atoms forming an almost regular tetrahedron, with Fe-SG distances ranging from 2.27 Å to 2.31 Å and angles varying from 103 ° to 115 °. The intramolecular hydrogen-onding pattern is quite comparable to those found in other proteins refined at high resolution. All the polar groups are involved in hydrogen bonds: intramolecular, intermolecular or with solvent molecules. The main structural differences from the other rubredoxins are in the nature and the distribution of some of the charged residues over the molecular surface. The possible influence of several structural factors on the intramolecular and intermolecular electron transfer properties such as the NH … SG bonds, the solvent exposure of the redox center, and the aromatic core is discussed. The conservation, during evolution, of a ring of acidic residues in the proximity of the FeSG4 center suggests that this ring may be implicated in the recognition processes between rubredoxins and their functional partners.

Generalized Poisson integrals and parametric head field equations

To calculate the head field distribution first we pursue the Poisson integral approach3 using Fourier series method retaining up to 200 terms in the series. The parameters related to the rational approximation to the field equation for both ring heads and pole-keeper heads have been obtained and tabulated. The idea of the Poisson integral is then extended, denoted as Poisson integral II, to enable us to calculate the head fields both within and without the head gap regions. The first Poisson integral II involves eight parameters and enables us to calculate head fields in reasonably good agreement with the direct Fourier series calculations. The second Poisson integral II involves a self-consistent calculation, which is reducible to the rational field equations with five parameters and is highly capable of producing head field distribution anywhere with great accuracy at low spacings. Results of a comparison with a finite element method calculation are excellent.

A comparison of the fast Fourier and trigonometric Fourier algorithms

The Cooley–Tukey fast Fourier algorithm and the factored trigonometric Fourier algorithm are compared for four typical crystallographic problems. With general PL/1 programs the execution speed of the fast Fourier algorithm ranged from 4.7 to 19 times faster than the trigonometric algorithm. In addition the PL/1 fast Fourier program was 1.8 times faster than a space group specific, fixed axial length trigonometric Fourier FORTRAN program. Our results indicate that the fast Fourier is the most efficient algorithm available for crystallographic Fourier series calculations, with efficiency increasing as one goes to larger problems.

Note on the Calculation of Fourier Series

Note on the Calculation of Fourier Series

Note on the calculation of Fourier series

Cooley and Tukey have recently presented an algorithm for the machine calculation of Fourier series [1]. In this connection mention should be made of the similar method described by Danielson and Lanezos [2]. Although the latter is less elegant and is phrased wholly in terms of real quantities, it yields the same results as the binary form of the Cooley-Tukey algorithm with a comparable number of arithmetical operations. A small-computer program has been written in this laboratory which uses the Danielson-Lanezos method with one minor modification, described below.* In this form cosine and sine series may be evaluated independently of one another, and as with the Cooley-Tukey process, the calculation can be performed by replacing input data with results, without any substantial storage requirement beyond that set by the original number of input coefficients. The procedure used is readily extensible to the computation of [(N/2) + 1] cosine and [(N/2) 1] sine sums, from an original sequence of N real numbers, where N is any power of 2 greater than the fourth. Two executions of this procedure will yield values for a set of N complex Fourier series of N terms each, and can be accomplished by N[log2 N + 3] real multiplications and 7N/2[10g2 N 23/7] real additions. For the same task the

An algorithm for the machine calculation of complex Fourier series

An efficient method for the calculation of the interactions of a 2' factorial ex- periment was introduced by Yates and is widely known by his name. The generaliza- tion to 3' was given by Box et al. (1). Good (2) generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series. In their full generality, Good's methods are applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices, where m is proportional to log N. This results inma procedure requiring a number of operations proportional to N log N rather than N2. These methods are applied here to the calculation of complex Fourier series. They are useful in situations where the number of data points is, or can be chosen to be, a highly composite number. The algorithm is here derived and presented in a rather different form. Attention is given to the choice of N. It is also shown how special advantage can be obtained in the use of a binary computer with N = 2' and how the entire calculation can be performed within the array of N data storage locations used for the given Fourier coefficients. Consider the problem of calculating the complex Fourier series N-1 (1) X(j) = EA(k)-Wjk, j = 0 1, * ,N- 1, k=0

X-ray crystallographic studies of compounds of biological interest.

In her review of the years 1944 to 1947, Dr. Crowfoot (1) commented upon the progress which had been made in our knowledge of organic structures since the first of the Annual Review series (1936) by Sponsler & Dore and called attention to the increasing productivity in recent years which has resulted from the growing power of x-ray analytical methods. This accelerated output of structure determinations has continued during the years 1948 to 1950. The first issue of Acta Crystallographica appeared in 1948, and already more than 60 articles dealing with the crystal structures of organic compounds have been published in its pages alone. But the increased number of structure determinations is probably less significant than the greatly increased accuracy with which they can now be made. This increased accuracy may be largely attributed to improved computational techniques for refining atomic coordinates by the calculation of Fourier series or by the method of least squares (2).