Fourier Series Coefficients Research Articles

Power-frequency relations in nonlinear two-poles

In the first part of the paper, some properties of the coefficients of the Fourier-series function f[v(t)] are presented, where f(v) is an arbitrary continuous function and v(t) is an almost-periodic function. In the second part of the paper, the power-frequency relations for nonlinear two-poles are obtained on the basis of these properties. These power-equilibrium equations are the generalizations of Groszkowski's and Manley-Rowe's equations. The presented, generalized form of these equations comprise as well the elements having loop-like characteristics, nonlinear inductance, capacitance with losses, and others.

Determination of Twin Fault Probabilities from the Diffraction Patterns of fcc Metals and Alloys

Twin faults may be detected from diffraction patterns of fcc metals and alloys by two existing methods: (1) from the sine coefficients of a one-dimensional Fourier series representing the peak, and (2) from the excess intensity between two peaks which are asymmetric in opposite directions, such as the 111 and 200. However, the contribution of twin faults to the particle size measured by Fourier analysis appears too large when the fault probabilities are determined by these methods. A third technique has been developed based on the displacement of the center of gravity of a peak from the peak maximum, ΔC.G. The displacements for the 111 and 200 peaks are related to the twin fault probability β: ΔC.G. (∘2θ)111=+11β tanθ111ΔC.G. (∘2θ)200=−14.6β tanθ200. By using two peaks, such as the 111 and 200, the difficult choice of background and instrumental effects can be minimized. A comparison of the three techniques is presented. As a consequence of the new method, it is clear that the probability of deformation stacking faults (α) should be measured from the displacements of peak maxima from those of an annealed specimen; if the C.G. is used and twin faults are present, a large error is possible.

Přibližný výpočet koeficientu Fourierovy řady periodické funkce

Přibližný výpočet koeficientu Fourierovy řady periodické funkce

Method for Numerical Evaluation of the Coefficients of Fourier Series

The coefficients a0, am, and bm of a Fourier series f(x) = a0 + Σmam cosmx + bm sinmx, where m = 1,2,3,⋯, ∞ is the harmonic order, can be evaluated by a simple formula if f(x) is a sequential trace of n straight lines. The formula makes use of the slopes Ai and the intercepts Bi of the 1 ≦i≦n lines contained within the period. If the line crossings are equidistant in the direction of the x axis, the formula will not only be more simplified by the direct use of these n sample values but the coefficients can be taken as the products am = Cmam* and bm = Cmbm*. The factor Cm decreases with increasing values of the integers m. The discrete values Cm appear within an envelope that is the picture of an attenuated square cosine wave. The harmonic order m and the subdivision number n of the period range are parameters in a tabulation of Cm. The second factors are introduced as periodic Fourier coefficients am* and bm*. These coefficients are an infinite repetition of the finite range 1 ≦ m ≦ n, if m runs from 1 to infinity. Assuming n is a multiple of 4, only n/4 periodic coefficients have to be evaluated. Multiplication with the tabulated values Cm result in an arbitrary series of coefficients am and bm. So far as any periodic function f(x)—given by an experimental observation, for instance—can be approximated by a sequential trace of straight lines, f(x) can be approximated by the Fourier series of the straight-line trace. The numerical evaluation of the coefficients of the trace is not restricted to a certain number. in the harmonic order, such as to the first 6 or first 12 harmonics, for instance. It can easily be programed for desk machine calculators or for computers. A simple schematic calculation can be made to yield either an arbitrary series or any particular order of coefficients. The method can be extended to numerical evaluation of the Fourier integral as well.

Proceedings of the Celestial Mechanics Conference: Numerical integration of ordinary differential equations at an interval which may be compared with some periods in the defining functions

view Abstract Citations (3) References (1) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Proceedings of the Celestial Mechanics Conference: Numerical integration of ordinary differential equations at an interval which may be compared with some periods in the defining functions Thomas, L. H. Abstract . The object of the paper is to consider the general solution of a system of linear differential equations the coefficients of which have definite periodicities. A method of numerical integration for such systems of equations is out- lined; its principal feature is the evaluation of the coefficients of Fourier series that represent the solution. Publication: The Astronomical Journal Pub Date: November 1958 DOI: 10.1086/107808 Bibcode: 1958AJ.....63..459T full text sources ADS |

Variations of characteristic impedance along short coaxial cables

A method is described for determining the local characteristic impedance at any point on a high-frequency coaxial cable. Measurements are made of the deviations from a harmonic series of the resonant frequencies of a short length of open or short-circuited cable. It is shown how these deviations can be used to calculate the coefficients of Fourier series describing the impedance variations along the length. Since the measurements are performed at microwave frequencies, the resolution of the impedance changes is considerably better than for lower-frequency methods.Measurements on an artificially discontinuous cable have allowed the experimental accuracy of the method to be determined, and close agreement has been obtained between the experimentally predicted and the actual impedance changes.Results from short cable samples indicate that the local variations of characteristic impedance are largely due to changes in the diameter over the dielectric filling.

Determining fourier series coefficients graphically

THE ACCURACY of the Fourier series coefficients when determined by the equation1 $B_{n} = {2 \over m} displaystyle\sum^{m}_{1} \left[y_{k} \sin n \left(k - {1 \over 2}\right){2 \pi \over m} \right]$ should receive some consideration.

Graphical determination of the Fourier series coefficients

ALMOST all of the many good textbooks on analysis of electric circuits include analysis of nonsinusoidal waves. Equations are usually given which determine the amplitude of each term of the Fourier trigonometric series. If the dependent variable can be expressed in terms of the independent variable, the familiar relations

On the limit of the coefficients of the eigenfunction series associated with a certain non-self-adjoint differential system

«»» — q(x)u = uxt — p(x)ut where u(x, t) must satisfy the conditions u(a, t)=u(b, /)=0 and u(x, 0) =F(x). In attempting to solve equation (1) by separation of variables, one is led to the consideration of expanding an arbitrary function F(x) in terms of the eigenfunctions, or nonzero solutions, un(x) of the equation: (2) (A + \B)u = 0 satisfying the conditions u(a)=u(b)=0, where A is the operator d2/dx2+q(x) and B is the operator — d/dx-\-p(x). The system adjoint to (2) is: (3) (A* + \B*)v « 0, v(a) = v(b) = 0 where A =A* and B*=d/dx+p(x). Conditions have been established [2], under which a function F(x) of bounded variation on (a, b) can be expanded in terms of un(x). However, in the expansion F(jc)=E-« anun(x) there are certain properties of the coefficients, an, which differ quite radically from the corresponding properties of the coefficients of certain well-known selfadjoint eigenfunction expansions. For example, if Bn are the Fourier coefficients of a function g(x), it is well known that limn,«, Bn = 0. However, in the expansion i(x)=E-» a„u»(x), it is found that limn_M an is not in general equal to zero;. Consequently, the series 22? °n> unlike the corresponding series of Fourier coefficients, does not in general converge.

経度方向に一次元フーリエ級為を用いた数値予報

The height change of 500mb isobaric surface in a barotropic atmosphere is calculated by making use of the coefficients of one-dimensional Fourier series along latitude circles with one-dimensional relaxation with respect to latitude.In this paper, the tendency equations concerning Fourier coefficients along latitude circles are derived as a function of latitude from a vorticity equation, and their finite difference expressions are given in order to simplify the calculation. The subjects as to the number of component-waves to be considered and the choice of their interactions were the most important. The problem of error in this method was also considered.

Analysis of Oval Rings by Fourier Series

The distinguishing feature of this method of analysis of thin oval rings is the representation of the shape of the ring by the use of Fourier series. By means of standard computational schemes, the coefficients of Fourier series of sine and/or cosine terms may be so selected as to cause the graph of the function represented by the series to approximate closely the shape of the ring under consideration.

Harmonic Analysis by Photographic Method

This paper describes a method for determining Fourier series coefficients from area measureurements on photographs of the function to be analyzed with the graph sheet formed into a series of semi-cyclinders.

VIII. The method of Symmetrical Components applied to harmonic analysis

Abstract A method of determining the Fourier series coefficients from a graphically known wave is presented. The special feature of this method is the use of Fortescue's sequence operator S m , which playssuch a prominent role in the theory of Symmetrical Components (l). The determination of the various coefficients is then reduced to a simple matrix multiplication scheme involving the sequence operators S m and a columnar matrix [u] formed from the ordinates of the graphically known wave.

The Electron Distribution of Magnesium Oxide

The intensity of $x$-rays reflected from powdered crystals of magnesium oxide was measured for all lines out to $\ensuremath{\theta}=45\ifmmode^\circ\else\textdegree\fi{}$. The values of the structure factor, $F$, were calculated and plotted against $sin \ensuremath{\theta}$. Using the values of $F$ from this curve as coefficients in Compton's Fourier series, the radial electron distribution for the atoms of magnesium and oxygen was determined. The electron distribution curves for these atoms are very similer to Havighurst's curves for sodium chloride and sodium fluoride. Although it has been generally conceded that crystals of the magnesium oxide type are polar in form, the data here included are more in line with their being non-polar crystals. However, it is not believed possible from x-ray reflection data to prove definitely this point.