Fourier transform value finiteness

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

Two-dimensional Fourier transformation of the signal with respect to the jc-and y-gradient values gives the spatial distribution of individual spectra. However, only a finite number of phase-encoding steps are acquired, and the point... 

In the simulation, the value of the F(t) function is calculated at discrete points in a finite time interval then discrete complex Fourier transform is performed on this array to obtain the simulated spectrum.101... 

The function Dxp is composed of Dira functions D = 2b gb 5b where 2b gb = 1 gb => 0 and the 5s are forbidden by Planck s law of the finiteness of the quantum of action (which may be formulated as the indeterminacy principle). This may be corrected for by the so-called Wigner transformation which transforms the expectation value linear form to U = / Hx x DXx where the coordinates x occur twice, independently, so that H and D become matrices. Since the Wigner transformation must lead to indeterminacy, it is closely related to a Fourier transformation and both matrices H and D become hermitian (Bopp, 1961). The dyads of hermitian matrices may be written j/x i//x and we see that their contribution u to the expected energy becomes u = / jj Hi//, therefore if we choose i//x as eigenvectors of H we see that we have in fact only discrete possible states in agreement with indeterminacy. 

Slowly-varying tails of the pair correlation function contribute to EXAFS data only at low k values. Sharp peaks in the pair correlation function, however, give rise to dominant features in the EXAFS signal W hich persist to high k values. As the data are Fourier transformed only in a finite range and the low k data of the EXAFS signal must be omitted in the Fourier transform, the broad tail in the atom pair correlation function is often lost in the anaivsis of the EXAFS data. A... 

A function that is compact in momentum space is equivalent to the band-limited Fourier transform of the function. Confinement of such a function to a finite volume in phase space is equivalent to a band-limited function with finite support. (The support of a function is the set for which the function is nonzero.) The accuracy of a representation of this function is assured by the Whittaker-Kotel nikov-Shannon sampling theorem (29-31). It states that a band-limited function with finite support is fully specified, if the functional values are given by a discrete, sufficiently dense set of equally spaced sampling points. The number of points is determined by Eq. (26). This implies that a value of the function at an intermediate point can be interpolated with any desired accuracy. This theorem also implies a faithful representation of the nth derivative of the function inside the interval of support. In other words, a finite set of well-chosen points yields arbitrary accuracy. 

I (q) is the intensity of dispersion for a structure with finite thickness, and pf iq) is the Fourier transform of h (r) in the reciprocal space. Because the width of the function h(r) should be small compared to the average regions of constant density (for a two-phase system), the width of the function H(q) will be considerably larger than that of the intensity. In this way, the intensity of dispersion is affected essentially only at large q values, that is, in the Porod s region. As a consequence, and using Equation 19.13, the result is... 

The principal errors in numerically approximating the transform integral to infinite time by a discrete Fourier transform (DFT), or series of a finite number of values at discrete points, are from aliasing and truncation. The magnitudes of the errors for the functions G(t) usually encountered can be readily assessed and kept... 

Because computerized Fourier Transforms are performed on discrete, digital data arrays of finite length, two well-known problems arise. The discreteness of the data array leads to a phenomenon referred to as "aliasing" in which frequencies which are higher than one-half the data point acquisition frequency (the Nyquist frequency) appear at values which are lower than the true frequency. This effect is illustrated in Figure 6 for the case of a sine wave. 

The Fourier transform can be explained as follows. Imagine we have a rectangular pulse waveform x t), which has values for a finite duration T, defined as follows ... 

The data contained in a digitally recorded image is an ordered finite array of discrete values of intensity (grayscale). To manipulate this data, the continuous integrals defining the Fourier transform and convolution must be expressed as approximating summations. For a series of discrete samples jc( t) of a continuous function x(t), a representation in the frequency domain can be written as... 

While in the disordered state the order parameter vanishes, it takes finite values in the ordered phases. The Fourier transform of the pair correlation function of the order parameter... 

IR spectra are given as products (< )/i(decadic linear absorption coefficient) as a function of wavenumber, time domain, i.e., each term of the correlation function C(t) is multiplied by a Gaussian function exp —0.5a(tlt, axf ), where is the length of the simulation, and a is usually chosen around a value of 10 for gas phase simulations. This convolution only has the purpose to remove the numerical noise arising from the finite length of the Fourier transform of (5). These calculations are performed with our home-made code. 

In practice, a diffractometer can only access a finite k-space range with a maximum cutoff value kmax- Provided that sufficiently small k-values can be accessed, a reciprocal-space function such as F(k) will therefore be truncated by a modification function given by M(k) = 1 for k < kmax and M(k) = 0 for k > kmax- lu consequence, the real-space information corresponding to F k) is obtained by the Fourier transform relation... 

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