Fourier convolution theorem method

There are three points to emphasize. First, the expressions for the concentration or concentration gradient distribution for non-sector-shaped centerpieces can be applied to other methods for obtaining MWD s, such as the Fourier convolution theorem method (JO, 15, 16), or to more recent methods developed by Gehatia and Wiff (38-40). The second point is that the method for the nonideal correction is general. Since these corrections are applied to the basic sedimentation equilibrium equation, the treatment is universal. The corrected sedimentation equilibrium equation (see Equation 78 or 83) forms the basis for any treatment of MWD s. Third, the Laplace transform method described here and elsewhere (11, 12) is not restricted to the three examples presented here. For those cases where the plots of F(n, u) vs. u will not fit the three cases described in Table I, it should still be possible to obtain an analytical expression for F(n, u) which is different from those in Table I. This expression for F (n, u) could then be used to obtain an equation in s using procedures described in the text (see Equations 39 and 44). Equation 39 would then be used to obtain the desired Laplace transform. 

The derivation of the electrostatic properties from the multipole coefficients given below follows the method of Su and Coppens (1992). It employs the Fourier convolution theorem used by Epstein and Swanton (1982) to evaluate the electric field gradient at the atomic nuclei. A direct-space method based on the Laplace expansion of 1/ RP — r has been described by Bentley (1981). 

Normally, discrete convolution involves shifting, adding, and multiplying —a laborious and time-consuming process, even in a large digital computer. The convolution theorem presents us with an alternative. It reveals the possibility of computing in the Fourier domain. What are the trade-offs between the two methods ... 

The entire discussion of relaxation methods was conducted without examining Fourier space consequences. Van Cittert s method is easy to study this way and has been treated by Burger and Van Cittert (1933), Bracewell and Roberts (1954), Sakai (1962), and Frieden (1975). By applying the convolution theorem to Eq. (14), we may write... 

To determine the function x(t), it is necessary to find the solution of the integral of Eq. (3.6) for a known h(t) and measured y(t) functions. One proposed method of calculating x(t) assumes that the shape of the function x(t) is known beforehand and that the transfer function h(t) can be experimentally determined.158 A more general solution uses Fourier transforms. If Eq. (3.6) is rewritten with the Fourier transforms of the functions x(t), y(t), and h(t - t), which are denoted by Y, X, and H, then, using the convolution theorem ... 

Fourier filters can be related to linear methods in Section 3.3 by an important principle called the convolution theorem as discussed in Section 3.5.3. 

As in the case of the waveform inversion method, we note that by using the convolution theorem and a reciprocity, we can write the expressions for the inverse Fourier transform (r,F) of the auxiliary field u (r,a ) as follows ... 

The second is that propagation effects must be dealt with if sample dimensions are an appreciable fraction of a wavelength, and this situation is not readily avoided at frequencies for which the method is otherwise useful. Both problems are better handled by use of onesided Fourier (Laplace) transforms, rather than direct time domain solutions, as a result of the convolution theorem for the former, and solution of the field equations in the frequency domain for the latter. 

Both methods for time domain (Fourier) and frequency domain (direct) filters are equivalent and are related by the convolution theorem, and both have similar aims, to improve the quality of spectroscopic or chromatographic or time series data. Two functions, /"and g, are said to be convoluted to give h, if... 

An alternative method for calculating the time correlation function, especially useful when its spectrum is also required, involves the fast Fourier transformation (FFT) algorithm and is based on the convolution theorem, which is a general property of the Fourier transformation. According to the convolution theorem, the Fourier transform of the correlation function C equals the product of the Fourier transforms of the correlated functions ... 

The FID obtained from the pulse method contains all the NMR data of a sample. Fourier transformation not only enables the transformation from the time domain, s t), to the frequency domain, 5(o)) or 5(v) but can also pretreat the time domain. The unnecessary data in the time sequence, such as noise, can be eliminated or trimmed before the transformation process. This would provide greater clarity of presentation and economy in labor. The pretransformation is carried out mathematically by the convolution theorem as follows Let r(t) be the function to pretreat the data function s(t). The convolution integral of the two functions is defined by... 

Convolution using Fourier Methods - The Convolution Theorem... 

This property is readily established from the definition of Fourier transform and convolution. In scattering theory this theorem is the basis of methods for the separation of (particle) size from distortions (Stokes [27], Warren-Averbach [28,29] lattice distortion, Ruland [30-34] misorientation of anisotropic structural entities) of the scattering pattern. 

The convolution and correlation theorems allow us to determine those functions using Fourier methods instead of direct integration. Before we discuss the two topics further, we choose to write all the key formulas here - in one place - so we can compare them easily. 

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