Fourier algorithm

As in the case of infrared, progress in computing and the development of powerful algorithms for Fourier transforms has made the development of pulse NMR possible. 

Low and High frequency can be restored by use of a deconvolution algorithm that enhances the resolution. We operate an improvement of the spectral bandwidth by Papoulis deconvolution based essentially on a non-linear adaptive extrapolation of the Fourier domain. 

One of the most efficient algorithms known for evaluating the Ewald sum is the Particle-mesh Ewald (PME) method of Darden et al. [8, 9]. The use of Ewald s trick of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fourier space can be adjusted within certain limits via a free parameter in the method, but one is still left with two distinct calculations. PME performs the real-space calculation in the conventional manner, evaluating the complementary error function within a cutoff... 

Furthermore, one may need to employ data transformation. For example, sometimes it might be a good idea to use the logarithms of variables instead of the variables themselves. Alternatively, one may take the square roots, or, in contrast, raise variables to the nth power. However, genuine data transformation techniques involve far more sophisticated algorithms. As examples, we shall later consider Fast Fourier Transform (FFT), Wavelet Transform and Singular Value Decomposition (SVD). 

Fast Fourier Transformation is widely used in many fields of science, among them chemoractrics. The Fast Fourier Transformation (FFT) algorithm transforms the data from the "wavelength" domain into the "frequency" domain. The method is almost compulsorily used in spectral analysis, e, g., when near-infrared spectroscopy data arc employed as independent variables. Next, the spectral model is built between the responses and the Fourier coefficients of the transformation, which substitute the original Y-matrix. 

Cooley J W and ] W Tukey 1965. An Algorithm for the Machine Calculation of Complex Fourier Series Aiathemalics of Computation 19 297-301. 

Obviously, the theory outhned above can be applied to two- and three-dimensional systems. In the case of a two-dimensional system the Fourier transforms of the two-particle function coefficients are carried out by using an algorithm, developed by Lado [85], that preserves orthogonality. A monolayer of adsorbed colloidal particles, having a continuous distribution of diameters, has been investigated by Lado. Specific calculations have been carried out for the system with the Schulz distribution [86]... 

A numerical algorithm for the solution of the system of Eqs. (15), (19) and (51) consists of the expansion of the two-particle functions into a Fourier-Bessel series. We omit all the details of the numerical method they can be found in Refs. 55-58, 85, 86. In Fig. 3 we show a comparison of the total... 

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

When performing optical simulations of laser beam propagation, using either the modal representation presented before, or fast Fourier transform algorithms, the available number of modes, or complex exponentials, is not inhnite, and this imposes a frequency cutoff in the simulations. All defects with frequencies larger than this cutoff frequency are not represented in the simulations, and their effects must be represented by scalar parameters. 

Omitting more details on this point, we refer the readers to the well-developed algorithm of the fast Fourier transform, in the framework of which Q arithmetic operations, Q fa 2N log. N, N = 2 , are necessary in connection with computations of these sums (instead of 0 N ) in the case of the usual summation), thus causing 0(nilog,- 2) arithmetic operations performed in the numerical solution of the Dirichlet problem (2) in a rectangle. 

In general, the topology of interprocessor communication reflects both the structure of the mathematical algorithms being employed and the way that the wave packet is distributed. For example, our very first implementation of parallel algorithms in a study of planar OH - - CO [47] used fast Fourier transforms (FFTs) to compute the action of 7, which also required all-to-all communication but in a topology that is very different from the simple ring-like structure shown in Fig. 5. 

J. W. Cooley and J.W. Tukey, An algorithm for the machine calculation of complex Fourier series. Math. Comput., 19 (1965) 297-301. 

Exponential decay often occurs in measurements of diffusion and spin-relaxation and both properties are sensitive probes of the electronic and molecular structure and of the dynamics. Such experiments and analysis of the decay as a spectrum of 7i or D, etc., are an analog of the one-dimensional Fourier spectroscopy in that the signal is measured as a function of one variable. The recent development of an efficient algorithm for two-dimensional Laplace inversion enables the two-dimensional spectroscopy using decaying functions to be made. These experiments are analogous to two-dimensional Fourier spectroscopy. 

Bricogne, G. (1993) Fourier transforms in crystallography theory, algorithms, and applications, In International Tables for Crystallography, Vol. B, Reciprocal Space, Shmueli, U. (Ed.), Dordrecht, Kluwer Academic Publishers, Holland, pp. 23-106. 

Obviously this is a little difficult to interpret, although with experience you can train yourself to extract all the frequencies by eye... (only kidding ) The FID is a time domain display but what we really need is a frequency domain display (with peaks rather than cosines). To bring about this magic, we make use of the work of Jean Baptiste Fourier (1768-1830) who was able to relate time-domain to frequency-domain data. These days, there are superfast algorithms to do this and it all happens in the background. It is worth knowing a little about this relationship as we will see later when we discuss some of the tricks that can be used to extract more information from the spectrum. 

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