Fourier back-transformation

Fig. 5 XANES region, -weighted Fourier transformed of the raw EXAFS functions and the corresponding first shell filtered, Fourier back transform (a, b and c, respectively) of TS-1 activated at 400 °C (full lines), after interaction with water (wet sample, dashed lines) and after interaction with NH3 (Pnh3 = 50 Torr, dotted lines). Adapted from [64] with permission. Copyright (2002) by the ACS...

Linear absorption factor Fourier transform -dimensional Fourier transform -dimensional Fourier back-transform Polarization factor... 

Visualizing a Set of Pure Distortion Profiles. After Fourier back-transformation, we retrieve a set of reduced profiles that are only determined by lattice-distortion... 

Complete information about the specimen would be available only by tomographic methods with a stepwise rotation of the sample (see e.g. Schroer, 2006) or using inherent symmetry properties of the sample. Under the assumption of fibre symmetry of the stretched specimen around the tensile axis, from the slices through the squared FT-structure the three-dimensional squared FT-structure in reciprocal space can be reconstructed and hence also the projection of the squared FT-structure in reciprocal space. The Fourier back-transformation of the latter delivers slices through the autocorrelation function of the initial structure. Stribeck pointed out that the chord distribution function (CDF) as Laplace transform of the autocorrelation function can be computed from the scattering intensity l(s) simply by multiplying I(s) by the factor L(s) = prior to the Fourier back-... 

It is essential to note that / and / are two different functions and not merely the same function depending on two different variables. For the sake of simplicity this distinction is not always reflected by the notation however, we will explicitly distinguish these two functions by the /-notation in this appendix. Furthermore, since all integrals in this appendix extend over the whole real line it is convenient to not explicitly write down the limits of integration, which has been done in the second step of Eq. (E.2). Given the transformed function / it is always possible to extract the original function / by a so-called Fourier back transformation (FBT) defined by... 

With the Fourier difference method, data obtained from single-crystal neutron diffraction provide a full view of the probability density of the H(D) atoms. For this purpose, once the crystal structure has been determined, Bragg peak intensities can be calculated for an ideal crystal in which the scattering cross-section of the H(D) atoms of the methyl groups is set to zero. The difference from the original pattern contains specific information on the methyl H(D) atoms. Further Fourier back-transformation gives the probability density distribution in direct space (see Figure 8.16). 

The theorems of Fourier transformation tell us that this Fourier back transform generates the autocorrelation function (r) of the density distribution p(r) ... 

A model-free approach to analyze the complete SAXS of a dilute system of particles is Clatter s indirect transformation method. The isotropic form of the forward Fourier transform corresponding to the Fourier back transform [5] can be written as... 

Another restriction we may often wish to place on the laser pulse is to limit the frequency range of the electric held in the pulse. One method that has been used to accomphsh this is simply to eliminate frequency components of the held that lie outside a specihed range [63]. Another possibility is to use a frequency hlter, such as the twentieth-order Butterworth bandpass hlter [64], which is a smoother way of imposing basically the same restrictions [41, 42]. In order to impose such restrictions on the frequency content of the pulse, the time-dependent electric held of the laser pulse must be Fourier transformed so as to obtain its frequency spectrum. After the frequency spectrum of the laser pulse has been passed through the hlter, it is back transformed to yield back a... 

Figure 9. Data reduction and data analysis in EXAFS spectroscopy. (A) EXAFS spectrum x(k) versus k after background removal. (B) The solid curve is the weighted EXAFS spectrum k3x(k) versus k (after multiplying (k) by k3). The dashed curve represents an attempt to fit the data with a two-distance model by the curve-fitting (CF) technique. (C) Fourier transformation (FT) of the weighted EXAFS spectrum in momentum (k) space into the radial distribution function p3(r ) versus r in distance space. The dashed curve is the window function used to filter the major peak in Fourier filtering (FF). (D) Fourier-filtered EXAFS spectrum k3x (k) versus k (solid curve) of the major peak in (C) after back-transforming into k space. The dashed curve attempts to fit the filtered data with a single-distance model. (From Ref. 25, with permission.)...

Such a function exhibits peaks (Fig. 9C) that correspond to interatomic distances but are shifted to smaller values (recall the distance correction mentioned above). This finding was a major breakthrough in the analysis of EXAFS data since it allowed ready visualization. However, because of the shift to shorter distances and the effects of truncation, such an approach is generally not employed for accurate distance determination. This approach, however, allows for the use of Fourier filtering techniques which make possible the isolation of individual coordination shells (the dashed line in Fig. 9C represents a Fourier filtering window that isolates the first coordination shell). After Fourier filtering, the data is back-transformed to k space (Fig. 9D), where it is fitted for amplitude and phase. The basic principle behind the curve-fitting analysis is to employ a parameterized function that will model the... 

It should be mentioned that when a peak from a Fourier transform is filtered and back-transformed to k space, the envelope represents the backscattering amplitude for the near neighbor involved. 

Figure 25. EXAFS data for K3[Fe(CN)6] (A) k2-weighted EXAFS (B) Fourier transform of (A) showing Fe—C and Fe— N peaks (C) Fourier-filtered back-transformation of the Fe—C peak. (From Ref. 97, with permission.)...

Desmearing. In practice, there are two pathways to desmear the measured image. The first is a simple result of the convolution theorem (cf. Sect. 2.7.8) which permits to carry out desmearing by means of Fourier transform, division and back-transformation (Stokes [27])... 

Figure 9.5 EXAFS of Rh/AKO, catalysts after reduction at 200 °C (left) and 400 °C (right) top the magnitude of the Fourier transform of the measured EXAFS signal, bottom the back transformed EXAFS corresponding to distances from Rh atoms of between 0.8 and 3.2 nm. The lower Fourier transform contains a dominant contribution from Rh nearest neighbors at 0.27 nm and a minor contribution from oxygen neighbors in the metal-support interface. After correction for the Rh-O phase shift, the oxygen ions are at a distance of 0.27 nm (from Koningsberger et at. 119]).

In order to restria attention to a smgle shell of scatterers, one selects a limited range of the R-space data for back-transformation to k-space, as illustrated m Figure 3B,C. In Ae ideA case, this procedure allows one to anAyze each shell separately, AAough in practice many shells cannot be adequately separated by Fourier tering (9). 

Figure 3. Continued. C) Back-transform (Fourier filter) of data. Upper trace corresponds...

The calculation thus consists of three steps (1) calculating the scattering factors of the analytical charge density functions (see appendix G for closed-form expressions), (2) Fourier transformation of the electrostatic operator, and (3) back transformation of the product of two Fourier transforms. 

Returning to the visual transforms of Figs. 2.7-2.10 each object (the sphere in Fig. 2.7, for instance) is the Fourier transform (the back-transform, if you wish) of its diffraction pattern. If we build a model that looks like the diffraction pattern on the right, and then obtain its diffraction pattern, we get an image of the object on the left. 

Plate 8 Relative amounts of information contained in reflection intensities and phases. The Fourier transforms of duck and cat are shown in (a) and ( >). In (c), the intensity or shading of the duck transform is combined with the phases or colors of the cat transform. The back-transform of (c), shown in (d), contains a clear image of the cat. Ironically, the phases, which are so difficult to obtain, provide more information than the easily obtained intensities. (For discussion, see Chapter 6.) Figure generously provided by Dr. Kevin Cowtan. 

The advantage of the transformed objects over the original ones in data reduction schemes lies in the order induced in the sequence of coefficients. This order is correlated with frequency while in original data the information is more or less uniformly distributed over all the sequence, in transformed object the first few low-frequency coefficients contain the information about the rough contours of the original object and the high-frequency coefficients describe the details. In both Fourier and Hadamard transforms the most important part of the information can be retained after back-transformation with the proper choice of coefficients. 

The amount of information is gradually withdrawn from the pattern as the number of coefficients for the back-transformation is reduced. The zero order coefficients, co of both transforms, Fourier and Hadamard, carry the sum (integral) of all elements of the original representation of the object and do not contribute any other information (ref. 5). Figure 5.2 shows the order of importance of other coefficients for both transformations. 

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