Curve fitting, Fourier transforms

Fourier domain fitting. The Fourier transform of the experimental elution curve is calculated. The parameters a and 3 are then determined using a fitting procedure in the Fourier domain that is equivalent to a least-squares criterion in the time domain. With Fourier domain estimation, model parameters are chosen to minimize the difference between the Fourier transforms of experimental and theoretical elution curves. The Fourier transform of a bounded, time varying response curve, f(t), is defined as... 

The details of the lamellar morphology, such as crystalline layer thickness l and amorphous layer thickness /, are quantitatively evaluated using SAXS [18]. For example, they are conveniently derived from the one-dimensional correlation function, that is, Fourier transform of SAXS curves, assuming an ideal lamellar morphology without any distribution for and l [19]. More detailed information on the lamellar morphology can be obtained by fitting theoretical scattering curves (or theoretical one-dimensional correlation functions) calculated from some appropriate model to SAXS curves (or Fourier transform of SAXS curves) experimentally obtained. The Hosemann model in reciprocal space [20] and the Vonk model in real space [7,21] are often employed for such purposes. 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

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... 

The data collected are subjected to Fourier transformation yielding a peak at the frequency of each sine wave component in the EXAFS. The sine wave frequencies are proportional to the absorber-scatterer (a-s) distance /7IS. Each peak in the display represents a particular shell of atoms. To answer the question of how many of what kind of atom, one must do curve fitting. This requires a reliance on chemical intuition, experience, and adherence to reasonable chemical bond distances expected for the molecule under study. In practice, two methods are used to determine what the back-scattered EXAFS data for a given system should look like. The first, an empirical method, compares the unknown system to known models the second, a theoretical method, calculates the expected behavior of the a-s pair. The empirical method depends on having information on a suitable model, whereas the theoretical method is dependent on having good wave function descriptions of both absorber and scatterer. 

By Fourier transforming the EXAFS oscillations, a radial structure function is obtained (2U). The peaks in the Fourier transform correspond to the different coordination shells and the position of these peaks gives the absorber-scatterer distances, but shifted to lower values due to the effect of the phase shift. The height of the peaks is related to the coordination number and to thermal (Debye-Waller smearing), as well as static disorder, and for systems, which contain only one kind of atoms at a given distance, the Fourier transform method may give reliable information on the local environment. However, for more accurate determinations of the coordination number N and the bond distance R, a more sophisticated curve-fitting analysis is required. 

In practice, the phase shift and the modulation ratio M are measured as a function of co. Curve fitting of the relevant plots (Figure 6.6) is performed using the theoretical expressions of the sine and cosine Fourier transforms of the b-pulse response and Eqs (6.23) and (6.24). In contrast to pulse Jluorometry, no deconvolution is required. 

Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

Table 10.2 Curve-fitting results of Fourier transformed EXAFS spectra (16 K) at V K-edge for a V precursor (L-leucine), its fresh supported V complex (3.4% V) and that treated with 2-naphthol the coordination number of V=0 was fixed as unity.

The simplest case, that of two large infrared lines, is shown in Fig. 31(a). A smooth curve was fitted to the base line as shown. A spline-fitting computer program developed by De Boor (1978) was used to obtain this fit very conveniently. After the fit was obtained, the data were adjusted to a flat base line, as shown in Fig. 31(b), and the data field was extended by padding with zeros to yield an overall data field of 28 = 256 points. Taking the Fourier transform, we obtained the interferogram function shown in Fig. 32. (Even though it was not obtained directly from the interferometer as recorded... 

Vanadium K-edge XANES measured in transmission mode at 15 K showed that the attached V complex (2) maintained its square pyramidal conformation with a V=0 bond (Figure 2.2a). Curve-fitting analysis of V K-edge EXAFS Fourier transforms (Figure 2.2b) provided local structure information on the supported V complex (2) with an unsaturated conformation, which differs from that of the V-monomer precursor (1). The EXAFS curve-fitting was performed in the R-space with two shells short V=0 and long V—O bonds. A V=0 bond was observed at 0.157 0.001 nm,... 

These maxima in the Fourier transform data, which correspond to the different chromium coordination shells, were isolated using a filter window function. The inverse transform of each peak was generated and fitted using a non-linear least squares program. The amplitude and phase functions were obtained from the theoretical curves reported by Teo and Lee (2 ). The parameters which were refined included a scale factor, the Debye-Waller factor, the interatomic distance, and the threshold energy difference. This process led to refined distances of 1.97(2) and 2.73(2) A which were attributed to Cr-0 and Cr-Cr distances, respectively. Our inability to resolve second nearest neighbor Cr-Cr distances may be a consequence of the limited domain size of the pillars. 

Mathematically, geometric parameters can be described by using the Fourier Series in polar coordinates (p,9). Thus, given a set of boundary points (x, y) from an object of interest, they can be transformed into the polar coordinates with respect to its geometric center (x, y). A curve fitting technique in polar coordinates can be used to fit this set of points into a Fourier Series such that any point p(0) on this boundary can be expressed by... 

Figure 16. Concept of using a Fourier Series to represent the boundary of an object. The (x,y) coordinates of the boundary points of an object is transformed to polar coordinates. Each point on the boundary p(0) can be expressed by a Fourier Series obtained from curve fitting of the boundary points. (X,Y) are the coordinates of the center of gravity.

The signal intensity at the start of the FID is proportional to the peak height after Fourier transformation, so we could make a plot of peak height versus x delay and fit the exponential decay to a theoretical curve to measure the T2 value. This is the T2 equivalent of the inversion-recovery experiment (Section 5.8) for measurement of T. ... 

This method gained a significant improvement with the introduction of the contemporary infrared technique with a Fourier transformer (FT-IR), permitting to obtain measurable values of adsorption of the infrared light even from single black foam films. The thickness of the aqueous core is derived from the adsorption at 3400 cm 1 which is related to the OH stretching vibration of the water molecules. Umemura et al. [114] have employed the polarised Fourier transformed infrared spectra for the study of the water content of NaDoS black films. The cell used to form films of ca. 2 cm2 area is illustrated in Fig. 2.19. By fitting the calculated curved of polarised FT-IR spectra to the respective experimentally obtained... 

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