Back transformation

Figure 4.38. Validation data for a RIA kit. (a) The average calibration curve is shown with the LOD and the LOQ if possible, the nearly linear portion is used which offers high sensitivity, (b) Estimate of the attained CVs the CV for the concentrations is tendentially higher than that obtained from QC-sample triplicates because the back transformation adds noise. Compare the CV-vs.-concentration function with the data in Fig. 4.6 (c) Presents the same data as (d), but on a run-by-run basis, (d) The 16 sets of calibration data were used to estimate the concentrations ( back-calculation ) the large variability at 0.1 pg/ml is due to the assumption of LOD =0.1.

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... 

Is as follows Apply the Pt-Pt phase shift (derived from Pt metal) to the 1st forward transform of Pt02 This will partially smear the Pt-0 peak. Then take the back transform of this smeared Pt-0 peak. Extract a new phase shift from this back transform using the known Pt-0 distance, 2.07 X. This phase shift can now be used on the catalysts to focus the Pt-0 peak region. 

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...

In Section 40.3.5 we concluded that the resolution (Av) in the frequency spectrum is equal to the reciprocal of the measurement time. The longer the measurement time in the time domain, the better the resolution is in the frequency domain. The opposite is also true the longer the measurement time in the frequency domain (e.g. in FTIR or FT NMR), the better is the separation of the peaks in the spectrum after the back-transform to the wavelength or chemical shift domain. 

Fig. 40.14. Effect of zero filling on the back transform of the pulse NMR signal given in Fig. 40.12. (a) before zero filling, (b) after zero filling.

Back-transform F(v) by which the undamaged signal jc) is estimated. The effect of deconvolution applied on a noise-free Gaussian peak is shown in... 

As a result of analytical measurements, signals are obtained and, in the case of instrumental measurements, signals functions, y = f(z). The record of the signal intensity as a function of the signal position, Fig. 3.8, represents a two-dimensional signal function which can be back-transformed into two-dimensional analytical information, x = /(Q). 

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.)...

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

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])... 

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... 

We have used the above analysis scheme for all single- and two-surface calculations. Thus, when the wave function is represented in polar coordinates, we have mapped the wave function, 4,ad(, t) to Tatime step to use in Eq. (17) and as the two surface calculations are performed in the diabatic representation, the wave function matrix is back transformed to the adiabatic representation in each time step as... 

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]).

After the calculation in the second rotating frame, a back transformation, in general, is needed to return to the conventional rotating frame... 

Then the scores have to be back-transformed to the original space, yielding an estimation j of x that is obtained using the / th PC A scores ... 

They are obtained by back-transformation of the normal coordinates (inverse of Eq. 3.8) ... 

Figure 21. Experimental Pt L3 EXAES data for a Pt/C electrode at 0.5 V vs RHE (solid line) and the back transformed Eourier filtered AXAES signal (dotted line). Eourier filtering parameters 0.5 < k < 8.5 A and 0.15 < r < 1.7 A.4 (Reproduced with permission from ref 47. Copyright 1998 Elsevier Sequoia S.A., Lausanne.)...

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...

As such, solving the reduced problem of Eq. (28), and back-transforming, ensures that the update, 5x, satisfies all the necessary conditions. 

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