Fourier-transform data

Data shown as examples in this review were typically acquired as 2K X 128 or 2K x 160 point files. Data were processed with linear prediction or zero-filling prior to the first Fourier transform. Data were uniformly linear predicted to 512 points in the second dimension followed by zero-filling to afford final data matrices that were 2K x IK points. 

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. 

Klisch, Belov, Schieder, Winnewisser and Herbst [157] have combined all of the data for NO to produce a current best set of molecular constants for three isotopomers, presented in table 10.15. The data used, apart from their own terahertz studies, included the A-doubling of Meerts and Dymanus [156, 158], the sub-millimetre transitions of 15N160 and 14NlsO, and Fourier transform data from Salek, Winnewisser and Yamada [159]. These last authors were able to study the magnetic dipole transitions between the two fine-structure states. The values of the spin orbit constant A for the less common isotopomers come from Amiot, Bacis and Guelachvih [160]. 

Data are experimental observations, for example the measurement of a time series of free induction decay prior to Fourier transformation. Data space contains a dataset for each experiment. 

In the 1970s, we were all impressed by the ability of the proton 2D /-resolved technique to provide (after tilting of the twice-Fourier-transformed data matrix) a proton-decoupled proton spectrum comprised of a singlet at each of the 1H chemical shifts, and resolved /-multiplets by taking slices parallel to the F1 axis. These /-multiplets displayed all of the coupling constants for a resonance (no chemical shift) and were remarkable in that they were sharper than the spin multiplets in ID H NMR spectra, due to the refocusing effect of the 180° pulse in the /-resolved sequence. 

Cd(II) plastocyanin does not, nnder the given experimental conditions. This is an illnstration of an application of PAC spectroscopy to the stndy of protem-protem interactions, and it is even possible to estimate the dissociation constant directly from the spectroscopic data. Note that in this example the pertnrbation function and not the Fourier transformed data, is displayed in the figure, as the perturbation function illustrates the effect of dynamics more clearly. Note that the data analysis is almost always carried out on the perturbation function. 

Figure 50.6. MRI signal, two dimensional Fourier transform and grey scale representation of the Fourier transformed data. Left grey scale representation of the single slice MRI of a spherical phantom filled with water. Center spectral representation of the Fourier transformed data. Right image represented in gray scale, the standard viewing method.

An alternative method consists on Fourier-transforming the step function that describes the experimental q range and convolute it with the theoretical r-space functions before comparing or fitting them to the Fourier-transformed data. ... 

With pseudo 2D NMR data consisting of a series of ID profiles, analysis by multivariate techniques is obvious, since the large number of potentially overlapping variables makes visual analysis very difficult and improved methods of analysis are already called for. Analysis of real 2D NMR data by multivariate techniques is less obvious, since a lot of information can already be extracted from the 2D Fourier-transformed data. However, if real 2D NMR data from a series of samples needs to be compared, the application of multivariate techniques is an obvious possibility. 

The spectral resolution of the Fourier transformed data is given by the wavelength corresponding to the maximum optical delay. We have. and therefore At = 1/6, . Therefore, if 0.1 cm spectral... 

Figure 23. (iXAFS results for Pu on tuff. Top The k1-weighted Pu-EXAFS spectra for one Pu-enriched region on the tuff. Bottom the complementary Fourier-transformed data of Pu-EXAFS spectra showing the pseudo-radial distribution of atoms around the Pu at the region examined (from Duff etal. 2001). 

Fourier transformation of the EXAFS data permits it to be replotted as a function of distance (Fig. 9). This aids visualization of the data, since each peak in the Fourier transform in principle represents a shell of atoms. However, because a phase shift of about 0.4 A [1] appears in the Fourier-transformed data, it is not possible to use transformed data to accurately determine M-L distances. Nonetheless, it is possible to use the transformed data to make an initial guess as to the radial distribution function of atoms surrounding the metal. The Fourier transform can also be used to check the background subtraction procedure and the noise level of the spectrum. The presence of peaks of significant intensity at very low R (A) suggests that errors were made in the subtraction process peaks at high R result from especially noisy data. 

The equality of transformations of the mixed time—frequency data and the completely Fourier transformed data is a consequence of Parseval s theorem (5.4) and has been previously discussed [10,11]. It can also be understood by taking into account that the spectral reconstruction is achieved by relating two direct dimensions via an indirect dimension, which in turn is discarded. Whether two frequency domains are correlated via a common time domain or a common frequency domain is therefore equivalent. It is also equivalent to Noda s model of relating two IR wavenumber dimensions via a common perturbation stemming from either time or sample space. From the matrix representation, it can be seen that Noda s synchronous matrix O, Eqs. (5.6) and (5.17) corresponds to the covariance map according to Eq. (5.16), if the data matrices yielding O are composed of... 

Fig. 5. AFM topography (A) and phase (B) images of PS-h-PFS thin films prepared from 1.0 wt% solution of polymer in toluene prior to any annealing. Some limited short-range order is present as indicated by the Fourier transform data shown as an insert in (A).

The Acquisition of Electrochemical Response Spectra hy On-Line East Fourier Transform Data Processing in Electrochemistry, Anal. Chem 48, 221A. 

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