Windowed Fourier transform

FbwO) is the Fourier transformation of effective beam width as a function of spatial frequency / Fuff) is the MTF of the XRll. Because of the XRll windows curvature, projection data must be transformed to obtain uniform pixel spacing, described by Errors in object centre... 

Figure 8.39 Fourier transformed Fe extended X-ray absorption fine structure (EXAFS) and retransformation, after applying a 0.9-3.5 A filter window, of (a) a rubredoxin, (b) a plant ferredoxin and (c) a bacterial ferredoxin, whose structures are also shown. (Reproduced, with permission, Ifom Teo, B. K. and Joy, D. C. (Eds), EXAFS Spectroscopy, p. 15, Plenum, New York, 1981)...

Transmission Fourier Transform Infrared Spectroscopy. The most straightforward method for the acquisition of in spectra of surface layers is standard transmission spectroscopy (35,36). This approach can only be used for samples which are partially in transparent or which can be diluted with an in transparent medium such as KBr and pressed into a transmissive pellet. The extent to which the in spectral region (typically ca 600 4000 cm ) is available for study depends on the in absorption characteristics of the soHd support material. Transmission ftir spectroscopy is most often used to study surface species on metal oxides. These soHds leave reasonably large spectral windows within which the spectral behavior of the surface species can be viewed. 

Figures Fourier transform (soiid curve), Osir ) versus r (A, without phase-shift correction), of the Mo K-edge EXAFS of Figure 5 for moiybdenum metal foii. The Fourier filtering window (dashed curve) is applied over the region -1.5-4.0 A to isolate the two nearest Mo-Mo peaks.

At the end of the 2D experiment, we will have acquired a set of N FIDs composed of quadrature data points, with N /2 points from channel A and points from channel B, acquired with sequential (alternate) sampling. How the data are processed is critical for a successful outcome. The data processing involves (a) dc (direct current) correction (performed automatically by the instrument software), (b) apodization (window multiplication) of the <2 time-domain data, (c) Fourier transformation and phase correction, (d) window multiplication of the t domain data and phase correction (unless it is a magnitude or a power-mode spectrum, in which case phase correction is not required), (e) complex Fourier transformation in Fu (f) coaddition of real and imaginary data (if phase-sensitive representation is required) to give a magnitude (M) or a power-mode (P) spectrum. Additional steps may be tilting, symmetrization, and calculation of projections. A schematic representation of the steps involved is presented in Fig. 3.5. 

The matrix obtained after the F Fourier transformation and rearrangement of the data set contains a number of spectra. If we look down the columns of these spectra parallel to h, we can see the variation of signal intensities with different evolution periods. Subdivision of the data matrix parallel to gives columns of data containing both the real and the imaginary parts of each spectrum. An equal number of zeros is now added and the data sets subjected to Fourier transformation along I,. This Fourier transformation may be either a Redfield transform, if the h data are acquired alternately (as on the Bruker instruments), or a complex Fourier transform, if the <2 data are collected as simultaneous A and B quadrature pairs (as on the Varian instruments). Window multiplication for may be with the same function as that employed for (e.g., in COSY), or it may be with a different function (e.g., in 2D /-resolved or heteronuclear-shift-correlation experiments). 

Fig, y. Resolution in scale space of (a) window Fourier transform and (b) wavelet transform. 

The growth and decay of all other species (including O3) were monitored by Fourier transform infrared (FT-IR) spectroscopy at a total pathlength of 460 meters and a spectral resolution of 1 cm". At this pathlength, the intense absorptions of H2O and CO limit the usable IR spectral windows to the approximate regions 750-1300, 2000-2300, and 2400-3000 cm". Each spectrum (700-3000 cm" ) was adequately covered by the response of the Cu Ge detector. Approximately 40 seconds were required to collect the 32 interferograms co-added for each spectrum. 

Ifourth(fd, 2 Q) was multiplied with a window function and then converted to a frequency-domain spectrum via Fourier transformation. The window function determined the wavenumber resolution of the transformed spectrum. Figure 6.3c presents the spectrum transformed with a resolution of 6cm as the fwhm. Negative, symmetrically shaped bands are present at 534, 558, 594, 620, and 683 cm in the real part, together with dispersive shaped bands in the imaginary part at the corresponding wavenumbers. The band shapes indicate the phase of the fourth-order field c() to be n. Cosine-like coherence was generated in the five vibrational modes by an impulsive stimulated Raman transition resonant to an electronic excitation. 

In single-scale filtering, basis functions are of a fixed resolution and all basis functions have the same localization in the time-frequency domain. For example, frequency domain filtering relies on basis functions localized in frequency but global in time, as shown in Fig. 7b. Other popular filters, such as those based on a windowed Fourier transform, mean filtering, and exponential smoothing, are localized in both time and frequency, but their resolution is fixed, as shown in Fig. 7c. Single-scale filters are linear because the measured data or basis function coefficients are transformed as their linear sum over a time horizon. A finite time horizon results infinite impulse response (FIR) and an infinite time horizon creates infinite impulse response (HR) filters. A linear filter can be represented as... 

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... 

Fig. 12 C -detected C CSA patterns of the SHPrP109 i22 fibril sample. The upper and lower traces correspond to the experimental and simulated spectra, respectively. Simulations correspond to the evolution of a one-spin system under the ROCSA sequence. The only variables are the chemical shift anisotropy and the asymmetry parameter. A Gaussian window function of 400 Hz was applied to the simulated spectmm before the Fourier transformation. (Figure and caption adapted from [164], Copyright (2007), with permission from Elsevier)...

Fig. 13. 13Ca-1HN planes from the HN(CO)CANH-TROSY (a) and HN(CO)CA-TROSY (b) spectra. Spectra were recorded on uniformly 15N, 13C, 2H enriched, 30.4 kDa protein Cel6A at 800 MHz at 277 K. The data were measured using identical parameters and conditions, using 8 transients per FID, 48, 32, 704 complex points corresponding acquisition times of 8, 12, and 64 ms in tly t2, and <3, respectively. A total acquisition time was 24 h per spectrum. The data were zero-filled to 128 x 128 x 2048 points before Fourier transform and phase-shifted squared sine-bell window functions were applied in all three dimensions. 

Instrumentation. Fourier transform infrared (FUR) spectra were recorded on a Nicolet 5DX using standard techniques. Spectra were measured from various sample supports, including KBR pellets, free polymer films and films cast on NaCl windows. Spectra for quantitative analysis were recorded in the absorbance mode. The height of the 639 cm 1 absorbance was measured after the spectrum was expanded or contracted such that the 829 cm 1 absorbance was a constant height. In some spectra an artifact due to instrumental response appeared near 2300 cm 1. 

FTIR is a natural for HPLC in that it (FTIR) is a technique that has been used mostly for liquids. The speed introduced by the Fourier transform technique allows, as was mentioned for GC, the recording of the complete IR spectrum of mixture components as they elute, thus allowing the IR photograph to be taken and interpreted for qualitative analysis. Of course, the mobile phase and its accompanying absorptions are ever present in such a technique and water must be absent if the NaCl windows are used, but IR holds great potential, at least for nonaqueous systems, as a detector for HPLC in the future. 

J.-H. Jiang, Y. Ozaki, M. Kleimann and H.W. Siesler, Resolution of two-way data from on-line Fourier-transform Raman spectroscopic monitoring of the anionic dispersion polymerization of styrene and 1,3-butadiene by parallel vector analysis (PVA) and window factor analysis (WFA), Chemom. Intell. Lab. Syst., 70, 83-92... 

Figure 3. X-ray absorption data analysis of Fe EXAFS data for Rieske-like Fe S cluster. A) EXAFS data B) Fourier transform of EXAFS data showing peaks for Fe—S and Fe-Fe scattering. Thin vertical lines indicate filter windows for first and second shell.

It is found that multiplication of the Fourier transform of the data by a carefully chosen window function is very effective in removing the artifacts around peaked functions. This process is called apodization. Apodization with the triangular window function is often applied to Fourier transform spectroscopy interferograms to remove the ringing around the infrared... 

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