Fourier filter window

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.

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... 

Figure 3A. Phase uncorrected radial distribution functions (solid line) and fourier filter window (dashed line) for bulk [Ru(v-bpy)3]+2.

Figure 4. Fourier transforms (solid curves) [s(t) vs. r, A (before phase shift correction)] of the background-subtracted EXAFS spectra in Figure 3 and Fourier-filtering windows (dashed curves). Key A, cis-Pt(NHs) Cl2 B, Pt(2-A-6-MPR)t C, Pt(6-MPR)2 Z>, Pd(6-MPR)2 E, Pd(2-A-6-MPR)2 and F, Pd(Guo)2Cl2.

Figure 8.39 shows some results of EXAFS following absorption by iron atoms in proteins with three prototype iron-sulphur active sites. In the example in Figure 8.39(a) application of a 0.9-3.5 A filter window before Fourier retransformation shows a single wave resulting... 

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)...

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.)...

Figure 7.19 Ni K-edge X-ray absorption spectra for as-isoiated (solid line) and NADH- and H2-reduced (dashed line) samples of the hydrogenase from R. eutropha HI6. (a) Edge region (b) Fourier-filtered EXAFS (backtransform window = I.I-2.6 A). Reprinted with permission from Gu, eta/. (1996) and the American Chemical Society.

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.

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. 

Figure 4. First coordination sphere (backtransform window = 1.1-2.7 A) Fourier-filtered Ni K-edge EXAFS spectra from redox-poised Thiocapsa ro-...

For the purposes of this study, the first two major maxima in the Fourier transform, corresponding to the M-N (M=Pt or Pd) and M-S or M-Cl distances, were Fourier filtered and back-transformed into k-space. The filtering windows are shown as dashed curves in Figure 4 and the back-transformed, filtered data in k-space are shown as dashed curves in Figure 3. 

Fig. 14. Fourier transform of the k k) spectrum shown in Fig. 12. The Fourier transform is resolved into three peaks. The dashed lines indicate the window used to generate the Fourier-filtered second peak, whose inverse Fourier transform is shown in Fig. 15. The apparent distance is shorter than the actual distance to a given neighboring atom because of the contribution of the phase shift to the frequency of the EXAFS wave, as explained in the text.

Pig. 3. Comparison of the Ar 2p- and Is-EXAFS (obtained at (N) = 750 (top) and N) = 400 (bottom), respectively), where both experimental and calculated EXAFS-signals are shown. The dashed lines represent the experimental data while the superimposed solid lines are the first-shell component, Fourier filtered over 2.2 < A < 4.4 A using a Hanning apodization window (see Refs. 24 and 25 for further details). 

Fig. 1. Simulations of the k -weighted k-space Fe EXAFS data from the Photosystem I core protein containing Fx (solid line -experimental dotted line - simulation). After background removal and weighting by k, data from k=3 to k=12 A-i were Fourier filtered with window limits at R =0.5 and R =3.3 A. The similation shown was performed by the method of Teo and Lee using two shells. The parameters for simulation (a) minic a [4Fe-4S] center and employ 4 S atoms at 2.27 A with a Debye-Wdler disorder parameter of 0.075 A and 3 Fe neighbors at 2.7 A with a disorder parameter of 0.1 A. The parameters for simulation (b) mimic a [2Fe-2S] center and employ 4 S atoms at 2.26 A with a Debye-Wdler disorder parameter of 0.08 A and 1 Fe neighbor at 2.7 A with a disorder parameter of 0.07 A.

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... 

Thus the data can be preprocessed by Fourier transforming V( ) 2, applying a window corresponding to the extent of the pupil function of the lens, and then inverse transforming to obtain a filtered V(m), which can then be used as the data for the Gerchberg-Saxton algorithm. 

Equation (9.5) can be viewed as first a modulation of the window to frequency cq thus producing a bandpass filter w(n)eJ an followed by a filtering of x(n) through this bandpass filter. The output is then demodulated back down to baseband. The temporal output of the filter bank can be interpreted as discrete sine waves that are both amplitude- and phase-modulated by the time-dependent Fourier transform. 

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