Data analysis Fourier transformation

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.

The Analysis Toolpak of Excel already includes a Fourier transform function, available through Tools => Data Analysis => Fourier Analysis. Unfortunately that function suffers from three serious limitations (1) it only accepts real inputs, (2) it does not properly scale its output, and (3) it generates its output in the form of labels, which need to be extracted using the = IMREAL() and = IMAGINARY)) functions before they canbe plotted or otherwise used in subsequent calculations. Although it is possible to work around those limitations, it is far easier to avoid them by starting afresh, and to include frequency and time scales at the same time. This is what we have done here. 

There are two main methods of analysis for the EXAFS data the Fourier transform and the curve fitting. The first method gives a radial distribution function Q(r) versus interatomic distances (Fig. 9) ... 

Slow ly-varying tails of the pair correlation function contribute to FXAI- S data only at low k values. Sharp peaks in the pair correlation function. however, give rise to dominant features in the EXAFS signal which persist to high k values. As the data are Fourier transformed only in a liniie range and the low k data of the FXAI- S signal must he omitted in the Fourier transform. Ihe broad tail in the atom pair correlation function is ol ien lost in the analysis ol ihe li.XAF S dala. A... 

In order to test the instrument and the data analysis procedures we made a number of experiments on dummy cells, for example the one shown in Fig.6. Here we made use of the possibility of time-domain averaging, the ac excitation was repeated 8 times and the data were averaged. The dummy cell was enclosed in a Faraday cage and the noise level was quite low, reducing the need for frequency-domain averaging, i.e., averaging of admittance data, after Fourier transformation and calculation of the admittance. 

Computer-Aided Experiments Computers have become common tools for the analysis of NMR data as Fourier transform spectrometers have replaced CW instruments. These computers perform the basic processing of data, including digital filtering, Fourier transformation, and phase correction. 

Fast Fourier Transformation is widely used in many fields of science, among them chemoractrics. The Fast Fourier Transformation (FFT) algorithm transforms the data from the "wavelength" domain into the "frequency" domain. The method is almost compulsorily used in spectral analysis, e, g., when near-infrared spectroscopy data arc employed as independent variables. Next, the spectral model is built between the responses and the Fourier coefficients of the transformation, which substitute the original Y-matrix. 

Most of the early vibration analysis was carried out using analog equipment, which necessitated the use of time-domain data. The reason for this is that it was difficult to convert time-domain data to frequency-domain data. Therefore, frequency-domain capability was not available until microprocessor-based analyzers incorporated a straightforward method (i.e.. Fast Fourier Transform, FFT) of transforming the time-domain spectmm into its frequency components. 

The measurement of the electrode impedance has also been ealled Faradaie impedanee method. Since measurements are possible by applying either an electrode potential modulated by an AC voltage of discrete frequeney (which is varied subsequently) or by applying a mix of frequencies (pink noise, white noise) followed by Fourier transform analysis, the former method is sometimes called AC impedance method. The optimization of this method for the use with ultramicroelectrodes has been described [91Barl]. (Data obtained with these methods are labelled IP.)... 

Lam, R. B., Wieboldt, R. C., and Isenhour, T. L., Practical Computation with Fourier Transforms for Data Analysis, Anal. Chem. 53, 1981, 889A-901A. 

The combination of PCA and LDA is often applied, in particular for ill-posed data (data where the number of variables exceeds the number of objects), e.g. Ref. [46], One first extracts a certain number of principal components, deleting the higher-order ones and thereby reducing to some degree the noise and then carries out the LDA. One should however be careful not to eliminate too many PCs, since in this way information important for the discrimination might be lost. A method in which both are merged in one step and which sometimes yields better results than the two-step procedure is reflected discriminant analysis. The Fourier transform is also sometimes used [14], and this is also the case for the wavelet transform (see Chapter 40) [13,16]. In that case, the information is included in the first few Fourier coefficients or in a restricted number of wavelet coefficients. 

Another advantage of frequency response analysis is that one can identify the process transfer function with experimental data. With either a frequency response experiment or a pulse experiment with proper Fourier transform, one can construct the Bode plot using the open-loop transfer functions and use the plot as the basis for controller design.1... 

IR reflectance allows the in situ analysis of the electrode-electrolyte interface [8, 9], The Fourier transform variant adds to this technique the advantage of very fast data collection [10],... 

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... 

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