Fourier transform filtered data

One solution to the problem is to increase the ionization probability. This can be done by choosing primary ions with heavy mass, for example, Bi+ or even Ccarbon atoms. The noise level can also be reduced by techniques of digital image processing. For example, a fast Fourier transform technique has been used to remove noise from the image. This technique transforms an image from a space domain to a reciprocal domain by sine and cosine functions. Noise can be readily filtered out in such domain. After a reverse Fourier transform, filtered data produces an image with much less noise. 

This filtering preprocessing method can be used whenever the variables are expressed as a continuous physical property. One example is dispersive or Fourier-Transform spectral data, where the spectral variables refer to a continuous series of wavelength or wavenumber values. In these cases, derivatives can serve a dual purpose (I) they can remove baseline offset variations between samples, and (2) they can improve the resolution of overlapped spectral features. 

Let the discrete spectrum, which consists of the coefficients of u(k) and v(k), be denoted by U(n) and V(n), respectively. The low-frequency spectral components U(n) are most often given by the most noise-free Fourier spectral components that have undergone inverse filtering. For these cases V(n) would then be the restored spectrum. However, for Fourier transform spectroscopy data, U(n) would be the finite number of samples that make up the interferogram. For these cases V(n) would then represent the interferogram extension. 

Fig. 6. Elimination of Ka3 4-satellites in the C lr-spectrum of paraffin. The raw data in Fig. 6 A are transformed with the help of the Fourier transform filtering procedure into the satellite-free spectrum in Fig. 6 B. For details see example in Fig. 7...

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. 

After applying filter functions, it is normal to Fourier transform the data. Sometimes for data recorded directly in the frequency domain, it is possible to perform Fourier self-deconvolution. This involves inverse transforming the spectrum or chromatogram back into a time domain, applying the filters, and then forward transforming to the frequency domain as illustrated in Figure 4. 

In Figure 13, a typical high resolution contact mode SFM scan of uniax-ially oriented POM is shown together with the corresponding 2-D fast Fourier transform filtered image (139). Since the polymer chain direction is in this particular case known a priori, the observed periodicities can be related to the well-established hexagonal crystal structure of POM in a straightforward manner. Uniaxially oriented or epitaxially crystallized specimens thus help in the analysis of the data, as has been discussed in recent review articles (140,141). 

Most modern infrared and NMR spectrometers collect data in the time domain with interferometers, and then the data are transformed to the familiar frequency domain by the Fourier transform. Filtering and signal enhancement in the Fourier domain before transformation is often an attractive approach to signal processing. 

In fig. 2 an ideal profile across a pipe is simulated. The unsharpness of the exposure rounds the edges. To detect these edges normally a differentiation is used. Edges are extrema in the second derivative. But a twofold numerical differentiation reduces the signal to noise ratio (SNR) of experimental data considerably. To avoid this a special filter procedure is used as known from Computerised Tomography (CT) /4/. This filter based on Fast Fourier transforms (1 dimensional FFT s) calculates a function like a second derivative based on the first derivative of the profile P (r) ... 

The solution is given by applying a linear filter f to the data and the Fourier transform of the solution writes ... 

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.)...

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.)...

By means of this procedure our problem is not only reduced from three to two dimensions, but also is the statistical noise in the scattering data considerably reduced. Multiplication by —4ns2 is equivalent to the 2D Laplacian89 in physical space. It is applied for the purpose of edge enhancement. Thereafter the 2D background is eliminated by spatial frequency filtering, and an interference function G(s 2,s ) is finally received. The process is demonstrated in Fig. 8.27. 2D Fourier transform of the interference function... 

The Fourier transforms were performed in the standard way. No smoothing nor filtering was employed. Subtraction of the data from the least squares fit removes the constant or linear term characterizing a Markovian process. Fourier transform of the differences from the linear fit suppresses the enhancement of both the power and amplitude spectra at low frequencies. 

The Co/Mo = 0.125 catalyst has all the cobalt atoms present as Co-Mo-S and, therefore, the EXAFS studies of this catalyst can give information about the molybdenum atoms in the Co-Mo-S structure. The Fourier transform (Figure 2c) of the Mo EXAFS of the above catalyst shows the presence of two distinct backscatterer peaks. A fit of the Fourier filtered EXAFS data using the phase and amplitude functions obtained for well-crystallized MoS2 shows (Table II) that the Mo-S and Mo-Mo bond lengths in the catalyst are identical (within 0.01 A) to those present in MoS2 (R =... 

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.

Fig. 4. Two-dimensional (2D) spectra of cyclo(Pro-Gly), 10 mM in 70/30 volume/volume DMSO/H2O mixture at CLio/27r = 500 MHz and T = 263 K. (A) TCX SY, t = 55 ms. (B) NOESY, Tm = 300 ms. (C) ROESY, = 300 ms, B, = 5 kHz. (D) T-ROESY, Tin = 300 ms, Bi = 10 kHz. Contours are plotted in the exponential mode with the increment of 1.41. Thus, a peak doubles its intensity every two contours. All spectra are recorded with 1024 data points, 8 scans per ti increment, 512 fi increments repetition time was 1.3 s and 90 = 8 ps 512x512 time domain data set was zero filled up to 1024 x 1024 data points, filtered by Lorentz to Gauss transformation in u>2 domain (GB = 0.03 LB = -3) and 80° skewed sin" in u), yielding a 2D Fourier transformation 1024 x 1024 data points real spectrum. (Continued on subsequent pages)...

We shall end this chapter with a few practical remarks concerning the calculation of the inverse-filtered spectrum. In this research the Fourier transform of the data is divided by the Fourier transform of the impulse response function for the low frequencies. Letting 6 denote the inverse-filtered estimate and n the discrete integral spectral variable, we would have for the inverse-filtered Fourier spectrum... 

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. 

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