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Chemometrics from basick to wavelet transform - Chau F.T

Chau F.T Chemometrics from basick to wavelet transform - Wiley publishing , 2004. - 333 p.
ISBN 0-471-20242-8
Download (direct link): chemometricsfrombasics2004.pdf
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Figure 5.22. The experimental chromatogram (a) and the smoothed results with the use of moving-average (b), Savitsky-Golay (c), and FFT (d) filtering.
application of wavelet transform in chemistry
of the peak height and peak area, which are very important parameters for signals analysis. The background always blurs the analytical signals. It is difficult or even impossible to analyze a signal with a strong background. In practice, an artificial baseline or background is usually drawn beneath the peak, although the error cannot be completely eliminated by this method. Therefore, most chemists prefer to find the exact shape of the baseline or background and then subtract it from the original signal. But it is not easy to obtain the ‘‘true’’ baseline or background because they are generally represented by curves instead of linear functions.
5.3.1. Principle and Algorithm
Baseline drift or background interference can be classified as a long-term noise. This property differs from that of common noise in that the frequency of the drifting baseline or background is always quite lower than the signals to be analyzed. In wavelet decomposition, the baseline or background component in an analytical signal should be easy to separate from the drifting signals. The removal procedure is similar to WT denoising and smoothing. The only difference is that the coefficients representing the lower-frequency components are suppressed. The following two methods are generally used.
Method A
1. Decompose the experiment data into discrete approximations cy and discrete details dy by Equations (5.1) and (5.2).
2. Examine the cy by visual inspection to find a cy that resembles the drifting baseline or background, and denote the scale j as ymax. The ymax can also be determined by examination of the dy, because there should be no information of the signal in djmax+1.
3. Reconstruct the signal by Equation (5.5) from only those dy with j < jmax, that is, set c^ to zeros.
Method B
1. Decompose the experiment data into discrete approximations cy and discrete details dy by Equations (5.32) and (5.33).
2. Examine the cy by visual inspection to find a cy (denote the scale y as ymax) that resembles the drifting baseline or background.
3. Subtract the selected cymax from the original signal by c0 - cymax or sometimes c0 - f x cymax, where f is an arbitrary factor.
baseline/background removal
5.3.2. Background Removal
Background removal is a universal problem in spectral studies, such as absorption spectroscopy and reflection spectroscopy. Example 5.8 describes a procedure to remove the background absorption from an EXAFS spectrum using the methods A and B, respectively.
Example 5.8: Background Removal of an EXAFS Spectrum. Extended X-ray absorption fine structure (EXAFS) is an X-ray absorption spectrum. A typical experimental absorption spectrum of copper with synchrotron radiation light source is shown in Figure 5.23a. For analyzing the EXAFS spectrum, we must separate the useful information (oscillation part) from the total raw spectrum. Then convert the oscillation part into k space by using the equation k = [0.263(E- E0)]1/2, where E0 is the absorption edge (the E0 for Cu is 8393.5 eV). At last, the oscillation signal is filtered by FFT and then fitted by the theoretical equation
where x (k) is the filtered oscillation signal multiplied by a factor k3, j is the number of the coordination shells, fj(k) is the amplitude value that can be obtained from handbook, <p(k) is the phase displacement of scattering, AE0 is the difference between the theoretical value and the experimental value of the E0, N is coordination number, r is coordination distance, a is Debye-Waller factor, and X is electron mean-free path. The aim is to obtain the structural parameter N, r, a, and X. Generally, the cubic spline interpolation method is used for the background removal and the least-squares method is used for the curve fitting. In our experience, it will take several hours to analyze one spectrum.
We can use method A for the background removal of an EXAFS spectrum. Figure 5.24a shows an experimental spectrum of a Cu sample, and panel (b) shows the spectrum in k space. In order to decompose the spectrum into its approximation and details, we can perform a WT on the k-space spectrum with Equations (5.1) and (5.2). Figure 5.25 shows the decomposed results, {c4, d4, d3, d2, di|, obtained with Daubechies8 (L = 8) filter and J = 4. It is clear that the information on the EXAFS oscillation is decomposed into the dj and that c4 represents the smooth background absorption. Therefore, if we reconstruct the spectrum from the dj components only, we will obtain the spectrum without the background absorption. The dotted line in Figure 5.24c shows the reconstructed
0.2625rj AE0
application of wavelet transform in chemistry
Figure 5.23. General procedures for analying an EXAFS spectrum: (a) raw spectrum; (b) oscillation part; (c) Filtered oscillation signal.
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