# Chemometrics from basick to wavelet transform - Chau F.T

ISBN 0-471-20242-8

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Chemical Shift / ppm

Figure 5.53. Detail coefficients of the simulated NMR spectrum obtained by WT decomposition.

difficult to select the appropriate detail coefficients when the noise level in the original signal is significant because the noise will also be decomposed into those low-scale detail coefficients for amplification. Figure 5.54, curve

(a) shows a simulated noisy NMR spectrum. The detail coefficients are shown in Figure 5.55. It can be seen that the d-i and d2 coefficients are noisy. If we multiply d1 and d2 by k1 = k2 = 55 as we did above, the noise level of the reconstructed spectrum will be increased as well, as is shown in Figure 5.54, curve (b). In such cases, we can only multiply the d2 or d3

i—'—i—1—i—1—i—1—i—1—i—1—i—1—i

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Chemical Shift / ppm

Figure 5.54. A simulated noisy NMR spectrum (a) and its reconstructed spectra by multiplying d1 and d2 with k1 = k2 = 55 (b), d2 with k2 = 60 (c), and d3 with k3 = 10 (d), respectively.

resolution enhancement

219

I 1-------------------------------------1--------------------1-------------------1-------------------1-------------------1--------------------1-------------------1-------------------'--------------------1-------------------1-------------------1--------------------'-------------------1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Chemical Shift / ppm

Figure 5.55. Detail coefficients of the simulated noisy NMR spectrum obtained by WT decomposition.

in order to avoid the effect of noise. Figure 5.54, curves (c) and (d) are obtained by multiplying d2 and d3, respectively, with k2 = 60 and k3 = 10. It is clear that the SNR of the results is improved.

Computational Details of Example 5.13

1. Generate the signal with 700 data points [Figure 5.52 (a)] using Lorentzian equations.

2. Make a wavelet filter—Symmlet4.

3. Set resolution level J = 4.

4. Perform WT to obtain the c and d components with the improved algorithm.

5. Perform reconstruction with multiplying the d1 and d2 with a factor 55.0.

6. Display Figures 5.52 and 5.53.

A more detailed discussion of Example 5.13 can be found in a paper published in Applied Spectroscopy [54:731-738 (2000)]. In this paper, an experimental NMR spectrum was also processed by the method. Figure 5.56 shows two enlarged parts of the experimental and the reconstructed spectra. It is clear that the resolution of the spectra is greatly improved by this method.

220 application of wavelet transform in chemistry

I 1--------------------------------------1-------------------1--------------------1-------------------1-------------------1--------------------1--------------------1-------------------1-------------------1 I--------------------'-------------------1-------------------'--------------------1--------------------'-------------------1-------------------'--------------------1-------------------'--------------------1

1.0 1.2 1.4 1.6 1.8 2.0 4.0 4.2 4.4 4.6 4.8 5.0

Chemical Shift / ppm Chemical Shift / ppm

Figure 5.56. An experimental NMR spectrum (a) and the reconstructed spectrum by multiplying d1 and d2 with k1 = k2 = 10 (b).

5.4.5. Resolution Enhancement by Using Wavelet Packet Transform

WPT differs from WT with respect to the decomposition tree. In WT, only the approximation coefficients on each scale are used for further decomposition, but in WPT, the further decomposition is applied to both the approximation and detail coefficients. Therefore, for resolution enhancement, the resolving ability of WPT should be stronger than that of WT, because there will be more decomposed components representing the information with different frequencies. Consequently, it is easy for us to select a component that represents the desired high-resolution information.

The procedures for resolution enhancement of analytical signals are almost the same as in methods A and B proposed above. The only difference is to use the Equations (5.42)-(5.44) instead of Equations (5.32)-

(5.34), for decomposition and reconstruction computation.

Figure 5.57 shows the experimental chromatograms of six samples containing six rare-earth ions. Concentrations of the samples are listed in Table 5.5. In order to extract the chromatographic information of each component from the overlapping chromatograms in Figure 5.57, we can subject them to WPT decomposition and obtain all wp first. Then we can select a coefficient component to represent the desired high-resolution information. Figure 5.58 shows the selected w3 coefficients of the six chromatograms. Finally, we can estimate a baseline by linking the minimum point at both sides of every peak. After subtracting the baseline, we can obtain the results shown in Figure 5.59. It can be seen that all six peaks in the six chromatograms are well resolved except for the second peak in

resolution enhancement

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0 ' 2 ' 4 ' 6 ' 8 ' 10 ' 12

Retention Time / min

Figure 5.57. Experimental chromatograms of mixed rare-earth solution samples.

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