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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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where x(1) represents the first derivative of the signal x, D2m and D2m represent two Daubechies wavelet functions, and Cj,D2m and Cj,Dm are the
X(1) = CjD2m - Cj,D2m m = m
(5.74)
202
application of wavelet transform in chemistry
jth-scale discrete approximations obtained by Equations (5.1) and (5.2) with the two wavelets, respectively. In practice applications, j = 1 is generally used. But for noisy signals, a higher value of j can be used to improve the SNR of the calculated result.
Higher-order derivative computation can be achieved by using the result obtained from the lower-derivative calculation as an input for WT calculation
x(n) = c(nn1) - c(nn1) m = m and n > 1 (5.75)
J ,D2m J ,D2m ' '
where x(n) represents the nth-order derivative and cjn—jand c(nD 1 represent the jth-scale approximation coefficients obtained by WT from the
(n - 1)th-order derivative with the Daubechies wavelet functions D2m and D2m.
Example 5.10: Approximate Derivative Calculation of Simulated Signals Using DWT. Figure 5.36a shows three types of typical analytical
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0 200 400 600 800 1000
0 500 1000 1500 2000
Data Point
Figure 5.36. A simulated signal (a) and its first- (a) and second- (c) order derivatives calculated by DWT method.
resolution enhancement
203
signals simulated by Gaussian, Lorentzian, and sigmoid functions
Xgauss = A) exp -4 ln(2)
^lorentz = A0 ( 1 + 4
t - to
Wi
/2 .
^sigmoid —
A0
1 + e-k(t-to)
(5.76)
(5.77)
(5.78)
where A0, t0, and W|/2. are the amplitude, position, and the full width at half maximum of the simulated peak, respectively, and k is a parameter to control the gradient of the curve. Figure 5.37a shows a noisy signal by the curve (a) in Figure 36 plus a random noise at SNR — 100.
Figure 5.37. A simulated noisy signal (a) and its first derivative calculated by DWT with j — 1 (b) and j — 3 (c), respectively.
204
application of wavelet transform in chemistry
Figure 5.36b,c shows the first- and second-order derivatives obtained by Equations (5.74) and (5.75), respectively, using j = 1 and Daubechies N = 18 and 8 as the filters (D2m and D2m). It can be seen that very good results can be obtained. The only drawback is that the number of data points is reduced to 2 for the first-order derivative and 4 for the second-order derivative.
Figure 5.37b shows the first derivative using the same method and the parameters as in Figure 5.36b. It can be seen that, when the signal is noisy, the SNR of the result will be even worse than that of the signal. In this case, we can use a higher value of j for the calculation. Figure 5.37c is obtained using j = 3, which is an acceptable result. But there is also the problem of the reduction of the data point number. The number of data point in Figure 5.37c is only 1th of the original length.
Although interpolation can solve the problem of the data point number, we can also use Equations (5.32) and (5.33) for calculation of the Cj,D2m and CjD2m in Equation (5.74) or cj'D-1n) and cjnD V in Equation (5.75). The benefit of using the two equations is that the number of data points in Cj does not change. Consequently, the length of the calculated derivatives will remain the same as that of the original signal. Figures 5.38 and 5.39 show the first- and the second-order derivatives of the simulated signals with the same filters as in the DWT method and j = 1 and j = 4, respectively. It can be seen that all the results are acceptable. Only the symmetry of the peaks is slightly distorted because of the effect of the noise in Figure 5.39c.
Computational Details of Example 5.10
1. Generate the signal with 2000 data points (Fig. 5.36a) by using Gaussian, Lorentzian, and sigmoid equations.
2. Extend the data point number to 2048.
3. Make two wavelet filters—Daubechies18 and 8.
4. Set resolution level J = 1.
5. Perform DWT with the two filters, respectively, to obtain the wavelet coefficients.
6. Calculate the first derivative by subtracting the approximate coefficients one from another.
7. Display Figure 5.36b.
8. Perform DWT on the coefficients obtained in step 5 with the two filters, respectively.
9. Calculate the second derivative from the approximate coefficients.
10. Display Figure 5.36c.
resolution enhancement
205
Note: The results in Figure 5.37 were obtained in a similar way with the only difference in J = 3 for the second-derivative calculation, and the results in Figures 5.38 and 5.39 were obtained by using the improved algorithm with J = 1 and 4, respectively.
5.4.2. Numerical Differentiation Using Continuous Wavelet Transform
CWT with some specific wavelet functions can also be used for approximate derivative calculation of analytical signals. The Haar wavelet function, for instance, is one of the appropriate wavelet bases because of its symmetric
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0 500 1000 1500 2000
0 ' 500 ' 1000 ' 1500 ' 2000
Data Point
Figure 5.38. A simulated signal (a) and its first- (b) and second- (c) order derivatives calculated by the improved WT algorithm with j = 1.
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