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

ISBN 0-471-20242-8

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xi=[0:.05:1.5];

yi=cos(xi)

ybad=yi+.2*(rand(size(xi))-.5); figure(2)

subplot(221),plot(xi,yi,ck:’,xi,ybad,‘kx’),grid on title(‘Original curve: dashed line; Noisey data: cross’) axis([0 1.5 0 1.2]) xlabel(‘Varibale (x)’) ylabel(‘Signal, (y)’)

yy1=csaps(xi,ybad,.9981,xi);

subplot(222),plot(xi,yi,ck:’,xi,ybad,ckx’,xi,yy1,‘k’), grid on title(‘Smoothed curve: solid line with p=9981’) axis([0 1.5 0 1.2])

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one-dimensional signal processing techniques in chemistry

xlabel(‘Varibale (x)’) ylabel(‘Signal, (y)’)

yy2=csaps(xi,ybad,.9756,xi);

subplot(223),plot(xi,yi,ck:J,xi,ybad,ckxJ,xi,yy2,‘kJ), grid on

title(‘Smoothed curve: solid line with p=9756’)

axis([0 1.5 0 1.2])

xlabel(‘Varibale (x)’)

ylabel(‘Signal, (y)’)

yy3=csaps(xi,ybad,.7856,xi);

subplot(224),plot(xi,yi,ck:J,xi,ybad,ckxJ,xi,yy3,‘kJ), grid on

title(‘Smoothed curve: solid line with p=7856’)

axis([0 1.5 0 1.2])

xlabel(‘Varibale (x)’)

ylabel(‘Signal, (y)’)

Usually, it is difficult to choose the best value for the parameter p without experimentation. If one has difficulty in doing this but has an idea of the noise level in Y, the MATLAB command spaps(X, Y, tol) may help. Select

Original curve: dashed line; Noisy data: cross

Smoothed curve: solid line with p=0.9981

Smoothed curve: solid line with p=0.9756

1

_ 0.8

1 0.6

D)

i'Si

0.4

0.2

0

(b)

*

51 Variable (x)

Smoothed curve: solid line with p=0.7856

Variable (x)

Variable (x)

Figure 2.4. Smoothing results obtained by a cubic spline smoother with different values of the parameter p: (a) the original curve and the raw noisy signals; (b) the smoothed curve with p = 0.9981; (c) the smoothed curve with p = 0.9756; (d) the smoothed curve with p = 0.7856.

0

transformation methods of analytical signals

39

a p value in such a way that

tol = ? (y - y)2 (2.24)

/

so as to produce the smoothest spline within an acceptable tolerance for the data.

2.2. TRANSFORMATION METHODS OF ANALYTICAL SIGNALS

Transformation is a very useful technique in pretreatment of analytical signals. Convolution, Hadamard, and Fourier transformation are just examples of this kind. In essence, wavelet analysis is also another kind of transformation technique. In this section the methods of convolution, Hadamard and Fourier transformation will be discussed in some detail.

2.2.1. Physical Meaning of the Convolution Algorithm

Convolution plays a very important role in statistics in treating analytical signals. An example from spectral measurement is now presented to illustrate how convolution works [6].

Suppose that the real spectrum of a sample is the one given by f(x) in the upper part of Figure 2.5. Now, a spectrometer with a slit is utilized to assist the acquisition of the spectrum. If the slit is infinitely narrow, the spectral signal recorded should be the same as that of f(x). In practice, any slit has certain width. Let the slit operation be expressed by function h(x) which is essential a triangular function (see the lower part of Fig. 2.5). From the plot of h(x), one can see how the slit function (triangular function) affects the intensity distribution of the lightbeam with respect to the location. The highest-intensity location appears at the central point of the slit. Thus, when the light ray passes this slit, the spectrum obtained [as expressed by the function g(x) in Fig. 2.5] by the spectrophotometer becomes broader than the real one f(x). The whole procedure as described above is called convolution in the field of signal processing.

From this plot (Fig. 2.5), it can be seen that the slit works somewhat like the Savitsky-Golay filter. The triangular function is essentially a weight function. That is why the Savitsky-Golay filter is originally called the polynomial convolution method. Since the spectrum g(x) obtained from the spectrophotometer is the convolution result of the original spectrum f(x) and the triangular function h(x), the term g(x) can be expressed

40 one-dimensional signal processing techniques in chemistry

Figure 2.5. Illustration of the physical meaning of convolution by the slit function. mathematically by the following formula:

m

g[x(/)] =J2 f(x)' h[x(') - x] (2.25)

i=-m

This formula is the discrete expression of the convolution operation through which one can see that N = 2m + 1 is the width of the slit. In Equation

(2.25), x(/) represents the intensity of the light of the measured spectrum at the central point. It should be noted that all the elements of the slit function h(x) outside the moving window have zero values. Thus, the continuous formula of convolution can be expressed as follows:

/+TO

f(x)h(x(/) - x)dx (2.26)

-TO

Let x(/) be represented by y. Then we have

/+TO

f(x)h(y - x)dx (2.27)

-TO

Here g(y) is usually called the convolution of functions f(y) and h(y) and is denoted by f (y) * h(y).

transformation methods of analytical signals

41

It should be mentioned that the convolution operation can be easily fulfilled by the Fourier transformation. Hence, the convolution operation is essentially a kind of transformation.

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