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ONE-DIMENSIONAL SIGNAL PROCESSING TECHNIQUES IN CHEMISTRY
In this chapter, some widely used methods for processing one-dimensional chemical signals are discussed. Chemical signals are usually recorded as chromatograms, spectra, voltammograms, kinetic curves, titration curves, and in other formats. Nowadays, almost all of them are acquired in the digital vector form. In order to denoise, compress, differentiate, and do other things on the signals acquired, it is always necessary to pretreat them first. The signal processing methods discussed in this chapter can be roughly divided into three classes including smoothing methods, transformation methods, and numerical treatment methods. They are described one by one below with illustrative examples.
2.1. DIGITAL SMOOTHING AND FILTERING METHODS
In general, averaging is widely used to improve the signal-to-noise ratio (SNR) for analytical signals. Through this treatment, the influence of noise can be reduced because the signals are often distributed normally on both positive and negative sides. In carrying out the averaging operation on a dataset xj, the mean x can be calculated by
X= -V X (2.1)
Notice that the variance of X is only 1/Vn of the original variable xj. Thus, the averaging operation can increase the SNR of analytical signals. This explains why the spectrum of a sample generated from an infrared instrument in your laboratory is often the mean spectrum from several measurements. It should be pointed out here that most methods discussed in the following subsections are based on this principle.
Chemometrjcs: From Basics To Wavelet Transform, edited by Foo-tim Chau, Yi-zeng Liang, Junbin Gao, and Xue-guang Shao. Chemical Analysis Series, Vol. 164.
ISBN 0-471-20242-8. Copyright © 2004 John Wiley & Sons, Inc.
one-dimensional signal processing techniques in chemistry
2.1.1. Moving-Window Average Smoothing Method
The moving-window average method is the classic and the simplest smoothing method. It can be utilized to enhance the SNR. The principle of this method is illustrated in Figure 2.1. Suppose that we have a raw signal vector, say, x = [x1, x2,... , xn-1, xn]. In practice, a window size of (2m +1) data points has to be specified first before doing any smoothing calculation. Here, an averaging filter of window size of 5(m = 2) is employed to illustrate the computing procedure. At the beginning, the first five data are used to find the first smoothed datum x3* via the following equation with i = 3 and m = 2:
x* = o--------7 y xi+j (2.2)
i 2m +1 A' i +j v '
In this equation, xf denotes the smoothed value while x + are the original raw data, where i and j are the running indices. It should be noted that the first two data points, x1 and x2, cannot be smoothed in the process. After finding x3, the next step is to move the window to the right by one datum to evaluate x4 (see Fig. 2.1). Then the procedure is repeated by moving the window successively along the equally spaced data until all the data are exhausted. As the width of the moving window is an important parameter
Figure 2.1. Moving-window-average filter for a window size of 2m + 1 = 5, that is, m = 2. It should be noted that for the extreme points, no smoothed data can be calculated because they are used for computing the first and the last averages. Top plot—the original raw signals; bottom plot-smoothed signals.
digital smoothing and filtering methods
in this smoothing process, it has to be defined before any calculation is performed. The guidelines for choosing a right value for this parameter will be discussed in Section 2.3.
The Savitsky-Golay filter is a smoothing filter based on polynomial regression [1-3]. Instead of simply using the averaging technique as mentioned previously, the Savitsky-Golay filter employs the regression fitting capacity to improve the smoothing results as depicted in Figure 2.2. From the plot, it can be seen that this method should perform better than the moving-window average method as mentioned in Section 2.1.1 because it takes advantage of the fitting ability of polynomial regression. However, the formulation of the Savitsky--Golay filter is quite similar to that of the averaging filter [Eq. (2.2)]. The major difference between the moving-window average method and the Savitsky--Golay filter is that the latter one is essentially a weighted average method in the form of