Savitzky-Golay filtering

Figure 3.11. Smoothing a noisy signal. The synthetic, noise-free signal is given at the top. After the addition of noise by means of the Monte Carlo technique, the panels in the second row are obtained (little noise, left, five times as much noise, right). A seven-point Savitzky-Golay filter of order 2 (third row) and a seven-point moving average (bottom row) filter are...

SMOOTH.dat A 26-point table of values interpolated from a figure in Ref. 162, to demonstrate the capability of the discussed extended Savitzky-Golay filter to provide a smoothed trace from the first to the last point in the time series. 

Polynomials do not play an important role in real chemical applications. Very few chemical data behave like polynomials. However, as a general data treatment tool, they are invaluable. Polynomials are used for empirical approximations of complex relationships, smoothing, differentiation and interpolation of data. Most of these applications have been introduced into chemistry by Savitzky and Golay and are known as Savitzky-Golay filters. Polynomial fitting is a linear, fast and explicit calculation, which, of course, explains the popularity. 

Figure 4-23. Savitzky-Golay filtering. A polynomial is fitted to a range of data points and the original point (x) is replaced by the value on the polynomial (o).

It is tempting to write a routine such as SavGo l bad. m, to perform the Savitzky-Golay filtering, but we will show its numerical weakness. F is built up by the appropriate range of x-values and used to calculate the polynomial coefficients as a=F y( i-n i+n), see e.g. equation (4.31). 

Very popular is the Savitzky-Golay filter As the method is used in almost any chromatographic data processing software package, the basic principles will be outlined hereafter. A least squares fit with a polynomial of the required order is performed over a window length. This is achieved by using a fixed convolution function. The shape of this function depends on the order of the chosen polynomial and the window length. The coefficients b of the convolution function are calculated from ... 

Figure 4.10 Graphical representation of (a) baseline correction, (b) smoothing and (c) simplified view of a Savitzky Golay filter (moving window, five points per window, linear interpolation).

Figure 18. Test data after filtering with a 21-point Savitzky-Golay filter...

In this work an original usage of Savitzky-Golay filter is proposed. An estimate for the local derivative is obtained from the interpolation process, which is used in a test that allows the discrimination of steady-states. 

Figure 4.4 Similar to the sliding polynomial smoothing (Savitzky Golay filter, the coefficients for 2nd order fit to a parabola) is the effect of Bromba Ziegler filters [Bromba and Ziegler, (1983c), coefficients fit to a triangle upper figure]. Both have bad low pass filter characteristics, as shown in the lower figure with the Fourier transforms of filters through 21 points each.

The profile of the 11-point Savitzky-Golay filters for performing smoothing... 

Figure 3.5 11-Point Savitzky-Golay filters for treating speactral data (a) smoothing,...

The filter coefficients Cj are tabulated in Table 3.1 for different filter widths. Figure 3.2, curve 3, demonstrates the effect of a Savitzky-Golay filter with a filter width of 5 points applied to the raw data. Compared to the 5-point moving-average filter, the obviously better fit can be seen. 

Table 3.1 Coefficients of the Savitzky-Golay filter for smoothing based on a quadratic/cubic polynomial according to Eg. (3.2). 

The Savitzky-Golay filter can also be used for derivation of signal curves. For this, appropriate filter coefficients are inserted as given in Table A.7 for the first derivative and in Table A.8 for the second derivative. 

At this point k, the filtered value is calculated up to the second derivative on the basis of a 5-point Savitzky-Golay filter. 

To characterize the corresponding noise, we consider the error propagation for the polynomial filter. For the Savitzky-Golay filter (see Eq. (3.2)), the result of error propagation (see Table 2.4) is expressed here by the standard deviation of the smoothed signal at point k, Sy ... 

Calculate the fourth smoothed value, for the signal data given in Example 3.1 by means of a Savitzky-Golay filter. 

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