Polynomial fitting Savitzky-Golay filter

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.

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).

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 ... 

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