Savitzky-Golay convolution functions

We turn now to the effect of using the Savitzky-Golay convolution functions. Table 57-1 presents a small subset of the convolutions from the tables. Since the tables were fairly extensive, the entries were scaled so that all of the coefficients could be presented as integers we have previously seen this. The nature of the values involved caused the entries to be difficult to compare directly, therefore we recomputed them to eliminate the normalization factors and using the actual direct coefficients, making the coefficients more easily comparable we present these in Table 57-2. For Table 57-2 we also computed the sums of the squares of the coefficients and present them in the last row. 

Our new method of determining nonlinearity (or showing linearity) is also related to our discussion of derivatives, particularly when using the Savitzky-Golay method of convolution functions, as we discussed recently [6], This last is not very surprising, once you consider that the Savitzky-Golay convolution functions are also (ultimately) derived from considerations of numerical analysis. 

Through the use of these formulas, Savitzky-Golay convolution coefficients could be computed for a convolution function using any odd number of data points for the convolution. 

Table 57-1 Some of the Savitzky-Golay convolution coefficients using a quadratic fitting function...

Table 57-2 The Savitzky-Golay convolution coefficients multiplied out. All coefficients are for a quadratic fitting function. See text for meaning of SSK...

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 ... 

The multiplication of the spectrum by a window is equivalent to a convolution in the time domain, and hence the approach is related to the Savitzky-Golay procedure. Indeed, by (4.10) this latter is also a convolution of the function values and the coefficients c /F ... 

Numerically the convolution of a step scan is merely the application of a sliding weighted mean (e.g. like the Savitzky-Golay method). The Fourier transform of the rectangular function has the shape of sin(nv)/(nv) (whereby n is inversely proportional to the width of the rectangle) and unfortunately approaches 0 only very slowly. To make do with a small number of points for a convolution, one must tolerate a compromise and renounce the ideal rectangular shape of the low pass filter (in the frequency domain). 

One way to improve the signal-to-noise ratio is through convolution of the spectrum with an appropriate function such as a boxcar, Lorentzian, or Gaussian function. The operation of spectral convolution has been presented in Section 2.3 Such operations tend to distort the spectrum, as the lineshape function is altered. The broader the convolution function, the greater is the distortion of the spectrum. The most common such convolution is the Savitzky-Golay smoothing algorithm [13]. 

Various other methods have been proposed to compute derivatives, including the use of suitable polynomial functions as suggested by Savitzky and Golay. The use of a suitable array of weighting coefficients as a smoothing function with which to convolute spectral data was described in... 

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