Savitzky-Golay smoothing algorithm

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]. 

First the signal was smoothed through a Savitzky-Golay algorithm, whose parameters were optimized considering the mean fringe number for a fixed temperature variation. 

The Savitzky-Golay algorithm could readily be adapted for polynomial interpolation. The computations are virtually identical to smoothing. In smoothing, a polynomial is fitted to a range of (x,y)-data pairs arranged around the x-value that needs to be smoothed. For polynomial smoothing, the polynomial is evaluated for a set number of data points around the desired x-value and the computed y-value at that x is the interpolated value. 

The data were smoothed using a 15 point cubic-quartic Savitzky and Golay (1A) algorithm, the x-ray satellites and a Shirley background were subtracted using computer routines available in the Vacuum Generators data analysis software. Only the treated data are presented here. 

When we have too few points to justify linearizing the function between adjacent points (as the trapezoidal integration does) we can use an algorithm based on a higher-order polynomial, which thereby can more faithfully represent the curvature of the function between adjacent measurement points. The Newton-Cotes method does just that for equidistant points, and is a moving polynomial method with fixed coefficients, just as the Savitzky-Golay method used for smoothing and differentation discussed in sections 8.5 and 8.8. For example, the formula for the area under the curve between x, and xn, is... 

In OPUS the Savitzky—Golay algorithm [1—3] is applied to smooth a spectrum. Possible values for smoothing points are 5 to 25. Open the smooth dialog box shown in Fig. 10.26, select the spectrum as usual, choose the number of smoothing points and click on the Smooth button to start the function. 

The Second Derivative method is typically applied in order to be independent of the baseline. As in the case of the Derivative function (see Section 10.11) the Savitzky-Golay algorithm is actually used to obtain the derivative. Again, the number of smoothing points used can be adjusted to suppress the effect of... 

Processing the previously acquired data by subjection to a mathematical treatment such as curve smoothing (e.g. with the Savitzky—Golay algorithm), derivation, calibration or spectral refining. Data can be also exported to be processed with other software (e.g. Excel spreadsheet). 

The most common algorithm is that of Savitzky and Golay [22, 23]. This polynomial incorporates a shortened least-square computation for data smoothing (see also Section 3.6.4.2). Some points must be considered when using this differentiation mode ... 

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