Convolution method for least-squares-fit smoothing and differentiation of digital data

TLDR: The use of least squares-fit-to-a-polynomial smoothing of uniformly spaced digital data by convoluting the data with a smoothing array is reviewed in this paper.

Abstract: The technique of least-squares-fit-to-a-polynominal smoothing of uniformly spaced digital data by convoluting the data with a smoothing array is reviewed. The use of digital computers for this type of numerical filtering and for determining smoothed derivatives was first discussed by Savitzky and Golay (Anal. Chem. 36, 1627 (1964)). The report presents methods for extending the widths of the convolution arrays beyond the 25-point-width maximum of Savitzky and Golay. It also gives corrections to errors in their paper. Three new algebraic equations are derived that can be used to determine the convolution array coefficients for determining the smoothed first, second, and third derivative values by least-squares-fitting to a quadratic/cubic polynominal. Two simple tests for determining errors in least-squares-fit convolution arrays are given. The use of these convolution arrays for processing digital data is illustrated by examples that make use of a Rutherford backscattering spectrum from a cobalt molybdate catalyst.