Comments on the Savitzky-Golay Convolution Method for Least-Squares Fit Smoothing and Differentiation of Digital Data

Sir: In 1964 a paper by Savitzky and Golay ( I ) describing a technique of smoothing and obtaining smoothed derivatives using convolution arrays derived from the coefficients of least-squares-fit formulas was published in this journal. That paper has become a classic judging by the number of citations @-well over 350 in more than 100 different journals-since its publication despite the fact that the paper contained numerous errors in the tables of convolution arrays. These errors were first pointed out and almost completely corrected by Steinier, Termonia, and Deltour (3) in 1972. The purpose of this letter is threefold: (1) to present two simple equations that may be used to check for errors in the arrays; (2) to note the simplicity involved in generating arrays wider than 25 points (the maximum width in the Savitzky-Golay tables), giving equations that may be used to easily expand the widths of all the convolution arrays discussed in the Savitzky-Golay paper; and (3) to call attention to examples that illustrate the beneficial use of smoothing techniques in extracting information other than simply peak position, height, and width from spectral data made up of peaks of known profile. More detailed discussions of least-squares-fit smoothing, than will be given here, may be found in the review papers of Nernst ( 4 ) and of Willson and Edwards ( 5 ) . The Savitzky-Golay convolution smoothing technique is based on fitting an array of n (= 2m + 1, with m a positive integer from 1 to 12) equally-spaced data points to a polynomial, U(X) = co + c1x + c2x2 + e* ’ + c,x’