General Least-Squares Smoothing and Differentiation by the Convolution (Savitzky-Golay) Method

Sm- and d#ferentlatkn of large data sets by plecewlse least-squares polynomlal fffllng are now wklely used tech- nlques. The calculation speed Is very greatly enhanced H a convolution formalism is used to perform the calcuiatlons. Prevlously tables of convolution weights for the center-pdnt least-squares evaluatlon of 2m + 1 points have been pres- ented. A major drawback of the technique Is that the end polnts of the data sets are kot (2m pohts for a 2m + 1 point fllter). Convdutlon weights have also been presented In the speclal case of Inltlai-point values. In this paper a sknple general procedure for calculatlng the convolution weights at all podtlons, for all polynomial orders, all fitter lengths, and any derlvatlve Is presented. The method, based on the re- cursive properties of Gram polynomials, enables the convo- lutlon technique to be extended to cover all points In the spectrum.