Least squares acceleration filtering for the estimation of signal derivatives and sharpness at extrema

A family of finite impulse-response (FIR) filters is derived which estimate the second derivative or "acceleration" of a digitized signal. The acceler ation is obtained from parabolas that are continuously fit to the signal us ing a least squares optimization criterion. A closed-form solution for the filter coefficients is obtained. The general approach is computationally si mple, can be performed in real-time, and is robust in the presence of noise . An important application of the method, that of measuring sharpness in bi ologic signals, is presented using the electroencephalogram (EEG) and elect rocardiogram (EKG) signals as examples. Furthermore, the design method is e xtended to derive FIR filters for estimating derivatives of arbitrary order in digital signals of biologic or other origins.