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 acceleration is obtained from parabolas that are continuously fit to the signal using a least squares optimization criterion. A closed-form solution for the filter coefficients is obtained. The general approach is computationally simple, 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 biologic signals, is presented using the electroencephalogram (EEG) and electrocardiogram (EKG) signals as examples. Furthermore, the design method is extended to derive FIR filters for estimating derivatives of arbitrary order in digital signals of biologic or other origins.