Higher derivative determination in digital filtering: a different cut-off frequency strategy [biomechanics application]

The present study investigated four different filtering and differentiation sequences for the calculation of the higher derivatives from noisy displacement data when using a second order Butterworth filter and first order finite differences. These were: (1) the conventional sequence (i.e. filtering the displacement data and then differentiating); (2) filtering the displacement with a different cut-off frequency depending upon optimal 0th, 1st and 2nd derivatives; (3) double filtering and differentiation (only for acceleration); and (4) differentiation and then filtering separately in each derivative domain i.e. treating the noisy higher derivatives as individual signals. Sixty levels of noise were generated and superimposed on 24 original signals, creating 1440 signals in which their original signal and added noise characteristics were known a priori. The results indicated that the conventional strategy has to be reconsidered and modified as the best results were obtained by the second strategy. The optimum cut-off frequency for acceleration was lower than that required for the velocity which in turn was lower than the optimum cut-off frequency for displacement. The findings of the present study will contribute to the development of existing and future automatic filtering techniques based on digital filtering.