Optimal digital filtering requires a different cut-off frequency strategy for the determination of the higher derivatives

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 cach derivative domain, i.e. treating the noisy higher derivatives as individual signals. Thirty levels of time domain and 30 levels of frequency domain computer-generated pure noise signals, were superimposed on 24 reference signals which simulated the medial-lateral, anterior-posterior and vertical displacement patterns of eight markers attached to the lower extremity segments during walking. The optimum cut-off frequency for the displacement, velocity and acceleration data was calculated as the one that produced the minimum root mean square error between the reference and noisy data in each derivative domain. 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.