Design and responses of Butterworth and critically damped digital filters

For many years the Butterworth lowpass filter has been used to smooth many kinds of biomechanical data, despite the fact that it is underdamped and therefore overshoots and/or undershoots data during rapid transitions. A comparison of the conventional Butterworth filter with a critically damped filter shows that the critically damped filter not only removes the undershooting and overshooting, but has a superior rise time during rapid transitions. While analog filters always create phase distortion, both the critically damped and Butterworth filters can be modified to become zero-lag filters when the data are processed in both the forward and reverse directions. In such cases little improvement is realized by applying multiple passes. The Butterworth filter has superior 'roll-off' (attenuation of noise above the cutoff frequency) than the critically damped filter, but by increasing the number of passes of the critically damped filter the same 'roll-off' can be achieved. In summary, the critically damped filter was shown to have superior performance in the time domain than the Butterworth filter, but for data that need to be double differentiated (e.g. displacement data) the Butterworth filter may still be the better choice.