A simple algorithm for a digital three-pole Butterworth filter of arbitrary cut-off frequency: application to digital electroencephalography

Algorithms for low-pass and high-pass three-pole recursive Butterworth filt ers of a given cut-off frequency have been developed. A band-pass filter ca n be implemented by sequential application of algorithms for low- and high- pass filters. The algorithms correspond to infinite impulse-response filter s that have been designed by applying the bilinear transformation to the tr ansfer functions of the corresponding analog filters, resulting in a recurs ive digital filter with seven real coefficients. Expressions for filter coe fficients as a function of the cut-off frequency and the sampling period ar e derived. Filter performance is evaluated and discussed. As in the case of their analog counterparts, their transfer function shows marked flattening over the pass band and gradually higher attenuation can be seen at frequen cies above or below the cut-off frequency, with a slope of around 60 dB/dec ade. There is a 3 dB attenuation at the cut-off frequency and a gradual inc rease in phase shift over one decade above or below the cut-off frequency. Low-pass filters show a maximum overshoot of 8% and high-pass filters show a maximum downwards overshoot of approximately 35%. The filter is mildly un der-damped, with a damping factor of 0.5. On an IBM 300GL personal computer at 600 MH with 128 MB RAM, filtering time with MATLAB 5.2 running under Wi ndows 98 is of the order of 50 ms for 60 000 samples. This will be adequate for on-line electroencephalography (EEG) applications. The simplicity of t he algorithm to calculate filter coefficients for an arbitrary cut-off freq uency can be useful to modern EEG laboratories and software designers for e lectrophysiological applications. (C) 2000 Elsevier Science B.V. All rights reserved.