From  of System Function, the general form of the discrete-time system function of the 2nd-order low-pass filter can be expressed as
The system function H(z) of the filter can be expressed in the following general form:
where X(z) = the z-transform of the input (raw) time series of the filter, and X'(z) = the z-transform of the output (filtered) time series. From  and :
From , and the linearity and time-shifting properties of the z-transform ( and  of z-Transform), the filter function of the Butterworth low-pass filter can be obtained as
where i = the frame number, x[i] = the input (raw) time series of the filter, and x'[i] = the output (filtered) time series. Thus, the final form of the filter function can be obtained as
Butterworth low-pass filter is a recursive filter since previously filtered coordinates, x'[i-1] and x'[i-2], are used in filtering the coordinate in the current frame.  in fact shows a weighted mean of the raw coordinates and the filtered coordinates of the current frame and the two immediately past frames. Thus, the sum of the coefficients must be equal to 1. Since x'[i] is affected by x'[i-1] & x'[i-2] and x'[i-1] is in turn affected by x'[i-2] & x'[i-3], etc., x'[i] is in fact a function of the coordinates of all the preceding frames. Therefore, the Butterworth low-pass filter is an IIR (infinite impulse response) filter.
The general form of a 4th-order low-pass filter is
Further generalizing  for any even-order Butterworth low-pass filter:
where N = 0, 2, 4, ....
Let the frequency ratio fr be the ratio of the sampling frequency to the cutoff frequency:
From  and  of System Function, the filter coefficients of any even-order Butterworth low-pass filter can be described as
k shown in  and  is the elementary 2nd-order filter number. For example, a 6th-order low-pass filter has 3 (6 divided by 2) elementary 2nd-order filters. Passing the data through these 3 2nd-order filters consecutively is the same to passing them through the 6th-order filter once.
The filter coefficients for N = 2 (), fs = 100 Hz, and fc = 10 Hz are
As shown in  and , the filter coefficients are basically determined by the frequency ratio and the order of the filter. The coefficients of the 2nd-order low-pass filter for different frequency ratios can be found in Winter (1990).
References & Related Literature
Oppenheim, A.V., & Schafer, R.W. (1989). Discrete-time signal processing. Englewood Cliffs, NJ: Prentice Hall.
Winter, D.A. (1990). Biomechanics and Motor Control of Human Movement (2nd Ed.). Toronto, Ontario: Wiley-Interscience.
© Young-Hoo Kwon, 1998-