
From [5] of System Function, the general form of the discretetime system function of the 2ndorder lowpass filter can be expressed as [1] where [2] The system function H(z) of the filter can be expressed in the following general form: , [3] where X(z) = the ztransform of the input (raw) time series of the filter, and X'(z) = the ztransform of the output (filtered) time series. From [1] and [3]: . [4] From [4], and the linearity and timeshifting properties of the ztransform ([6] and [9] of zTransform), the filter function of the Butterworth lowpass filter can be obtained as , [5] since , [6] 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 . [7] Butterworth lowpass filter is a recursive filter since previously filtered coordinates, x'[i1] and x'[i2], are used in filtering the coordinate in the current frame. [7] 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'[i1] & x'[i2] and x'[i1] is in turn affected by x'[i2] & x'[i3], etc., x'[i] is in fact a function of the coordinates of all the preceding frames. Therefore, the Butterworth lowpass filter is an IIR (infinite impulse response) filter. The general form of a 4thorder lowpass filter is . [8] Further generalizing [8] for any evenorder Butterworth lowpass filter: , [9] where N = 0, 2, 4, .... Let the frequency ratio f_{r} be the ratio of the sampling frequency to the cutoff frequency: . [10] From [7] and [8] of System Function, the filter coefficients of any evenorder Butterworth lowpass filter can be described as , [11] where . [12] k shown in [11] and [12] is the elementary 2ndorder filter number. For example, a 6thorder lowpass filter has 3 (6 divided by 2) elementary 2ndorder filters. Passing the data through these 3 2ndorder filters consecutively is the same to passing them through the 6thorder filter once. The filter coefficients for N = 2 ([7]), f_{s} = 100 Hz, and f_{c} = 10 Hz are [13] As shown in [11] and [12], the filter coefficients are basically determined by the frequency ratio and the order of the filter. The coefficients of the 2ndorder lowpass filter for different frequency ratios can be found in Winter (1990). References & Related Literature Oppenheim, A.V., & Schafer, R.W. (1989). Discretetime signal processing. Englewood Cliffs, NJ: Prentice Hall. Winter, D.A. (1990). Biomechanics and Motor Control of Human Movement (2^{nd} Ed.). Toronto, Ontario: WileyInterscience. 
© YoungHoo Kwon, 1998 