The filter function of the Butterworth low-pass filter has the following general form:
where ak & bk = filter coefficients, x[i] = raw coordinate of frame i, and x'[i] = filtered coordinate. A second order filter, thus, can be expressed as
 shown above has two main problems:
In other words, certain initial guess is necessary to use  in filtering the position data in motion analysis, such as
It was assumed that the filtered coordinates are equal to the raw coordinates in the first two frames. Since the filtered coordinates of the first two frames are now available,  can be used for the third frame, etc. Generalizing :
where k = 0, ..., N - 1, and N = order of the filter.
One problem of this approach is the error propagation from the first N frames to the rest of the frames. Since a Butterworth low-pass filter is an IIR (infinite impulse response) filter, the coordinates of the first N frames are continuously involved in the filtering of the coordinates of the rest of the frames. Thus, errors included in the filtered coordinates of the first N frames will be propagated to the rest of the frames. The effects of the first N frames gradually diminish as the filtering progresses from one frame to the next.
There are three possible ways to reduce the error propagation from the first N frames to the rest of the frames:
Both methods 1 and 2 above are based on the fact that the effects of the first N frames gradually diminish as the filtering progresses from one frame to the next. In either case, it is important to digitized the frames accurately at the beginning of the trial since the amount of error in the first N frames will affect the rest of the frames and the data padding is based on these frames.
Data padding is the process of generating additional frame data numerically at the boundary of the frame range, based on the digitized coordinates. The typical data padding methods include the point-symmetric padding method and the cubic padding method. Point-symmetric padding was originally called as reflection padding but it is not an accurate term.
© Young-Hoo Kwon, 1998-