Frequency limitations of the two-point central difference differentiation algorithm

Abstract

A two-point central difference algorithm is often used to calculate the derivative of a function. This estimate is only valid over a limited frequency range. Therefore, the algorithm can be modeled as an ideal differentiator in series with a low-pass filter. The filter cutoff frequency is a function of the time between the points. We discuss the accuracy and limitations of using this algorithm on human saccadic eye movement data. To calculate the velocity of saccadic eye movements the algorithm should have a cutoff frequency of 74 Hz or above.

The effect of sampling rate and lowpass filters on saccades – A modeling approach

“…Finally, the reduction in peak velocities reported here is a best-case estimate as we made sure that the actual time of the peak velocity coincides with a sample. Since this time is a one-point measure, it can easily be missed by an eye tracker during a real measurement (Andersson, Nystrom, & Holmqvist, 2010), resulting in further lowered peak velocities (Bahill et al, 1982). Therefore the reduction in peak velocities for lower sampling rates reported here might even be worse for real data.…”

“…It has been shown that a one-sample, two-point central difference (2pCD) method is ideal for this purpose (Bahill, Kallman, & Lieberman, 1982). Importantly, this method produces no lag in the velocity channel with respect to the position channel.…”

“…Importantly, this method produces no lag in the velocity channel with respect to the position channel. In addition, it implicitly applies a lowpass filter that reduces the high frequency noise introduced by the differentiation process (Bahill et al, 1982;Inchingolo & Spanio, 1985). However, the computed peak velocity is also affected by the sampling rate of the eye tracker itself.…”

Different effects of eyelid blinks and target blanking on saccadic suppression of displacement

“…A two-point central difference differentiation algorithm (Bahill, Kallman, & Lieberman, 1982) continuously estimated the momentary eye velocity and generated a trigger signal indicating the occurrence of a saccade once the velocity exceeded 15º/sec. We verified that the trigger-induced target blanking or target shift consistently occurred in the first third of the duration of the saccade.…”

Gaze Control during Horizontal and Vertical Target Tracking.

“…Thus the oculometer did not limit the frequency content of these smooth pursuit eye movements. However an FFT analysis of saccadic eye movements has shown information up to 40 Hz (Bahill, et al, 1982).…”