Frequency Limitations and Optimal Step Size for the Two-Point Central Difference Derivative Algorithm with Applications to Human Eye Movement Data

Abstract
There are many algorithms for calculating derivatives. The two-point central difference algorithm is the simplest. Besides simplicity, the two most important characteristics of this algorithm are accuracy and frequency response. The frequency content of the data prescribes a lower limit on the sampling rate. The smoothness and accuracy of the data determine the optimal step size. We discuss the low-pass filter characteristics of this algorithm and derive the optimal step size for two types of human eye movement data. To calculate the velocity of fast (saccadic) eye movements, the algorithm should have a cutoff frequency of 74 Hz. For typical slow (smooth pursuit) eye movements, a step size of 25 or 50 ms is optimal.