Tao General Differential and Difference: Theory and Application

Modern numerical analysis is executed on discrete data, of which numerical difference computation is one of the cores and is indispensable. Nevertheless, difference algorithms have a critical weakness in their sensitivity to noise, which has long posed a challenge in various fields including signal processing. Difference is an extension or generalization of differential in the discrete domain. However, due to the finite interval in discrete calculation, there is a failure in meeting the most fundamental definition of differential, where dy and dx are both infinitesimal (Leibniz) or the limit of dx is 0 (Cauchy). In this regard, the generalization of differential to difference does not hold. To address this issue, we depart from the original derivative approach, construct a finite interval-based differential, and further generalize it to obtain the difference by convolution. Based on this theory, we present a variety of difference operators suitable for practical signal processing. Experimental results demonstrate that these difference operators possess exceptional signal processing capabilities, including high noise immunity.