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.

翻译：现代数值分析基于离散数据执行，其中数值差分计算是核心且不可或缺的一环。然而，差分算法存在对噪声敏感的关键缺陷，长期以来在信号处理等多个领域构成挑战。差分是微分在离散域中的推广或泛化。然而，由于离散计算中有限区间的限制，无法满足微分最基础的定义——dy和dx均为无穷小（莱布尼茨定义）或dx的极限趋于0（柯西定义）。因此，微分向差分的推广在此意义上并不成立。为解决此问题，本文从原始导数方法出发，构建了基于有限区间的微分，并进一步通过卷积推广得到差分。基于该理论，我们提出了一系列适用于实际信号处理的差分算子。实验结果表明，这些差分算子展现出卓越的信号处理能力，包括高抗噪性。