Error analysis for the first-order Gaussian recursive filter operator

Nowadays, recursive filters (RFs) are frequently used in several research fields. More in particular, Gaussian RFs offer a more efficient way for computing approximate Gaussian filters and Gaussian-based convolutions. The use of such recursive filters introduces many sources of errors. Among them, here we consider the filter truncation error, that is the error due to the transition from the starting filter operator to the RF approximating it. Since input and output signals have infinite dimensions, the analysis of the related filter operator involves infinite matrices. In this paper, starting from a summary of the comprehensive mathematical background, we consider the case of the first-order Gaussian recursive filter. Then, taking into account the matrix form of the related operator, we perform the error analysis and provide theoretical results that estimate the filter truncation error.