Algebraic Characterizations of Nonlinear Digital Filters

The book, Fundamentals of Nonlinear Digital Filtering, by Jaako Astola and Pauli Kuosmanen presents both a useful selection guide for practitioners seeking a signal processing solution and a valuable panoramic perspective for theoreticians interested in the underlying principles on which nonlinear digital filters are based. For example, the results presented in their book make it clear that most popular nonlinear filters exhibit homogeneous scaling behavior, and are mostly based on non-smooth (i.e., “median-like”) functions, rendering the Taylor series expansions popular in many engineering disciplines largely useless. What is less obvious is that these two characteristics are closely related, a result that comes from the theory of functional equations. This paper examines a range of algebraic ideas (e.g., categories, groupoids, clones, etc.) that can be useful in characterizing and designing nonlinear filters, building in part on our past joint work with Jaakko Astola.