A Functional Composition Approach to Filter Sharpening and Modular Filter Design

Designing and implementing systems as an interconnection of smaller subsystems is a common practice for modularity and standardization of components and design algorithms. Although not typically cast in this framework, many of these approaches can be viewed within the mathematical context of functional composition. This paper re-interprets and generalizes within the functional composition framework one such approach known as filter sharpening, i.e., interconnecting filter modules which have significant approximation error in order to obtain improved filter characteristics. More specifically, filter sharpening is approached by determining the composing polynomial to minimize the infinity-norm of the approximation error, utilizing the First Algorithm of Remez. This is applied both to sharpening for FIR, even-symmetric filters and for the more general case of subfilters that have complex-valued frequency responses including causal IIR filters and for continuous-time filters. Within the framework of functional composition, this paper also explores the use of functional decomposition to approximate a desired system as a composition of simpler functions based on a two-norm on the approximation error. Among the potential advantages of this decomposition is the ability for modular implementation in which the inner component of the functional decomposition represents the subfilters and the outer the interconnection.