Optimum piecewise-linear transcoders Part 1. Weighted minimax piecewise-linear approximation and minimax decomposition of piecewise functions

Abstract

This paper mainly concerns the following mathematical problem: an initial single-argument single-valued function F is known only through a set of points Pi(Xi; Yi Wi, with Yi = F(Xi), and with application-dependent weights Wi. An optimum (or almost-optimum in some cases) approximating function/should be derived from these points Pi f searched within a predefined class of functions. Various such classes are successively considered. They consist of subsets of piecewise-linear functions. The approximation criterion used to derivef from points Pi consists of determining an approximating function which minimizes an overall error. This error is typically defined as the maximum among local weighted errors associated with each point Pi Beyond piecewise-linear approximation, this paper also presents algorithms for optimizing the domains of operation of the subfunctions of any type of piecewise function acccording to a possibly-weighted minimax criterion. This investigation is motivated by an industrial application, i.e. automatic TV tuner alignment. This application is outlined in this paper and detailed in a companion paper (see Deville 1994a), which shows that the proposed approach applies to a wide class of systems, including active filters and phase shifters.