OPTIMUM PIECEWISE-LINEAR TRANSCODERS .1. WEIGHTED MINIMAX PIECEWISE-LINEAR APPROXIMATION AND MINIMAX DECOMPOSITION OF PIECEWISE FUNCTIONS

This paper mainly concerns the following mathematical problem: an initial single-argument single-valued function F is known only through a set of points P-i(X(l), Y-i,W-i), with Y-i = F(X(i)), and with application-dependent weights W-i. An optimum (or almost-optimum in some cases) approximating function f should be derived from these points P-i. f is 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 derive f from points P-i 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 P-i. 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 1994 a), which shows that the proposed approach applies to a wide class of systems, including active filters and phase shifters.