Distortion

Abstract

We now turn to quantification of various notions of the distortion between random variables, vectors and processes. A distortion measure is not a “measure” in the sense used so far; it is an assignment of a nonnegative real number which indicates how bad an approximation one symbol or random object is of another; the smaller the distortion, the better the approximation. If the two objects correspond to the input and output of a communication system, then the distortion provides a measure of the performance of the system. Distortion measures need not have metric properties such as the triangle inequality and symmetry, but such properties can be exploited when available. We shall encounter several notions of distortion and a diversity of applications, with eventually the most important application being a measure of the performance of a communications system by an average distortion between the input and output. Other applications include extensions of finite memory channels to channels which approximate finite memory channels and different characterizations of the optimal performance of communications systems.