The Identification of Nonlinear Discrete-Time FadingiMemory Systems Using Neural Network Models

A fading-memory system is a system that tends to forget its input asymptotically over time. It has been shown that discrete-time fading-memory systems can be uniformly approximated arbitrarily closely over a set of bounded input sequences simply by uniformly approximating sufficiently closely either the external or internal representation of the system. In other words, the problem of uniformly approximating a fading-memory system reduces to the problem of uniformly approximating continuous real-valued functions on compact sets. The perceptron is a parametric model that realizes a set of continuous real-valued functions that is uniformly dense in the set of all continuous real-valued functions. Using the perceptron to uniformly approximate the external and internal representations of a discrete-time fading-memory system results, respectively, in simple finite-memory and infinite-memory parametric system models. Algorithms for estimating the model parameters that yield a best approximation to a given fading-memory system are discussed. An application to nonlinear noise cancellation in telephone systems is presented.