Iterated function system models of digital channels

Abstract

This paper introduces a new class of models of digital communications channels. Physically, these models take account of the digital nature of the input. Mathematically, they are iterated function systems. As a consequence of making explicit assumptions about the role of discreteness in the models, it is possible to make general statements about the behaviour of these channels without needing to assume that they are linear. We provide the mathematical background necessary to understand the behaviour of these models and prove a number of results about their observability. We also provide a number of examples intended to demonstrate their connection with linear state space models, and to suggest how the nonlinear theory might be developed towards applications.

Left invertibility of discrete systems with finite inputs and quantized output

“…The main tool used in the paper is the theory of Iterated Function Systems (IFS). IFS have been already used as a model in different fields ( [5], [24]). One can construct a natural map in the space of compact subsets of the euclidean space, simply by mapping a set in the union of the images of all contractions forming the IFS.…”

Finite-dimensional behaviour and observability in a randomly forced PDE

“…(The measurement is determined completely by the state of the system when the measurement is made, but note that, because of effects such as dispersion, the state-and the measurement-will typically be influenced by all the symbols that have been transmitted up to the time the measurement is made.) It is important to know whether the states of the system are fixed uniquely by finite sequences of these values (a situation known as 'observability'); in [5,16] we give observability results for the case where the state space is finite-dimensional. It is assumed that the dynamics of the channel (and so the maps in the IFS) are differentiable, but it is not assumed that they are linear.…”

“…In [5] we describe such an approach relevant to the particular area of digital communications; 'digital' here implies that the message to be communicated consists of a sequence of symbols, all taken from a fixed, finite set (or 'alphabet'). The ideas described in [5] are motivated by a fundamentally physical view of signal transmission: a signal is sent by creating an excitation at some location (the transmitter) in a physical medium, and allowing it to propagate to some destination (the receiver). (For example, the medium in wireless radio transmission is the electromagnetic field in the atmosphere or free space; in sonar the medium is the ocean.)…”

“…[5] treats the channel as a dynamical system. The external forcing means the dynamics are non-autonomous, but the fact that the signal is digital means that the input to the channel consists of sequences of elementary inputs, one for each symbol.…”

“…In earlier work [5], aimed at developing an approach to signal processing that can be applied as well to nonlinear systems as linear ones, we produced mathematical models of digital communications channels that took the form of iterated function systems (IFS). For finite-dimensional systems these models have observability properties indicating they could be used for signal processing applications.…”