Signal processing with Levy information

TL;DR

This paper develops a theory of Levy information processes, modeling noise in communication channels using Levy processes and enabling signal detection and enhancement through measure transformations.

Contribution

It introduces a new framework for Levy information processes, expanding the understanding of noise modeling and signal extraction in various Levy process types.

Findings

Developed a general theory of Levy information processes.

Worked out detailed examples for multiple Levy process types.

Provided schemes for signal detection and enhancement.

Abstract

Levy processes, which have stationary independent increments, are ideal for modelling the various types of noise that can arise in communication channels. If a Levy process admits exponential moments, then there exists a parametric family of measure changes called Esscher transformations. If the parameter is replaced with an independent random variable, the true value of which represents a "message", then under the transformed measure the original Levy process takes on the character of an "information process". In this paper we develop a theory of such Levy information processes. The underlying Levy process, which we call the fiducial process, represents the "noise type". Each such noise type is capable of carrying a message of a certain specification. A number of examples are worked out in detail, including information processes of the Brownian, Poisson, gamma, variance gamma, negative binomial, inverse Gaussian, and normal inverse Gaussian type. Although in general there is no additive decomposition of information into signal and noise, one is led nevertheless for each noise type to a well-defined scheme for signal detection and enhancement relevant to a variety of practical situations.