Talk
Abstract:
Fractional Gaussian Noise in Linear Systems
Tyrone E. Duncan
University of Kansas
Fractional Gaussian noise is a stochastic process that is the
formal derivative of a fractional Brownian motion. Fractional
Brownian motions are a family of Gaussian processes that are indexed
by the Hurst parameter, H, in the interval (0, 1). These processes
have a property of self similarity of the probability laws under
a scale change in time. For H in (1/2, 1) these processes have
a property of long range dependence that is characterized by the
relatively slow decay of the correlation (or covariance) function.
These properties of self similarity and long range dependence
have been observed empirically, initially in hydrology by Hurst,
subsequently in economic data, and most recently in telecommunications.
This presentation will focus on the fractional Brownian motions
with a long range dependence, that is, H in (1/2, 1) and will
contrast the results with those for Brownian motion, that is,
H = _. To analyze systems with fractional Gaussian noise, it is
important to have an effective stochastic calculus for fractional
Brownian motion by analogy to the use of white Gaussian noise
in systems via a stochastic calculus for Brownian motion. Various
properties of linear systems excited by fractional Gaussian noise
are considered. Explicit solutions for the linear systems are
given, conditions for stability and instability are described
and limiting distributions are characterized. Some Radon-Nikodym
derivatives, that are important for statistical applications,
are described. Some applications to linear filtering and the calculation
of mutual information are given. Some relations to the spectral
description of a fractional Gaussian noise are given.
Modeling
and Analysis of Noise in Integrated Circuits and Systems "IMA
"Hot Topics" Workshops
|
