Ambiguity domain definitions and covariance function estimation for
nonstationary random processes in discrete time
Johan Sandberg
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2008
ISSN 1404028X

Abstract:

The ambiguity domain plays a central role in estimating the timevarying
spectrum of a nonstationary random process in continuous time, since
multiplication in this domain is equivalent with estimating the covariance
function of the random process using an intuitively appealing estimator.
For processes in discrete time there exists a corresponding covariance function
estimator. The ambiguity domain was originally defined for processes in
continuous time and by its construction it is not trivial to

define a similar concept for processes in discrete time. Several different
definitions have been proposed. In Paper A we examine three of the most
frequently used definitions and prove that only one of them has the important
property that multiplication is equivalent with the mentioned covariance
function estimator. Another useful property of the continuous ambiguity domain
is that the mean square error optimal covariance function estimator has an
attractive formulation in this domain. In Paper B we prove that none of the
three examined ambiguity domain definitions for discrete processes has this
property. However, we prove that the optimal estimator can be computed without
the use of the ambiguity domain for processes in discrete time. In Paper
C we prove that the mean square error

optimal covariance function estimator of the form discussed in this thesis,
can be computed for any parameterized family of random processes as the solution
to a system of linear equations. Examples of families and their corresponding
optimal estimators are given.



