Ambiguity Domain Definitions and Covariance Function Estimation for Non-Stationary Random Processes in Discrete Time

Abstract: The ambiguity domain plays a central role in estimating the time-varying spectrum of a non-stationary 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.