The harmonizable representation of complex-valued nonstationary random processes

Abstract

In this thesis we study the second-order statistical moment functions that characterizes complex-valued harmonizable processes. A real-valued harmonizable process has four Hermitian second-order functions. These functions are equivalent representations of the second-order statistical quantities of the process. For a complex-valued harmonizable process, however, we need the complementary functions in addition to the Hermitian functions to completely describe the second-order statistical behavior of the process. We define and discuss the Hermitian and the complementary functions of complex-valued harmonizable processes. Exact expressions for the Hermitian and complementary second-order moment functions for some important sub-classes of complex-valued harmonizable processes are derived and discussed. We introduce, test, and characterize numerical estimators of these functions. Numerically generated data, and a real-world earthquake data set, are employed to demonstrate that the estimators work in practice. Based on the concept of widely linear mean square estimation, we propose a novel generalized measure of coherence for harmonizable random processes. This measure generalizes a recently proposed measure of coherence based on linear mean square estimation. Finally, we argue that our alternative coherence measure may in fact be beneficial for most nonstationary processes, even for real-valued harmonizable processes.