Complex time-frequency and dual-frequency spectra of harmonizable processes

Harmonizable processes form a huge and useful class of nonstationary random processes. In this paper, we will discuss the properties of, and some consequences of employing a natural choice of complex valued densities to characterize the harmonizable class. In particular, we will discuss the Hilbert space geometry of the resulting complex time-frequency description (related to the Rihaczek distribution), and the dual-frequency description (related to the Loève spectrum). We will demonstrate that useful normalized nonstationary generalizations of coherence emerge from this picture. Finally, we extend the formalism to random fields and to higher-order cases, and we will discuss possible estimators.