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There are three equivalent ways of representing two jointly observedreal-valued signals: as a bivariate vector signal, as a single complex-valuedsignal, or as two analytic signals known as the rotary components. Eachrepresentation has unique advantages depending on the system of interest andthe application goals. In this paper we provide a joint framework for all threerepresentations in the context of frequency-domain stochastic modeling. Thisframework allows us to extend many established statistical procedures forbivariate vector time series to complex-valued and rotary representations.These include procedures for parametrically modeling signal coherence,estimating model parameters using the Whittle likelihood, performingsemi-parametric modeling, and choosing between classes of nested models usingmodel choice. We also provide a new method of testing for impropriety incomplex-valued signals, which tests for noncircular or anisotropic second-orderstatistical structure when the signal is represented in the complex plane.Finally, we demonstrate the usefulness of our methodology in capturing theanisotropic structure of signals observed from fluid dynamic simulations ofturbulence.

Frequency-Domain Stochastic Modeling of Stationary Bivariate or Complex-Valued Signals