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

TL;DR

This paper develops a unified frequency-domain stochastic modeling framework for bivariate, complex-valued, and rotary signals, enabling advanced analysis and testing of their statistical properties, with applications to turbulence data.

Contribution

It introduces a comprehensive framework that extends statistical procedures to all three signal representations and proposes a new impropriety test for complex signals.

Findings

Framework successfully models coherence and anisotropy in turbulence signals.

New impropriety test detects noncircular structures in complex signals.

Method enhances analysis of multivariate and complex-valued time series.

Abstract

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