Modelling Improper Complex-Valued Signals using a Stochastic Differential Equation Approach.

Complex-valued signals are often observed to be improper, meaning that the complementary covariance or complementary spectrum of the signal is non-zero. Stochastic models for improper signals are often represented as widely linear filters of discrete-time noise processes. In this paper we propose an alternative perspective and model the signal in continuous time using a stochastic differential equation (SDE) approach. Specifically, we propose a first order SDE representation of a complex-valued signal which generates impropriety in the form of elliptical oscillations in the signal's trajectory. The key benefit of our approach is that elliptical trajectories can be generated using one simple first order SDE, whereas the alternative of bivariate modelling requires more complicated vectorised or higher order SDE representations. The second key benefit is that parameter estimation can be performed directly using only the power spectral density of the complex-valued signal, without having to compute cross spectra of individual signal components. Our proposed model can be interpreted as a widely linear version of the complex Ornstein-Uhlenbeck (OU) process. We determine properties of the model including the conditions for stationarity, and the geometrical structure of the elliptical oscillations. We apply the model to measure periodic and elliptical properties of Earth's polar motion.