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Modelling improper complex-valued signals using a stochastic differential equation approach

Modelling improper complex-valued signals using a stochastic differential equation approach
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.
arXiv
Sykulski, Adam
cdd48194-6105-4501-8ac3-462adefa75b3
Olhede, S.C.
b0f12dd2-270c-440e-8c01-86a2f36c390e
Sykulska-Lawrence, Hanna
844512fc-78fb-420c-8d16-fefa2ce35d26
Sykulski, Adam
cdd48194-6105-4501-8ac3-462adefa75b3
Olhede, S.C.
b0f12dd2-270c-440e-8c01-86a2f36c390e
Sykulska-Lawrence, Hanna
844512fc-78fb-420c-8d16-fefa2ce35d26

Sykulski, Adam, Olhede, S.C. and Sykulska-Lawrence, Hanna (2020) Modelling improper complex-valued signals using a stochastic differential equation approach arXiv 20pp.

Record type: Monograph (Working Paper)

Abstract

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.

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2001.05965 - Version of Record
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Published date: 16 January 2020

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Local EPrints ID: 445124
URI: http://eprints.soton.ac.uk/id/eprint/445124
PURE UUID: 42dd5d16-30d6-4837-9860-0ee23e2d8220

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Date deposited: 20 Nov 2020 17:31
Last modified: 20 Nov 2020 17:31

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Contributors

Author: Adam Sykulski
Author: S.C. Olhede

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