[Paper Review] Stochastic Modeling and Estimation of Stationary Complex-Valued Signals

Adam M. Sykulski, Sofia C. Olhede+2 more|arXiv (Cornell University)|Jun 25, 2013

Blind Source Separation Techniques3 citations

TL;DR

This paper introduces a stochastic modeling framework for stationary complex-valued signals using rotary components to capture improper or anisotropic signal structures. By extending the Whittle likelihood to complex signals and enabling frequency-domain parameter estimation, it enables efficient inference, model selection, and testing for impropriety in fluid dynamics turbulence data.

ABSTRACT

This paper provides a stochastic modeling framework for the power spectral representations of stationary complex-valued signals. We specify how complex-valued signals can be modeled stochastically in terms of their rotary components, which decompose a bivariate signal according to direction of rotation. The necessary relationships are provided to map between complex-rotary and bivariate-Cartesian representations. We demonstrate how by modeling in rotary components we can infer useful features from application datasets---in particular for capturing the improper or anisotropic structure of a signal---by implementing our methodology on fluid dynamic simulations of turbulence. In addition, we detail how parameters of a chosen stochastic model can be efficiently estimated in the frequency domain, by extending the Whittle likelihood to complex-valued signals. We also provide a new method of testing for complex structure such as impropriety, as well as procedures for model choice and semi-parametric modeling.

Motivation & Objective

• To develop a stochastic modeling approach for stationary complex-valued signals that captures improper or anisotropic signal structures.
• To establish mathematical mappings between rotary (complex) and bivariate-Cartesian representations of signals.
• To enable efficient frequency-domain parameter estimation via an extension of the Whittle likelihood to complex-valued signals.
• To provide a new statistical test for detecting complex signal structure, such as impropriety.
• To support model choice and semi-parametric modeling in the context of complex-valued signal analysis.

Proposed method

• Modeling complex-valued signals through their rotary components, which represent directional rotation in bivariate signal space.
• Deriving mathematical relationships to map between rotary and bivariate-Cartesian signal representations.
• Extending the Whittle likelihood to complex-valued signals for efficient frequency-domain parameter estimation.
• Proposing a new test statistic to detect impropriety in complex-valued signals, indicating non-circularity or anisotropy.
• Integrating procedures for model selection and semi-parametric modeling within the proposed framework.
• Applying the framework to fluid dynamic simulations of turbulence to demonstrate practical utility.

Experimental results

Research questions

• RQ1 How can complex-valued signals be stochastically modeled using rotary components to capture directional and improper signal characteristics?
• RQ2 What is the appropriate mapping between rotary and bivariate-Cartesian representations of complex signals?
• RQ3 How can the Whittle likelihood be adapted for efficient parameter estimation in the frequency domain for complex-valued signals?
• RQ4 What statistical test can reliably detect impropriety or anisotropy in complex-valued signal data?
• RQ5 How can model choice and semi-parametric modeling be effectively integrated into the stochastic framework for complex signals?

Key findings

• The rotary component representation enables effective modeling of improper or anisotropic signal structures in complex-valued signals.
• The extended Whittle likelihood provides a computationally efficient method for frequency-domain parameter estimation in complex-valued signal models.
• A new statistical test for impropriety is introduced, allowing detection of non-circularity in complex signals.
• The framework supports model choice and semi-parametric modeling, enhancing flexibility for real-world applications.
• Application to fluid dynamic turbulence simulations confirms the method’s ability to infer meaningful signal features from complex-valued data.