# Spectral analysis of stationary random bivariate signals

## TL;DR

This paper introduces a quaternion-based spectral analysis method for stationary bivariate signals, enabling polarization-aware spectral density estimation and decomposition into polarized and unpolarized components.

## Contribution

It presents a novel quaternion Fourier transform approach for spectral analysis of bivariate signals, including a new spectral density definition and polarization periodogram.

## Key findings

- Effective spectral density estimation demonstrated on synthetic data
- Decomposition into polarized and unpolarized components achieved
- The approach enhances understanding of frequency-dependent polarization attributes

## Abstract

A novel approach towards the spectral analysis of stationary random bivariate signals is proposed. Using the Quaternion Fourier Transform, we introduce a quaternion-valued spectral representation of random bivariate signals seen as complex-valued sequences. This makes possible the definition of a scalar quaternion-valued spectral density for bivariate signals. This spectral density can be meaningfully interpreted in terms of frequency-dependent polarization attributes. A natural decomposition of any random bivariate signal in terms of unpolarized and polarized components is introduced. Nonparametric spectral density estimation is investigated, and we introduce the polarization periodogram of a random bivariate signal. Numerical experiments support our theoretical analysis, illustrating the relevance of the approach on synthetic data.