A complete framework for linear filtering of bivariate signals

TL;DR

This paper introduces a comprehensive quaternion Fourier transform-based framework for linear filtering of bivariate signals, enabling physically interpretable filter design and advanced spectral synthesis with polarization properties.

Contribution

It presents a novel, complete LTI filtering framework for bivariate signals using quaternion Fourier transforms, allowing direct eigenproperty-based filter design and insights into polarization filtering.

Findings

Efficient spectral synthesis method developed.

Framework supports polarization property control.

Numerical experiments validate theoretical results.

Abstract

A complete framework for the linear time-invariant (LTI) filtering theory of bivariate signals is proposed based on a tailored quaternion Fourier transform. This framework features a direct description of LTI filters in terms of their eigenproperties enabling compact calculus and physically interpretable filtering relations in the frequency domain. The design of filters exhibiting fondamental properties of polarization optics (birefringence, diattenuation) is straightforward. It yields an efficient spectral synthesis method and new insights on Wiener filtering for bivariate signals with prescribed frequency-dependent polarization properties. This generic framework facilitates original descriptions of bivariate signals in two components with specific geometric or statistical properties. Numerical experiments support our theoretical analysis and illustrate the relevance of the approach on synthetic data.