# Simultaneous diagonalisation of the covariance and complementary covariance matrices in quaternion widely linear signal processing

## TL;DR

This paper introduces novel quaternion matrix decomposition techniques to simultaneously diagonalize covariance and complementary covariance matrices, enhancing widely linear signal processing capabilities for improper quaternion signals.

## Contribution

It develops a quaternion Takagi factorisation-based method for simultaneous diagonalization of multiple covariance matrices, advancing quaternion uncorrelating transforms.

## Key findings

- Effective diagonalization of quaternion covariance matrices demonstrated
- Improved quaternion uncorrelating transforms validated on synthetic data
- Enhanced processing of improper quaternion signals shown in simulations

## Abstract

Recent developments in quaternion-valued widely linear processing have established that the exploitation of complete second-order statistics requires consideration of both the standard covariance and the three complementary covariance matrices. Although such matrices have a tremendous amount of structure and their decomposition is a powerful tool in a variety of applications, the non-commutative nature of the quaternion product has been prohibitive to the development of quaternion uncorrelating transforms. To this end, we introduce novel techniques for a simultaneous decomposition of the covariance and complementary covariance matrices in the quaternion domain, whereby the quaternion version of the Takagi factorisation is explored to diagonalise symmetric quaternion-valued matrices. This gives new insights into the quaternion uncorrelating transform (QUT) and forms a basis for the proposed quaternion approximate uncorrelating transform (QAUT) which simultaneously diagonalises all four covariance matrices associated with improper quaternion signals. The effectiveness of the proposed uncorrelating transforms is validated by simulations on both synthetic and real-world quaternion-valued signals.