Simultaneous diagonalisation of the covariance and complementary covariance matrices in quaternion widely linear signal processing
For researchers in quaternion signal processing, this provides a method to exploit complete second-order statistics, addressing a known bottleneck due to non-commutative quaternion algebra.
The paper introduces novel techniques for simultaneous diagonalisation of covariance and complementary covariance matrices in quaternion widely linear signal processing, enabling the quaternion approximate uncorrelating transform (QAUT) that diagonalises all four covariance matrices for improper quaternion signals. Simulations on synthetic and real-world data validate its effectiveness.
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.