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

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.Comment: 41 pages, single column, 10 figure

On the unitary diagonalisation of a special class of quaternion matrices

AbstractWe propose a unitary diagonalisation of a special class of quaternion matrices, the so-called η-Hermitian matrices A=AηH,η∈{ı,j,κ} arising in widely linear modelling. In 1915, Autonne exploited the symmetric structure of a matrix A=AT to propose its corresponding factorisation (also known as the Takagi factorisation) in the complex domain C. Similarly, we address the factorisation of an ‘augmented’ class of quaternion matrices, by taking advantage of their structures unique to the quaternion domain H. Applications of such unitary diagonalisation include independent component analysis and convergence analysis in statistical signal processing

On the unitary diagonalisation of a special class of quaternion matrices

We propose a unitary diagonalisation of a special class of quaternion matrices, the so-called η-Hermitian matrices A=A , η∈,{l,j,κ} arising in widely linear modelling. In 1915, Autonne exploited the symmetric structure of a matrix A=A to propose its corresponding factorisation (also known as the Takagi factorisation) in the complex domain C. Similarly, we address the factorisation of an 'augmented' class of quaternion matrices, by taking advantage of their structures unique to the quaternion domain H. Applications of such unitary diagonalisation include independent component analysis and convergence analysis in statistical signal processing. © 2011 Elsevier Ltd. All rights reserved