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

Collaborative adaptive filtering in the complex domain

A novel hybrid filter combining the complex least mean square (CLMS) and augmented CLMS (ACLMS) algorithms for complex domain adaptive filtering is introduced. The ACLMS has been shown to have improved performance in terms of prediction of non–circular complex data compared to that of the CLMS. By taking advantage of this along with the faster convergence of the CLMS, the hybrid filter is shown to give improved performance over both algorithms for both cir-cular and non–circular data. Simulations on complex–valued synthetic and real world data support the effectiveness of this approach. 1

Modelling of brain consciousness based on collaborative adaptive filters

A novel method for the discrimination between discrete states of brain consciousness is proposed, achieved through examination of nonlinear features within the electroencephalogram (EEG). To allow for real time modes of operation, a collaborative adaptive filtering architecture, using a convex combination of adaptive filters is implemented. The evolution of the mixing parameter within this structure is then used as an indication of the predominant nature of the EEG recordings. Simulations based upon a number of different filter combinations illustrate the suitability of this approach to differentiate between the coma and quasi-brain-death states based upon fundamental signal characteristics