Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing

We present a new approach for vector-sensor signal modelling and processing, based on the use of quaternion algebra. We introduce the concept of quaternionic signal and give some primary tools to characterize it. We then study the problem of vector-sensor array signals, and introduce a subspace method for wave separation on such arrays. For this purpose, we expose the extension of the Singular Value Decomposition (SVD) algorithm to the field of quaternions. We discuss in more details Singular Value Decomposition for matrices of Quaternions (SVDQ) and linear algebra over quaternions field. The SVDQ allows to calculate the best rank- α approximation of a quaternion matrix and can be used in subspace method for wave separation over vector-sensor array. As the SVDQ takes into account the relationship between components, we show that SVDQ is more efficient than classical SVD in polarized wave separation problem.