Algorithms and techniques for polynomial matrix decompositions

Abstract

The concept of polynomial matrices is introduced and the potential application of polynomial matrix decompositions is discussed within the general context of multi-channel digital signal processing. A recently developed technique, known as the second order sequential rotation algorithm (SBR2), for performing the eigenvalue decomposition of a para-Hermitian polynomial matrix (PEVD) is presented. The potential benefit of using the SBR2 algorithm to impose strong decorrelation on the signals received by a broadband sensor array is demonstrated by means of a suitable numerical simulation. This demonstrates how the polynomial matrices produced as a result of the PEVD can be of unnecessarily high order. This is undesirable for many practical applications and slows down the iterative computational procedure. An effective truncation technique for controlling the growth in order of these polynomial matrices is proposed. Depending on the choice of truncation parameters, it provides an excellent compromise between reduced order polynomial matrix factors and accuracy of the resulting decomposition. This is demonstrated by means of a set of numerical simulations performed by applying the modified SBR2 algorithm with a variety of truncation parameters to a representative set of test matrices. Three new polynomial matrix decompositions are then introduced - one for implementing a polynomial matrix QR decomposition (PQRD) and two for implementing a polynomial matrix singular value decomposition (PSVD). Several variants of the PQRD algorithm (including polynomial order reduction) are proposed and compared by numerical simulation using an appropriate set of test matrices. The most effective variant w.r.t. computational speed, order of the polynomial matrix factors and accuracy of the resulting decomposition is identified. The PSVD can be computed using either the PEVD technique, based on the SBR2 algorithm, or the new algorithm proposed for implementing the PQRD. These two approaches are also compared by means of computer simulations which demonstrate that the method based on the PQRD is numerically superior. The potential application of the preferred PQRD and PSVD algorithms to multiple input multiple output (MIMO) communications for the purpose of counteracting both co-channel interference and inter-symbol interference (multi-channel equalisation) is demonstrated in terms of reduced bit error rate by means of representative computer simulations