Initial results on an MMSE precoding and equalisation approach to MIMO PLC channels

This paper addresses some initial experiments using polynomial matrix decompositions to construct MMSE precoders and equalisers for MIMO power line communications (PLC) channels. The proposed scheme is based on a Wiener formulation based on polynomial matrices, and recent results to design and implement such systems with polynomial matrix tools. Applied to the MIMO PLC channel, the strong spectral dynamics of the PLC system together with the long impulse responses contained in the MIMO system result in problems, such that diagonlisation and spectral majorisation is mostly achieved in bands of high energy, while low-energy bands can resist any diagonalisation efforts. We introduce the subband approach in order to deal with this problem. A representative example using a simulated MIMO PLC channel is presented

Comparative study for broadband direction of arrival estimation techniques

This paper reviews and compares three different linear algebraic signal subspace techniques for broadband direction of arrival estimation --- (i) the coherent signal subspace approach, (ii) eigenanalysis of the parameterised spatial correlation matrix, and (iii) a polynomial version of the multiple signal classification algorithm. Simulation results comparing the accuracy of these methods are presented

Relevance of polynomial matrix decompositions to broadband blind signal separation

The polynomial matrix EVD (PEVD) is an extension of the conventional eigenvalue decomposition (EVD) to polynomial matrices. The purpose of this article is to provide a review of the theoretical foundations of the PEVD and to highlight practical applications in the area of broadband blind source separation (BSS). Based on basic definitions of polynomial matrix terminology such as parahermitian and paraunitary matrices, strong decorrelation and spectral majorization, the PEVD and its theoretical foundations will be briefly outlined. The paper then focuses on the applicability of the PEVD and broadband subspace techniques — enabled by the diagonalization and spectral majorization capabilities of PEVD algorithms—to define broadband BSS solutions that generalise well-known narrowband techniques based on the EVD. This is achieved through the analysis of new results from three exemplar broadband BSS applications — underwater acoustics, radar clutter suppression, and domain-weighted broadband beamforming — and their comparison with classical broadband methods

Shortening of paraunitary matrices obtained by polynomial eigenvalue decomposition algorithms

This paper extends the analysis of the recently introduced row-shift corrected truncation method for paraunitary matrices to those produced by the state-of-the-art sequential matrix diagonalisation (SMD) family of polynomial eigenvalue decomposition (PEVD) algorithms. The row-shift corrected truncation method utilises the ambiguity in the paraunitary matrices to reduce their order. The results presented in this paper compare the effect a simple change in PEVD method can have on the performance of the paraunitary truncation. In the case of the SMD algorithm the benefits of the new approach are reduced compared to what has been seen before however there is still a reduction in both reconstruction error and paraunitary matrix order

On the existence and uniqueness of the eigenvalue decomposition of a parahermitian matrix

This paper addresses the extension of the factorisation of a Hermitian matrix by an eigenvalue decomposition (EVD) to the case of a parahermitian matrix that is analytic at least on an annulus containing the unit circle. Such parahermitian matrices contain polynomials or rational functions in the complex variable z, and arise e.g. as cross spectral density matrices in broadband array problems. Specifically, conditions for the existence and uniqueness of eigenvalues and eigenvectors of a parahermitian matrix EVD are given, such that these can be represented by a power or Laurent series that is absolutely convergent, at least on the unit circle, permitting a direct realisation in the time domain. Based on an analysis on the unit circle, we prove that eigenvalues exist as unique and convergent but likely infinite-length Laurent series. The eigenvectors can have an arbitrary phase response, and are shown to exist as convergent Laurent series if eigenvalues are selected as analytic functions on the unit circle, and if the phase response is selected such that the eigenvectors are Hölder continuous with α>½ on the unit circle. In the case of a discontinuous phase response or if spectral majorisation is enforced for intersecting eigenvalues, an absolutely convergent Laurent series solution for the eigenvectors of a parahermitian EVD does not exist. We provide some examples, comment on the approximation of a parahermitian matrix EVD by Laurent polynomial factors, and compare our findings to the solutions provided by polynomial matrix EVD algorithms

Cyclic-by-row approximation of iterative polynomial EVD algorithms

A recent class of sequential matrix diagonalisation (SMD) algorithms have been demonstrated to provide a fast converging solution to iteratively approximating the polynomial eigenvalue decomposition of a parahermitian matrix. However, the calculation of an EVD, and the application of a full unitary matrix to every time lag of the parahermitian matrix in the SMD algorithm results in a high numerical cost. In this paper, we replace the EVD with a limited number of Givens rotations forming a cyclic-by-row Jacobi sweep. Simulations indicate that a considerable reduction in computational complexity compared to SMD can be achieved with a negligible sacrifice in diagonalisation performance, such that the benefits in applying the SMD are maintained