Polynomial matrix decompositions and paraunitary filter banks

Abstract

There are an increasing number of problems that can be solved using paraunitary filter banks. The design of optimal orthonormal filter banks for the efficient coding of signals has received considerable interest over the years. In contrast, very little attention has been given to the problem of constructing paraunitary matrices for the purpose of broadband signal subspace estimation. This thesis begins by relating these two areas of research. A frequency-domain method of diagonalising parahermitian polynomial matrices is proposed and shown to have fundamental limitations. Then the thesis focuses on the development of a novel time-domain technique that extends the eigenvalue decomposition to polynomial matrices, referred to as the second order sequential best rotation (SBR2) algorithm. This technique imposes strong decorrelation on its input signals by applying a sequence of elementary paraunitary matrices which constitutes a generalisation of the classical Jacobi algorithm to the field of polynomial matrices. It is shown to be highly applicable to the problems of broadband signal subspace estimation and data compression. Variations on the algorithm are presented which give a significant improvement in subspace estimation accuracy and data compression performance.

Discussions are then mainly concerned with the application of the SBR2 algorithm to the problem of data compression, particularly the adaptation of the SBR2 algorithm to subband coding. The relevance of the algorithm to traditional orthonormal filter bank design methods is examined, highlighting that these techniques are based on an implicit assumption regarding the statistics of the input signal. This provides motivation for the development of a method of exploiting this knowledge for use with the SBR2 algorithm. The resulting algorithm can design orthonormal filter banks for subband coding. The suboptimality, in the sense of maximising the coding gain, of the filter bank constructed becomes negligible as the number of algorithm iterations increases. The technique is shown to compare favourably to the state-of-the-art on a set of benchmark problems.

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