1. Algorithmic enhancements to polynomial matrix factorisations
- Author
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Coutts, Fraser Kenneth, Weiss, Stephan, and Marshall, Stephen
- Subjects
621.3 - Abstract
In broadband array processing applications, an extension of the eigenvalue decomposition (EVD) to parahermitian Laurent polynomial matrices - named the polynomial matrix EVD (PEVD) - has proven to be a useful tool for the decomposition of spacetime covariance matrices and their associated cross-spectral density matrices. Existing PEVD methods typically operate in the time domain and utilise iterative frameworks established by the second-order sequential best rotation (SBR2) or sequential matrix diagonalisation (SMD) algorithms. However, motivated by recent discoveries that establish the existence of an analytic PEVD - which is rarely recovered by SBR2 or SMD - alternative algorithms that better meet analyticity by operating in the discrete Fourier transform (DFT)-domain have received increasing attention. While offering promising results in applications including broadband MIMO and beamforming, the PEVD has seen limited deployment in hardware due to its high computational complexity. If the PEVD is to be fully utilised, overcoming this bottleneck is paramount. This thesis therefore seeks to reduce the computational cost of iterative PEVD algorithms - with particular emphasis on SMD - through the development of several novel algorithmic improvements. While these are effective, the complexity of the optimised algorithms still grows rapidly with the spatial dimensions of the decomposition. Steps are therefore taken to convert the sequential form of SMD to a novel reduced dimensionality and partially parallelisable divide-and-conquer architecture. The resulting algorithms are shown to converge an order of magnitude faster than existing methods for large spatial dimensions, and are well-suited to application scenarios with many sensors. Further in this thesis, an investigation into DFT-based algorithms highlights their potential to offer compact, analytic solutions to the PEVD. Subsequently, two novel DFT-based algorithms improve upon an existing method by reducing decomposition error and eliminating a priori knowledge requirements. Finally, an innovative strategy is shown to be capable of extracting a minimum-order solution to the PEVD.
- Published
- 2019
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