Thesis
Advanced polynomial matrix factorization techniques
- Creator
- Rights statement
- Awarding institution
- University of Strathclyde
- Date of award
- 2024
- Thesis identifier
- T16925
- Person Identifier (Local)
- 202171539
- Qualification Level
- Qualification Name
- Department, School or Faculty
- Abstract
- The advancements in polynomial matrix algebra have significantly bolstered the formulation of broadband array problems, enabling more effective solutions. This is particularly evident through the utilization of polynomial matrix factorization techniques such as the polynomial eigenvalue decomposition (PEVD), the polynomial singular value decomposition (PSVD), and the polynomial QR decomposition (PQRD), which have notably revolutionized the approach to solving these complex problems. Therefore, these polynomial matrix factorization methods have received significant attention over the last decades. Still, the polynomial order of the decompositions, complexity scaling with spatial and temporal dimensions of the matrix, and issues regarding parallelizability remain pertinent. Therefore, this thesis address these issues in these three mentioned polynomial factorization methods. This thesis demonstrates that in most practical situations, eigen- and singular values will be spectrally majorised. We exploit this property in the algorithms proposed in this thesis. The first and foremost algorithm is for applications such as data compaction or dominant component extraction, where the power method is extended to the para- Hermitian polynomial matrices for the extraction of a dominant eigenvector. This approach prevents computing an entire PEVD of a para-Hermitian matrix. Later, this extension is combined with a deflation approach to compute the PEVD of a low rank polynomial matrix. Perturbation bounds are computed which show that these bounds increase with repeated deflations. Ensemble tests reveal its superior performance over state-of-the-art PEVD algorithms. To accommodate non-para-Hermitian polynomial matrices, the polynomial power method is extended to general polynomial matrices for the extraction of dominant left and right singular vectors and the corresponding dominant singular value. To reduce iterations in the polynomial power method to just one, the provided polynomial matrix is decomposed into a sum of rank-one matrices using a computationally inexpensive approach in the discrete Fourier transform (DFT) domain. Subsequently, each rank-one term is subjected to a single iteration of the polynomial power method, resulting in the corresponding eigenpair or the left- and right singular vectors and the associated singular value. For the PQRD, rank one terms are obtained by computing QRDs in the DFT bins. To obtain any column of the paraunitary matrix and the corresponding row of the upper-right triangular matrix, any column of the respective rank one matrix is normalized to unit norm on the unit circle. This rank one decomposition based method is termed as unified-I algorithm which is highly parallelizable and outperforms all state-of-the-art algorithms in accuracy of the decomposition and execution time. Lastly, this thesis demonstrates that a further reduction in the order of any of the above decompositions can be achieved via assessing the auto- and cross-correlation terms of eigen- or singular vectors. A spectral factorisation of these terms can then lead to the compact polynomial order factors. This spectral factorization is obtained via polynomial root finding method to avoid the positive definite restriction of a Laurent polynomial on the unit circle. Ensemble results show its superior performance over state-of-the-art algorithms including the rank decomposition method proposed in this thesis.
- Advisor / supervisor
- Weiss, Stephan, 1968-
- Resource Type
- Note
- Previously held under moratorium from 29th April 2024 until 22nd May 2025.
- DOI
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PDF of thesis T16925 | 2024-05-29 | Public | Download |