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.
| Date of Award | 24 Apr 2024 |
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| Original language | English |
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| Awarding Institution | - University Of Strathclyde
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| Sponsors | University of Strathclyde |
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| Supervisor | Stephan Weiss (Supervisor) & Vladimir Stankovic (Supervisor) |
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