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dc.contributor.advisorSowa, Artur
dc.contributor.advisorBabyn, Paul
dc.creatorVlahu, Izabela 1986-
dc.date.accessioned2018-04-26T15:04:43Z
dc.date.available2018-04-26T15:04:43Z
dc.date.created2018-06
dc.date.issued2018-04-26
dc.date.submittedJune 2018
dc.identifier.urihttp://hdl.handle.net/10388/8523
dc.description.abstractIn my work I establish and extend the theory of finite D-matrices for the purposes of signal processing applications in the finite, digital setting. Finite D-matrices are obtained by truncating infinite D-matrices to upper-left corners. I show that finite D-matrices are furnished with a number-theoretical structure that is not present in their infinite counterparts. In particular, I show that the columns of every finite D-matrix of size $N\times N$ admits a natural, non-trivial, Orthogonal Band Decomposition, induced by the Floor Band Decomposition on the finite set $\{1,2,\dots,N\}$. When the D-matrix is invertible, its Orthogonal Band Decomposition induces a non-trivial resolution of the identity. Furthermore, for every finite D-matrix $A$, I show that the sum $P$ of the orthogonal projections corresponding to each band of $A$ admits the following sparse representation $P=A\Lambda^{-1} A^*$, where $\Lambda$ is a special diagonal matrix and $A^\star$ is the Hermitian adjoint of $A$. I also show that the matrix $P$ and its inverse induce another non-trivial resolution of the identity. Being a sum of projection matrices, I call the matrix $P$ the associated P-matrix of $A$. Both the finite D-matrices and their associated P-matrices can be applied in the processing of digital signals. For example, given a D-matrix $A$, its associated P-matrix allows us to pass from a signal representation in the Fourier basis to a representation, as a sum of projections, in the basis induced by the Orthogonal Band Decomposition of $A$. Preliminary experiments suggest that the error of approximating signals with partial sums of projections might offer a more suitable metric to choose D-matrix representations in specific applications. Significantly, computations with finite D-matrices and P-matrices can be carried out via fast algorithms, which makes these transforms computationally competitive.
dc.format.mimetypeapplication/pdf
dc.subjectFinite Dirichlet Matrix
dc.subjectNon-trivial factorizations
dc.subjectresolutions of identity
dc.titleThe Orthogonal Band Decomposition of the Finite Dirichlet Matrix and its Applications
dc.typeThesis
dc.date.updated2018-04-26T15:04:44Z
thesis.degree.departmentMathematics and Statistics
thesis.degree.disciplineMathematics
thesis.degree.grantorUniversity of Saskatchewan
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy (Ph.D.)
dc.type.materialtext
dc.contributor.committeeMemberBui, Francis
dc.contributor.committeeMemberFrank, Cameron
dc.contributor.committeeMemberSoteros, Christine


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