The Orthogonal Band Decomposition of the Finite Dirichlet Matrix and its Applications
| dc.contributor.advisor | Sowa, Artur | |
| dc.contributor.advisor | Babyn, Paul | |
| dc.contributor.committeeMember | Bui, Francis | |
| dc.contributor.committeeMember | Frank, Cameron | |
| dc.contributor.committeeMember | Soteros, Christine | |
| dc.creator | Vlahu, Izabela 1986- | |
| dc.date.accessioned | 2018-04-26T15:04:43Z | |
| dc.date.available | 2018-04-26T15:04:43Z | |
| dc.date.created | 2018-06 | |
| dc.date.issued | 2018-04-26 | |
| dc.date.submitted | June 2018 | |
| dc.date.updated | 2018-04-26T15:04:44Z | |
| dc.description.abstract | In 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.mimetype | application/pdf | |
| dc.identifier.uri | http://hdl.handle.net/10388/8523 | |
| dc.subject | Finite Dirichlet Matrix | |
| dc.subject | Non-trivial factorizations | |
| dc.subject | resolutions of identity | |
| dc.title | The Orthogonal Band Decomposition of the Finite Dirichlet Matrix and its Applications | |
| dc.type | Thesis | |
| dc.type.material | text | |
| thesis.degree.department | Mathematics and Statistics | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | University of Saskatchewan | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy (Ph.D.) |