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      • HARVEST
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      The Orthogonal Band Decomposition of the Finite Dirichlet Matrix and its Applications

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      VLAHU-DISSERTATION-2018.pdf (1.453Mb)
      Date
      2018-04-26
      Author
      Vlahu, Izabela 1986-
      Type
      Thesis
      Degree Level
      Doctoral
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      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.
      Degree
      Doctor of Philosophy (Ph.D.)
      Department
      Mathematics and Statistics
      Program
      Mathematics
      Supervisor
      Sowa, Artur; Babyn, Paul
      Committee
      Bui, Francis; Frank, Cameron; Soteros, Christine
      Copyright Date
      June 2018
      URI
      http://hdl.handle.net/10388/8523
      Subject
      Finite Dirichlet Matrix
      Non-trivial factorizations
      resolutions of identity
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