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Reconstruction of Functions From Non-uniformly Distributed Sampled Data in Shift-Invariant Frame Subspaces
The focus of this research is to study and implement efficient iterative reconstruction algorithms. Iterative reconstruction algorithms are used to reconstruct bandlimited signals in shift-invariant L2 subspaces from a set of non-uniformly distributed sampled data. The Shannon-Whittaker reconstructi...
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| Format: | Thesis |
| Language: | English |
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Department of Electrical Engineering
2019
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| _version_ | 1867613334858104832 |
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| access_status_str | Open Access |
| author | Mkhaliphi, Mkhuseli Bruce |
| author2 | Baghai-Wadji, Alireza |
| author_browse | Baghai-Wadji, Alireza Mkhaliphi, Mkhuseli Bruce |
| author_facet | Baghai-Wadji, Alireza Mkhaliphi, Mkhuseli Bruce |
| author_sort | Mkhaliphi, Mkhuseli Bruce |
| collection | Thesis |
| description | The focus of this research is to study and implement efficient iterative reconstruction algorithms. Iterative reconstruction algorithms are used to reconstruct bandlimited signals in shift-invariant L2 subspaces from a set of non-uniformly distributed sampled data. The Shannon-Whittaker reconstruction formula commonly used in uniform sampling problems is insufficient in reconstructing function from non-uniformly distributed sampled data. Therefore new techniques are required. There are many traditional approaches for non-uniform sampling and reconstruction methods where the Adaptive Weights (AW) algorithm is considered to be the most efficient. Recently, the Partitions of Unity (PoU) algorithm has been suggested to outperform the AW although there has been much literature covering its numerical performance. A study and analysis of the implementation of the Adaptive Weights (AW) and Partitions of Unity (PoU) reconstruction methods is conducted. The algorithms consider the missing data problem, defined as reconstructing continuous-time (CT) signals from non-uniform samples which resulted from missing samples on a uniform grid. Mainly, the algorithms convert the non-uniform grid to a uniform grid. The implemented iterative methods construct CT bandlimited functions in frame subspaces. Bandlimited functions are considered to be a superposition of basis functions, named frames. PoU is a variation of AW, they differ by the choice of frame because each frame produces a different approximation operator and convergence rate. If efficiency is defined as the norm convergence and computational time of an algorithm, then among the two methods, discussed, the PoU method is more efficient. The AW method is slow and converged to a higher error than that of the PoU. However, AW compensates for its slowness and less accuracy by being convergent and robust for large sampling gaps and less sensitive to the sampling irregularities. The impact of additive white Gaussian noise on the performance of the two algorithms is also investigated. The numerical tools utilized in this research consist of the theory of discrete irregular sampling, frames, and iterative techniques. The developed software provides a platform for sampling signals under non-ideal conditions with real devices. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/30079 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:34:28.941Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2019 |
| publishDateRange | 2019 |
| publishDateSort | 2019 |
| publisher | Department of Electrical Engineering |
| publisherStr | Department of Electrical Engineering |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/30079 Reconstruction of Functions From Non-uniformly Distributed Sampled Data in Shift-Invariant Frame Subspaces Mkhaliphi, Mkhuseli Bruce Baghai-Wadji, Alireza Engineering The focus of this research is to study and implement efficient iterative reconstruction algorithms. Iterative reconstruction algorithms are used to reconstruct bandlimited signals in shift-invariant L2 subspaces from a set of non-uniformly distributed sampled data. The Shannon-Whittaker reconstruction formula commonly used in uniform sampling problems is insufficient in reconstructing function from non-uniformly distributed sampled data. Therefore new techniques are required. There are many traditional approaches for non-uniform sampling and reconstruction methods where the Adaptive Weights (AW) algorithm is considered to be the most efficient. Recently, the Partitions of Unity (PoU) algorithm has been suggested to outperform the AW although there has been much literature covering its numerical performance. A study and analysis of the implementation of the Adaptive Weights (AW) and Partitions of Unity (PoU) reconstruction methods is conducted. The algorithms consider the missing data problem, defined as reconstructing continuous-time (CT) signals from non-uniform samples which resulted from missing samples on a uniform grid. Mainly, the algorithms convert the non-uniform grid to a uniform grid. The implemented iterative methods construct CT bandlimited functions in frame subspaces. Bandlimited functions are considered to be a superposition of basis functions, named frames. PoU is a variation of AW, they differ by the choice of frame because each frame produces a different approximation operator and convergence rate. If efficiency is defined as the norm convergence and computational time of an algorithm, then among the two methods, discussed, the PoU method is more efficient. The AW method is slow and converged to a higher error than that of the PoU. However, AW compensates for its slowness and less accuracy by being convergent and robust for large sampling gaps and less sensitive to the sampling irregularities. The impact of additive white Gaussian noise on the performance of the two algorithms is also investigated. The numerical tools utilized in this research consist of the theory of discrete irregular sampling, frames, and iterative techniques. The developed software provides a platform for sampling signals under non-ideal conditions with real devices. 2019-05-15T07:18:33Z 2019-05-15T07:18:33Z 2018 2019-05-14T11:34:48Z Master Thesis Masters MSc http://hdl.handle.net/11427/30079 eng application/pdf Department of Electrical Engineering Faculty of Engineering and the Built Environment |
| spellingShingle | Engineering Mkhaliphi, Mkhuseli Bruce Reconstruction of Functions From Non-uniformly Distributed Sampled Data in Shift-Invariant Frame Subspaces |
| thesis_degree_str | Master's |
| title | Reconstruction of Functions From Non-uniformly Distributed Sampled Data in Shift-Invariant Frame Subspaces |
| title_full | Reconstruction of Functions From Non-uniformly Distributed Sampled Data in Shift-Invariant Frame Subspaces |
| title_fullStr | Reconstruction of Functions From Non-uniformly Distributed Sampled Data in Shift-Invariant Frame Subspaces |
| title_full_unstemmed | Reconstruction of Functions From Non-uniformly Distributed Sampled Data in Shift-Invariant Frame Subspaces |
| title_short | Reconstruction of Functions From Non-uniformly Distributed Sampled Data in Shift-Invariant Frame Subspaces |
| title_sort | reconstruction of functions from non uniformly distributed sampled data in shift invariant frame subspaces |
| topic | Engineering |
| url | http://hdl.handle.net/11427/30079 |
| work_keys_str_mv | AT mkhaliphimkhuselibruce reconstructionoffunctionsfromnonuniformlydistributedsampleddatainshiftinvariantframesubspaces |