Discrete sampling, discrete generalizations of the Nyquist-Shannon sampling theorem
Abstract/Contents
- Abstract
- The Nyquist-Shannon sampling theorem is ubiquitiously used in telecommunications, control, audio, imaging, and speech applications. This dissertation generalizes Shannon's sampling theorem in the finite dimensional setting. Namely, we ask when it is possible to interpolate a signal from a limited number of samples when the signal is drawn from a general finite dimensional vector space. Which vector spaces will admit such an interpolation equation? What algebraic properties must be satisfied? When can we find an orthogonal interpolating basis? We first establish a simple theoretical foundation, which we call discrete sampling, that answers these questions. We then apply this theory to bandlimited spaces, in which certain frequencies are activated, and others are not. We present algorithms for constructing all possible Nyquist-Shannon-style interpolation equations for a given bandlimited space, yielding a sampling dictionary. Many patterns are revealed in these dictionaries, establishing connections with group theory, number theory, and graph theory; specifically orbit counting, prime factorization, cliques and perfect graphs, difference sets, vanishing sums of roots of unity, and the DFT. Lastly, we conduct a similar investigation of discrete Haar wavelet spaces, in which certain wavelet basis vectors are activated. The problem of constructing interpolation equations then reduces to an analysis of posets and Hasse diagrams. We conclude with open problems and ideas for future research.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2010 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Wu, William David |
|---|---|
| Associated with | Stanford University, Department of Electrical Engineering. |
| Primary advisor | Osgood, Brad |
| Thesis advisor | Osgood, Brad |
| Thesis advisor | Cover, T. M, 1938-2012 |
| Thesis advisor | Gill, John T III |
| Advisor | Cover, T. M, 1938-2012 |
| Advisor | Gill, John T III |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | William David Wu. |
|---|---|
| Note | Submitted to the Department of Electrical Engineering. |
| Thesis | Thesis (Ph. D.)--Stanford University, 2010. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2010 by William David Wu
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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