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Title page for ETD etd-03292007-180005


Type of Document Dissertation
Author Leonetti, Casey Clark
Author's Email Address casey.leonetti@alumni.vanderbilt.edu
URN etd-03292007-180005
Title Reconstruction from Error-Affected Sampled Data in Shift-Invariant Spaces
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Akram Aldroubi Committee Chair
Benoit Dawant Committee Member
Douglas P. Hardin Committee Member
Guoliang Yu Committee Member
Larry Schumaker Committee Member
Keywords
  • jitter
  • noise
  • sampling
  • shift-invariant space
  • frame
  • fourier
Date of Defense 2007-03-28
Availability unrestricted
Abstract
In the following chapters we provide error estimates for signals reconstructed from corrupt data. Two different types of error are considered. First, we address the problem of reconstructing a continuous function defined on Rd from a countable collection of samples corrupted by noise. The additive noise is assumed to be i.i.d. with mean 0 and variance σ2. We sample the continuous function f on the uniform lattice (1/m)Zd and show for large enough m that the variance of the error between the frame reconstruction fε from noisy samples of f and the function f satisfy var(fε (x)-f(x))≈ (σ2/md)Cx. Second, we address the problem of non-uniform sampling and reconstruction in the presence of jitter. In sampling applications, the set X={xj: j ∈ J} on which a signal f is sampled is not precisely known. Two main questions are considered. First, if sampling a function f on the countable set X leads to unique and stable reconstruction of f, then when does sampling on the set X'={xjj: jJ} also lead to unique and stable reconstruction? Second, if we attempt to recover a sampled function f using the reconstruction operator corresponding to the sampling set X (because the precise sample points are unknown), is the recovered function a good approximation of the original f?
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