Stability Bounds for Reconstruction from Sampling Erasures

Abstract

Frame and sampling theory are often used in the analysis of digital signals. We can view digital signals as vectors (or functions) in a Hilbert space, and utilize properties associated with these spaces for computation. Frame theory and its applications in signal processing have been studied in great detail ([1, 6, 7, 10, 11, 15]), and there are vast amounts of literature, in both theoretical and applied mathematics, that relate to analysis of digital signals ([3, 4, 5, 8, 13, 17]). This thesis is primarily focused on applications of sampling reconstruction methods that allow signals to be reconstructed when part of the received signal is lost (or erased). This, of course, has many real world applications, which will be demonstrated within. The Shannon-Whittaker Sampling Theorem states that a frequency bounded signal can be completely determined by its sampled values at a countable number of points. Thus, the theorem allows us to convert analog signals to digital signals by sampling (or evaluating) the signal at these points. In prior work, it was shown that if a signal is oversampled, and if some of the sampled values are lost when transmitting the signal, then it is still possible to reconstruct the signal. However, in certain situations, the reconstruction algorithm is very unstable. The goal of this project is to provide stability bounds on the reconstruction algorithm and to determine when it is not feasible to perform the reconstruction.