Mathematical properties of coarse quantization schemes in signal analysis with new applications.

摘要

This thesis consists of two parts. In the first part1 we investigate stability and robustness properties of a family of algorithms used to “coarsely quantize” bandlimited functions. The algorithms we will consider are one-bit second-order sigma-delta quantization schemes and some modified versions of these. We prove that there exists a bounded region that remains invariant under the two-dimensional piecewise-affine discrete dynamical system associated with each of these quantizers. Moreover, this bounded region can be constructed so that it is robust under small changes in the quantizer. We also show some interesting properties of the resulting binary sequences.; The second part is on coarse quantization of redundant representations, in particular Weyl-Heisenberg frame expansions. We introduce two algorithms—that are inspired by sigma-delta quantization algorithms for bandlimited functions—to quantize Weyl-Heisenberg frame expansions of certain classes of square-integrable functions. One of the two algorithms, TFΣΔ-I, is translation invariant; however it produces a weak type approximation. The other algorithm, TFΣΔ-II produces an approximation in L2; however the algorithm is not translation invariant and the class of functions that can be quantized by TFΣΔ-II is smaller than the class of functions that can be quantized by TFΣΔ-I. We discuss these and various other properties of each algorithm in detail.; 1The first part of this thesis is submitted for publication [1].