Sigma Delta Modulation and Correlation Criteria for the Construction of Finite Frames Arising in Communication Theory

Abstract

In this dissertation we first consider a problem in analog to digital (A/D) conversion. We compute the power spectra of the error arising from an A/D conversion. We then design various higher dimensional analogs of A/D schemes, and compare these schemes to a standard error diffusion scheme in digital halftoning.

Secondly, we study finite frames. We classify certain finite frames that are constructed as orbits of a group. These frames are seen to have subtle symmetry properties. We also study Grassmannian frames which are frames with minimal correlation. Grassmannian frames have an important intersection with spherical codes, erasure channel models, and communication theory. This is the main part of the dissertation, and we introduce new theory and algorithms to decrease the maximum frame correlation and hence construct specific examples of Grassmannian frames.

A connection has been drawn between the two parts of this thesis, namely A/D conversion and finite frames. In particular, finite frames are used to expand vectors in $\RR^d$, and then different quantization schemes are applied to the coefficients of these expansions. The advantage is that all possible outcomes of quantization can be considered because of the finite dimensionality.