Harmonic analysis in finite phase space

Abstract

The Wigner distribution and linear canonical transforms are important tools for optics, signal processing, quantum mechanics, and mathematics. In this thesis, we study the discrete versions of Wigner distributions and linear canonical transforms. In the definition of a discrete entity we focus on two aspects: structural analogy and continuum approximation and/or limits. Based on this framework, the tradeoffs are analyzed and a compromise for a discrete Wigner distribution that meets both objectives to a high degree is presented by consolidating sampling theory and the algebraic approach. Such a compromise is necessary since it is impossible to meet the conditions to the highest possible degree. The differences between discrete and continuous time-frequency analysis are also discussed in a group theoretical perspective. In the second part of the thesis, the discrete versions of linear canonical transforms are reviewed and their connections to the continuous theory is established. As a special case the discrete fractional Fourier transform is defined and its properties are derived.