Time-Frequency Analysis and the Dark Side of Representation Theory
The group of operators on generated by a single translation and a single modulation is a familiar object of study in signal analysis. It forms a unitary representation of the so-called discrete Heisenberg group. As such, its analysis yields easily to classical methods when is rational. But when is irrational, it provides instructive concrete examples of several pathological representation-theoretic phenomena. We shall introduce the relevant concepts from representation theory, discuss the rational case briefly, and then explore the irrational case in more detail.