Gabor Time-Frequency Lattices and the Wexler-Raz Identity

Abstract

Gabor time-frequency lattices are sets of functions of the form $g_{m \alpha , n \beta} (t) =e^{-2 \pi i \alpha m t}g(t-n \beta)$ generated from a given function $g(t)$ by discrete translations in time and frequency. They are potential tools for the decomposition and handling of signals that, like speech or music, seem over short intervals to have well-defined frequencies that, however, change with time. It was recently observed that the behavior of a lattice $(m \alpha , n \beta )$ can be connected to that of a dual lattice $(m/ \beta , n /\alpha ).$ Here we establish this interesting relationship and study its properties. We then clarify the results by applying the theory of von Neumann algebras. One outcome is a simple proof that for $g_{m \alpha , n \beta}$ to span $L^2,$ the lattice $(m \alpha , n \beta )$ must have at least unit density. Finally, we exploit the connection between the two lattices to construct expansions having improved convergence and localization properties.