Approximation of dual Gabor frames, window decay, and wireless communications
Thomas Strohmer
TL;DR
The paper advances three interconnected problems in Gabor analysis: practical finite-dimensional approximations of dual Gabor frames, rigorous transfer of decay from a window to its dual via Laurent operators, and the design of time–frequency localized OFDM pulses for doubly dispersive channels. It shows that dual windows can be efficiently approximated by finite sections with guaranteed convergence, provides exponential-rate convergence under exponential decay, and establishes decay preservation for duals under a GRS-weight in the Wiener algebra framework. The results bridge harmonic analysis and wireless communications, offering a solid theoretical basis for constructing well-localized OFDM pulses and for understanding how window decay properties endure under dualization. Practically, the work informs pulse design and stability considerations in OFDM systems while clarifying the limits imposed by time–frequency localization principles.
Abstract
We consider three problems for Gabor frames that have recently received much attention. The first problem concerns the approximation of dual Gabor frames in $L_2(R)$ by finite-dimensional methods. Utilizing Wexler-Raz type duality relations we derive a method to approximate the dual Gabor frame, that is much simpler than previously proposed techniques. Furthermore it enables us to give estimates for the approximation rate when the dimension of the finite model approaches infinity. The second problem concerns the relation between the decay of the window function $g$ and its dual $γ$. Based on results on commutative Banach algebras and Laurent operators we derive a general condition under which the dual $γ$ inherits the decay properties of $g$. The third problem concerns the design of pulse shapes for orthogonal frequency division multiplex (OFDM) systems for time- and frequency dispersive channels. In particular, we provide a theoretical foundation for a recently proposed algorithm to construct orthogonal transmission functions that are well localized in the time-frequency plane.