Approximation of Dual Gabor Frames, Window Decay, and Wireless Communications

We consider two problems involving Gabor frames that have recently received much attention. The first problem concerns the approximation of dual Gabor frames in L2(R) by finite-dimensional methods. Utilizing the duality relations for Gabor frames 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 canonical dual window γ=S−1g as well as its canonical tight window ψ=S−1/2g. Based on results on commutative Banach algebras and Laurent operators we derive a general condition under which γ and h inherit the decay properties of g. These derivations are of relevance in the context of wireless communications. More precisely, our results provide a theoretical foundation for a recently proposed method for the design of time-frequency well-localized pulse shapes for orthogonal frequency division multiplexing systems.