Double preconditioning for Gabor frame operators: Algebraic, functional analytic and numerical aspects

Abstract

This paper provides algebraic and analytic, as well as numerical arguments why and how double preconditioning of the Gabor frame operator yields an efficient method to compute approximate dual (respectively tight) Gabor atoms for a given time-frequency lattice. We extend the definition of the approach to the continuous setting, making use of the so-called Banach Gelfand Triple, based on the Segal algebra $\left(S_0\right(R^d),{\Vert \text{ ⋅ }\Vert }_{S_0})$ and show the continuous dependency of the double preconditioning operators on their parameters. The generalization allows to investigate the influence of the order of the two main single preconditioners (diagonal and convolutional). In the applied section we demonstrate the quality of double preconditioning over all possible lattices and adapt the method to approximate the canonical tight Gabor window, which yields a significant generalization of the FAB-method used in OFDM-applications. Finally, we demonstrate that our approach provides a way to efficiently compute approximate dual families for Gabor families which arise from a slowly varying pattern instead of a regular lattice.