Equivalence of two methods for constructing tight Gabor frames

Recently, in the context of Orthogonal Frequency Division Multiplexing (OFDM), a new method (FAB-method) [11] for constructing tight Gabor frames (with redundancy 2) from a (non-tight) Gaussian g was proposed. In this letter, we prove that the FAB-method yields the tight window function canonically associated to the Gaussian. We furthermore provide a necessary and sufficient condition on the initial window function g in the Zak transform domain for the FAB-method to yield a tight Gabor frame. This yields a characterization of all initial window functions g for which the FAB-method works.