Gabor orthonormal bases generated by the unit cubes

We consider the problem in determining the countable sets Λ in the time-frequency plane such that the Gabor system generated by the time-frequency shifts of the window χ[0,1]dχ[0,1]d associated with Λ forms a Gabor orthonormal basis for L2(Rd)L2(Rd). We show that, if this is the case, the translates by elements Λ of the unit cube in R2dR2d must tile the time-frequency space R2dR2d. By studying the possible structure of such tiling sets, we completely classify all such admissible sets Λ of time-frequency shifts when d=1,2d=1,2. Moreover, an inductive procedure for constructing such sets Λ in dimension d≥3d≥3 is also given. An interesting and surprising consequence of our results is the existence, for d≥2d≥2, of discrete sets Λ with G(χ[0,1]d,Λ)G(χ[0,1]d,Λ) forming a Gabor orthonormal basis but with the associated “time”-translates of the window χ[0,1]dχ[0,1]d having significant overlaps.