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 $χ_{[0,1]^d}$ associated with $Λ$ forms a Gabor orthonormal basis for $L^2({\Bbb R}^d)$. We show that, if this is the case, the translates by elements $Λ$ of the unit cube in ${\Bbb R}^{2d}$ must tile the time-frequency space ${\Bbb R}^{2d}$. By studying the possible structure of such tiling sets, we completely classify all such admissible sets $Λ$ of time-frequency shifts when $d=1,2$. Moreover, an inductive procedure for constructing such sets $Λ$ in dimension $d\ge 3$ is also given. An interesting and surprising consequence of our results is the existence, for $d\geq 2$, of discrete sets $Λ$ with ${\mathcal G}(χ_{[0,1]^d},Λ)$ forming a Gabor orthonormal basis but with the associated "time"-translates of the window $χ_{[0,1]^d}$ having significant overlaps.

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