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On the construction of discrete orthonormal Gabor bases on finite dimensional spaces
Applied and Computational Harmonic Analysis  (IF3.055),  Pub Date : 2021-06-07, DOI: 10.1016/j.acha.2021.06.001
Weiqi Zhou

We show that orthonormality of a discrete Gabor bases on ${\mathbb{C}}^{n}$ hinges heavily on the following pattern of its support set $\mathrm{\Gamma }\subset {\mathbb{Z}}_{n}×{\mathbb{Z}}_{n}$: (i) Γ is itself a subgroup of order n, or (ii) Γ is the quotient of such a subgroup, i.e., there exists an order n subgroup $H◁{\mathbb{Z}}_{n}×{\mathbb{Z}}_{n}$ such that Γ takes precisely one element from each coset of H (i.e., ${\mathbb{Z}}_{n}×{\mathbb{Z}}_{n}=H×\mathrm{\Gamma }$). If n is a prime number, then Γ satisfying (i) automatically implies that it satisfies (ii), and the condition is both sufficient and necessary. If n is a composite number, then (i) and (ii) do not necessarily imply each other, and the condition is sufficient (whether it is also necessary is unknown yet). Main contributions of this article are (a) necessity of the condition for prime n; (b) sufficiency of (i) for composite n; (c) the characterization that if Γ is an order n subgroup, then its corresponding discrete time-frequency shifts mutually commute.