We show that orthonormality of a discrete Gabor bases on ${\mathbb{C}}^n$ hinges heavily on the following pattern of its support set $\mathrm{\Gamma }\subset {\mathbb{Z}}_n\times {\mathbb{Z}}_n$: (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 $H◁{\mathbb{Z}}_n\times {\mathbb{Z}}_n$ such that Γ takes precisely one element from each coset of H (i.e., ${\mathbb{Z}}_n\times {\mathbb{Z}}_n=H\times \mathrm{\Gamma }$). 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.

我们证明离散 Gabor 的正交性基于 ${\mathbb{C}}^n$ 很大程度上取决于其支持集的以下模式 $\mathrm{\Gamma }\subset {\mathbb{Z}}_n\times {\mathbb{Z}}_n$: (i) Γ 本身是n阶子群，或 (ii) Γ 是这样一个子群的商，即存在一个n阶子群$H◁{\mathbb{Z}}_n\times {\mathbb{Z}}_n$使得 Γ 从H 的每个陪集中精确地取一个元素（即，${\mathbb{Z}}_n\times {\mathbb{Z}}_n=H\times \mathrm{\Gamma }$）。如果n是素数，那么满足 (i) 的 Γ 自动意味着它满足 (ii)，并且条件是充分必要的。如果n是合数，则(i)和(ii)不一定互指，条件是充分的（是否也有必要尚不可知）。本文的主要贡献是 (a) 素数n的条件的必要性；(b) (i) 对于复合n 的充分性；(c) 如果Γ是n阶子群，则其对应的离散时频位移互易交换的表征。