On the construction of discrete orthonormal Gabor bases on finite dimensional spaces

We show that orthonormality of a discrete Gabor bases on C-n hinges heavily on the following pattern of its support set Gamma subset of Z(n) x Z(n): (i) Gamma is itself a subgroup of order n, or (ii) Gamma is the quotient of such a subgroup, i.e., there exists an order n subgroup H (sic) Z(n) x Z(n) such that Gamma takes precisely one element from each coset of H (i.e., Z(n) x Z(n) = H x Gamma). If n is a prime number, then Gamma 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 Gamma is an order n subgroup, then its corresponding discrete time-frequency shifts mutually commute. (C) 2021 Elsevier Inc. All rights reserved.