We give a simple argument which shows that Gabor systems consisting of odd
variables and symplectic lattices of density
cannot constitute a Gabor frame. In the one-dimensional,
separable case, this follows from a more general result of Lyubarskii and
Nes [‘Gabor frames with rational density’, Appl. Comput. Harmon.
Anal.34(3) (2013), 488–494]. We use a different
approach exploiting the algebraic relation between the ambiguity function
and the Wigner distribution as well as their relation given by the
(symplectic) Fourier transform. Also, we do not need the assumption that the
lattice is separable and, hence, new restrictions are added to the full
frame set of odd functions.