The theory of Gabor frames of functions defined on finite abelian groups was initially
developed in order to better understand the properties of Gabor frames of functions
defined over the reals. However, during the last twenty years the topic has acquired
an interest of its own. One of the fundamental questions asked in this finite setting is
on the existence of full spark Gabor frames. In a previous paper, we proved the
existence of such frames when the underlying group is finite cyclic, and constructed
some examples. In this paper, we resolve the noncyclic case; in particular, we show
that there can be no full spark Gabor frames of windows defined on finite
abelian noncyclic groups. We also prove that all eigenvectors of certain unitary
matrices in the Clifford group in odd dimensions generate spark deficient Gabor
frames. Finally, similarities between the uncertainty principles concerning the
finite-dimensional Fourier transform and the short-time Fourier transform are
discussed.
