Gabor frames in ${\mathrm{\ell }}^2(\mathbf{Z})$ and linear dependence

We prove that an overcomplete Gabor frame in ${\mathrm{\ell }}^2(\mathbf{Z})$ by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in ${\mathrm{\ell }}^2(\mathbf{Z})$ with modulation parameter $1\mathrm{/}M$ and translation parameter $N$ for some $M,N\in \mathbf{N},$ and generated by a finite sequence $g$ in ${\mathrm{\ell }}^2(\mathbf{Z})$ with $K$ nonzero entries.