Gabor Frames in $${\ell }^2({\mathbb {Z}})$$ ℓ 2 ( Z ) and Linear Dependence

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We prove that an overcomplete Gabor frame in $${\ell }^2({\mathbb {Z}})$$ generated 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 $${\ell }^2({\mathbb {Z}})$$ with modulation parameter 1 / M and translation parameter N for some $$M,N\in {\mathbb {N}},$$ and generated by a finite sequence g in $${\ell }^2({\mathbb {Z}})$$ with K nonzero entries.