Gabor's signal expansion and the Gabor transform on a rectangular lattice have been introduced, along with the Fourier transform of the array of expansion coefficients and the Zak transforms of the signal and the window functions. Based on these Fourier and Zak transforms, the sum-of-products forms for the Gabor expansion and the Gabor transform, which hold in the rationally oversampled case, have been derived.
We have then studied Gabor's signal expansion and the Gabor transform based on a non-orthogonal sampling geometry. We have done this by considering the non-orthogonal lattice as a sub-lattice of an orthogonal lattice. This procedure allows us to use all the formulas that hold for the orthogonal sampling geometry. In particular we can use the sum-of-products forms that hold in the case of a rationally oversampled rectangular lattice.