Gabor's discrete signal expansion and the Gabor transform on a non-separable lattice

Gabors discrete signal expansion and the discrete Gabor transform are formulated on a general, non-separable time-frequency lattice instead of on the traditional rectangular lattice. The representation of the general lattice is based on the rectangular lattice via a shear operation, which corresponds to a description of the general lattice by means of a lattice generator matrix that has the Hermite normal form. The shear operation on the lattice is associated with simple operations on the signal, on the synthesis and the analysis window, and on Gabor's expansion coefficients; these operations consist of multiplications by quadratic phase terms. This procedure makes it possible to reuse algorithms, which are designed for a rectangular lattice only, to calculate the analysis window, Gabor's expansion coefficients and Gabor's expansion on a general non-separable lattice.