Gabor’s Signal Expansion and the Gabor Transform for a General, Non-Separable Sampling Geometry

Gabor’s signal expansion and the Gabor transform are formulated on a general, nonseparable 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. Following this procedure, the well-known biorthogonality condition for the window functions in the rectangular sampling geometry, can be directly translated to the general case. In the same way, a modified Zak transform can be defined for the non-separable case, with the help of which Gabor’s signal expansion and the Gabor transform can be brought into product forms that are identical to the ones that are well known for the rectangular sampling geometry.