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

Gabor's signal expansion and the Gabor transform are formulated on a genera l, nonseparable time-frequency lattice instead of on the traditional rectan gular lattice. The representation of the general lattice is based on the re ctangular 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 multipl ications by quadratic phase terms. Following this procedure, the well-known bi-orthogonality condition for the window functions in the rectangular sam pling 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? w ith 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. (C) 2000 The Franklin Institut e. Published by Elsevier Science Ltd. All rights reserved.