# Gabor's signal expansion and the Gabor transform for a general, non-separable sampling geometry

## Martin J. Bastiaans and Arno J. van Leest

Recently a new sampling lattice - the quincunx lattice - has been introduced  as a sampling geometry in the Gabor scheme, which geometry is different from the traditional rectangular sampling geometry. The quincunx lattice is just one example of the general class of non-separable time-frequency lattices. In this paper we will show how results that hold for rectangular sampling (see, for instance, [2,3]) can be transformed to the general, non-separable case.

Gabor's signal expansion and the 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. 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.

1. P. Prinz, "Calculating the dual Gabor window for general sampling sets," IEEE Trans. Signal Processing, 44 (1996) 2078-2082.
2. M.J. Bastiaans, "Gabor's signal expansion and its relation to sampling of the sliding-window spectrum," in Advanced Topics in Shannon Sampling and Interpolation Theory, R.J. Marks II, Ed., Springer, New York (1993) 1-35.
3. H.G. Feichtinger and T. Strohmer, Eds., Gabor Analysis and Algorithms: Theory and Applications. Berlin: Birkhauser (1998).

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